Human-AI-Powered Hypothesis Testing: Cost-Aware Selective AI Scoring and Sequential Human Escalation

# Cost-Aware Selective AI Scoring and Sequential Human Escalation
Source: [https://arxiv.org/html/2609.28859](https://arxiv.org/html/2609.28859)
## Human–AI\-Powered Hypothesis Testing: Cost\-Aware Selective AI Scoring and Sequential Human Escalation

###### Abstract

Large language models are increasingly used as inexpensive judges to evaluate outputs, label data, and assess whether a system meets a desired quality standard\. Yet using AI judgments for formal statistical inference is fundamentally different from simply treating them as ground\-truth labels: AI evaluations can be biased or noisy, and rigorous hypothesis testing requires explicit control of type\-I and type\-II errors\. We study how to use AI judgments, together with selective human verification, to conduct a valid hypothesis test at minimum cost\. We consider a population of items with hidden binary labels\. After choosing a fixed pool of items, the decision maker can selectively query AI, send an item directly to a human, escalate an AI\-scored item to a human after observing the AI report, or stop once sufficient evidence has accumulated\. We derive an information\-theoretic lower bound that captures the minimum cost of achieving prescribed testing errors and characterizes the value of AI information and human verification through a report\-dependent information frontier\. Motivated by this characterization, we develop SCALE, a sequential cost\-aware policy that combines selective AI scoring with adaptive human escalation\. SCALE is valid at finite sample sizes and matches the lower bound to first order as the target error probabilities vanish\. We further extend the framework to an unknown AI\-output model using paired AI–human pilot data\. Numerically, SCALE approaches Human\-only or AI\-only testing when one source clearly dominates, while achieving its largest savings when inexpensive AI judgments and selective human verification are both valuable\.

††runningauthor:Ham et al\.††authors:University of Michigan, Ann Arbor, MI 48109, USA,
daewoong@umich\.edu University of North Carolina at Charlotte, Charlotte, NC 28223, USA,
xzhao19@charlotte\.edu University of Michigan, Ann Arbor, MI 48109, USA,
sjasin@umich\.edu, yfenghua@umich\.edu††affiliation:††affiliation:††affiliation:††affiliation:††affiliation:††affiliation:## 1Introduction

Large language models are increasingly being used not only to generate content, but also to evaluate it\. This is particularly attractive in settings where the ultimate goal is to make a population\-level quality statement from many individual evaluations\. Three application domains illustrate this opportunity particularly clearly\.

First, consider software reliability\. A company deploying an AI coding assistant may want to certify that the fraction of generated programs that are semantically correct exceeds a prescribed reliability threshold\. When exhaustive test cases are unavailable, expert human review provides a natural gold\-standard assessment, but conducting such review at scale can be costly\. A growing literature therefore studies LLMs themselves as code judges\.[Tong and Zhang \(2024\)](https://arxiv.org/html/2609.28859#bib.bib41)develop CodeJudge, which uses LLMs to assess the semantic correctness of generated code without requiring test cases\.[Zhao et al\. \(2025\)](https://arxiv.org/html/2609.28859#bib.bib45)introduce CodeJudge\-Eval, which asks LLMs to determine whether submitted code solutions are correct across different error types and compilation issues\. More recently,[Jiang et al\. \(2026\)](https://arxiv.org/html/2609.28859#bib.bib17)benchmark 26 LLM judges across code generation, code repair, and unit\-test generation\. Their results also illustrate why a statistical treatment is needed: LLM code judges can be sensitive to seemingly irrelevant changes such as response ordering, variable names, and misleading comments\. Thus, an LLM judge can provide an inexpensive and scalable signal of code correctness, but its judgment cannot automatically be treated as ground truth\.

Second, similar ideas are already being used to screen large collections of clinical documents\. A health system may, for example, want to assess whether the prevalence of biased or stigmatizing language in its clinical notes exceeds an acceptable level\. Manual review by clinical experts provides the natural reference standard but is difficult to perform at the scale of a large electronic\-health\-record system\.[Apakama et al\. \(2025\)](https://arxiv.org/html/2609.28859#bib.bib4)apply GPT\-4 to 50,000 emergency\-department medical and nursing notes to identify several categories of biased language and use human reviewers to verify the model’s detections\.[Zhang et al\. \(2025\)](https://arxiv.org/html/2609.28859#bib.bib44)compare ChatGPT\-4 with human expert annotations for identifying stigmatizing language in electronic health records and find that performance varies across categories and is sensitive to prompt design\.[Sethi et al\. \(2026\)](https://arxiv.org/html/2609.28859#bib.bib35)study LLM\-based detection of stigmatizing language using more than 77,000 ICU notes and externally validate the approach on a substantially larger health\-system data set\. These studies demonstrate both sides of the opportunity: AI can make large\-scale screening feasible, while expert human labels remain important for establishing what is actually present in the underlying records\.

Third, automated judging has become central to AI safety and content moderation\. An AI provider may want to certify that the fraction of model responses violating a safety policy remains below a prescribed threshold\. Reviewing every generated response by hand is prohibitively expensive, so large\-scale safety evaluations increasingly rely on automated judges or moderation classifiers\.[Mazeika et al\. \(2024\)](https://arxiv.org/html/2609.28859#bib.bib24)develop HarmBench, a standardized framework for large\-scale evaluation of harmful model behavior that uses automated evaluation classifiers calibrated against human\-labeled examples\.[Movva et al\. \(2024\)](https://arxiv.org/html/2609.28859#bib.bib26)directly compare GPT\-4 safety annotations with human judgments of conversational safety\.[Chen and Goldfarb\-Tarrant \(2025\)](https://arxiv.org/html/2609.28859#bib.bib10)evaluate 11 LLM judges used to assess the safety of generated content and show that seemingly superficial artifacts, including apologetic and verbose phrasing, can substantially change the resulting verdicts\. Here again, automated evaluation is valuable precisely because it is scalable, but its errors become consequential when the objective is to make a formal statement about an underlying safety rate\.

Across these applications, the same basic structure emerges: there is a population of items with latent ground\-truth labels, an AI system provides inexpensive but imperfect assessments of those labels, and human review provides costly ground truth\. Much of the emerging literature focuses on improving the quality of the AI judge itself\. Researchers develop more effective judging procedures, richer evaluation rubrics, better prompting and reasoning mechanisms, specialized evaluation models, robustness benchmarks, and aggregation schemes intended to bring automated judgments closer to human ground truth\. Related work in clinical applications similarly evaluates and refines AI screening procedures by comparing their outputs with expert human annotations\. These efforts are important: a better AI judge can clearly reduce the amount of expensive human review that is needed\. However, improving the judge does not eliminate the underlying statistical problem\. Even a highly accurate and carefully calibrated AI evaluator remains an imperfect measurement of the ground truth, and high average agreement with humans does not by itself guarantee valid type\-I and type\-II error control for a population\-level hypothesis test\. If the ultimate goal is rigorous population\-level statistical inference, then building a better AI judge is not enough: its residual errors must be incorporated into a formal inferential procedure with explicit statistical guarantees, so that conclusions about the underlying population are scientifically defensible rather than merely reflections of the judge’s average agreement with humans\.

One natural approach is to combine inexpensive AI judgments with selective human verification\. Rather than treating AI as a complete substitute for human evaluation, a decision maker can use AI as a cheap source of preliminary information and purchase a human ground\-truth label only when the additional information is worth its cost\. This idea is related to prediction\-powered inference \(PPI\), which combines machine predictions with a smaller number of gold\-standard labels to obtain valid statistical inference\([Angelopoulos et al\. 2023](https://arxiv.org/html/2609.28859#bib.bib2)\)\. Our setting adds an explicitly operational dimension: rather than taking AI predictions and human labels as given, we jointly decide how each acquired item should be evaluated and when sufficient evidence has been collected to stop the test\. This leads to the central question of the paper:*How should AI and human evaluations be combined to reach a statistically valid conclusion at minimum cost?*

We study this question in a simple hypothesis\-testing model for a population proportion\. Each acquired item has an unobserved binary ground\-truth labelXi∈\{0,1\}X\_\{i\}\\in\\\{0,1\\\}, and the objective is to distinguish

H0:p=p0fromH1:p=p1,0<p0<p1<1,H\_\{0\}:p=p\_\{0\}\\qquad\\text\{from\}\\qquad H\_\{1\}:p=p\_\{1\},\\qquad 0<p\_\{0\}<p\_\{1\}<1,whereppis the population fraction of positive labels\. A human query reveals the true label, whereas an AI query returns a potentially noisy finite\-valued report that may contain a predicted label, confidence score, or other metadata\. Acquiring an item, querying the AI, and querying a human all carry potentially different costs\. The policy chooses a fixed pool of items in advance, but information acquisition within that pool is adaptive: it may query AI, query a human directly, query a human after observing an AI report on the same item, or stop and decide\. We minimize worst\-case expected total cost subject to prescribed type\-I and type\-II error guarantees\.

Two features are central\. First,*AI scoring itself is selective*: the decision maker need not run AI on every acquired item\. When direct human information is sufficiently valuable, it may be preferable to skip the AI and query a human immediately\. Conversely, when AI information is inexpensive and sufficiently informative, an AI report may by itself provide useful statistical evidence\. Second, human review after an AI report is a*nested information action*\. Once reportrrhas been observed, a subsequent human label provides only the residual information in the ground truth conditional on that report, and this residual value can vary substantially across reports\. Some AI reports may already be highly informative, whereas others may leave considerable uncertainty and therefore be particularly valuable to verify\. Moreover, because the test is sequential, the desirable mix of AI and human information also depends on the likelihood evidence accumulated across previous items\. The policy and our paper therefore study the joint decision of*when AI is worth querying, when an AI judgment is worth verifying, and when the test should stop*\.

Contributions and main results\.Our first contribution is a lower bound on the minimum cost that any valid human–AI testing policy must incur\. The basic idea is simple: to control type\-I and type\-II errors, every policy must collect enough statistical evidence to distinguish the two hypotheses\. We use KL\-divergence arguments to quantify how much evidence can be obtained from an AI judgment, from a human label, and from a human label obtained after an AI judgment has already been observed\. We also account for the fact that the number of items must be chosen in advance\. Combining these requirements gives a computable lower bound on the best possible cost of any policy; this lower bound is developed in Section[4](https://arxiv.org/html/2609.28859#S4)\.

Our second contribution is SCALE, the*Sequential Cost\-Aware Likelihood\-Guided Escalation*policy\. SCALE builds on the basic logic of the classical sequential probability ratio test \(SPRT\)\([Wald 1945](https://arxiv.org/html/2609.28859#bib.bib43)\): it continually tracks the likelihood ratio between the two hypotheses and stops once the accumulated evidence is sufficiently strong\. The key difference is that, in a classical SPRT, each new observation is drawn from a fixed information source\. In our setting, the policy must also decide*how*to obtain the next piece of evidence\. SCALE therefore chooses whether the next item should be evaluated by AI only, by a human directly, or by AI first followed by selective human verification based on the observed AI report\. In this sense, SCALE extends the SPRT from a stopping rule with a fixed observation channel to a joint sensing\-and\-stopping rule for a human–AI system\. Because the number of items is acquired in advance, SCALE also includes a pre\-specified fallback test in case the pool is exhausted before either likelihood\-ratio boundary is reached, which guarantees the desired type\-I and type\-II errors at every finite sample size\. Our main theoretical result shows that, as the target error probabilities become small, SCALE achieves the lower bound to first order\. In other words, no other valid adaptive policy can have a meaningfully lower leading\-order cost\. Section[5](https://arxiv.org/html/2609.28859#S5)develops the policy and proves this result\.

Our third contribution considers the practically important case in which the AI judge’s error behavior is not known in advance\. In practice, one typically has to learn how the AI’s judgments relate to ground truth from a calibration sample\. We therefore collect a paired pilot sample in which each item is evaluated by both the AI and a human, use it to estimate the AI\-output model, and then run a guarded plug\-in version of SCALE\. The guardrails account for the fact that the estimated AI model is itself uncertain, so that this estimation error does not undermine the statistical guarantees of the test\. We show that, when the pilot is sufficiently informative, the resulting procedure achieves the same first\-order cost as if the AI model were known from the outset\. We also characterize when the additional cost of collecting a fresh pilot is small enough not to change this first\-order benchmark\.

Finally, our numerical experiments show when combining AI and human judgments is most valuable\. When human review is inexpensive, SCALE behaves almost like Human\-only testing; when human review is very expensive, it approaches AI\-only testing\. The largest gains arise in the intermediate regime, where AI provides useful low\-cost information but selective human verification is still worth purchasing\. This is precisely the regime in which committing in advance to either a Human\-only or an AI\-only architecture is most costly\. In our benchmark instance, SCALE reduces cost by as much as36\.37%36\.37\\%relative to the cheaper single\-source baseline as the human\-review cost varies, and its savings reach39\.16%39\.16\\%as the testing requirements become more stringent\. These results illustrate the main operational value of the hybrid design: AI is useful not only because it can replace some human review, but because it helps determine*which*items still warrant costly human verification\. Section[5\.4](https://arxiv.org/html/2609.28859#S5.SS4)reports the full numerical results\.

Relation to existing approaches\.Our paper is most closely related to prediction\-powered inference \(PPI\), two\-phase sampling with selective gold\-standard verification, and active hypothesis testing; Section[2](https://arxiv.org/html/2609.28859#S2)provides a detailed review\. The connection to prediction\-powered inference is especially close: both settings combine inexpensive machine\-generated information with a smaller amount of costly ground truth to obtain valid statistical inference\([Angelopoulos et al\. 2023](https://arxiv.org/html/2609.28859#bib.bib2)\)\. Our main distinction is operational\. AI and human information are not taken as given; the policy decides how each item should be evaluated, whether an AI\-scored item should be escalated to a human, and when to stop\. Our nested AI\-then\-human structure is also related to classical two\-phase sampling, where a cheap first\-stage measurement is followed by selective gold\-standard verification\([Tenenbein 1970](https://arxiv.org/html/2609.28859#bib.bib39),[Begg and Greenes 1983](https://arxiv.org/html/2609.28859#bib.bib6)\)\. In our setting, however, the first\-stage AI measurement is itself optional and the verification decision depends on both the observed AI report and the accumulated evidence\. Finally, SCALE is related to active hypothesis testing and controlled sensing, which adaptively choose among information sources as evidence accumulates\([Chernoff 1959](https://arxiv.org/html/2609.28859#bib.bib11),[Naghshvar and Javidi 2013a](https://arxiv.org/html/2609.28859#bib.bib28),[Nitinawarat et al\. 2013](https://arxiv.org/html/2609.28859#bib.bib31)\)\. The distinctive feature here is that querying AI creates a report\-dependent option to reveal the same item’s ground\-truth label through human verification\. This nested action structure is central to both our lower bound and our policy\.

Organization of the paper\.The remainder of the paper is organized as follows\. Section[2](https://arxiv.org/html/2609.28859#S2)reviews the related literature in greater detail\. Section[3](https://arxiv.org/html/2609.28859#S3)introduces the model, policy class, cost objective, and information quantities\. Section[4](https://arxiv.org/html/2609.28859#S4)derives the lower bound on the optimal cost\. Section[5](https://arxiv.org/html/2609.28859#S5)develops SCALE, establishes its finite\-sample feasibility and first\-order optimality, and contains the numerical experiments in Section[5\.4](https://arxiv.org/html/2609.28859#S5.SS4)\. Section[6](https://arxiv.org/html/2609.28859#S6)develops the pilot\-calibrated procedure for an unknown AI\-output model\. Section[7](https://arxiv.org/html/2609.28859#S7)concludes the paper\.

## 2Literature Review

Our paper lies at the intersection of three statistical literature\. The first uses machine predictions to economize on expensive gold\-standard labels while preserving valid inference\. The second studies two\-phase designs that combine broadly available but fallible measurements with selectively acquired exact outcomes\. The third studies active hypothesis testing, in which observation channels and stopping decisions are chosen as evidence accumulates\. At an architectural level, selective prediction and learning to defer also motivate endogenous human escalation, but those models primarily optimize item\-level prediction or delegation loss\([Madras et al\. 2018](https://arxiv.org/html/2609.28859#bib.bib22),[Mozannar and Sontag 2020](https://arxiv.org/html/2609.28859#bib.bib27)\); our terminal decision concerns a population hypothesis\. We therefore organize the review around the three statistical streams that map most directly to our inferential target, nested observation structure, and adaptive policy\.

### 2\.1Prediction\-Powered and Active Statistical Inference

A growing statistical literature uses machine predictions to reduce the need for expensive labels while retaining inferential validity\.[Angelopoulos et al\. \(2023\)](https://arxiv.org/html/2609.28859#bib.bib2)introduce prediction\-powered inference \(PPI\), which combines abundant machine predictions with a smaller labeled sample to correct prediction error\.[Kossen et al\. \(2021\)](https://arxiv.org/html/2609.28859#bib.bib19)select which test examples to label for sample\-efficient model evaluation, while[Zrnic and Candès \(2024\)](https://arxiv.org/html/2609.28859#bib.bib46)assign observation\-dependent labeling probabilities and construct valid confidence intervals and hypothesis tests\. Recent work learns data\-adaptive labeling policies or develops anytime\-valid and sequential prediction\-assisted tests\([Ma and Candès 2026](https://arxiv.org/html/2609.28859#bib.bib21),[Csillag et al\. 2025](https://arxiv.org/html/2609.28859#bib.bib13),[Tenzer et al\. 2026](https://arxiv.org/html/2609.28859#bib.bib40)\)\. Most closely related operationally,[Angelopoulos et al\. \(2025\)](https://arxiv.org/html/2609.28859#bib.bib3)derive cost\-aware allocations between a cheap weak rater and an expensive strong rater to estimate the mean strong rating accurately under an annotation budget\.

The main distinction is our objective and action space\. Most work in this stream evaluates estimator variance, confidence\-interval width, or testing power under a labeling or annotation budget\. We instead study a simple population hypothesis test in which AI scoring is itself optional and costly, direct human review is an alternative sensing action, and the acquired pool is chosen jointly with the within\-pool policy\. A non\-escalated AI report remains direct likelihood evidence, whereas a human label obtained after that report contributes only report\-conditional residual information\. Furthermore, in the PPI literature there is no action to further escalate a machine prediction to a “human” label as we explicitly model\. We minimize worst\-case expected data, AI, and human cost subject to separate type\-I and type\-II error constraints\. Consequently, labeling decisions depend on both the global likelihood\-ratio direction and direction\-specific residual KL information, rather than only on predictive uncertainty, influence, or conditional squared error\.

### 2\.2Two\-Phase Sampling and Selective Gold\-Standard Verification

A longstanding literature studies two\-phase or double\-sampling designs in which an inexpensive but fallible measurement is collected broadly and an expensive gold\-standard outcome is obtained for a validation subsample\.[Tenenbein \(1970\)](https://arxiv.org/html/2609.28859#bib.bib39)estimate a binomial proportion using a fallible classifier on a first\-stage sample and an exact classifier on a subsample, explicitly optimizing the two sample sizes against cost and precision\.[Begg and Greenes \(1983\)](https://arxiv.org/html/2609.28859#bib.bib6)show that selective verification based on the initial screen must be incorporated into inference to avoid verification bias\. Subsequent work develops efficient prevalence\-survey designs and inference or tests under partial validation\([Shrout and Newman 1989](https://arxiv.org/html/2609.28859#bib.bib36),[McNamee 2003](https://arxiv.org/html/2609.28859#bib.bib25),[Pepe 1992](https://arxiv.org/html/2609.28859#bib.bib33),[Alonzo et al\. 2003](https://arxiv.org/html/2609.28859#bib.bib1),[Tang et al\. 2012](https://arxiv.org/html/2609.28859#bib.bib37)\)\. Design\-oriented contributions choose second\-phase sampling fractions using relative costs, pilot information, or semiparametric efficiency criteria\([Reilly 1996](https://arxiv.org/html/2609.28859#bib.bib34),[Tao et al\. 2020](https://arxiv.org/html/2609.28859#bib.bib38)\); recent work similarly targets expensive manual chart review using noisy electronic\-health\-record phenotypes and covariates\([Marks\-Anglin et al\. 2025](https://arxiv.org/html/2609.28859#bib.bib23)\)\.

This literature provides the closest static analogue to our nested AI\-then\-human observation structure\. However, conventional two\-phase designs generally collect the phase\-one measurement for the full cohort and choose a validation sample through prespecified strata or observation\-dependent sampling probabilities, with the goal of correcting bias or improving estimation precision\. In our model, even the phase\-one AI measurement is optional: an acquired item may receive direct human review, AI scoring alone, AI scoring followed by human verification, or no query\. Moreover, the follow\-up rule changes with the accumulated likelihood ratio and values a human label by its directional residual KL information\. Thus the validation design is not merely tailored to an estimand; it is embedded in an adaptive hypothesis test\. This additional history dependence leads to the active\-testing literature discussed next\.

### 2\.3Active Hypothesis Testing and Controlled Sensing

Active hypothesis testing and controlled sensing study how a decision maker should choose among observation channels while learning which hypothesis is true\. Wald’s sequential probability ratio test endogenizes stopping under a fixed observation channel\([Wald 1945](https://arxiv.org/html/2609.28859#bib.bib43)\), whereas[Chernoff \(1959\)](https://arxiv.org/html/2609.28859#bib.bib11)allows the experiment itself to be selected adaptively from past observations\.[Naghshvar and Javidi \(2013a\)](https://arxiv.org/html/2609.28859#bib.bib28)derive information\-acquisition bounds and asymptotically optimal sensing policies under sampling costs and wrong\-decision penalties; companion work separates the gains from sequential stopping and adaptive experiment selection\([Naghshvar and Javidi 2013b](https://arxiv.org/html/2609.28859#bib.bib29)\)\.[Nitinawarat et al\. \(2013\)](https://arxiv.org/html/2609.28859#bib.bib31)analyze controlled sensing in both fixed\-sample and sequential multihypothesis testing, deriving error\-exponent bounds and Chernoff\-type policies under decision\-risk constraints\.[Kartik et al\. \(2022\)](https://arxiv.org/html/2609.28859#bib.bib18)study fixed\-horizon active tests with adaptive experiment selection and the option of an inconclusive decision\. Extensions allow controlled Markovian observations and nonuniform control costs\([Nitinawarat and Veeravalli 2015](https://arxiv.org/html/2609.28859#bib.bib32)\); more recently,[Vershinin et al\. \(2026\)](https://arxiv.org/html/2609.28859#bib.bib42)study heterogeneous action costs and show that the relevant efficiency criterion is expected information gain divided by expected cost\.

These studies generally model each action as selecting an experiment\-level observation law\. In the standard binary fixed\-sample model, a stationary open\-loop control can already attain the optimal error exponent\([Nitinawarat et al\. 2013](https://arxiv.org/html/2609.28859#bib.bib31)\); the value of adaptivity in our setting instead comes from sequential stopping, direction\-dependent cost efficiency, and report\-contingent follow\-up\. Our model has a hybrid fixed\-pool and sequential structure: the number of items is chosen and paid for in advance, but sensing and stopping within that pool remain adaptive\. The sensing technologies are also nested within an item\. An AI query first generates a noisy report, after which the policy may purchase an exact human label on the same item\. The current likelihood ratio selects a direction\-specific mixture of direct\-human and AI\-first sensing, while the realized AI report determines human follow\-up through a direction\-specific residual\-information frontier\. We minimize worst\-case expected acquisition, AI, and human cost subject to separate type\-I and type\-II error constraints\. This structure creates a full\-label pool\-size floor and a capacity\-aware transcript\-KL lower bound, and it supports a finite\-sample\-valid policy that matches the lower bound to first order\.

## 3Model Setup

We study a hypothesis testing problem in which each data itemiicarries a binary labelXi∈\{0,1\}X\_\{i\}\\in\\\{0,1\\\}with population meanpp\. Although the binary label assumption may appear restrictive, it covers a wide range of practically important settings\. A binary label naturally captures any pass/fail, accept/reject, or present/absent decision, which are among the most common output types in AI\-assisted workflows\. Examples include automated correctness checking of documents, code, and mathematical proofs\([Kryściński et al\. 2020](https://arxiv.org/html/2609.28859#bib.bib20),[Tong and Zhang 2024](https://arxiv.org/html/2609.28859#bib.bib41),[Dekoninck et al\. 2026](https://arxiv.org/html/2609.28859#bib.bib14)\), AI\-assisted legal adjudication of decisions such as “guilty” or “innocent”\([Imai et al\. 2023](https://arxiv.org/html/2609.28859#bib.bib16)\), and medical screening of patient records for a binary health outcome\.

Formally, we test the simple hypotheses

H0:p=p0,H1:p=p1,0<p0<p1<1\.H\_\{0\}:p=p\_\{0\},\\qquad H\_\{1\}:p=p\_\{1\},\\qquad 0<p\_\{0\}<p\_\{1\}<1\.\(1\)The orderingp0<p1p\_\{0\}<p\_\{1\}is adopted for concreteness; the casep1<p0p\_\{1\}<p\_\{0\}is symmetric and can be analyzed analogously\. To fix ideas, consider a company that wants to assess whether an AI coding assistant produces correct code more thanp0p\_\{0\}fraction of the time\. The company can collect a batch of code submissions but even collecting each submission carries a cost \(e\.g\., compute time, storage, or API fees\), and the correct labelXi∈\{0,1\}X\_\{i\}\\in\\\{0,1\\\}of each submission is not immediately known\. To obtain a label, the company can either ask an AI checker to review the submission \(fast and cheap, but potentially inaccurate\) or hire a human expert to verify it \(slow and expensive, but exact\)\. The central question is: can we design a policy that intelligently mixes AI and human queries \(e\.g\., escalating to a human only when the AI evidence is insufficient\) and still retain the same statistical guarantees as full human labeling, but at a fraction of the cost?

To make this precise, for any positive integermmwe write\[m\]:=\{1,2,…,m\}\[m\]:=\\\{1,2,\\ldots,m\\\}\. A policyπ\\pifirst acquiresNπN^\{\\pi\}i\.i\.d\. items\. Each itemi∈\[Nπ\]i\\in\[N^\{\\pi\}\]has a hidden binary labelXi∈\{0,1\}X\_\{i\}\\in\\\{0,1\\\}; underHhH\_\{h\}the labels are distributed asXi∼Bernoulli⁡\(ph\)X\_\{i\}\\sim\\operatorname\{Bernoulli\}\(p\_\{h\}\), but they are not observed unless the policy explicitly queries a human on itemii\. A costcdata\>0c\_\{\\mathrm\{data\}\}\>0is incurred for each acquired item regardless of whether it is subsequently queried\. Importantly, we focus on a*fixed\-sample*design: the number of acquired itemsNπN^\{\\pi\}is chosen in advance and does not depend on the data, in contrast to sequential testing where the sample size is itself an adaptive stopping decision\. Given the fixed pool ofNπN^\{\\pi\}items, the policy may then query the AI, query a human, both, or neither on each item\. The central decision is therefore how to jointly chooseNπN^\{\\pi\}and how to allocate AI and human queries across the acquired items, so as to minimize total cost while meeting the target type\-I and type\-II error constraints \(to be specified below\)\. We formalize the AI output model next\.

### 3\.1AI Outputs

If the policy queries the AI on itemii, it receives a reportRiR\_\{i\}, which may encode a predicted label, a confidence score, or any other finite metadata\. For example, in the code correctness setting, a typical AI response might beRi=\(1,0\.99\)R\_\{i\}=\(1,0\.99\), indicating that the AI predictsXi=1X\_\{i\}=1\(correct\) with99%99\\%confidence\. We allowRiR\_\{i\}to be as flexible as the practitioner requires; the only constraint is that its range is finite\. Formally, letℛ\\mathcal\{R\}be the finite set of all possible AI outputs\. Conditional onXi=xX\_\{i\}=x, the report takes valuer∈ℛr\\in\\mathcal\{R\}with probability

fx​\(r\):=ℙ⁡\(Ri=r∣Xi=x\),r∈ℛ,x∈\{0,1\}\.f\_\{x\}\(r\):=\\mathbb\{P\}\(R\_\{i\}=r\\mid X\_\{i\}=x\),\\qquad r\\in\\mathcal\{R\},\\quad x\\in\\\{0,1\\\}\.We assume throughout the main analysis \(relaxed later in Section[6](https://arxiv.org/html/2609.28859#S6)\) that these conditional distributions are known:

\{assumption\}

\[Known strictly positive finite\-alphabet AI\-output model\] The conditional probability mass functionsf0f\_\{0\}andf1f\_\{1\}are known\. They satisfyfx​\(r\)\>0,∀r∈ℛ,x∈\{0,1\},f\_\{x\}\(r\)\>0,\\,\\forall r\\in\\mathcal\{R\},\\,x\\in\\\{0,1\\\},and∑r∈ℛfx​\(r\)=1,∀x∈\{0,1\}\.\\sum\_\{r\\in\\mathcal\{R\}\}f\_\{x\}\(r\)=1,\\,\\forall x\\in\\\{0,1\\\}\.

The strict positivity conditionfx​\(r\)\>0f\_\{x\}\(r\)\>0rules out report values that are structurally impossible under one of the labels, which simplifies the analysis without materially restricting the model\. In the simplest case whereRiR\_\{i\}is just a predicted labelX^i∈\{0,1\}\\hat\{X\}\_\{i\}\\in\\\{0,1\\\}, Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)reduces to knowing the AI’s confusion matrix: the probability of predicting11when the true label is11, and the probability of predicting00when the true label is00\. More generally, knowingfx​\(r\)f\_\{x\}\(r\)means knowing the full conditional distribution of the AI output given the true label, which characterizes how informative and reliable the AI is\.

The known\-ffassumption provides a useful benchmark for understanding how AI and human information should be combined when the AI’s accuracy profile is known\. In practice, however,f0f\_\{0\}andf1f\_\{1\}generally must be estimated from data for which both the AI report and the human\-verified label are observed \(e\.g\., using pilot data\)\. For completeness, we return to this issue in Section[6](https://arxiv.org/html/2609.28859#S6), where we relax the known\-ffassumption, estimatef0f\_\{0\}andf1f\_\{1\}from an independent paired pilot sample, and study how estimation uncertainty affects both statistical validity and cost\.

In addition to Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1), we also impose the standard assumption that items are independent and identically distributed \(i\.i\.d\.\)\.

\{assumption\}

\[Independent items\] Under each hypothesisHhH\_\{h\}, the pairs\(X1,R1\),…,\(XNCLOSE,\(X\_\{1\},R\_\{1\}\),\\ldots,\(X\_\{N\},OPENRN\)R\_\{N\}\)are independent and identically distributed\.

Letℙh\\mathbb\{P\}\_\{h\}denote the joint distribution of\(Xi,Ri\)\(X\_\{i\},R\_\{i\}\)underHhH\_\{h\}\. The marginal probability that the AI report takes valuerrunderHhH\_\{h\}is

gh​\(r\):=ℙh​\(Ri=r\)=ph​f1​\(r\)\+\(1−ph\)​f0​\(r\),r∈ℛ,g\_\{h\}\(r\):=\\mathbb\{P\}\_\{h\}\(R\_\{i\}=r\)=p\_\{h\}f\_\{1\}\(r\)\+\(1\-p\_\{h\}\)f\_\{0\}\(r\),\\qquad r\\in\\mathcal\{R\},\(2\)where the second equality follows by the law of total probability, conditioning onXi∈\{0,1\}X\_\{i\}\\in\\\{0,1\\\}\. By Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)andph∈\(0,1\)p\_\{h\}\\in\(0,1\), we havegh​\(r\)\>0g\_\{h\}\(r\)\>0for everyr∈ℛr\\in\\mathcal\{R\}and bothh∈\{0,1\}h\\in\\\{0,1\\\}\.

After observing the AI reportRi=rR\_\{i\}=r, the posterior probability that the hidden label equals one underHhH\_\{h\}follows from Bayes’ rule:

qh​\(r\):=ℙh​\(Xi=1∣Ri=r\)=ph​f1​\(r\)ph​f1​\(r\)\+\(1−ph\)​f0​\(r\)=ph​f1​\(r\)gh​\(r\)\.q\_\{h\}\(r\):=\\mathbb\{P\}\_\{h\}\(X\_\{i\}=1\\mid R\_\{i\}=r\)=\\frac\{p\_\{h\}f\_\{1\}\(r\)\}\{p\_\{h\}f\_\{1\}\(r\)\+\(1\-p\_\{h\}\)f\_\{0\}\(r\)\}=\\frac\{p\_\{h\}f\_\{1\}\(r\)\}\{g\_\{h\}\(r\)\}\.\(3\)Conditional on the AI reportRi=rR\_\{i\}=r, a subsequently revealed human labelXiX\_\{i\}is therefore Bernoulli with success probabilityqh​\(r\)q\_\{h\}\(r\), with mass function

ρh​\(x∣r\):=qh​\(r\)x​\(1−qh​\(r\)\)1−x,x∈\{0,1\},r∈ℛ\.\\rho\_\{h\}\(x\\mid r\):=q\_\{h\}\(r\)^\{x\}\(1\-q\_\{h\}\(r\)\)^\{1\-x\},\\qquad x\\in\\\{0,1\\\},\\quad r\\in\\mathcal\{R\}\.\(4\)Throughout the paper, all quantities carrying a subscripthhdepend on whether the true proportion isp0p\_\{0\}orp1p\_\{1\}, and our analysis proceeds under each hypothesis separately\.

### 3\.2Feasible Policy

We consider a general setting where a policy can be a dynamic, data\-dependent, and possibly randomized decision rule\. At each time stept=1,2,…t=1,2,\\dots, the policy observes the history of past actions and outcomes, and chooses the next action, stopping at some timet≤2​Nπ\+1t\\leq 2N^\{\\pi\}\+1\. The upper bound2​Nπ\+12N^\{\\pi\}\+1arises as follows\. Each of theNπN^\{\\pi\}acquired items can be queried at most once by the AI and at most once by a human, giving at most2​Nπ2N^\{\\pi\}paid queries in total\. The additional\+1\+1accounts for the final stopping decision, at which the policy rejects or acceptsH0H\_\{0\}\. The policy is free to stop at any time and act on the items in any order: for instance, it may query the AI on itemi=3i=3att=1t=1, then query a human on itemi=1i=1att=2t=2, and stop and decide att=3t=3without ever querying the remaining items\. We now formalize this\.

GivenNπN^\{\\pi\}acquired items, at each time stepttthe policy chooses one of three actions:

1. \(i\)query the AI on itemi∈\[Nπ\]i\\in\[N^\{\\pi\}\], denotedAt=AI⁡\(i\)A\_\{t\}=\\mathrm\{AI\}\(i\), which returns the AI reportOt=Ri∈ℛO\_\{t\}=R\_\{i\}\\in\\mathcal\{R\};
2. \(ii\)query a human evaluator on itemi∈\[Nπ\]i\\in\[N^\{\\pi\}\], denotedAt=H⁡\(i\)A\_\{t\}=\\mathrm\{H\}\(i\), which reveals the exact labelOt=Xi∈\{0,1\}O\_\{t\}=X\_\{i\}\\in\\\{0,1\\\};
3. \(iii\)stop, denotedAt=STOPA\_\{t\}=\\text\{STOP\}, and emit a final decisionδπ∈\{0,1\}\\delta^\{\\pi\}\\in\\\{0,1\\\}, whereδπ=1\\delta^\{\\pi\}=1means rejectH0H\_\{0\}\.

Each item may be AI\-queried at most once and human\-queried at most once\. The policy may query the AI on itemiibefore or after querying a human on the same item\. However, once the human reveals the exact labelXiX\_\{i\}, any subsequent AI query on itemiiyields no additional information about the hypothesis and wastes costcAIc\_\{\\mathrm\{AI\}\}\. As we show formally in Section[4](https://arxiv.org/html/2609.28859#S4), such queries contribute zero statistical evidence and are therefore never used by an optimal policy\.

To formalize the information available to the policy at each time step, let

ℋt:=σ⁡\(A1,O1,…,At,Ot\)\\mathcal\{H\}\_\{t\}:=\\sigma\(A\_\{1\},O\_\{1\},\\ldots,A\_\{t\},O\_\{t\}\)be the filtration generated by the actions and observations up to the end of epochtt, withℋ0=∅\\mathcal\{H\}\_\{0\}=\\emptyset\. This filtration is defined for allttsuch thatAt′≠STOPA\_\{t^\{\\prime\}\}\\neq\\text\{STOP\}for allt′≤tt^\{\\prime\}\\leq t\. To allow for randomized policies, letU=\(Ut\)t=12​Nπ\+1U=\(U\_\{t\}\)\_\{t=1\}^\{2N^\{\\pi\}\+1\}be a vector of i\.i\.d\. random seeds, independent of\(Xi,Ri\)i=1Nπ\(X\_\{i\},R\_\{i\}\)\_\{i=1\}^\{N^\{\\pi\}\}under both hypotheses, with common lawμ\\muthat does not depend on the hypothesis\. The seedUtU\_\{t\}is used at epochttto randomize the policy’s action\.

At each epochtt, the set of actions available to the policy is

𝒜t\(ℋt−1\):=\{\\displaystyle\\mathcal\{A\}\_\{t\}\(\\mathcal\{H\}\_\{t\-1\}\):=\\\{AI\(i\):i∈\[Nπ\],inot yet AI\-queried\}\\displaystyle\\mathrm\{AI\}\(i\):i\\in\[N^\{\\pi\}\],i\\text\{ not yet AI\-queried\}\\\}∪\{H\(i\):i∈\[Nπ\],inot yet human\-queried\}\\displaystyle\\cup\\,\\\{\\mathrm\{H\}\(i\):i\\in\[N^\{\\pi\}\],i\\text\{ not yet human\-queried\}\\\}∪\{STOP\},\\displaystyle\\cup\\,\\\{\\text\{STOP\}\\\},which encodes the constraint that each item may be AI\-queried at most once and human\-queried at most once\. We now formally define an admissible policy\.

###### Definition 3\.1\(Admissible policy\)

An admissible policy is a sequence of measurable functionsπ=\(π1,π2,…\)\\pi=\(\\pi\_\{1\},\\pi\_\{2\},\\ldots\)such thatAt=STOPA\_\{t\}=\\text\{STOP\}for somet∈\[2​Nπ\+1\]t\\in\[2N^\{\\pi\}\+1\], and

πt:ℋt−1×Ut→At∈𝒜t​\(ℋt−1\),if​At′≠STOPfor all​t′<t,πt:ℋt−1×Ut→δπ∈\{0,1\},if​At=STOP\.\\begin\{split\}\\pi\_\{t\}:\\mathcal\{H\}\_\{t\-1\}\\times U\_\{t\}\\to A\_\{t\}\\in\\mathcal\{A\}\_\{t\}\(\\mathcal\{H\}\_\{t\-1\}\),&\\quad\\text\{if \}A\_\{t^\{\\prime\}\}\\neq\\text\{STOP\}\\text\{ for all \}t^\{\\prime\}<t,\\\\ \\pi\_\{t\}:\\mathcal\{H\}\_\{t\-1\}\\times U\_\{t\}\\to\\delta^\{\\pi\}\\in\\\{0,1\\\},&\\quad\\text\{if \}A\_\{t\}=\\text\{STOP\}\.\\end\{split\}

Definition[3\.1](https://arxiv.org/html/2609.28859#S3.Thmtheorem1)captures the key requirements of a valid policy: actions are chosen based only on the observed history and the current random seed, the policy is guaranteed to stop within2​Nπ\+12N^\{\\pi\}\+1steps, and upon stopping it emits a binary decision to reject or acceptH0H\_\{0\}\.

Not all admissible policies have good statistical properties\. We restrict attention to policies that simultaneously control both type\-I and type\-II errors at prescribed levels\. LetTstopπ:=min⁡\{t:At=STOP\}T^\{\\pi\}\_\{\\text\{stop\}\}:=\\min\\\{t:A\_\{t\}=\\text\{STOP\}\\\}be the stopping time of the policy, and letℙhπ\\mathbb\{P\}^\{\\pi\}\_\{h\}and𝔼hπ\\mathbb\{E\}^\{\\pi\}\_\{h\}denote probability and expectation under hypothesisHhH\_\{h\}and policyπ\\pi\.

###### Definition 3\.2\(Feasible policy\)

A policyπ\\piis feasible for target errors\(α,β\)\(\\alpha,\\beta\)with0<α<10<\\alpha<1,0<β<10<\\beta<1, andα\+β<1\\alpha\+\\beta<1, if it is admissible and satisfies

ℙ0π​\(δπ=1\)⏟type\-I error≤α,ℙ1π​\(δπ=0\)⏟type\-II error≤β\.\\underbrace\{\\mathbb\{P\}^\{\\pi\}\_\{0\}\(\\delta^\{\\pi\}=1\)\}\_\{\\text\{type\-I error\}\}\\leq\\alpha,\\qquad\\underbrace\{\\mathbb\{P\}^\{\\pi\}\_\{1\}\(\\delta^\{\\pi\}=0\)\}\_\{\\text\{type\-II error\}\}\\leq\\beta\.\(5\)We letℱ⁡\(α,β\)\\mathcal\{F\}\(\\alpha,\\beta\)denote the class of all feasible policies for target errors\(α,β\)\(\\alpha,\\beta\)\.

By restricting toℱ⁡\(α,β\)\\mathcal\{F\}\(\\alpha,\\beta\), we ensure that the policies we consider provide meaningful statistical guarantees, and our goal is to find the one among them that minimizes cost\.

### 3\.3Optimal Cost

There are three cost components: a data acquisition costcdata\>0c\_\{\\mathrm\{data\}\}\>0per acquired item, an AI query costcAI\>0c\_\{\\mathrm\{AI\}\}\>0per AI query, and a human query costcH\>0c\_\{\\mathrm\{H\}\}\>0per human query\. While in most practical settings we havecH≫cAIc\_\{\\mathrm\{H\}\}\\gg c\_\{\\mathrm\{AI\}\}, we do not impose this ordering in our analysis; our results hold for any positive cost parameters\. Let

NAItot:=∑t=12​Nπ\+1𝟏\{At=AI\(i\)for somei∈\[Nπ\]\}N^\{\\mathrm\{tot\}\}\_\{\\mathrm\{AI\}\}:=\\sum\_\{t=1\}^\{2N^\{\\pi\}\+1\}\\mathbf\{1\}\\\{A\_\{t\}=\\mathrm\{AI\}\(i\)\\text\{ for some \}i\\in\[N^\{\\pi\}\]\\\}be the total number of AI queries, and

NHtot:=∑t=12​Nπ\+1𝟏\{At=H\(i\)for somei∈\[Nπ\]\}N^\{\\mathrm\{tot\}\}\_\{\\mathrm\{H\}\}:=\\sum\_\{t=1\}^\{2N^\{\\pi\}\+1\}\\mathbf\{1\}\\\{A\_\{t\}=\\mathrm\{H\}\(i\)\\text\{ for some \}i\\in\[N^\{\\pi\}\]\\\}be the total number of human queries\. The total cost incurred by policyπ\\piis

Cπ:=cdata​Nπ\+cAI​NAItot\+cH​NHtot\.C^\{\\pi\}:=c\_\{\\mathrm\{data\}\}N^\{\\pi\}\+c\_\{\\mathrm\{AI\}\}N^\{\\mathrm\{tot\}\}\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}N^\{\\mathrm\{tot\}\}\_\{\\mathrm\{H\}\}\.
Since the true hypothesis is unknown, we evaluate a policy by its worst\-case expected cost over the two hypotheses as typically done in the hypothesis testing literature\([Wald 1945](https://arxiv.org/html/2609.28859#bib.bib43),[Baraud 2002](https://arxiv.org/html/2609.28859#bib.bib5)\)\. Specifically, the optimal cost is

C∗​\(α,β\):=infπ∈ℱ⁡\(α,β\)max⁡\{𝔼0π​\[Cπ\],𝔼1π​\[Cπ\]\}\.C^\{\*\}\(\\alpha,\\beta\):=\\inf\_\{\\pi\\in\\mathcal\{F\}\(\\alpha,\\beta\)\}\\max\\bigl\\\{\\mathbb\{E\}^\{\\pi\}\_\{0\}\[C^\{\\pi\}\],\\mathbb\{E\}^\{\\pi\}\_\{1\}\[C^\{\\pi\}\]\\bigr\\\}\.\(6\)
An exact closed\-form solution of \([6](https://arxiv.org/html/2609.28859#S3.E6)\) is generally intractable\. Thus, we will study the asymptotic regime in which the target errorsα\\alphaandβ\\betavanish\. This regime is practically relevant because in many high\-stakes applications \(such as medical diagnosis, legal adjudication, or quality control\) decision makers often require stringent error guarantees, and it is precisely in this low\-error regime that the structure of the optimal policy becomes clear and analytically tractable\. In this regime, we seek a policy that is asymptotically optimal, i\.e\., a feasible policyπ∈ℱ⁡\(α,β\)\\pi\\in\\mathcal\{F\}\(\\alpha,\\beta\)satisfying

max⁡\{𝔼0π​\[Cπ\],𝔼1π​\[Cπ\]\}C∗​\(α,β\)→1as​α,β→0\.\\frac\{\\max\\\{\\mathbb\{E\}^\{\\pi\}\_\{0\}\[C^\{\\pi\}\],\\mathbb\{E\}^\{\\pi\}\_\{1\}\[C^\{\\pi\}\]\\\}\}\{C^\{\*\}\(\\alpha,\\beta\)\}\\to 1\\qquad\\text\{as \}\\alpha,\\beta\\to 0\.In other words, the policy achieves the same leading\-order cost as the best possible feasible policy, with only lower\-order terms left unmatched\.

We construct such a policy in Section[5](https://arxiv.org/html/2609.28859#S5)\. To guide its construction and to certify its optimality, we first derive in Section[4](https://arxiv.org/html/2609.28859#S4)a lower bound onC∗​\(α,β\)C^\{\*\}\(\\alpha,\\beta\)that any feasible policy must satisfy\.

## 4Lower Bound on the Optimal Cost

In this section, we derive a lower bound on the optimal costC∗​\(α,β\)C^\{\*\}\(\\alpha,\\beta\)defined in \([6](https://arxiv.org/html/2609.28859#S3.E6)\)\. The analysis of the lower bound has two components\. First, in Section[4\.1](https://arxiv.org/html/2609.28859#S4.SS1), we show that any feasible policy must acquire at leastNfixed,H​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)items, the minimum number of items needed to test \([1](https://arxiv.org/html/2609.28859#S3.E1)\) even with full access to all labels\. Second, in Sections[4\.2](https://arxiv.org/html/2609.28859#S4.SS2)and[4\.4](https://arxiv.org/html/2609.28859#S4.SS4), we use information\-theoretic arguments to show that any feasible policy must spend enough on AI and human queries to accumulate sufficient statistical evidence to distinguishH0H\_\{0\}fromH1H\_\{1\}\. Each item can contribute evidence in one of three informative ways: through an AI query alone, through a human query alone, or through both an AI query and a human query on the same item\. Each of these contributes a quantifiable amount of statistical evidence at a certain cost, and the lower bound captures the minimum cost of assembling enough evidence to meet the error constraints\. Together, these two components yield the lower boundLB⁡\(α,β\)\\mathrm\{LB\}\(\\alpha,\\beta\), which we show in Section[5](https://arxiv.org/html/2609.28859#S5)is achievable to first order by our proposed policy\.

### 4\.1Lower Bound on Number of Acquired Samples

We begin by establishing a fundamental lower bound on the number of items any feasible policy must acquire\. The benchmark isNfixed,H​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\), the minimum number of items needed to test \([1](https://arxiv.org/html/2609.28859#S3.E1)\) even in the idealized setting where all labels are observed directly, with no AI or human query costs\. Any feasible policy in our setting, which has access to strictly less information per item than the full label, cannot possibly require fewer items\.

###### Definition 4\.1\(Full\-label fixed\-sample\-size benchmark\)

For0<α<10<\\alpha<1and0<β<10<\\beta<1, letNfixed,HN\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(α,β\)\(\\alpha,\\beta\)be the smallest integerN∈ℕN\\in\\mathbb\{N\}for which there exists a randomized testϕN:\{0,1\}N→\[0,1\]\\phi\_\{N\}:\\\{0,1\\\}^\{N\}\\to\[0,1\]satisfying

𝔼0​\[ϕN​\(X1,…,XN\)\]≤α,𝔼1​\[1−ϕN​\(X1,…,XN\)\]≤β,\\mathbb\{E\}\_\{0\}\[\\phi\_\{N\}\(X\_\{1\},\\ldots,X\_\{N\}\)\]\\leq\\alpha,\\qquad\\mathbb\{E\}\_\{1\}\[1\-\\phi\_\{N\}\(X\_\{1\},\\ldots,X\_\{N\}\)\]\\leq\\beta,whereϕN​\(x\)∈\[0,1\]\\phi\_\{N\}\(x\)\\in\[0,1\]is the probability of rejectingH0H\_\{0\}upon observing the full label vectorx∈\{0,1\}Nx\\in\\\{0,1\\\}^\{N\}, and𝔼h\\mathbb\{E\}\_\{h\}denotes expectation underHhH\_\{h\}\.

We first show that any feasible policyπ∈ℱ⁡\(α,β\)\\pi\\in\\mathcal\{F\}\(\\alpha,\\beta\)must acquire at leastNfixed,H​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)items\.

###### Lemma 4\.2\(Full\-label data\-pool lower bound\)

Suppose Assumptions[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)and[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)hold\. If a feasible policyπ∈ℱ⁡\(α,β\)\\pi\\in\\mathcal\{F\}\(\\alpha,\\beta\)acquiresNNitems, thenN≥Nfixed,H​\(α,β\)\.N\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)\.

The intuition behind Lemma[4\.2](https://arxiv.org/html/2609.28859#S4.Thmtheorem2)is straightforward\. Even if a policy could somehow observe allNNtrue labels at no cost, it would still need at leastNfixed,H​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)items to meet the error constraints\. A policy in our setting, which must pay for each label it observes and receives only noisy AI reports on some items, has access to no more information than the full\-label benchmark\. It therefore cannot satisfy the same error constraints with fewer items\.

We now describe how to computeNfixed,H​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)explicitly\. This construction is also needed for the fallback test in Section[5](https://arxiv.org/html/2609.28859#S5)\. By the Neyman–Pearson lemma\([Neyman and Pearson 1933](https://arxiv.org/html/2609.28859#bib.bib30)\), for any fixed sample sizeNN, the most powerful test at type\-I levelα\\alphais the likelihood\-ratio test\. Since the likelihood ratio is strictly increasing in the label sumLN:=∑i=1NXiL\_\{N\}:=\\sum\_\{i=1\}^\{N\}X\_\{i\}\(as we show below\), this reduces to a simple threshold test onLNL\_\{N\}\. We can therefore computeNfixed,H​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)by increasingNNfrom11upward and checking whether the threshold test meets the type\-II targetβ\\beta\.

FixN∈ℕN\\in\\mathbb\{N\}and suppose the full label vectorX=\(X1,…,XN\)X=\(X\_\{1\},\\ldots,X\_\{N\}\)is observed\. UnderHhH\_\{h\},LNL\_\{N\}has a binomial distribution with parametersNNandphp\_\{h\}\. The likelihood ratio underH1H\_\{1\}versusH0H\_\{0\}is

ΛN​\(X\)=∏i=1Np1Xi​\(1−p1\)1−Xip0Xi​\(1−p0\)1−Xi=\(p1p0\)LN​\(1−p11−p0\)N−LN\.\\displaystyle\\Lambda\_\{N\}\(X\)=\\prod\_\{i=1\}^\{N\}\\frac\{p\_\{1\}^\{X\_\{i\}\}\(1\-p\_\{1\}\)^\{1\-X\_\{i\}\}\}\{p\_\{0\}^\{X\_\{i\}\}\(1\-p\_\{0\}\)^\{1\-X\_\{i\}\}\}=\\left\(\\frac\{p\_\{1\}\}\{p\_\{0\}\}\\right\)^\{L\_\{N\}\}\\left\(\\frac\{1\-p\_\{1\}\}\{1\-p\_\{0\}\}\\right\)^\{N\-L\_\{N\}\}\.Sincep1\>p0p\_\{1\}\>p\_\{0\}, the log\-likelihood ratio

log⁡ΛN​\(X\)=LN​log⁡p1p0\+\(N−LN\)​log⁡1−p11−p0\\log\\Lambda\_\{N\}\(X\)=L\_\{N\}\\log\\frac\{p\_\{1\}\}\{p\_\{0\}\}\+\(N\-L\_\{N\}\)\\log\\frac\{1\-p\_\{1\}\}\{1\-p\_\{0\}\}has a strictly positive coefficient onLNL\_\{N\}, soΛN​\(X\)\\Lambda\_\{N\}\(X\)is strictly increasing inLNL\_\{N\}\. The optimal test therefore rejectsH0H\_\{0\}whenLNL\_\{N\}is large\.

For a target type\-I levelα\\alpha, definecNc\_\{N\}to be the smallest integer in\{0,1,…,N\}\\\{0,1,\\ldots,N\\\}such thatℙ0​\(LN\>cN\)≤α\\mathbb\{P\}\_\{0\}\(L\_\{N\}\>c\_\{N\}\)\\leq\\alpha, and set

γN:=α−ℙ0​\(LN\>cN\)ℙ0​\(LN=cN\)\.\\gamma\_\{N\}:=\\frac\{\\alpha\-\\mathbb\{P\}\_\{0\}\(L\_\{N\}\>c\_\{N\}\)\}\{\\mathbb\{P\}\_\{0\}\(L\_\{N\}=c\_\{N\}\)\}\.The denominator is positive because0<p0<10<p\_\{0\}<1, andγN∈\[0,1\]\\gamma\_\{N\}\\in\[0,1\]by the definition ofcNc\_\{N\}\. This gives rise to the following test\.

###### Definition 4\.3\(Neyman–Pearson test\([Casella and Berger 2024](https://arxiv.org/html/2609.28859#bib.bib9)\)\)

The randomized threshold test

ϕN∗​\(x\)=\{1,∑i=1Nxi\>cN,γN,∑i=1Nxi=cN,0,∑i=1Nxi<cN,\\phi\_\{N\}^\{\*\}\(x\)=\\begin\{cases\}1,&\\sum^\{N\}\_\{i=1\}x\_\{i\}\>c\_\{N\},\\\\ \\gamma\_\{N\},&\\sum^\{N\}\_\{i=1\}x\_\{i\}=c\_\{N\},\\\\ 0,&\\sum^\{N\}\_\{i=1\}x\_\{i\}<c\_\{N\},\\end\{cases\}wherex=\(x1,…,xN\)∈\{0,1\}Nx=\(x\_\{1\},\\ldots,x\_\{N\}\)\\in\\\{0,1\\\}^\{N\}, is called the Neyman–Pearson test\.

By construction,ϕN∗\\phi\_\{N\}^\{\*\}meets the type\-I constraint exactly:𝔼0​\[ϕN∗​\(X\)\]=α\\mathbb\{E\}\_\{0\}\[\\phi\_\{N\}^\{\*\}\(X\)\]=\\alpha\. Its type\-II error is

𝔼1​\[1−ϕN∗​\(X\)\]=ℙ1​\(LN<cN\)\+\(1−γN\)​ℙ1​\(LN=cN\)\.\\mathbb\{E\}\_\{1\}\[1\-\\phi\_\{N\}^\{\*\}\(X\)\]=\\mathbb\{P\}\_\{1\}\(L\_\{N\}<c\_\{N\}\)\+\(1\-\\gamma\_\{N\}\)\\mathbb\{P\}\_\{1\}\(L\_\{N\}=c\_\{N\}\)\.\(7\)The Neyman–Pearson lemma guarantees that this test is optimal, which we state formally below\.

###### Lemma 4\.4\(Neyman–Pearson lemma\([Neyman and Pearson 1933](https://arxiv.org/html/2609.28859#bib.bib30)\)\)

The testϕN∗​\(X\)\\phi\_\{N\}^\{\*\}\(X\)has the smallest type\-II error among all testsϕ⁡\(X\)\\phi\(X\)satisfying𝔼0​\[ϕ⁡\(X\)\]≤α\\mathbb\{E\}\_\{0\}\[\\phi\(X\)\]\\leq\\alpha\. That is,ϕN∗\\phi\_\{N\}^\{\*\}is the most powerful test at significance levelα\\alpha\.

It follows thatNfixed,H​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)can be computed exactly by increasingNNfrom11upward and checking whether \([7](https://arxiv.org/html/2609.28859#S4.E7)\) is at mostβ\\beta\. The logic follows from the Neyman–Pearson lemma: at eachNN,ϕN∗\\phi\_\{N\}^\{\*\}has the smallest type\-II error of any test with type\-I error at mostα\\alpha\. So ifϕN∗\\phi\_\{N\}^\{\*\}fails to meet the type\-II targetβ\\beta, no test at that sample size can\. Conversely, ifϕN∗\\phi\_\{N\}^\{\*\}does meetβ\\beta, it is itself a valid test meeting both error targets\. The firstNNat which \([7](https://arxiv.org/html/2609.28859#S4.E7)\) is at mostβ\\betais thereforeNfixed,H​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)\.

### 4\.2Information Quantities

At its core, any feasible policy must accumulate enough statistical evidence to reliably distinguishH0H\_\{0\}fromH1H\_\{1\}\. The fundamental currency of this evidence is KL divergence: it quantifies how distinguishable the distribution of observations is underH1H\_\{1\}versusH0H\_\{0\}, and therefore measures how much each query contributes toward meeting the error constraints\. We now define the KL\-divergence quantities associated with each possible query type\.

We use two standard KL\-divergence quantities throughout\. Fora,b∈\(0,1\)a,b\\in\(0,1\), the Bernoulli KL divergence is given by

kl\(a∥b\):=alogab\+\(1−a\)log1−a1−b\.\\operatorname\{kl\}\(a\\\|b\):=a\\log\\frac\{a\}\{b\}\+\(1\-a\)\\log\\frac\{1\-a\}\{1\-b\}\.For probability mass functionsPPandQQon a common finite setSS, the KL divergence is

D\(P∥Q\):=∑s∈SP\(s\)logP⁡\(s\)Q⁡\(s\),D\(P\\\|Q\):=\\sum\_\{s\\in S\}P\(s\)\\log\\frac\{P\(s\)\}\{Q\(s\)\},with conventions0​log⁡\(0/q\)=00\\log\(0/q\)=0andp​log⁡\(p/0\)=\+∞p\\log\(p/0\)=\+\\inftyforp\>0p\>0\.

##### AI query only\.

When the policy queries only the AI on itemii, it observes the reportRiR\_\{i\}with marginal distributionGhG\_\{h\}\(with probability mass functionghg\_\{h\}\) underHhH\_\{h\}\. Since the two error constraints \([5](https://arxiv.org/html/2609.28859#S3.E5)\) are asymmetric \(one applies whenH0H\_\{0\}is true and the other whenH1H\_\{1\}is true\) we need to track the discriminating power of an AI query under each hypothesis separately\. Specifically,IR\(1\)I\_\{R\}^\{\(1\)\}measures how informative the AI report is for distinguishingH1H\_\{1\}fromH0H\_\{0\}whenH1H\_\{1\}is the true hypothesis, andIR\(0\)I\_\{R\}^\{\(0\)\}measures the same whenH0H\_\{0\}is true:

IR\(1\):=D\(G1∥G0\)=∑r∈ℛg1\(r\)logg1​\(r\)g0​\(r\),I\_\{R\}^\{\(1\)\}:=D\(G\_\{1\}\\\|G\_\{0\}\)=\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{1\}\(r\)\\log\\frac\{g\_\{1\}\(r\)\}\{g\_\{0\}\(r\)\},\(8\)IR\(0\):=D\(G0∥G1\)=∑r∈ℛg0\(r\)logg0​\(r\)g1​\(r\)\.I\_\{R\}^\{\(0\)\}:=D\(G\_\{0\}\\\|G\_\{1\}\)=\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{0\}\(r\)\\log\\frac\{g\_\{0\}\(r\)\}\{g\_\{1\}\(r\)\}\.\(9\)In generalIR\(1\)≠IR\(0\)I\_\{R\}^\{\(1\)\}\\neq I\_\{R\}^\{\(0\)\}, reflecting the fact that the AI report may be more informative under one hypothesis than the other\.

##### Human query only\.

If the policy queries a human on itemiiwith no prior AI query, the revealed labelXiX\_\{i\}is distributed asBernoulli⁡\(ph\)\\operatorname\{Bernoulli\}\(p\_\{h\}\)underHhH\_\{h\}\. By the same reasoning as above, we track the discriminating power of a human query under each hypothesis separately\. Specifically,JX\(1\)J\_\{X\}^\{\(1\)\}measures how informative a directly revealed label is for distinguishingH1H\_\{1\}fromH0H\_\{0\}whenH1H\_\{1\}is true, andJX\(0\)J\_\{X\}^\{\(0\)\}measures the same whenH0H\_\{0\}is true:

JX\(1\):=kl\(p1∥p0\),JX\(0\):=kl\(p0∥p1\),J\_\{X\}^\{\(1\)\}:=\\operatorname\{kl\}\(p\_\{1\}\\\|p\_\{0\}\),\\qquad J\_\{X\}^\{\(0\)\}:=\\operatorname\{kl\}\(p\_\{0\}\\\|p\_\{1\}\),\(10\)both of which are strictly positive because0<p0<p1<10<p\_\{0\}<p\_\{1\}<1\. Note thatJX\(1\)≠JX\(0\)J\_\{X\}^\{\(1\)\}\\neq J\_\{X\}^\{\(0\)\}in general, for the same reason as above\.

##### Human query followed by AI query\.

If itemiiis first queried by a human, revealing the exact labelXiX\_\{i\}, and is then queried by the AI, the AI reportRiR\_\{i\}carries no additional information: the true labelXiX\_\{i\}is already known, so the AI output is redundant\. This query ordering therefore contributes zero additional KL divergence and is never used by an optimal policy\.

##### AI query followed by human query\.

If the policy first queries the AI on itemii, receiving reportRi=rR\_\{i\}=r, and then queries a human on the same item, the human reveals the exact labelXiX\_\{i\}\. Conditional onRi=rR\_\{i\}=r, the labelXiX\_\{i\}is distributed asBernoulli⁡\(qh​\(r\)\)\\operatorname\{Bernoulli\}\(q\_\{h\}\(r\)\)underHhH\_\{h\}by \([4](https://arxiv.org/html/2609.28859#S3.E4)\)\. The information this human label carries for discriminatingH1H\_\{1\}fromH0H\_\{0\}, beyond what the AI reportrralready provided, depends on which hypothesis is true\. WhenH1H\_\{1\}is true, the additional information is

d\(1\)\(r\):=D\(ρ1\(⋅∣r\)∥ρ0\(⋅∣r\)\)=kl\(q1\(r\)∥q0\(r\)\),d^\{\(1\)\}\(r\):=D\(\\rho\_\{1\}\(\\cdot\\mid r\)\\\|\\rho\_\{0\}\(\\cdot\\mid r\)\)=\\operatorname\{kl\}\(q\_\{1\}\(r\)\\\|q\_\{0\}\(r\)\),\(11\)and whenH0H\_\{0\}is true it is

d\(0\)\(r\):=D\(ρ0\(⋅∣r\)∥ρ1\(⋅∣r\)\)=kl\(q0\(r\)∥q1\(r\)\)\.d^\{\(0\)\}\(r\):=D\(\\rho\_\{0\}\(\\cdot\\mid r\)\\\|\\rho\_\{1\}\(\\cdot\\mid r\)\)=\\operatorname\{kl\}\(q\_\{0\}\(r\)\\\|q\_\{1\}\(r\)\)\.\(12\)Note thatd\(h\)​\(r\)d^\{\(h\)\}\(r\)depends on the AI reportrr: a more informative AI report leaves less residual uncertainty aboutXiX\_\{i\}, and hence contributes less additional information when the human is subsequently queried\.

### 4\.3Supporting Results for the Lower Bound

To derive a lower bound onC∗​\(α,β\)C^\{\*\}\(\\alpha,\\beta\), we need to understand how much statistical evidence any admissible policy can accumulate, and at what cost\. The key insight is that any feasible policy must accumulate enough statistical evidence to distinguishH0H\_\{0\}fromH1H\_\{1\}, and this evidence can only be obtained by paying for AI queries, human queries, or both\. We formalize this by measuring the statistical evidence in terms of KL divergence between the distributions of the policy’s actions and observations under the two hypotheses\.

For a policyπ\\piwith stopping timeTstopπT^\{\\pi\}\_\{\\text\{stop\}\}, we call the realized sequence

\(A1,O1,…,ATstopπ−1,OTstopπ−1,ATstopπ,δπ\)\(A\_\{1\},O\_\{1\},\\ldots,A\_\{T^\{\\pi\}\_\{\\text\{stop\}\}\-1\},O\_\{T^\{\\pi\}\_\{\\text\{stop\}\}\-1\},A\_\{T^\{\\pi\}\_\{\\text\{stop\}\}\},\\delta^\{\\pi\}\)the*transcript*of the policy\. LetPhπP^\{\\pi\}\_\{h\}denote the distribution of the transcript underHhH\_\{h\}\. The KL divergenceD\(P1π∥P0π\)D\(P\_\{1\}^\{\\pi\}\\\|P\_\{0\}^\{\\pi\}\)is the expected log\-likelihood ratio of the transcript distribution underH1H\_\{1\}, measuring how muchP1πP\_\{1\}^\{\\pi\}favorsH1H\_\{1\}overH0H\_\{0\}\. Similarly,D\(P0π∥P1π\)D\(P\_\{0\}^\{\\pi\}\\\|P\_\{1\}^\{\\pi\}\)is the expected log\-likelihood ratio underH0H\_\{0\}, measuring how muchP0πP\_\{0\}^\{\\pi\}favorsH0H\_\{0\}overH1H\_\{1\}\. Both must be large enough for the policy to reliably distinguish the two hypotheses and meet the error constraints \([5](https://arxiv.org/html/2609.28859#S3.E5)\)\.

###### Lemma 4\.5\(Testing errors imply transcript KL requirements\)

Every feasible policyπ∈ℱ⁡\(α,β\)\\pi\\in\\mathcal\{F\}\(\\alpha,\\beta\)satisfies

D\(P1π∥P0π\)≥A,D\(P0π∥P1π\)≥B,D\(P\_\{1\}^\{\\pi\}\\\|P\_\{0\}^\{\\pi\}\)\\geq A,\\qquad D\(P\_\{0\}^\{\\pi\}\\\|P\_\{1\}^\{\\pi\}\)\\geq B,where

A:=kl\(1−β∥α\),B:=kl\(1−α∥β\)\.A:=\\operatorname\{kl\}\(1\-\\beta\\\|\\alpha\),\\qquad B:=\\operatorname\{kl\}\(1\-\\alpha\\\|\\beta\)\.

Lemma[4\.5](https://arxiv.org/html/2609.28859#S4.Thmtheorem5)says that meeting the error constraints forces the transcript distributionsP1πP\_\{1\}^\{\\pi\}andP0πP\_\{0\}^\{\\pi\}to be sufficiently separated:P1πP\_\{1\}^\{\\pi\}must be at leastAAunits of KL divergence away fromP0πP\_\{0\}^\{\\pi\}, andP0πP\_\{0\}^\{\\pi\}must be at leastBBunits away fromP1πP\_\{1\}^\{\\pi\}\. We next characterize exactly how this total KL information accumulates across the policy’s query decisions\.

For a policyπ\\pi, letℐAIπ\\mathcal\{I\}\_\{\\mathrm\{AI\}\}^\{\\pi\}be the set of items whose first query is an AI query \(which may or may not be followed by a human query\),ℐHπ\\mathcal\{I\}\_\{\\mathrm\{H\}\}^\{\\pi\}the set of items whose first query is a human query with no preceding AI query, andℐAIHπ\\mathcal\{I\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}^\{\\pi\}the set of items that are human\-queried after their AI report has been observed\. LetNAIdir,π:=\|ℐAIπ\|N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}:=\|\\mathcal\{I\}\_\{\\mathrm\{AI\}\}^\{\\pi\}\|,NHdir,π:=\|ℐHπ\|N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}:=\|\\mathcal\{I\}\_\{\\mathrm\{H\}\}^\{\\pi\}\|, andNAIHπ:=\|ℐAIHπ\|N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}:=\|\\mathcal\{I\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}^\{\\pi\}\|be the corresponding random counts\. Note that these counts are random and their expectations underH0H\_\{0\}andH1H\_\{1\}differ in general\. An AI query on an item that has already received a human query contributes zero KL information, since the report lawfxf\_\{x\}does not depend on the hypothesis once the labelxxis known\.

###### Theorem 4\.6\(Adaptive KL decomposition\)

Suppose Assumptions[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)and[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)hold\. For every admissible policyπ\\piusingNπN^\{\\pi\}items,

D\(P1π∥P0π\)\\displaystyle D\(P\_\{1\}^\{\\pi\}\\\|P\_\{0\}^\{\\pi\}\)=𝔼1π​\[NAIdir,π​IR\(1\)\+NHdir,π​JX\(1\)\+∑i∈ℐAIHπd\(1\)​\(Ri\)\],\\displaystyle=\\mathbb\{E\}^\{\\pi\}\_\{1\}\\\!\\left\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}I\_\{R\}^\{\(1\)\}\+N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}J\_\{X\}^\{\(1\)\}\+\\sum\_\{i\\in\\mathcal\{I\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}^\{\\pi\}\}d^\{\(1\)\}\(R\_\{i\}\)\\right\],\(13\)D\(P0π∥P1π\)\\displaystyle D\(P\_\{0\}^\{\\pi\}\\\|P\_\{1\}^\{\\pi\}\)=𝔼0π​\[NAIdir,π​IR\(0\)\+NHdir,π​JX\(0\)\+∑i∈ℐAIHπd\(0\)​\(Ri\)\]\.\\displaystyle=\\mathbb\{E\}^\{\\pi\}\_\{0\}\\\!\\left\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}I\_\{R\}^\{\(0\)\}\+N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}J\_\{X\}^\{\(0\)\}\+\\sum\_\{i\\in\\mathcal\{I\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}^\{\\pi\}\}d^\{\(0\)\}\(R\_\{i\}\)\\right\]\.\(14\)

Theorem[4\.6](https://arxiv.org/html/2609.28859#S4.Thmtheorem6)shows that the transcript KL divergence decomposes exactly into three per\-item contributions: each AI\-first item contributesIR\(h\)I\_\{R\}^\{\(h\)\}, each human\-first item contributesJX\(h\)J\_\{X\}^\{\(h\)\}, and each AI\-first item that is subsequently human\-queried contributes an additionald\(h\)​\(Ri\)d^\{\(h\)\}\(R\_\{i\}\)depending on its realized reportRiR\_\{i\}\. The escalation term∑i∈ℐAIHπd\(h\)​\(Ri\)\\sum\_\{i\\in\\mathcal\{I\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}^\{\\pi\}\}d^\{\(h\)\}\(R\_\{i\}\)depends on the realized reports, making it difficult to work with directly\. We therefore define, fors∈\[0,1\]s\\in\[0,1\],

Ψh​\(s\):=infλ≥0\{λ​s\+∑r∈ℛgh​\(r\)​\[d\(h\)​\(r\)−λ\]\+\},\\Psi\_\{h\}\(s\):=\\inf\_\{\\lambda\\geq 0\}\\left\\\{\\lambda s\+\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\left\[d^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}\\right\\\},\(15\)which provides a deterministic upper bound on the expected escalation term, as shown in Lemma[4\.9](https://arxiv.org/html/2609.28859#S4.Thmtheorem9)below\. The following lemma establishes the key properties ofΨh\\Psi\_\{h\}, including a dual representation that givesΨh​\(s\)\\Psi\_\{h\}\(s\)a natural interpretation\.

###### Lemma 4\.7\(Properties of the follow\-up frontier\)

For eachh∈\{0,1\}h\\in\\\{0,1\\\},Ψh\\Psi\_\{h\}is nondecreasing, concave, and continuous on\[0,1\]\[0,1\]\. By linear programming duality, it admits the equivalent dual representation

Ψh\(s\)=maxη:ℛ→\[0,1\]\{∑r∈ℛgh\(r\)η\(r\)d\(h\)\(r\):∑r∈ℛgh\(r\)η\(r\)≤s\}\.\\Psi\_\{h\}\(s\)=\\max\_\{\\eta:\\mathcal\{R\}\\to\[0,1\]\}\\left\\\{\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\eta\(r\)d^\{\(h\)\}\(r\):\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\eta\(r\)\\leq s\\right\\\}\.\(16\)
Moreover,

Ψh​\(1\)=JX\(h\)−IR\(h\)\.\\Psi\_\{h\}\(1\)=J\_\{X\}^\{\(h\)\}\-I\_\{R\}^\{\(h\)\}\.\(17\)

The dual representation \([16](https://arxiv.org/html/2609.28859#S4.E16)\) reveals the interpretation ofΨh​\(s\)\\Psi\_\{h\}\(s\)\. The variableη⁡\(r\)∈\[0,1\]\\eta\(r\)\\in\[0,1\]can be interpreted as the probability of querying a human on an AI\-first item with reportrr\. UnderHhH\_\{h\}, the report equalsrrwith probabilitygh​\(r\)g\_\{h\}\(r\), so the expected KL information gained from human follow\-up queries under ruleη\\etais∑r∈ℛgh​\(r\)​η​\(r\)​d\(h\)​\(r\)\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\eta\(r\)d^\{\(h\)\}\(r\), and the expected number of follow\-up queries per AI\-first item is∑r∈ℛgh​\(r\)​η​\(r\)\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\eta\(r\)\. ThusΨh​\(s\)\\Psi\_\{h\}\(s\)is the maximum expected KL information extractable per AI\-first item, over all report\-dependent follow\-up rules, when the expected follow\-up rate is at mostss\. Settings=0s=0forbids any human follow\-up and givesΨh​\(0\)=0\\Psi\_\{h\}\(0\)=0; settings=1s=1permits following up every AI\-first item\.

Identity \([17](https://arxiv.org/html/2609.28859#S4.E17)\) then has a clean interpretation: when every AI\-first item is also followed up by a human \(s=1s=1\), the policy observes both the AI report and the true label\. The total KL information from observing both equalsJX\(h\)J\_\{X\}^\{\(h\)\}by the KL chain rule, of whichIR\(h\)I\_\{R\}^\{\(h\)\}was already contributed by the AI report alone\. The remainingJX\(h\)−IR\(h\)J\_\{X\}^\{\(h\)\}\-I\_\{R\}^\{\(h\)\}is the additional information the human follow\-up provides, which is exactlyΨh​\(1\)\\Psi\_\{h\}\(1\)\.

It is worth noting here that the optimalη⁡\(r\)\\eta\(r\)achievingΨh​\(s\)\\Psi\_\{h\}\(s\)provides an implementable report\-dependent escalation rule that we actually use to construct the matching policy in Section[5](https://arxiv.org/html/2609.28859#S5)\. Specifically, to construct the frontier optimizer, without loss of generality we can order the reports asr1\(h\),…,r\|ℛ\|\(h\)r^\{\(h\)\}\_\{1\},\\ldots,r^\{\(h\)\}\_\{\|\\mathcal\{R\}\|\}so that

d\(h\)​\(r1\(h\)\)≥d\(h\)​\(r2\(h\)\)≥⋯≥d\(h\)​\(r\|ℛ\|\(h\)\),d^\{\(h\)\}\(r^\{\(h\)\}\_\{1\}\)\\geq d^\{\(h\)\}\(r^\{\(h\)\}\_\{2\}\)\\geq\\cdots\\geq d^\{\(h\)\}\(r^\{\(h\)\}\_\{\|\\mathcal\{R\}\|\}\),with deterministic tie\-breaking\. Let

W0\(h\):=0,Wj\(h\):=∑ℓ=1jgh\(rℓ\(h\)\),j=1,…,\|ℛ\|\.W^\{\(h\)\}\_\{0\}:=0,\\qquad W^\{\(h\)\}\_\{j\}:=\\sum\_\{\\ell=1\}^\{j\}g\_\{h\}\(r^\{\(h\)\}\_\{\\ell\}\),\\qquad j=1,\\ldots,\|\\mathcal\{R\}\|\.Then the optimal solution to \([16](https://arxiv.org/html/2609.28859#S4.E16)\) is to escalate reports in decreasing order of conditional information and then randomize only at the marginal report value, given by Proposition[4\.8](https://arxiv.org/html/2609.28859#S4.Thmtheorem8)below\.

###### Proposition 4\.8

For everys∈\[0,1\]s\\in\[0,1\], let

ηh∗​\(rj\(h\),s\):=\{1,Wj\(h\)≤s,s−Wj−1\(h\)gh​\(rj\(h\)\),Wj−1\(h\)<s<Wj\(h\),0,Wj−1\(h\)≥s\.\\eta^\{\*\}\_\{h\}\(r^\{\(h\)\}\_\{j\};s\):=\\begin\{cases\}1,&W^\{\(h\)\}\_\{j\}\\leq s,\\\\\[4\.30554pt\] \\displaystyle\\frac\{s\-W^\{\(h\)\}\_\{j\-1\}\}\{g\_\{h\}\(r^\{\(h\)\}\_\{j\}\)\},&W^\{\(h\)\}\_\{j\-1\}<s<W^\{\(h\)\}\_\{j\},\\\\\[8\.61108pt\] 0,&W^\{\(h\)\}\_\{j\-1\}\\geq s\.\\end\{cases\}\(18\)Then\{ηh∗​\(r,s\)\}r∈ℛ\\\{\\eta^\{\*\}\_\{h\}\(r;s\)\\\}\_\{r\\in\\mathcal\{R\}\}is an optimal solution to \([16](https://arxiv.org/html/2609.28859#S4.E16)\) under directionhh\.

We can now bound the expected escalation term in Theorem[4\.6](https://arxiv.org/html/2609.28859#S4.Thmtheorem6)by a deterministic quantity\.

###### Lemma 4\.9\(Selective escalation information bound\)

Suppose Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)holds\. Fixh∈\{0,1\}h\\in\\\{0,1\\\}\. For any admissible policyπ\\pi, we have:

𝔼hπ​\[∑i∈ℐAIHπd\(h\)​\(Ri\)\]≤𝔼hπ​\[NAIdir,π\]​Ψh​\(𝔼hπ​\[NAIHπ\]𝔼hπ​\[NAIdir,π\]\)\.\\mathbb\{E\}^\{\\pi\}\_\{h\}\\\!\\left\[\\sum\_\{i\\in\\mathcal\{I\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}^\{\\pi\}\}d^\{\(h\)\}\(R\_\{i\}\)\\right\]\\leq\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\\,\\Psi\_\{h\}\\\!\\left\(\\frac\{\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]\}\{\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\}\\right\)\.\(19\)
If𝔼hπ​\[NAIdir,π\]=0\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]=0, then𝔼hπ​\[NAIHπ\]=0\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]=0and the right side is defined as00\.

With all the above results in hand, we are ready to derive the lower bound\. Lemma[4\.5](https://arxiv.org/html/2609.28859#S4.Thmtheorem5)tells us that any feasible policy must achieve KL divergences of at leastAAandBBin the two directions; Theorem[4\.6](https://arxiv.org/html/2609.28859#S4.Thmtheorem6)decomposes these KL divergences exactly into per\-item contributions from AI queries, human queries, and human follow\-up queries; Lemma[4\.9](https://arxiv.org/html/2609.28859#S4.Thmtheorem9)then bounds the human follow\-up contribution by a deterministic quantity involvingΨh\\Psi\_\{h\}\. Combining Lemma[4\.5](https://arxiv.org/html/2609.28859#S4.Thmtheorem5), Theorem[4\.6](https://arxiv.org/html/2609.28859#S4.Thmtheorem6), and Lemma[4\.9](https://arxiv.org/html/2609.28859#S4.Thmtheorem9), any feasible policyπ∈ℱ⁡\(α,β\)\\pi\\in\\mathcal\{F\}\(\\alpha,\\beta\)satisfies, for eachh∈\{0,1\}h\\in\\\{0,1\\\},

Th≤D\(Phπ∥P1−hπ\)≤𝔼hπ\[NHdir,π\]JX\(h\)\+𝔼hπ\[NAIdir,π\]IR\(h\)\+𝔼hπ\[NAIdir,π\]Ψh\(𝔼hπ​\[NAIHπ\]𝔼hπ​\[NAIdir,π\]\),T\_\{h\}\\leq D\(P\_\{h\}^\{\\pi\}\\\|P\_\{1\-h\}^\{\\pi\}\)\\leq\\mathbb\{E\}\_\{h\}^\{\\pi\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}\]\\,J\_\{X\}^\{\(h\)\}\+\\mathbb\{E\}\_\{h\}^\{\\pi\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\\,I\_\{R\}^\{\(h\)\}\+\\mathbb\{E\}\_\{h\}^\{\\pi\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\\,\\Psi\_\{h\}\\\!\\left\(\\frac\{\\mathbb\{E\}\_\{h\}^\{\\pi\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]\}\{\\mathbb\{E\}\_\{h\}^\{\\pi\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\}\\right\),\(20\)
whereT1:=AT\_\{1\}:=AandT0:=BT\_\{0\}:=B\. Crucially, the right\-hand side depends only on the expected query counts, reducing a stochastic constraint on the policy to a deterministic feasibility condition, which we exploit in the next subsection to derive the lower boundLB⁡\(α,β\)\\mathrm\{LB\}\(\\alpha,\\beta\)\.

### 4\.4Information\-Theoretic Lower Bound

We are now ready to derive the lower bound onC∗​\(α,β\)C^\{\*\}\(\\alpha,\\beta\)\. The three results in Section[4\.3](https://arxiv.org/html/2609.28859#S4.SS3)together imply that any feasible policy must incur a minimum cost to generate enough KL information to distinguishH0H\_\{0\}fromH1H\_\{1\}\. We formalize this as a cost minimization program\.

Forh∈\{0,1\}h\\in\\\{0,1\\\},T≥0T\\geq 0, and integerN≥1N\\geq 1, define

Γh​\(T,N\):=minnH,nAI,nesc≥0\\displaystyle\\Gamma\_\{h\}\(T,N\):=\\min\_\{n\_\{\\mathrm\{H\}\},\\,n\_\{\\mathrm\{AI\}\},\\,n\_\{\\mathrm\{esc\}\}\\geq 0\}cH​nH\+cAI​nAI\+cH​nesc\\displaystyle c\_\{\\mathrm\{H\}\}n\_\{\\mathrm\{H\}\}\+c\_\{\\mathrm\{AI\}\}n\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}n\_\{\\mathrm\{esc\}\}\(21\)subject to\\displaystyle\\text\{subject to\}nH\+nAI≤N,\\displaystyle n\_\{\\mathrm\{H\}\}\+n\_\{\\mathrm\{AI\}\}\\leq N,0≤nesc≤nAI,\\displaystyle 0\\leq n\_\{\\mathrm\{esc\}\}\\leq n\_\{\\mathrm\{AI\}\},nH​JX\(h\)\+nAI​IR\(h\)\+nAI​Ψh​\(nesc/nAI\)≥T\.\\displaystyle n\_\{\\mathrm\{H\}\}J\_\{X\}^\{\(h\)\}\+n\_\{\\mathrm\{AI\}\}I\_\{R\}^\{\(h\)\}\+n\_\{\\mathrm\{AI\}\}\\Psi\_\{h\}\(n\_\{\\mathrm\{esc\}\}/n\_\{\\mathrm\{AI\}\}\)\\geq T\.
HerenHn\_\{\\mathrm\{H\}\},nAIn\_\{\\mathrm\{AI\}\}, andnescn\_\{\\mathrm\{esc\}\}are continuous optimization variables representing, respectively, the expected number of human\-first items, AI\-first items, and human follow\-up \(escalation\) queries in directionhh\. The objective is the total expected cost:cHc\_\{\\mathrm\{H\}\}per human\-first item,cAIc\_\{\\mathrm\{AI\}\}per AI\-first item, andcHc\_\{\\mathrm\{H\}\}per human follow\-up query\. The first constraint reflects that human\-first and AI\-first items are drawn from the same pool ofNNacquired items\. The second constraint ensures that human follow\-up queries can only be applied to AI\-first items\. The information constraint, justified by \([20](https://arxiv.org/html/2609.28859#S4.E20)\), requires the total KL information from all three query types to meet the targetTT, whereTTwill be set toAAwhenh=1h=1and toBBwhenh=0h=0in the lower bound \([22](https://arxiv.org/html/2609.28859#S4.E22)\) below\.

Problem \([21](https://arxiv.org/html/2609.28859#S4.E21)\) can be interpreted as follows: given a pool ofNNitems and a KL information targetTT, what is the cheapest mix of human\-first items, AI\-first items, and human follow\-up queries that meets the target? The answer depends on the relative costscHc\_\{\\mathrm\{H\}\},cAIc\_\{\\mathrm\{AI\}\}, and the information yieldsJX\(h\)J\_\{X\}^\{\(h\)\},IR\(h\)I\_\{R\}^\{\(h\)\},Ψh\\Psi\_\{h\}\. For example, if AI queries are cheap and informative, the optimizer will favor AI\-first items over human\-first items\. The minimum in \([21](https://arxiv.org/html/2609.28859#S4.E21)\) is attained whenever the feasible set is nonempty, since all variables are bounded to\[0,N\]\[0,N\]and the objective is continuous and bounded below by zero\. If no feasible triple satisfies the information constraint, we setΓh​\(T,N\)=\+∞\\Gamma\_\{h\}\(T,N\)=\+\\infty\.

By Theorem[4\.6](https://arxiv.org/html/2609.28859#S4.Thmtheorem6), Lemma[4\.9](https://arxiv.org/html/2609.28859#S4.Thmtheorem9), and Lemma[4\.5](https://arxiv.org/html/2609.28859#S4.Thmtheorem5), the expected query counts of any feasible policyπ\\piunderHhH\_\{h\}satisfy all constraints of \([21](https://arxiv.org/html/2609.28859#S4.E21)\) withT=AT=Awhenh=1h=1andT=BT=Bwhenh=0h=0\. SinceΓh​\(T,Nπ\)\\Gamma\_\{h\}\(T,N^\{\\pi\}\)is the minimum cost over all such feasible triples, the expected cost ofπ\\piunderHhH\_\{h\}satisfies

𝔼hπ​\[Cπ\]≥Nπ​cdata\+Γh​\(T,Nπ\)\.\\mathbb\{E\}^\{\\pi\}\_\{h\}\[C^\{\\pi\}\]\\geq N^\{\\pi\}c\_\{\\mathrm\{data\}\}\+\\Gamma\_\{h\}\(T,N^\{\\pi\}\)\.Since we minimize the worst\-case expected cost over both hypotheses, we take the maximum of the two direction\-wise lower bounds\. Combining with the sample size floor from Lemma[4\.2](https://arxiv.org/html/2609.28859#S4.Thmtheorem2), we get

LB⁡\(α,β\):=minN≥Nfixed,H​\(α,β\)⁡\{N​cdata\+max⁡\{Γ1​\(A,N\),Γ0​\(B,N\)\}\}\.\\mathrm\{LB\}\(\\alpha,\\beta\):=\\min\_\{N\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)\}\\left\\\{Nc\_\{\\mathrm\{data\}\}\+\\max\\\{\\Gamma\_\{1\}\(A,N\),\\Gamma\_\{0\}\(B,N\)\\\}\\right\\\}\.\(22\)The outer minimum searches over all integersN≥Nfixed,H​\(α,β\)N\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)\. The minimum in \([22](https://arxiv.org/html/2609.28859#S4.E22)\) is attained because only finitely many sample sizes need to be checked: once a finite valueV0V\_\{0\}is achieved at someN0N\_\{0\}, everyN\>V0/cdataN\>V\_\{0\}/c\_\{\\mathrm\{data\}\}has objective value at leastN​cdata\>V0Nc\_\{\\mathrm\{data\}\}\>V\_\{0\}and cannot be optimal\. Moreover,LB⁡\(α,β\)\\mathrm\{LB\}\(\\alpha,\\beta\)is computationally tractable, since each inner programΓh​\(T,N\)\\Gamma\_\{h\}\(T,N\)is convex\.

###### Proposition 4\.10

For eachh∈\{0,1\}h\\in\\\{0,1\\\},T≥0T\\geq 0, and integerN≥1N\\geq 1, the optimization \([21](https://arxiv.org/html/2609.28859#S4.E21)\) is convex in\(nH,nAI,nesc\)\(n\_\{\\mathrm\{H\}\},n\_\{\\mathrm\{AI\}\},n\_\{\\mathrm\{esc\}\}\)\. Thus,LB⁡\(α,β\)\\mathrm\{LB\}\(\\alpha,\\beta\)can be computed by solving a finite sequence of convex programs\.

The following theorem formalizes the validity ofLB⁡\(α,β\)\\mathrm\{LB\}\(\\alpha,\\beta\)as a lower bound on the optimal cost\.

###### Theorem 4\.11\(Selective\-scoring cost lower bound\)

Under Assumptions[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)and[3\.1](https://arxiv.org/html/2609.28859#S3.SS1), we haveC∗​\(α,β\)≥LB⁡\(α,β\)\.C^\{\*\}\(\\alpha,\\beta\)\\geq\\mathrm\{LB\}\(\\alpha,\\beta\)\.

The lower boundLB⁡\(α,β\)\\mathrm\{LB\}\(\\alpha,\\beta\)serves as the benchmark for our proposed policy\. We show in Section[5](https://arxiv.org/html/2609.28859#S5)that our policy achievesLB⁡\(α,β\)\\mathrm\{LB\}\(\\alpha,\\beta\)to first order asα,β→0\\alpha,\\beta\\to 0, establishing its asymptotic optimality\. The gap betweenLB⁡\(α,β\)\\mathrm\{LB\}\(\\alpha,\\beta\)andC∗​\(α,β\)C^\{\*\}\(\\alpha,\\beta\)at finite\(α,β\)\(\\alpha,\\beta\)arises from the information\-theoretic relaxation in Lemma[4\.5](https://arxiv.org/html/2609.28859#S4.Thmtheorem5), and vanishes asymptotically\.

## 5A Sequential Cost\-Aware Policy

Section[4](https://arxiv.org/html/2609.28859#S4)established that any feasible policy must incur cost at leastLB⁡\(α,β\)\\mathrm\{LB\}\(\\alpha,\\beta\)\. We now construct a policy, which we call SCALE \(Sequential Cost\-Aware Likelihood\-Guided Escalation\) policy, that matches this lower bound asymptotically\. Specifically, we consider a sequence of problem instances indexed byk=1,2,…k=1,2,\\ldots, with type\-I and type\-II error levelsαk↓0\\alpha\_\{k\}\\downarrow 0andβk↓0\\beta\_\{k\}\\downarrow 0ask→∞k\\to\\infty\(we refer to this as first\-order asymptotics\)\. For eachkk, our proposed policy, which we refer to asπk\\pi\_\{k\}, is constructed with respect to\(αk,βk\)\(\\alpha\_\{k\},\\beta\_\{k\}\)\. We describeπk\\pi\_\{k\}at a high level in Section[5\.1](https://arxiv.org/html/2609.28859#S5.SS1)\. In Section[5\.2](https://arxiv.org/html/2609.28859#S5.SS2), we give its formal construction without specifying the values for the tuning parameters and show that the policy is feasible regardless of those values\. In Section[5\.3](https://arxiv.org/html/2609.28859#S5.SS3), we discuss the choice of tuning parameters and establish the asymptotics ofπk\\pi\_\{k\}with those choices\. For interested readers, we lay out the main technical arguments in Appendix[10\.1](https://arxiv.org/html/2609.28859#S10.SS1)\.

### 5\.1High Level Description of SCALE

The policy proceeds in two stages\. The first is the sequential main stage\. In this stage, the policy maintains a running log\-likelihood\-ratio statistic and uses a Wald\-style sequential probability ratio test\([Wald 1945](https://arxiv.org/html/2609.28859#bib.bib43)\)whose increments are generated by the three types of queries introduced in Section[4](https://arxiv.org/html/2609.28859#S4)\. At the beginning of each epoch of the main stage, the policy uses the running statistic to determine a sensing rule, which in turn specifies the probability of querying a human versus an AI, and the probability of escalating an AI\-scored item to human review\. The main stage stops as soon as the statistic crosses\+ak\+a\_\{k\}\(decideH1H\_\{1\}\) or−bk\-b\_\{k\}\(decideH0H\_\{0\}\)\. If the main stage ends without a crossing, the policy enters the second stage, a fallback stage that employs a fixed\-sample\-size test using either full human labels or full AI labels, depending on a comparison of the two tests’ costs\.

### 5\.2Policy Construction

We now formalize the two\-stage procedure described in Section[5\.1](https://arxiv.org/html/2609.28859#S5.SS1)\. To start, we split the type\-I and type\-II budgets into two parts: one for the sequential main stage, and one for the fallback\. Letffb,k∈\(0,1\)f\_\{\\mathrm\{fb\},k\}\\in\(0,1\)be the fraction of type\-I and type\-II budgets allocated to the fallback\. As classically done through the union bound, the type\-I and type\-II budgets allocated to sequential main stage and fallback are:

α1,k:=\(1−ffb,k\)​αk,α2,k:=ffb,k​αk,\\alpha\_\{1,k\}:=\(1\-f\_\{\\mathrm\{fb\},k\}\)\\alpha\_\{k\},\\qquad\\alpha\_\{2,k\}:=f\_\{\\mathrm\{fb\},k\}\\alpha\_\{k\},\(23\)β1,k:=\(1−ffb,k\)​βk,β2,k:=ffb,k​βk\.\\beta\_\{1,k\}:=\(1\-f\_\{\\mathrm\{fb\},k\}\)\\beta\_\{k\},\\qquad\\beta\_\{2,k\}:=f\_\{\\mathrm\{fb\},k\}\\beta\_\{k\}\.\(24\)Thenα1,k\+α2,k=αk\\alpha\_\{1,k\}\+\\alpha\_\{2,k\}=\\alpha\_\{k\}andβ1,k\+β2,k=βk\\beta\_\{1,k\}\+\\beta\_\{2,k\}=\\beta\_\{k\}, and the likelihood\-ratio boundaries for the main stage sequential test are

ak:=log⁡1α1,k,bk:=log⁡1β1,k\.a\_\{k\}:=\\log\\frac\{1\}\{\\alpha\_\{1,k\}\},\\qquad b\_\{k\}:=\\log\\frac\{1\}\{\\beta\_\{1,k\}\}\.\(25\)
Additionally, we also acquireNmain,kN\_\{\\mathrm\{main\},k\}data items\. As the fallback test revisits a fixed\-sample\-size test with type\-I and type\-II guaranteesα2,k\\alpha\_\{2,k\}andβ2,k\\beta\_\{2,k\}, we requireNmain,k≥Nfixed,H​\(α2,k,β2,k\)N\_\{\\mathrm\{main\},k\}\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)to make sure we have enough data items for finite\-sample feasibility\. In other words, the conditionNmain,k≥Nfixed,H​\(α2,k,β2,k\)N\_\{\\mathrm\{main\},k\}\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)is only required so that our fallback stage has the proper type\-I and type\-II error guarantees while the main sequential stage immediately guarantees the rightα1,k,β1,k\\alpha\_\{1,k\},\\beta\_\{1,k\}errors through the sequential boundariesak,bka\_\{k\},b\_\{k\}\.

#### Sequential main stage\.

This main stage processes the fixed pool in order1,2,…,Nmain,k1,2,\\ldots,N\_\{\\mathrm\{main\},k\}\. Items are sensed one at a time, in this order, until a boundary is crossed or the pool is exhausted\. In other words, the number of sensed items is a stopping time determined by the likelihood\-ratio path\. The running statistic starts atS0=0S\_\{0\}=0\. At the beginning of each epoch during the sequential main stage, we compare the running statistic with the boundaryzkz\_\{k\}and−zk\-z\_\{k\}to determine the sensing rule for that epoch\. We refer tozkz\_\{k\}as the “hypothesis boundary” as these determine whether we focus onH0H\_\{0\}orH1H\_\{1\}\. Specifically, for itemii, letSi−1S\_\{i\-1\}be the statistic before itemiiis sensed and set

Ji:=\{1,Si−1\>zk,0,Si−1<−zk,∗,\|Si−1\|≤zk\.J\_\{i\}:=\\begin\{cases\}1,&S\_\{i\-1\}\>z\_\{k\},\\\\ 0,&S\_\{i\-1\}<\-z\_\{k\},\\\\ \*,&\|S\_\{i\-1\}\|\\leq z\_\{k\}\.\\end\{cases\}\(26\)Then we end up with three types of sensing rules for each itemii: one favoringH0H\_\{0\}\(i\.e\.Ji=0J\_\{i\}=0\), one favoringH1H\_\{1\}\(i\.e\.Ji=1J\_\{i\}=1\), and another undecided \(i\.e\.Ji=∗J\_\{i\}=\*\)\. We note thatJiJ\_\{i\}is only used to determine which hypothesis directionh=0,1h=0,1to focus on\. As shown in Sections[4\.3](https://arxiv.org/html/2609.28859#S4.SS3)\-[4\.4](https://arxiv.org/html/2609.28859#S4.SS4), many important quantities relevant for our policy \(e\.g\.,Ψh​\(s\),ηh∗​\(rj\(h\),s\)\\Psi\_\{h\}\(s\),\\eta^\{\*\}\_\{h\}\(r^\{\(h\)\}\_\{j\};s\)\) differ according to the nullh=0h=0or the alternativeh=1h=1\. HenceJiJ\_\{i\}aims to determine whichhh\-direction to focus on sequentially\.

Based on the sensing ruleJi=j∈\{0,1,∗\}J\_\{i\}=j\\in\\\{0,1,\*\\\}, we choose to query human, query AI, or escalate the item to human after querying AI\. WithJi=jJ\_\{i\}=j, the policy directly queries a human with probabilityηj,kH∈\[0,1\]\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\\in\[0,1\]; otherwise it queries the AI with probability1−ηj,kH1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}, observes the reportRiR\_\{i\}, and escalates with probabilityηj,kesc​\(Ri\)∈\[0,1\]\\eta^\{\\mathrm\{esc\}\}\_\{j,k\}\(R\_\{i\}\)\\in\[0,1\]\. At this point, we emphasize thatzkz\_\{k\},ηh,kH\\eta^\{\\mathrm\{H\}\}\_\{h,k\}andηh,kesc\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}are all tuning parameters that do not affect the feasibility and can be tuned to improve the finite sample and asymptotic performance of the algorithm\. We will discuss the choice of these tuning parameters in Section[5\.3](https://arxiv.org/html/2609.28859#S5.SS3)\.

After each AI or human query, we update our running statistic\. If a human is queried directly, before any AI query on the same item, we increment the running statistic by the single\-observation log\-likelihood ratio for the observed labelX=xX=xunderH1H\_\{1\}versusH0H\_\{0\}:

ℓX​\(x\):=log⁡p1x​\(1−p1\)1−xp0x​\(1−p0\)1−x,x∈\{0,1\}\.\\ell\_\{X\}\(x\):=\\log\\frac\{p\_\{1\}^\{x\}\(1\-p\_\{1\}\)^\{1\-x\}\}\{p\_\{0\}^\{x\}\(1\-p\_\{0\}\)^\{1\-x\}\},\\qquad x\\in\\\{0,1\\\}\.\(27\)After a direct AI query withR=rR=r, we increment the running statistic by the single\-observation log\-likelihood ratio ofR=rR=runderH1H\_\{1\}versusH0H\_\{0\}:

ℓR​\(r\):=log⁡g1​\(r\)g0​\(r\),r∈ℛ,\\ell\_\{R\}\(r\):=\\log\\frac\{g\_\{1\}\(r\)\}\{g\_\{0\}\(r\)\},\\qquad r\\in\\mathcal\{R\},\(28\)If the query is a human query preceded by an AI query on the same item, we increment the running statistic by the conditional log\-likelihood ratio ofX=xX=xgiven the previously observed AI reportR=rR=runderH1H\_\{1\}versusH0H\_\{0\}:

ℓH​\(x,r\):=log⁡ρ1​\(x∣r\)ρ0​\(x∣r\),x∈\{0,1\},r∈ℛ\.\\ell\_\{H\}\(x,r\):=\\log\\frac\{\\rho\_\{1\}\(x\\mid r\)\}\{\\rho\_\{0\}\(x\\mid r\)\},\\qquad x\\in\\\{0,1\\\},\\ r\\in\\mathcal\{R\}\.\(29\)ThusℓX​\(x\)\\ell\_\{X\}\(x\)andℓR​\(r\)\\ell\_\{R\}\(r\)capture the evidence provided by a human label and AI label respectively, andℓH​\(x,r\)\\ell\_\{H\}\(x,r\)captures the additional evidence provided by the human label beyond the evidence already contained in the AI report\. After each increment, the policy stops and rejectsH0H\_\{0\}ifSi≥akS\_\{i\}\\geq a\_\{k\}, and stops and acceptsH0H\_\{0\}ifSi≤−bkS\_\{i\}\\leq\-b\_\{k\}\.

#### Fallback benchmarks\.

The main sequential stage does not guarantee that we reach a terminal decision to reject or accept the null hypothesis \(i\.e\., we may never cross eitheraka\_\{k\}or−bk\-b\_\{k\}\)\. The fallback stage guarantees, with potentially more queries, that we conclude our hypothesis test with a rejection/acceptance decision\. The human fallback uses the exact full\-label fixed\-sample\-size benchmarkNfixed,HN\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}from Definition[4\.1](https://arxiv.org/html/2609.28859#S4.Thmtheorem1)\. The AI fallback uses the analogous exact benchmarkNfixed,AIN\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}for i\.i\.d\. AI reports\.

###### Definition 5\.1\(Exact AI\-report sample\-size benchmark\)

For0<α<10<\\alpha<1and0<β<10<\\beta<1, we defineNfixed,AI​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha,\\beta\)as the smallestN∈ℕN\\in\\mathbb\{N\}for which there exists a randomized testψN:ℛN→\[0,1\]\\psi\_\{N\}:\\mathcal\{R\}^\{N\}\\to\[0,1\]satisfying

𝔼0​\[ψN​\(R1,…,RN\)\]≤α,𝔼1​\[1−ψN​\(R1,…,RN\)\]≤β,\\mathbb\{E\}\_\{0\}\[\\psi\_\{N\}\(R\_\{1\},\\ldots,R\_\{N\}\)\]\\leq\\alpha,\\qquad\\mathbb\{E\}\_\{1\}\[1\-\\psi\_\{N\}\(R\_\{1\},\\ldots,R\_\{N\}\)\]\\leq\\beta,where, underHhH\_\{h\}, the reportsR1,…,RNR\_\{1\},\\ldots,R\_\{N\}are i\.i\.d\. with mass functionghg\_\{h\}\. If no finiteNNexists, setNfixed,AI​\(α,β\)=\+∞N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha,\\beta\)=\+\\infty\.

IfG0≠G1G\_\{0\}\\neq G\_\{1\}, the Neyman–Pearson lemma applied to the report likelihood ratio gives a finiteNfixed,AI​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha,\\beta\)\. IfG0=G1G\_\{0\}=G\_\{1\}andα\+β<1\\alpha\+\\beta<1, report\-only observations have the same law under the two hypotheses, so no report\-only test can satisfy both error constraints andNfixed,AI​\(α,β\)=\+∞N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha,\\beta\)=\+\\infty\. As the computation ofNfixed,AI​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha,\\beta\)follows similarly as that ofNfixed,H​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)according to Neyman\-Pearson lemma, we refer the readers to Appendix[9\.1](https://arxiv.org/html/2609.28859#S9.SS1)for more details on computation ofNfixed,AI​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha,\\beta\)\.

#### Pre\-committed fallback\.

If no boundary is crossed by the time the pool is exhausted, the policy uses a single fallback completion chosen before any data are observed\. The choice is made by computing a non\-data dependent deterministic cost of running only AI\-based or only human\-based Neyman Pearson tests\. Formally, the human\-only completion cost upper bound is

C¯H,k:=cH​Nfixed,H​\(α2,k,β2,k\)\.\\overline\{C\}\_\{\\mathrm\{H\},k\}:=c\_\{\\mathrm\{H\}\}N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\.The AI\-only completion is available only ifNfixed,AI​\(α2,k,β2,k\)<\+∞N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)<\+\\inftyandNfixed,AI​\(α2,k,β2,k\)≤Nmain,kN\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\\leq N\_\{\\mathrm\{main\},k\}, because this section keeps the fixed\-pool convention\. If it is available, its primitive cost upper bound is

C¯AI,k:=cAI​Nfixed,AI​\(α2,k,β2,k\)\.\\overline\{C\}\_\{\\mathrm\{AI\},k\}:=c\_\{\\mathrm\{AI\}\}N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\.IfNfixed,AI​\(α2,k,β2,k\)=\+∞N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)=\+\\inftyorNfixed,AI​\(α2,k,β2,k\)\>Nmain,kN\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\>N\_\{\\mathrm\{main\},k\}, setC¯AI,k:=\+∞\\overline\{C\}\_\{\\mathrm\{AI\},k\}:=\+\\infty\. The pre\-committed fallback mode is

𝖥k:=\{H,C¯H,k≤C¯AI,k,AI,C¯AI,k<C¯H,k\.\\mathsf\{F\}\_\{k\}:=\\begin\{cases\}\\mathrm\{H\},&\\overline\{C\}\_\{\\mathrm\{H\},k\}\\leq\\overline\{C\}\_\{\\mathrm\{AI\},k\},\\\\ \\mathrm\{AI\},&\\overline\{C\}\_\{\\mathrm\{AI\},k\}<\\overline\{C\}\_\{\\mathrm\{H\},k\}\.\\end\{cases\}\(30\)The value of𝖥k\\mathsf\{F\}\_\{k\}is a deterministic function of the primitives, error targets, costs, and precomputed sample sizes\. It is not a function of any realized label, AI report, randomization seed, stopping event, or terminal statistic, thus can be computed before collecting any data\.

If𝖥k=H\\mathsf\{F\}\_\{k\}=\\mathrm\{H\}, the fallback uses the fixed index set

𝒥H,k:=\{1,2,…,Nfixed,H​\(α2,k,β2,k\)\}\.\\mathcal\{J\}\_\{\\mathrm\{H\},k\}:=\\\{1,2,\\ldots,N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\\\}\.It queries a human on every item in𝒥H,k\\mathcal\{J\}\_\{\\mathrm\{H\},k\}whose label has not already been revealed by human, and then applies a Neyman–Pearson full\-label testϕNfixed,H​\(α2,k,β2,k\)∗\\phi^\{\*\}\_\{N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\}satisfying

𝔼0πk​\[ϕNfixed,H​\(α2,k,β2,k\)∗\]≤α2,k,𝔼1πk​\[1−ϕNfixed,H​\(α2,k,β2,k\)∗\]≤β2,k\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\phi^\{\*\}\_\{N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\}\]\\leq\\alpha\_\{2,k\},\\qquad\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[1\-\\phi^\{\*\}\_\{N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\}\]\\leq\\beta\_\{2,k\}\.\(31\)The test is applied to\(Xi\)i∈𝒥H,k\(X\_\{i\}\)\_\{i\\in\\mathcal\{J\}\_\{\\mathrm\{H\},k\}\}\. The set is fixed in advance, so these labels are i\.i\.d\. Bernoulli under each hypothesis\. If𝖥k=AI\\mathsf\{F\}\_\{k\}=\\mathrm\{AI\}, by design, we must haveNfixed,AI​\(α2,k,β2,k\)≤Nmain,kN\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\\leq N\_\{\\mathrm\{main\},k\}\. The fallback uses the fixed report set\{R1,…,RNfixed,AI​\(α2,k,β2,k\)\}\\\{R\_\{1\},\\ldots,R\_\{N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\}\\\}, querying any missing reports in that set, and applies a Neyman–Pearson report testψNfixed,AI​\(α2,k,β2,k\)∗\\psi^\{\*\}\_\{N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\}satisfying

𝔼0πk​\[ψNfixed,AI​\(α2,k,β2,k\)∗\]≤α2,k,𝔼1πk​\[1−ψNfixed,AI​\(α2,k,β2,k\)∗\]≤β2,k\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\psi^\{\*\}\_\{N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\}\]\\leq\\alpha\_\{2,k\},\\qquad\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[1\-\\psi^\{\*\}\_\{N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\}\]\\leq\\beta\_\{2,k\}\.\(32\)Again the set is fixed in advance, so the reports are i\.i\.d\. with massGhG\_\{h\}underHhH\_\{h\}\. The full description of SCALE is summarized in Algorithm[1](https://arxiv.org/html/2609.28859#alg1)\.

#### Feasibility\.

We show in Theorem[5\.2](https://arxiv.org/html/2609.28859#S5.Thmtheorem2)thatπk\\pi\_\{k\}remains feasible regardless of the choice of the tuning parameters\. We give the proof of Theorem[5\.2](https://arxiv.org/html/2609.28859#S5.Thmtheorem2)in Appendix[9\.2](https://arxiv.org/html/2609.28859#S9.SS2)\.

###### Theorem 5\.2\(Feasibility of SCALE\)

Suppose Assumptions[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)and[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)hold\. SCALE \(formally presented as policyπk\\pi\_\{k\}in Algorithm[1](https://arxiv.org/html/2609.28859#alg1)\) is feasible for everykk:

ℙ0πk​\(δπk=1\)≤αk,ℙ1πk​\(δπk=0\)≤βk\.\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\(\\delta^\{\\pi\_\{k\}\}=1\)\\leq\\alpha\_\{k\},\\qquad\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(\\delta^\{\\pi\_\{k\}\}=0\)\\leq\\beta\_\{k\}\.

Feasibility follows from a two\-step argument\. The policy terminates either by crossing a boundary during the sequential main stage or by invoking the fallback test\. LetE\+E\_\{\+\}andE−E\_\{\-\}denote the events of stopping at the upper and lower boundaries respectively, andEfbE\_\{\\mathrm\{fb\}\}the event of reaching fallback stage\. Analogous to Wald’s sequential hypothesis testing, we show thatℙ0πk​\(E\+\)≤α1,k\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\(E\_\{\+\}\)\\leq\\alpha\_\{1,k\}andℙ1πk​\(E−\)≤β1,k,\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(E\_\{\-\}\)\\leq\\beta\_\{1,k\},while the fallback test either satisfies \([31](https://arxiv.org/html/2609.28859#S5.E31)\) or satisfies \([32](https://arxiv.org/html/2609.28859#S5.E32)\), so

ℙ0πk​\(Efb∩\{fallback rejects\}\)≤α2,k,ℙ1πk​\(Efb∩\{fallback accepts\}\)≤β2,k\.\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\\bigl\(E\_\{\\mathrm\{fb\}\}\\cap\\\{\\text\{fallback rejects\}\\\}\\bigr\)\\leq\\alpha\_\{2,k\},\\qquad\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\\bigl\(E\_\{\\mathrm\{fb\}\}\\cap\\\{\\text\{fallback accepts\}\\\}\\bigr\)\\leq\\beta\_\{2,k\}\.Since these routes are mutually exclusive and exhaustive,ℙ0πk​\(δπk=1\)≤αk\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\(\\delta^\{\\pi\_\{k\}\}=1\)\\leq\\alpha\_\{k\}andℙ1πk​\(δπk=0\)≤βk\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(\\delta^\{\\pi\_\{k\}\}=0\)\\leq\\beta\_\{k\}\.

Theorem[5\.2](https://arxiv.org/html/2609.28859#S5.Thmtheorem2)implies that we can choose the values of the tuning parameters to improve its finite sample and asymptotic performance ofπk\\pi\_\{k\}without affecting its feasibility\. In the next section, we discuss one specific choice of the tuning parameters which leads to asymptotic optimality ofπk\\pi\_\{k\}\.

Algorithm 1SCALE — Sequential Cost\-Aware Likelihood\-Guided Escalation \(πk\\pi\_\{k\}\)Input:Primitives

\(p0,p1,f0,f1,cdata,cAI,cH\)\(p\_\{0\},p\_\{1\},f\_\{0\},f\_\{1\},c\_\{\\mathrm\{data\}\},c\_\{\\mathrm\{AI\}\},c\_\{\\mathrm\{H\}\}\), error targets

\(αk,βk\)\(\\alpha\_\{k\},\\beta\_\{k\}\), and tuning parameters

\(ffb,k,zk,Nmain,k\(≥Nfixed,H\(α2,k,β2,k\)\),\{ηh,kH\}h∈\{0,1,∗\},\{ηh,kesc\}h∈\{0,1,∗\}\)\(f\_\{\\mathrm\{fb\},k\},z\_\{k\},N\_\{\\mathrm\{main\},k\}\\,\(\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\),\\\{\\eta^\{\\mathrm\{H\}\}\_\{h,k\}\\\}\_\{h\\in\\\{0,1,\*\\\}\},\\\{\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}\\\}\_\{h\\in\\\{0,1,\*\\\}\}\)\.

1Compute the split \([23](https://arxiv.org/html/2609.28859#S5.E23)\)–\([24](https://arxiv.org/html/2609.28859#S5.E24)\) and boundaries \([25](https://arxiv.org/html/2609.28859#S5.E25)\)\.

2Pre\-commit to

𝖥k∈\{H,AI\},\\mathsf\{F\}\_\{k\}\\in\\\{\\mathrm\{H\},\\mathrm\{AI\}\\\},by \([30](https://arxiv.org/html/2609.28859#S5.E30)\)\.

3Acquire

Nmain,kN\_\{\\mathrm\{main\},k\}items and set

S←0S\\leftarrow 0\.

4for*i=1i=1toNmain,kN\_\{\\mathrm\{main\},k\}*do

5Set

J←1J\\leftarrow 1if

S\>zkS\>z\_\{k\},

J←0J\\leftarrow 0if

S<−zkS<\-z\_\{k\}, and

J←∗J\\leftarrow\*if

\|S\|≤zk\|S\|\\leq z\_\{k\}\.

6Draw

Ui∼Uniform⁡\[0,1\]U\_\{i\}\\sim\\mathrm\{Uniform\}\[0,1\]\.

7if*Ui≤ηJ,kHU\_\{i\}\\leq\\eta^\{\\mathrm\{H\}\}\_\{J,k\}*then

8Query a human on item

ii, observe

XiX\_\{i\}, and set

S←S\+ℓX​\(Xi\)S\\leftarrow S\+\\ell\_\{X\}\(X\_\{i\}\)\.

9if*S≥akS\\geq a\_\{k\}*then

10reject

H0H\_\{0\}and stop\.

11if*S≤−bkS\\leq\-b\_\{k\}*then

12accept

H0H\_\{0\}and stop\.

13else

14Query the AI on item

ii, observe

RiR\_\{i\}, and set

S←S\+ℓR​\(Ri\)S\\leftarrow S\+\\ell\_\{R\}\(R\_\{i\}\)\.

15if*S≥akS\\geq a\_\{k\}*then

16reject

H0H\_\{0\}and stop\.

17if*S≤−bkS\\leq\-b\_\{k\}*then

18accept

H0H\_\{0\}and stop\.

19Draw

Vi∼Uniform⁡\[0,1\]V\_\{i\}\\sim\\mathrm\{Uniform\}\[0,1\]\.

20if*Vi≤ηJ,kesc​\(Ri\)V\_\{i\}\\leq\\eta^\{\\mathrm\{esc\}\}\_\{J,k\}\(R\_\{i\}\)*then

21Query a human on item

ii, observe

XiX\_\{i\}, and set

S←S\+ℓH​\(Xi,Ri\)S\\leftarrow S\+\\ell\_\{H\}\(X\_\{i\},R\_\{i\}\)\.

22if*S≥akS\\geq a\_\{k\}*then

23reject

H0H\_\{0\}and stop\.

24if*S≤−bkS\\leq\-b\_\{k\}*then

25accept

H0H\_\{0\}and stop\.

26If no boundary has been crossed, execute the pre\-committed fallback mode

𝖥k\\mathsf\{F\}\_\{k\}and output the corresponding fallback test decision\.

### 5\.3Discussion on Tuning Parameters with Asymptotics Analysis

In the previous subsection we construct a feasible algorithm without giving details about the specific choice of the parameters\. In this subsection, we suggest a particular set of parameter choices that guarantees the first order asymptotics ofπk\\pi\_\{k\}\. Recall our policyπk\\pi\_\{k\}is indexed byk=1,2,…k=1,2,\\ldots, with type\-I and type\-II error levelsαk↓0\\alpha\_\{k\}\\downarrow 0andβk↓0\\beta\_\{k\}\\downarrow 0ask→∞k\\to\\infty\. We tune the parameters to guarantee

max⁡\{𝔼0πk​\[Cπk\],𝔼1πk​\[Cπk\]\}LB⁡\(αk,βk\)→1as​k→∞,\\frac\{\\max\\\{\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[C^\{\\pi\_\{k\}\}\],\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[C^\{\\pi\_\{k\}\}\]\\\}\}\{\\mathrm\{LB\}\(\\alpha\_\{k\},\\beta\_\{k\}\)\}\\to 1\\qquad\\text\{as \}k\\to\\infty,and consequently

max⁡\{𝔼0πk​\[Cπk\],𝔼1πk​\[Cπk\]\}C∗​\(αk,βk\)→1as​k→∞\.\\frac\{\\max\\\{\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[C^\{\\pi\_\{k\}\}\],\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[C^\{\\pi\_\{k\}\}\]\\\}\}\{C^\{\*\}\(\\alpha\_\{k\},\\beta\_\{k\}\)\}\\to 1\\qquad\\text\{as \}k\\to\\infty\.In particular, define:

Lk:=max⁡\{log⁡1αk,log⁡1βk\}\.L\_\{k\}:=\\max\\\!\\left\\\{\\log\\frac\{1\}\{\\alpha\_\{k\}\},\\,\\log\\frac\{1\}\{\\beta\_\{k\}\}\\right\\\}\.
We will show in Theorem[5\.3](https://arxiv.org/html/2609.28859#S5.Thmtheorem3)that the lower boundLBk:=LB⁡\(αk,βk\)\\mathrm\{LB\}\_\{k\}:=\\mathrm\{LB\}\(\\alpha\_\{k\},\\beta\_\{k\}\)scales asLBk=Θ⁡\(Lk\)\\mathrm\{LB\}\_\{k\}=\\Theta\(L\_\{k\}\), so first\-order optimality is equivalent to matchingLBk\\mathrm\{LB\}\_\{k\}up to ano⁡\(Lk\)o\(L\_\{k\}\)additive term\. Throughout the asymptotic analysis that follows, we impose the balanced small\-error regime given by the following assumption\.

\{assumption\}

\[Balanced small\-error regime\]log⁡\(1/αk\)=Θ⁡\(Lk\)\\log\\left\(1/\\alpha\_\{k\}\\right\)=\\Theta\(L\_\{k\}\)andlog⁡\(1/βk\)=Θ⁡\(Lk\)\\log\\left\(1/\\beta\_\{k\}\\right\)=\\Theta\(L\_\{k\}\)\.

Assumption 3 requires the logarithmic type\-I and type\-II error levels,log⁡\(1/αk\)\\log\(1/\\alpha\_\{k\}\)andlog⁡\(1/βk\)\\log\(1/\\beta\_\{k\}\), to be of the same order\. In particular, it does not requireαk\\alpha\_\{k\}andβk\\beta\_\{k\}to be of the same order\. For example,αk=e−k\\alpha\_\{k\}=e^\{\-k\}andβk=e−2​k\\beta\_\{k\}=e^\{\-2k\}satisfy the assumption, even thoughβk/αk→0\\beta\_\{k\}/\\alpha\_\{k\}\\to 0\. More generally, the assumption allowsβk=αkq\\beta\_\{k\}=\\alpha\_\{k\}^\{q\}for any fixedq\>0q\>0\. Thus, it accommodates a broad range of asymmetric sequences of vanishing type\-I and type\-II error levels\.

Recall that the policy is equipped with the following set of tunable parameter sequences: \(i\) a fixed pool sizeNmain,kN\_\{\\mathrm\{main\},k\}as the total number of items to acquire, \(ii\) direction\-specific sensing rulesηh,kH∈\[0,1\]\\eta^\{\\mathrm\{H\}\}\_\{h,k\}\\in\[0,1\], the probability to query human, andηh,kesc∈\[0,1\]\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}\\in\[0,1\], the probability to escalate the item to the human given that the item has already been AI\-queried before under directionhh, \(iii\) a fallback\-budget fractionffb,kf\_\{\\mathrm\{fb\},k\}that is bounded away from zero and one, and \(iv\) the hypothesis boundaryzkz\_\{k\}\. In what follows, we pick specific values for these parameters\.

#### Fixed pool size\.

We chooseNmain,kN\_\{\\mathrm\{main\},k\}to be the minimum number of samples required to meet some information targets\(T1,k,T0,k\)\(T\_\{1,k\},T\_\{0,k\}\), andNmain,kN\_\{\\mathrm\{main\},k\}is inspired by \([22](https://arxiv.org/html/2609.28859#S4.E22)\) with\(T1,k,T0,k\)\(T\_\{1,k\},T\_\{0,k\}\)\. Specifically, letT1,k:=ak\+ΔkT\_\{1,k\}:=a\_\{k\}\+\\Delta\_\{k\}andT0,k:=bk\+ΔkT\_\{0,k\}:=b\_\{k\}\+\\Delta\_\{k\}be the buffered direction targets whereΔk\\Delta\_\{k\}is the buffer with which we inflate the boundary on the sequential main stage to calculate the pool size\. By inflating the boundary byΔk\\Delta\_\{k\}, we ensure that, failure to cross the boundary is a large\-deviation event that occurs with vanishing probability\. The fallback test therefore contributes onlyo⁡\(Lk\)o\(L\_\{k\}\)to the expected cost and does not affect first\-order optimality\.

For integerNN, define the buffered design value

FkΔ​\(N\):=N​cdata\+max⁡\{Γ1​\(T1,k,N\),Γ0​\(T0,k,N\)\}\.F\_\{k\}^\{\\Delta\}\(N\):=Nc\_\{\\mathrm\{data\}\}\+\\max\\\{\\Gamma\_\{1\}\(T\_\{1,k\},N\),\\Gamma\_\{0\}\(T\_\{0,k\},N\)\\\}\.\(33\)
Choose

N¯main,k∈arg​minN∈ℤ\+,N≥Nfixed,H​\(α2,k,β2,k\)⁡FkΔ​\(N\)\.\\overline\{N\}\_\{\\mathrm\{main\},k\}\\in\\operatorname\*\{arg\\,min\}\_\{N\\in\\mathbb\{Z\}\_\{\+\},N\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\}F\_\{k\}^\{\\Delta\}\(N\)\.\(34\)We then take the fixed pool size as

Nmain,k:=N¯main,k\+1\.N\_\{\\mathrm\{main\},k\}:=\\overline\{N\}\_\{\\mathrm\{main\},k\}\+1\.\(35\)In the above, we add an additive one\-item cushion\. We explain the reason of doing so subsequently\.

#### Direction\-specific sensing rules\.

GivenNmain,kN\_\{\\mathrm\{main\},k\}, we now choose the human\-query and escalation probabilities\(ηh,kH,ηh,kesc\)\(\\eta^\{\\mathrm\{H\}\}\_\{h,k\},\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}\)to match the minimal cost plan identified by the cost minimization problem \([21](https://arxiv.org/html/2609.28859#S4.E21)\)\. For each directionh∈\{0,1\}h\\in\\\{0,1\\\}, choose an optimizer

\(nH,h,k∗,nAI,h,k∗,nesc,h,k∗\)∈arg​minnH,nAI,nesc≥0cH​nH\+cAI​nAI\+cH​nescsubject tonH\+nAI≤N¯main,k,0≤nesc≤nAI,nH​JX\(h\)\+nAI​IR\(h\)\+nAI​Ψh​\(nesc/nAI\)≥Th,k\.\\begin\{split\}\(n^\{\*\}\_\{\\mathrm\{H\},h,k\},n^\{\*\}\_\{\\mathrm\{AI\},h,k\},n^\{\*\}\_\{\\mathrm\{esc\},h,k\}\)\\in\\operatorname\*\{arg\\,min\}\_\{n\_\{\\mathrm\{H\}\},\\,n\_\{\\mathrm\{AI\}\},\\,n\_\{\\mathrm\{esc\}\}\\geq 0\}\\quad&c\_\{\\mathrm\{H\}\}n\_\{\\mathrm\{H\}\}\+c\_\{\\mathrm\{AI\}\}n\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}n\_\{\\mathrm\{esc\}\}\\\\ \\text\{subject to\}\\quad&n\_\{\\mathrm\{H\}\}\+n\_\{\\mathrm\{AI\}\}\\leq\\overline\{N\}\_\{\\mathrm\{main\},k\},\\\\ &0\\leq n\_\{\\mathrm\{esc\}\}\\leq n\_\{\\mathrm\{AI\}\},\\\\ &n\_\{\\mathrm\{H\}\}J\_\{X\}^\{\(h\)\}\+n\_\{\\mathrm\{AI\}\}I\_\{R\}^\{\(h\)\}\+n\_\{\\mathrm\{AI\}\}\\Psi\_\{h\}\(n\_\{\\mathrm\{esc\}\}/n\_\{\\mathrm\{AI\}\}\)\\geq T\_\{h,k\}\.\\end\{split\}\(36\)
The variables are named to match the interpretation of the programΓh\\Gamma\_\{h\}:nH,h,k∗n^\{\*\}\_\{\\mathrm\{H\},h,k\}is the direct\-human count,nAI,h,k∗n^\{\*\}\_\{\\mathrm\{AI\},h,k\}is the AI\-scored count, andnesc,h,k∗n^\{\*\}\_\{\\mathrm\{esc\},h,k\}is the escalation count among AI\-scored items\. BecausenH,h,k∗\+nAI,h,k∗≤N¯main,kn^\{\*\}\_\{\\mathrm\{H\},h,k\}\+n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\leq\\overline\{N\}\_\{\\mathrm\{main\},k\}in the programΓh\\Gamma\_\{h\}, and since⌈x⌉\+⌈y⌉≤⌈x\+y⌉\+1\\lceil x\\rceil\+\\lceil y\\rceil\\leq\\lceil x\+y\\rceil\+1for any realsx,y≥0x,y\\geq 0, the cushion in \([35](https://arxiv.org/html/2609.28859#S5.E35)\) gives

⌈nH,h,k∗⌉\+⌈nAI,h,k∗⌉≤⌈nH,h,k∗\+nAI,h,k∗⌉\+1≤⌈N¯main,k⌉\+1=Nmain,k,\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil\\leq\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\+n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil\+1\\leq\\lceil\\overline\{N\}\_\{\\mathrm\{main\},k\}\\rceil\+1=N\_\{\\mathrm\{main\},k\},\(37\)
which ensures we have acquired enough items to make⌈nH,h,k∗⌉\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceilhuman queries and⌈nAI,h,k∗⌉\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceilAI queries for the sequential main stage\.

The direct\-human fraction of the direction\-hhrule is

ηh,kH:=\{⌈nH,h,k∗⌉/\(⌈nH,h,k∗⌉\+⌈nAI,h,k∗⌉\),⌈nH,h,k∗⌉\+⌈nAI,h,k∗⌉\>0,0,⌈nH,h,k∗⌉\+⌈nAI,h,k∗⌉=0\.\\eta^\{\\mathrm\{H\}\}\_\{h,k\}:=\\begin\{cases\}\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil/\(\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil\),&\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil\>0,\\\\ 0,&\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil=0\.\\end\{cases\}\(38\)For all largekk,⌈nH,h,k∗⌉\+⌈nAI,h,k∗⌉\>0\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil\>0becauseTh,k\>0T\_\{h,k\}\>0and no zero\-item rule can supply positive information\.

IfnAI,h,k∗\>0n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\>0, define the planned escalation ratesh,k:=nesc,h,k∗/nAI,h,k∗∈\[0,1\]s\_\{h,k\}:=n^\{\*\}\_\{\\mathrm\{esc\},h,k\}/n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\in\[0,1\]\. IfnAI,h,k∗=0n^\{\*\}\_\{\\mathrm\{AI\},h,k\}=0, setsh,k:=0s\_\{h,k\}:=0\. To construct the frontier optimizer, following Proposition[4\.8](https://arxiv.org/html/2609.28859#S4.Thmtheorem8)we defineηh,kesc:ℛ→\[0,1\]\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}:\\mathcal\{R\}\\to\[0,1\]by

ηh,kesc​\(rj\(h\)\):=\{1,Wj\(h\)≤sh,k,sh,k−Wj−1\(h\)gh​\(rj\(h\)\),Wj−1\(h\)<sh,k<Wj\(h\),0,Wj−1\(h\)≥sh,k\.\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}\(r^\{\(h\)\}\_\{j\}\):=\\begin\{cases\}1,&W^\{\(h\)\}\_\{j\}\\leq s\_\{h,k\},\\\\\[4\.30554pt\] \\displaystyle\\frac\{s\_\{h,k\}\-W^\{\(h\)\}\_\{j\-1\}\}\{g\_\{h\}\(r^\{\(h\)\}\_\{j\}\)\},&W^\{\(h\)\}\_\{j\-1\}<s\_\{h,k\}<W^\{\(h\)\}\_\{j\},\\\\\[8\.61108pt\] 0,&W^\{\(h\)\}\_\{j\-1\}\\geq s\_\{h,k\}\.\\end\{cases\}\(39\)Thenηh,kesc\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}is the optimal solution of \([16](https://arxiv.org/html/2609.28859#S4.E16)\) withs=sh,ks=s\_\{h,k\}\.

In the dead\-zone\[−zk,zk\]\[\-z\_\{k\},z\_\{k\}\]the sign of the likelihood\-ratio statistic is ambiguous\. The policy therefore uses the averaged dead\-zone rule

η∗,kH:=η0,kH\+η1,kH2,η∗,kesc​\(r\):=η0,kesc​\(r\)\+η1,kesc​\(r\)2,r∈ℛ\.\\eta^\{\\mathrm\{H\}\}\_\{\*,k\}:=\\frac\{\\eta^\{\\mathrm\{H\}\}\_\{0,k\}\+\\eta^\{\\mathrm\{H\}\}\_\{1,k\}\}\{2\},\\qquad\\eta^\{\\mathrm\{esc\}\}\_\{\*,k\}\(r\):=\\frac\{\\eta^\{\\mathrm\{esc\}\}\_\{0,k\}\(r\)\+\\eta^\{\\mathrm\{esc\}\}\_\{1,k\}\(r\)\}\{2\},\\quad r\\in\\mathcal\{R\}\.\(40\)The average of two numbers in\[0,1\]\[0,1\]again lies in\[0,1\]\[0,1\], so the rule is a valid randomized sensing rule\. The specific values ofη∗,kH\\eta^\{\\mathrm\{H\}\}\_\{\*,k\}andη∗,kesc​\(r\),r∈ℛ\\eta^\{\\mathrm\{esc\}\}\_\{\*,k\}\(r\),r\\in\\mathcal\{R\}will not affect the asymptotics ofπk\\pi\_\{k\}; therefore, we choose the averaged rule here for simplicity\.

#### Fallback\-budget fractionffb,kf\_\{\\mathrm\{fb\},k\}\.

According to Theorem[5\.3](https://arxiv.org/html/2609.28859#S5.Thmtheorem3)below,ffb,kf\_\{\\mathrm\{fb\},k\}only affects the cost ofπk\\pi\_\{k\}up to some constant, so we can chooseffb,kf\_\{\\mathrm\{fb\},k\}to be any constant that is bounded away from00and11; for example, we can letffb,k=1/2f\_\{\\mathrm\{fb\},k\}=1/2\.

#### BufferΔk\\Delta\_\{k\}and hypothesis boundaryzkz\_\{k\}\.

Additionally, to ensure that policyπk\\pi\_\{k\}has asymptotically converging cost toLBk\\mathrm\{LB\}\_\{k\}, we require that the remaining parameters\(zk,Δk\)\(z\_\{k\},\\Delta\_\{k\}\)to satisfy the following conditions:

1. \(i\)Δk→∞,Δk=o⁡\(Lk\),and​Δk2Lk→∞;\\Delta\_\{k\}\\to\\infty,\\,\\Delta\_\{k\}=o\(L\_\{k\}\),\\text\{ and \}\\frac\{\\Delta\_\{k\}^\{2\}\}\{L\_\{k\}\}\\to\\infty;\(41\)
2. \(ii\)zk≥0​and​zk\+1Δk→0\.z\_\{k\}\\geq 0\\text\{ and \}\\frac\{z\_\{k\}\+1\}\{\\Delta\_\{k\}\}\\to 0\.\(42\)

Intuitively, the accumulated likelihood\-ratio statistic has variance of orderLkL\_\{k\}, and hence fluctuations of orderLk\\sqrt\{L\_\{k\}\}\. Thus,Δk2/Lk→∞\\Delta\_\{k\}^\{2\}/L\_\{k\}\\to\\inftyensures that the buffer dominates these fluctuations and makes the probability of reaching fallback vanish, whileΔk=o⁡\(Lk\)\\Delta\_\{k\}=o\(L\_\{k\}\)keeps the cost of this buffer lower order\. The condition\(zk\+1\)/Δk→0\(z\_\{k\}\+1\)/\\Delta\_\{k\}\\to 0similarly ensures that the additional cost incurred while the statistic lies in the ambiguous region\[−zk,zk\]\[\-z\_\{k\},z\_\{k\}\]is negligible\. We summarize the tuning parameters and their choices in Table[1](https://arxiv.org/html/2609.28859#S5.T1)\.

Table 1:Tuning parameters of the policyπk\\pi\_\{k\}in Algorithm[1](https://arxiv.org/html/2609.28859#alg1)\.Following the above choices ofNmain,kN\_\{\\mathrm\{main\},k\},\(ηh,kH,ηh,kesc\)\(\\eta^\{\\mathrm\{H\}\}\_\{h,k\},\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}\),ffb,k,zkf\_\{\\mathrm\{fb\},k\},z\_\{k\}andΔk\\Delta\_\{k\}, we now establish first\-order asymptotic optimality ofπk\\pi\_\{k\}in Theorem[5\.3](https://arxiv.org/html/2609.28859#S5.Thmtheorem3)\. Before stating the theorem, we remark that all choices of tuning parameters introduced in this section are neither unique nor finite\-sample optimal\. They should be viewed as reasonable sufficient conditions to ensure theoretical first order optimality as shown by the subsequent theorem\. For practical implementation, we recommend practitioners to computationally tune these parameters for their own applications\.

###### Theorem 5\.3\(Parameterized ratio upper bound\)

Suppose Assumptions[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)\-[5\.3](https://arxiv.org/html/2609.28859#S5.SS3)hold\. The following hold:

1. \(i\)LBk=Θ⁡\(Lk\)\\mathrm\{LB\}\_\{k\}=\\Theta\(L\_\{k\}\);
2. \(ii\)SupposeNmain,kN\_\{\\mathrm\{main\},k\}is given by \([35](https://arxiv.org/html/2609.28859#S5.E35)\),\(ηh,kH,ηh,kesc\)\(\\eta^\{\\mathrm\{H\}\}\_\{h,k\},\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}\)is given by \([38](https://arxiv.org/html/2609.28859#S5.E38)\) and \([39](https://arxiv.org/html/2609.28859#S5.E39)\), and\(η∗,kH,η∗,kesc\)\(\\eta^\{\\mathrm\{H\}\}\_\{\*,k\},\\eta^\{\\mathrm\{esc\}\}\_\{\*,k\}\)is given by \([40](https://arxiv.org/html/2609.28859#S5.E40)\)\. SupposeΔk,zk\\Delta\_\{k\},z\_\{k\}satisfy \([41](https://arxiv.org/html/2609.28859#S5.E41)\) and \([42](https://arxiv.org/html/2609.28859#S5.E42)\) andffb,kf\_\{\\mathrm\{fb\},k\}is any constant bounded away from 0 and 1\. Then for all largekk, max⁡\{𝔼0πk​\[Cπk\],𝔼1πk​\[Cπk\]\}LBk=1\+o⁡\(1\)\.\\begin\{split\}\\frac\{\\max\\\{\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[C^\{\\pi\_\{k\}\}\],\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[C^\{\\pi\_\{k\}\}\]\\\}\}\{\\mathrm\{LB\}\_\{k\}\}=1\+o\(1\)\.\\end\{split\}\(43\)

As specified in \([41](https://arxiv.org/html/2609.28859#S5.E41)\)–\([42](https://arxiv.org/html/2609.28859#S5.E42)\), we impose only rate conditions on these tuning parameters\. To state explicit convergence rates, letνh∗\\nu\_\{h\}^\{\*\}denote the least active\-item consumption among the cost\-minimizing unit\-information plans in directionhh:

νh∗:=min\(nH,nAI,nesc\)∈ℳh⁡\(nH\+nAI\)\.\\nu\_\{h\}^\{\*\}:=\\min\_\{\(n\_\{\\mathrm\{H\}\},n\_\{\\mathrm\{AI\}\},n\_\{\\mathrm\{esc\}\}\)\\in\\mathcal\{M\}\_\{h\}\}\(n\_\{\\mathrm\{H\}\}\+n\_\{\\mathrm\{AI\}\}\)\.\(44\)whereℳh\\mathcal\{M\}\_\{h\}denotes the set of minimizers ofminnH,nAI,nesc≥0\{cHnH\+cAInAI\+cHnesc:nHJX\(h\)\+nAIIR\(h\)\+nAIΨh\(nesc/nAI\)≥1,0≤nesc≤nAI\}\\min\_\{n\_\{\\mathrm\{H\}\},n\_\{\\mathrm\{AI\}\},n\_\{\\mathrm\{esc\}\}\\geq 0\}\\\{c\_\{\\mathrm\{H\}\}n\_\{\\mathrm\{H\}\}\+c\_\{\\mathrm\{AI\}\}n\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}n\_\{\\mathrm\{esc\}\}:n\_\{\\mathrm\{H\}\}J\_\{X\}^\{\(h\)\}\+n\_\{\\mathrm\{AI\}\}I\_\{R\}^\{\(h\)\}\+n\_\{\\mathrm\{AI\}\}\\Psi\_\{h\}\(n\_\{\\mathrm\{esc\}\}/n\_\{\\mathrm\{AI\}\}\)\\geq 1,0\\leq n\_\{\\mathrm\{esc\}\}\\leq n\_\{\\mathrm\{AI\}\}\\\}, which computes the minimum sensing cost required to generate one unit of information in directionhh\. Define

Gk:=Nfixed,H​\(αk,βk\)−max⁡\{ν1∗​ak,ν0∗​bk\},G\_\{k\}:=N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)\-\\max\\\{\\nu\_\{1\}^\{\*\}a\_\{k\},\\nu\_\{0\}^\{\*\}b\_\{k\}\\\},\(45\)and writeGk\+:=\[Gk\]\+G^\{\+\}\_\{k\}:=\[G\_\{k\}\]\_\{\+\}\. Thus,Gk\+G\_\{k\}^\{\+\}measures the unused item reserve supplied by the mandatory full\-label fallback floor\. Corollary[5\.4](https://arxiv.org/html/2609.28859#S5.Thmtheorem4)provides explicit convergence rates with exact parameter choices\.

###### Corollary 5\.4\(Parameter choices and rates\)

Suppose Assumptions[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)\-[5\.3](https://arxiv.org/html/2609.28859#S5.SS3)hold\. Letffb,k=1/2f\_\{\\mathrm\{fb\},k\}=1/2andzk=1z\_\{k\}=1, let the optimizer in \([36](https://arxiv.org/html/2609.28859#S5.E36)\) be selected with least active\-item consumptionnH,h,k∗\+nAI,h,k∗n^\{\*\}\_\{\\mathrm\{H\},h,k\}\+n^\{\*\}\_\{\\mathrm\{AI\},h,k\}among the cost minimizers \(cf\. \([44](https://arxiv.org/html/2609.28859#S5.E44)\)\)\. Then the following hold for all largekk:

1. \(i\)\(Baseline, no condition on the reserve\.\)LettingΔk=κ​Lk​log⁡Lk\\Delta\_\{k\}=\\kappa\\sqrt\{L\_\{k\}\\log L\_\{k\}\}for a sufficiently large constantκ\\kappadepending only on the primitives, then regardless of the value ofGkG\_\{k\}, maxh⁡𝔼hπk​\[Cπk\]LBk≤1\+O~\(Lk−1/2\)\.\\frac\{\\max\_\{h\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[C^\{\\pi\_\{k\}\}\]\}\{\\mathrm\{LB\}\_\{k\}\}\\leq 1\+\\widetilde\{O\}\(L\_\{k\}^\{\-1/2\}\)\.
2. \(ii\)\(Improvement when the reserve is generous\.\)IfGk\+≥C​Lk​log⁡LkG\_\{k\}^\{\+\}\\geq C\\sqrt\{L\_\{k\}\\log L\_\{k\}\}for a sufficiently large primitive constantCC, takingΔk\\Delta\_\{k\}polylogarithmic sharpens part \(i\) to maxh⁡𝔼hπk​\[Cπk\]LBk≤1\+O~​\(1/Lk\)\.\\frac\{\\max\_\{h\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[C^\{\\pi\_\{k\}\}\]\}\{\\mathrm\{LB\}\_\{k\}\}\\leq 1\+\\widetilde\{O\}\(1/L\_\{k\}\)\.

### 5\.4Numerical Simulations

We evaluate the finite\-sample cost performance of SCALE under the known AI\-output model\. The experiments examine how the benefit of combining AI queries with selective human verification changes with the human query cost and the required testing accuracy\. We compare SCALE with implementable Human\-only and AI\-only policies under the same error constraints and cost accounting\. We consider independent labelsXi∼Bernoulli⁡\(ph\)X\_\{i\}\\sim\\operatorname\{Bernoulli\}\(p\_\{h\}\)underHhH\_\{h\}, withp0=0\.10p\_\{0\}=0\.10andp1=0\.20p\_\{1\}=0\.20\. Human queries revealXiX\_\{i\}exactly, whereas the AI returns a binary reportRi∈\{0,1\}R\_\{i\}\\in\\\{0,1\\\}with known sensitivity and specificityℙ⁡\(Ri=1∣Xi=1\)=ℙ⁡\(Ri=0∣Xi=0\)=0\.80\.\\mathbb\{P\}\(R\_\{i\}=1\\mid X\_\{i\}=1\)=\\mathbb\{P\}\(R\_\{i\}=0\\mid X\_\{i\}=0\)=0\.80\.We setcdata=cAI=1c\_\{\\mathrm\{data\}\}=c\_\{\\mathrm\{AI\}\}=1\. Each policy pays for its entire sample pool upfront, including items that remain unqueried when the sequential test stops\. Reported costs include data acquisition and all AI and human queries, including those required by fallback\.

Our preliminary numerical exploration suggested that total cost was relatively insensitive to the parameterszzandffbf\_\{\\mathrm\{fb\}\}over the ranges considered\. We therefore use a coarse tuning scheme for these two parameters: at each setting, we fixz=1z=1and search overffb∈\{0\.2,0\.5,0\.8\}f\_\{\\mathrm\{fb\}\}\\in\\\{0\.2,0\.5,0\.8\\\}\. We tune the integer pool sizeNmainN\_\{\\mathrm\{main\}\}in unit increments over

Nfixed,H​\(ffb​α,ffb​β\)≤Nmain≤Nmax,N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(f\_\{\\mathrm\{fb\}\}\\alpha,f\_\{\\mathrm\{fb\}\}\\beta\)\\leq N\_\{\\mathrm\{main\}\}\\leq N\_\{\\mathrm\{max\}\},whereNmaxN\_\{\\mathrm\{max\}\}is the upper bound derived from the single\-source policy\. We also tune the three direct\-human probabilitiesηjH∈\[0,1\]\\eta^\{\\mathrm\{H\}\}\_\{j\}\\in\[0,1\]and the six escalation probabilitiesηjesc​\(r\)∈\[0,1\]\\eta^\{\\mathrm\{esc\}\}\_\{j\}\(r\)\\in\[0,1\],j∈\{0,1,∗\}j\\in\\\{0,1,\*\\\}andr∈\{0,1\}r\\in\\\{0,1\\\}, using a grid search with steps of0\.10\.1\. We retain SCALE’s pre\-committed fallback rule\. No separate bufferΔ\\Deltais introduced, sinceNmainN\_\{\\mathrm\{main\}\}and the escalation probabilities are tuned directly\. Parameters are selected using a simulation\-based cost criterion, without a claim of global optimality\.

The Human\-only and AI\-only baselines use a truncated sequential probability ratio test \(SPRT\), followed, if no boundary is crossed, by a same\-source Neyman–Pearson \(NP\) test\. Each baseline uses its designated source in both stages\. For a fixedffbf\_\{\\mathrm\{fb\}\}, its pool size is set toNs=Nfixed,s​\(ffb​α,ffb​β\)N\_\{s\}=N\_\{\\mathrm\{fixed\},s\}\(f\_\{\\mathrm\{fb\}\}\\alpha,f\_\{\\mathrm\{fb\}\}\\beta\),s∈\{H,AI\},s\\in\\\{\\mathrm\{H\},\\mathrm\{AI\}\\\},computed using the exact randomized binomial NP test\. This pool size is analytically optimal within the baseline class for the givenffbf\_\{\\mathrm\{fb\}\}: a larger pool increases the upfront acquisition cost without reducing the expected number of sequential queries\. If fallback is reached, all observations required by the terminal NP test have already been queried, so the terminal decision incurs no additional acquisition or query cost\. For each baseline,ffbf\_\{\\mathrm\{fb\}\}is selected over the same three\-point grid by exact expected\-cost evaluation\.

After all parameters are fixed, we evaluate each policy usingM=106M=10^\{6\}independent Monte Carlo trajectories under each hypothesis, independently of the tuning simulations\. The plotted absolute cost is the estimated worst\-case expected cost

𝒞^π:=maxh∈\{0,1\}⁡1M​∑m=1MCh,mπ,\\widehat\{\\mathcal\{C\}\}\_\{\\pi\}:=\\max\_\{h\\in\\\{0,1\\\}\}\\frac\{1\}\{M\}\\sum\_\{m=1\}^\{M\}C^\{\\pi\}\_\{h,m\},whereCh,mπC^\{\\pi\}\_\{h,m\}is the total cost in replicationmmunderHhH\_\{h\}\. To quantify relative performance, we report

Gap\(%\):=100\[𝒞^SCALEmin⁡\{𝒞^H,𝒞^AI\}−1\]\.\\operatorname\{Gap\}\(\\%\):=100\\left\[\\frac\{\\widehat\{\\mathcal\{C\}\}\_\{\\mathrm\{SCALE\}\}\}\{\\min\\\{\\widehat\{\\mathcal\{C\}\}\_\{\\mathrm\{H\}\},\\widehat\{\\mathcal\{C\}\}\_\{\\mathrm\{AI\}\}\\\}\}\-1\\right\]\.\(46\)A negative gap indicates a cost saving relative to the cheaper of the two evaluated single\-source baselines\.

We conduct two parameter sweeps\. The first variescH∈\{2,5,10,20,50\}c\_\{\\mathrm\{H\}\}\\in\\\{2,5,10,20,50\\\}withα=β=0\.01\\alpha=\\beta=0\.01\. The second variesα=β∈\{0\.05,0\.01,0\.0001\}\\alpha=\\beta\\in\\\{0\.05,0\.01,0\.0001\\\}withcH=10c\_\{\\mathrm\{H\}\}=10\. Figures[1](https://arxiv.org/html/2609.28859#S5.F1)and[2](https://arxiv.org/html/2609.28859#S5.F2)report the absolute costs and percentage gaps, respectively\. Both horizontal axes use logarithmic spacing, and the error targets decrease from left to right\.

AVaryingcHc\_\{\\mathrm\{H\}\};α=β=0\.01\\alpha=\\beta=0\.01\.BVaryingα=β\\alpha=\\beta;cH=10c\_\{\\mathrm\{H\}\}=10\.
Figure 1:Worst\-case expected total costs of SCALE, Human\-only, and AI\-only\. Each point is the maximum of the two estimated hypothesis\-specific mean costs, based on10610^\{6\}Monte Carlo trajectories per hypothesis\. Costs include the full upfront acquisition cost and all subsequent queries\.As the human query cost increases, both Human\-only and SCALE become more expensive, while AI\-only remains constant at2020\.602020\.60\(Figure[1A](https://arxiv.org/html/2609.28859#S5.F1.sf1)\)\. The percentage gap exhibits a U\-shaped pattern across the evaluated human costs \(Figure[2A](https://arxiv.org/html/2609.28859#S5.F2.sf1)\)\. AtcH=2c\_\{\\mathrm\{H\}\}=2, SCALE costs613\.73613\.73, compared with612\.83612\.83for Human\-only, giving a small positive gap of0\.15%0\.15\\%\. AtcH=5c\_\{\\mathrm\{H\}\}=5,1010, and2020, SCALE reduces cost relative to the cheaper baseline by19\.74%19\.74\\%,36\.37%36\.37\\%, and20\.96%20\.96\\%, respectively\. The largest observed saving occurs atcH=10c\_\{\\mathrm\{H\}\}=10: SCALE costs1,093\.181,093\.18, compared with1,718\.001,718\.00for Human\-only and2,020\.602,020\.60for AI\-only, saving624\.82624\.82cost units relative to the cheaper baseline\. AtcH=50c\_\{\\mathrm\{H\}\}=50, SCALE costs2,020\.462,020\.46and essentially matches AI\-only; the estimated gap of−0\.01%\-0\.01\\%is smaller than the Monte Carlo uncertainty\.

This pattern is consistent with the selected routing policies\. SCALE relies predominantly on direct human queries when human queries are inexpensive, uses AI queries with selective human escalation at intermediate costs, and becomes nearly AI\-only when human queries are expensive\. AtcH=50c\_\{\\mathrm\{H\}\}=50, the selected pool contains13241324items and supports an AI\-only terminal NP test\. Thus, the largest observed gains arise at intermediate human costs, where selective verification offers the greatest advantage over committing to a single source\.

AVaryingcHc\_\{\\mathrm\{H\}\};α=β=0\.01\\alpha=\\beta=0\.01\.BVaryingα=β\\alpha=\\beta;cH=10c\_\{\\mathrm\{H\}\}=10\.
Figure 2:SCALE’s percentage cost gap relative to the cheaper evaluated Human\-only or AI\-only policy, as defined in \([46](https://arxiv.org/html/2609.28859#S5.E46)\)\. Negative values indicate cost savings\. Panel A shows a U\-shaped pattern across human query costs; Panel B shows increasing relative savings as the error targets become more stringent\.Tighter error targets increase the absolute costs of all three policies \(Figure[1B](https://arxiv.org/html/2609.28859#S5.F1.sf2)\)\. Asα=β\\alpha=\\betadecreases from0\.050\.05to0\.010\.01and then to0\.00010\.0001, SCALE’s cost increases from714\.35714\.35to1,093\.181,093\.18and2,081\.232,081\.23, while its selected pool size increases from269269to410410and835835\. Human\-only remains the cheaper single\-source baseline throughout this sweep, with corresponding costs of1,101\.571,101\.57,1,718\.001,718\.00, and3,420\.773,420\.77\. SCALE’s relative savings increase from35\.15%35\.15\\%to36\.37%36\.37\\%and39\.16%39\.16\\%\(Figure[2B](https://arxiv.org/html/2609.28859#S5.F2.sf2)\)\. At the most stringent target, SCALE saves1,339\.541,339\.54cost units relative to Human\-only\. Across these three error targets, stronger error control therefore requires greater expenditure while increasing the relative benefit of combining AI queries with selective human verification\.

## 6Pilot\-Calibrated Design with Unknown AI Accuracy

So far, our results have assumed that the conditional AI\-output lawsf0f\_\{0\}andf1f\_\{1\}are known, as stated in Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)\. This assumption can be reasonable in settings where the same AI system has been repeatedly evaluated on a stable population and its performance has been estimated from a large historical labeled data set\. In such cases, the AI\-output model may be treated as a known characteristic of the deployed system\. Moreover, whenf0f\_\{0\}andf1f\_\{1\}are known, an analyst can in principle testH0H\_\{0\}versusH1H\_\{1\}using AI reports alone, because the systematic error in the AI output can be accounted for statistically, although doing so need not be cost optimal\.

In other settings, however, the AI system may be new, the target population may differ from the population on which it was previously evaluated, or sufficiently reliable labeled calibration data may simply be unavailable\. In these cases,f0f\_\{0\}andf1f\_\{1\}must be estimated from observations containing both the AI report and the human\-verified label\. Accordingly, in this section we estimatef0f\_\{0\}andf1f\_\{1\}from an independent paired pilot sample and construct a plug\-in policy based onf^0\\hat\{f\}\_\{0\}andf^1\\hat\{f\}\_\{1\}\. We add guardrail terms that account for estimation error and preserve finite\-sample type\-I and type\-II error control\. We then show that, when the pilot sample is sufficiently large, the resulting policy retains the first\-order asymptotic optimality established in Theorem[5\.3](https://arxiv.org/html/2609.28859#S5.Thmtheorem3)\.

### 6\.1Pilot Sample and Smoothed Plug\-In Model

In order to estimatef0,f1f\_\{0\},f\_\{1\}, we first need to obtain both pilot samples withXipilot=0X^\{\\mathrm\{pilot\}\}\_\{i\}=0labels andXipilot=1X^\{\\mathrm\{pilot\}\}\_\{i\}=1labels\. With a slight buase of notation, letmmbe a positive integer, the minimal number of samples for bothXipilot=0X^\{\\mathrm\{pilot\}\}\_\{i\}=0andXipilot=1X^\{\\mathrm\{pilot\}\}\_\{i\}=1labels we target to obtain\. A sufficiently largemmensures we have accurate estimates forf0f\_\{0\}andf1f\_\{1\}\. In order to obtain enough samples for both labels, we consider the following pilot sampling procedure: we sample each pilot item and reveal its true label using a human query sequentially; we stop sampling after we have collectedmmsamples for both labels\. WritingCx\(n\):=∑i=1n𝟏\{Xipilot=x\}C\_\{x\}\(n\):=\\sum^\{n\}\_\{i=1\}\\mathbf\{1\}\\\{X^\{\\mathrm\{pilot\}\}\_\{i\}=x\\\}for the number of label\-xxsamples among the firstnn, the total number of random samples we get is the stopping timeMtot:=min⁡\{n≥1:minx∈\{0,1\}⁡Cx​\(n\)≥m\}\.M\_\{\\mathrm\{tot\}\}:=\\min\\\{n\\geq 1:\\min\_\{x\\in\\\{0,1\\\}\}C\_\{x\}\(n\)\\geq m\\\}\.The random numbers ofXipilot=0X^\{\\mathrm\{pilot\}\}\_\{i\}=0andXipilot=1X^\{\\mathrm\{pilot\}\}\_\{i\}=1samples are respectivelyM0:=C0​\(Mtot\)M\_\{0\}:=C\_\{0\}\(M\_\{\\mathrm\{tot\}\}\)andM1:=C1​\(Mtot\)M\_\{1\}:=C\_\{1\}\(M\_\{\\mathrm\{tot\}\}\), and we haveM0\+M1=Mtot≥2​mM\_\{0\}\+M\_\{1\}=M\_\{\\mathrm\{tot\}\}\\geq 2mandmin⁡\{M0,M1\}=m\\min\\\{M\_\{0\},M\_\{1\}\\\}=m\.

For each pilot sampleXipilotX^\{\\mathrm\{pilot\}\}\_\{i\}, we also obtain its AI reportRipilotR^\{\\mathrm\{pilot\}\}\_\{i\}, generated by the same AI system on the same population as in the main stage; this is what makes the pilot informative aboutf0f\_\{0\}andf1f\_\{1\}\. We assume that, conditional on the pilot labels, the AI reports are independent across samples and are drawn fromfxf\_\{x\}within the label\-xxstratum\. We summarize this formally below:

\{assumption\}

\[Independent pilot\] UnderHhH\_\{h\}, the pilot pairs\(Xipilot,Ripilot\)i≥1\(X\_\{i\}^\{\\mathrm\{pilot\}\},R\_\{i\}^\{\\mathrm\{pilot\}\}\)\_\{i\\geq 1\}are i\.i\.d\. withℙh​\(Xipilot=x\)=phx​\(1−ph\)1−x\\mathbb\{P\}\_\{h\}\(X\_\{i\}^\{\\mathrm\{pilot\}\}=x\)=p\_\{h\}^\{x\}\(1\-p\_\{h\}\)^\{1\-x\}andℙh​\(Ripilot=r\|Xipilot=x\)=fx​\(r\),for​x∈\{0,1\},r∈ℛ\.\\mathbb\{P\}\_\{h\}\(R\_\{i\}^\{\\mathrm\{pilot\}\}=r\|X\_\{i\}^\{\\mathrm\{pilot\}\}=x\)=f\_\{x\}\(r\),\\text\{ for \}x\\in\\\{0,1\\\},r\\in\\mathcal\{R\}\.The entire pilot stream is independent of the main\-stage items and policy randomization\.

We denote the full pilot sample as:

𝒟m:=\{\(0,R0,jpilot\):1≤j≤M0\}∪\{\(1,R1,jpilot\):1≤j≤M1\}\.\\mathcal\{D\}\_\{m\}:=\\\{\(0,R^\{\\mathrm\{pilot\}\}\_\{0,j\}\):1\\leq j\\leq M\_\{0\}\\\}\\cup\\\{\(1,R^\{\\mathrm\{pilot\}\}\_\{1,j\}\):1\\leq j\\leq M\_\{1\}\\\}\.Next, we use a smoothed empirical distribution to estimatef0,f1f\_\{0\},f\_\{1\}\. Forx∈\{0,1\}x\\in\\\{0,1\\\}andr∈ℛr\\in\\mathcal\{R\}, define

Mx\(r\):=∑j=1Mx𝟏\{Rx,jpilot=r\},f^x\(r\):=Mx​\(r\)\+λmMx\+\|ℛ\|​λm,λm:=m−2\.M\_\{x\}\(r\):=\\sum\_\{j=1\}^\{M\_\{x\}\}\\mathbf\{1\}\\\{R^\{\\mathrm\{pilot\}\}\_\{x,j\}=r\\\},\\qquad\\hat\{f\}\_\{x\}\(r\):=\\frac\{M\_\{x\}\(r\)\+\\lambda\_\{m\}\}\{M\_\{x\}\+\|\\mathcal\{R\}\|\\lambda\_\{m\}\},\\qquad\\lambda\_\{m\}:=m^\{\-2\}\.\(47\)HereMx​\(r\)M\_\{x\}\(r\)counts how many pilot reports in the label\-xxstratum equalrr, so the natural estimate offx​\(r\)f\_\{x\}\(r\)is the empirical frequencyMx​\(r\)/MxM\_\{x\}\(r\)/M\_\{x\}, which is well defined becauseMx≥m≥1M\_\{x\}\\geq m\\geq 1\. We add the smoothing termλm\\lambda\_\{m\}to avoid the edge caseMx​\(r\)=0M\_\{x\}\(r\)=0, which would make the plug\-in increments \(see \([52](https://arxiv.org/html/2609.28859#S6.E52)\)\) infinite\. The resulting estimate satisfiesf^x​\(r\)\>0\\hat\{f\}\_\{x\}\(r\)\>0and∑r∈ℛf^x​\(r\)=1\.\\sum\_\{r\\in\\mathcal\{R\}\}\\hat\{f\}\_\{x\}\(r\)=1\.The choiceλm=m−2\\lambda\_\{m\}=m^\{\-2\}makes the smoothing displace the empirical frequency by at most\|ℛ\|​λm/Mx≤\|ℛ\|​m−3\|\\mathcal\{R\}\|\\lambda\_\{m\}/M\_\{x\}\\leq\|\\mathcal\{R\}\|m^\{\-3\}, which is negligible relative to the sampling error of orderm−1/2m^\{\-1/2\}recorded inrmr\_\{m\}below; the correction therefore does not affect first\-order asymptotic optimality\.

Next, we aim to control the error of our estimatesf^x​\(r\)\\hat\{f\}\_\{x\}\(r\)\. Letδm:=m−2\\delta\_\{m\}:=m^\{\-2\}be a target failure probability, and set

tm:=12​m​log⁡\(4​\|ℛ\|δm\),rm:=min⁡\{1,tm\+\|ℛ\|​λmm\}\.t\_\{m\}:=\\sqrt\{\\frac\{1\}\{2m\}\\log\\\!\\left\(\\frac\{4\|\\mathcal\{R\}\|\}\{\\delta\_\{m\}\}\\right\)\},\\qquad r\_\{m\}:=\\min\\left\\\{1,\\;t\_\{m\}\+\\frac\{\|\\mathcal\{R\}\|\\lambda\_\{m\}\}\{m\}\\right\\\}\.
We then define the “good pilot event”

ℰpilot,m:=\{maxx∈\{0,1\},r∈ℛ\|f^x\(r\)−fx\(r\)\|≤rm\},\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}:=\\left\\\{\\max\_\{x\\in\\\{0,1\\\},\\,r\\in\\mathcal\{R\}\}\|\\hat\{f\}\_\{x\}\(r\)\-f\_\{x\}\(r\)\|\\leq r\_\{m\}\\right\\\},\(48\)
on which every plug\-in report probability is withinrmr\_\{m\}of its true value, uniformly inxxandrr\. In Appendix[11\.1](https://arxiv.org/html/2609.28859#S11.SS1)we show thatℙh​\(ℰpilot,m\)≥1−δm\\mathbb\{P\}\_\{h\}\(\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}\)\\geq 1\-\\delta\_\{m\}forh∈\{0,1\}h\\in\\\{0,1\\\}\.

Once we havef^0,f^1\\hat\{f\}\_\{0\},\\hat\{f\}\_\{1\}, we form the plug\-in analogue of every object in Sections[3](https://arxiv.org/html/2609.28859#S3)–[5](https://arxiv.org/html/2609.28859#S5)that depends on the report channel, by substitutingf^x\\hat\{f\}\_\{x\}forfxf\_\{x\}whereverfxf\_\{x\}appears and leaving all quantities that depend only on the known primitives\(p0,p1,cdata,cAI,cH\)\(p\_\{0\},p\_\{1\},c\_\{\\mathrm\{data\}\},c\_\{\\mathrm\{AI\}\},c\_\{\\mathrm\{H\}\}\)unchanged:

g^h\(r\):=phf^1\(r\)\+\(1−ph\)f^0\(r\),q^h\(r\):=ph​f^1​\(r\)g^h​\(r\),ρ^h\(x∣r\):=q^h\(r\)x\(1−q^h\(r\)\)1−x,d^\(h\)\(r\):=kl\(q^h\(r\)∥q^1−h\(r\)\),I^R\(h\):=∑r∈ℛg^h\(r\)logg^h​\(r\)g^1−h​\(r\)\.\\begin\{split\}\\hat\{g\}\_\{h\}\(r\):=p\_\{h\}\\hat\{f\}\_\{1\}\(r\)\+\(1\-p\_\{h\}\)\\hat\{f\}\_\{0\}\(r\),\\qquad\\hat\{q\}\_\{h\}\(r\):=\\frac\{p\_\{h\}\\hat\{f\}\_\{1\}\(r\)\}\{\\hat\{g\}\_\{h\}\(r\)\},\\qquad\\hat\{\\rho\}\_\{h\}\(x\\mid r\):=\\hat\{q\}\_\{h\}\(r\)^\{x\}\(1\-\\hat\{q\}\_\{h\}\(r\)\)^\{1\-x\},\\\\ \\hat\{d\}^\{\(h\)\}\(r\):=\\operatorname\{kl\}\(\\hat\{q\}\_\{h\}\(r\)\\\|\\hat\{q\}\_\{1\-h\}\(r\)\),\\qquad\\hat\{I\}\_\{R\}^\{\(h\)\}:=\\sum\_\{r\\in\\mathcal\{R\}\}\\hat\{g\}\_\{h\}\(r\)\\log\\frac\{\\hat\{g\}\_\{h\}\(r\)\}\{\\hat\{g\}\_\{1\-h\}\(r\)\}\.\\end\{split\}\(49\)The full\-label informationJX\(h\)J\_\{X\}^\{\(h\)\}is unchanged becausep0,p1p\_\{0\},p\_\{1\}remain known\. Fors∈\[0,1\]s\\in\[0,1\], let

Ψ^h\(s\):=maxη:ℛ→\[0,1\]\\displaystyle\\hat\{\\Psi\}\_\{h\}\(s\):=\\max\_\{\\eta:\\mathcal\{R\}\\to\[0,1\]\}∑r∈ℛg^h​\(r\)​η​\(r\)​d^\(h\)​\(r\)\\displaystyle\\sum\_\{r\\in\\mathcal\{R\}\}\\hat\{g\}\_\{h\}\(r\)\\eta\(r\)\\hat\{d\}^\{\(h\)\}\(r\)\(50\)s\.t\.\\displaystyle\\text\{s\.t\.\}∑r∈ℛg^h​\(r\)​η​\(r\)≤s,\\displaystyle\\sum\_\{r\\in\\mathcal\{R\}\}\\hat\{g\}\_\{h\}\(r\)\\eta\(r\)\\leq s,and define the plug\-in sensing program

Γ^h​\(T,N\):=minnH,nAI,nesc≥0\\displaystyle\\hat\{\\Gamma\}\_\{h\}\(T,N\):=\\min\_\{n\_\{\\mathrm\{H\}\},n\_\{\\mathrm\{AI\}\},n\_\{\\mathrm\{esc\}\}\\geq 0\}cH​nH\+cAI​nAI\+cH​nesc\\displaystyle c\_\{\\mathrm\{H\}\}n\_\{\\mathrm\{H\}\}\+c\_\{\\mathrm\{AI\}\}n\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}n\_\{\\mathrm\{esc\}\}\(51\)s\.t\.\\displaystyle\\text\{s\.t\.\}nH\+nAI≤N,0≤nesc≤nAI,\\displaystyle n\_\{\\mathrm\{H\}\}\+n\_\{\\mathrm\{AI\}\}\\leq N,\\qquad 0\\leq n\_\{\\mathrm\{esc\}\}\\leq n\_\{\\mathrm\{AI\}\},nH​JX\(h\)\+nAI​I^R\(h\)\+nAI​Ψ^h​\(nesc/nAI\)≥T,\\displaystyle n\_\{\\mathrm\{H\}\}J\_\{X\}^\{\(h\)\}\+n\_\{\\mathrm\{AI\}\}\\hat\{I\}\_\{R\}^\{\(h\)\}\+n\_\{\\mathrm\{AI\}\}\\hat\{\\Psi\}\_\{h\}\(n\_\{\\mathrm\{esc\}\}/n\_\{\\mathrm\{AI\}\}\)\\geq T,with the same the convention as in \([21](https://arxiv.org/html/2609.28859#S4.E21)\) \(i\.e\.Ψ^h​\(nesc/nAI\)=0\\hat\{\\Psi\}\_\{h\}\(n\_\{\\mathrm\{esc\}\}/n\_\{\\mathrm\{AI\}\}\)=0whenevernAI=nesc=0n\_\{\\mathrm\{AI\}\}=n\_\{\\mathrm\{esc\}\}=0\)\. The plug\-in increments are

ℓ^R​\(r\):=log⁡g^1​\(r\)g^0​\(r\),ℓ^H​\(x,r\):=log⁡ρ^1​\(x∣r\)ρ^0​\(x∣r\)\.\\hat\{\\ell\}\_\{R\}\(r\):=\\log\\frac\{\\hat\{g\}\_\{1\}\(r\)\}\{\\hat\{g\}\_\{0\}\(r\)\},\\qquad\\hat\{\\ell\}\_\{H\}\(x,r\):=\\log\\frac\{\\hat\{\\rho\}\_\{1\}\(x\\mid r\)\}\{\\hat\{\\rho\}\_\{0\}\(x\\mid r\)\}\.\(52\)

### 6\.2The Guarded Plug\-In Policy

Section[6\.1](https://arxiv.org/html/2609.28859#S6.SS1)estimatesf0,f1f\_\{0\},f\_\{1\}and replaces all relevant known\-ffobjects with their plug\-in versions, e\.g\.,Ψ^h​\(s\),Γ^h​\(T,N\)\\hat\{\\Psi\}\_\{h\}\(s\),\\hat\{\\Gamma\}\_\{h\}\(T,N\), etc\. Running Algorithm[1](https://arxiv.org/html/2609.28859#alg1)on these objects alone, however, would not be valid, since the statistic built from the plug\-in increments \([52](https://arxiv.org/html/2609.28859#S6.E52)\) is not the exact log\-likelihood ratio, and the plug\-in sensing program may overstate the information an allocation delivers\. To finish defining our policy, we therefore add two guardrail terms, calibrated to the estimation errorrmr\_\{m\}and hence valid onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}: one inflates the information targets, the other widens the stopping boundaries\.

The first guard is an information\-target guardrail, where instead of solvingΓh​\(T,N\)\\Gamma\_\{h\}\(T,N\)atT=Th,kT=T\_\{h,k\}we add an extra buffer because the plug\-in expected drift of the log\-likelihood statistic may be overstated by the randomness in our estimates\. Specifically, we use the following guarded information\-target:

T^h,m,kpilot​\(N\):=Th,k\+\(N\+1\)​εmdr,\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{h,m,k\}\(N\):=T\_\{h,k\}\+\(N\+1\)\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\},\(53\)whereεmdr:=3​\|ℛ\|​Bℓ​rm\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}:=3\|\\mathcal\{R\}\|B\_\{\\ell\}r\_\{m\}, withBℓ:=max⁡\{log⁡\(p1/p0\),log⁡\(\(1−p0\)/\(1−p1\)\)\}B\_\{\\ell\}:=\\max\\\{\\log\(p\_\{1\}/p\_\{0\}\),\\;\\log\(\(1\-p\_\{0\}\)/\(1\-p\_\{1\}\)\)\\\}, is the upper bound on the deviation between the true\-channel and plug\-in\-channel expected one\-item drift of the plug\-in log\-likelihood statistic conditional on the pilot good eventℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}\(see details in Appendix[13\.1](https://arxiv.org/html/2609.28859#S13.SS1)\)\.

The guarded analogue of the buffered design value \([33](https://arxiv.org/html/2609.28859#S5.E33)\) is then

F^m,kpilot​\(N\):=N​cdata\+maxh∈\{0,1\}⁡Γ^h​\(T^h,m,kpilot​\(N\),N\)\.\\widehat\{F\}^\{\\mathrm\{pilot\}\}\_\{m,k\}\(N\):=Nc\_\{\\mathrm\{data\}\}\+\\max\_\{h\\in\\\{0,1\\\}\}\\hat\{\\Gamma\}\_\{h\}\(\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{h,m,k\}\(N\),N\)\.\(54\)
Call an integerN≥Nfixed,H​\(α2,k,β2,k\)N\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)admissible ifΓ^h​\(T^h,m,kpilot​\(N\),N\)<∞\\hat\{\\Gamma\}\_\{h\}\(\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{h,m,k\}\(N\),N\)<\\inftyfor bothh∈\{0,1\}h\\in\\\{0,1\\\}, that is, if some allocation within a pool of sizeNNmeets both guarded targets\. If at least one admissibleNNexists, choose

N¯^main,m,k∈arg​minN∈ℤ\+,N≥Nfixed,H​\(α2,k,β2,k\)⁡F^m,kpilot​\(N\)\.\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\\in\\operatorname\*\{arg\\,min\}\_\{N\\in\\mathbb\{Z\}\_\{\+\},N\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\}\\widehat\{F\}^\{\\mathrm\{pilot\}\}\_\{m,k\}\(N\)\.\(55\)and letN^main,m,k:=N¯^main,m,k\+1\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}:=\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\+1\. The remainder of the procedure is structurally identical to Algorithm[1](https://arxiv.org/html/2609.28859#alg1), with plug\-in quantities and the guardrails\. Formally, for eachhh, choose an optimizer\(n^H,h,m,k∗,n^AI,h,m,k∗,n^esc,h,m,k∗\)\(\\hat\{n\}^\{\*\}\_\{\\mathrm\{H\},h,m,k\},\\hat\{n\}^\{\*\}\_\{\\mathrm\{AI\},h,m,k\},\\hat\{n\}^\{\*\}\_\{\\mathrm\{esc\},h,m,k\}\)of problem \([51](https://arxiv.org/html/2609.28859#S6.E51)\) atN=N¯^main,m,kN=\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}andT=T^h,m,kpilot​\(N¯^main,m,k\)T=\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{h,m,k\}\(\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\)\. Set

η^h,m,kH:=\{⌈n^H,h,m,k∗⌉/\(⌈n^H,h,m,k∗⌉\+⌈n^AI,h,m,k∗⌉\),⌈n^H,h,m,k∗⌉\+⌈n^AI,h,m,k∗⌉\>0,0,⌈n^H,h,m,k∗⌉\+⌈n^AI,h,m,k∗⌉=0,\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{h,m,k\}:=\\begin\{cases\}\\lceil\\hat\{n\}^\{\*\}\_\{\\mathrm\{H\},h,m,k\}\\rceil/\(\\lceil\\hat\{n\}^\{\*\}\_\{\\mathrm\{H\},h,m,k\}\\rceil\+\\lceil\\hat\{n\}^\{\*\}\_\{\\mathrm\{AI\},h,m,k\}\\rceil\),&\\lceil\\hat\{n\}^\{\*\}\_\{\\mathrm\{H\},h,m,k\}\\rceil\+\\lceil\\hat\{n\}^\{\*\}\_\{\\mathrm\{AI\},h,m,k\}\\rceil\>0,\\\\ 0,&\\lceil\\hat\{n\}^\{\*\}\_\{\\mathrm\{H\},h,m,k\}\\rceil\+\\lceil\\hat\{n\}^\{\*\}\_\{\\mathrm\{AI\},h,m,k\}\\rceil=0,\\end\{cases\}\(56\)and

s^h,m,k:=\{n^esc,h,m,k∗/n^AI,h,m,k∗,n^AI,h,m,k∗\>0,0,n^AI,h,m,k∗=0\.\\hat\{s\}\_\{h,m,k\}:=\\begin\{cases\}\\hat\{n\}^\{\*\}\_\{\\mathrm\{esc\},h,m,k\}/\\hat\{n\}^\{\*\}\_\{\\mathrm\{AI\},h,m,k\},&\\hat\{n\}^\{\*\}\_\{\\mathrm\{AI\},h,m,k\}\>0,\\\\ 0,&\\hat\{n\}^\{\*\}\_\{\\mathrm\{AI\},h,m,k\}=0\.\\end\{cases\}
Chooseη^h,m,kesc\\hat\{\\eta\}^\{\\mathrm\{esc\}\}\_\{h,m,k\}to be any optimizer of \([50](https://arxiv.org/html/2609.28859#S6.E50)\) ats=s^h,m,ks=\\hat\{s\}\_\{h,m,k\}; equivalently, sort the reports byd^\(h\)​\(r\)\\hat\{d\}^\{\(h\)\}\(r\)and use the fractional\-knapsack rule of Proposition[4\.8](https://arxiv.org/html/2609.28859#S4.Thmtheorem8)\. In the dead zone, use

η^∗,m,kH:=η^0,m,kH\+η^1,m,kH2,η^∗,m,kesc​\(r\):=η^0,m,kesc​\(r\)\+η^1,m,kesc​\(r\)2\.\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{\*,m,k\}:=\\frac\{\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{0,m,k\}\+\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{1,m,k\}\}\{2\},\\qquad\\hat\{\\eta\}^\{\\mathrm\{esc\}\}\_\{\*,m,k\}\(r\):=\\frac\{\\hat\{\\eta\}^\{\\mathrm\{esc\}\}\_\{0,m,k\}\(r\)\+\\hat\{\\eta\}^\{\\mathrm\{esc\}\}\_\{1,m,k\}\(r\)\}\{2\}\.
If no admissibleNNexists, so that the guarded outer problem \([55](https://arxiv.org/html/2609.28859#S6.E55)\) is infeasible, we fall back on the safe defaultN^main,m,k:=Nfixed,H​\(α2,k,β2,k\)\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}:=N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)to query a human on every item\.

Next, we define the second guardrail, on the crossing boundariesaka\_\{k\}and−bk\-b\_\{k\}\. The policy now accumulatesS^\\hat\{S\}, built from the plug\-in increments \([52](https://arxiv.org/html/2609.28859#S6.E52)\), rather than the exact log\-likelihood ratioStS\_\{t\}, so crossingaka\_\{k\}no longer certifies the levelα1,k\\alpha\_\{1,k\}; we therefore widen the boundaries by enough to cover the gap\|S^−S\|\|\\hat\{S\}\-S\|\. To that end, letεmLR\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}defined in Appendix[12\.1](https://arxiv.org/html/2609.28859#S12.SS1)\([149](https://arxiv.org/html/2609.28859#S12.E149)\) be an upper bound on\|ℓ^R​\(r\)−ℓR​\(r\)\|\|\\hat\{\\ell\}\_\{R\}\(r\)\-\\ell\_\{R\}\(r\)\|onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}\(see details in Appendix[12\.1](https://arxiv.org/html/2609.28859#S12.SS1)\)\. If the guarded plug\-in programs are feasible then we set the guardrailωm,k:=N^main,m,k​εmLR\\omega\_\{m,k\}:=\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}, and the two stopping boundaries are given by

ak\+ωm,k​and−\(bk\+ωm,k\)\.a\_\{k\}\+\\omega\_\{m,k\}\\text\{ and \}\-\(b\_\{k\}\+\\omega\_\{m,k\}\)\.If the guarded plug\-in programs are infeasible we set the guardrailωm,k=0\\omega\_\{m,k\}=0, since the human\-only safe default accumulates the exact incrementsℓX\\ell\_\{X\}, which depend only on the knownp0,p1p\_\{0\},p\_\{1\}\. If neither boundary is reached, we use a fallback stage and reveal every still\-unknown label in the fixed set

𝒥^H,k:=\{1,…,Nfixed,H​\(α2,k,β2,k\)\}\\hat\{\\mathcal\{J\}\}\_\{\\mathrm\{H\},k\}:=\\\{1,\\ldots,N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\\\}and apply the exact randomized full\-label Neyman–Pearson test at levels\(α2,k,β2,k\)\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\. Denote the resulting policy byπ^m,k\\hat\{\\pi\}\_\{m,k\}corresponding tomm\.

Table 2:Parameters of the guarded plug\-in policyπ^m,k\\hat\{\\pi\}\_\{m,k\}in Algorithm[2](https://arxiv.org/html/2609.28859#alg2)\.We summarize the calculations of parameters needed for policyπ^m,k\\hat\{\\pi\}\_\{m,k\}in Table[2](https://arxiv.org/html/2609.28859#S6.T2), and give the policy for estimatedf0,f1f\_\{0\},f\_\{1\}in Algorithm[2](https://arxiv.org/html/2609.28859#alg2)\. We then show the feasibility and first\-order optimality, both conditional on the good pilot eventℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}\.

Algorithm 2Pilot\-Calibrated Guarded Plug\-In Policy \(π^m,k\\hat\{\\pi\}\_\{m,k\}\)Input:Primitives\(p0,p1,cdata,cAI,cH\)\(p\_\{0\},p\_\{1\},c\_\{\\mathrm\{data\}\},c\_\{\\mathrm\{AI\}\},c\_\{\\mathrm\{H\}\}\), error targets\(αk,βk\)\(\\alpha\_\{k\},\\beta\_\{k\}\), admissible parameters\(Δk,ffb,k,zk\)\(\\Delta\_\{k\},f\_\{\\mathrm\{fb\},k\},z\_\{k\}\), and pilot data𝒟m\\mathcal\{D\}\_\{m\}\.

1Compute the error\-budget split \([23](https://arxiv.org/html/2609.28859#S5.E23)\)–\([24](https://arxiv.org/html/2609.28859#S5.E24)\), boundaries

\(ak,bk\)\(a\_\{k\},b\_\{k\}\)in \([25](https://arxiv.org/html/2609.28859#S5.E25)\), and buffered targets

\(T1,k,T0,k\)=\(ak\+Δk,bk\+Δk\)\(T\_\{1,k\},T\_\{0,k\}\)=\(a\_\{k\}\+\\Delta\_\{k\},b\_\{k\}\+\\Delta\_\{k\}\)\.

2Compute

f^0,f^1\\hat\{f\}\_\{0\},\\hat\{f\}\_\{1\}, all plug\-in quantities in \([49](https://arxiv.org/html/2609.28859#S6.E49)\)–\([52](https://arxiv.org/html/2609.28859#S6.E52)\), and

rm,εmLR,εmdrr\_\{m\},\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\},\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}\.

3Form the guarded targets and outer objective in \([53](https://arxiv.org/html/2609.28859#S6.E53)\)–\([54](https://arxiv.org/html/2609.28859#S6.E54)\)\.

4if*the guarded outer problem \([55](https://arxiv.org/html/2609.28859#S6.E55)\) is feasible*then

5Solve \([55](https://arxiv.org/html/2609.28859#S6.E55)\) for

N¯^main,m,k\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}and set

N^main,m,k←N¯^main,m,k\+1\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\\leftarrow\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\+1; compute

η^h,m,kH\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{h,m,k\},

η^h,m,kesc\\hat\{\\eta\}^\{\\mathrm\{esc\}\}\_\{h,m,k\}, and their dead\-zone averages; set

ωm,k←N^main,m,k​εmLR\\omega\_\{m,k\}\\leftarrow\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}\.

6else

7Set

N^main,m,k←Nfixed,H​\(α2,k,β2,k\)\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\\leftarrow N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)and

ωm,k←0\\omega\_\{m,k\}\\leftarrow 0\.

8For every

j∈\{0,1,∗\}j\\in\\\{0,1,\*\\\}and

r∈ℛr\\in\\mathcal\{R\}, set

η^j,m,kH←1\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{j,m,k\}\\leftarrow 1and

η^j,m,kesc​\(r\)←0\\hat\{\\eta\}^\{\\mathrm\{esc\}\}\_\{j,m,k\}\(r\)\\leftarrow 0\.

9Set the guarded boundaries to

ak\+ωm,ka\_\{k\}\+\\omega\_\{m,k\}and

−\(bk\+ωm,k\)\-\(b\_\{k\}\+\\omega\_\{m,k\}\)\.

10Acquire

N^main,m,k\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}items and initialize

S^←0\\hat\{S\}\\leftarrow 0\.

11for*i=1i=1toN^main,m,k\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}*do

12Set

J←1J\\leftarrow 1if

S^\>zk\\hat\{S\}\>z\_\{k\},

J←0J\\leftarrow 0if

S^<−zk\\hat\{S\}<\-z\_\{k\}, and

J←∗J\\leftarrow\*otherwise\.

13With probability

η^J,m,kH\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{J,m,k\}, query a human and add

ℓX​\(Xi\)\\ell\_\{X\}\(X\_\{i\}\)to

S^\\hat\{S\}\.

14Otherwise query the AI and add

ℓ^R​\(Ri\)\\hat\{\\ell\}\_\{R\}\(R\_\{i\}\); then, with probability

η^J,m,kesc​\(Ri\)\\hat\{\\eta\}^\{\\mathrm\{esc\}\}\_\{J,m,k\}\(R\_\{i\}\), query a human and add

ℓ^H​\(Xi,Ri\)\\hat\{\\ell\}\_\{H\}\(X\_\{i\},R\_\{i\}\)\.

15After every update, reject if

S^≥ak\+ωm,k\\hat\{S\}\\geq a\_\{k\}\+\\omega\_\{m,k\}and accept if

S^≤−\(bk\+ωm,k\)\\hat\{S\}\\leq\-\(b\_\{k\}\+\\omega\_\{m,k\}\)\.

16If neither boundary is crossed, execute the fixed\-set full\-label fallback on

𝒥^H,k\\hat\{\\mathcal\{J\}\}\_\{\\mathrm\{H\},k\}\.

###### Theorem 6\.1\(Pilot\-conditional exact validity\)

Suppose Assumptions[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)and[6\.1](https://arxiv.org/html/2609.28859#S6.SS1)hold\. For everym≥2m\\geq 2, everyk≥1k\\geq 1, and every choice of the admissible parameters\(Δk,ffb,k,zk\)\(\\Delta\_\{k\},f\_\{\\mathrm\{fb\},k\},z\_\{k\}\), the following hold almost surely onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}:

ℙ0π^m,k​\(δπ^m,k=1∣𝒟m\)≤αk,ℙ1π^m,k​\(δπ^m,k=0∣𝒟m\)≤βk\.\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{0\}\(\\delta^\{\\hat\{\\pi\}\_\{m,k\}\}=1\\mid\\mathcal\{D\}\_\{m\}\)\\leq\\alpha\_\{k\},\\qquad\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\(\\delta^\{\\hat\{\\pi\}\_\{m,k\}\}=0\\mid\\mathcal\{D\}\_\{m\}\)\\leq\\beta\_\{k\}\.

Theorem[6\.1](https://arxiv.org/html/2609.28859#S6.Thmtheorem1)is the plug\-in counterpart of Theorem[5\.2](https://arxiv.org/html/2609.28859#S5.Thmtheorem2): the error targets are met exactly, at every pilot size and without any rate condition, provided the pilot realization is good\. Averaging over pilot realizations removes the conditioning at the cost of the failure probability ofℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}\.

###### Corollary 6\.2\(Unconditional validity\)

Suppose Assumptions[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)and[6\.1](https://arxiv.org/html/2609.28859#S6.SS1)hold\. For everym≥2m\\geq 2andk≥1k\\geq 1, and every choice of the admissible parameters\(Δk,ffb,k,zk\)\(\\Delta\_\{k\},f\_\{\\mathrm\{fb\},k\},z\_\{k\}\),

ℙ0π^m,k​\(δπ^m,k=1\)≤αk\+δm,ℙ1π^m,k​\(δπ^m,k=0\)≤βk\+δm\.\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{0\}\(\\delta^\{\\hat\{\\pi\}\_\{m,k\}\}=1\)\\leq\\alpha\_\{k\}\+\\delta\_\{m\},\\qquad\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\(\\delta^\{\\hat\{\\pi\}\_\{m,k\}\}=0\)\\leq\\beta\_\{k\}\+\\delta\_\{m\}\.

For first\-order cost optimality of the policy, we focus on the balanced logarithmic regime under Assumption[5\.3](https://arxiv.org/html/2609.28859#S5.SS3)\. Under this regime,min⁡\{T0,k,T1,k\}=Θ⁡\(Lk\)\\min\\\{T\_\{0,k\},T\_\{1,k\}\\\}=\\Theta\(L\_\{k\}\), so factors ofmin⁡\{T0,k,T1,k\}/Lk\\min\\\{T\_\{0,k\},T\_\{1,k\}\\\}/L\_\{k\}are absorbed into primitive constants below\. Theorem[6\.3](https://arxiv.org/html/2609.28859#S6.Thmtheorem3)then states the convergence rate ofπ^m,k\\hat\{\\pi\}\_\{m,k\}\.

###### Theorem 6\.3\(First\-order optimality with pilot\-estimated AI accuracy\)

Suppose Assumptions[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)\-[6\.1](https://arxiv.org/html/2609.28859#S6.SS1)hold,ffb,kf\_\{\\mathrm\{fb\},k\}is bounded away from zero and one, and\(Δk,zk\)\(\\Delta\_\{k\},z\_\{k\}\)satisfy \([41](https://arxiv.org/html/2609.28859#S5.E41)\) and \([42](https://arxiv.org/html/2609.28859#S5.E42)\)\. Letm=mpilot,k→∞m=m\_\{\\mathrm\{pilot\},k\}\\to\\inftysatisfy

Lk​rmpilot,k=o⁡\(Δk\)\.L\_\{k\}r\_\{m\_\{\\mathrm\{pilot\},k\}\}=o\(\\Delta\_\{k\}\)\.\(57\)Then, onℰpilot,mpilot,k\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\_\{\\mathrm\{pilot\},k\}\}and for all sufficiently largekk,

maxh⁡𝔼hπ^mpilot,k,k​\[Cπ^mpilot,k,k∣𝒟mpilot,k\]LBk=1\+o⁡\(1\)\.\\displaystyle\\frac\{\\max\_\{h\}\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\[C^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\\mid\\mathcal\{D\}\_\{m\_\{\\mathrm\{pilot\},k\}\}\]\}\{\\mathrm\{LB\}\_\{k\}\}=1\+o\(1\)\.

In words, not knowingf0,f1f\_\{0\},f\_\{1\}does not impact our first\-order asymptotic optimality once the pilot is large enough relative toLkL\_\{k\}, though the cost of collecting that pilot is not yet charged; we return to it in Section[6\.3](https://arxiv.org/html/2609.28859#S6.SS3)\.

### 6\.3Pilot\-Size Tradeoffs and Calibration Cost

In this subsection, we further unpack Theorem[6\.3](https://arxiv.org/html/2609.28859#S6.Thmtheorem3)and condition \([57](https://arxiv.org/html/2609.28859#S6.E57)\)\. By construction,rm=O⁡\(log⁡\(m\)/m\)r\_\{m\}=O\(\\sqrt\{\\log\{\(m\)\}/m\}\), and thus condition \([57](https://arxiv.org/html/2609.28859#S6.E57)\) is implied by

mpilot,klog⁡mpilot,k​\(ΔkLk\)2⟶∞\.\\frac\{m\_\{\\mathrm\{pilot\},k\}\}\{\\log m\_\{\\mathrm\{pilot\},k\}\}\\left\(\\frac\{\\Delta\_\{k\}\}\{L\_\{k\}\}\\right\)^\{2\}\\longrightarrow\\infty\.\(58\)Therefore, there is a tradeoff between pilot size and a larger statistical buffer\. We summarize this and give specific conditions on tuning parameters to maintain the first\-order optimality in Theorem[6\.3](https://arxiv.org/html/2609.28859#S6.Thmtheorem3)in the following corollary:

###### Corollary 6\.4\(Concrete pilot and buffer rates\)

Suppose Assumptions[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)\-[6\.1](https://arxiv.org/html/2609.28859#S6.SS1)hold\. Assumeffb,k=1/2f\_\{\\mathrm\{fb\},k\}=1/2andzk=1z\_\{k\}=1\. LetΔk=Lk2/3\\Delta\_\{k\}=L\_\{k\}^\{2/3\}andmpilot,k=⌈Lk2/3​\(log⁡Lk\)2⌉m\_\{\\mathrm\{pilot\},k\}=\\lceil L\_\{k\}^\{2/3\}\(\\log L\_\{k\}\)^\{2\}\\rceil\. Then we have thatmpilot,k/log⁡mpilot,k≫Lk2/3m\_\{\\mathrm\{pilot\},k\}/\\log m\_\{\\mathrm\{pilot\},k\}\\gg L\_\{k\}^\{2/3\}, and onℰpilot,mpilot,k\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\_\{\\mathrm\{pilot\},k\}\}and for all sufficiently largekk,

maxh⁡𝔼hπ^mpilot,k,k​\[Cπ^mpilot,k,k∣𝒟mpilot,k\]LBk≤1\+O\(Lk−1/3\)\.\\frac\{\\max\_\{h\}\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\[C^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\\mid\\mathcal\{D\}\_\{m\_\{\\mathrm\{pilot\},k\}\}\]\}\{\\mathrm\{LB\}\_\{k\}\}\\leq 1\+O\(L^\{\-1/3\}\_\{k\}\)\.

Corollary[6\.4](https://arxiv.org/html/2609.28859#S6.Thmtheorem4)does not account for the cost for the pilot items\. If, instead, a fresh paired pilot is collected solely for thekkth test, its acquisition cost must also be charged to that test\. Since each pilot item is acquired and labeled by both the AI and the human, it has a total cost ofcdata\+cAI\+cHc\_\{\\mathrm\{data\}\}\+c\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}\. To obtain a minimum ofmmpilot samples for both labels, recall we needMtotM\_\{\\mathrm\{tot\}\}total pilot samples\. Thus the total cost for pilot samples isCmpilot:=Mtot​\(cdata\+cAI\+cH\)C\_\{m\}^\{\\mathrm\{pilot\}\}:=M\_\{\\mathrm\{tot\}\}\(c\_\{\\mathrm\{data\}\}\+c\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}\)\. The maximum expected cost including the pilot data collection cost over two hypotheses ismaxh⁡𝔼hπ^m,k​\[Cmpilot\+Cπ^m,k\]\\max\_\{h\}\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m,k\}\}\[C\_\{m\}^\{\\mathrm\{pilot\}\}\+C^\{\\hat\{\\pi\}\_\{m,k\}\}\]\. Corollary[6\.5](https://arxiv.org/html/2609.28859#S6.Thmtheorem5)verifies this total cost under the choice ofΔk\\Delta\_\{k\}given by Corollary[6\.4](https://arxiv.org/html/2609.28859#S6.Thmtheorem4)\.

###### Corollary 6\.5\(Charging the one\-time pilot\)

Suppose Assumptions[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)\-[6\.1](https://arxiv.org/html/2609.28859#S6.SS1)hold, and letffb,k=1/2f\_\{\\mathrm\{fb\},k\}=1/2andzk=1z\_\{k\}=1\. Then the choice in Corollary[6\.4](https://arxiv.org/html/2609.28859#S6.Thmtheorem4)satisfiesLk​rmpilot,k=o⁡\(Δk\)L\_\{k\}r\_\{m\_\{\\mathrm\{pilot\},k\}\}=o\(\\Delta\_\{k\}\)andmpilot,k=o⁡\(Lk\)m\_\{\\mathrm\{pilot\},k\}=o\(L\_\{k\}\), and gives that for all sufficiently largekk,

maxh⁡𝔼hπ^mpilot,k,k​\[Cmpilot,kpilot\+Cπ^mpilot,k,k\]LBk≤1\+O~\(Lk−1/3\)\.\\frac\{\\max\_\{h\}\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\[C\_\{m\_\{\\mathrm\{pilot\},k\}\}^\{\\mathrm\{pilot\}\}\+C^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\]\}\{\\mathrm\{LB\}\_\{k\}\}\\leq 1\+\\widetilde\{O\}\(L\_\{k\}^\{\-1/3\}\)\.

Corollary[6\.5](https://arxiv.org/html/2609.28859#S6.Thmtheorem5)closes the gap between the statistical and operational costs of learning the AI\-output model\. It tells us that even when the paired pilot must be collected specifically for the current test and its full data, AI, and human costs are charged to the procedure, the resulting total cost remains first\-order optimal\. Thus the known\-ffbenchmark is asymptotically attainable without assuming that calibration data are available for free\.

## 7Conclusion

This paper studies how to combine inexpensive but imperfect AI information with costly human verification when the goal is to conduct a statistically valid hypothesis test at minimum cost\. The key operational feature is selectivity: after acquiring a fixed pool of items, the decision maker can choose whether to query the AI, query a human directly, escalate an AI\-scored item to a human, or stop once sufficient evidence has accumulated\. This creates a joint statistical and operational design problem in which the value of a query depends not only on its cost and information content, but also on the information already collected\.

We derive an information\-theoretic lower bound that captures the minimum cost required to satisfy the testing errors while accounting for data acquisition, AI scoring, direct human review, and selective escalation\. We then develop SCALE, a sequential cost\-aware policy that dynamically combines these actions as evidence accumulates\. SCALE is finite\-sample valid and matches the lower bound to first order as the target errors vanish\. We also extend the analysis to the practically important case in which the AI\-output model is unknown and must be estimated from paired AI–human pilot data\. A guarded plug\-in version of SCALE remains first\-order optimal when the pilot is sufficiently accurate\.

Several directions remain open\. We have focused on a binary label, two simple hypotheses, a single AI source, and a finite AI\-report alphabet\. Extending the framework to composite hypotheses, multiple AI systems with heterogeneous costs and accuracies, and richer or continuous report spaces would broaden its applicability\. Another natural direction is to learn the AI\-output model during the main experiment rather than through a separate pilot, thereby jointly deciding when information should be used for calibration and when it should be used for the hypothesis test itself\. More broadly, the analysis suggests that the relevant question in human–AI inference is not simply whether AI should replace human judgment\. Rather, the operational value of AI comes from deciding when inexpensive machine information is sufficient and when the remaining uncertainty is valuable enough to justify human verification\.

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\\ECSwitch

## 8Additional Materials for Section[4](https://arxiv.org/html/2609.28859#S4)

### 8\.1Proof of Lemma[4\.2](https://arxiv.org/html/2609.28859#S4.Thmtheorem2)

###### Proof 8\.1

Proof of Lemma[4\.2](https://arxiv.org/html/2609.28859#S4.Thmtheorem2)Suppose, for a contradiction, that there exists a feasible policyπ∈ℱ⁡\(α,β\)\\pi\\in\\mathcal\{F\}\(\\alpha,\\beta\)usingN<Nfixed,H​\(α,β\)N<N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)acquired items\. We will construct a randomized full\-label testϕ¯:\{0,1\}N→\[0,1\]\\bar\{\\phi\}:\\\{0,1\\\}^\{N\}\\to\[0,1\]with the same type\-I and type\-II error probabilities asπ\\pi\. Hereϕ¯​\(x\)\\bar\{\\phi\}\(x\)denotes the probability of rejectingH0H\_\{0\}given the full label vectorxx\. This will contradict the definition ofNfixed,H​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)as the minimum sample size for a full\-label randomized test satisfying the target errors\.

Represent any policy randomization by a seedUUindependent of the data and with the same law under both hypotheses\. Conditional onU=uU=u, the policy is deterministic\. Given a full label vectorx∈\{0,1\}Nx\\in\\\{0,1\\\}^\{N\}, a potential report vectorr∈ℛNr\\in\\mathcal\{R\}^\{N\}, and a seeduu, simulate the policy as follows\. Whenever the simulated policy queries the AI on itemii, revealrir\_\{i\}to the simulation\. Whenever it queries a human on itemii, revealxix\_\{i\}\. Continue until the simulated policy stops, and letϕ~​\(x,r,u\)∈\{0,1\}\\tilde\{\\phi\}\(x,r,u\)\\in\\\{0,1\\\}be the resulting decision\. Thus each item is AI\-queried at most once and human\-queried at most once, so the simulation reveals eachrir\_\{i\}and eachxix\_\{i\}at most once\.

Under the actual experiment,ϕ~​\(X,R,U\)=δπ\\tilde\{\\phi\}\(X,R,U\)=\\delta^\{\\pi\}almost surely, because the simulation reveals exactly the observations that the policy would receive\. Hence

ℙ0π​\(ϕ~​\(X,R,U\)=1\)≤α,ℙ1π​\(ϕ~​\(X,R,U\)=0\)≤β\.\\mathbb\{P\}^\{\\pi\}\_\{0\}\(\\tilde\{\\phi\}\(X,R,U\)=1\)\\leq\\alpha,\\qquad\\mathbb\{P\}^\{\\pi\}\_\{1\}\(\\tilde\{\\phi\}\(X,R,U\)=0\)\\leq\\beta\.\(59\)
Conditional onX=xX=x, the potential reportsR1,…,RNR\_\{1\},\\ldots,R\_\{N\}have product law∏ifxi​\(ri\)\\prod\_\{i\}f\_\{x\_\{i\}\}\(r\_\{i\}\)by Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)\. This conditional law does not depend on whetherH0H\_\{0\}orH1H\_\{1\}is true\. The seed law also does not depend on the hypothesis\. Define

ϕ¯​\(x\):=𝔼⁡\[ϕ~​\(x,R,U\)∣X=x\],x∈\{0,1\}N,\\bar\{\\phi\}\(x\):=\\mathbb\{E\}\[\\tilde\{\\phi\}\(x,R,U\)\\mid X=x\],\\qquad x\\in\\\{0,1\\\}^\{N\},where the expectation is taken over the common conditional law of\(R,U\)\(R,U\)givenX=xX=x\. Becauseϕ~∈\{0,1\}\\tilde\{\\phi\}\\in\\\{0,1\\\}, its conditional expectation satisfiesϕ¯​\(x\)∈\[0,1\]\\bar\{\\phi\}\(x\)\\in\[0,1\]; thusϕ¯\\bar\{\\phi\}is a valid randomized full\-label test\. By the tower property of conditional expectation,

𝔼h​\[ϕ¯​\(X\)\]=𝔼h​\[𝔼⁡\[ϕ~​\(X,R,U\)∣X\]\]=𝔼h​\[ϕ~​\(X,R,U\)\],h∈\{0,1\}\.\\mathbb\{E\}\_\{h\}\[\\bar\{\\phi\}\(X\)\]=\\mathbb\{E\}\_\{h\}\[\\mathbb\{E\}\[\\tilde\{\\phi\}\(X,R,U\)\\mid X\]\]=\\mathbb\{E\}\_\{h\}\[\\tilde\{\\phi\}\(X,R,U\)\],\\qquad h\\in\\\{0,1\\\}\.Together with \([59](https://arxiv.org/html/2609.28859#S8.E59)\), this implies

𝔼0​\[ϕ~¯​\(X\)\]≤α,𝔼1​\[1−ϕ~¯​\(X\)\]≤β\.\\mathbb\{E\}\_\{0\}\[\\bar\{\\tilde\{\\phi\}\}\(X\)\]\\leq\\alpha,\\qquad\\mathbb\{E\}\_\{1\}\[1\-\\bar\{\\tilde\{\\phi\}\}\(X\)\]\\leq\\beta\.We have therefore constructed a full\-label randomized test satisfying the target errors with onlyN<Nfixed,H​\(α,β\)N<N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)labels\. This contradicts the minimality in Definition[4\.1](https://arxiv.org/html/2609.28859#S4.Thmtheorem1)\. Hence every feasible policy must satisfyN≥Nfixed,H​\(α,β\)N\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)\.

### 8\.2Transcript Distributions

In this section we formally define the transcript and derive the transcript distribution, which will be used in the proof of subsequent results\.

Recall that a \(full\) transcript is the realization of a sequence of actions and observations up to stopping time, concatenated with the final decisionδ\\delta:𝒯=\(A1,O1,A2,O2,…,ATstopπ−1,OTstopπ−1,ATstopπ,δ\)\\mathcal\{T\}=\(A\_\{1\},O\_\{1\},A\_\{2\},O\_\{2\},\\ldots,A\_\{T^\{\\pi\}\_\{\\text\{stop\}\}\-1\},O\_\{T^\{\\pi\}\_\{\\text\{stop\}\}\-1\},A\_\{T^\{\\pi\}\_\{\\text\{stop\}\}\},\\delta\)\. Also letτ=\(a1,o1,a2,o2,…,atstopπ−1,otstopπ−1,atstopπ,δ~\)\\tau=\(a\_\{1\},o\_\{1\},a\_\{2\},o\_\{2\},\\ldots,a\_\{t^\{\\pi\}\_\{\\text\{stop\}\}\-1\},o\_\{t^\{\\pi\}\_\{\\text\{stop\}\}\-1\},a\_\{t^\{\\pi\}\_\{\\text\{stop\}\}\},\\tilde\{\\delta\}\)be the realization of𝒯\\mathcal\{T\}, withtstopπt^\{\\pi\}\_\{\\text\{stop\}\},δ~\\tilde\{\\delta\},ata\_\{t\},oto\_\{t\}being the realizations ofTstopπT^\{\\pi\}\_\{\\text\{stop\}\},δ\\delta,AtA\_\{t\}andOtO\_\{t\}for eachtt, respectively\. We also define the partial transcript𝒯t\\mathcal\{T\}\_\{t\}as the realization of a sequence of actions and observations at somet<Tstopπt<T^\{\\pi\}\_\{\\text\{stop\}\}:

𝒯t=\(A1,O1,A2,O2,…,At,Ot\)\.\\mathcal\{T\}\_\{t\}=\(A\_\{1\},O\_\{1\},A\_\{2\},O\_\{2\},\\ldots,A\_\{t\},O\_\{t\}\)\.\(60\)Its realization is denoted asτt=\(a1,o1,a2,o2,…,at,ot\)\\tau\_\{t\}=\(a\_\{1\},o\_\{1\},a\_\{2\},o\_\{2\},\\ldots,a\_\{t\},o\_\{t\}\)\. We letτ0=∅\\tau\_\{0\}=\\emptyset\. In other words,τt\\tau\_\{t\}is the realization of historyℋt\\mathcal\{H\}\_\{t\}\.

Let𝔗N\\mathfrak\{T\}\_\{N\}be the set of valid transcripts induced by an admissible policyπ\\pi\. Specifically,𝔗N:=\{τ:at∈𝒜t\(τt−1\)for allt∈\[tstopπ\],ot∈ℛifat=AI\(i\)for somei∈\[Nπ\],ot∈\{0,1\}ifat=H\(i\)for somei∈\[Nπ\],andat=STOPif and only ift=tstopπ\}\\mathfrak\{T\}\_\{N\}:=\\\{\\tau:a\_\{t\}\\in\\mathcal\{A\}\_\{t\}\(\\tau\_\{t\-1\}\)\\text\{ for all \}t\\in\[t^\{\\pi\}\_\{\\text\{stop\}\}\],o\_\{t\}\\in\\mathcal\{R\}\\text\{ if \}a\_\{t\}=\\mathrm\{AI\}\(i\)\\text\{ for some \}i\\in\[N^\{\\pi\}\],o\_\{t\}\\in\\\{0,1\\\}\\text\{ if \}a\_\{t\}=\\mathrm\{H\}\(i\)\\text\{ for some \}i\\in\[N^\{\\pi\}\],\\text\{ and \}a\_\{t\}=\\text\{STOP\}\\text\{ if and only if \}t=t^\{\\pi\}\_\{\\text\{stop\}\}\\\}\. We note that𝔗N\\mathfrak\{T\}\_\{N\}has finite cardinality, as eachata\_\{t\},oto\_\{t\}andδ~\\tilde\{\\delta\}takes only finite values, andtstopπ≤2​Nπ\+1t^\{\\pi\}\_\{\\text\{stop\}\}\\leq 2N^\{\\pi\}\+1\. We letPhπP^\{\\pi\}\_\{h\}be the distribution of transcript underHhH\_\{h\}, andPhπ​\(τ\)=ℙhπ​\(𝒯=τ\)P^\{\\pi\}\_\{h\}\(\\tau\)=\\mathbb\{P\}^\{\\pi\}\_\{h\}\(\\mathcal\{T\}=\\tau\)is the transcript PMF \(Probability Mass Function\) atτ\\tau\. We have that for a feasibleπ∈ℱ⁡\(α,β\)\\pi\\in\\mathcal\{F\}\(\\alpha,\\beta\), the support ofPhπP^\{\\pi\}\_\{h\}has:supp​\(Phπ\)⊆𝔗N\\text\{supp\}\(P^\{\\pi\}\_\{h\}\)\\subseteq\\mathfrak\{T\}\_\{N\}forh∈\{0,1\}h\\in\\\{0,1\\\}\.

We letφh​\(ot\|at,τt−1\)=ℙhπ​\(Ot=ot\|At=at,𝒯t−1=τt−1\)\\varphi\_\{h\}\(o\_\{t\}\|a\_\{t\},\\tau\_\{t\-1\}\)=\\mathbb\{P\}^\{\\pi\}\_\{h\}\(O\_\{t\}=o\_\{t\}\|A\_\{t\}=a\_\{t\},\\mathcal\{T\}\_\{t\-1\}=\\tau\_\{t\-1\}\)be the conditional probability mass of observingoto\_\{t\}at epochtt, given the partial transcriptτt−1\\tau\_\{t\-1\}and the actionata\_\{t\}\. Then we have that

φh​\(ot\|at,τt−1\)=\{gh​\(ot\),ot∈ℛ,at=AI\(i\),ihas not been human\-queried beforet,fx​\(ot\),ot∈ℛ,at=AI\(i\),ihas been human\-queried beforetwith observationx,phot​\(1−ph\)1−ot,ot∈\{0,1\},at=H\(i\),ihas not been AI\-queried beforet,ρh​\(ot\|r\),ot∈\{0,1\},at=H\(i\),ihas been AI\-queried beforetwith observationr,0,otherwise\.\\begin\{split\}&\\,\\varphi\_\{h\}\(o\_\{t\}\|a\_\{t\},\\tau\_\{t\-1\}\)\\\\ =&\\begin\{cases\}g\_\{h\}\(o\_\{t\}\),&o\_\{t\}\\in\\mathcal\{R\},a\_\{t\}=\\mathrm\{AI\}\(i\),i\\text\{ has not been human\-queried before \}t,\\\\ f\_\{x\}\(o\_\{t\}\),&o\_\{t\}\\in\\mathcal\{R\},a\_\{t\}=\\mathrm\{AI\}\(i\),i\\text\{ has been human\-queried before \}t\\text\{ with observation \}x,\\\\ p^\{o\_\{t\}\}\_\{h\}\(1\-p\_\{h\}\)^\{1\-o\_\{t\}\},&o\_\{t\}\\in\\\{0,1\\\},a\_\{t\}=\\mathrm\{H\}\(i\),i\\text\{ has not been AI\-queried before \}t,\\\\ \\rho\_\{h\}\(o\_\{t\}\|r\),&o\_\{t\}\\in\\\{0,1\\\},a\_\{t\}=\\mathrm\{H\}\(i\),i\\text\{ has been AI\-queried before \}t\\text\{ with observation \}r,\\\\ 0,&\\text\{otherwise\}\.\\end\{cases\}\\end\{split\}\(61\)
We letχπ​\(τ\)\\chi^\{\\pi\}\(\\tau\)represent the common policy\-kernel factor of transcriptτ\\tau\. Specifically, letχπ​\(τ\)=∏t=1tstopπℙhπ​\(At=at\|𝒯t−1=τt−1\)​ℙhπ​\(δπ=δ~\|𝒯tstopπ−1=τtstopπ−1,Atstopπ=STOP\)\\chi^\{\\pi\}\(\\tau\)=\\prod^\{t^\{\\pi\}\_\{\\text\{stop\}\}\}\_\{t=1\}\\mathbb\{P\}^\{\\pi\}\_\{h\}\(A\_\{t\}=a\_\{t\}\|\\mathcal\{T\}\_\{t\-1\}=\\tau\_\{t\-1\}\)\\mathbb\{P\}^\{\\pi\}\_\{h\}\(\\delta^\{\\pi\}=\\tilde\{\\delta\}\|\\mathcal\{T\}\_\{t^\{\\pi\}\_\{\\text\{stop\}\}\-1\}=\\tau\_\{t^\{\\pi\}\_\{\\text\{stop\}\}\-1\},A\_\{t^\{\\pi\}\_\{\\text\{stop\}\}\}=\\text\{STOP\}\)\. Because the policy and its randomization law are hypothesis\-independent, the conditional distributions of the action and of the terminal decision, given the same realized history, are identical underH0H\_\{0\}andH1H\_\{1\}\. That is,ℙ0π​\(At=at\|𝒯t−1=τt−1\)=ℙ1π​\(At=at\|𝒯t−1=τt−1\)\\mathbb\{P\}^\{\\pi\}\_\{0\}\(A\_\{t\}=a\_\{t\}\|\\mathcal\{T\}\_\{t\-1\}=\\tau\_\{t\-1\}\)=\\mathbb\{P\}^\{\\pi\}\_\{1\}\(A\_\{t\}=a\_\{t\}\|\\mathcal\{T\}\_\{t\-1\}=\\tau\_\{t\-1\}\)for alltt, and thatχπ​\(τ\)\\chi^\{\\pi\}\(\\tau\)is the same acrossH0H\_\{0\}andH1H\_\{1\}\. We are now ready to characterize the transcript distribution\.

###### Lemma 8\.2\(Transcript probability mass\)

For every admissible policyπ\\pi, everyh∈\{0,1\}h\\in\\\{0,1\\\}, and everyτ∈𝔗N\\tau\\in\\mathfrak\{T\}\_\{N\},

Phπ​\(τ\)=χπ​\(τ\)​∏t=1tstopπ−1φh​\(ot∣at,τt−1\)\.P\_\{h\}^\{\\pi\}\(\\tau\)=\\chi^\{\\pi\}\(\\tau\)\\prod\_\{t=1\}^\{t^\{\\pi\}\_\{\\text\{stop\}\}\-1\}\\varphi\_\{h\}\(o\_\{t\}\\mid a\_\{t\},\\tau\_\{t\-1\}\)\.\(62\)Additionally,supp​\(P1π\)=supp​\(P0π\)\\text\{supp\}\(P^\{\\pi\}\_\{1\}\)=\\text\{supp\}\(P^\{\\pi\}\_\{0\}\), and that forτ∈supp​\(Phπ\)\\tau\\in\\text\{supp\}\(P^\{\\pi\}\_\{h\}\),

log⁡P1π​\(τ\)P0π​\(τ\)=∑t=1tstopπ−1log⁡φ1​\(ot∣at,τt−1\)φ0​\(ot∣at,τt−1\)\.\\log\\frac\{P\_\{1\}^\{\\pi\}\(\\tau\)\}\{P\_\{0\}^\{\\pi\}\(\\tau\)\}=\\sum\_\{t=1\}^\{t^\{\\pi\}\_\{\\text\{stop\}\}\-1\}\\log\\frac\{\\varphi\_\{1\}\(o\_\{t\}\\mid a\_\{t\},\\tau\_\{t\-1\}\)\}\{\\varphi\_\{0\}\(o\_\{t\}\\mid a\_\{t\},\\tau\_\{t\-1\}\)\}\.\(63\)

###### Proof 8\.3

Proof of Lemma[8\.2](https://arxiv.org/html/2609.28859#S8.Thmtheorem2)\([62](https://arxiv.org/html/2609.28859#S8.E62)\) follows from the chain rule for joint probability mass functions:

Phπ​\(τ\)\\displaystyle P\_\{h\}^\{\\pi\}\(\\tau\)=\(∏t=1tstopπ−1ℙhπ​\(Ot=ot\|At=at,𝒯t−1=τt−1\)\)​\(∏t=1tstopπℙhπ​\(At=at\|𝒯t−1=τt−1\)\)\\displaystyle=\\left\(\\prod^\{t^\{\\pi\}\_\{\\text\{stop\}\}\-1\}\_\{t=1\}\\mathbb\{P\}^\{\\pi\}\_\{h\}\(O\_\{t\}=o\_\{t\}\|A\_\{t\}=a\_\{t\},\\mathcal\{T\}\_\{t\-1\}=\\tau\_\{t\-1\}\)\\right\)\\left\(\\prod^\{t^\{\\pi\}\_\{\\text\{stop\}\}\}\_\{t=1\}\\mathbb\{P\}^\{\\pi\}\_\{h\}\(A\_\{t\}=a\_\{t\}\|\\mathcal\{T\}\_\{t\-1\}=\\tau\_\{t\-1\}\)\\right\)⋅ℙhπ​\(δπ=δ~\|𝒯tstopπ−1=τtstopπ−1,Atstopπ=STOP\)\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\\cdot\\mathbb\{P\}^\{\\pi\}\_\{h\}\(\\delta^\{\\pi\}=\\tilde\{\\delta\}\|\\mathcal\{T\}\_\{t^\{\\pi\}\_\{\\text\{stop\}\}\-1\}=\\tau\_\{t^\{\\pi\}\_\{\\text\{stop\}\}\-1\},A\_\{t^\{\\pi\}\_\{\\text\{stop\}\}\}=\\text\{STOP\}\)=χπ​\(τ\)​∏t=1tstopπ−1φh​\(ot\|at,τt−1\),∀τ∈𝔗N,\\displaystyle=\\chi^\{\\pi\}\(\\tau\)\\prod^\{t^\{\\pi\}\_\{\\text\{stop\}\}\-1\}\_\{t=1\}\\varphi\_\{h\}\(o\_\{t\}\|a\_\{t\},\\tau\_\{t\-1\}\),\\forall\\tau\\in\\mathfrak\{T\}\_\{N\},andPhπ​\(τ\)=0P\_\{h\}^\{\\pi\}\(\\tau\)=0ifτ∉𝔗N\\tau\\notin\\mathfrak\{T\}\_\{N\}\.

Now we verify that for anyτ\\tau,P0π​\(τ\)\>0P^\{\\pi\}\_\{0\}\(\\tau\)\>0if and only ifP1π​\(τ\)\>0P^\{\\pi\}\_\{1\}\(\\tau\)\>0\. In particular,χπ​\(τ\)\\chi^\{\\pi\}\(\\tau\)is the same acrossH0H\_\{0\}andH1H\_\{1\}, and by definition ofφh​\(ot\|at,τt−1\)\\varphi\_\{h\}\(o\_\{t\}\|a\_\{t\},\\tau\_\{t\-1\}\),φ0​\(ot\|at,τt−1\)\>0\\varphi\_\{0\}\(o\_\{t\}\|a\_\{t\},\\tau\_\{t\-1\}\)\>0if and only ifφ1​\(ot\|at,τt−1\)\>0\\varphi\_\{1\}\(o\_\{t\}\|a\_\{t\},\\tau\_\{t\-1\}\)\>0\. Then \([63](https://arxiv.org/html/2609.28859#S8.E63)\) is well defined and follows straightforwardly from \([62](https://arxiv.org/html/2609.28859#S8.E62)\)\.

### 8\.3Proof of Lemma[4\.5](https://arxiv.org/html/2609.28859#S4.Thmtheorem5)

To prove Lemma[4\.5](https://arxiv.org/html/2609.28859#S4.Thmtheorem5), we need auxiliary Lemmas[8\.4](https://arxiv.org/html/2609.28859#S8.Thmtheorem4)\-[8\.7](https://arxiv.org/html/2609.28859#S8.Thmtheorem7)\.

###### Lemma 8\.4\(Theorem 2\.5\.3 of[Cover and Thomas \(2006\)](https://arxiv.org/html/2609.28859#bib.bib12)\)

Let\(Y,Z\)\(Y,Z\)take values in a finite or countably infinite set\. Under probability lawsPPandQQ, letPY,ZP\_\{Y,Z\}andQY,ZQ\_\{Y,Z\}denote the joint laws of\(Y,Z\)\(Y,Z\), letPYP\_\{Y\}andQYQ\_\{Y\}denote the marginal laws ofYY, and letPZ\|Y=yP\_\{Z\\mid Y=y\}andQZ\|Y=yQ\_\{Z\\mid Y=y\}denote the conditional laws ofZZgivenY=yY=y, respectively\. Then

D\(PY,Z∥QY,Z\)=D\(PY∥QY\)\+∑y∈𝒴PY\(y\)D\(PZ\|Y=y∥QZ\|Y=y\),D\(P\_\{Y,Z\}\\\|Q\_\{Y,Z\}\)=D\(P\_\{Y\}\\\|Q\_\{Y\}\)\+\\sum\_\{y\\in\\mathcal\{Y\}\}P\_\{Y\}\(y\)D\(P\_\{Z\\mid Y=y\}\\\|Q\_\{Z\\mid Y=y\}\),\(64\)with the usual extended\-value convention\. In particular, forgettingZZcannot increase KL divergence:

D\(PY∥QY\)≤D\(PY,Z∥QY,Z\)\.D\(P\_\{Y\}\\\|Q\_\{Y\}\)\\leq D\(P\_\{Y,Z\}\\\|Q\_\{Y,Z\}\)\.\(65\)

###### Lemma 8\.5\(Binary coarsening\)

LetPPandQQbe probability mass functions on the same finite or countably infinite setSS, and letE⊆SE\\subseteq S\. Then

D\(P∥Q\)≥kl\(P\(E\)∥Q\(E\)\)\.D\(P\\\|Q\)\\geq\\operatorname\{kl\}\(P\(E\)\\\|Q\(E\)\)\.\(66\)

###### Proof 8\.6

Proof of Lemma[8\.5](https://arxiv.org/html/2609.28859#S8.Thmtheorem5)LetZZdenote the originalSS\-valued random outcome and defineY:=𝟏\{Z∈E\}Y:=\\mathbf\{1\}\\\{Z\\in E\\\}\. SinceYYis a deterministic function ofZZ, the joint variable\(Y,Z\)\(Y,Z\)contains exactly the same information asZZ, and thereforeD\(PY,Z∥QY,Z\)=D\(P∥Q\)D\(P\_\{Y,Z\}\\\|Q\_\{Y,Z\}\)=D\(P\\\|Q\)\. The marginal law ofYYunderPPis Bernoulli with success probabilityP⁡\(E\)P\(E\), and the marginal law ofYYunderQQis Bernoulli with success probabilityQ⁡\(E\)Q\(E\)\. Applying Lemma[8\.4](https://arxiv.org/html/2609.28859#S8.Thmtheorem4)and then forgettingZZgives

D\(P∥Q\)=D\(PY,Z∥QY,Z\)≥D\(PY∥QY\)=kl\(P\(E\)∥Q\(E\)\),D\(P\\\|Q\)=D\(P\_\{Y,Z\}\\\|Q\_\{Y,Z\}\)\\geq D\(P\_\{Y\}\\\|Q\_\{Y\}\)=\\operatorname\{kl\}\(P\(E\)\\\|Q\(E\)\),which proves \([66](https://arxiv.org/html/2609.28859#S8.E66)\)\.

###### Lemma 8\.7\(Monotonicity of Bernoulli KL in the testing region\)

If0<b<a<10<b<a<1, thenkl\(a∥b\)\\operatorname\{kl\}\(a\\\|b\)is increasing inaafor fixedbband decreasing inbbfor fixedaa\.

###### Proof 8\.8

Proof of Lemma[8\.7](https://arxiv.org/html/2609.28859#S8.Thmtheorem7)Expanding the logarithms in the definition ofkl\(a∥b\)\\operatorname\{kl\}\(a\\\|b\),

kl\(a∥b\)=aloga−alogb\+\(1−a\)log\(1−a\)−\(1−a\)log\(1−b\)\.\\operatorname\{kl\}\(a\\\|b\)=a\\log a\-a\\log b\+\(1\-a\)\\log\(1\-a\)\-\(1\-a\)\\log\(1\-b\)\.
*Derivative inaa\.*Differentiating term by term with respect toaa, withbbheld fixed:

∂∂a​\[a​log⁡a\]\\displaystyle\\tfrac\{\\partial\}\{\\partial a\}\[a\\log a\]=log⁡a\+1\\displaystyle=\\log a\+1\(product rule\),\\displaystyle\\text\{\(product rule\)\},∂∂a​\[−a​log⁡b\]\\displaystyle\\tfrac\{\\partial\}\{\\partial a\}\[\-a\\log b\]=−log⁡b,\\displaystyle=\-\\log b,∂∂a​\[\(1−a\)​log⁡\(1−a\)\]\\displaystyle\\tfrac\{\\partial\}\{\\partial a\}\[\(1\-a\)\\log\(1\-a\)\]=−log⁡\(1−a\)−1\\displaystyle=\-\\log\(1\-a\)\-1\(product and chain rules\),\\displaystyle\\text\{\(product and chain rules\)\},∂∂a​\[−\(1−a\)​log⁡\(1−b\)\]\\displaystyle\\tfrac\{\\partial\}\{\\partial a\}\[\-\(1\-a\)\\log\(1\-b\)\]=log⁡\(1−b\)\.\\displaystyle=\\log\(1\-b\)\.Summing the four lines, the constants\+1\+1and−1\-1cancel, leaving

∂∂akl\(a∥b\)=loga−logb−log\(1−a\)\+log\(1−b\)=loga⁡\(1−b\)b⁡\(1−a\)\.\\tfrac\{\\partial\}\{\\partial a\}\\operatorname\{kl\}\(a\\\|b\)=\\log a\-\\log b\-\\log\(1\-a\)\+\\log\(1\-b\)=\\log\\frac\{a\(1\-b\)\}\{b\(1\-a\)\}\.Sincea\>ba\>b, we havea⁡\(1−b\)\>b⁡\(1−a\)a\(1\-b\)\>b\(1\-a\)\(equivalent toa\>ba\>bafter expanding\), so the argument of the logarithm exceeds one and the derivative is strictly positive\. Hencekl\(a∥b\)\\operatorname\{kl\}\(a\\\|b\)is strictly increasing inaaon the regiona\>ba\>b\.

*Derivative inbb\.*Differentiating term by term with respect tobb, withaaheld fixed:

∂∂b​\[a​log⁡a\]\\displaystyle\\tfrac\{\\partial\}\{\\partial b\}\[a\\log a\]=0,\\displaystyle=0,∂∂b​\[−a​log⁡b\]\\displaystyle\\tfrac\{\\partial\}\{\\partial b\}\[\-a\\log b\]=−ab,\\displaystyle=\-\\frac\{a\}\{b\},∂∂b​\[\(1−a\)​log⁡\(1−a\)\]\\displaystyle\\tfrac\{\\partial\}\{\\partial b\}\[\(1\-a\)\\log\(1\-a\)\]=0,\\displaystyle=0,∂∂b​\[−\(1−a\)​log⁡\(1−b\)\]\\displaystyle\\tfrac\{\\partial\}\{\\partial b\}\[\-\(1\-a\)\\log\(1\-b\)\]=1−a1−b\\displaystyle=\\frac\{1\-a\}\{1\-b\}\(chain rule\)\.\\displaystyle\\text\{\(chain rule\)\}\.Summing,

∂∂bkl\(a∥b\)=−ab\+1−a1−b=−a⁡\(1−b\)\+b⁡\(1−a\)b⁡\(1−b\)=b−ab⁡\(1−b\)\.\\tfrac\{\\partial\}\{\\partial b\}\\operatorname\{kl\}\(a\\\|b\)=\-\\frac\{a\}\{b\}\+\\frac\{1\-a\}\{1\-b\}=\\frac\{\-a\(1\-b\)\+b\(1\-a\)\}\{b\(1\-b\)\}=\\frac\{b\-a\}\{b\(1\-b\)\}\.Sincea\>ba\>bandb⁡\(1−b\)\>0b\(1\-b\)\>0, the numerator is negative and the denominator is positive, so the derivative is strictly negative\. Hencekl\(a∥b\)\\operatorname\{kl\}\(a\\\|b\)is strictly decreasing inbbon the regiona\>ba\>b\.

###### Proof 8\.9

Proof of Lemma[4\.5](https://arxiv.org/html/2609.28859#S4.Thmtheorem5)LetE=\{τ:δ~=1\}E=\\\{\\tau:\\tilde\{\\delta\}=1\\\}be the set of rejection transcripts\. By Lemma[8\.5](https://arxiv.org/html/2609.28859#S8.Thmtheorem5),

D\(P1π∥P0π\)≥kl\(P1π\(E\)∥P0π\(E\)\)\.D\(P\_\{1\}^\{\\pi\}\\\|P\_\{0\}^\{\\pi\}\)\\geq\\operatorname\{kl\}\(P\_\{1\}^\{\\pi\}\(E\)\\\|P\_\{0\}^\{\\pi\}\(E\)\)\.The type\-II constraint givesP1π​\(E\)=ℙ1π​\(δπ=1\)≥1−βP\_\{1\}^\{\\pi\}\(E\)=\\mathbb\{P\}^\{\\pi\}\_\{1\}\(\\delta^\{\\pi\}=1\)\\geq 1\-\\beta, and the type\-I constraint givesP0π​\(E\)=ℙ0π​\(δπ=1\)≤αP\_\{0\}^\{\\pi\}\(E\)=\\mathbb\{P\}^\{\\pi\}\_\{0\}\(\\delta^\{\\pi\}=1\)\\leq\\alpha\. Sinceα\+β<1\\alpha\+\\beta<1, we have1−β\>α1\-\\beta\>\\alpha\. By Lemma[8\.2](https://arxiv.org/html/2609.28859#S8.Thmtheorem2),P0πP\_\{0\}^\{\\pi\}andP1πP\_\{1\}^\{\\pi\}have common support; together with the error constraints, this implies0<P0π​\(E\)<P1π​\(E\)<10<P\_\{0\}^\{\\pi\}\(E\)<P\_\{1\}^\{\\pi\}\(E\)<1\. Lemma[8\.7](https://arxiv.org/html/2609.28859#S8.Thmtheorem7)gives

kl\(P1π\(E\)∥P0π\(E\)\)≥kl\(1−β∥α\)=A\.\\operatorname\{kl\}\(P\_\{1\}^\{\\pi\}\(E\)\\\|P\_\{0\}^\{\\pi\}\(E\)\)\\geq\\operatorname\{kl\}\(1\-\\beta\\\|\\alpha\)=A\.This proves the forward KL requirement\.

For the reverse direction, letF=\{τ:δ~=0\}F=\\\{\\tau:\\tilde\{\\delta\}=0\\\}be the set of acceptance transcripts\. Lemma[8\.5](https://arxiv.org/html/2609.28859#S8.Thmtheorem5)gives

D\(P0π∥P1π\)≥kl\(P0π\(F\)∥P1π\(F\)\)\.D\(P\_\{0\}^\{\\pi\}\\\|P\_\{1\}^\{\\pi\}\)\\geq\\operatorname\{kl\}\(P\_\{0\}^\{\\pi\}\(F\)\\\|P\_\{1\}^\{\\pi\}\(F\)\)\.The type\-I constraint givesP0π​\(F\)≥1−αP\_\{0\}^\{\\pi\}\(F\)\\geq 1\-\\alpha, and the type\-II constraint givesP1π​\(F\)≤βP\_\{1\}^\{\\pi\}\(F\)\\leq\\beta\. Since1−α\>β1\-\\alpha\>\\beta, Lemma[8\.7](https://arxiv.org/html/2609.28859#S8.Thmtheorem7)gives

kl\(P0π\(F\)∥P1π\(F\)\)≥kl\(1−α∥β\)=B\.\\operatorname\{kl\}\(P\_\{0\}^\{\\pi\}\(F\)\\\|P\_\{1\}^\{\\pi\}\(F\)\)\\geq\\operatorname\{kl\}\(1\-\\alpha\\\|\\beta\)=B\.

### 8\.4Proof of Theorem[4\.6](https://arxiv.org/html/2609.28859#S4.Thmtheorem6)

###### Proof 8\.10

Proof of Theorem[4\.6](https://arxiv.org/html/2609.28859#S4.Thmtheorem6)

We prove \([13](https://arxiv.org/html/2609.28859#S4.E13)\)\. The proof of \([14](https://arxiv.org/html/2609.28859#S4.E14)\) is obtained by interchanging the roles ofH0H\_\{0\}andH1H\_\{1\}\.

According to Lemma[8\.2](https://arxiv.org/html/2609.28859#S8.Thmtheorem2),P0πP^\{\\pi\}\_\{0\}andP1πP^\{\\pi\}\_\{1\}have common support, and

log⁡P1π​\(τ\)P0π​\(τ\)=∑t=1tstopπ−1log⁡φ1​\(ot∣at,τt−1\)φ0​\(ot∣at,τt−1\)\.\\log\\frac\{P\_\{1\}^\{\\pi\}\(\\tau\)\}\{P\_\{0\}^\{\\pi\}\(\\tau\)\}=\\sum\_\{t=1\}^\{t^\{\\pi\}\_\{\\text\{stop\}\}\-1\}\\log\\frac\{\\varphi\_\{1\}\(o\_\{t\}\\mid a\_\{t\},\\tau\_\{t\-1\}\)\}\{\\varphi\_\{0\}\(o\_\{t\}\\mid a\_\{t\},\\tau\_\{t\-1\}\)\}\.
Define the one\-step log\-likelihood\-ratio increment

Lt:=\{log⁡φ1​\(Ot∣At,𝒯t−1\)φ0​\(Ot∣At,𝒯t−1\),t<Tstopπ,0,t≥Tstopπ,L\_\{t\}:=\\begin\{cases\}\\log\\dfrac\{\\varphi\_\{1\}\(O\_\{t\}\\mid A\_\{t\},\\mathcal\{T\}\_\{t\-1\}\)\}\{\\varphi\_\{0\}\(O\_\{t\}\\mid A\_\{t\},\\mathcal\{T\}\_\{t\-1\}\)\},&t<T^\{\\pi\}\_\{\\text\{stop\}\},\\\\\[8\.61108pt\] 0,&t\\geq T^\{\\pi\}\_\{\\text\{stop\}\},\\end\{cases\}Then, sinceTstopπ≤2​Nπ\+1T^\{\\pi\}\_\{\\text\{stop\}\}\\leq 2N^\{\\pi\}\+1almost surely, we have that

D\(P1π∥P0π\)=𝔼1π\[logP1π​\(𝒯\)P0π​\(𝒯\)\]=𝔼1π\[∑t=1Tstopπ−1logφ1​\(Ot∣At,𝒯t−1\)φ0​\(Ot∣At,𝒯t−1\)\]=∑t=12​Nπ𝔼1π\[Lt\]\.D\(P\_\{1\}^\{\\pi\}\\\|P\_\{0\}^\{\\pi\}\)=\\mathbb\{E\}^\{\\pi\}\_\{1\}\\\!\\left\[\\log\\frac\{P\_\{1\}^\{\\pi\}\(\\mathcal\{T\}\)\}\{P\_\{0\}^\{\\pi\}\(\\mathcal\{T\}\)\}\\right\]=\\mathbb\{E\}^\{\\pi\}\_\{1\}\\left\[\\sum\_\{t=1\}^\{T^\{\\pi\}\_\{\\text\{stop\}\}\-1\}\\log\\frac\{\\varphi\_\{1\}\(O\_\{t\}\\mid A\_\{t\},\\mathcal\{T\}\_\{t\-1\}\)\}\{\\varphi\_\{0\}\(O\_\{t\}\\mid A\_\{t\},\\mathcal\{T\}\_\{t\-1\}\)\}\\right\]=\\sum\_\{t=1\}^\{2N^\{\\pi\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\[L\_\{t\}\]\.\(67\)
It remains to evaluate𝔼1π​\[Lt\]\\mathbb\{E\}^\{\\pi\}\_\{1\}\[L\_\{t\}\]\. Let

ℱt=σ⁡\(A1,O1,⋯,At−1,Ot−1,At\)\\mathcal\{F\}\_\{t\}=\\sigma\(A\_\{1\},O\_\{1\},\\cdots,A\_\{t\-1\},O\_\{t\-1\},A\_\{t\}\)be the filtration generated by the historyℋt−1=σ⁡\(A1,O1,⋯,At−1,Ot−1\)\\mathcal\{H\}\_\{t\-1\}=\\sigma\(A\_\{1\},O\_\{1\},\\cdots,A\_\{t\-1\},O\_\{t\-1\}\)and actionAtA\_\{t\}\. Then\{t<Tstopπ\}∈ℱt\\\{t<T^\{\\pi\}\_\{\\text\{stop\}\}\\\}\\in\\mathcal\{F\}\_\{t\}\. By law of iterated expectation,

D\(P1π∥P0π\)=∑t=12​Nπ𝔼1π\[Lt\]=∑t=12​Nπ𝔼1π\[𝔼1π\[Lt\|ℱt\]\]=∑t=12​Nπ𝔼1π\[𝟏\{t<Tstopπ\}𝔼1π\[Lt\|ℱt\]\],D\(P\_\{1\}^\{\\pi\}\\\|P\_\{0\}^\{\\pi\}\)=\\sum\_\{t=1\}^\{2N^\{\\pi\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\[L\_\{t\}\]=\\sum\_\{t=1\}^\{2N^\{\\pi\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\[\\mathbb\{E\}^\{\\pi\}\_\{1\}\[L\_\{t\}\|\\mathcal\{F\}\_\{t\}\]\]=\\sum\_\{t=1\}^\{2N^\{\\pi\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\[\\mathbf\{1\}\\\{t<T^\{\\pi\}\_\{\\text\{stop\}\}\\\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\[L\_\{t\}\|\\mathcal\{F\}\_\{t\}\]\],where the third equality follows from definition ofLtL\_\{t\}and that\{t<Tstopπ\}∈ℱt\\\{t<T^\{\\pi\}\_\{\\text\{stop\}\}\\\}\\in\\mathcal\{F\}\_\{t\}\.

We now compute𝔼1π​\[Lt∣ℱt\]\\mathbb\{E\}^\{\\pi\}\_\{1\}\[L\_\{t\}\\mid\\mathcal\{F\}\_\{t\}\]fort<Tstopπt<T^\{\\pi\}\_\{\\text\{stop\}\}\. By definition of an admissible policy, for allt<Tstopπt<T^\{\\pi\}\_\{\\text\{stop\}\}we have that∑s=1t𝟏\{As=AI\(i\)\}≤1\\sum^\{t\}\_\{s=1\}\\mathbf\{1\}\\\{A\_\{s\}=\\mathrm\{AI\}\(i\)\\\}\\leq 1and∑s=1t𝟏\{As=H\(i\)\}≤1\\sum^\{t\}\_\{s=1\}\\mathbf\{1\}\\\{A\_\{s\}=\\mathrm\{H\}\(i\)\\\}\\leq 1for allii\. Thus we have four cases of the filtrationℱt\\mathcal\{F\}\_\{t\}:

*Case 1: AI query not preceded by a human query\.*On the event\{t<Tstopπ\}∩\{At=AI\(i\),∑s=1t−1𝟏\{As=H\(i\)\}=0\}\\\{t<T^\{\\pi\}\_\{\\text\{stop\}\}\\\}\\cap\\\{A\_\{t\}=\\mathrm\{AI\}\(i\),\\,\\sum^\{t\-1\}\_\{s=1\}\\mathbf\{1\}\\\{A\_\{s\}=\\mathrm\{H\}\(i\)\\\}=0\\\},

𝔼1π\[Lt∣ℱt\]=∑r∈ℛg1\(r\)logg1​\(r\)g0​\(r\)=D\(G1∥G0\)=IR\(1\),\\mathbb\{E\}^\{\\pi\}\_\{1\}\[L\_\{t\}\\mid\\mathcal\{F\}\_\{t\}\]=\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{1\}\(r\)\\log\\frac\{g\_\{1\}\(r\)\}\{g\_\{0\}\(r\)\}=D\(G\_\{1\}\\\|G\_\{0\}\)=I\_\{R\}^\{\(1\)\},by the definition of KL divergence and the definition \([8](https://arxiv.org/html/2609.28859#S4.E8)\) ofIR\(1\)I\_\{R\}^\{\(1\)\}\.

*Case 2: direct human query\.*On the event\{t<Tstopπ\}∩\{At=H\(i\),∑s=1t−1𝟏\{As=AI\(i\)\}=0\}\\\{t<T^\{\\pi\}\_\{\\text\{stop\}\}\\\}\\cap\\\{A\_\{t\}=\\mathrm\{H\}\(i\),\\,\\sum^\{t\-1\}\_\{s=1\}\\mathbf\{1\}\\\{A\_\{s\}=\\mathrm\{AI\}\(i\)\\\}=0\\\},

𝔼1π\[Lt∣ℱt\]=p1logp1p0\+\(1−p1\)log1−p11−p0=kl\(p1∥p0\)=JX\(1\)\.\\mathbb\{E\}^\{\\pi\}\_\{1\}\[L\_\{t\}\\mid\\mathcal\{F\}\_\{t\}\]=p\_\{1\}\\log\\frac\{p\_\{1\}\}\{p\_\{0\}\}\+\(1\-p\_\{1\}\)\\log\\frac\{1\-p\_\{1\}\}\{1\-p\_\{0\}\}=\\operatorname\{kl\}\(p\_\{1\}\\\|p\_\{0\}\)=J\_\{X\}^\{\(1\)\}\.
*Case 3: human query after an AI report\.*On the event\{t<Tstopπ\}∩\{At=H\(i\),∑s=1t−1𝟏\{As=AI\(i\)\}=1\}\\\{t<T^\{\\pi\}\_\{\\text\{stop\}\}\\\}\\cap\\\{A\_\{t\}=\\mathrm\{H\}\(i\),\\,\\sum^\{t\-1\}\_\{s=1\}\\mathbf\{1\}\\\{A\_\{s\}=\\mathrm\{AI\}\(i\)\\\}=1\\\}\. We have for eachot∈\{0,1\}o\_\{t\}\\in\\\{0,1\\\},

log⁡ρ1​\(ot∣r\)ρ0​\(ot∣r\)=ot​log⁡q1​\(r\)q0​\(r\)\+\(1−ot\)​log⁡1−q1​\(r\)1−q0​\(r\),\\log\\frac\{\\rho\_\{1\}\(o\_\{t\}\\mid r\)\}\{\\rho\_\{0\}\(o\_\{t\}\\mid r\)\}=o\_\{t\}\\log\\frac\{q\_\{1\}\(r\)\}\{q\_\{0\}\(r\)\}\+\(1\-o\_\{t\}\)\\log\\frac\{1\-q\_\{1\}\(r\)\}\{1\-q\_\{0\}\(r\)\},and𝔼1π​\[Ot∣ℱt\]=q1​\(Ri\)\\mathbb\{E\}^\{\\pi\}\_\{1\}\[O\_\{t\}\\mid\\mathcal\{F\}\_\{t\}\]=q\_\{1\}\(R\_\{i\}\)on this event, so

𝔼1π\[Lt∣ℱt\]=q1\(Ri\)logq1​\(Ri\)q0​\(Ri\)\+\(1−q1\(Ri\)\)log1−q1​\(Ri\)1−q0​\(Ri\)=kl\(q1\(Ri\)∥q0\(Ri\)\)=d\(1\)\(Ri\)\.\\mathbb\{E\}^\{\\pi\}\_\{1\}\[L\_\{t\}\\mid\\mathcal\{F\}\_\{t\}\]=q\_\{1\}\(R\_\{i\}\)\\log\\frac\{q\_\{1\}\(R\_\{i\}\)\}\{q\_\{0\}\(R\_\{i\}\)\}\+\(1\-q\_\{1\}\(R\_\{i\}\)\)\\log\\frac\{1\-q\_\{1\}\(R\_\{i\}\)\}\{1\-q\_\{0\}\(R\_\{i\}\)\}=\\operatorname\{kl\}\(q\_\{1\}\(R\_\{i\}\)\\\|q\_\{0\}\(R\_\{i\}\)\)=d^\{\(1\)\}\(R\_\{i\}\)\.
*Case 4: AI query after the item’s human label has been revealed\.*On the event\{t<Tstopπ\}∩\{At=AI\(i\),∑s=1t−1𝟏\{As=H\(i\)\}=1\}\\\{t<T^\{\\pi\}\_\{\\text\{stop\}\}\\\}\\cap\\\{A\_\{t\}=\\mathrm\{AI\}\(i\),\\,\\sum^\{t\-1\}\_\{s=1\}\\mathbf\{1\}\\\{A\_\{s\}=\\mathrm\{H\}\(i\)\\\}=1\\\}, forOt=r∈ℛO\_\{t\}=r\\in\\mathcal\{R\}andXi=xX\_\{i\}=x,φ1​\(Ot\|At,𝒯t−1\)=φ0​\(Ot\|At,𝒯t−1\)=fx​\(r\)\\varphi\_\{1\}\(O\_\{t\}\|A\_\{t\},\\mathcal\{T\}\_\{t\-1\}\)=\\varphi\_\{0\}\(O\_\{t\}\|A\_\{t\},\\mathcal\{T\}\_\{t\-1\}\)=f\_\{x\}\(r\)almost surely, and thus𝔼1π​\[Lt∣ℱt\]=0\\mathbb\{E\}^\{\\pi\}\_\{1\}\[L\_\{t\}\\mid\\mathcal\{F\}\_\{t\}\]=0\.

Combining the above four cases, we have that

D\(P1π∥P0π\)\\displaystyle D\(P\_\{1\}^\{\\pi\}\\\|P\_\{0\}^\{\\pi\}\)=∑t=12​Nπ𝔼1π\[𝟏\{t<Tstopπ\}𝔼1π\[Lt\|ℱt\]\]\\displaystyle=\\sum\_\{t=1\}^\{2N^\{\\pi\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\[\\mathbf\{1\}\\\{t<T^\{\\pi\}\_\{\\text\{stop\}\}\\\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\[L\_\{t\}\|\\mathcal\{F\}\_\{t\}\]\]=∑t=12​Nπ𝔼1π\[∑i=1Nπ𝟏\{t<Tstopπ,At=AI\(i\),∑s=1t−1𝟏\{As=H\(i\)\}=0\}𝔼1π\[Lt\|ℱt\]\]\\displaystyle=\\sum\_\{t=1\}^\{2N^\{\\pi\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\\Big\[\\sum^\{N^\{\\pi\}\}\_\{i=1\}\\mathbf\{1\}\\\{t<T^\{\\pi\}\_\{\\text\{stop\}\},A\_\{t\}=\\mathrm\{AI\}\(i\),\\sum^\{t\-1\}\_\{s=1\}\\mathbf\{1\}\\\{A\_\{s\}=\\mathrm\{H\}\(i\)\\\}=0\\\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\[L\_\{t\}\|\\mathcal\{F\}\_\{t\}\]\\Big\]\+∑t=12​Nπ𝔼1π\[∑i=1Nπ𝟏\{t<Tstopπ,At=H\(i\),∑s=1t−1𝟏\{As=AI\(i\)\}=0\}𝔼1π\[Lt\|ℱt\]\]\\displaystyle\\qquad\\qquad\+\\sum\_\{t=1\}^\{2N^\{\\pi\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\\Big\[\\sum^\{N^\{\\pi\}\}\_\{i=1\}\\mathbf\{1\}\\\{t<T^\{\\pi\}\_\{\\text\{stop\}\},A\_\{t\}=\\mathrm\{H\}\(i\),\\sum^\{t\-1\}\_\{s=1\}\\mathbf\{1\}\\\{A\_\{s\}=\\mathrm\{AI\}\(i\)\\\}=0\\\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\[L\_\{t\}\|\\mathcal\{F\}\_\{t\}\]\\Big\]\+∑t=12​Nπ𝔼1π\[∑i=1Nπ𝟏\{t<Tstopπ,At=H\(i\),∑s=1t−1𝟏\{As=AI\(i\)\}=1\}𝔼1π\[Lt\|ℱt\]\]\\displaystyle\\qquad\\qquad\+\\sum\_\{t=1\}^\{2N^\{\\pi\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\\Big\[\\sum^\{N^\{\\pi\}\}\_\{i=1\}\\mathbf\{1\}\\\{t<T^\{\\pi\}\_\{\\text\{stop\}\},A\_\{t\}=\\mathrm\{H\}\(i\),\\sum^\{t\-1\}\_\{s=1\}\\mathbf\{1\}\\\{A\_\{s\}=\\mathrm\{AI\}\(i\)\\\}=1\\\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\[L\_\{t\}\|\\mathcal\{F\}\_\{t\}\]\\Big\]\+∑t=12​Nπ𝔼1π\[∑i=1Nπ𝟏\{t<Tstopπ,At=AI\(i\),∑s=1t−1𝟏\{As=H\(i\)\}=1\}𝔼1π\[Lt\|ℱt\]\]\\displaystyle\\qquad\\qquad\+\\sum\_\{t=1\}^\{2N^\{\\pi\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\\Big\[\\sum^\{N^\{\\pi\}\}\_\{i=1\}\\mathbf\{1\}\\\{t<T^\{\\pi\}\_\{\\text\{stop\}\},A\_\{t\}=\\mathrm\{AI\}\(i\),\\sum^\{t\-1\}\_\{s=1\}\\mathbf\{1\}\\\{A\_\{s\}=\\mathrm\{H\}\(i\)\\\}=1\\\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\[L\_\{t\}\|\\mathcal\{F\}\_\{t\}\]\\Big\]=∑t=12​Nπ𝔼1π\[∑i=1Nπ𝟏\{t<Tstopπ,At=AI\(i\),∑s=1t−1𝟏\{As=H\(i\)\}=0\}IR\(1\)\]\\displaystyle=\\sum\_\{t=1\}^\{2N^\{\\pi\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\\Big\[\\sum^\{N^\{\\pi\}\}\_\{i=1\}\\mathbf\{1\}\\\{t<T^\{\\pi\}\_\{\\text\{stop\}\},A\_\{t\}=\\mathrm\{AI\}\(i\),\\sum^\{t\-1\}\_\{s=1\}\\mathbf\{1\}\\\{A\_\{s\}=\\mathrm\{H\}\(i\)\\\}=0\\\}I^\{\(1\)\}\_\{R\}\\Big\]\+∑t=12​Nπ𝔼1π\[∑i=1Nπ𝟏\{t<Tstopπ,At=H\(i\),∑s=1t−1𝟏\{As=AI\(i\)\}=0\}JX\(1\)\]\\displaystyle\\qquad\\qquad\+\\sum\_\{t=1\}^\{2N^\{\\pi\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\\Big\[\\sum^\{N^\{\\pi\}\}\_\{i=1\}\\mathbf\{1\}\\\{t<T^\{\\pi\}\_\{\\text\{stop\}\},A\_\{t\}=\\mathrm\{H\}\(i\),\\sum^\{t\-1\}\_\{s=1\}\\mathbf\{1\}\\\{A\_\{s\}=\\mathrm\{AI\}\(i\)\\\}=0\\\}J^\{\(1\)\}\_\{X\}\\Big\]\+∑t=12​Nπ𝔼1π\[∑i=1Nπ𝟏\{t<Tstopπ,At=H\(i\),∑s=1t−1𝟏\{As=AI\(i\)\}=1\}d\(1\)\(Ri\)\]\\displaystyle\\qquad\\qquad\+\\sum\_\{t=1\}^\{2N^\{\\pi\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\\Big\[\\sum^\{N^\{\\pi\}\}\_\{i=1\}\\mathbf\{1\}\\\{t<T^\{\\pi\}\_\{\\text\{stop\}\},A\_\{t\}=\\mathrm\{H\}\(i\),\\sum^\{t\-1\}\_\{s=1\}\\mathbf\{1\}\\\{A\_\{s\}=\\mathrm\{AI\}\(i\)\\\}=1\\\}d^\{\(1\)\}\(R\_\{i\}\)\\Big\]=𝔼1π​\[NAIdir,π​IR\(1\)\+NHdir,π​JX\(1\)\+∑i∈ℐAIHπd\(1\)​\(Ri\)\],\\displaystyle=\\mathbb\{E\}^\{\\pi\}\_\{1\}\\left\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}I\_\{R\}^\{\(1\)\}\+N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}J\_\{X\}^\{\(1\)\}\+\\sum\_\{i\\in\\mathcal\{I\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}^\{\\pi\}\}d^\{\(1\)\}\(R\_\{i\}\)\\right\],where the last equality follows by the definition ofNAIdir,π,NHdir,πN^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\},N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}andℐAIHπ\\mathcal\{I\}^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\. Then \([14](https://arxiv.org/html/2609.28859#S4.E14)\) follows by interchanging the roles ofH0H\_\{0\}andH1H\_\{1\}\.

### 8\.5Proof of Lemma[4\.7](https://arxiv.org/html/2609.28859#S4.Thmtheorem7)

###### Proof 8\.11

Proof of Lemma[4\.7](https://arxiv.org/html/2609.28859#S4.Thmtheorem7)We first prove the dual representation \([16](https://arxiv.org/html/2609.28859#S4.E16)\)\. We start by deriving the dual program of \([16](https://arxiv.org/html/2609.28859#S4.E16)\), and show that the dual of \([16](https://arxiv.org/html/2609.28859#S4.E16)\) is \([15](https://arxiv.org/html/2609.28859#S4.E15)\)\. Then, according to Theorem 4\.1 of[Bertsimas and Tsitsiklis \(1997\)](https://arxiv.org/html/2609.28859#bib.bib7), which states that for linear programming problem, the dual of dual is the primal, we conclude that \([16](https://arxiv.org/html/2609.28859#S4.E16)\) is the dual of \([15](https://arxiv.org/html/2609.28859#S4.E15)\)\.

Letλ≥0\\lambda\\geq 0be the dual variable for the constraint∑rgh​\(r\)​η​\(r\)≤s\\sum\_\{r\}g\_\{h\}\(r\)\\eta\(r\)\\leq s, and for eachr∈ℛr\\in\\mathcal\{R\}, letμ⁡\(r\)≥0\\mu\(r\)\\geq 0be the dual variable for the constraintη⁡\(r\)≤1\\eta\(r\)\\leq 1\. Following the recipe in Section 4\.2 of[Bertsimas and Tsitsiklis \(1997\)](https://arxiv.org/html/2609.28859#bib.bib7), the linear programming \(LP\) dual problem for \([16](https://arxiv.org/html/2609.28859#S4.E16)\) is:

minλ≥0,μλ​s\+∑r∈ℛμ⁡\(r\)s\.t\.λgh\(r\)\+μ\(r\)≥gh\(r\)d\(h\)\(r\),∀r∈ℛ,μ\(r\)≥0,∀r∈ℛ\.\\begin\{split\}\\min\_\{\\lambda\\geq 0,\\mu\}\\quad&\\lambda s\+\\sum\_\{r\\in\\mathcal\{R\}\}\\mu\(r\)\\\\ \\text\{s\.t\.\}\\quad&\\lambda g\_\{h\}\(r\)\+\\mu\(r\)\\geq g\_\{h\}\(r\)d^\{\(h\)\}\(r\),\\qquad\\forall r\\in\\mathcal\{R\},\\\\ &\\mu\(r\)\\geq 0,\\qquad\\forall r\\in\\mathcal\{R\}\.\\end\{split\}\(68\)For each fixedλ≥0\\lambda\\geq 0, the constraintμ⁡\(r\)≥gh​\(r\)​\(d\(h\)​\(r\)−λ\)\\mu\(r\)\\geq g\_\{h\}\(r\)\(d^\{\(h\)\}\(r\)\-\\lambda\)together withμ⁡\(r\)≥0\\mu\(r\)\\geq 0implies that

μ⁡\(r\)=gh​\(r\)​\[d\(h\)​\(r\)−λ\]\+\\mu\(r\)=g\_\{h\}\(r\)\\left\[d^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}minimizes the objective in \([68](https://arxiv.org/html/2609.28859#S8.E68)\)\. Substitutingμ⁡\(r\)=gh​\(r\)​\[d\(h\)​\(r\)−λ\]\+\\mu\(r\)=g\_\{h\}\(r\)\\left\[d^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}into the dual objective givesλ​s\+∑rgh​\(r\)​\[d\(h\)​\(r\)−λ\]\+\\lambda s\+\\sum\_\{r\}g\_\{h\}\(r\)\\left\[d^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}\. Additionally, since the LP problem \([16](https://arxiv.org/html/2609.28859#S4.E16)\) is feasible and bounded above, strong LP duality \(Theorem 4\.4 of[Bertsimas and Tsitsiklis \(1997\)](https://arxiv.org/html/2609.28859#bib.bib7)\) gives

Ψh​\(s\)=infλ≥0\{λ​s\+∑r∈ℛgh​\(r\)​\[d\(h\)​\(r\)−λ\]\+\},\\Psi\_\{h\}\(s\)=\\inf\_\{\\lambda\\geq 0\}\\left\\\{\\lambda s\+\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\left\[d^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}\\right\\\},which is \([15](https://arxiv.org/html/2609.28859#S4.E15)\)\.

Fixh∈\{0,1\}h\\in\\\{0,1\\\}\. The maximization in \([16](https://arxiv.org/html/2609.28859#S4.E16)\) is a finite\-dimensional linear program in the variables\{η⁡\(r\)\}r∈ℛ\\\{\\eta\(r\)\\\}\_\{r\\in\\mathcal\{R\}\}, parameterized byss\. Its feasible region is nonempty \(η≡0\\eta\\equiv 0is feasible\) and bounded \(0≤η⁡\(r\)≤10\\leq\\eta\(r\)\\leq 1\), so the maximum is attained for everys∈\[0,1\]s\\in\[0,1\]\. Monotonicity ofΨh​\(s\)\\Psi\_\{h\}\(s\)inssfollows because increasingssenlarges the feasible set\. According to Section 5\.2 of[Bertsimas and Tsitsiklis \(1997\)](https://arxiv.org/html/2609.28859#bib.bib7),Ψh​\(s\)\\Psi\_\{h\}\(s\)is concave and piecewise linear inss, hence continuous on\[0,1\]\[0,1\]\.

It remains to prove \([17](https://arxiv.org/html/2609.28859#S4.E17)\)\. Ats=1s=1,η⁡\(r\)=1\\eta\(r\)=1for everyr∈ℛr\\in\\mathcal\{R\}is feasible \(because∑r∈ℛgh​\(r\)=1\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)=1\) and optimal \(becaused\(h\)​\(r\)≥0d^\{\(h\)\}\(r\)\\geq 0for allr∈ℛr\\in\\mathcal\{R\}\) for \([16](https://arxiv.org/html/2609.28859#S4.E16)\)\. Thus

Ψh​\(1\)=∑r∈ℛgh​\(r\)​d\(h\)​\(r\)\.\\Psi\_\{h\}\(1\)=\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)d^\{\(h\)\}\(r\)\.
Forh∈\{0,1\}h\\in\\\{0,1\\\}, letPhX,RP\_\{h\}^\{X,R\}denote the joint pmf of\(Xi,Ri\)\(X\_\{i\},R\_\{i\}\)underHhH\_\{h\}, withPhX,R​\(x,r\)=phx​\(1−ph\)1−x​fx​\(r\)P\_\{h\}^\{X,R\}\(x,r\)=p\_\{h\}^\{x\}\(1\-p\_\{h\}\)^\{1\-x\}f\_\{x\}\(r\)\. Forh=1h=1, we computeD\(P1X,R∥P0X,R\)D\(P\_\{1\}^\{X,R\}\\\|P\_\{0\}^\{X,R\}\)in two different ways and equate the two expressions\.

On one hand, we have that

D\(P1X,R∥P0X,R\)\\displaystyle D\(P\_\{1\}^\{X,R\}\\\|P\_\{0\}^\{X,R\}\)=∑x∈\{0,1\}∑r∈ℛP1X,R​\(x,r\)​log⁡P1X,R​\(x,r\)P0X,R​\(x,r\)\\displaystyle=\\sum\_\{x\\in\\\{0,1\\\}\}\\sum\_\{r\\in\\mathcal\{R\}\}P\_\{1\}^\{X,R\}\(x,r\)\\log\\frac\{P\_\{1\}^\{X,R\}\(x,r\)\}\{P\_\{0\}^\{X,R\}\(x,r\)\}=∑x∈\{0,1\}∑r∈ℛp1x​\(1−p1\)1−x​fx​\(r\)​log⁡p1x​\(1−p1\)1−xp0x​\(1−p0\)1−x\\displaystyle=\\sum\_\{x\\in\\\{0,1\\\}\}\\sum\_\{r\\in\\mathcal\{R\}\}p\_\{1\}^\{x\}\(1\-p\_\{1\}\)^\{1\-x\}f\_\{x\}\(r\)\\log\\frac\{p\_\{1\}^\{x\}\(1\-p\_\{1\}\)^\{1\-x\}\}\{p\_\{0\}^\{x\}\(1\-p\_\{0\}\)^\{1\-x\}\}=∑x∈\{0,1\}p1x​\(1−p1\)1−x​log⁡p1x​\(1−p1\)1−xp0x​\(1−p0\)1−x\\displaystyle=\\sum\_\{x\\in\\\{0,1\\\}\}p\_\{1\}^\{x\}\(1\-p\_\{1\}\)^\{1\-x\}\\log\\frac\{p\_\{1\}^\{x\}\(1\-p\_\{1\}\)^\{1\-x\}\}\{p\_\{0\}^\{x\}\(1\-p\_\{0\}\)^\{1\-x\}\}=kl\(p1\|\|p0\)\\displaystyle=\\operatorname\{kl\}\(p\_\{1\}\|\|p\_\{0\}\)=JX\(1\)\.\\displaystyle=J^\{\(1\)\}\_\{X\}\.In the above equations, the second equality follows from the definition ofPhX,R​\(x,r\)P\_\{h\}^\{X,R\}\(x,r\), and the third follows since∑rP1X,R​\(x,r\)=p1x​\(1−p1\)1−x\\sum\_\{r\}P\_\{1\}^\{X,R\}\(x,r\)=p\_\{1\}^\{x\}\(1\-p\_\{1\}\)^\{1\-x\}\. Moreover, Lemma[8\.4](https://arxiv.org/html/2609.28859#S8.Thmtheorem4)gives

D\(P1X,R∥P0X,R\)=D\(G1∥G0\)\+∑r∈ℛg1\(r\)D\(ρ1\(⋅∣r\)∥ρ0\(⋅∣r\)\)=IR\(1\)\+∑r∈ℛg1\(r\)d\(1\)\(r\),D\(P\_\{1\}^\{X,R\}\\\|P\_\{0\}^\{X,R\}\)=D\(G\_\{1\}\\\|G\_\{0\}\)\+\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{1\}\(r\)D\(\\rho\_\{1\}\(\\cdot\\mid r\)\\\|\\rho\_\{0\}\(\\cdot\\mid r\)\)=I\_\{R\}^\{\(1\)\}\+\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{1\}\(r\)d^\{\(1\)\}\(r\),where the second equality uses \([8](https://arxiv.org/html/2609.28859#S4.E8)\) and \([11](https://arxiv.org/html/2609.28859#S4.E11)\)\.

Equating the two expressions forD\(P1X,R∥P0X,R\)D\(P\_\{1\}^\{X,R\}\\\|P\_\{0\}^\{X,R\}\)gives

JX\(1\)=IR\(1\)\+∑r∈ℛg1​\(r\)​d\(1\)​\(r\)=IR\(1\)\+Ψ1​\(1\)\.J\_\{X\}^\{\(1\)\}=I\_\{R\}^\{\(1\)\}\+\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{1\}\(r\)d^\{\(1\)\}\(r\)=I\_\{R\}^\{\(1\)\}\+\\Psi\_\{1\}\(1\)\.The proof forh=0h=0is identical\.

### 8\.6Proof of Proposition[4\.8](https://arxiv.org/html/2609.28859#S4.Thmtheorem8)

###### Proof 8\.12

Proof of Proposition[4\.8](https://arxiv.org/html/2609.28859#S4.Thmtheorem8)Fixh∈\{0,1\}h\\in\\\{0,1\\\}ands∈\[0,1\]s\\in\[0,1\], and abbreviate

gj:=gh​\(rj\(h\)\),dj:=d\(h\)​\(rj\(h\)\),Wj:=Wj\(h\)\.g\_\{j\}:=g\_\{h\}\(r^\{\(h\)\}\_\{j\}\),\\qquad d\_\{j\}:=d^\{\(h\)\}\(r^\{\(h\)\}\_\{j\}\),\\qquad W\_\{j\}:=W^\{\(h\)\}\_\{j\}\.By Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1),gj\>0g\_\{j\}\>0for everyjj, and by construction,

d1≥d2≥⋯≥d\|ℛ\|≥0,Wj=∑ℓ=1jgℓ,W\|ℛ\|=1\.d\_\{1\}\\geq d\_\{2\}\\geq\\cdots\\geq d\_\{\|\\mathcal\{R\}\|\}\\geq 0,\\qquad W\_\{j\}=\\sum\_\{\\ell=1\}^\{j\}g\_\{\\ell\},\\qquad W\_\{\|\\mathcal\{R\}\|\}=1\.
We first verify feasibility\. Ifs=0s=0, the rule in \([18](https://arxiv.org/html/2609.28859#S4.E18)\) sets every component equal to zero\. Ifs∈\(0,1\]s\\in\(0,1\], letq:=min⁡\{j∈\[\|ℛ\|\]:Wj≥s\}q:=\\min\\\{j\\in\[\|\\mathcal\{R\}\|\]:W\_\{j\}\\geq s\\\}\. Then the same rule sets

ηh∗​\(rj\(h\),s\)=\{1,j<q,\(s−Wq−1\)/gq,j=q,0,j\>q\.\\eta^\{\*\}\_\{h\}\(r^\{\(h\)\}\_\{j\};s\)=\\begin\{cases\}1,&j<q,\\\\\[4\.30554pt\] \(s\-W\_\{q\-1\}\)/g\_\{q\},&j=q,\\\\\[4\.30554pt\] 0,&j\>q\.\\end\{cases\}This representation also coverss=Wqs=W\_\{q\}, in which case the middle component equals one\. SinceWq−1<s≤Wq=Wq−1\+gqW\_\{q\-1\}<s\\leq W\_\{q\}=W\_\{q\-1\}\+g\_\{q\}, every component belongs to\[0,1\]\[0,1\], and

∑j=1\|ℛ\|gj​ηh∗​\(rj\(h\),s\)=Wq−1\+gq​s−Wq−1gq=s\.\\sum\_\{j=1\}^\{\|\\mathcal\{R\}\|\}g\_\{j\}\\eta^\{\*\}\_\{h\}\(r^\{\(h\)\}\_\{j\};s\)=W\_\{q\-1\}\+g\_\{q\}\\frac\{s\-W\_\{q\-1\}\}\{g\_\{q\}\}=s\.Thusηh∗:=\{ηh∗​\(rj\(h\),s\)\}j∈\[\|ℛ\|\]\\eta^\{\*\}\_\{h\}:=\\\{\\eta^\{\*\}\_\{h\}\(r^\{\(h\)\}\_\{j\};s\)\\\}\_\{j\\in\[\|\\mathcal\{R\}\|\]\}is feasible for \([16](https://arxiv.org/html/2609.28859#S4.E16)\)\.

It remains to prove optimality\. The cases=0s=0is immediate because strict positivity of thegjg\_\{j\}’s forces every feasible component to be zero\. Supposes∈\(0,1\]s\\in\(0,1\], letqqbe as above, and consider any feasible ruleη⁡\(⋅,s\):ℛ→\[0,1\]\\eta\(\\cdot;s\):\\mathcal\{R\}\\to\[0,1\]\. Using∑jgj​η​\(rj\(h\),s\)≤s=∑jgj​ηh∗​\(rj\(h\),s\)\\sum\_\{j\}g\_\{j\}\\eta\(r^\{\(h\)\}\_\{j\};s\)\\leq s=\\sum\_\{j\}g\_\{j\}\\eta^\{\*\}\_\{h\}\(r^\{\(h\)\}\_\{j\};s\), we obtain

∑j=1\|ℛ\|gj​dj​\(η⁡\(rj\(h\),s\)−ηh∗​\(rj\(h\),s\)\)\\displaystyle\\sum\_\{j=1\}^\{\|\\mathcal\{R\}\|\}g\_\{j\}d\_\{j\}\\bigl\(\\eta\(r^\{\(h\)\}\_\{j\};s\)\-\\eta^\{\*\}\_\{h\}\(r^\{\(h\)\}\_\{j\};s\)\\bigr\)=∑j=1\|ℛ\|gj​\(dj−dq\)​\(η⁡\(rj\(h\),s\)−ηh∗​\(rj\(h\),s\)\)\+dq​∑j=1\|ℛ\|gj​\(η⁡\(rj\(h\),s\)−ηh∗​\(rj\(h\),s\)\)≤0\.\\displaystyle\\quad=\\sum\_\{j=1\}^\{\|\\mathcal\{R\}\|\}g\_\{j\}\(d\_\{j\}\-d\_\{q\}\)\\bigl\(\\eta\(r^\{\(h\)\}\_\{j\};s\)\-\\eta^\{\*\}\_\{h\}\(r^\{\(h\)\}\_\{j\};s\)\\bigr\)\+d\_\{q\}\\sum\_\{j=1\}^\{\|\\mathcal\{R\}\|\}g\_\{j\}\\bigl\(\\eta\(r^\{\(h\)\}\_\{j\};s\)\-\\eta^\{\*\}\_\{h\}\(r^\{\(h\)\}\_\{j\};s\)\\bigr\)\\leq 0\.Indeed, the last sum is nonpositive anddq≥0d\_\{q\}\\geq 0\. Forj<qj<q,dj−dq≥0d\_\{j\}\-d\_\{q\}\\geq 0whileη⁡\(rj\(h\),s\)−ηh∗​\(rj\(h\),s\)=η⁡\(rj\(h\),s\)−1≤0\\eta\(r^\{\(h\)\}\_\{j\};s\)\-\\eta^\{\*\}\_\{h\}\(r^\{\(h\)\}\_\{j\};s\)=\\eta\(r^\{\(h\)\}\_\{j\};s\)\-1\\leq 0\. Forj\>qj\>q,dj−dq≤0d\_\{j\}\-d\_\{q\}\\leq 0whileη⁡\(rj\(h\),s\)−ηh∗​\(rj\(h\),s\)=η⁡\(rj\(h\),s\)≥0\\eta\(r^\{\(h\)\}\_\{j\};s\)\-\\eta^\{\*\}\_\{h\}\(r^\{\(h\)\}\_\{j\};s\)=\\eta\(r^\{\(h\)\}\_\{j\};s\)\\geq 0\. Thej=qj=qterm vanishes becausedj−dq=0d\_\{j\}\-d\_\{q\}=0\. Hence every feasibleη\\etahas objective value no larger than that ofηh∗\\eta^\{\*\}\_\{h\}, proving thatηh∗\\eta^\{\*\}\_\{h\}is optimal for \([16](https://arxiv.org/html/2609.28859#S4.E16)\)\.

### 8\.7Proof of Lemma[4\.9](https://arxiv.org/html/2609.28859#S4.Thmtheorem9)

###### Proof 8\.13

Proof of Lemma[4\.9](https://arxiv.org/html/2609.28859#S4.Thmtheorem9)By definition,ℐAIHπ⊆ℐAIπ\\mathcal\{I\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}^\{\\pi\}\\subseteq\\mathcal\{I\}\_\{\\mathrm\{AI\}\}^\{\\pi\}, and in particularNAIHπ≤NAIdir,πN^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\\leq N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}on every sample path\. If𝔼hπ​\[NAIdir,π\]=0\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]=0, thenNAIdir,π=0N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}=0almost surely, sinceNAIdir,πN^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}is a nonnegative integer with zero mean\. ByNAIHπ≤NAIdir,πN^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\\leq N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}, alsoNAIHπ=0N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}=0almost surely, so both sides of \([19](https://arxiv.org/html/2609.28859#S4.E19)\) are zero\. Assume𝔼hπ​\[NAIdir,π\]\>0\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\>0for the remainder\.

For each itemi∈\[Nπ\]i\\in\[N^\{\\pi\}\], let

Bi:=𝟏\{itemiis AI\-queried before any human query on it\},B\_\{i\}:=\\mathbf\{1\}\\\{\\text\{item \}i\\text\{ is AI\-queried before any human query on it\}\\\},equivalently, that itemii’s first paid action is an AI query, and let

Ei:=𝟏\{itemiis human\-queried after its AI report is observed\}\.E\_\{i\}:=\\mathbf\{1\}\\\{\\text\{item \}i\\text\{ is human\-queried after its AI report is observed\}\\\}\.ThenNAIdir,π=∑i=1NπBiN^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}=\\sum\_\{i=1\}^\{N^\{\\pi\}\}B\_\{i\}andNAIHπ=∑i=1NπEiN^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}=\\sum\_\{i=1\}^\{N^\{\\pi\}\}E\_\{i\}, so𝔼hπ​\[NAIdir,π\]=∑i𝔼hπ​\[Bi\]\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]=\\sum\_\{i\}\\mathbb\{E\}^\{\\pi\}\_\{h\}\[B\_\{i\}\]and𝔼hπ​\[NAIHπ\]=∑i𝔼hπ​\[Ei\]\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]=\\sum\_\{i\}\\mathbb\{E\}^\{\\pi\}\_\{h\}\[E\_\{i\}\]\. SinceℐAIHπ⊆ℐAIπ\\mathcal\{I\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}^\{\\pi\}\\subseteq\\mathcal\{I\}\_\{\\mathrm\{AI\}\}^\{\\pi\},Ei=1E\_\{i\}=1impliesBi=1B\_\{i\}=1, so0≤Ei≤Bi≤10\\leq E\_\{i\}\\leq B\_\{i\}\\leq 1\. SinceℐAIHπ=\{i:Ei=1\}\\mathcal\{I\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}^\{\\pi\}=\\\{i:E\_\{i\}=1\\\},

∑i∈ℐAIHπd\(h\)​\(Ri\)=∑i=1NπEi​d\(h\)​\(Ri\)\.\\sum\_\{i\\in\\mathcal\{I\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}^\{\\pi\}\}d^\{\(h\)\}\(R\_\{i\}\)=\\sum\_\{i=1\}^\{N^\{\\pi\}\}E\_\{i\}\\,d^\{\(h\)\}\(R\_\{i\}\)\.
Fixλ≥0\\lambda\\geq 0\. For every itemii,

Ei​d\(h\)​\(Ri\)≤λ​Ei\+Ei​\[d\(h\)​\(Ri\)−λ\]\+≤λ​Ei\+Bi​\[d\(h\)​\(Ri\)−λ\]\+\.E\_\{i\}\\,d^\{\(h\)\}\(R\_\{i\}\)\\leq\\lambda E\_\{i\}\+E\_\{i\}\\left\[d^\{\(h\)\}\(R\_\{i\}\)\-\\lambda\\right\]\_\{\+\}\\leq\\lambda E\_\{i\}\+B\_\{i\}\\left\[d^\{\(h\)\}\(R\_\{i\}\)\-\\lambda\\right\]\_\{\+\}\.\(69\)Summing \([69](https://arxiv.org/html/2609.28859#S8.E69)\) overii, taking expectation underHhH\_\{h\}, and using∑i𝔼hπ​\[Ei\]=𝔼hπ​\[NAIHπ\]\\sum\_\{i\}\\mathbb\{E\}^\{\\pi\}\_\{h\}\[E\_\{i\}\]=\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\],

𝔼hπ​\[∑i=1NπEi​d\(h\)​\(Ri\)\]≤λ​𝔼hπ​\[NAIHπ\]\+∑i=1Nπ𝔼hπ​\[Bi​\[d\(h\)​\(Ri\)−λ\]\+\]\.\\mathbb\{E\}^\{\\pi\}\_\{h\}\\left\[\\sum\_\{i=1\}^\{N^\{\\pi\}\}E\_\{i\}\\,d^\{\(h\)\}\(R\_\{i\}\)\\right\]\\leq\\lambda\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]\+\\sum\_\{i=1\}^\{N^\{\\pi\}\}\\mathbb\{E\}^\{\\pi\}\_\{h\}\\left\[B\_\{i\}\\left\[d^\{\(h\)\}\(R\_\{i\}\)\-\\lambda\\right\]\_\{\+\}\\right\]\.\(70\)
We claimBiB\_\{i\}is independent ofRiR\_\{i\}underHhH\_\{h\}\. The indicatorBiB\_\{i\}records whether itemii’s first paid action is an AI query\. That decision is made from the history available just before itemiiis first queried, and at that moment neitherRiR\_\{i\}\(revealed only by an AI query onii\) norXiX\_\{i\}\(revealed only by a human query onii\) has been observed\. HenceBiB\_\{i\}is a function of the randomization seed and the other items’ variables\{\(Xj,Rj\):j≠i\}\\\{\(X\_\{j\},R\_\{j\}\):j\\neq i\\\}alone\. By Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)the family\{\(Xj,Rj\):j≠i\}\\\{\(X\_\{j\},R\_\{j\}\):j\\neq i\\\}is independent ofRiR\_\{i\}, and the seed is independent of all data; thereforeBi⟂RiB\_\{i\}\\perp R\_\{i\}underHhH\_\{h\}\. The product rule for independent random variables gives the first equality below, and the marginal lawRi∼ghR\_\{i\}\\sim g\_\{h\}underHhH\_\{h\}\(by \([2](https://arxiv.org/html/2609.28859#S3.E2)\)\) gives the second:

𝔼hπ​\[Bi​\[d\(h\)​\(Ri\)−λ\]\+\]=𝔼hπ​\[Bi\]​𝔼hπ​\[\[d\(h\)​\(Ri\)−λ\]\+\]=𝔼hπ​\[Bi\]​∑r∈ℛgh​\(r\)​\[d\(h\)​\(r\)−λ\]\+\.\\mathbb\{E\}^\{\\pi\}\_\{h\}\\left\[B\_\{i\}\\left\[d^\{\(h\)\}\(R\_\{i\}\)\-\\lambda\\right\]\_\{\+\}\\right\]=\\mathbb\{E\}^\{\\pi\}\_\{h\}\[B\_\{i\}\]\\,\\mathbb\{E\}^\{\\pi\}\_\{h\}\\left\[\\left\[d^\{\(h\)\}\(R\_\{i\}\)\-\\lambda\\right\]\_\{\+\}\\right\]=\\mathbb\{E\}^\{\\pi\}\_\{h\}\[B\_\{i\}\]\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\left\[d^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}\.Summing overiiand using∑i𝔼hπ​\[Bi\]=𝔼hπ​\[NAIdir,π\]\\sum\_\{i\}\\mathbb\{E\}^\{\\pi\}\_\{h\}\[B\_\{i\}\]=\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\],

∑i=1Nπ𝔼hπ​\[Bi​\[d\(h\)​\(Ri\)−λ\]\+\]=𝔼hπ​\[NAIdir,π\]​∑r∈ℛgh​\(r\)​\[d\(h\)​\(r\)−λ\]\+\.\\sum\_\{i=1\}^\{N^\{\\pi\}\}\\mathbb\{E\}^\{\\pi\}\_\{h\}\\left\[B\_\{i\}\\left\[d^\{\(h\)\}\(R\_\{i\}\)\-\\lambda\\right\]\_\{\+\}\\right\]=\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\left\[d^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}\.
Substituting into \([70](https://arxiv.org/html/2609.28859#S8.E70)\) and factoring out𝔼hπ​\[NAIdir,π\]\>0\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\>0,

𝔼hπ​\[∑i=1NπEi​d\(h\)​\(Ri\)\]≤𝔼hπ​\[NAIdir,π\]​\[λ​𝔼hπ​\[NAIHπ\]𝔼hπ​\[NAIdir,π\]\+∑r∈ℛgh​\(r\)​\[d\(h\)​\(r\)−λ\]\+\]\.\\mathbb\{E\}^\{\\pi\}\_\{h\}\\left\[\\sum\_\{i=1\}^\{N^\{\\pi\}\}E\_\{i\}\\,d^\{\(h\)\}\(R\_\{i\}\)\\right\]\\leq\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\\left\[\\lambda\\,\\frac\{\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]\}\{\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\}\+\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\left\[d^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}\\right\]\.The left side does not depend onλ\\lambda, so the inequality holds with the right side replaced by its infimum overλ≥0\\lambda\\geq 0\. Writes:=𝔼hπ​\[NAIHπ\]/𝔼hπ​\[NAIdir,π\]s:=\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]/\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]; thens∈\[0,1\]s\\in\[0,1\]because0≤𝔼hπ​\[NAIHπ\]≤𝔼hπ​\[NAIdir,π\]0\\leq\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]\\leq\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\. By \([15](https://arxiv.org/html/2609.28859#S4.E15)\),

infλ≥0\{λ​s\+∑r∈ℛgh​\(r\)​\[d\(h\)​\(r\)−λ\]\+\}=Ψh​\(s\)\.\\inf\_\{\\lambda\\geq 0\}\\left\\\{\\lambda s\+\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\left\[d^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}\\right\\\}=\\Psi\_\{h\}\(s\)\.Combining the last two displays with the identity∑i∈ℐAIHπd\(h\)​\(Ri\)=∑iEi​d\(h\)​\(Ri\)\\sum\_\{i\\in\\mathcal\{I\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}^\{\\pi\}\}d^\{\(h\)\}\(R\_\{i\}\)=\\sum\_\{i\}E\_\{i\}\\,d^\{\(h\)\}\(R\_\{i\}\)yields

𝔼hπ​\[∑i∈ℐAIHπd\(h\)​\(Ri\)\]≤𝔼hπ​\[NAIdir,π\]​Ψh​\(𝔼hπ​\[NAIHπ\]𝔼hπ​\[NAIdir,π\]\),\\mathbb\{E\}^\{\\pi\}\_\{h\}\\left\[\\sum\_\{i\\in\\mathcal\{I\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}^\{\\pi\}\}d^\{\(h\)\}\(R\_\{i\}\)\\right\]\\leq\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\\,\\Psi\_\{h\}\\\!\\left\(\\frac\{\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]\}\{\\mathbb\{E\}^\{\\pi\}\_\{h\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\}\\right\),which is \([19](https://arxiv.org/html/2609.28859#S4.E19)\)\.

### 8\.8Proof of Proposition[4\.10](https://arxiv.org/html/2609.28859#S4.Thmtheorem10)

###### Proof 8\.14

Proof of Proposition[4\.10](https://arxiv.org/html/2609.28859#S4.Thmtheorem10)We show that Problem \([21](https://arxiv.org/html/2609.28859#S4.E21)\) is a convex optimization problem\. We first notice that the first and second constraints of \([21](https://arxiv.org/html/2609.28859#S4.E21)\) are linear constraints in the decision variables\(nH,nAI,nesc\)\(n\_\{\\mathrm\{H\}\},n\_\{\\mathrm\{AI\}\},n\_\{\\mathrm\{esc\}\}\)\. The third constraint involves nonlinear functionnAI​Ψh​\(nesc/nAI\)n\_\{\\mathrm\{AI\}\}\\Psi\_\{h\}\(n\_\{\\mathrm\{esc\}\}/n\_\{\\mathrm\{AI\}\}\); we show that it is jointly concave in\(nAI,nesc\)\(n\_\{\\mathrm\{AI\}\},n\_\{\\mathrm\{esc\}\}\)\. Notice thatΨh​\(s\)\\Psi\_\{h\}\(s\)is concave inss\(see Theorem 5\.1 of[Bertsimas and Tsitsiklis \(1997\)](https://arxiv.org/html/2609.28859#bib.bib7)\)\. Additionally,Ψhpersp​\(nAI,nesc\):=nAI​Ψh​\(nesc/nAI\)\\Psi\_\{h\}^\{\\mathrm\{persp\}\}\(n\_\{\\mathrm\{AI\}\},n\_\{\\mathrm\{esc\}\}\):=n\_\{\\mathrm\{AI\}\}\\Psi\_\{h\}\(n\_\{\\mathrm\{esc\}\}/n\_\{\\mathrm\{AI\}\}\)is the perspective function ofΨh\\Psi\_\{h\}with domain\{\(nAI,nesc\):0≤nesc≤nAI\}\\\{\(n\_\{\\mathrm\{AI\}\},n\_\{\\mathrm\{esc\}\}\):0\\leq n\_\{\\mathrm\{esc\}\}\\leq n\_\{\\mathrm\{AI\}\}\\\}and therefore is also concave \(Section 3\.2\.6 of[Boyd and Vandenberghe \(2004\)](https://arxiv.org/html/2609.28859#bib.bib8)\)\. Finally the feasible set of \([21](https://arxiv.org/html/2609.28859#S4.E21)\) is a superlevel set of a concave function intersected with half\-spaces formed by linear inequalities\. Thus the feasible set is convex\. The objective is linear\. Therefore \([21](https://arxiv.org/html/2609.28859#S4.E21)\) is a convex optimization problem\.

### 8\.9Proof of Theorem[4\.11](https://arxiv.org/html/2609.28859#S4.Thmtheorem11)

###### Proof 8\.15

Proof of Theorem[4\.11](https://arxiv.org/html/2609.28859#S4.Thmtheorem11)Fix any feasible policyπ\\piusingNπN^\{\\pi\}items\. We first show that𝔼1π​\[Cπ\]≥Nπ​cdata\+Γ1​\(A,Nπ\)\\mathbb\{E\}^\{\\pi\}\_\{1\}\[C^\{\\pi\}\]\\geq N^\{\\pi\}c\_\{\\mathrm\{data\}\}\+\\Gamma\_\{1\}\(A,N^\{\\pi\}\), and that𝔼0π​\[Cπ\]≥Nπ​cdata\+Γ0​\(B,Nπ\)\\mathbb\{E\}^\{\\pi\}\_\{0\}\[C^\{\\pi\}\]\\geq N^\{\\pi\}c\_\{\\mathrm\{data\}\}\+\\Gamma\_\{0\}\(B,N^\{\\pi\}\)can be derived analogously\. Recall

𝔼1π​\[Cπ\]=cdata​Nπ\+𝔼1π​\[cAI​NAItot\+cH​NHtot\]\\displaystyle\\mathbb\{E\}^\{\\pi\}\_\{1\}\[C^\{\\pi\}\]=c\_\{\\mathrm\{data\}\}N^\{\\pi\}\+\\mathbb\{E\}^\{\\pi\}\_\{1\}\[c\_\{\\mathrm\{AI\}\}N^\{\\mathrm\{tot\}\}\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}N^\{\\mathrm\{tot\}\}\_\{\\mathrm\{H\}\}\]≥cdata​Nπ\+cAI​𝔼1π​\[NAIdir,π\]\+cH​\(𝔼1π​\[NHdir,π\]\+𝔼1π​\[NAIHπ\]\),\\displaystyle\\geq c\_\{\\mathrm\{data\}\}N^\{\\pi\}\+c\_\{\\mathrm\{AI\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\+c\_\{\\mathrm\{H\}\}\(\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}\]\+\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]\),where the inequality follows since each item counted byNAIdir,πN^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}is AI\-queried once, soNAIdir,π≤NAItotN^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\\leq N^\{\\mathrm\{tot\}\}\_\{\\mathrm\{AI\}\}, and every direct human query and every AI\-after\-report human escalation is a human query, soNHdir,π\+NAIHπ≤NHtotN^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}\+N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\\leq N^\{\\mathrm\{tot\}\}\_\{\\mathrm\{H\}\}\. Thus it is sufficient to show thatcAI​𝔼1π​\[NAIdir,π\]\+cH​\(𝔼1π​\[NHdir,π\]\+𝔼1π​\[NAIHπ\]\)≥Γ1​\(A,Nπ\)c\_\{\\mathrm\{AI\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\+c\_\{\\mathrm\{H\}\}\(\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}\]\+\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]\)\\geq\\Gamma\_\{1\}\(A,N^\{\\pi\}\)\.

We showcAI​𝔼1π​\[NAIdir,π\]\+cH​\(𝔼1π​\[NHdir,π\]\+𝔼1π​\[NAIHπ\]\)≥Γ1​\(A,Nπ\)c\_\{\\mathrm\{AI\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\+c\_\{\\mathrm\{H\}\}\(\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}\]\+\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]\)\\geq\\Gamma\_\{1\}\(A,N^\{\\pi\}\)by noting that\(𝔼1π​\[NHdir,π\],𝔼1π​\[NAIdir,π\],𝔼1π​\[NAIHπ\]\)\(\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}\],\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\],\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]\)is feasible forΓ1​\(A,Nπ\)\\Gamma\_\{1\}\(A,N^\{\\pi\}\)\. Specifically, by Lemma[4\.5](https://arxiv.org/html/2609.28859#S4.Thmtheorem5)and Theorem[4\.6](https://arxiv.org/html/2609.28859#S4.Thmtheorem6),

A≤𝔼1π​\[NAIdir,π​IR\(1\)\+NHdir,π​JX\(1\)\+∑i∈ℐAIHπd\(1\)​\(Ri\)\]\.A\\leq\\mathbb\{E\}^\{\\pi\}\_\{1\}\\left\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}I\_\{R\}^\{\(1\)\}\+N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}J\_\{X\}^\{\(1\)\}\+\\sum\_\{i\\in\\mathcal\{I\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}^\{\\pi\}\}d^\{\(1\)\}\(R\_\{i\}\)\\right\]\.Since an item cannot be both direct\-human\-before\-AI and AI\-before\-human, we haveNHdir,π\+NAIdir,π≤NπN^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}\+N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\\leq N^\{\\pi\}on every sample path\. Thus𝔼1π​\[NHdir,π\]\+𝔼1π​\[NAIdir,π\]≤Nπ\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}\]\+\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\\leq N^\{\\pi\}\. Also0≤𝔼1π​\[NAIHπ\]≤𝔼1π​\[NAIdir,π\]0\\leq\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]\\leq\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\. By Lemma[4\.9](https://arxiv.org/html/2609.28859#S4.Thmtheorem9),

𝔼1π​\[∑i∈ℐAIHπd\(1\)​\(Ri\)\]≤𝔼1π​\[NAIdir,π\]​Ψ1​\(𝔼1π​\[NAIHπ\]/𝔼1π​\[NAIdir,π\]\),\\mathbb\{E\}^\{\\pi\}\_\{1\}\\left\[\\sum\_\{i\\in\\mathcal\{I\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}^\{\\pi\}\}d^\{\(1\)\}\(R\_\{i\}\)\\right\]\\leq\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\\Psi\_\{1\}\(\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]/\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\),where𝔼1π​\[NAIdir,π\]​Ψ1​\(𝔼1π​\[NAIHπ\]/𝔼1π​\[NAIdir,π\]\)=0\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\\Psi\_\{1\}\(\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]/\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\)=0when𝔼1π​\[NAIdir,π\]=0\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]=0, as in \([21](https://arxiv.org/html/2609.28859#S4.E21)\)\. Therefore

𝔼1π​\[NHdir,π\]​JX\(1\)\+𝔼1π​\[NAIdir,π\]​IR\(1\)\+𝔼1π​\[NAIdir,π\]​Ψ1​\(𝔼1π​\[NAIHπ\]/𝔼1π​\[NAIdir,π\]\)≥A\.\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}\]J\_\{X\}^\{\(1\)\}\+\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]I\_\{R\}^\{\(1\)\}\+\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\\Psi\_\{1\}\(\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]/\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\)\\geq A\.So\(𝔼1π​\[NHdir,π\],𝔼1π​\[NAIdir,π\],𝔼1π​\[NAIHπ\]\)\(\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}\],\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\],\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]\)is feasible forΓ1​\(A,Nπ\)\\Gamma\_\{1\}\(A,N^\{\\pi\}\), and we have that

cAI​𝔼1π​\[NAIdir,π\]\+cH​\(𝔼1π​\[NHdir,π\]\+𝔼1π​\[NAIHπ\]\)≥Γ1​\(A,Nπ\),c\_\{\\mathrm\{AI\}\}\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{AI\}\}\]\+c\_\{\\mathrm\{H\}\}\(\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\mathrm\{dir\},\\pi\}\_\{\\mathrm\{H\}\}\]\+\\mathbb\{E\}^\{\\pi\}\_\{1\}\[N^\{\\pi\}\_\{\\mathrm\{AI\}\\mathrm\{H\}\}\]\)\\geq\\Gamma\_\{1\}\(A,N^\{\\pi\}\),and therefore

𝔼1π​\[Cπ\]≥Nπ​cdata\+Γ1​\(A,Nπ\)\.\\mathbb\{E\}^\{\\pi\}\_\{1\}\[C^\{\\pi\}\]\\geq N^\{\\pi\}c\_\{\\mathrm\{data\}\}\+\\Gamma\_\{1\}\(A,N^\{\\pi\}\)\.\(71\)
By similar arguments, we have that

𝔼0π​\[Cπ\]≥Nπ​cdata\+Γ0​\(B,Nπ\)\\mathbb\{E\}^\{\\pi\}\_\{0\}\[C^\{\\pi\}\]\\geq N^\{\\pi\}c\_\{\\mathrm\{data\}\}\+\\Gamma\_\{0\}\(B,N^\{\\pi\}\)\(72\)Combining \([71](https://arxiv.org/html/2609.28859#S8.E71)\) and \([72](https://arxiv.org/html/2609.28859#S8.E72)\),

max⁡\{𝔼0π​\[Cπ\],𝔼1π​\[Cπ\]\}≥Nπ​cdata\+max⁡\{Γ1​\(A,Nπ\),Γ0​\(B,Nπ\)\}\.\\max\\\{\\mathbb\{E\}^\{\\pi\}\_\{0\}\[C^\{\\pi\}\],\\mathbb\{E\}^\{\\pi\}\_\{1\}\[C^\{\\pi\}\]\\\}\\geq N^\{\\pi\}c\_\{\\mathrm\{data\}\}\+\\max\\\{\\Gamma\_\{1\}\(A,N^\{\\pi\}\),\\Gamma\_\{0\}\(B,N^\{\\pi\}\)\\\}\.SinceNπ≥Nfixed,H​\(α,β\)N^\{\\pi\}\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)by Lemma[4\.2](https://arxiv.org/html/2609.28859#S4.Thmtheorem2), the right side is at least its minimum over integersN≥Nfixed,H​\(α,β\)N\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\), which isLB⁡\(α,β\)\\mathrm\{LB\}\(\\alpha,\\beta\)\. Taking the infimum over all feasible policies proves

C∗​\(α,β\)≥LB⁡\(α,β\)\.C^\{\*\}\(\\alpha,\\beta\)\\geq\\mathrm\{LB\}\(\\alpha,\\beta\)\.

## 9Additional Materials for Section[5\.2](https://arxiv.org/html/2609.28859#S5.SS2)

### 9\.1Computation ofNfixed,AI​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha,\\beta\)

We now describe how to computeNfixed,AI​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha,\\beta\), together with the cutoff and randomization of the associated report test, mirroring the computation ofNfixed,H​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)in Section[4\.1](https://arxiv.org/html/2609.28859#S4.SS1)\. As in Definition[5\.1](https://arxiv.org/html/2609.28859#S5.Thmtheorem1), all expectations and probabilities below refer to the i\.i\.d\. report model: underHhH\_\{h\}the reportsR1,…,RNR\_\{1\},\\ldots,R\_\{N\}are i\.i\.d\. with mass functionghg\_\{h\}\. By the Neyman–Pearson lemma\([Neyman and Pearson 1933](https://arxiv.org/html/2609.28859#bib.bib30)\), for any fixed sample sizeNNthe most powerful report test at type\-I levelα\\alphais the likelihood\-ratio test\. In the full\-label case the likelihood ratio is strictly increasing in the label sum, which reduces the test to a count threshold; for a general report alphabetℛ\\mathcal\{R\}no scalar count is sufficient, so the test thresholds the log\-likelihood\-ratio statistic itself\.

FixN∈ℕN\\in\\mathbb\{N\}and suppose the report vectorR=\(R1,…,RN\)R=\(R\_\{1\},\\ldots,R\_\{N\}\)is observed\. Fort=1,…,Nt=1,\\ldots,N, the running log\-likelihood ratio underH1H\_\{1\}versusH0H\_\{0\}is, by \([28](https://arxiv.org/html/2609.28859#S5.E28)\),

ℒtseq\(R1:t\):=log∏i=1tg1​\(Ri\)g0​\(Ri\)=∑i=1tℓR\(Ri\)\.\\mathcal\{L\}\_\{t\}^\{\\mathrm\{seq\}\}\(R\_\{1:t\}\):=\\log\\prod\_\{i=1\}^\{t\}\\frac\{g\_\{1\}\(R\_\{i\}\)\}\{g\_\{0\}\(R\_\{i\}\)\}=\\sum\_\{i=1\}^\{t\}\\ell\_\{R\}\(R\_\{i\}\)\.\(73\)Sinceℛ\\mathcal\{R\}is finite, the terminal statisticℒNseq\\mathcal\{L\}\_\{N\}^\{\\mathrm\{seq\}\}takes finitely many values: writingnr:=\|\{i≤N:Ri=r\}\|n\_\{r\}:=\|\\\{i\\leq N:R\_\{i\}=r\\\}\|for the report counts,ℒNseq=∑r∈ℛnr​ℓR​\(r\)\\mathcal\{L\}\_\{N\}^\{\\mathrm\{seq\}\}=\\sum\_\{r\\in\\mathcal\{R\}\}n\_\{r\}\\,\\ell\_\{R\}\(r\), and\(nr\)r∈ℛ\(n\_\{r\}\)\_\{r\\in\\mathcal\{R\}\}is multinomial with parameters\(N,gh\)\(N,g\_\{h\}\)underHhH\_\{h\}\. The distribution ofℒNseq\\mathcal\{L\}\_\{N\}^\{\\mathrm\{seq\}\}under each hypothesis is therefore computable exactly\. Let𝒮N\\mathcal\{S\}\_\{N\}denote the finite set of values ofℒNseq\\mathcal\{L\}\_\{N\}^\{\\mathrm\{seq\}\}; sincegh​\(r\)\>0g\_\{h\}\(r\)\>0for everyr∈ℛr\\in\\mathcal\{R\}and bothh∈\{0,1\}h\\in\\\{0,1\\\}, every point of𝒮N\\mathcal\{S\}\_\{N\}has positive probability under both hypotheses\.

For a target type\-I levelα\\alpha, define the cutoffcNAIc^\{\\mathrm\{AI\}\}\_\{N\}to be the smallest point of𝒮N\\mathcal\{S\}\_\{N\}such thatℙ0​\(ℒNseq\>cNAI\)≤α\\mathbb\{P\}\_\{0\}\(\\mathcal\{L\}\_\{N\}^\{\\mathrm\{seq\}\}\>c^\{\\mathrm\{AI\}\}\_\{N\}\)\\leq\\alpha, and set

γNAI:=α−ℙ0​\(ℒNseq\>cNAI\)ℙ0​\(ℒNseq=cNAI\),\\gamma^\{\\mathrm\{AI\}\}\_\{N\}:=\\frac\{\\alpha\-\\mathbb\{P\}\_\{0\}\(\\mathcal\{L\}\_\{N\}^\{\\mathrm\{seq\}\}\>c^\{\\mathrm\{AI\}\}\_\{N\}\)\}\{\\mathbb\{P\}\_\{0\}\(\\mathcal\{L\}\_\{N\}^\{\\mathrm\{seq\}\}=c^\{\\mathrm\{AI\}\}\_\{N\}\)\},\(74\)withγNAI=0\\gamma^\{\\mathrm\{AI\}\}\_\{N\}=0ifℙ0​\(ℒNseq\>cNAI\)=α\\mathbb\{P\}\_\{0\}\(\\mathcal\{L\}\_\{N\}^\{\\mathrm\{seq\}\}\>c^\{\\mathrm\{AI\}\}\_\{N\}\)=\\alpha\. The denominator is positive becausecNAI∈𝒮Nc^\{\\mathrm\{AI\}\}\_\{N\}\\in\\mathcal\{S\}\_\{N\}, andγNAI∈\[0,1\]\\gamma^\{\\mathrm\{AI\}\}\_\{N\}\\in\[0,1\]by the choice ofcNAIc^\{\\mathrm\{AI\}\}\_\{N\}\. The Neyman–Pearson report test is the randomized threshold test

ψN∗\(R\)=\{1,ℒNseq\(R1:N\)\>cAIN,γNAI,ℒNseq\(R1:N\)=cAIN,0,ℒNseq\(R1:N\)<cAIN\.\\psi\_\{N\}^\{\*\}\(R\)=\\begin\{cases\}1,&\\mathcal\{L\}\_\{N\}^\{\\mathrm\{seq\}\}\(R\_\{1:N\}\)\>c^\{\\mathrm\{AI\}\}\_\{N\},\\\\ \\gamma^\{\\mathrm\{AI\}\}\_\{N\},&\\mathcal\{L\}\_\{N\}^\{\\mathrm\{seq\}\}\(R\_\{1:N\}\)=c^\{\\mathrm\{AI\}\}\_\{N\},\\\\ 0,&\\mathcal\{L\}\_\{N\}^\{\\mathrm\{seq\}\}\(R\_\{1:N\}\)<c^\{\\mathrm\{AI\}\}\_\{N\}\.\\end\{cases\}\(75\)By construction,ψN∗\\psi\_\{N\}^\{\*\}meets the type\-I constraint exactly:𝔼0​\[ψN∗​\(R\)\]=α\\mathbb\{E\}\_\{0\}\[\\psi\_\{N\}^\{\*\}\(R\)\]=\\alpha\. Its type\-II error is

𝔼1​\[1−ψN∗​\(R\)\]=ℙ1​\(ℒNseq<cNAI\)\+\(1−γNAI\)​ℙ1​\(ℒNseq=cNAI\)\.\\mathbb\{E\}\_\{1\}\[1\-\\psi\_\{N\}^\{\*\}\(R\)\]=\\mathbb\{P\}\_\{1\}\(\\mathcal\{L\}\_\{N\}^\{\\mathrm\{seq\}\}<c^\{\\mathrm\{AI\}\}\_\{N\}\)\+\(1\-\\gamma^\{\\mathrm\{AI\}\}\_\{N\}\)\\,\\mathbb\{P\}\_\{1\}\(\\mathcal\{L\}\_\{N\}^\{\\mathrm\{seq\}\}=c^\{\\mathrm\{AI\}\}\_\{N\}\)\.\(76\)It follows, exactly as forNfixed,H​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha,\\beta\)in Section[4\.1](https://arxiv.org/html/2609.28859#S4.SS1), thatNfixed,AI​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha,\\beta\)can be computed by increasingNNfrom11upward and checking whether \([76](https://arxiv.org/html/2609.28859#S9.E76)\) is at mostβ\\beta: ifψN∗\\psi\_\{N\}^\{\*\}fails the type\-II target, no report test onNNreports can meet it; if it succeeds, it is itself a valid test meeting both error targets\. The firstNNat which \([76](https://arxiv.org/html/2609.28859#S9.E76)\) is at mostβ\\betais thereforeNfixed,AI​\(α,β\)N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha,\\beta\)\.

### 9\.2Proof of Theorem[5\.2](https://arxiv.org/html/2609.28859#S5.Thmtheorem2)

To prove Theorem[5\.2](https://arxiv.org/html/2609.28859#S5.Thmtheorem2), we need auxiliary Lemmas[9\.1](https://arxiv.org/html/2609.28859#S9.Thmtheorem1)and[9\.3](https://arxiv.org/html/2609.28859#S9.Thmtheorem3)\. We present the proof of Theorem[5\.2](https://arxiv.org/html/2609.28859#S5.Thmtheorem2)in the end of this subsection\. We letE\+E\_\{\+\}be the event that Algorithm[1](https://arxiv.org/html/2609.28859#alg1)stops before fallback by crossing the upper boundaryaka\_\{k\}, and letE−E\_\{\-\}be the event that it stops before fallback by crossing the lower boundary−bk\-b\_\{k\}\.

###### Lemma 9\.1\(Boundary\-route error control\)

ℙ0πk​\(E\+\)≤α1,k\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\(E\_\{\+\}\)\\leq\\alpha\_\{1,k\}andℙ1πk​\(E−\)≤β1,k\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(E\_\{\-\}\)\\leq\\beta\_\{1,k\}\.

###### Proof 9\.2

Proof of Lemma[9\.1](https://arxiv.org/html/2609.28859#S9.Thmtheorem1)Index the likelihood\-ratio updates byt=1,2,…t=1,2,\\ldots, where each update corresponds to one paid observation revealed by Algorithm[1](https://arxiv.org/html/2609.28859#alg1): a direct\-human label, an AI report, or a post\-report escalation label\. Recallℋt:=σ⁡\(A1,O1,…,At,Ot\)\\mathcal\{H\}\_\{t\}:=\\sigma\\bigl\(A\_\{1\},O\_\{1\},\\ldots,A\_\{t\},O\_\{t\}\\bigr\)\(withℋ0\\mathcal\{H\}\_\{0\}the trivialσ\\sigma\-algebra\{∅,Ω\}\\\{\\emptyset,\\Omega\\\}\) is the filtration generated by the actions and observations up to and including updatett\. Letξt\\xi\_\{t\}be the exactH1H\_\{1\}\-versus\-H0H\_\{0\}log\-likelihood increment contributed by the observation at updatett:

ξt:=\{ℓX​\(Xi\),At=H⁡\(i\)​with iteminot previously AI\-queried,ℓR​\(Ri\),At=AI⁡\(i\),ℓH​\(Xi,Ri\),At=H⁡\(i\)​with reportRialready observed,\\xi\_\{t\}:=\\begin\{cases\}\\ell\_\{X\}\(X\_\{i\}\),&A\_\{t\}=\\mathrm\{H\}\(i\)\\text\{ with item $i$ not previously AI\-queried\},\\\\ \\ell\_\{R\}\(R\_\{i\}\),&A\_\{t\}=\\mathrm\{AI\}\(i\),\\\\ \\ell\_\{H\}\(X\_\{i\},R\_\{i\}\),&A\_\{t\}=\\mathrm\{H\}\(i\)\\text\{ with report $R\_\{i\}$ already observed\},\\end\{cases\}\(77\)whereℓX,ℓR,ℓH\\ell\_\{X\},\\ell\_\{R\},\\ell\_\{H\}are the increments \([27](https://arxiv.org/html/2609.28859#S5.E27)\)–\([29](https://arxiv.org/html/2609.28859#S5.E29)\)\. The running statistic and its exponential are

St:=∑s=1tξs,Λt:=exp⁡\(St\)=∏s=1teξs,Λ0:=1\.S\_\{t\}:=\\sum\_\{s=1\}^\{t\}\\xi\_\{s\},\\qquad\\Lambda\_\{t\}:=\\exp\(S\_\{t\}\)=\\prod\_\{s=1\}^\{t\}e^\{\\xi\_\{s\}\},\\qquad\\Lambda\_\{0\}:=1\.\(78\)The primitivesp0,p1,f0,f1p\_\{0\},p\_\{1\},f\_\{0\},f\_\{1\}do not depend onkk:p0,p1∈\(0,1\)p\_\{0\},p\_\{1\}\\in\(0,1\)are fixed constants by \([1](https://arxiv.org/html/2609.28859#S3.E1)\), andf0,f1f\_\{0\},f\_\{1\}are strictly positive on the finite setℛ\\mathcal\{R\}by Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)\. Hence all four primitives are bounded away from00and11, so the incrementsξt\\xi\_\{t\}are uniformly bounded; since the number of updates is at most2​Nmain,k2N\_\{\\mathrm\{main\},k\}, eachΛt\\Lambda\_\{t\}is bounded and hence integrable\. Next we show that\(Λt\)t≥0\(\\Lambda\_\{t\}\)\_\{t\\geq 0\}is a nonnegative\(ℋt\)\(\\mathcal\{H\}\_\{t\}\)\-martingale underH0H\_\{0\}with𝔼0πk​\[Λt\]=Λ0=1\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\Lambda\_\{t\}\]=\\Lambda\_\{0\}=1\.

*Martingale property underH0H\_\{0\}\.*Fix an update timettand condition on the pair\(ℋt−1,At\)\(\\mathcal\{H\}\_\{t\-1\},A\_\{t\}\)\. The actionAtA\_\{t\}is a measurable function ofℋt−1\\mathcal\{H\}\_\{t\-1\}and of a fresh randomization seed whose law does not depend on the hypothesis and which is independent of the not\-yet\-revealed observationOtO\_\{t\}; conditioning onAtA\_\{t\}therefore leaves theH0H\_\{0\}\-law ofOtO\_\{t\}unchanged\. We evaluate𝔼0πk\[eξt∣ℋt−1,At\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[e^\{\\xi\_\{t\}\}\\mid\\mathcal\{H\}\_\{t\-1\},A\_\{t\}\]in each case\.

IfAt=H⁡\(i\)A\_\{t\}=\\mathrm\{H\}\(i\)with itemiinot previously queried, then underH0H\_\{0\}the labelXi∼Bernoulli⁡\(p0\)X\_\{i\}\\sim\\operatorname\{Bernoulli\}\(p\_\{0\}\)is independent ofℋt−1\\mathcal\{H\}\_\{t\-1\}, so by \([27](https://arxiv.org/html/2609.28859#S5.E27)\)

𝔼0πk\[eξt∣ℋt−1,At\]=∑x∈\{0,1\}p0x\(1−p0\)1−xp1x​\(1−p1\)1−xp0x​\(1−p0\)1−x=∑x∈\{0,1\}p1x\(1−p1\)1−x=1\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\\\!\\left\[e^\{\\xi\_\{t\}\}\\mid\\mathcal\{H\}\_\{t\-1\},A\_\{t\}\\right\]=\\sum\_\{x\\in\\\{0,1\\\}\}p\_\{0\}^\{x\}\(1\-p\_\{0\}\)^\{1\-x\}\\,\\frac\{p\_\{1\}^\{x\}\(1\-p\_\{1\}\)^\{1\-x\}\}\{p\_\{0\}^\{x\}\(1\-p\_\{0\}\)^\{1\-x\}\}=\\sum\_\{x\\in\\\{0,1\\\}\}p\_\{1\}^\{x\}\(1\-p\_\{1\}\)^\{1\-x\}=1\.\(79\)IfAt=AI⁡\(i\)A\_\{t\}=\\mathrm\{AI\}\(i\), then underH0H\_\{0\}the reportRi∼G0R\_\{i\}\\sim G\_\{0\}is independent ofℋt−1\\mathcal\{H\}\_\{t\-1\}, so by \([28](https://arxiv.org/html/2609.28859#S5.E28)\)

𝔼0πk\[eξt∣ℋt−1,At\]=∑r∈ℛg0\(r\)g1​\(r\)g0​\(r\)=∑r∈ℛg1\(r\)=1\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\\\!\\left\[e^\{\\xi\_\{t\}\}\\mid\\mathcal\{H\}\_\{t\-1\},A\_\{t\}\\right\]=\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{0\}\(r\)\\,\\frac\{g\_\{1\}\(r\)\}\{g\_\{0\}\(r\)\}=\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{1\}\(r\)=1\.\(80\)IfAt=H⁡\(i\)A\_\{t\}=\\mathrm\{H\}\(i\)with reportRi=rR\_\{i\}=ralready observed \(soRiR\_\{i\}isℋt−1\\mathcal\{H\}\_\{t\-1\}\-measurable\), then underH0H\_\{0\}the escalated label has conditional massρ0\(⋅∣r\)\\rho\_\{0\}\(\\cdot\\mid r\)by \([4](https://arxiv.org/html/2609.28859#S3.E4)\), so by \([29](https://arxiv.org/html/2609.28859#S5.E29)\)

𝔼0πk\[eξt∣ℋt−1,At\]=∑x∈\{0,1\}ρ0\(x∣r\)ρ1​\(x∣r\)ρ0​\(x∣r\)=∑x∈\{0,1\}ρ1\(x∣r\)=1\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\\\!\\left\[e^\{\\xi\_\{t\}\}\\mid\\mathcal\{H\}\_\{t\-1\},A\_\{t\}\\right\]=\\sum\_\{x\\in\\\{0,1\\\}\}\\rho\_\{0\}\(x\\mid r\)\\,\\frac\{\\rho\_\{1\}\(x\\mid r\)\}\{\\rho\_\{0\}\(x\\mid r\)\}=\\sum\_\{x\\in\\\{0,1\\\}\}\\rho\_\{1\}\(x\\mid r\)=1\.\(81\)In every case𝔼0πk\[eξt∣ℋt−1,At\]=1\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[e^\{\\xi\_\{t\}\}\\mid\\mathcal\{H\}\_\{t\-1\},A\_\{t\}\]=1\. Averaging over theℋt−1\\mathcal\{H\}\_\{t\-1\}\-conditional law ofAtA\_\{t\}gives𝔼0πk​\[eξt∣ℋt−1\]=1\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[e^\{\\xi\_\{t\}\}\\mid\\mathcal\{H\}\_\{t\-1\}\]=1, and sinceΛt−1\\Lambda\_\{t\-1\}isℋt−1\\mathcal\{H\}\_\{t\-1\}\-measurable,

𝔼0πk​\[Λt∣ℋt−1\]=Λt−1​𝔼0πk​\[eξt∣ℋt−1\]=Λt−1\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\\\!\\left\[\\Lambda\_\{t\}\\mid\\mathcal\{H\}\_\{t\-1\}\\right\]=\\Lambda\_\{t\-1\}\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\\\!\\left\[e^\{\\xi\_\{t\}\}\\mid\\mathcal\{H\}\_\{t\-1\}\\right\]=\\Lambda\_\{t\-1\}\.\(82\)Hence\(Λt\)t≥0\(\\Lambda\_\{t\}\)\_\{t\\geq 0\}is a nonnegative\(ℋt\)\(\\mathcal\{H\}\_\{t\}\)\-martingale underH0H\_\{0\}with𝔼0πk​\[Λt\]=Λ0=1\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\Lambda\_\{t\}\]=\\Lambda\_\{0\}=1\.

Letτb\\tau\_\{\\mathrm\{b\}\}be the index of the last paid update in the main stage: the update at which a boundary is first crossed, or, if no crossing occurs, the last paid update before entering fallback\. Thenτb≤2​Nmain,k\\tau\_\{\\mathrm\{b\}\}\\leq 2N\_\{\\mathrm\{main\},k\}\. The optional stopping theorem gives

𝔼0πk​\[Λτb\]=𝔼0πk​\[Λ0\]=1\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\Lambda\_\{\\tau\_\{\\mathrm\{b\}\}\}\]=\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\Lambda\_\{0\}\]=1\.OnE\+E\_\{\+\},Sτb≥akS\_\{\\tau\_\{\\mathrm\{b\}\}\}\\geq a\_\{k\}, henceΛτb≥eak=1/α1,k\\Lambda\_\{\\tau\_\{\\mathrm\{b\}\}\}\\geq e^\{a\_\{k\}\}=1/\\alpha\_\{1,k\}\. Thus

ℙ0πk​\(E\+\)=𝔼0πk​\[𝟏E\+\]≤α1,k​𝔼0πk​\[Λτb​𝟏E\+\]≤α1,k​𝔼0πk​\[Λτb\]=α1,k​𝔼0πk​\[Λ0\]≤α1,k\.\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\(E\_\{\+\}\)=\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\mathbf\{1\}\_\{E\_\{\+\}\}\]\\leq\\alpha\_\{1,k\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\Lambda\_\{\\tau\_\{\\mathrm\{b\}\}\}\\mathbf\{1\}\_\{E\_\{\+\}\}\]\\leq\\alpha\_\{1,k\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\Lambda\_\{\\tau\_\{\\mathrm\{b\}\}\}\]=\\alpha\_\{1,k\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\Lambda\_\{0\}\]\\leq\\alpha\_\{1,k\}\.
The lower\-boundary statement follows from the identical argument applied underH1H\_\{1\}toΛt−1:=exp⁡\(−St\)\\Lambda\_\{t\}^\{\-1\}:=\\exp\(\-S\_\{t\}\), with the roles of\(p0,p1\)\(p\_\{0\},p\_\{1\}\),\(g0,g1\)\(g\_\{0\},g\_\{1\}\), and\(ρ0\(⋅∣r\),ρ1\(⋅∣r\)\)\(\\rho\_\{0\}\(\\cdot\\mid r\),\\rho\_\{1\}\(\\cdot\\mid r\)\)interchanged in each of the three cases above: the same computation shows𝔼1πk\[e−ξt∣ℋt−1,At\]=1\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[e^\{\-\\xi\_\{t\}\}\\mid\\mathcal\{H\}\_\{t\-1\},A\_\{t\}\]=1in every case, so\(Λt−1\)t≥0\(\\Lambda\_\{t\}^\{\-1\}\)\_\{t\\geq 0\}is a nonnegative\(ℋt\)\(\\mathcal\{H\}\_\{t\}\)\-martingale underH1H\_\{1\}with𝔼1πk​\[Λt−1\]=Λ0−1=1\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\Lambda\_\{t\}^\{\-1\}\]=\\Lambda\_\{0\}^\{\-1\}=1\. Bounded optional stopping applied toτb\\tau\_\{\\mathrm\{b\}\}underH1H\_\{1\}gives𝔼1πk​\[Λτb−1\]=1\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\Lambda\_\{\\tau\_\{\\mathrm\{b\}\}\}^\{\-1\}\]=1\. OnE−E\_\{\-\},Sτb≤−bkS\_\{\\tau\_\{\\mathrm\{b\}\}\}\\leq\-b\_\{k\}, henceΛτb−1≥ebk=1/β1,k\\Lambda\_\{\\tau\_\{\\mathrm\{b\}\}\}^\{\-1\}\\geq e^\{b\_\{k\}\}=1/\\beta\_\{1,k\}, and

ℙ1πk​\(E−\)≤β1,k​𝔼1πk​\[Λτb−1​𝟏E−\]≤β1,k​𝔼1πk​\[Λτb−1\]=β1,k\.\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(E\_\{\-\}\)\\leq\\beta\_\{1,k\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\Lambda\_\{\\tau\_\{\\mathrm\{b\}\}\}^\{\-1\}\\mathbf\{1\}\_\{E\_\{\-\}\}\]\\leq\\beta\_\{1,k\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\Lambda\_\{\\tau\_\{\\mathrm\{b\}\}\}^\{\-1\}\]=\\beta\_\{1,k\}\.

LetEfbE\_\{\\mathrm\{fb\}\}be the event that no boundary is crossed before the fixed pool is exhausted\. Lemma[9\.3](https://arxiv.org/html/2609.28859#S9.Thmtheorem3)bounds the fallback probabilities\.

###### Lemma 9\.3\(Fallback\-route error control\)

ℙ0πk​\(Efb∩\{fallback rejects​H0\}\)≤α2,k,ℙ1πk​\(Efb∩\{fallback accepts​H0\}\)≤β2,k\.\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\(E\_\{\\mathrm\{fb\}\}\\cap\\\{\\text\{fallback rejects \}H\_\{0\}\\\}\)\\leq\\alpha\_\{2,k\},\\qquad\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(E\_\{\\mathrm\{fb\}\}\\cap\\\{\\text\{fallback accepts \}H\_\{0\}\\\}\)\\leq\\beta\_\{2,k\}\.

###### Proof 9\.4

Proof of Lemma[9\.3](https://arxiv.org/html/2609.28859#S9.Thmtheorem3)

There are two cases, according to the deterministic value of𝖥k\\mathsf\{F\}\_\{k\}\.

If𝖥k=H\\mathsf\{F\}\_\{k\}=\\mathrm\{H\}, writeNH:=Nfixed,H​\(α2,k,β2,k\)N\_\{H\}:=N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\. The fallback test is the full\-label testϕNH∗\\phi^\{\*\}\_\{N\_\{H\}\}on the fixed index set𝒥H,k\\mathcal\{J\}\_\{\\mathrm\{H\},k\}\. By \([31](https://arxiv.org/html/2609.28859#S5.E31)\), its unconditional type\-I error is at mostα2,k\\alpha\_\{2,k\}\. Then

ℙ0πk​\(Efb∩\{fallback rejects\}\)≤ℙ0πk​\(\{fallback rejects\}\)=𝔼0πk​\[ϕNH∗\]≤α2,k\.\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\(E\_\{\\mathrm\{fb\}\}\\cap\\\{\\text\{fallback rejects\}\\\}\)\\leq\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\(\\\{\\text\{fallback rejects\}\\\}\)=\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\phi^\{\*\}\_\{N\_\{H\}\}\]\\leq\\alpha\_\{2,k\}\.The type\-II statement follows the same way:

ℙ1πk​\(Efb∩\{fallback accepts\}\)≤ℙ1πk​\(\{fallback accepts\}\)=𝔼1πk​\[1−ϕNH∗\]≤β2,k\.\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(E\_\{\\mathrm\{fb\}\}\\cap\\\{\\text\{fallback accepts\}\\\}\)\\leq\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(\\\{\\text\{fallback accepts\}\\\}\)=\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[1\-\\phi^\{\*\}\_\{N\_\{H\}\}\]\\leq\\beta\_\{2,k\}\.
If𝖥k=AI\\mathsf\{F\}\_\{k\}=\\mathrm\{AI\}, write𝒥AI,k:=\{1,…,Nfixed,AI​\(α2,k,β2,k\)\}\\mathcal\{J\}\_\{\\mathrm\{AI\},k\}:=\\\{1,\\ldots,N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\\\}, so the fallback test is the report testψNfixed,AI​\(α2,k,β2,k\)∗\\psi^\{\*\}\_\{N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\}applied to the fixed reports\{Ri:i∈𝒥AI,k\}\\\{R\_\{i\}:i\\in\\mathcal\{J\}\_\{\\mathrm\{AI\},k\}\\\}\. Then \([32](https://arxiv.org/html/2609.28859#S5.E32)\) and𝟏Efb≤1\\mathbf\{1\}\_\{E\_\{\\mathrm\{fb\}\}\}\\leq 1, gives

ℙ0πk​\(Efb∩\{fallback rejects\}\)≤𝔼0πk​\[ψNfixed,AI​\(α2,k,β2,k\)∗\]≤α2,k,\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\(E\_\{\\mathrm\{fb\}\}\\cap\\\{\\text\{fallback rejects\}\\\}\)\\leq\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\psi^\{\*\}\_\{N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\}\]\\leq\\alpha\_\{2,k\},and

ℙ1πk​\(Efb∩\{fallback accepts\}\)≤𝔼1πk​\[1−ψNfixed,AI​\(α2,k,β2,k\)∗\]≤β2,k\.\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(E\_\{\\mathrm\{fb\}\}\\cap\\\{\\text\{fallback accepts\}\\\}\)\\leq\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[1\-\\psi^\{\*\}\_\{N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\}\]\\leq\\beta\_\{2,k\}\.

###### Proof 9\.5

Proof of Theorem[5\.2](https://arxiv.org/html/2609.28859#S5.Thmtheorem2)By construction, Algorithm[1](https://arxiv.org/html/2609.28859#alg1)terminates through exactly one of three mutually exclusive routes: it crosses the upper boundary \(eventE\+E\_\{\+\}\), crosses the lower boundary \(eventE−E\_\{\-\}\), or reaches the end of the fixed pool without crossing either boundary and executes the pre\-committed fallback \(eventEfbE\_\{\\mathrm\{fb\}\}\); in particularE\+,E−,EfbE\_\{\+\},E\_\{\-\},E\_\{\\mathrm\{fb\}\}are pairwise disjoint\.

Consequently the event that the policy rejectsH0H\_\{0\}is the disjoint union of the upper boundary eventE\+E\_\{\+\}and the fallback\-rejection eventEfb∩\{fallback rejects\}⊆EfbE\_\{\\mathrm\{fb\}\}\\cap\\\{\\text\{fallback rejects\}\\\}\\subseteq E\_\{\\mathrm\{fb\}\}, so by countable additivity of probability over disjoint events,

ℙ0πk​\(δπk=1\)=ℙ0πk​\(E\+\)\+ℙ0πk​\(Efb∩\{fallback rejects\}\)\.\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\(\\delta^\{\\pi\_\{k\}\}=1\)=\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\(E\_\{\+\}\)\+\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\\bigl\(E\_\{\\mathrm\{fb\}\}\\cap\\\{\\text\{fallback rejects\}\\\}\\bigr\)\.By Lemmas[9\.1](https://arxiv.org/html/2609.28859#S9.Thmtheorem1)and[9\.3](https://arxiv.org/html/2609.28859#S9.Thmtheorem3), the two terms on the right are at mostα1,k\\alpha\_\{1,k\}andα2,k\\alpha\_\{2,k\}respectively, so

ℙ0πk​\(δπk=1\)≤α1,k\+α2,k=αk,\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\(\\delta^\{\\pi\_\{k\}\}=1\)\\leq\\alpha\_\{1,k\}\+\\alpha\_\{2,k\}=\\alpha\_\{k\},where the equality is the budget identity following \([23](https://arxiv.org/html/2609.28859#S5.E23)\)\.

Likewise, the event that the policy acceptsH0H\_\{0\}underH1H\_\{1\}is the disjoint union of the lower boundary eventE−E\_\{\-\}and the fallback\-acceptance eventEfb∩\{fallback accepts\}E\_\{\\mathrm\{fb\}\}\\cap\\\{\\text\{fallback accepts\}\\\}, so

ℙ1πk​\(δπk=0\)=ℙ1πk​\(E−\)\+ℙ1πk​\(Efb∩\{fallback accepts\}\)\.\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(\\delta^\{\\pi\_\{k\}\}=0\)=\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(E\_\{\-\}\)\+\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\\bigl\(E\_\{\\mathrm\{fb\}\}\\cap\\\{\\text\{fallback accepts\}\\\}\\bigr\)\.The same two lemmas and \([24](https://arxiv.org/html/2609.28859#S5.E24)\) give

ℙ1πk​\(δπk=0\)≤β1,k\+β2,k=βk\.\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(\\delta^\{\\pi\_\{k\}\}=0\)\\leq\\beta\_\{1,k\}\+\\beta\_\{2,k\}=\\beta\_\{k\}\.

## 10Additional Materials for Sections[5\.3](https://arxiv.org/html/2609.28859#S5.SS3)

Throughout, we letT1,k′:=kl\(1−βk\|\|αk\)T^\{\\prime\}\_\{1,k\}:=\\operatorname\{kl\}\(1\-\\beta\_\{k\}\|\|\\alpha\_\{k\}\),T0,k′:=kl\(1−αk\|\|βk\)T^\{\\prime\}\_\{0,k\}:=\\operatorname\{kl\}\(1\-\\alpha\_\{k\}\|\|\\beta\_\{k\}\)for the information thresholds\. Recall thatT1,k=ak\+ΔkT\_\{1,k\}=a\_\{k\}\+\\Delta\_\{k\}andT0,k=bk\+ΔkT\_\{0,k\}=b\_\{k\}\+\\Delta\_\{k\}whereak=log⁡\(1/\(\(1−ffb,k\)​αk\)\)a\_\{k\}=\\log\(1/\(\(1\-f\_\{\\mathrm\{fb\},k\}\)\\alpha\_\{k\}\)\)andbk=log⁡\(1/\(\(1−ffb,k\)​βk\)\)b\_\{k\}=\\log\(1/\(\(1\-f\_\{\\mathrm\{fb\},k\}\)\\beta\_\{k\}\)\)\.

### 10\.1Proof of Theorem[5\.3](https://arxiv.org/html/2609.28859#S5.Thmtheorem3)

To prove Theorem[5\.3](https://arxiv.org/html/2609.28859#S5.Thmtheorem3)\(i\), we first state an auxiliary Lemma[10\.1](https://arxiv.org/html/2609.28859#S10.Thmtheorem1)together with its proof\.

###### Lemma 10\.1\(Binary\-KL thresholds are logarithmic\)

Under Assumption[5\.3](https://arxiv.org/html/2609.28859#S5.SS3), ask→∞k\\to\\infty,

T1,k′=log⁡1αk\+o⁡\(1\),T0,k′=log⁡1βk\+o⁡\(1\)\.T^\{\\prime\}\_\{1,k\}=\\log\\frac\{1\}\{\\alpha\_\{k\}\}\+o\(1\),\\qquad T^\{\\prime\}\_\{0,k\}=\\log\\frac\{1\}\{\\beta\_\{k\}\}\+o\(1\)\.Consequently, sinceak=log⁡\(1/α1,k\)=log⁡\(1/αk\)\+log⁡\(1/\(1−ffb,k\)\)a\_\{k\}=\\log\(1/\\alpha\_\{1,k\}\)=\\log\(1/\\alpha\_\{k\}\)\+\\log\(1/\(1\-f\_\{\\mathrm\{fb\},k\}\)\)andbk=log⁡\(1/β1,k\)=log⁡\(1/βk\)\+log⁡\(1/\(1−ffb,k\)\)b\_\{k\}=\\log\(1/\\beta\_\{1,k\}\)=\\log\(1/\\beta\_\{k\}\)\+\\log\(1/\(1\-f\_\{\\mathrm\{fb\},k\}\)\)by \([23](https://arxiv.org/html/2609.28859#S5.E23)\)–\([24](https://arxiv.org/html/2609.28859#S5.E24)\),

ak=T1,k′\+log⁡\(1/\(1−ffb,k\)\)\+o⁡\(1\),bk=T0,k′\+log⁡\(1/\(1−ffb,k\)\)\+o⁡\(1\)\.a\_\{k\}=T^\{\\prime\}\_\{1,k\}\+\\log\(1/\(1\-f\_\{\\mathrm\{fb\},k\}\)\)\+o\(1\),\\qquad b\_\{k\}=T^\{\\prime\}\_\{0,k\}\+\\log\(1/\(1\-f\_\{\\mathrm\{fb\},k\}\)\)\+o\(1\)\.

###### Proof 10\.2

Proof of Lemma[10\.1](https://arxiv.org/html/2609.28859#S10.Thmtheorem1)We prove the expansion forT1,k′T^\{\\prime\}\_\{1,k\}; the proof forT0,k′T^\{\\prime\}\_\{0,k\}is symmetric\. By the definition of Bernoulli KL divergence,

T1,k′\\displaystyle T^\{\\prime\}\_\{1,k\}=\(1−βk\)​log⁡1−βkαk\+βk​log⁡βk1−αk\\displaystyle=\(1\-\\beta\_\{k\}\)\\log\\frac\{1\-\\beta\_\{k\}\}\{\\alpha\_\{k\}\}\+\\beta\_\{k\}\\log\\frac\{\\beta\_\{k\}\}\{1\-\\alpha\_\{k\}\}=log⁡1αk−βk​log⁡1αk\+\(1−βk\)​log⁡\(1−βk\)\+βk​log⁡βk−βk​log⁡\(1−αk\)\.\\displaystyle=\\log\\frac\{1\}\{\\alpha\_\{k\}\}\-\\beta\_\{k\}\\log\\frac\{1\}\{\\alpha\_\{k\}\}\+\(1\-\\beta\_\{k\}\)\\log\(1\-\\beta\_\{k\}\)\+\\beta\_\{k\}\\log\\beta\_\{k\}\-\\beta\_\{k\}\\log\(1\-\\alpha\_\{k\}\)\.\(83\)Every correction term is bounded in absolute value by a primitive constant multiple ofβk​Lk\\beta\_\{k\}L\_\{k\}: the first directly; the second because\(1−βk\)​log⁡\(1−βk\)=O⁡\(βk\)\(1\-\\beta\_\{k\}\)\\log\(1\-\\beta\_\{k\}\)=O\(\\beta\_\{k\}\); the third because\|βk​log⁡βk\|≤βk​Lk\|\\beta\_\{k\}\\log\\beta\_\{k\}\|\\leq\\beta\_\{k\}L\_\{k\}; and the fourth because−βk​log⁡\(1−αk\)=O⁡\(αk​βk\)\-\\beta\_\{k\}\\log\(1\-\\alpha\_\{k\}\)=O\(\\alpha\_\{k\}\\beta\_\{k\}\)\. Under Assumption[5\.3](https://arxiv.org/html/2609.28859#S5.SS3),

min⁡\{log⁡1αk,log⁡1βk\}=Θ⁡\(Lk\),\\min\\\!\\left\\\{\\log\\frac\{1\}\{\\alpha\_\{k\}\},\\log\\frac\{1\}\{\\beta\_\{k\}\}\\right\\\}=\\Theta\(L\_\{k\}\),soβk≤e−Θ⁡\(Lk\)\\beta\_\{k\}\\leq e^\{\-\\Theta\(L\_\{k\}\)\}and thereforeβk​Lk→0\\beta\_\{k\}L\_\{k\}\\to 0\. It follows thatT1,k′=log⁡\(1/αk\)\+o⁡\(1\)T^\{\\prime\}\_\{1,k\}=\\log\(1/\\alpha\_\{k\}\)\+o\(1\)\. Interchangingαk\\alpha\_\{k\}andβk\\beta\_\{k\}givesT0,k′=log⁡\(1/βk\)\+o⁡\(1\)T^\{\\prime\}\_\{0,k\}=\\log\(1/\\beta\_\{k\}\)\+o\(1\)\. The final two relations follow straightforwardly\.

###### Proof 10\.3

Proof of Theorem[5\.3](https://arxiv.org/html/2609.28859#S5.Thmtheorem3)

Part \(i\)\. We first show thatLBk=O⁡\(Lk\)\\mathrm\{LB\}\_\{k\}=O\(L\_\{k\}\)and thenLBk=Ω⁡\(Lk\)\\mathrm\{LB\}\_\{k\}=\\Omega\(L\_\{k\}\)\.

*Step 1:LBk=O⁡\(Lk\)\\mathrm\{LB\}\_\{k\}=O\(L\_\{k\}\)\.*By definition ofLBk\\mathrm\{LB\}\_\{k\}, we have that

LBk\\displaystyle\\mathrm\{LB\}\_\{k\}=minN≥Nfixed,H​\(αk,βk\)⁡N​cdata\+max⁡\{Γ1​\(T1,k′,N\),Γ0​\(T0,k′,N\)\}\.\\displaystyle=\\min\_\{N\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)\}Nc\_\{\\mathrm\{data\}\}\+\\max\\\{\\Gamma\_\{1\}\(T^\{\\prime\}\_\{1,k\},N\),\\Gamma\_\{0\}\(T^\{\\prime\}\_\{0,k\},N\)\\\}\.To showLBk=O⁡\(Lk\)\\mathrm\{LB\}\_\{k\}=O\(L\_\{k\}\), it suffices to find an integer solutionNk1N^\{1\}\_\{k\}to \([22](https://arxiv.org/html/2609.28859#S4.E22)\) that satisfiesNk1≥Nfixed,H​\(αk,βk\)N^\{1\}\_\{k\}\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)and thatNk1​cdata\+max⁡\{Γ1​\(T1,k′,Nk1\),Γ0​\(T0,k′,Nk1\)\}=O⁡\(Lk\)N^\{1\}\_\{k\}c\_\{\\mathrm\{data\}\}\+\\max\\\{\\Gamma\_\{1\}\(T^\{\\prime\}\_\{1,k\},N^\{1\}\_\{k\}\),\\Gamma\_\{0\}\(T^\{\\prime\}\_\{0,k\},N^\{1\}\_\{k\}\)\\\}=O\(L\_\{k\}\)\.

*Bounding the floor\.*We first showNfixed,H​\(αk,βk\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)isO⁡\(Lk\)O\(L\_\{k\}\)by exhibiting a cheap full\-label test\. Consider the sample\-mean test that rejectsH0H\_\{0\}iffX¯N≥p¯\\bar\{X\}\_\{N\}\\geq\\bar\{p\}, whereX¯N:=N−1​∑i=1NXi\\bar\{X\}\_\{N\}:=N^\{\-1\}\\sum\_\{i=1\}^\{N\}X\_\{i\}andp¯:=\(p0\+p1\)/2\\bar\{p\}:=\(p\_\{0\}\+p\_\{1\}\)/2\. UnderHhH\_\{h\}the labels are i\.i\.d\.Bernoulli⁡\(ph\)\\operatorname\{Bernoulli\}\(p\_\{h\}\)by Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1), with meanp0<p¯p\_\{0\}<\\bar\{p\}underH0H\_\{0\}andp1\>p¯p\_\{1\}\>\\bar\{p\}underH1H\_\{1\}\. WritingcHfd:=\(\(p1−p0\)/2\)2\>0c\_\{\\mathrm\{Hfd\}\}:=\(\(p\_\{1\}\-p\_\{0\}\)/2\)^\{2\}\>0, Hoeffding’s inequality gives

ℙ0πk​\(X¯N≥p¯\)≤e−2​N​cHfd,ℙ1πk​\(X¯N<p¯\)≤e−2​N​cHfd\.\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\(\\bar\{X\}\_\{N\}\\geq\\bar\{p\}\)\\leq e^\{\-2Nc\_\{\\mathrm\{Hfd\}\}\},\\qquad\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(\\bar\{X\}\_\{N\}<\\bar\{p\}\)\\leq e^\{\-2Nc\_\{\\mathrm\{Hfd\}\}\}\.AtNk0:=⌈Lk/\(2​cHfd\)⌉N\_\{k\}^\{0\}:=\\lceil L\_\{k\}/\(2c\_\{\\mathrm\{Hfd\}\}\)\\rceil, we haveℙ0πk​\(X¯Nk0≥p¯\)≤αk\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{0\}\(\\bar\{X\}\_\{N^\{0\}\_\{k\}\}\\geq\\bar\{p\}\)\\leq\\alpha\_\{k\}andℙ1πk​\(X¯Nk0<p¯\)≤βk\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(\\bar\{X\}\_\{N^\{0\}\_\{k\}\}<\\bar\{p\}\)\\leq\\beta\_\{k\}\. It follows by Definition[4\.1](https://arxiv.org/html/2609.28859#S4.Thmtheorem1)thatNfixed,H​\(αk,βk\)≤Nk0=O⁡\(Lk\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)\\leq N\_\{k\}^\{0\}=O\(L\_\{k\}\)\.

*Choosing the evaluation point\.*We now choose the sample sizeNk1N^\{1\}\_\{k\}as:

Nk1:=max⁡\{Nk0,⌈T1,k′/JX\(1\)⌉,⌈T0,k′/JX\(0\)⌉\}\.N\_\{k\}^\{1\}:=\\max\\\!\\left\\\{N^\{0\}\_\{k\},\\lceil T^\{\\prime\}\_\{1,k\}/J\_\{X\}^\{\(1\)\}\\rceil,\\lceil T^\{\\prime\}\_\{0,k\}/J\_\{X\}^\{\(0\)\}\\rceil\\right\\\}\.In particular,Nk1≥Nk0N\_\{k\}^\{1\}\\geq N^\{0\}\_\{k\}ensures thatNk1≥Nfixed,H​\(αk,βk\)N^\{1\}\_\{k\}\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\);Nk1≥⌈T1,k′/JX\(1\)⌉N\_\{k\}^\{1\}\\geq\\lceil T^\{\\prime\}\_\{1,k\}/J\_\{X\}^\{\(1\)\}\\rceilandNk1≥⌈T0,k′/JX\(0\)⌉N\_\{k\}^\{1\}\\geq\\lceil T^\{\\prime\}\_\{0,k\}/J\_\{X\}^\{\(0\)\}\\rceilensure thatNk1N^\{1\}\_\{k\}direct\-human queries supplyT1,k′T^\{\\prime\}\_\{1,k\}andT0,k′T^\{\\prime\}\_\{0,k\}units of information under directionh=1h=1andh=0h=0, respectively, since each direct\-human query supplyJX\(h\)J\_\{X\}^\{\(h\)\}information\.

*Bounding the objective atNk1N\_\{k\}^\{1\}\.*We now show thatNk1​cdata\+max⁡\{Γ1​\(T1,k′,Nk1\),Γ0​\(T0,k′,Nk1\)\}=O⁡\(Lk\)N^\{1\}\_\{k\}c\_\{\\mathrm\{data\}\}\+\\max\\\{\\Gamma\_\{1\}\(T^\{\\prime\}\_\{1,k\},N^\{1\}\_\{k\}\),\\Gamma\_\{0\}\(T^\{\\prime\}\_\{0,k\},N^\{1\}\_\{k\}\)\\\}=O\(L\_\{k\}\)\. First noticeNk1=O⁡\(Lk\)N\_\{k\}^\{1\}=O\(L\_\{k\}\):Nk0=O⁡\(Lk\)N\_\{k\}^\{0\}=O\(L\_\{k\}\), andT1,k′,T0,k′=O⁡\(Lk\)T^\{\\prime\}\_\{1,k\},T^\{\\prime\}\_\{0,k\}=O\(L\_\{k\}\)by Lemma[10\.1](https://arxiv.org/html/2609.28859#S10.Thmtheorem1)withJX\(0\),JX\(1\)J\_\{X\}^\{\(0\)\},J\_\{X\}^\{\(1\)\}constant\.

Additionally, for each directionhh, the all\-direct\-human triple\(nH,nAI,nesc\):=\(⌈Th,k′/JX\(h\)⌉,0,0\)\(n\_\{\\mathrm\{H\}\},n\_\{\\mathrm\{AI\}\},n\_\{\\mathrm\{esc\}\}\):=\(\\lceil T^\{\\prime\}\_\{h,k\}/J\_\{X\}^\{\(h\)\}\\rceil,0,0\)is feasible for \([21](https://arxiv.org/html/2609.28859#S4.E21)\) atT=Th,k′,N=Nk1T=T^\{\\prime\}\_\{h,k\},N=N\_\{k\}^\{1\}: its item count⌈Th,k′/JX\(h\)⌉≤Nk1\\lceil T^\{\\prime\}\_\{h,k\}/J\_\{X\}^\{\(h\)\}\\rceil\\leq N\_\{k\}^\{1\}satisfies the capacity constraint by the choice ofNk1N\_\{k\}^\{1\}; its escalation term is zero by thenAI=0n\_\{\\mathrm\{AI\}\}=0convention; and its information⌈Th,k′/JX\(h\)⌉​JX\(h\)≥Th,k′\\lceil T^\{\\prime\}\_\{h,k\}/J\_\{X\}^\{\(h\)\}\\rceil\\,J\_\{X\}^\{\(h\)\}\\geq T^\{\\prime\}\_\{h,k\}meets the information constraint\. Its objective value iscH​⌈Th,k′/JX\(h\)⌉=O⁡\(Lk\)c\_\{\\mathrm\{H\}\}\\lceil T^\{\\prime\}\_\{h,k\}/J\_\{X\}^\{\(h\)\}\\rceil=O\(L\_\{k\}\), soΓh​\(Th,k′,Nk1\)=O⁡\(Lk\)\\Gamma\_\{h\}\(T^\{\\prime\}\_\{h,k\},N\_\{k\}^\{1\}\)=O\(L\_\{k\}\)for eachhh\. Therefore

LBk≤Nk1​cdata\+max⁡\{Γ1​\(T1,k′,Nk1\),Γ0​\(T0,k′,Nk1\)\}=O⁡\(Lk\)\.\\mathrm\{LB\}\_\{k\}\\leq N\_\{k\}^\{1\}c\_\{\\mathrm\{data\}\}\+\\max\\\{\\Gamma\_\{1\}\(T^\{\\prime\}\_\{1,k\},N\_\{k\}^\{1\}\),\\Gamma\_\{0\}\(T^\{\\prime\}\_\{0,k\},N\_\{k\}^\{1\}\)\\\}=O\(L\_\{k\}\)\.
*Step 2:LBk=Ω⁡\(Lk\)\\mathrm\{LB\}\_\{k\}=\\Omega\(L\_\{k\}\)\.*It suffices to showNfixed,H​\(αk,βk\)=Ω⁡\(Lk\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)=\\Omega\(L\_\{k\}\), and the argument follows by noting that any feasible solutionNNto \([22](https://arxiv.org/html/2609.28859#S4.E22)\) must satisfyN≥Nfixed,H​\(αk,βk\)N\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)and therefore must have costLBk≥Nfixed,H​\(αk,βk\)​cdata=Ω⁡\(Lk\)\\mathrm\{LB\}\_\{k\}\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)c\_\{\\mathrm\{data\}\}=\\Omega\(L\_\{k\}\)\. Letπf\\pi\_\{f\}be a feasible full\-label test usingNf:=Nfixed,H​\(αk,βk\)N\_\{f\}:=N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)labels \(for example we can letπf\\pi\_\{f\}be the Neyman\-Pearson testϕNf∗\\phi^\{\*\}\_\{N\_\{f\}\}; see the details in Section[4\.1](https://arxiv.org/html/2609.28859#S4.SS1)\); its transcript is the label vector\(X1,…,XNf\)\(X\_\{1\},\\ldots,X\_\{N\_\{f\}\}\)with no AI queries, and underHhH\_\{h\}these labels are i\.i\.d\.Bernoulli⁡\(ph\)\\operatorname\{Bernoulli\}\(p\_\{h\}\)by Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)\. Applying the finite chain rule \(Lemma[8\.4](https://arxiv.org/html/2609.28859#S8.Thmtheorem4)\) inductively across theNfN\_\{f\}independent coordinates, each conditional term reduces to the per\-label divergencekl\(p1∥p0\)\\operatorname\{kl\}\(p\_\{1\}\\\|p\_\{0\}\), so

D\(P1πf∥P0πf\)=Nfkl\(p1∥p0\)=NfJX\(1\),D\(P\_\{1\}^\{\\pi\_\{f\}\}\\\|P\_\{0\}^\{\\pi\_\{f\}\}\)=N\_\{f\}\\,\\operatorname\{kl\}\(p\_\{1\}\\\|p\_\{0\}\)=N\_\{f\}J\_\{X\}^\{\(1\)\},the last equality by the definition \([10](https://arxiv.org/html/2609.28859#S4.E10)\) ofJX\(1\)J\_\{X\}^\{\(1\)\}\. Sinceπf\\pi\_\{f\}is feasible, Lemma[4\.5](https://arxiv.org/html/2609.28859#S4.Thmtheorem5)givesD\(P1πf∥P0πf\)≥T1,k′D\(P\_\{1\}^\{\\pi\_\{f\}\}\\\|P\_\{0\}^\{\\pi\_\{f\}\}\)\\geq T^\{\\prime\}\_\{1,k\}, soNf≥T1,k′/JX\(1\)N\_\{f\}\\geq T^\{\\prime\}\_\{1,k\}/J\_\{X\}^\{\(1\)\}\. By Lemma[10\.1](https://arxiv.org/html/2609.28859#S10.Thmtheorem1),T1,k′=log⁡\(1/αk\)\+o⁡\(1\)T^\{\\prime\}\_\{1,k\}=\\log\(1/\\alpha\_\{k\}\)\+o\(1\), henceNf=Ω⁡\(log⁡\(1/αk\)\)N\_\{f\}=\\Omega\(\\log\(1/\\alpha\_\{k\}\)\)\. The same argument in the reverse direction, withD\(P0πf∥P1πf\)=NfJX\(0\)≥T0,k′D\(P\_\{0\}^\{\\pi\_\{f\}\}\\\|P\_\{1\}^\{\\pi\_\{f\}\}\)=N\_\{f\}J\_\{X\}^\{\(0\)\}\\geq T^\{\\prime\}\_\{0,k\}andT0,k′=log⁡\(1/βk\)\+o⁡\(1\)T^\{\\prime\}\_\{0,k\}=\\log\(1/\\beta\_\{k\}\)\+o\(1\), and thusNf=Ω⁡\(log⁡\(1/βk\)\)N\_\{f\}=\\Omega\(\\log\(1/\\beta\_\{k\}\)\)\. Taking the larger,Nf=Ω⁡\(max⁡\{log⁡\(1/αk\),log⁡\(1/βk\)\}\)=Ω⁡\(Lk\)N\_\{f\}=\\Omega\(\\max\\\{\\log\(1/\\alpha\_\{k\}\),\\log\(1/\\beta\_\{k\}\)\\\}\)=\\Omega\(L\_\{k\}\)\. Combining Step 1 and Step 2 givesLBk=Θ⁡\(Lk\)\\mathrm\{LB\}\_\{k\}=\\Theta\(L\_\{k\}\)\.

Part \(ii\)\. Fix a sufficiently largekk\. UnderHhH\_\{h\}, the cost decomposes as

𝔼hπk​\[Cπk\]=cdata​Nmain,k\+𝔼hπk​\[Ckmain\]\+𝔼hπk​\[Ckfb\],\\mathbb\{E\}\_\{h\}^\{\\pi\_\{k\}\}\[C^\{\\pi\_\{k\}\}\]=c\_\{\\mathrm\{data\}\}N\_\{\\mathrm\{main\},k\}\+\\mathbb\{E\}\_\{h\}^\{\\pi\_\{k\}\}\[C\_\{k\}^\{\\mathrm\{main\}\}\]\+\\mathbb\{E\}\_\{h\}^\{\\pi\_\{k\}\}\[C\_\{k\}^\{\\mathrm\{fb\}\}\],whereCkmainC\_\{k\}^\{\\mathrm\{main\}\}is the sensing cost incurred during the sequential main stage andCkfbC\_\{k\}^\{\\mathrm\{fb\}\}is the additional cost of the pre\-committed fallback, equal to zero when the main stage stops at a boundary\.

We first control how often the policy uses a rule other than the one corresponding to the true hypothesis\. The following lemma bounds the expected number of items processed with the wrong\-direction or dead\-zone rule\.

###### Lemma 10\.4\(Direction tracking\)

LetW1,kW\_\{1,k\}be the number of sensed items processed whileSi−1≤zkS\_\{i\-1\}\\leq z\_\{k\}underH1H\_\{1\}, before the policy stops or exhausts the pool\. LetW0,kW\_\{0,k\}be the number of sensed items processed whileSi−1≥−zkS\_\{i\-1\}\\geq\-z\_\{k\}underH0H\_\{0\}\. Then there is a constantCW<∞C\_\{W\}<\\inftysuch that

𝔼1πk​\[W1,k\]\+𝔼0πk​\[W0,k\]≤CW​\(zk\+1\)\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[W\_\{1,k\}\]\+\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[W\_\{0,k\}\]\\leq C\_\{W\}\(z\_\{k\}\+1\)\.\(84\)

The proof of Lemma[10\.4](https://arxiv.org/html/2609.28859#S10.Thmtheorem4)is given in Appendix[10\.3](https://arxiv.org/html/2609.28859#S10.SS3)\. Since the sensing cost of each item is bounded by a primitive constant, this result limits the expected cost of wrong\-direction and dead\-zone items toO⁡\(zk\+1\)O\(z\_\{k\}\+1\)\.

We next bound the main\-stage cost\. For items processed with the correct direction rule, the planned sensing cost and information yield are linked by the optimizer ofΓh\\Gamma\_\{h\}\. The bounded likelihood\-ratio overshoot then bounds their expected total cost byΓh​\(Th,k,N¯main,k\)\+O⁡\(1\)\\Gamma\_\{h\}\(T\_\{h,k\},\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\+O\(1\)\. Combining this bound with Lemma[10\.4](https://arxiv.org/html/2609.28859#S10.Thmtheorem4)gives the following result\.

###### Lemma 10\.5\(Main\-stage sensing cost\)

There is a constantCM<∞C\_\{M\}<\\inftysuch that, forh=0,1h=0,1,

𝔼hπk​\[Ckmain\]≤Γh​\(Th,k,N¯main,k\)\+CM​\(zk\+1\)\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[C^\{\\mathrm\{main\}\}\_\{k\}\]\\leq\\Gamma\_\{h\}\(T\_\{h,k\},\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\+C\_\{M\}\(z\_\{k\}\+1\)\.\(85\)

The proof of Lemma[10\.5](https://arxiv.org/html/2609.28859#S10.Thmtheorem5)is given in Appendix[10\.4](https://arxiv.org/html/2609.28859#S10.SS4)\.

To control the expected fallback cost, we next bound the probability that the main stage exhausts the pool without crossing a boundary\.

###### Lemma 10\.6\(Fallback probability\)

LetEfbE\_\{\\mathrm\{fb\}\}be the event that the fixed pool is exhausted before either boundary is crossed\. There are constantsCfb<∞C\_\{\\mathrm\{fb\}\}<\\inftyandcfb\>0c\_\{\\mathrm\{fb\}\}\>0such that, forh=0,1h=0,1and all largekk,

ℙhπk​\(Efb\)≤Cfb​zk\+1Δk\+Cfb​exp⁡\{−cfb​Δk2Lk\}\.\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{h\}\(E\_\{\\mathrm\{fb\}\}\)\\leq C\_\{\\mathrm\{fb\}\}\\frac\{z\_\{k\}\+1\}\{\\Delta\_\{k\}\}\+C\_\{\\mathrm\{fb\}\}\\exp\\left\\\{\-c\_\{\\mathrm\{fb\}\}\\frac\{\\Delta\_\{k\}^\{2\}\}\{L\_\{k\}\}\\right\\\}\.\(86\)

The proof of Lemma[10\.6](https://arxiv.org/html/2609.28859#S10.Thmtheorem6)is given in Appendix[10\.5](https://arxiv.org/html/2609.28859#S10.SS5)\.

The pre\-committed fallback requires at mostO⁡\(Lk\)O\(L\_\{k\}\)additional cost whenever it is invoked, sinceNmain,k=O⁡\(Lk\)N\_\{\\mathrm\{main\},k\}=O\(L\_\{k\}\)by Lemma[10\.11](https://arxiv.org/html/2609.28859#S10.Thmtheorem11)and either completion mode queries at mostNmain,kN\_\{\\mathrm\{main\},k\}items\. Multiplying this deterministic cost bound by the probability in Lemma[10\.6](https://arxiv.org/html/2609.28859#S10.Thmtheorem6)yields the next lemma\.

###### Lemma 10\.7\(Fallback expected cost\)

There is a constantCF<∞C\_\{F\}<\\inftysuch that, forh=0,1h=0,1,

𝔼hπk​\[Ckfb\]≤CF​Lk​\[zk\+1Δk\+exp⁡\{−cfb​Δk2Lk\}\]\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[C^\{\\mathrm\{fb\}\}\_\{k\}\]\\leq C\_\{F\}L\_\{k\}\\left\[\\frac\{z\_\{k\}\+1\}\{\\Delta\_\{k\}\}\+\\exp\\left\\\{\-c\_\{\\mathrm\{fb\}\}\\frac\{\\Delta\_\{k\}^\{2\}\}\{L\_\{k\}\}\\right\\\}\\right\]\.\(87\)

The proof of Lemma[10\.7](https://arxiv.org/html/2609.28859#S10.Thmtheorem7)is given in Appendix[10\.6](https://arxiv.org/html/2609.28859#S10.SS6)\.

It remains to compare the acquisition cost and the buffered sensing plan with the lower boundLBk\\mathrm\{LB\}\_\{k\}\. The following perturbation result bounds the cost of increasing the information targets and imposing the fallback sample\-size floor\.

###### Lemma 10\.8\(Perturbation from the buffered design to the lower bound\)

There is a constantCP<∞C\_\{P\}<\\inftysuch that

Nmain,k​cdata\+maxh⁡Γh​\(Th,k,N¯main,k\)≤LBk\+CP​Δk\+O⁡\(log⁡Lk\+log⁡\(1ffb,k\)\+log⁡\(11−ffb,k\)\)\.N\_\{\\mathrm\{main\},k\}c\_\{\\mathrm\{data\}\}\+\\max\_\{h\}\\Gamma\_\{h\}\(T\_\{h,k\},\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\\leq\\mathrm\{LB\}\_\{k\}\+C\_\{P\}\\Delta\_\{k\}\+O\\\!\\left\(\\log L\_\{k\}\+\\log\\left\(\\frac\{1\}\{f\_\{\\mathrm\{fb\},k\}\}\\right\)\+\\log\\left\(\\frac\{1\}\{1\-f\_\{\\mathrm\{fb\},k\}\}\\right\)\\right\)\.\(88\)

The proof of Lemma[10\.8](https://arxiv.org/html/2609.28859#S10.Thmtheorem8)is given in Appendix[10\.2](https://arxiv.org/html/2609.28859#S10.SS2)\.

Combining the cost decomposition with Lemmas[10\.5](https://arxiv.org/html/2609.28859#S10.Thmtheorem5),[10\.7](https://arxiv.org/html/2609.28859#S10.Thmtheorem7), and[10\.8](https://arxiv.org/html/2609.28859#S10.Thmtheorem8), we obtain

maxh⁡𝔼hπk​\[Cπk\]≤\\displaystyle\\max\_\{h\}\\mathbb\{E\}\_\{h\}^\{\\pi\_\{k\}\}\[C^\{\\pi\_\{k\}\}\]\\leq\{\}LBk\+CP​Δk\+CM​\(zk\+1\)\+CF​Lk​\[zk\+1Δk\+exp⁡\{−cfb​Δk2Lk\}\]\\displaystyle\\mathrm\{LB\}\_\{k\}\+C\_\{P\}\\Delta\_\{k\}\+C\_\{M\}\(z\_\{k\}\+1\)\+C\_\{F\}L\_\{k\}\\left\[\\frac\{z\_\{k\}\+1\}\{\\Delta\_\{k\}\}\+\\exp\\left\\\{\-c\_\{\\mathrm\{fb\}\}\\frac\{\\Delta\_\{k\}^\{2\}\}\{L\_\{k\}\}\\right\\\}\\right\]\+O⁡\(log⁡Lk\+log⁡1ffb,k\+log⁡11−ffb,k\)\.\\displaystyle\+O\\\!\\left\(\\log L\_\{k\}\+\\log\\frac\{1\}\{f\_\{\\mathrm\{fb\},k\}\}\+\\log\\frac\{1\}\{1\-f\_\{\\mathrm\{fb\},k\}\}\\right\)\.SinceΔk=o⁡\(Lk\)\\Delta\_\{k\}=o\(L\_\{k\}\), the termCM​\(zk\+1\)C\_\{M\}\(z\_\{k\}\+1\)can be absorbed into a constant multiple ofLk​\(zk\+1\)/ΔkL\_\{k\}\(z\_\{k\}\+1\)/\\Delta\_\{k\}for all largekk\. Consequently, there are primitive constantsc\>0c\>0andC1,C2,C3,C4<∞C\_\{1\},C\_\{2\},C\_\{3\},C\_\{4\}<\\inftysuch that

maxh⁡𝔼hπk​\[Cπk\]≤\\displaystyle\\max\_\{h\}\\mathbb\{E\}\_\{h\}^\{\\pi\_\{k\}\}\[C^\{\\pi\_\{k\}\}\]\\leq\{\}LBk\+C1​Δk\+C2​\(zk\+1\)​LkΔk\+C3​Lk​exp⁡\{−c​Δk2Lk\}\\displaystyle\\mathrm\{LB\}\_\{k\}\+C\_\{1\}\\Delta\_\{k\}\+C\_\{2\}\\frac\{\(z\_\{k\}\+1\)L\_\{k\}\}\{\\Delta\_\{k\}\}\+C\_\{3\}L\_\{k\}\\exp\\left\\\{\-c\\frac\{\\Delta\_\{k\}^\{2\}\}\{L\_\{k\}\}\\right\\\}\(89\)\+C4​\(log⁡Lk\+log⁡1ffb,k\+log⁡11−ffb,k\)\.\\displaystyle\+C\_\{4\}\\left\(\\log L\_\{k\}\+\\log\\frac\{1\}\{f\_\{\\mathrm\{fb\},k\}\}\+\\log\\frac\{1\}\{1\-f\_\{\\mathrm\{fb\},k\}\}\\right\)\.By part \(i\),LBk=Θ⁡\(Lk\)\\mathrm\{LB\}\_\{k\}=\\Theta\(L\_\{k\}\)\. Dividing \([89](https://arxiv.org/html/2609.28859#S10.E89)\) byLBk\\mathrm\{LB\}\_\{k\}, the conditions \([41](https://arxiv.org/html/2609.28859#S5.E41)\) and \([42](https://arxiv.org/html/2609.28859#S5.E42)\) give

ΔkLk→0,zk\+1Δk→0,exp⁡\{−c​Δk2Lk\}→0\.\\frac\{\\Delta\_\{k\}\}\{L\_\{k\}\}\\to 0,\\qquad\\frac\{z\_\{k\}\+1\}\{\\Delta\_\{k\}\}\\to 0,\\qquad\\exp\\left\\\{\-c\\frac\{\\Delta\_\{k\}^\{2\}\}\{L\_\{k\}\}\\right\\\}\\to 0\.The remaining logarithmic contribution also vanishes becauseffb,kf\_\{\\mathrm\{fb\},k\}is bounded away from zero and one\. Hence

maxh⁡𝔼hπk​\[Cπk\]LBk≤1\+o⁡\(1\)\.\\frac\{\\max\_\{h\}\\mathbb\{E\}\_\{h\}^\{\\pi\_\{k\}\}\[C^\{\\pi\_\{k\}\}\]\}\{\\mathrm\{LB\}\_\{k\}\}\\leq 1\+o\(1\)\.Finally, feasibility ofπk\\pi\_\{k\}and Theorem[4\.11](https://arxiv.org/html/2609.28859#S4.Thmtheorem11)implymaxh⁡𝔼hπk​\[Cπk\]≥C∗​\(αk,βk\)≥LBk\\max\_\{h\}\\mathbb\{E\}\_\{h\}^\{\\pi\_\{k\}\}\[C^\{\\pi\_\{k\}\}\]\\geq C^\{\*\}\(\\alpha\_\{k\},\\beta\_\{k\}\)\\geq\\mathrm\{LB\}\_\{k\}\. The two bounds together prove part \(ii\)\.

### 10\.2Proof of Lemma[10\.8](https://arxiv.org/html/2609.28859#S10.Thmtheorem8)

We advance the proof of Lemma[10\.8](https://arxiv.org/html/2609.28859#S10.Thmtheorem8)as the proofs of Lemmas[10\.4](https://arxiv.org/html/2609.28859#S10.Thmtheorem4)\-[10\.7](https://arxiv.org/html/2609.28859#S10.Thmtheorem7)rely on the conclusions from Lemma[10\.8](https://arxiv.org/html/2609.28859#S10.Thmtheorem8), while the proof of Lemma[10\.8](https://arxiv.org/html/2609.28859#S10.Thmtheorem8)does not rely on the conclusions from Lemmas[10\.4](https://arxiv.org/html/2609.28859#S10.Thmtheorem4)\-[10\.7](https://arxiv.org/html/2609.28859#S10.Thmtheorem7)\.

To prove Lemma[10\.8](https://arxiv.org/html/2609.28859#S10.Thmtheorem8), we need auxiliary Lemma[10\.9](https://arxiv.org/html/2609.28859#S10.Thmtheorem9)\. We relegate the proof of Lemma[10\.9](https://arxiv.org/html/2609.28859#S10.Thmtheorem9)to Appendix[10\.2\.1](https://arxiv.org/html/2609.28859#S10.SS2.SSS1)\.

###### Lemma 10\.9\(Safe\-floor stability at first order\)

\(i\)Nfixed,H​\(αk,βk\)=O⁡\(Lk\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)=O\(L\_\{k\}\);

\(ii\)

0≤Nfixed,H​\(α2,k,β2,k\)−Nfixed,H​\(αk,βk\)=O⁡\(log⁡Lk\+log⁡\(1ffb,k\)\)\.0\\leq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\-N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)=O\\\!\\left\(\\log L\_\{k\}\+\\log\\left\(\\frac\{1\}\{f\_\{\\mathrm\{fb\},k\}\}\\right\)\\right\)\.\(90\)

We are now ready to prove Lemma[10\.8](https://arxiv.org/html/2609.28859#S10.Thmtheorem8)\.

###### Proof 10\.10

Proof of Lemma[10\.8](https://arxiv.org/html/2609.28859#S10.Thmtheorem8)To show \([88](https://arxiv.org/html/2609.28859#S10.E88)\) holds, we find an integerN~k\\tilde\{N\}\_\{k\}withN~k≥Nfixed,H​\(α2,k,β2,k\)\\tilde\{N\}\_\{k\}\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)andFkΔ​\(N~k\)≤LBk\+CP​Δk\+O⁡\(log⁡Lk\+log⁡\(1/ffb,k\)\+log⁡\(1/\(1−ffb,k\)\)\)F^\{\\Delta\}\_\{k\}\(\\tilde\{N\}\_\{k\}\)\\leq\\mathrm\{LB\}\_\{k\}\+C\_\{P\}\\Delta\_\{k\}\+O\(\\log L\_\{k\}\+\\log\(1/f\_\{\\mathrm\{fb\},k\}\)\+\\log\(1/\(1\-f\_\{\\mathrm\{fb\},k\}\)\)\), and thus by construction ofN¯main,k\\overline\{N\}\_\{\\mathrm\{main\},k\}, we have thatFkΔ​\(N¯main,k\)≤FkΔ​\(N~k\)≤LBk\+CP​Δk\+O⁡\(log⁡Lk\+log⁡\(1/ffb,k\)\+log⁡\(1/\(1−ffb,k\)\)\)F^\{\\Delta\}\_\{k\}\(\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\\leq F^\{\\Delta\}\_\{k\}\(\\tilde\{N\}\_\{k\}\)\\leq\\mathrm\{LB\}\_\{k\}\+C\_\{P\}\\Delta\_\{k\}\+O\(\\log L\_\{k\}\+\\log\(1/f\_\{\\mathrm\{fb\},k\}\)\+\\log\(1/\(1\-f\_\{\\mathrm\{fb\},k\}\)\)\)\.

*Step 1: Construction ofN~k\\tilde\{N\}\_\{k\}\.*LetNk∗N\_\{k\}^\{\*\}be an optimizer in \([22](https://arxiv.org/html/2609.28859#S4.E22)\) for\(αk,βk\)\(\\alpha\_\{k\},\\beta\_\{k\}\)\. That is,

Nk∗∈arg⁡minN≥Nfixed,H​\(αk,βk\)​N​cdata\+max⁡\{Γ1​\(T1,k′,N\),Γ0​\(T0,k′,N\)\}\.N^\{\*\}\_\{k\}\\in\\arg\\min\_\{N\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)\}Nc\_\{\\mathrm\{data\}\}\+\\max\\\{\\Gamma\_\{1\}\(T^\{\\prime\}\_\{1,k\},N\),\\Gamma\_\{0\}\(T^\{\\prime\}\_\{0,k\},N\)\\\}\.ThusLBk=Nk∗​cdata\+maxh⁡Γh​\(Th,k′,Nk∗\)\\mathrm\{LB\}\_\{k\}=N\_\{k\}^\{\*\}c\_\{\\mathrm\{data\}\}\+\\max\_\{h\}\\Gamma\_\{h\}\(T^\{\\prime\}\_\{h,k\},N\_\{k\}^\{\*\}\)\. Also define the*reserve*

Rk:=⌈\[ak\+Δk−T1,k′\]\+JX\(1\)\+\[bk\+Δk−T0,k′\]\+JX\(0\)\+Nfixed,H​\(α2,k,β2,k\)−Nfixed,H​\(αk,βk\)⌉\.R\_\{k\}:=\\left\\lceil\\frac\{\\left\[a\_\{k\}\+\\Delta\_\{k\}\-T^\{\\prime\}\_\{1,k\}\\right\]\_\{\+\}\}\{J\_\{X\}^\{\(1\)\}\}\+\\frac\{\\left\[b\_\{k\}\+\\Delta\_\{k\}\-T^\{\\prime\}\_\{0,k\}\\right\]\_\{\+\}\}\{J\_\{X\}^\{\(0\)\}\}\+N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\-N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)\\right\\rceil\.\(91\)Let

N~k:=⌈Nk∗\+Rk⌉\.\\tilde\{N\}\_\{k\}:=\\lceil N^\{\*\}\_\{k\}\+R\_\{k\}\\rceil\.
*Step 2: show thatFkΔ​\(N~k\)≤LBk\+CP​Δk\+O⁡\(log⁡Lk\+log⁡\(1/ffb,k\)\+log⁡\(1/\(1−ffb,k\)\)\)F^\{\\Delta\}\_\{k\}\(\\tilde\{N\}\_\{k\}\)\\leq\\mathrm\{LB\}\_\{k\}\+C\_\{P\}\\Delta\_\{k\}\+O\(\\log L\_\{k\}\+\\log\(1/f\_\{\\mathrm\{fb\},k\}\)\+\\log\(1/\(1\-f\_\{\\mathrm\{fb\},k\}\)\)\)\.*We first notice thatN~k≥Nk∗\+Rk≥Nfixed,H​\(αk,βk\)\+Nfixed,H​\(α2,k,β2,k\)−Nfixed,H​\(αk,βk\)=Nfixed,H​\(α2,k,β2,k\)\\tilde\{N\}\_\{k\}\\geq N^\{\*\}\_\{k\}\+R\_\{k\}\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)\+N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\-N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)=N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\), usingNk∗≥Nfixed,H​\(αk,βk\)N\_\{k\}^\{\*\}\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)\(Step 1\)\. We now boundFkΔ​\(Nk∗\+Rk\)F\_\{k\}^\{\\Delta\}\(N\_\{k\}^\{\*\}\+R\_\{k\}\)from above\.

Fixh∈\{0,1\}h\\in\\\{0,1\\\}\. SinceLBk<∞\\mathrm\{LB\}\_\{k\}<\\infty, we also have thatΓh​\(Th,k′,Nk∗\)<∞\\Gamma\_\{h\}\(T^\{\\prime\}\_\{h,k\},N\_\{k\}^\{\*\}\)<\\inftyforh∈\{0,1\}h\\in\\\{0,1\\\}, so there is an optimal triple\(nH∗,nAI∗,nesc∗\)\(n\_\{\\mathrm\{H\}\}^\{\*\},n\_\{\\mathrm\{AI\}\}^\{\*\},n\_\{\\mathrm\{esc\}\}^\{\*\}\)forΓh​\(Th,k′,Nk∗\)\\Gamma\_\{h\}\(T^\{\\prime\}\_\{h,k\},N\_\{k\}^\{\*\}\); being feasible,\(nH∗,nAI∗,nesc∗\)\(n\_\{\\mathrm\{H\}\}^\{\*\},n\_\{\\mathrm\{AI\}\}^\{\*\},n\_\{\\mathrm\{esc\}\}^\{\*\}\)forΓh​\(Th,k′,Nk∗\)\\Gamma\_\{h\}\(T^\{\\prime\}\_\{h,k\},N\_\{k\}^\{\*\}\)satisfies:nH∗\+nAI∗≤Nk∗n\_\{\\mathrm\{H\}\}^\{\*\}\+n\_\{\\mathrm\{AI\}\}^\{\*\}\\leq N\_\{k\}^\{\*\},0≤nesc∗≤nAI∗0\\leq n\_\{\\mathrm\{esc\}\}^\{\*\}\\leq n\_\{\\mathrm\{AI\}\}^\{\*\}and thatnH∗​JX\(h\)\+nAI∗​IR\(h\)\+nAI∗​Ψh​\(nesc∗/nAI∗\)≥Th,k′n\_\{\\mathrm\{H\}\}^\{\*\}J^\{\(h\)\}\_\{X\}\+n\_\{\\mathrm\{AI\}\}^\{\*\}I^\{\(h\)\}\_\{R\}\+n\_\{\\mathrm\{AI\}\}^\{\*\}\\Psi\_\{h\}\(n\_\{\\mathrm\{esc\}\}^\{\*\}/n\_\{\\mathrm\{AI\}\}^\{\*\}\)\\geq T^\{\\prime\}\_\{h,k\}\. We construct the perturbed triple\(n^H,n^AI,n^esc\):=\(nH∗\+Rk,nAI∗,nesc∗\)\(\\hat\{n\}\_\{\\mathrm\{H\}\},\\hat\{n\}\_\{\\mathrm\{AI\}\},\\hat\{n\}\_\{\\mathrm\{esc\}\}\):=\(n\_\{\\mathrm\{H\}\}^\{\*\}\+R\_\{k\},\\,n\_\{\\mathrm\{AI\}\}^\{\*\},\\,n\_\{\\mathrm\{esc\}\}^\{\*\}\), which addsRkR\_\{k\}direct\-human items\. We check it is feasible for \([21](https://arxiv.org/html/2609.28859#S4.E21)\) withT=Th,kT=T\_\{h,k\}andN=N~kN=\\tilde\{N\}\_\{k\}:

- •n^H\+n^AI=\(nH∗\+Rk\)\+nAI∗≤Nk∗\+Rk=N~k\\hat\{n\}\_\{\\mathrm\{H\}\}\+\\hat\{n\}\_\{\\mathrm\{AI\}\}=\(n\_\{\\mathrm\{H\}\}^\{\*\}\+R\_\{k\}\)\+n\_\{\\mathrm\{AI\}\}^\{\*\}\\leq N\_\{k\}^\{\*\}\+R\_\{k\}=\\tilde\{N\}\_\{k\}, sincenH∗\+nAI∗≤Nk∗n\_\{\\mathrm\{H\}\}^\{\*\}\+n\_\{\\mathrm\{AI\}\}^\{\*\}\\leq N\_\{k\}^\{\*\}\.
- •n^AI=nAI∗,n^esc=nesc∗\\hat\{n\}\_\{\\mathrm\{AI\}\}=n\_\{\\mathrm\{AI\}\}^\{\*\},\\hat\{n\}\_\{\\mathrm\{esc\}\}=n\_\{\\mathrm\{esc\}\}^\{\*\}, so0≤n^esc≤n^AI0\\leq\\hat\{n\}\_\{\\mathrm\{esc\}\}\\leq\\hat\{n\}\_\{\\mathrm\{AI\}\}holds since0≤nesc∗≤nAI∗0\\leq n^\{\*\}\_\{\\mathrm\{esc\}\}\\leq n^\{\*\}\_\{\\mathrm\{AI\}\}\.
- •n^H​JX\(h\)\+n^AI​IR\(h\)\+n^AI​Ψh​\(n^esc/n^AI\)=\(nH∗\+Rk\)​JX\(h\)\+nAI∗​IR\(h\)\+nAI∗​Ψh​\(nesc∗/nAI∗\)≥Th,k′\+Rk​JX\(h\)\\hat\{n\}\_\{\\mathrm\{H\}\}J^\{\(h\)\}\_\{X\}\+\\hat\{n\}\_\{\\mathrm\{AI\}\}I^\{\(h\)\}\_\{R\}\+\\hat\{n\}\_\{\\mathrm\{AI\}\}\\Psi\_\{h\}\(\\hat\{n\}\_\{\\mathrm\{esc\}\}/\\hat\{n\}\_\{\\mathrm\{AI\}\}\)=\(n\_\{\\mathrm\{H\}\}^\{\*\}\+R\_\{k\}\)J^\{\(h\)\}\_\{X\}\+n\_\{\\mathrm\{AI\}\}^\{\*\}I^\{\(h\)\}\_\{R\}\+n\_\{\\mathrm\{AI\}\}^\{\*\}\\Psi\_\{h\}\(n\_\{\\mathrm\{esc\}\}^\{\*\}/n\_\{\\mathrm\{AI\}\}^\{\*\}\)\\geq T^\{\\prime\}\_\{h,k\}\+R\_\{k\}J^\{\(h\)\}\_\{X\}\. By \([91](https://arxiv.org/html/2609.28859#S10.E91)\),Rk≥\[Th,k−Th,k′\]\+/JX\(h\)R\_\{k\}\\geq\[T\_\{h,k\}\-T^\{\\prime\}\_\{h,k\}\]\_\{\+\}/J\_\{X\}^\{\(h\)\}for allh∈\{0,1\}h\\in\\\{0,1\\\}, soRk​JX\(h\)≥\[Th,k−Th,k′\]\+R\_\{k\}J\_\{X\}^\{\(h\)\}\\geq\\left\[T\_\{h,k\}\-T^\{\\prime\}\_\{h,k\}\\right\]\_\{\+\}\. Thusn^H​JX\(h\)\+n^AI​IR\(h\)\+n^AI​Ψh​\(n^esc/n^AI\)≥Th,k′\+\[Th,k−Th,k′\]\+≥Th,k\\hat\{n\}\_\{\\mathrm\{H\}\}J^\{\(h\)\}\_\{X\}\+\\hat\{n\}\_\{\\mathrm\{AI\}\}I^\{\(h\)\}\_\{R\}\+\\hat\{n\}\_\{\\mathrm\{AI\}\}\\Psi\_\{h\}\(\\hat\{n\}\_\{\\mathrm\{esc\}\}/\\hat\{n\}\_\{\\mathrm\{AI\}\}\)\\geq T^\{\\prime\}\_\{h,k\}\+\\left\[T\_\{h,k\}\-T^\{\\prime\}\_\{h,k\}\\right\]\_\{\+\}\\geq T\_\{h,k\}\.

Then, since\(n^H,n^AI,n^esc\)\(\\hat\{n\}\_\{\\mathrm\{H\}\},\\hat\{n\}\_\{\\mathrm\{AI\}\},\\hat\{n\}\_\{\\mathrm\{esc\}\}\)is feasible for \([21](https://arxiv.org/html/2609.28859#S4.E21)\) withT=Th,kT=T\_\{h,k\}andN=N~kN=\\tilde\{N\}\_\{k\}, we have that

Γh​\(Th,k,N~k\)\\displaystyle\\Gamma\_\{h\}\(T\_\{h,k\},\\tilde\{N\}\_\{k\}\)≤cH​n^H\+cAI​n^AI\+cH​n^esc\\displaystyle\\leq c\_\{\\mathrm\{H\}\}\\hat\{n\}\_\{\\mathrm\{H\}\}\+c\_\{\\mathrm\{AI\}\}\\hat\{n\}\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}\\hat\{n\}\_\{\\mathrm\{esc\}\}=cH​\(nH∗\+Rk\)\+cAI​nAI∗\+cH​nesc∗\\displaystyle=c\_\{\\mathrm\{H\}\}\(n^\{\*\}\_\{\\mathrm\{H\}\}\+R\_\{k\}\)\+c\_\{\\mathrm\{AI\}\}n^\{\*\}\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}n^\{\*\}\_\{\\mathrm\{esc\}\}≤Γh​\(Th,k′,Nk∗\)\+cH​Rk\.\\displaystyle\\leq\\Gamma\_\{h\}\(T^\{\\prime\}\_\{h,k\},N\_\{k\}^\{\*\}\)\+c\_\{\\mathrm\{H\}\}R\_\{k\}\.Thus

FkΔ​\(N~k\)=\\displaystyle F\_\{k\}^\{\\Delta\}\(\\tilde\{N\}\_\{k\}\)=⌈Nk∗\+Rk⌉​cdata\+max⁡\{Γ1​\(T1,k,N~k\),Γ0​\(T0,k,N~k\)\}\\displaystyle\\lceil N^\{\*\}\_\{k\}\+R\_\{k\}\\rceil c\_\{\\mathrm\{data\}\}\+\\max\\\{\\Gamma\_\{1\}\(T\_\{1,k\},\\tilde\{N\}\_\{k\}\),\\Gamma\_\{0\}\(T\_\{0,k\},\\tilde\{N\}\_\{k\}\)\\\}≤\\displaystyle\\leq\(Nk∗​cdata\+maxh⁡Γh​\(Th,k′,Nk∗\)\)\+Rk​\(cdata\+cH\)\+cdata\\displaystyle\\Bigl\(N\_\{k\}^\{\*\}c\_\{\\mathrm\{data\}\}\+\\max\_\{h\}\\Gamma\_\{h\}\(T^\{\\prime\}\_\{h,k\},N\_\{k\}^\{\*\}\)\\Bigr\)\+R\_\{k\}\(c\_\{\\mathrm\{data\}\}\+c\_\{\\mathrm\{H\}\}\)\+c\_\{\\mathrm\{data\}\}=\\displaystyle=LBk\+Rk​\(cdata\+cH\)\+cdata,\\displaystyle\\mathrm\{LB\}\_\{k\}\+R\_\{k\}\(c\_\{\\mathrm\{data\}\}\+c\_\{\\mathrm\{H\}\}\)\+c\_\{\\mathrm\{data\}\},where the last equality usesLBk=Nk∗​cdata\+maxh⁡Γh​\(Th,k′,Nk∗\)\\mathrm\{LB\}\_\{k\}=N\_\{k\}^\{\*\}c\_\{\\mathrm\{data\}\}\+\\max\_\{h\}\\Gamma\_\{h\}\(T^\{\\prime\}\_\{h,k\},N\_\{k\}^\{\*\}\)from Step 1\. It remains to boundRk​\(cdata\+cH\)R\_\{k\}\(c\_\{\\mathrm\{data\}\}\+c\_\{\\mathrm\{H\}\}\)\. By Lemma[10\.1](https://arxiv.org/html/2609.28859#S10.Thmtheorem1),\[ak−T1,k′\]\+=log⁡\(1/\(1−ffb,k\)\)\+o⁡\(1\)\\left\[a\_\{k\}\-T^\{\\prime\}\_\{1,k\}\\right\]\_\{\+\}=\\log\(1/\(1\-f\_\{\\mathrm\{fb\},k\}\)\)\+o\(1\)and\[bk−T0,k′\]\+=log⁡\(1/\(1−ffb,k\)\)\+o⁡\(1\)\\left\[b\_\{k\}\-T^\{\\prime\}\_\{0,k\}\\right\]\_\{\+\}=\\log\(1/\(1\-f\_\{\\mathrm\{fb\},k\}\)\)\+o\(1\), hence\[ak\+Δk−T1,k′\]\+≤Δk\+log⁡\(1/\(1−ffb,k\)\)\+o⁡\(1\)\\left\[a\_\{k\}\+\\Delta\_\{k\}\-T^\{\\prime\}\_\{1,k\}\\right\]\_\{\+\}\\leq\\Delta\_\{k\}\+\\log\(1/\(1\-f\_\{\\mathrm\{fb\},k\}\)\)\+o\(1\)and likewise for thebb\-term; and by Lemma[10\.9](https://arxiv.org/html/2609.28859#S10.Thmtheorem9),Nfixed,H​\(α2,k,β2,k\)−Nfixed,H​\(αk,βk\)=O⁡\(log⁡Lk\+log⁡\(1/ffb,k\)\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\-N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)=O\\\!\\left\(\\log L\_\{k\}\+\\log\(1/f\_\{\\mathrm\{fb\},k\}\)\\right\)\. Substituting these estimates into \([91](https://arxiv.org/html/2609.28859#S10.E91)\) \(the\+1\+1from the ceiling isO⁡\(1\)O\(1\)\) gives

Rk≤\(1/JX\(1\)\+1/JX\(0\)\)​Δk\+O⁡\(log⁡Lk\+log⁡\(1/ffb,k\)\+log⁡\(1/\(1−ffb,k\)\)\)\.R\_\{k\}\\leq\(1/J\_\{X\}^\{\(1\)\}\+1/\{J\_\{X\}^\{\(0\)\}\}\)\\Delta\_\{k\}\+O\\\!\\left\(\\log L\_\{k\}\+\\log\(1/f\_\{\\mathrm\{fb\},k\}\)\+\\log\(1/\(1\-f\_\{\\mathrm\{fb\},k\}\)\)\\right\)\.Therefore, settingCP:=\(cdata\+cH\)​\(1/JX\(1\)\+1/JX\(0\)\)C\_\{P\}:=\(c\_\{\\mathrm\{data\}\}\+c\_\{\\mathrm\{H\}\}\)\\bigl\(1/J\_\{X\}^\{\(1\)\}\+1/J\_\{X\}^\{\(0\)\}\\bigr\),

FkΔ​\(N~k\)≤LBk\+Rk​\(cdata\+cH\)\+cdata≤LBk\+CP​Δk\+O⁡\(log⁡Lk\+log⁡\(1/ffb,k\)\+log⁡\(1/\(1−ffb,k\)\)\)\.F\_\{k\}^\{\\Delta\}\(\\tilde\{N\}\_\{k\}\)\\leq\\mathrm\{LB\}\_\{k\}\+R\_\{k\}\(c\_\{\\mathrm\{data\}\}\+c\_\{\\mathrm\{H\}\}\)\+c\_\{\\mathrm\{data\}\}\\leq\\mathrm\{LB\}\_\{k\}\+C\_\{P\}\\Delta\_\{k\}\+O\\\!\\left\(\\log L\_\{k\}\+\\log\(1/f\_\{\\mathrm\{fb\},k\}\)\+\\log\(1/\(1\-f\_\{\\mathrm\{fb\},k\}\)\)\\right\)\.Finally, sinceN¯main,k\\overline\{N\}\_\{\\mathrm\{main\},k\}minimizesFkΔF\_\{k\}^\{\\Delta\}overN≥Nfixed,H​\(α2,k,β2,k\)N\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)andN~k≥Nfixed,H​\(α2,k,β2,k\)\\tilde\{N\}\_\{k\}\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)lies in its feasible range,FkΔ​\(N¯main,k\)≤FkΔ​\(N~k\)≤LBk\+CP​Δk\+O⁡\(log⁡Lk\+log⁡\(1/ffb,k\)\+log⁡\(1/\(1−ffb,k\)\)\)F^\{\\Delta\}\_\{k\}\(\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\\leq F^\{\\Delta\}\_\{k\}\(\\tilde\{N\}\_\{k\}\)\\leq\\mathrm\{LB\}\_\{k\}\+C\_\{P\}\\Delta\_\{k\}\+O\\\!\\left\(\\log L\_\{k\}\+\\log\(1/f\_\{\\mathrm\{fb\},k\}\)\+\\log\(1/\(1\-f\_\{\\mathrm\{fb\},k\}\)\)\\right\)\.

Lemma[10\.11](https://arxiv.org/html/2609.28859#S10.Thmtheorem11)is an immediate consequence of Lemma[10\.8](https://arxiv.org/html/2609.28859#S10.Thmtheorem8)\.

###### Lemma 10\.11

N¯main,k=O⁡\(Lk\)\\overline\{N\}\_\{\\mathrm\{main\},k\}=O\(L\_\{k\}\)andNmain,k=O⁡\(Lk\)N\_\{\\mathrm\{main\},k\}=O\(L\_\{k\}\)\.

###### Proof 10\.12

Proof of Lemma[10\.11](https://arxiv.org/html/2609.28859#S10.Thmtheorem11)From the proof of Lemma[10\.8](https://arxiv.org/html/2609.28859#S10.Thmtheorem8)we have shown thatFkΔ​\(N¯main,k\)≤FkΔ​\(N~k\)≤LBk\+CP​Δk\+O⁡\(log⁡Lk\+log⁡\(1/ffb,k\)\+log⁡\(1/\(1−ffb,k\)\)\)=O⁡\(Lk\)F^\{\\Delta\}\_\{k\}\(\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\\leq F^\{\\Delta\}\_\{k\}\(\\tilde\{N\}\_\{k\}\)\\leq\\mathrm\{LB\}\_\{k\}\+C\_\{P\}\\Delta\_\{k\}\+O\\\!\\left\(\\log L\_\{k\}\+\\log\(1/f\_\{\\mathrm\{fb\},k\}\)\+\\log\(1/\(1\-f\_\{\\mathrm\{fb\},k\}\)\)\\right\)=O\(L\_\{k\}\)\. In the meanwhile, sinceFkΔ​\(N¯main,k\)≥N¯main,k​cdataF\_\{k\}^\{\\Delta\}\(\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\\geq\\overline\{N\}\_\{\\mathrm\{main\},k\}c\_\{\\mathrm\{data\}\}, we getN¯main,k=O⁡\(Lk\)\\overline\{N\}\_\{\\mathrm\{main\},k\}=O\(L\_\{k\}\), andNmain,k=N¯main,k\+1=O⁡\(Lk\)N\_\{\\mathrm\{main\},k\}=\\overline\{N\}\_\{\\mathrm\{main\},k\}\+1=O\(L\_\{k\}\)\.

#### 10\.2\.1Proof of Lemma[10\.9](https://arxiv.org/html/2609.28859#S10.Thmtheorem9)

In this subsection, we prove Lemma[10\.9](https://arxiv.org/html/2609.28859#S10.Thmtheorem9)\. We first present an auxiliary Lemma[10\.13](https://arxiv.org/html/2609.28859#S10.Thmtheorem13), and then present the proof of Lemma[10\.9](https://arxiv.org/html/2609.28859#S10.Thmtheorem9)at the end of this subsection\.

To introduce Lemma[10\.13](https://arxiv.org/html/2609.28859#S10.Thmtheorem13), we motivate the quantityV⁡\(u,v\)V\(u,v\)that will appear in Lemma[10\.13](https://arxiv.org/html/2609.28859#S10.Thmtheorem13)\. Consider the family of threshold tests that rejectH0H\_\{0\}whenX¯n:=n−1​∑i=1nXi≥ϑ\\bar\{X\}\_\{n\}:=n^\{\-1\}\\sum\_\{i=1\}^\{n\}X\_\{i\}\\geq\\varthetafor someϑ∈\(p0,p1\)\\vartheta\\in\(p\_\{0\},p\_\{1\}\)\. Forp∈\(0,1\)p\\in\(0,1\), letℙp\\mathbb\{P\}\_\{p\}denote the probability law under whichX1,X2,…X\_\{1\},X\_\{2\},\\ldotsare i\.i\.d\.Bernoulli⁡\(p\)\\operatorname\{Bernoulli\}\(p\), and let𝔼p\\mathbb\{E\}\_\{p\}denote expectation with respect toℙp\\mathbb\{P\}\_\{p\}\. By the Chernoff bound for Bernoulli sums,

ℙp0​\(X¯n≥ϑ\)≤e−n​D0​\(ϑ\),ℙp1​\(X¯n≤ϑ\)≤e−n​D1​\(ϑ\),\\mathbb\{P\}\_\{p\_\{0\}\}\\\!\\left\(\\bar\{X\}\_\{n\}\\geq\\vartheta\\right\)\\leq e^\{\-nD\_\{0\}\(\\vartheta\)\},\\qquad\\mathbb\{P\}\_\{p\_\{1\}\}\\\!\\left\(\\bar\{X\}\_\{n\}\\leq\\vartheta\\right\)\\leq e^\{\-nD\_\{1\}\(\\vartheta\)\},whereD0\(ϑ\):=kl\(ϑ∥p0\)D\_\{0\}\(\\vartheta\):=\\text\{kl\}\(\\vartheta\\,\\\|\\,p\_\{0\}\)andD1\(ϑ\):=kl\(ϑ∥p1\)D\_\{1\}\(\\vartheta\):=\\text\{kl\}\(\\vartheta\\,\\\|\\,p\_\{1\}\)\. Letuk:=log⁡\(1/αk\)u\_\{k\}:=\\log\(1/\\alpha\_\{k\}\),vk:=log⁡\(1/βk\)v\_\{k\}:=\\log\(1/\\beta\_\{k\}\)\. To satisfy the type\-I error targete−uk=αke^\{\-u\_\{k\}\}=\\alpha\_\{k\}, it suffices thatn​D0​\(ϑ\)≥uknD\_\{0\}\(\\vartheta\)\\geq u\_\{k\}, i\.e\.,n≥uk/D0​\(ϑ\)n\\geq u\_\{k\}/D\_\{0\}\(\\vartheta\)\. Symmetrically, the type\-II error targete−vk=βke^\{\-v\_\{k\}\}=\\beta\_\{k\}requiresn≥vk/D1​\(ϑ\)n\\geq v\_\{k\}/D\_\{1\}\(\\vartheta\)\. For a fixed thresholdϑ\\vartheta, the smallest sample size at which both Chernoff bounds simultaneously deliver the target errors is therefore

max⁡\{ukD0​\(ϑ\),vkD1​\(ϑ\)\},\\max\\\!\\left\\\{\\frac\{u\_\{k\}\}\{D\_\{0\}\(\\vartheta\)\},\\frac\{v\_\{k\}\}\{D\_\{1\}\(\\vartheta\)\}\\right\\\},and optimizing overϑ∈\(p0,p1\)\\vartheta\\in\(p\_\{0\},p\_\{1\}\)yields the quantity

V⁡\(u,v\):=infϑ∈\(p0,p1\)max⁡\{uD0​\(ϑ\),vD1​\(ϑ\)\}\.V\(u,v\):=\\inf\_\{\\vartheta\\in\(p\_\{0\},p\_\{1\}\)\}\\max\\\!\\left\\\{\\frac\{u\}\{D\_\{0\}\(\\vartheta\)\},\\frac\{v\}\{D\_\{1\}\(\\vartheta\)\}\\right\\\}\.\(92\)This construction immediately gives the upper boundNfixed,H​\(αk,βk\)≤V⁡\(uk,vk\)\+1N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)\\leq V\(u\_\{k\},v\_\{k\}\)\+1by exhibiting an explicit feasible test\. The next lemma asserts that this Chernoff\-style bound is in fact first\-order tight: no test on i\.i\.d\. Bernoulli labels can achieve the target errors with substantially fewer samples\. Recall thatLk:=max⁡\{uk,vk\}L\_\{k\}:=\\max\\\{u\_\{k\},v\_\{k\}\\\}\.

###### Lemma 10\.13\(First\-order sample\-size identity for the full\-label benchmark\)

Letαk,βk↓0\\alpha\_\{k\},\\beta\_\{k\}\\downarrow 0satisfy Assumption[5\.3](https://arxiv.org/html/2609.28859#S5.SS3)\. ThenV⁡\(uk,vk\)=Θ⁡\(Lk\)V\(u\_\{k\},v\_\{k\}\)=\\Theta\(L\_\{k\}\)and

V⁡\(uk,vk\)−O⁡\(log⁡Lk\)≤Nfixed,H​\(e−uk,e−vk\)≤V⁡\(uk,vk\)\+1\.V\(u\_\{k\},v\_\{k\}\)\-O\(\\log L\_\{k\}\)\\leq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(e^\{\-u\_\{k\}\},e^\{\-v\_\{k\}\}\)\\leq V\(u\_\{k\},v\_\{k\}\)\+1\.\(93\)

###### Proof 10\.14

Proof of Lemma[10\.13](https://arxiv.org/html/2609.28859#S10.Thmtheorem13)The proof proceeds in five steps\. Step 1 establishes the monotonicity ofD0D\_\{0\}andD1D\_\{1\}and shows thatV⁡\(uk,vk\)=Θ⁡\(Lk\)V\(u\_\{k\},v\_\{k\}\)=\\Theta\(L\_\{k\}\)\. Step 2 applies Chernoff bounds to an optimally chosen sample\-mean threshold test, yieldingNfixed,H≤V⁡\(uk,vk\)\+1N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\\leq V\(u\_\{k\},v\_\{k\}\)\+1\. Step 3 derives a lower bound on Bernoulli type\-class probabilitiesℙp​\(∑i=1nXi=j\)\\mathbb\{P\}\_\{p\}\(\\sum\_\{i=1\}^\{n\}X\_\{i\}=j\)\. Step 4 shows that, for any feasible test, no type class can have probability exceeding the respective error levels under both hypotheses\. Step 5 combines Steps 3–4 to show that a sample size belowV⁡\(uk−δk,vk−δk\)V\(u\_\{k\}\-\\delta\_\{k\},v\_\{k\}\-\\delta\_\{k\}\), whereδk=O⁡\(log⁡Lk\)\\delta\_\{k\}=O\(\\log L\_\{k\}\), would violate this incompatibility\. Assumption[5\.3](https://arxiv.org/html/2609.28859#S5.SS3)then givesNfixed,H≥V⁡\(uk,vk\)−O⁡\(log⁡Lk\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\\geq V\(u\_\{k\},v\_\{k\}\)\-O\(\\log L\_\{k\}\)\.

##### Step 1: properties ofD0,D1,VD\_\{0\},D\_\{1\},V\.

Differentiating gives, forϑ∈\(0,1\)\\vartheta\\in\(0,1\),

D0′​\(ϑ\)=log⁡ϑ⁡\(1−p0\)p0​\(1−ϑ\),D1′​\(ϑ\)=log⁡ϑ⁡\(1−p1\)p1​\(1−ϑ\)\.D\_\{0\}^\{\\prime\}\(\\vartheta\)=\\log\\frac\{\\vartheta\(1\-p\_\{0\}\)\}\{p\_\{0\}\(1\-\\vartheta\)\},\\qquad D\_\{1\}^\{\\prime\}\(\\vartheta\)=\\log\\frac\{\\vartheta\(1\-p\_\{1\}\)\}\{p\_\{1\}\(1\-\\vartheta\)\}\.Hence on\(p0,p1\)\(p\_\{0\},p\_\{1\}\),D0D\_\{0\}is strictly increasing andD1D\_\{1\}is strictly decreasing, so

0<D0​\(ϑ\)<D0​\(p1\)=JX\(1\),0<D1​\(ϑ\)<D1​\(p0\)=JX\(0\),0<D\_\{0\}\(\\vartheta\)<D\_\{0\}\(p\_\{1\}\)=J^\{\(1\)\}\_\{X\},\\qquad 0<D\_\{1\}\(\\vartheta\)<D\_\{1\}\(p\_\{0\}\)=J^\{\(0\)\}\_\{X\},whereJX\(1\)=kl\(p1∥p0\)\>0J^\{\(1\)\}\_\{X\}=\\operatorname\{kl\}\(p\_\{1\}\\\|p\_\{0\}\)\>0andJX\(0\)=kl\(p0∥p1\)\>0J^\{\(0\)\}\_\{X\}=\\operatorname\{kl\}\(p\_\{0\}\\\|p\_\{1\}\)\>0\.

Takingϑ¯:=\(p0\+p1\)/2∈\(p0,p1\)\\bar\{\\vartheta\}:=\(p\_\{0\}\+p\_\{1\}\)/2\\in\(p\_\{0\},p\_\{1\}\), bothD0​\(ϑ¯\),D1​\(ϑ¯\)D\_\{0\}\(\\bar\{\\vartheta\}\),D\_\{1\}\(\\bar\{\\vartheta\}\)are positive constants, so

V⁡\(u,v\)≤max⁡\{uD0​\(ϑ¯\),vD1​\(ϑ¯\)\}≤CV​max⁡\{u,v\}V\(u,v\)\\leq\\max\\\!\\left\\\{\\frac\{u\}\{D\_\{0\}\(\\bar\{\\vartheta\}\)\},\\frac\{v\}\{D\_\{1\}\(\\bar\{\\vartheta\}\)\}\\right\\\}\\leq C\_\{V\}\\max\\\{u,v\\\}\(94\)for a finite constantCVC\_\{V\}depending only onp0,p1p\_\{0\},p\_\{1\}\. Also,

V⁡\(u,v\)≥max⁡\{uJX\(1\),vJX\(0\)\},V\(u,v\)\\geq\\max\\\!\\left\\\{\\frac\{u\}\{J^\{\(1\)\}\_\{X\}\},\\frac\{v\}\{J^\{\(0\)\}\_\{X\}\}\\right\\\},\(95\)becauseD0​\(ϑ\)≤JX\(1\)D\_\{0\}\(\\vartheta\)\\leq J^\{\(1\)\}\_\{X\}andD1​\(ϑ\)≤JX\(0\)D\_\{1\}\(\\vartheta\)\\leq J^\{\(0\)\}\_\{X\}on\(p0,p1\)\(p\_\{0\},p\_\{1\}\)\. Together, \([94](https://arxiv.org/html/2609.28859#S10.E94)\) and \([95](https://arxiv.org/html/2609.28859#S10.E95)\) implyV⁡\(uk,vk\)=Θ⁡\(Lk\)V\(u\_\{k\},v\_\{k\}\)=\\Theta\(L\_\{k\}\)sinceLk=max⁡\{uk,vk\}→∞L\_\{k\}=\\max\\\{u\_\{k\},v\_\{k\}\\\}\\to\\infty\.

##### Step 2: Chernoff upper bound\.

We prove the bound

Nfixed,H​\(e−uk,e−vk\)≤V⁡\(uk,vk\)\+1\.N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(e^\{\-u\_\{k\}\},e^\{\-v\_\{k\}\}\)\\leq V\(u\_\{k\},v\_\{k\}\)\+1\.\(96\)
Forϑ∈\(p0,p1\)\\vartheta\\in\(p\_\{0\},p\_\{1\}\), Chernoff’s bound for Bernoulli sums gives

ℙp0​\(X¯n≥ϑ\)≤e−n​D0​\(ϑ\),ℙp1​\(X¯n≤ϑ\)≤e−n​D1​\(ϑ\)\.\\mathbb\{P\}\_\{p\_\{0\}\}\\\!\\left\(\\bar\{X\}\_\{n\}\\geq\\vartheta\\right\)\\leq e^\{\-nD\_\{0\}\(\\vartheta\)\},\\qquad\\mathbb\{P\}\_\{p\_\{1\}\}\\\!\\left\(\\bar\{X\}\_\{n\}\\leq\\vartheta\\right\)\\leq e^\{\-nD\_\{1\}\(\\vartheta\)\}\.\(97\)
The infimum definingV⁡\(uk,vk\)V\(u\_\{k\},v\_\{k\}\)in \([92](https://arxiv.org/html/2609.28859#S10.E92)\) is attained\. Indeed,f⁡\(ϑ\):=max⁡\{uk/D0​\(ϑ\),vk/D1​\(ϑ\)\}f\(\\vartheta\):=\\max\\\{u\_\{k\}/D\_\{0\}\(\\vartheta\),v\_\{k\}/D\_\{1\}\(\\vartheta\)\\\}is continuous on\(p0,p1\)\(p\_\{0\},p\_\{1\}\); asϑ↓p0\\vartheta\\downarrow p\_\{0\}we haveD0​\(ϑ\)↓0D\_\{0\}\(\\vartheta\)\\downarrow 0, and asϑ↑p1\\vartheta\\uparrow p\_\{1\}we haveD1​\(ϑ\)↓0D\_\{1\}\(\\vartheta\)\\downarrow 0, sof⁡\(ϑ\)→\+∞f\(\\vartheta\)\\to\+\\inftyat both endpoints\. Extendingffby\+∞\+\\inftyto the endpoints makes it lower semicontinuous and coercive on the compact interval\[p0,p1\]\[p\_\{0\},p\_\{1\}\]; it therefore attains its minimum, necessarily at an interior pointϑ⋆∈\(p0,p1\)\\vartheta^\{\\star\}\\in\(p\_\{0\},p\_\{1\}\), and that minimum value isV⁡\(uk,vk\)V\(u\_\{k\},v\_\{k\}\)by \([92](https://arxiv.org/html/2609.28859#S10.E92)\)\.

Definenk:=⌈V⁡\(uk,vk\)⌉n\_\{k\}:=\\lceil V\(u\_\{k\},v\_\{k\}\)\\rceil\. Thennk≥V⁡\(uk,vk\)≥uk/D0​\(ϑ⋆\)n\_\{k\}\\geq V\(u\_\{k\},v\_\{k\}\)\\geq u\_\{k\}/D\_\{0\}\(\\vartheta^\{\\star\}\)andnk≥V⁡\(uk,vk\)≥vk/D1​\(ϑ⋆\)n\_\{k\}\\geq V\(u\_\{k\},v\_\{k\}\)\\geq v\_\{k\}/D\_\{1\}\(\\vartheta^\{\\star\}\), sonk​D0​\(ϑ⋆\)≥ukn\_\{k\}D\_\{0\}\(\\vartheta^\{\\star\}\)\\geq u\_\{k\}andnk​D1​\(ϑ⋆\)≥vkn\_\{k\}D\_\{1\}\(\\vartheta^\{\\star\}\)\\geq v\_\{k\}\.

By the definition ofNfixed,HN\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(the smallest sample size at which*some*test on i\.i\.d\. Bernoulli labels achieves the target errors\), to proveNfixed,H​\(αk,βk\)≤nkN\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)\\leq n\_\{k\}it suffices to exhibit one feasible test onnkn\_\{k\}labels\. We use the sample\-mean test that rejectsH0H\_\{0\}whenX¯nk≥ϑ⋆\\bar\{X\}\_\{n\_\{k\}\}\\geq\\vartheta^\{\\star\}\. By \([97](https://arxiv.org/html/2609.28859#S10.E97)\),

ℙp0​\(X¯nk≥ϑ⋆\)≤e−nk​D0​\(ϑ⋆\)≤e−uk=αk,\\mathbb\{P\}\_\{p\_\{0\}\}\(\\bar\{X\}\_\{n\_\{k\}\}\\geq\\vartheta^\{\\star\}\)\\leq e^\{\-n\_\{k\}D\_\{0\}\(\\vartheta^\{\\star\}\)\}\\leq e^\{\-u\_\{k\}\}=\\alpha\_\{k\},ℙp1​\(X¯nk≤ϑ⋆\)≤e−nk​D1​\(ϑ⋆\)≤e−vk=βk\.\\mathbb\{P\}\_\{p\_\{1\}\}\(\\bar\{X\}\_\{n\_\{k\}\}\\leq\\vartheta^\{\\star\}\)\\leq e^\{\-n\_\{k\}D\_\{1\}\(\\vartheta^\{\\star\}\)\}\\leq e^\{\-v\_\{k\}\}=\\beta\_\{k\}\.This test is feasible at\(αk,βk\)\(\\alpha\_\{k\},\\beta\_\{k\}\), so

Nfixed,H​\(αk,βk\)≤nk=⌈V⁡\(uk,vk\)⌉≤V⁡\(uk,vk\)\+1\.N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)\\leq n\_\{k\}=\\lceil V\(u\_\{k\},v\_\{k\}\)\\rceil\\leq V\(u\_\{k\},v\_\{k\}\)\+1\.

##### Step 3: type\-class point\-mass lower bound\.

We prove the elementary inequality: Forj∈\{0,…,n\}j\\in\\\{0,\\ldots,n\\\},

ℙp\(∑i=1nXi=j\)≥1n\+1exp\{−nkl\(j/n∥p\)\}\.\\mathbb\{P\}\_\{p\}\(\\sum\_\{i=1\}^\{n\}X\_\{i\}=j\)\\geq\\frac\{1\}\{n\+1\}\\exp\\\{\-n\\operatorname\{kl\}\(j/n\\\|p\)\\\}\.\(98\)Leth⁡\(ϑ\):=−ϑ​log⁡ϑ−\(1−ϑ\)​log⁡\(1−ϑ\)h\(\\vartheta\):=\-\\vartheta\\log\\vartheta\-\(1\-\\vartheta\)\\log\(1\-\\vartheta\)\(binary entropy, with0​log⁡0=00\\log 0=0\)\. It suffices to prove

\(nj\)≥1n\+1​en​h​\(j/n\)\.\\binom\{n\}\{j\}\\geq\\frac\{1\}\{n\+1\}e^\{nh\(j/n\)\}\.\(99\)Indeed, multiplying \([99](https://arxiv.org/html/2609.28859#S10.E99)\) bypj​\(1−p\)n−jp^\{j\}\(1\-p\)^\{n\-j\}gives

ℙp​\(∑i=1nXi=j\)=\(nj\)​pj​\(1−p\)n−j≥1n\+1​exp⁡\{n​h​\(j/n\)\+j​log⁡p\+\(n−j\)​log⁡\(1−p\)\}\.\\mathbb\{P\}\_\{p\}\(\\sum\_\{i=1\}^\{n\}X\_\{i\}=j\)=\\binom\{n\}\{j\}p^\{j\}\(1\-p\)^\{n\-j\}\\geq\\frac\{1\}\{n\+1\}\\exp\\bigl\\\{nh\(j/n\)\+j\\log p\+\(n\-j\)\\log\(1\-p\)\\bigr\\\}\.Writingϑ:=j/n\\vartheta:=j/n, the exponent inside the braces becomesn⁡\[h⁡\(ϑ\)\+ϑ​log⁡p\+\(1−ϑ\)​log⁡\(1−p\)\]n\[h\(\\vartheta\)\+\\vartheta\\log p\+\(1\-\\vartheta\)\\log\(1\-p\)\], and a direct expansion ofkl\(ϑ∥p\)=ϑlog\(ϑ/p\)\+\(1−ϑ\)log\(\(1−ϑ\)/\(1−p\)\)\\operatorname\{kl\}\(\\vartheta\\\|p\)=\\vartheta\\log\(\\vartheta/p\)\+\(1\-\\vartheta\)\\log\(\(1\-\\vartheta\)/\(1\-p\)\)gives the identityh\(ϑ\)\+ϑlogp\+\(1−ϑ\)log\(1−p\)=−kl\(ϑ∥p\)h\(\\vartheta\)\+\\vartheta\\log p\+\(1\-\\vartheta\)\\log\(1\-p\)=\-\\operatorname\{kl\}\(\\vartheta\\\|p\)\. Substituting this yields \([98](https://arxiv.org/html/2609.28859#S10.E98)\)\.

The proof of \([99](https://arxiv.org/html/2609.28859#S10.E99)\) uses an auxiliary Bernoulli computation with parameterϑ:=j/n\\vartheta:=j/n\. LetY1,…,YnY\_\{1\},\\ldots,Y\_\{n\}be i\.i\.d\.Bernoulli⁡\(ϑ\)\\operatorname\{Bernoulli\}\(\\vartheta\), and letTn:=∑iYiT\_\{n\}:=\\sum\_\{i\}Y\_\{i\}, soTn∼Binomial⁡\(n,ϑ\)T\_\{n\}\\sim\\operatorname\{Binomial\}\(n,\\vartheta\)\. We will show thatℙ⁡\(Tn=j\)≥1/\(n\+1\)\\mathbb\{P\}\(T\_\{n\}=j\)\\geq 1/\(n\+1\), and then rearrange this to get \([99](https://arxiv.org/html/2609.28859#S10.E99)\)\.

*Mode atjj\.*The ratio of consecutive binomial masses is

ℙ⁡\(Tn=r\+1\)ℙ⁡\(Tn=r\)=n−rr\+1⋅ϑ1−ϑ,r∈\{0,1,…,n−1\}\.\\frac\{\\mathbb\{P\}\(T\_\{n\}=r\+1\)\}\{\\mathbb\{P\}\(T\_\{n\}=r\)\}=\\frac\{n\-r\}\{r\+1\}\\cdot\\frac\{\\vartheta\}\{1\-\\vartheta\},\\qquad r\\in\\\{0,1,\\ldots,n\-1\\\}\.Solving\(n−r\)​ϑ≥\(r\+1\)​\(1−ϑ\)\(n\-r\)\\vartheta\\geq\(r\+1\)\(1\-\\vartheta\)givesr≤ϑ⁡\(n\+1\)−1r\\leq\\vartheta\(n\+1\)\-1, and withϑ=j/n\\vartheta=j/n, the bound becomesr≤j−1\+j/nr\\leq j\-1\+j/n\. For integerrrandj∈\{1,…,n−1\}j\\in\\\{1,\\ldots,n\-1\\\}, this is equivalent tor≤j−1r\\leq j\-1\. Hence the masses are nondecreasing asrrgoes from00tojj, and strictly decreasing asrrgoes fromjjtonn\. Soℙ⁡\(Tn=j\)\\mathbb\{P\}\(T\_\{n\}=j\)is the maximum of then\+1n\+1massesℙ⁡\(Tn=0\),…,ℙ⁡\(Tn=n\)\\mathbb\{P\}\(T\_\{n\}=0\),\\ldots,\\mathbb\{P\}\(T\_\{n\}=n\)\.

*Lower bound on the mode\.*Then\+1n\+1binomial masses are nonnegative and sum to11\. The maximum ofn\+1n\+1nonnegative numbers summing to11is at least1/\(n\+1\)1/\(n\+1\), so

ℙ⁡\(Tn=j\)=\(nj\)​ϑj​\(1−ϑ\)n−j≥1n\+1\.\\mathbb\{P\}\(T\_\{n\}=j\)=\\binom\{n\}\{j\}\\vartheta^\{j\}\(1\-\\vartheta\)^\{n\-j\}\\geq\\frac\{1\}\{n\+1\}\.
*Rearrangement\.*Using the identityϑj​\(1−ϑ\)n−j=exp⁡\{j​log⁡ϑ\+\(n−j\)​log⁡\(1−ϑ\)\}=exp⁡\{−n​h​\(ϑ\)\}\\vartheta^\{j\}\(1\-\\vartheta\)^\{n\-j\}=\\exp\\\{j\\log\\vartheta\+\(n\-j\)\\log\(1\-\\vartheta\)\\\}=\\exp\\\{\-nh\(\\vartheta\)\\\}\(valid forϑ=j/n\\vartheta=j/nby direct expansion ofhh\), the previous display gives

\(nj\)≥1n\+1⋅exp⁡\{n​h​\(ϑ\)\},\\binom\{n\}\{j\}\\geq\\frac\{1\}\{n\+1\}\\cdot\\exp\\\{nh\(\\vartheta\)\\\},which is \([99](https://arxiv.org/html/2609.28859#S10.E99)\)\.

##### Step 4: type\-class incompatibility for feasible tests\.

We prove the following fact, used in Step 5 to derive the lower bound onNfixed,HN\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}: if there exists a test onnnBernoulli labels with type\-I error at moste−ue^\{\-u\}and type\-II error at moste−ve^\{\-v\}, then for everyj∈\{0,…,n\}j\\in\\\{0,\\ldots,n\\\},

ℙp0​\(∑i=1nXi=j\)≤2​e−uorℙp1​\(∑i=1nXi=j\)≤2​e−v\.\\mathbb\{P\}\_\{p\_\{0\}\}\(\\sum\_\{i=1\}^\{n\}X\_\{i\}=j\)\\leq 2e^\{\-u\}\\quad\\text\{or\}\\quad\\mathbb\{P\}\_\{p\_\{1\}\}\(\\sum\_\{i=1\}^\{n\}X\_\{i\}=j\)\\leq 2e^\{\-v\}\.\(100\)
Fixn,u,vn,u,vand suppose there is a \(possibly randomized\) testϕ:\{0,1\}n→\[0,1\]\\phi:\\\{0,1\\\}^\{n\}\\to\[0,1\], whereϕ⁡\(x\)\\phi\(x\)denotes the conditional probability of rejectingH0H\_\{0\}given the observationxx, with type\-I error at moste−ue^\{\-u\}underBernoulli⁡\(p0\)\\operatorname\{Bernoulli\}\(p\_\{0\}\)and type\-II error at moste−ve^\{\-v\}underBernoulli⁡\(p1\)\\operatorname\{Bernoulli\}\(p\_\{1\}\)\. For eachj∈\{0,…,n\}j\\in\\\{0,\\ldots,n\\\}, define

ϕ¯\(j\):=1\(nj\)∑x:∑ixi=jϕ\(x\),\\bar\{\\phi\}\(j\):=\\frac\{1\}\{\\binom\{n\}\{j\}\}\\sum\_\{x:\\sum\_\{i\}x\_\{i\}=j\}\\phi\(x\),the average ofϕ⁡\(x\)\\phi\(x\)over the type class\{x∈\{0,1\}n:∑ixi=j\}\\\{x\\in\\\{0,1\\\}^\{n\}:\\sum\_\{i\}x\_\{i\}=j\\\}\.

Conditional on∑i=1nXi=j\\sum\_\{i=1\}^\{n\}X\_\{i\}=j, the label vector\(X1,…,Xn\)\(X\_\{1\},\\ldots,X\_\{n\}\)is uniform on the type class under either Bernoulli law\. Indeed,ℙp​\(X1=x1,…,Xn=xn\)=pj​\(1−p\)n−j\\mathbb\{P\}\_\{p\}\(X\_\{1\}=x\_\{1\},\\ldots,X\_\{n\}=x\_\{n\}\)=p^\{j\}\(1\-p\)^\{n\-j\}for everyx∈\{0,1\}nx\\in\\\{0,1\\\}^\{n\}with∑ixi=j\\sum\_\{i\}x\_\{i\}=j, so all suchxxare equiprobable\. Sinceϕ¯​\(j\)\\bar\{\\phi\}\(j\)is the average ofϕ⁡\(x\)\\phi\(x\)over the type class,

𝔼p​\[ϕ⁡\(X\)∣∑i=1nXi=j\]=ϕ¯​\(j\)under either​p∈\{p0,p1\}\.\\mathbb\{E\}\_\{p\}\[\\phi\(X\)\\mid\\sum\_\{i=1\}^\{n\}X\_\{i\}=j\]=\\bar\{\\phi\}\(j\)\\qquad\\text\{under either \}p\\in\\\{p\_\{0\},p\_\{1\}\\\}\.By the tower property,

𝔼p0​\[ϕ⁡\(X\)\]=𝔼p0​\[𝔼p0​\[ϕ⁡\(X\)∣∑i=1nXi\]\]=𝔼p0​\[ϕ¯​\(∑i=1nXi\)\]=∑j=0nℙp0​\(∑i=1nXi=j\)​ϕ¯​\(j\),\\mathbb\{E\}\_\{p\_\{0\}\}\[\\phi\(X\)\]=\\mathbb\{E\}\_\{p\_\{0\}\}\\\!\\left\[\\mathbb\{E\}\_\{p\_\{0\}\}\[\\phi\(X\)\\mid\\sum\_\{i=1\}^\{n\}X\_\{i\}\]\\right\]=\\mathbb\{E\}\_\{p\_\{0\}\}\[\\bar\{\\phi\}\(\\sum\_\{i=1\}^\{n\}X\_\{i\}\)\]=\\sum\_\{j=0\}^\{n\}\\mathbb\{P\}\_\{p\_\{0\}\}\(\\sum\_\{i=1\}^\{n\}X\_\{i\}=j\)\\bar\{\\phi\}\(j\),and similarly

𝔼p1​\[1−ϕ⁡\(X\)\]=∑j=0nℙp1​\(∑i=1nXi=j\)​\(1−ϕ¯​\(j\)\)\.\\mathbb\{E\}\_\{p\_\{1\}\}\[1\-\\phi\(X\)\]=\\sum\_\{j=0\}^\{n\}\\mathbb\{P\}\_\{p\_\{1\}\}\(\\sum\_\{i=1\}^\{n\}X\_\{i\}=j\)\(1\-\\bar\{\\phi\}\(j\)\)\.Sinceϕ¯​\(j\)∈\[0,1\]\\bar\{\\phi\}\(j\)\\in\[0,1\], each summand in these two sums is nonnegative\. The type\-I error bound𝔼p0​\[ϕ⁡\(X\)\]≤e−u\\mathbb\{E\}\_\{p\_\{0\}\}\[\\phi\(X\)\]\\leq e^\{\-u\}then forces each individual summand to be at moste−ue^\{\-u\}:

ℙp0​\(∑i=1nXi=j\)​ϕ¯​\(j\)≤e−ufor every​j∈\{0,…,n\}\.\\mathbb\{P\}\_\{p\_\{0\}\}\(\\sum\_\{i=1\}^\{n\}X\_\{i\}=j\)\\bar\{\\phi\}\(j\)\\leq e^\{\-u\}\\qquad\\text\{for every \}j\\in\\\{0,\\ldots,n\\\}\.Similarly,ℙp1​\(∑i=1nXi=j\)​\(1−ϕ¯​\(j\)\)≤e−v\\mathbb\{P\}\_\{p\_\{1\}\}\(\\sum\_\{i=1\}^\{n\}X\_\{i\}=j\)\(1\-\\bar\{\\phi\}\(j\)\)\\leq e^\{\-v\}for everyjj\. Now suppose, for somejj, bothℙp0​\(∑i=1nXi=j\)\>2​e−u\\mathbb\{P\}\_\{p\_\{0\}\}\(\\sum\_\{i=1\}^\{n\}X\_\{i\}=j\)\>2e^\{\-u\}andℙp1​\(∑i=1nXi=j\)\>2​e−v\\mathbb\{P\}\_\{p\_\{1\}\}\(\\sum\_\{i=1\}^\{n\}X\_\{i\}=j\)\>2e^\{\-v\}\. Then the displayed bounds force

ϕ¯​\(j\)<e−u2​e−u=12,1−ϕ¯​\(j\)<e−v2​e−v=12,\\bar\{\\phi\}\(j\)<\\frac\{e^\{\-u\}\}\{2e^\{\-u\}\}=\\frac\{1\}\{2\},\\qquad 1\-\\bar\{\\phi\}\(j\)<\\frac\{e^\{\-v\}\}\{2e^\{\-v\}\}=\\frac\{1\}\{2\},which together giveϕ¯​\(j\)\+\(1−ϕ¯​\(j\)\)<1\\bar\{\\phi\}\(j\)\+\(1\-\\bar\{\\phi\}\(j\)\)<1, a contradiction\. Hence every feasible test satisfies, for everyjj,

ℙp0​\(∑i=1nXi=j\)≤2​e−uorℙp1​\(∑i=1nXi=j\)≤2​e−v\.\\mathbb\{P\}\_\{p\_\{0\}\}\(\\sum\_\{i=1\}^\{n\}X\_\{i\}=j\)\\leq 2e^\{\-u\}\\quad\\text\{or\}\\quad\\mathbb\{P\}\_\{p\_\{1\}\}\(\\sum\_\{i=1\}^\{n\}X\_\{i\}=j\)\\leq 2e^\{\-v\}\.\(101\)

##### Step 5: lower bound via anη\\eta\-free shifted estimate\.

Letnk:=Nfixed,H​\(e−uk,e−vk\)n\_\{k\}:=N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(e^\{\-u\_\{k\}\},e^\{\-v\_\{k\}\}\)\. By Step 2 and \([94](https://arxiv.org/html/2609.28859#S10.E94)\),nk=O⁡\(Lk\)n\_\{k\}=O\(L\_\{k\}\), so there is a constantCNC\_\{N\}withnk≤CN​Lkn\_\{k\}\\leq C\_\{N\}L\_\{k\}for all largekk\. We prove

nk≥V⁡\(uk,vk\)−O⁡\(log⁡Lk\)as​k→∞\.n\_\{k\}\\geq V\(u\_\{k\},v\_\{k\}\)\-O\(\\log L\_\{k\}\)\\quad\\text\{as \}k\\to\\infty\.\(102\)Letmk:=min⁡\{uk,vk\}m\_\{k\}:=\\min\\\{u\_\{k\},v\_\{k\}\\\}\. Thenmk=Θ⁡\(Lk\)m\_\{k\}=\\Theta\(L\_\{k\}\)by Assumption[5\.3](https://arxiv.org/html/2609.28859#S5.SS3)\.

*Setup\.*LetI:=\[p0/2,\(1\+p1\)/2\]⊂\(0,1\)I:=\[p\_\{0\}/2,\(1\+p\_\{1\}\)/2\]\\subset\(0,1\)\. SinceD0,D1D\_\{0\},D\_\{1\}are continuously differentiable on the interior of\[0,1\]\[0,1\], their derivatives are bounded onII; letBD<∞B\_\{D\}<\\inftysatisfy\|D0′​\(ϑ\)\|≤BD\|D\_\{0\}^\{\\prime\}\(\\vartheta\)\|\\leq B\_\{D\}and\|D1′​\(ϑ\)\|≤BD\|D\_\{1\}^\{\\prime\}\(\\vartheta\)\|\\leq B\_\{D\}for allϑ∈I\\vartheta\\in I\. For eachkkset

δk:=BD\+1\+log⁡\{2​\(nk\+1\)\}\.\\delta\_\{k\}:=B\_\{D\}\+1\+\\log\\\{2\(n\_\{k\}\+1\)\\\}\.Sincenk≤CN​Lkn\_\{k\}\\leq C\_\{N\}L\_\{k\},δk=O⁡\(log⁡Lk\)\\delta\_\{k\}=O\(\\log L\_\{k\}\)\. Sincemk=Θ⁡\(Lk\)m\_\{k\}=\\Theta\(L\_\{k\}\),δk/mk=O⁡\(\(log⁡Lk\)/Lk\)→0\\delta\_\{k\}/m\_\{k\}=O\(\(\\log L\_\{k\}\)/L\_\{k\}\)\\to 0, soδk<mk=min⁡\{uk,vk\}\\delta\_\{k\}<m\_\{k\}=\\min\\\{u\_\{k\},v\_\{k\}\\\}for all largekk; in particularuk−δk\>0u\_\{k\}\-\\delta\_\{k\}\>0andvk−δk\>0v\_\{k\}\-\\delta\_\{k\}\>0\. We also note thatnk→∞n\_\{k\}\\to\\infty: by Lemma[4\.5](https://arxiv.org/html/2609.28859#S4.Thmtheorem5)applied to a feasible test onnkn\_\{k\}labels,nkJX\(1\)≥kl\(1−βk∥αk\)n\_\{k\}J^\{\(1\)\}\_\{X\}\\geq\\operatorname\{kl\}\(1\-\\beta\_\{k\}\\\|\\alpha\_\{k\}\), and the right\-hand side tends to infinity asαk,βk→0\\alpha\_\{k\},\\beta\_\{k\}\\to 0; this makes the grid spacing1/\(2​nk\)→01/\(2n\_\{k\}\)\\to 0used in the Discretizing paragraph below\.

*Anη\\eta\-free shifted lower bound\.*The core estimate is

nk≥V⁡\(uk−δk,vk−δk\)for all sufficiently large​k\.n\_\{k\}\\geq V\(u\_\{k\}\-\\delta\_\{k\},v\_\{k\}\-\\delta\_\{k\}\)\\qquad\\text\{for all sufficiently large \}k\.\(103\)We prove \([103](https://arxiv.org/html/2609.28859#S10.E103)\) by contradiction: suppose it fails along an infinite subsequence\{kℓ\}ℓ≥1\\\{k\_\{\\ell\}\\\}\_\{\\ell\\geq 1\}, so that

nkℓ<V⁡\(ukℓ−δkℓ,vkℓ−δkℓ\)\.n\_\{k\_\{\\ell\}\}<V\(u\_\{k\_\{\\ell\}\}\-\\delta\_\{k\_\{\\ell\}\},v\_\{k\_\{\\ell\}\}\-\\delta\_\{k\_\{\\ell\}\}\)\.\(104\)We will derive a contradiction by exhibiting, for eachℓ\\elllarge, a type classjkℓ∈\{0,1,…,nkℓ\}j\_\{k\_\{\\ell\}\}\\in\\\{0,1,\\ldots,n\_\{k\_\{\\ell\}\}\\\}for whichℙp0​\(∑i=1nkℓXi=jkℓ\)\>2​e−ukℓ\\mathbb\{P\}\_\{p\_\{0\}\}\(\\sum^\{n\_\{k\_\{\\ell\}\}\}\_\{i=1\}X\_\{i\}=j\_\{k\_\{\\ell\}\}\)\>2e^\{\-u\_\{k\_\{\\ell\}\}\}andℙp1​\(∑i=1nkℓXi=jkℓ\)\>2​e−vkℓ\\mathbb\{P\}\_\{p\_\{1\}\}\(\\sum^\{n\_\{k\_\{\\ell\}\}\}\_\{i=1\}X\_\{i\}=j\_\{k\_\{\\ell\}\}\)\>2e^\{\-v\_\{k\_\{\\ell\}\}\}, violating the type\-class incompatibility \([100](https://arxiv.org/html/2609.28859#S10.E100)\)\.

*Finding a goodϑkℓ∈\(p0,p1\)\\vartheta\_\{k\_\{\\ell\}\}\\in\(p\_\{0\},p\_\{1\}\)\.*We claim that \([104](https://arxiv.org/html/2609.28859#S10.E104)\) implies the existence ofϑkℓ∈\(p0,p1\)\\vartheta\_\{k\_\{\\ell\}\}\\in\(p\_\{0\},p\_\{1\}\)with

nkℓ​D0​\(ϑkℓ\)<ukℓ−δkℓ,nkℓ​D1​\(ϑkℓ\)<vkℓ−δkℓ\.n\_\{k\_\{\\ell\}\}D\_\{0\}\(\\vartheta\_\{k\_\{\\ell\}\}\)<u\_\{k\_\{\\ell\}\}\-\\delta\_\{k\_\{\\ell\}\},\\qquad n\_\{k\_\{\\ell\}\}D\_\{1\}\(\\vartheta\_\{k\_\{\\ell\}\}\)<v\_\{k\_\{\\ell\}\}\-\\delta\_\{k\_\{\\ell\}\}\.\(105\)The strategy is to define two open subsets of\(p0,p1\)\(p\_\{0\},p\_\{1\}\), one where the first inequality of \([105](https://arxiv.org/html/2609.28859#S10.E105)\) holds and one where the second holds, show that they cover\(p0,p1\)\(p\_\{0\},p\_\{1\}\), and use connectedness to conclude they must overlap\.

By the definition ofVVas an infimum,nkℓ<V⁡\(ukℓ−δkℓ,vkℓ−δkℓ\)n\_\{k\_\{\\ell\}\}<V\(u\_\{k\_\{\\ell\}\}\-\\delta\_\{k\_\{\\ell\}\},v\_\{k\_\{\\ell\}\}\-\\delta\_\{k\_\{\\ell\}\}\)means: for everyϑ∈\(p0,p1\)\\vartheta\\in\(p\_\{0\},p\_\{1\}\),

max⁡\{ukℓ−δkℓD0​\(ϑ\),vkℓ−δkℓD1​\(ϑ\)\}\>nkℓ\.\\max\\\!\\left\\\{\\frac\{u\_\{k\_\{\\ell\}\}\-\\delta\_\{k\_\{\\ell\}\}\}\{D\_\{0\}\(\\vartheta\)\},\\frac\{v\_\{k\_\{\\ell\}\}\-\\delta\_\{k\_\{\\ell\}\}\}\{D\_\{1\}\(\\vartheta\)\}\\right\\\}\>n\_\{k\_\{\\ell\}\}\.This is equivalent to: for everyϑ∈\(p0,p1\)\\vartheta\\in\(p\_\{0\},p\_\{1\}\),D0​\(ϑ\)<\(ukℓ−δkℓ\)/nkℓD\_\{0\}\(\\vartheta\)<\(u\_\{k\_\{\\ell\}\}\-\\delta\_\{k\_\{\\ell\}\}\)/n\_\{k\_\{\\ell\}\}orD1​\(ϑ\)<\(vkℓ−δkℓ\)/nkℓD\_\{1\}\(\\vartheta\)<\(v\_\{k\_\{\\ell\}\}\-\\delta\_\{k\_\{\\ell\}\}\)/n\_\{k\_\{\\ell\}\}\. Define the two sets

U:=\{ϑ∈\(p0,p1\):D0​\(ϑ\)<ukℓ−δkℓnkℓ\},U:=\\left\\\{\\vartheta\\in\(p\_\{0\},p\_\{1\}\):D\_\{0\}\(\\vartheta\)<\\frac\{u\_\{k\_\{\\ell\}\}\-\\delta\_\{k\_\{\\ell\}\}\}\{n\_\{k\_\{\\ell\}\}\}\\right\\\},V′:=\{ϑ∈\(p0,p1\):D1​\(ϑ\)<vkℓ−δkℓnkℓ\}\.V^\{\\prime\}:=\\left\\\{\\vartheta\\in\(p\_\{0\},p\_\{1\}\):D\_\{1\}\(\\vartheta\)<\\frac\{v\_\{k\_\{\\ell\}\}\-\\delta\_\{k\_\{\\ell\}\}\}\{n\_\{k\_\{\\ell\}\}\}\\right\\\}\.The displayed equivalence above says exactlyU∪V′=\(p0,p1\)U\\cup V^\{\\prime\}=\(p\_\{0\},p\_\{1\}\)\. Any pointϑkℓ∈U∩V′\\vartheta\_\{k\_\{\\ell\}\}\\in U\\cap V^\{\\prime\}satisfies both inequalities in \([105](https://arxiv.org/html/2609.28859#S10.E105)\); we now showU∩V′≠∅U\\cap V^\{\\prime\}\\neq\\emptyset\.

The setUUis open in\(p0,p1\)\(p\_\{0\},p\_\{1\}\)becauseD0D\_\{0\}is continuous; it contains a right\-neighborhood ofp0p\_\{0\}becauseD0​\(p0\)=0<\(ukℓ−δkℓ\)/nkℓD\_\{0\}\(p\_\{0\}\)=0<\(u\_\{k\_\{\\ell\}\}\-\\delta\_\{k\_\{\\ell\}\}\)/n\_\{k\_\{\\ell\}\}andD0D\_\{0\}is continuous atp0p\_\{0\}, soD0​\(ϑ\)<\(ukℓ−δkℓ\)/nkℓD\_\{0\}\(\\vartheta\)<\(u\_\{k\_\{\\ell\}\}\-\\delta\_\{k\_\{\\ell\}\}\)/n\_\{k\_\{\\ell\}\}forϑ\\varthetasufficiently close top0p\_\{0\}\. In particular,UUis nonempty\. Similarly,V′V^\{\\prime\}is open and contains a left\-neighborhood ofp1p\_\{1\}\(sinceD1​\(p1\)=0D\_\{1\}\(p\_\{1\}\)=0\), soV′V^\{\\prime\}is nonempty\.

IfU∩V′=∅U\\cap V^\{\\prime\}=\\emptyset, thenUUandV′V^\{\\prime\}would be two disjoint nonempty open subsets of\(p0,p1\)\(p\_\{0\},p\_\{1\}\)whose union is\(p0,p1\)\(p\_\{0\},p\_\{1\}\)\. This contradicts connectedness of the interval\(p0,p1\)\(p\_\{0\},p\_\{1\}\)\. HenceU∩V′≠∅U\\cap V^\{\\prime\}\\neq\\emptyset; anyϑkℓ∈U∩V′\\vartheta\_\{k\_\{\\ell\}\}\\in U\\cap V^\{\\prime\}satisfies \([105](https://arxiv.org/html/2609.28859#S10.E105)\)\.

*Discretizing\.*The point\-mass lower bound \([98](https://arxiv.org/html/2609.28859#S10.E98)\) is stated for binomial point masses at integer valuesjj\. To apply it, we replaceϑkℓ\\vartheta\_\{k\_\{\\ell\}\}by a nearby grid point of the formj/nkℓj/n\_\{k\_\{\\ell\}\}\.

Choosejkℓ∈\{0,1,…,nkℓ\}j\_\{k\_\{\\ell\}\}\\in\\\{0,1,\\ldots,n\_\{k\_\{\\ell\}\}\\\}with\|jkℓ/nkℓ−ϑkℓ\|≤1/\(2​nkℓ\)\|j\_\{k\_\{\\ell\}\}/n\_\{k\_\{\\ell\}\}\-\\vartheta\_\{k\_\{\\ell\}\}\|\\leq 1/\(2n\_\{k\_\{\\ell\}\}\)\(roundingϑkℓ⋅nkℓ\\vartheta\_\{k\_\{\\ell\}\}\\cdot n\_\{k\_\{\\ell\}\}to the nearest integer\), and writeϑ~kℓ:=jkℓ/nkℓ\\widetilde\{\\vartheta\}\_\{k\_\{\\ell\}\}:=j\_\{k\_\{\\ell\}\}/n\_\{k\_\{\\ell\}\}\. Sinceϑkℓ∈\(p0,p1\)\\vartheta\_\{k\_\{\\ell\}\}\\in\(p\_\{0\},p\_\{1\}\), its distance to the boundary ofII\(atp0/2p\_\{0\}/2and\(1\+p1\)/2\(1\+p\_\{1\}\)/2\) is bounded below by the positive constantmin⁡\{p0/2,\(1−p1\)/2\}\\min\\\{p\_\{0\}/2,\(1\-p\_\{1\}\)/2\\\}\. Since1/\(2​nkℓ\)→01/\(2n\_\{k\_\{\\ell\}\}\)\\to 0\(becausenkℓ→∞n\_\{k\_\{\\ell\}\}\\to\\infty\), for allℓ\\elllarge,\|ϑ~kℓ−ϑkℓ\|<min⁡\{p0/2,\(1−p1\)/2\}\|\\widetilde\{\\vartheta\}\_\{k\_\{\\ell\}\}\-\\vartheta\_\{k\_\{\\ell\}\}\|<\\min\\\{p\_\{0\}/2,\(1\-p\_\{1\}\)/2\\\}, so bothϑkℓ\\vartheta\_\{k\_\{\\ell\}\}andϑ~kℓ\\widetilde\{\\vartheta\}\_\{k\_\{\\ell\}\}lie inII\.

By the mean value theorem applied toDhD\_\{h\}onIIand the derivative bound\|Dh′​\(ϑ\)\|≤BD\|D\_\{h\}^\{\\prime\}\(\\vartheta\)\|\\leq B\_\{D\}forϑ∈I\\vartheta\\in I,

\|Dh​\(ϑ~kℓ\)−Dh​\(ϑkℓ\)\|≤BD⋅\|ϑ~kℓ−ϑkℓ\|≤BD2​nkℓ,h∈\{0,1\}\.\|D\_\{h\}\(\\widetilde\{\\vartheta\}\_\{k\_\{\\ell\}\}\)\-D\_\{h\}\(\\vartheta\_\{k\_\{\\ell\}\}\)\|\\leq B\_\{D\}\\cdot\|\\widetilde\{\\vartheta\}\_\{k\_\{\\ell\}\}\-\\vartheta\_\{k\_\{\\ell\}\}\|\\leq\\frac\{B\_\{D\}\}\{2n\_\{k\_\{\\ell\}\}\},\\qquad h\\in\\\{0,1\\\}\.Multiplying bynkℓn\_\{k\_\{\\ell\}\}givesnkℓ​\|Dh​\(ϑ~kℓ\)−Dh​\(ϑkℓ\)\|≤BD/2n\_\{k\_\{\\ell\}\}\|D\_\{h\}\(\\widetilde\{\\vartheta\}\_\{k\_\{\\ell\}\}\)\-D\_\{h\}\(\\vartheta\_\{k\_\{\\ell\}\}\)\|\\leq B\_\{D\}/2, sonkℓ​Dh​\(ϑ~kℓ\)≤nkℓ​Dh​\(ϑkℓ\)\+BD/2n\_\{k\_\{\\ell\}\}D\_\{h\}\(\\widetilde\{\\vartheta\}\_\{k\_\{\\ell\}\}\)\\leq n\_\{k\_\{\\ell\}\}D\_\{h\}\(\\vartheta\_\{k\_\{\\ell\}\}\)\+B\_\{D\}/2\. Combining with the strict inequalities in \([105](https://arxiv.org/html/2609.28859#S10.E105)\) and substitutingδkℓ=BD\+1\+log⁡\{2​\(nkℓ\+1\)\}\\delta\_\{k\_\{\\ell\}\}=B\_\{D\}\+1\+\\log\\\{2\(n\_\{k\_\{\\ell\}\}\+1\)\\\},

nkℓ​D0​\(ϑ~kℓ\)\\displaystyle n\_\{k\_\{\\ell\}\}D\_\{0\}\(\\widetilde\{\\vartheta\}\_\{k\_\{\\ell\}\}\)<ukℓ−δkℓ\+BD2\\displaystyle<u\_\{k\_\{\\ell\}\}\-\\delta\_\{k\_\{\\ell\}\}\+\\frac\{B\_\{D\}\}\{2\}=ukℓ−\(BD\+1\+log⁡\{2​\(nkℓ\+1\)\}\)\+BD2\\displaystyle=u\_\{k\_\{\\ell\}\}\-\\bigl\(B\_\{D\}\+1\+\\log\\\{2\(n\_\{k\_\{\\ell\}\}\+1\)\\\}\\bigr\)\+\\frac\{B\_\{D\}\}\{2\}=ukℓ−log⁡\{2​\(nkℓ\+1\)\}−1−BD2\\displaystyle=u\_\{k\_\{\\ell\}\}\-\\log\\\{2\(n\_\{k\_\{\\ell\}\}\+1\)\\\}\-1\-\\frac\{B\_\{D\}\}\{2\}<ukℓ−log⁡\{2​\(nkℓ\+1\)\},\\displaystyle<u\_\{k\_\{\\ell\}\}\-\\log\\\{2\(n\_\{k\_\{\\ell\}\}\+1\)\\\},where the last strict inequality uses−1−BD/2<0\-1\-B\_\{D\}/2<0\(sinceBD≥0B\_\{D\}\\geq 0\)\. Symmetrically,

nkℓ​D1​\(ϑ~kℓ\)<vkℓ−log⁡\{2​\(nkℓ\+1\)\}\.n\_\{k\_\{\\ell\}\}D\_\{1\}\(\\widetilde\{\\vartheta\}\_\{k\_\{\\ell\}\}\)<v\_\{k\_\{\\ell\}\}\-\\log\\\{2\(n\_\{k\_\{\\ell\}\}\+1\)\\\}\.
*Contradiction\.*By the point\-mass bound \([98](https://arxiv.org/html/2609.28859#S10.E98)\) applied withp=p0p=p\_\{0\},n=nkℓn=n\_\{k\_\{\\ell\}\},j=jkℓj=j\_\{k\_\{\\ell\}\},

ℙp0​\(∑i=1nkℓXi=jkℓ\)≥1nkℓ\+1​e−nkℓ​D0​\(ϑ~kℓ\)\.\\mathbb\{P\}\_\{p\_\{0\}\}\(\\sum^\{n\_\{k\_\{\\ell\}\}\}\_\{i=1\}X\_\{i\}=j\_\{k\_\{\\ell\}\}\)\\geq\\frac\{1\}\{n\_\{k\_\{\\ell\}\}\+1\}e^\{\-n\_\{k\_\{\\ell\}\}D\_\{0\}\(\\widetilde\{\\vartheta\}\_\{k\_\{\\ell\}\}\)\}\.The strict inequalitynkℓ​D0​\(ϑ~kℓ\)<ukℓ−log⁡\{2​\(nkℓ\+1\)\}n\_\{k\_\{\\ell\}\}D\_\{0\}\(\\widetilde\{\\vartheta\}\_\{k\_\{\\ell\}\}\)<u\_\{k\_\{\\ell\}\}\-\\log\\\{2\(n\_\{k\_\{\\ell\}\}\+1\)\\\}from the Discretizing paragraph gives, after negation,−nkℓ​D0​\(ϑ~kℓ\)\>−ukℓ\+log⁡\{2​\(nkℓ\+1\)\}\-n\_\{k\_\{\\ell\}\}D\_\{0\}\(\\widetilde\{\\vartheta\}\_\{k\_\{\\ell\}\}\)\>\-u\_\{k\_\{\\ell\}\}\+\\log\\\{2\(n\_\{k\_\{\\ell\}\}\+1\)\\\}\. Exponentiating,

e−nkℓ​D0​\(ϑ~kℓ\)\>e−ukℓ\+log⁡\{2​\(nkℓ\+1\)\}=2​\(nkℓ\+1\)​e−ukℓ\.e^\{\-n\_\{k\_\{\\ell\}\}D\_\{0\}\(\\widetilde\{\\vartheta\}\_\{k\_\{\\ell\}\}\)\}\>e^\{\-u\_\{k\_\{\\ell\}\}\+\\log\\\{2\(n\_\{k\_\{\\ell\}\}\+1\)\\\}\}=2\(n\_\{k\_\{\\ell\}\}\+1\)\\,e^\{\-u\_\{k\_\{\\ell\}\}\}\.Dividing bynkℓ\+1n\_\{k\_\{\\ell\}\}\+1,

1nkℓ\+1​e−nkℓ​D0​\(ϑ~kℓ\)\>2​e−ukℓ\.\\frac\{1\}\{n\_\{k\_\{\\ell\}\}\+1\}e^\{\-n\_\{k\_\{\\ell\}\}D\_\{0\}\(\\widetilde\{\\vartheta\}\_\{k\_\{\\ell\}\}\)\}\>2e^\{\-u\_\{k\_\{\\ell\}\}\}\.Combining with the point\-mass bound,ℙp0​\(∑i=1nkℓXi=jkℓ\)\>2​e−ukℓ\\mathbb\{P\}\_\{p\_\{0\}\}\(\\sum^\{n\_\{k\_\{\\ell\}\}\}\_\{i=1\}X\_\{i\}=j\_\{k\_\{\\ell\}\}\)\>2e^\{\-u\_\{k\_\{\\ell\}\}\}\. Symmetrically,ℙp1​\(∑i=1nkℓXi=jkℓ\)\>2​e−vkℓ\\mathbb\{P\}\_\{p\_\{1\}\}\(\\sum^\{n\_\{k\_\{\\ell\}\}\}\_\{i=1\}X\_\{i\}=j\_\{k\_\{\\ell\}\}\)\>2e^\{\-v\_\{k\_\{\\ell\}\}\}\.

By the definition ofnkℓ=Nfixed,H​\(e−ukℓ,e−vkℓ\)n\_\{k\_\{\\ell\}\}=N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(e^\{\-u\_\{k\_\{\\ell\}\}\},e^\{\-v\_\{k\_\{\\ell\}\}\}\), some test onnkℓn\_\{k\_\{\\ell\}\}Bernoulli labels achieves errors at most\(e−ukℓ,e−vkℓ\)\(e^\{\-u\_\{k\_\{\\ell\}\}\},e^\{\-v\_\{k\_\{\\ell\}\}\}\)\. Applying the type\-class incompatibility \([100](https://arxiv.org/html/2609.28859#S10.E100)\) to this test atj=jkℓj=j\_\{k\_\{\\ell\}\}yields eitherℙp0​\(∑i=1nkℓXi=jkℓ\)≤2​e−ukℓ\\mathbb\{P\}\_\{p\_\{0\}\}\(\\sum^\{n\_\{k\_\{\\ell\}\}\}\_\{i=1\}X\_\{i\}=j\_\{k\_\{\\ell\}\}\)\\leq 2e^\{\-u\_\{k\_\{\\ell\}\}\}orℙp1​\(∑i=1nkℓXi=jkℓ\)≤2​e−vkℓ\\mathbb\{P\}\_\{p\_\{1\}\}\(\\sum^\{n\_\{k\_\{\\ell\}\}\}\_\{i=1\}X\_\{i\}=j\_\{k\_\{\\ell\}\}\)\\leq 2e^\{\-v\_\{k\_\{\\ell\}\}\}, contradicting both strict inequalities we just derived\. The contradiction shows the contradiction hypothesis was false, so \([103](https://arxiv.org/html/2609.28859#S10.E103)\) holds\.

*From the shifted bound to \([102](https://arxiv.org/html/2609.28859#S10.E102)\)\.*We haveδk<mk\\delta\_\{k\}<m\_\{k\}anduk,vk≥mku\_\{k\},v\_\{k\}\\geq m\_\{k\}, so

uk−δk≥\(1−δkmk\)​uk,vk−δk≥\(1−δkmk\)​vk\.u\_\{k\}\-\\delta\_\{k\}\\geq\\Bigl\(1\-\\frac\{\\delta\_\{k\}\}\{m\_\{k\}\}\\Bigr\)u\_\{k\},\\qquad v\_\{k\}\-\\delta\_\{k\}\\geq\\Bigl\(1\-\\frac\{\\delta\_\{k\}\}\{m\_\{k\}\}\\Bigr\)v\_\{k\}\.By the positive homogeneityV⁡\(c​u,c​v\)=c​V​\(u,v\)V\(cu,cv\)=cV\(u,v\)and coordinatewise monotonicity ofVV,

V⁡\(uk−δk,vk−δk\)≥\(1−δkmk\)​V​\(uk,vk\),V\(u\_\{k\}\-\\delta\_\{k\},v\_\{k\}\-\\delta\_\{k\}\)\\geq\\Bigl\(1\-\\frac\{\\delta\_\{k\}\}\{m\_\{k\}\}\\Bigr\)V\(u\_\{k\},v\_\{k\}\),so, usingV⁡\(uk,vk\)≤CV​LkV\(u\_\{k\},v\_\{k\}\)\\leq C\_\{V\}L\_\{k\}from \([94](https://arxiv.org/html/2609.28859#S10.E94)\),

0≤V⁡\(uk,vk\)−V⁡\(uk−δk,vk−δk\)≤δkmk​V​\(uk,vk\)≤δkmk​CV​Lk\.0\\leq V\(u\_\{k\},v\_\{k\}\)\-V\(u\_\{k\}\-\\delta\_\{k\},v\_\{k\}\-\\delta\_\{k\}\)\\leq\\frac\{\\delta\_\{k\}\}\{m\_\{k\}\}V\(u\_\{k\},v\_\{k\}\)\\leq\\frac\{\\delta\_\{k\}\}\{m\_\{k\}\}\\,C\_\{V\}L\_\{k\}\.Combining with \([103](https://arxiv.org/html/2609.28859#S10.E103)\),

nk≥V⁡\(uk−δk,vk−δk\)≥V⁡\(uk,vk\)−δkmk​CV​Lk\.n\_\{k\}\\geq V\(u\_\{k\}\-\\delta\_\{k\},v\_\{k\}\-\\delta\_\{k\}\)\\geq V\(u\_\{k\},v\_\{k\}\)\-\\frac\{\\delta\_\{k\}\}\{m\_\{k\}\}\\,C\_\{V\}L\_\{k\}\.Sinceδk=O⁡\(log⁡Lk\)\\delta\_\{k\}=O\(\\log L\_\{k\}\)andmk=Θ⁡\(Lk\)m\_\{k\}=\\Theta\(L\_\{k\}\), the last term isO⁡\(\(log⁡Lk\)​Lk/Lk\)=O⁡\(log⁡Lk\)O\(\(\\log L\_\{k\}\)L\_\{k\}/L\_\{k\}\)=O\(\\log L\_\{k\}\)\. This proves \([102](https://arxiv.org/html/2609.28859#S10.E102)\), which together with the upper bound \([96](https://arxiv.org/html/2609.28859#S10.E96)\) proves \([93](https://arxiv.org/html/2609.28859#S10.E93)\)\.

###### Proof 10\.15

Proof of Lemma[10\.9](https://arxiv.org/html/2609.28859#S10.Thmtheorem9)

Part \(i\) directly follows from the proof of Theorem[5\.3](https://arxiv.org/html/2609.28859#S5.Thmtheorem3)\(i\)\. We thus omit its proof\. We prove Part \(ii\) now\. Letαk,βk↓0\\alpha\_\{k\},\\beta\_\{k\}\\downarrow 0withuk,vk,Lku\_\{k\},v\_\{k\},L\_\{k\}as in Lemma[10\.13](https://arxiv.org/html/2609.28859#S10.Thmtheorem13), and writemk:=min⁡\{uk,vk\}=Θ⁡\(Lk\)m\_\{k\}:=\\min\\\{u\_\{k\},v\_\{k\}\\\}=\\Theta\(L\_\{k\}\)\. Letck:=log⁡\(1/ffb,k\)≥0c\_\{k\}:=\\log\(1/f\_\{\\mathrm\{fb\},k\}\)\\geq 0, soα2,k=e−\(uk\+ck\)\\alpha\_\{2,k\}=e^\{\-\(u\_\{k\}\+c\_\{k\}\)\}andβ2,k=e−\(vk\+ck\)\\beta\_\{2,k\}=e^\{\-\(v\_\{k\}\+c\_\{k\}\)\}\. Adding the common shiftckc\_\{k\}to both coordinates preserves balance:min⁡\{uk\+ck,vk\+ck\}=mk\+ck=Θ⁡\(Lk\+ck\)\\min\\\{u\_\{k\}\+c\_\{k\},v\_\{k\}\+c\_\{k\}\\\}=m\_\{k\}\+c\_\{k\}=\\Theta\(L\_\{k\}\+c\_\{k\}\), so the strengthened Lemma[10\.13](https://arxiv.org/html/2609.28859#S10.Thmtheorem13)\(theO\(log⋅\)O\(\\log\\cdot\)form \([93](https://arxiv.org/html/2609.28859#S10.E93)\)\) applies at both levels:

Nfixed,H​\(αk,βk\)=V⁡\(uk,vk\)\+O⁡\(log⁡Lk\),N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)=V\(u\_\{k\},v\_\{k\}\)\+O\(\\log L\_\{k\}\),\(106\)Nfixed,H​\(α2,k,β2,k\)=V⁡\(uk\+ck,vk\+ck\)\+O⁡\(log⁡\(Lk\+ck\)\)\.N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)=V\(u\_\{k\}\+c\_\{k\},v\_\{k\}\+c\_\{k\}\)\+O\(\\log\(L\_\{k\}\+c\_\{k\}\)\)\.\(107\)Subtracting,

Nfixed,H​\(α2,k,β2,k\)−Nfixed,H​\(αk,βk\)=\[V⁡\(uk\+ck,vk\+ck\)−V⁡\(uk,vk\)\]\+O⁡\(log⁡\(Lk\+ck\)\)\.N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\-N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)=\\bigl\[V\(u\_\{k\}\+c\_\{k\},v\_\{k\}\+c\_\{k\}\)\-V\(u\_\{k\},v\_\{k\}\)\\bigr\]\+O\(\\log\(L\_\{k\}\+c\_\{k\}\)\)\.
*Frontier shift\.*For everyϑ∈\(p0,p1\)\\vartheta\\in\(p\_\{0\},p\_\{1\}\), usinguk≥mku\_\{k\}\\geq m\_\{k\}andvk≥mkv\_\{k\}\\geq m\_\{k\},

uk\+ck≤\(1\+ckmk\)​uk,vk\+ck≤\(1\+ckmk\)​vk\.u\_\{k\}\+c\_\{k\}\\leq\\left\(1\+\\frac\{c\_\{k\}\}\{m\_\{k\}\}\\right\)u\_\{k\},\\qquad v\_\{k\}\+c\_\{k\}\\leq\\left\(1\+\\frac\{c\_\{k\}\}\{m\_\{k\}\}\\right\)v\_\{k\}\.Dividing byD0​\(ϑ\)\>0D\_\{0\}\(\\vartheta\)\>0andD1​\(ϑ\)\>0D\_\{1\}\(\\vartheta\)\>0, taking the maximum, and then the infimum overϑ∈\(p0,p1\)\\vartheta\\in\(p\_\{0\},p\_\{1\}\)\(the positive factor1\+ck/mk1\+c\_\{k\}/m\_\{k\}pulls out of the infimum\),

V⁡\(uk\+ck,vk\+ck\)≤\(1\+ckmk\)​V​\(uk,vk\)\.V\(u\_\{k\}\+c\_\{k\},v\_\{k\}\+c\_\{k\}\)\\leq\\left\(1\+\\frac\{c\_\{k\}\}\{m\_\{k\}\}\\right\)V\(u\_\{k\},v\_\{k\}\)\.By coordinatewise monotonicity ofVV\(immediate from \([92](https://arxiv.org/html/2609.28859#S10.E92)\), sinceuk\+ck≥uku\_\{k\}\+c\_\{k\}\\geq u\_\{k\}andvk\+ck≥vkv\_\{k\}\+c\_\{k\}\\geq v\_\{k\}\),V⁡\(uk\+ck,vk\+ck\)≥V⁡\(uk,vk\)V\(u\_\{k\}\+c\_\{k\},v\_\{k\}\+c\_\{k\}\)\\geq V\(u\_\{k\},v\_\{k\}\)\. Combining,

0≤V⁡\(uk\+ck,vk\+ck\)−V⁡\(uk,vk\)≤ckmk​V​\(uk,vk\)\.0\\leq V\(u\_\{k\}\+c\_\{k\},v\_\{k\}\+c\_\{k\}\)\-V\(u\_\{k\},v\_\{k\}\)\\leq\\frac\{c\_\{k\}\}\{m\_\{k\}\}V\(u\_\{k\},v\_\{k\}\)\.\(108\)
*Conclusion\.*By Lemma[10\.13](https://arxiv.org/html/2609.28859#S10.Thmtheorem13),V⁡\(uk,vk\)=Θ⁡\(Lk\)V\(u\_\{k\},v\_\{k\}\)=\\Theta\(L\_\{k\}\); hence the right\-hand side of \([108](https://arxiv.org/html/2609.28859#S10.E108)\) isckmk​V​\(uk,vk\)=O⁡\(ck\)=O⁡\(log⁡\(1/ffb,k\)\)\\frac\{c\_\{k\}\}\{m\_\{k\}\}V\(u\_\{k\},v\_\{k\}\)=O\(c\_\{k\}\)=O\(\\log\(1/f\_\{\\mathrm\{fb\},k\}\)\)\. Combining with the subtraction display andO⁡\(log⁡\(Lk\+ck\)\)=O⁡\(log⁡Lk\+log⁡\(1/ffb,k\)\)O\(\\log\(L\_\{k\}\+c\_\{k\}\)\)=O\(\\log L\_\{k\}\+\\log\(1/f\_\{\\mathrm\{fb\},k\}\)\),

0≤Nfixed,H​\(α2,k,β2,k\)−Nfixed,H​\(αk,βk\)≤log⁡\(1/ffb,k\)mk​V​\(uk,vk\)\+O⁡\(log⁡Lk\)=O⁡\(log⁡Lk\+log⁡\(1ffb,k\)\),0\\leq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\-N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)\\leq\\frac\{\\log\(1/f\_\{\\mathrm\{fb\},k\}\)\}\{m\_\{k\}\}V\(u\_\{k\},v\_\{k\}\)\+O\(\\log L\_\{k\}\)=O\\\!\\left\(\\log L\_\{k\}\+\\log\\left\(\\frac\{1\}\{f\_\{\\mathrm\{fb\},k\}\}\\right\)\\right\),which is \([90](https://arxiv.org/html/2609.28859#S10.E90)\)\. The lower bound holds becauseNfixed,HN\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}is nonincreasing in each error budget andα2,k=ffb,k​αk<αk\\alpha\_\{2,k\}=f\_\{\\mathrm\{fb\},k\}\\alpha\_\{k\}<\\alpha\_\{k\},β2,k=ffb,k​βk<βk\\beta\_\{2,k\}=f\_\{\\mathrm\{fb\},k\}\\beta\_\{k\}<\\beta\_\{k\}\.

### 10\.3Proof of Lemma[10\.4](https://arxiv.org/html/2609.28859#S10.Thmtheorem4)

We define some notations that will be used throughout Appendices[10\.3](https://arxiv.org/html/2609.28859#S10.SS3)\-[10\.7](https://arxiv.org/html/2609.28859#S10.SS7)\. LetZj,kZ\_\{j,k\}denote the one\-itemH1H\_\{1\}\-versus\-H0H\_\{0\}log\-likelihood increment produced by rulej∈\{0,1,∗\}j\\in\\\{0,1,\*\\\}on a fresh item before boundary stopping\. According to Algorithm[1](https://arxiv.org/html/2609.28859#alg1), under rulejj, the item is first sent to a human with probabilityηj,kH\\eta^\{\\mathrm\{H\}\}\_\{j,k\}and otherwise sent to the AI, after which it is escalated to a human with probabilityηj,kesc​\(R\)\\eta^\{\\mathrm\{esc\}\}\_\{j,k\}\(R\)\. WritingU,V∼Uniform⁡\[0,1\]U,V\\sim\\mathrm\{Uniform\}\[0,1\]for the two independent randomization seeds, the increment is

Zj,k:=𝟏\{U≤ηj,kH\}ℓX\(X\)\+𝟏\{U\>ηj,kH\}\(ℓR\(R\)\+𝟏\{V≤ηj,kesc\(R\)\}ℓH\(X,R\)\),Z\_\{j,k\}:=\\mathbf\{1\}\\\{U\\leq\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\\\}\\,\\ell\_\{X\}\(X\)\+\\mathbf\{1\}\\\{U\>\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\\\}\\Bigl\(\\ell\_\{R\}\(R\)\+\\mathbf\{1\}\\\{V\\leq\\eta^\{\\mathrm\{esc\}\}\_\{j,k\}\(R\)\\\}\\,\\ell\_\{H\}\(X,R\)\\Bigr\),\(109\)whereℓX,ℓR,ℓH\\ell\_\{X\},\\ell\_\{R\},\\ell\_\{H\}are the increments \([27](https://arxiv.org/html/2609.28859#S5.E27)\)–\([29](https://arxiv.org/html/2609.28859#S5.E29)\)\. Writeσ1:=\+1\\sigma\_\{1\}:=\+1andσ0:=−1\\sigma\_\{0\}:=\-1, so the signed drift underHhH\_\{h\}is𝔼hπk​\[σh​Zj,k\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{j,k\}\]\.

Throughout Appendices[10\.3](https://arxiv.org/html/2609.28859#S10.SS3)\-[10\.6](https://arxiv.org/html/2609.28859#S10.SS6), we index the process at item\-completion times\. Work with the completed\-item continuation on which the fresh tuple\(Xi,Ri,Ui,Vi\)\(X\_\{i\},R\_\{i\},U\_\{i\},V\_\{i\}\)is generated for every acquired item and each AI\-first item is completed according to the escalation rule selected at its start; if the actual policy has already stopped, this continuation is only counterfactual\. With some abuse of notations we letℋ0\\mathcal\{H\}\_\{0\}be the trivial sigma\-field and letℋi\\mathcal\{H\}\_\{i\}be the sigma\-field generated by the sensing decisions, randomization seeds, and observations through the completion of itemii\. SetS0:=0S\_\{0\}:=0and recursively chooseJiJ\_\{i\}fromSi−1S\_\{i\-1\}by \([26](https://arxiv.org/html/2609.28859#S5.E26)\), and define

Yi:=\\displaystyle Y\_\{i\}:=\{\}𝟏\{Ui≤ηHJi,k\}ℓX\(Xi\)\+𝟏\{Ui\>ηHJi,k\}\(ℓR\(Ri\)\+𝟏\{Vi≤ηescJi,k\(Ri\)\}ℓH\(Xi,Ri\)\)\.\\displaystyle\\mathbf\{1\}\\\{U\_\{i\}\\leq\\eta^\{\\mathrm\{H\}\}\_\{J\_\{i\},k\}\\\}\\,\\ell\_\{X\}\(X\_\{i\}\)\+\\mathbf\{1\}\\\{U\_\{i\}\>\\eta^\{\\mathrm\{H\}\}\_\{J\_\{i\},k\}\\\}\\Bigl\(\\ell\_\{R\}\(R\_\{i\}\)\+\\mathbf\{1\}\\\{V\_\{i\}\\leq\\eta^\{\\mathrm\{esc\}\}\_\{J\_\{i\},k\}\(R\_\{i\}\)\\\}\\,\\ell\_\{H\}\(X\_\{i\},R\_\{i\}\)\\Bigr\)\.Note that the conditional law ofYiY\_\{i\}givenℋi−1\\mathcal\{H\}\_\{i\-1\}is exactly the unconditional law of the single\-item incrementZJi,kZ\_\{J\_\{i\},k\}under the \(now fixed\) ruleJiJ\_\{i\}\. With some abuse of notations we letSi:=∑ℓ=1iYℓS\_\{i\}:=\\sum\_\{\\ell=1\}^\{i\}Y\_\{\\ell\}denote the cumulative log\-likelihood statistic after completing the first i items\. BecauseJiJ\_\{i\}isℋi−1\\mathcal\{H\}\_\{i\-1\}\-measurable and the tuple for itemiiis fresh, the conditional law ofYiY\_\{i\}givenℋi−1\\mathcal\{H\}\_\{i\-1\}is the law of the generic incrementZJi,kZ\_\{J\_\{i\},k\}\. In particular,SiS\_\{i\}is the cumulative log\-likelihood ratio at the end of itemii, whileSi−1S\_\{i\-1\}andJiJ\_\{i\}areℋi−1\\mathcal\{H\}\_\{i\-1\}\-measurable\. UnderH0H\_\{0\}we apply the same item\-level construction to the reflected increments−Yi\-Y\_\{i\}and the reflected statistic−Si\-S\_\{i\}\.

To prove Lemma[10\.4](https://arxiv.org/html/2609.28859#S10.Thmtheorem4), we need auxiliary Lemma[10\.16](https://arxiv.org/html/2609.28859#S10.Thmtheorem16), and we present the proof of Lemma[10\.4](https://arxiv.org/html/2609.28859#S10.Thmtheorem4)in the end of this subsection\.

###### Lemma 10\.16\(Finite\-alphabet increment control and target\-scale drift\)

There are primitive constantsBinc<∞B\_\{\\mathrm\{inc\}\}<\\infty,c×\>0c\_\{\\times\}\>0,μ0\>0\\mu\_\{0\}\>0, andCV<∞C\_\{V\}<\\inftysuch that the following statements hold\.

\(i\) For everykk,j∈\{0,1,∗\}j\\in\\\{0,1,\*\\\}, andh∈\{0,1\}h\\in\\\{0,1\\\},

\|Zj,k\|≤Binc\.\|Z\_\{j,k\}\|\\leq B\_\{\\mathrm\{inc\}\}\.\(110\)
\(ii\) For everykk,h∈\{0,1\}h\\in\\\{0,1\\\}, andj∈\{0,1,∗\}j\\in\\\{0,1,\*\\\},

𝔼hπk​\[σh​Zj,k\]≥c×​𝔼1−hπk​\[σ1−h​Zj,k\]\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{j,k\}\]\\geq c\_\{\\times\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\-h\}\[\\sigma\_\{1\-h\}Z\_\{j,k\}\]\.\(111\)
\(iii\) For all sufficiently largekk,

𝔼hπk\[σhZj,k\]≥μ0\(h∈\{0,1\},j∈\{0,1,∗\}\)\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{j,k\}\]\\geq\\mu\_\{0\}\\qquad\(h\\in\\\{0,1\\\},\\ j\\in\\\{0,1,\*\\\}\)\.\(112\)
\(iv\) For all sufficiently largekk,h∈\{0,1\}h\\in\\\{0,1\\\}, andj∈\{0,1,∗\}j\\in\\\{0,1,\*\\\},

Varh⁡\(Zj,k\)≤CV​𝔼hπk​\[σh​Zj,k\]\.\\operatorname\{Var\}\_\{h\}\(Z\_\{j,k\}\)\\leq C\_\{V\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{j,k\}\]\.\(113\)

###### Proof 10\.17

Proof of Lemma[10\.16](https://arxiv.org/html/2609.28859#S10.Thmtheorem16)\(i\) We notice thatℓX​\(X\),ℓR​\(R\),ℓH​\(X,R\)\\ell\_\{X\}\(X\),\\ell\_\{R\}\(R\),\\ell\_\{H\}\(X,R\)all have bounded absolute values since the support sets are finite and common under the two hypotheses,p0,p1∈\(0,1\)p\_\{0\},p\_\{1\}\\in\(0,1\), andg1g\_\{1\}andg0g\_\{0\}are strictly positive onℛ\\mathcal\{R\}\. This proves \([110](https://arxiv.org/html/2609.28859#S10.E110)\)\.

\(ii\) We establish the reverse comparison \([111](https://arxiv.org/html/2609.28859#S10.E111)\), starting with a term\-by\-term derivation of the one\-item signed expectation\. Taking expectations in \([109](https://arxiv.org/html/2609.28859#S10.E109)\) termwise gives, forh=1h=1,

𝔼1πk​\[Zj,k\]=ηj,kH​𝔼1πk​\[ℓX​\(X\)\]\+\(1−ηj,kH\)​𝔼1πk​\[ℓR​\(R\)\]\+\(1−ηj,kH\)​∑r∈ℛg1​\(r\)​ηj,kesc​\(r\)​𝔼1πk​\[ℓH​\(X,r\)∣R=r\],\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{j,k\}\]=\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\ell\_\{X\}\(X\)\]\+\(1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\)\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\ell\_\{R\}\(R\)\]\+\(1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\)\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{1\}\(r\)\\,\\eta^\{\\mathrm\{esc\}\}\_\{j,k\}\(r\)\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\ell\_\{H\}\(X,r\)\\mid R=r\],and, by the identical computation withH0H\_\{0\}in place ofH1H\_\{1\},

𝔼0πk​\[Zj,k\]=ηj,kH​𝔼0πk​\[ℓX​\(X\)\]\+\(1−ηj,kH\)​𝔼0πk​\[ℓR​\(R\)\]\+\(1−ηj,kH\)​∑r∈ℛg0​\(r\)​ηj,kesc​\(r\)​𝔼0πk​\[ℓH​\(X,r\)∣R=r\]\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[Z\_\{j,k\}\]=\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\ell\_\{X\}\(X\)\]\+\(1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\)\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\ell\_\{R\}\(R\)\]\+\(1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\)\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{0\}\(r\)\\,\\eta^\{\\mathrm\{esc\}\}\_\{j,k\}\(r\)\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\ell\_\{H\}\(X,r\)\\mid R=r\]\.By \([27](https://arxiv.org/html/2609.28859#S5.E27)\) and \([10](https://arxiv.org/html/2609.28859#S4.E10)\),𝔼1πk​\[ℓX​\(X\)\]=JX\(1\)\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\ell\_\{X\}\(X\)\]=J\_\{X\}^\{\(1\)\}and𝔼0πk​\[ℓX​\(X\)\]=−JX\(0\)\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\ell\_\{X\}\(X\)\]=\-J\_\{X\}^\{\(0\)\}; by \([28](https://arxiv.org/html/2609.28859#S5.E28)\) and \([8](https://arxiv.org/html/2609.28859#S4.E8)\)–\([9](https://arxiv.org/html/2609.28859#S4.E9)\),𝔼1πk​\[ℓR​\(R\)\]=IR\(1\)\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\ell\_\{R\}\(R\)\]=I\_\{R\}^\{\(1\)\}and𝔼0πk​\[ℓR​\(R\)\]=−IR\(0\)\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\ell\_\{R\}\(R\)\]=\-I\_\{R\}^\{\(0\)\}; and by \([29](https://arxiv.org/html/2609.28859#S5.E29)\) and \([11](https://arxiv.org/html/2609.28859#S4.E11)\)–\([12](https://arxiv.org/html/2609.28859#S4.E12)\),𝔼1πk​\[ℓH​\(X,r\)∣R=r\]=d\(1\)​\(r\)\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\ell\_\{H\}\(X,r\)\\mid R=r\]=d^\{\(1\)\}\(r\)and𝔼0πk​\[ℓH​\(X,r\)∣R=r\]=−d\(0\)​\(r\)\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{0\}\[\\ell\_\{H\}\(X,r\)\\mid R=r\]=\-d^\{\(0\)\}\(r\)\. Substituting into the two displays above and usingσ1=1\\sigma\_\{1\}=1andσ0=−1\\sigma\_\{0\}=\-1, the one\-item signed expectation of rulejjunderHhH\_\{h\}is

𝔼hπk\[σhZj,k\]=ηj,kHJX\(h\)\+\(1−ηj,kH\)IR\(h\)\+\(1−ηj,kH\)∑r∈ℛgh\(r\)ηj,kesc\(r\)d\(h\)\(r\),h∈\{0,1\},j∈\{0,1,∗\}\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{j,k\}\]=\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\\,J\_\{X\}^\{\(h\)\}\+\(1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\)\\,I\_\{R\}^\{\(h\)\}\+\(1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\)\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\,\\eta^\{\\mathrm\{esc\}\}\_\{j,k\}\(r\)\\,d^\{\(h\)\}\(r\),\\qquad h\\in\\\{0,1\\\},\\ j\\in\\\{0,1,\*\\\}\.\(114\)The human probabilityηj,kH\\eta^\{\\mathrm\{H\}\}\_\{j,k\}and the escalation ruleηj,kesc\\eta^\{\\mathrm\{esc\}\}\_\{j,k\}are fixed numbers that do not depend on the hypothesis, so𝔼hπk​\[σh​Zj,k\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{j,k\}\]and𝔼1−hπk​\[σ1−h​Zj,k\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\-h\}\[\\sigma\_\{1\-h\}Z\_\{j,k\}\]differ only through the information coefficients and the report weightsgh​\(r\)g\_\{h\}\(r\)\.

We compare \([114](https://arxiv.org/html/2609.28859#S10.E114)\) forhhagainst1−h1\-htermwise\. Because the primitives\(p0,p1,f0,f1\)\(p\_\{0\},p\_\{1\},f\_\{0\},f\_\{1\}\)are fixed, there are only finitely many component KL pairs: the direct\-label pair\(JX\(1\),JX\(0\)\)\(J\_\{X\}^\{\(1\)\},J\_\{X\}^\{\(0\)\}\), the report pair\(IR\(1\),IR\(0\)\)\(I\_\{R\}^\{\(1\)\},I\_\{R\}^\{\(0\)\}\), and the\|ℛ\|\|\\mathcal\{R\}\|escalation pairs\(d\(1\)​\(r\),d\(0\)​\(r\)\)\(d^\{\(1\)\}\(r\),d^\{\(0\)\}\(r\)\)\. For any two lawsP,QP,Qon a common finite support,D\(P∥Q\)=0D\(P\\\|Q\)=0if and only ifP=QP=Q, in which caseD\(Q∥P\)=0D\(Q\\\|P\)=0as well; otherwise both are strictly positive\. Hence each component pair is either zero in both directions or strictly positive in both, and

ccmp:=min\{JX\(h\)JX\(1−h\),IR\(h\)IR\(1−h\),d\(h\)​\(r\)d\(1−h\)​\(r\):h∈\{0,1\},r∈ℛ,JX\(1−h\),IR\(1−h\),d\(1−h\)\(r\)\>0\}c\_\{\\mathrm\{cmp\}\}:=\\min\\Bigl\\\{\\tfrac\{J\_\{X\}^\{\(h\)\}\}\{J\_\{X\}^\{\(1\-h\)\}\},\\,\\tfrac\{I\_\{R\}^\{\(h\)\}\}\{I\_\{R\}^\{\(1\-h\)\}\},\\,\\tfrac\{d^\{\(h\)\}\(r\)\}\{d^\{\(1\-h\)\}\(r\)\}\\ :\\ h\\in\\\{0,1\\\},\\ r\\in\\mathcal\{R\},\\ J\_\{X\}^\{\(1\-h\)\},I\_\{R\}^\{\(1\-h\)\},d^\{\(1\-h\)\}\(r\)\>0\\Bigr\\\}is a strictly positive constant depending only on the primitives, and every component obeys the two\-sided bound

JX\(h\)≥ccmp​JX\(1−h\),IR\(h\)≥ccmp​IR\(1−h\),d\(h\)​\(r\)≥ccmp​d\(1−h\)​\(r\),∀h∈\{0,1\},r∈ℛ,J\_\{X\}^\{\(h\)\}\\geq c\_\{\\mathrm\{cmp\}\}\\,J\_\{X\}^\{\(1\-h\)\},\\quad I\_\{R\}^\{\(h\)\}\\geq c\_\{\\mathrm\{cmp\}\}\\,I\_\{R\}^\{\(1\-h\)\},\\quad d^\{\(h\)\}\(r\)\\geq c\_\{\\mathrm\{cmp\}\}\\,d^\{\(1\-h\)\}\(r\),\\quad\\forall h\\in\\\{0,1\\\},r\\in\\mathcal\{R\},\(115\)where each inequality is trivial when its right\-hand side vanishes\.

The escalation term additionally carries the report weightgh​\(r\)g\_\{h\}\(r\), which differs across hypotheses\. Sinceg0g\_\{0\}andg1g\_\{1\}are fixed and strictly positive on the finite setℛ\\mathcal\{R\}\(by \([2](https://arxiv.org/html/2609.28859#S3.E2)\) and Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)\), their likelihood ratio is bounded:mg:=minr∈ℛ,h∈\{0,1\}⁡gh​\(r\)g1−h​\(r\)∈\(0,∞\)m\_\{g\}:=\\min\_\{r\\in\\mathcal\{R\},\\ h\\in\\\{0,1\\\}\}\\frac\{g\_\{h\}\(r\)\}\{g\_\{1\-h\}\(r\)\}\\in\(0,\\infty\)\.

Combininggh​\(r\)≥mg​g1−h​\(r\)≥0g\_\{h\}\(r\)\\geq m\_\{g\}\\,g\_\{1\-h\}\(r\)\\geq 0with the escalation bound in \([115](https://arxiv.org/html/2609.28859#S10.E115)\), and multiplying the two nonnegative\-termed inequalities together, for everyr∈ℛr\\in\\mathcal\{R\}

gh​\(r\)​d\(h\)​\(r\)≥mg​ccmp​g1−h​\(r\)​d\(1−h\)​\(r\);g\_\{h\}\(r\)\\,d^\{\(h\)\}\(r\)\\geq m\_\{g\}\\,c\_\{\\mathrm\{cmp\}\}\\;g\_\{1\-h\}\(r\)\\,d^\{\(1\-h\)\}\(r\);multiplying byηj,kesc​\(r\)∈\[0,1\]\\eta^\{\\mathrm\{esc\}\}\_\{j,k\}\(r\)\\in\[0,1\], summing overrr\(both operations preserve the inequality, since they act identically on both sides\), and then multiplying by the nonnegative weight\(1−ηj,kH\)\(1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\)gives

\(1−ηj,kH\)​∑r∈ℛgh​\(r\)​ηj,kesc​\(r\)​d\(h\)​\(r\)≥mg​ccmp​\(1−ηj,kH\)​∑r∈ℛg1−h​\(r\)​ηj,kesc​\(r\)​d\(1−h\)​\(r\)\.\(1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\)\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\,\\eta^\{\\mathrm\{esc\}\}\_\{j,k\}\(r\)\\,d^\{\(h\)\}\(r\)\\geq m\_\{g\}c\_\{\\mathrm\{cmp\}\}\\,\(1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\)\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{1\-h\}\(r\)\\,\\eta^\{\\mathrm\{esc\}\}\_\{j,k\}\(r\)\\,d^\{\(1\-h\)\}\(r\)\.Additionally, multiplying \([115](https://arxiv.org/html/2609.28859#S10.E115)\)ηj,kH\\eta^\{\\mathrm\{H\}\}\_\{j,k\}and1−ηj,kH1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}on both sides gives

ηj,kH​JX\(h\)≥ccmp​ηj,kH​JX\(1−h\),\(1−ηj,kH\)​IR\(h\)≥ccmp​\(1−ηj,kH\)​IR\(1−h\)\.\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\\,J\_\{X\}^\{\(h\)\}\\geq c\_\{\\mathrm\{cmp\}\}\\,\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\\,J\_\{X\}^\{\(1\-h\)\},\\qquad\(1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\)\\,I\_\{R\}^\{\(h\)\}\\geq c\_\{\\mathrm\{cmp\}\}\\,\(1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\)\\,I\_\{R\}^\{\(1\-h\)\}\.
Setc×:=ccmp​min⁡\{1,mg\}\>0c\_\{\\times\}:=c\_\{\\mathrm\{cmp\}\}\\min\\\{1,m\_\{g\}\\\}\>0\. Then we have

𝔼hπk​\[σh​Zj,k\]\\displaystyle\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{j,k\}\]≥c×\[ηj,kHJX\(1−h\)\+\(1−ηj,kH\)IR\(1−h\)\\displaystyle\\geq c\_\{\\times\}\\Bigl\[\\eta^\{\\mathrm\{H\}\}\_\{j,k\}J\_\{X\}^\{\(1\-h\)\}\+\(1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\)I\_\{R\}^\{\(1\-h\)\}\+\(1−ηj,kH\)∑r∈ℛg1−h\(r\)ηj,kesc\(r\)d\(1−h\)\(r\)\]\\displaystyle\\hskip 76\.82243pt\+\(1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\)\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{1\-h\}\(r\)\\eta^\{\\mathrm\{esc\}\}\_\{j,k\}\(r\)d^\{\(1\-h\)\}\(r\)\\Bigr\]=c×​𝔼1−hπk​\[σ1−h​Zj,k\]\.\\displaystyle=c\_\{\\times\}\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\-h\}\[\\sigma\_\{1\-h\}Z\_\{j,k\}\]\.forh∈\{0,1\},j∈\{0,1,∗\},h\\in\\\{0,1\\\},\\ j\\in\\\{0,1,\*\\\},which is \([111](https://arxiv.org/html/2609.28859#S10.E111)\)\. The argument applies verbatim to the dead\-zone rulej=∗j=\*, since \([114](https://arxiv.org/html/2609.28859#S10.E114)\) holds for its parameters\(η∗,kH,η∗,kesc\)\(\\eta^\{\\mathrm\{H\}\}\_\{\*,k\},\\eta^\{\\mathrm\{esc\}\}\_\{\*,k\}\)as well; the constantc×c\_\{\\times\}depends only on the primitives and not on the rule\.

\(iii\) We have that𝔼hπk​\[σh​Zj,k\]=ηj,kH​JX\(h\)\+\(1−ηj,kH\)​IR\(h\)\+\(1−ηj,kH\)​∑r∈ℛgh​\(r\)​ηj,kesc​\(r\)​d\(h\)​\(r\)\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{j,k\}\]=\\eta^\{\\mathrm\{H\}\}\_\{j,k\}J\_\{X\}^\{\(h\)\}\+\(1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\)I\_\{R\}^\{\(h\)\}\+\(1\-\\eta^\{\\mathrm\{H\}\}\_\{j,k\}\)\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\eta^\{\\mathrm\{esc\}\}\_\{j,k\}\(r\)d^\{\(h\)\}\(r\)\.

*Correct rulej=hj=h\.*By construction \([39](https://arxiv.org/html/2609.28859#S5.E39)\), the escalation ruleηh,kesc\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}is the fractional\-knapsack maximizer in the dual \([16](https://arxiv.org/html/2609.28859#S4.E16)\) at budgetsh,ks\_\{h,k\}, so it attains the frontier exactly:

∑r∈ℛgh​\(r\)​ηh,kesc​\(r\)=sh,k,∑r∈ℛgh​\(r\)​ηh,kesc​\(r\)​d\(h\)​\(r\)=Ψh​\(sh,k\)\.\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}\(r\)=s\_\{h,k\},\\qquad\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}\(r\)d^\{\(h\)\}\(r\)=\\Psi\_\{h\}\(s\_\{h,k\}\)\.\(116\)Multiplying the drift expansion by⌈nH,h,k∗⌉\+⌈nAI,h,k∗⌉\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceiland usingηh,kH=⌈nH,h,k∗⌉/\(⌈nH,h,k∗⌉\+⌈nAI,h,k∗⌉\)\\eta^\{\\mathrm\{H\}\}\_\{h,k\}=\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil/\(\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil\),1−ηh,kH=⌈nAI,h,k∗⌉/\(⌈nH,h,k∗⌉\+⌈nAI,h,k∗⌉\)1\-\\eta^\{\\mathrm\{H\}\}\_\{h,k\}=\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil/\(\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil\)from \([38](https://arxiv.org/html/2609.28859#S5.E38)\), together with \([116](https://arxiv.org/html/2609.28859#S10.E116)\),

\(⌈nH,h,k∗⌉\+⌈nAI,h,k∗⌉\)​𝔼hπk​\[σh​Zh,k\]=⌈nH,h,k∗⌉​JX\(h\)\+⌈nAI,h,k∗⌉​IR\(h\)\+⌈nAI,h,k∗⌉​Ψh​\(sh,k\)\.\(\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil\)\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{h,k\}\]=\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil J\_\{X\}^\{\(h\)\}\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil I\_\{R\}^\{\(h\)\}\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil\\Psi\_\{h\}\(s\_\{h,k\}\)\.Given⌈nH,h,k∗⌉≥nH,h,k∗\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil\\geq n^\{\*\}\_\{\\mathrm\{H\},h,k\}and⌈nAI,h,k∗⌉≥nAI,h,k∗\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil\\geq n^\{\*\}\_\{\\mathrm\{AI\},h,k\}, andΨh≥0\\Psi\_\{h\}\\geq 0, so the right\-hand side is at least

nH,h,k∗​JX\(h\)\+nAI,h,k∗​IR\(h\)\+nAI,h,k∗​Ψh​\(nesc,h,k∗nAI,h,k∗\)≥Th,k,n^\{\*\}\_\{\\mathrm\{H\},h,k\}J\_\{X\}^\{\(h\)\}\+n^\{\*\}\_\{\\mathrm\{AI\},h,k\}I\_\{R\}^\{\(h\)\}\+n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\Psi\_\{h\}\\\!\\Bigl\(\\tfrac\{n^\{\*\}\_\{\\mathrm\{esc\},h,k\}\}\{n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\}\\Bigr\)\\geq T\_\{h,k\},the last inequality being the information constraint ofΓh\\Gamma\_\{h\}in \([21](https://arxiv.org/html/2609.28859#S4.E21)\), met by the optimizer \([36](https://arxiv.org/html/2609.28859#S5.E36)\) \(ifnAI,h,k∗=0n^\{\*\}\_\{\\mathrm\{AI\},h,k\}=0thensh,k=0s\_\{h,k\}=0,Ψh​\(0\)=0\\Psi\_\{h\}\(0\)=0, and the escalation term vanishes on both sides\)\. Hence

\(⌈nH,h,k∗⌉\+⌈nAI,h,k∗⌉\)​𝔼hπk​\[σh​Zh,k\]≥Th,k\.\(\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil\)\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{h,k\}\]\\geq T\_\{h,k\}\.\(117\)By construction,⌈nH,h,k∗⌉\+⌈nAI,h,k∗⌉≤⌈nH,h,k∗\+nAI,h,k∗⌉\+1≤Nmain,k=O⁡\(Lk\)\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil\\leq\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\+n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil\+1\\leq N\_\{\\mathrm\{main\},k\}=O\(L\_\{k\}\)\. Additionally,Th,k=Θ⁡\(Lk\)T\_\{h,k\}=\\Theta\(L\_\{k\}\)\. It thus follows there is a primitive constantcdrift\>0c\_\{\\mathrm\{drift\}\}\>0such that, for all sufficiently largekk,

𝔼hπk​\[σh​Zh,k\]≥Th,k⌈nH,h,k∗⌉\+⌈nAI,h,k∗⌉≥cdrift\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{h,k\}\]\\geq\\frac\{T\_\{h,k\}\}\{\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceil\}\\geq c\_\{\\mathrm\{drift\}\}\.\(118\)
*Wrong rulej=1−hj=1\-h\.*The reverse comparison \([111](https://arxiv.org/html/2609.28859#S10.E111)\) gives𝔼hπk​\[σh​Z1−h,k\]≥c×​𝔼1−hπk​\[σ1−h​Z1−h,k\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{1\-h,k\}\]\\geq c\_\{\\times\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\-h\}\[\\sigma\_\{1\-h\}Z\_\{1\-h,k\}\], and \([118](https://arxiv.org/html/2609.28859#S10.E118)\) applied to direction1−h1\-hgives𝔼1−hπk​\[σ1−h​Z1−h,k\]≥cdrift\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\-h\}\[\\sigma\_\{1\-h\}Z\_\{1\-h,k\}\]\\geq c\_\{\\mathrm\{drift\}\}\. Therefore𝔼hπk​\[σh​Z1−h,k\]≥c×​cdrift\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{1\-h,k\}\]\\geq c\_\{\\times\}c\_\{\\mathrm\{drift\}\}\.

*Dead\-zone rulej=∗j=\*\.*Its parameters \([40](https://arxiv.org/html/2609.28859#S5.E40)\) areη∗,kH=12​\(η0,kH\+η1,kH\)\\eta^\{\\mathrm\{H\}\}\_\{\*,k\}=\\tfrac\{1\}\{2\}\(\\eta^\{\\mathrm\{H\}\}\_\{0,k\}\+\\eta^\{\\mathrm\{H\}\}\_\{1,k\}\)andη∗,kesc=12​\(η0,kesc\+η1,kesc\)\\eta^\{\\mathrm\{esc\}\}\_\{\*,k\}=\\tfrac\{1\}\{2\}\(\\eta^\{\\mathrm\{esc\}\}\_\{0,k\}\+\\eta^\{\\mathrm\{esc\}\}\_\{1,k\}\), hence

η∗,kH≥12​ηh,kH,1−η∗,kH≥12​\(1−ηh,kH\),η∗,kesc​\(r\)≥12​ηh,kesc​\(r\)\.\\eta^\{\\mathrm\{H\}\}\_\{\*,k\}\\geq\\tfrac\{1\}\{2\}\\eta^\{\\mathrm\{H\}\}\_\{h,k\},\\qquad 1\-\\eta^\{\\mathrm\{H\}\}\_\{\*,k\}\\geq\\tfrac\{1\}\{2\}\(1\-\\eta^\{\\mathrm\{H\}\}\_\{h,k\}\),\\qquad\\eta^\{\\mathrm\{esc\}\}\_\{\*,k\}\(r\)\\geq\\tfrac\{1\}\{2\}\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}\(r\)\.Substituting these inequalities into the nonnegative terms of the drift expansion for𝔼hπk​\[σh​Z∗,k\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{\*,k\}\],

𝔼hπk​\[σh​Z∗,k\]≥12​ηh,kH​JX\(h\)\+12​\(1−ηh,kH\)​IR\(h\)\+14​\(1−ηh,kH\)​∑r∈ℛgh​\(r\)​ηh,kesc​\(r\)​d\(h\)​\(r\),\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{\*,k\}\]\\geq\\tfrac\{1\}\{2\}\\eta^\{\\mathrm\{H\}\}\_\{h,k\}J\_\{X\}^\{\(h\)\}\+\\tfrac\{1\}\{2\}\(1\-\\eta^\{\\mathrm\{H\}\}\_\{h,k\}\)I\_\{R\}^\{\(h\)\}\+\\tfrac\{1\}\{4\}\(1\-\\eta^\{\\mathrm\{H\}\}\_\{h,k\}\)\\\!\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}\(r\)d^\{\(h\)\}\(r\),where the escalation factor14\\tfrac\{1\}\{4\}uses\(1−η∗,kH\)​η∗,kesc​\(r\)≥14​\(1−ηh,kH\)​ηh,kesc​\(r\)\(1\-\\eta^\{\\mathrm\{H\}\}\_\{\*,k\}\)\\eta^\{\\mathrm\{esc\}\}\_\{\*,k\}\(r\)\\geq\\tfrac\{1\}\{4\}\(1\-\\eta^\{\\mathrm\{H\}\}\_\{h,k\}\)\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}\(r\)\(the product of the two nonnegative\-termed bounds above\)\. Since12≥14\\tfrac\{1\}\{2\}\\geq\\tfrac\{1\}\{4\}, the first two coefficients may be further weakened to14\\tfrac\{1\}\{4\}, so comparing with the expansion \([114](https://arxiv.org/html/2609.28859#S10.E114)\) for𝔼hπk​\[σh​Zh,k\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{h,k\}\],

𝔼hπk​\[σh​Z∗,k\]≥14​\[ηh,kH​JX\(h\)\+\(1−ηh,kH\)​IR\(h\)\+\(1−ηh,kH\)​∑r∈ℛgh​\(r\)​ηh,kesc​\(r\)​d\(h\)​\(r\)\]=14​𝔼hπk​\[σh​Zh,k\]\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{\*,k\}\]\\geq\\tfrac\{1\}\{4\}\\Bigl\[\\eta^\{\\mathrm\{H\}\}\_\{h,k\}J\_\{X\}^\{\(h\)\}\+\(1\-\\eta^\{\\mathrm\{H\}\}\_\{h,k\}\)I\_\{R\}^\{\(h\)\}\+\(1\-\\eta^\{\\mathrm\{H\}\}\_\{h,k\}\)\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\eta^\{\\mathrm\{esc\}\}\_\{h,k\}\(r\)d^\{\(h\)\}\(r\)\\Bigr\]=\\tfrac\{1\}\{4\}\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{h,k\}\]\.With \([118](https://arxiv.org/html/2609.28859#S10.E118)\) this gives𝔼hπk​\[σh​Z∗,k\]≥cdrift/4\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{\*,k\}\]\\geq c\_\{\\mathrm\{drift\}\}/4\.

Takingμ0:=min⁡\{cdrift,c×​cdrift,cdrift/4\}\>0\\mu\_\{0\}:=\\min\\\{c\_\{\\mathrm\{drift\}\},\\,c\_\{\\times\}c\_\{\\mathrm\{drift\}\},\\,c\_\{\\mathrm\{drift\}\}/4\\\}\>0proves \([112](https://arxiv.org/html/2609.28859#S10.E112)\)\.

\(iv\) By parts \(i\) and \(iii\), for all sufficiently largekk,

Varh⁡\(Zj,k\)≤𝔼hπk​\[Zj,k2\]≤Binc2≤Binc2μ0​𝔼hπk​\[σh​Zj,k\]\.\\operatorname\{Var\}\_\{h\}\(Z\_\{j,k\}\)\\leq\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[Z\_\{j,k\}^\{2\}\]\\leq B\_\{\\mathrm\{inc\}\}^\{2\}\\leq\\frac\{B\_\{\\mathrm\{inc\}\}^\{2\}\}\{\\mu\_\{0\}\}\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\sigma\_\{h\}Z\_\{j,k\}\]\.Thus \([113](https://arxiv.org/html/2609.28859#S10.E113)\) holds withCV:=Binc2/μ0C\_\{V\}:=B\_\{\\mathrm\{inc\}\}^\{2\}/\\mu\_\{0\}, which depends only on the primitives\.

###### Proof 10\.18

Proof of Lemma[10\.4](https://arxiv.org/html/2609.28859#S10.Thmtheorem4)Fix a sufficiently largekk\. We prove the bound forW1,kW\_\{1,k\}underH1H\_\{1\}; the bound forW0,kW\_\{0,k\}underH0H\_\{0\}follows by applying the same argument to the reflected statistic−Si\-S\_\{i\}, since by Lemma[10\.16](https://arxiv.org/html/2609.28859#S10.Thmtheorem16)\(iii\) every deployed rule also has drift at leastμ0\\mu\_\{0\}toward the correct lower boundary underH0H\_\{0\}\.

Use the item\-level process\(Yi,ℋi,Si\)\(Y\_\{i\},\\mathcal\{H\}\_\{i\},S\_\{i\}\)defined at the beginning of this subsection\. The ruleJiJ\_\{i\}isℋi−1\\mathcal\{H\}\_\{i\-1\}\-measurable, being determined by the sign test \([26](https://arxiv.org/html/2609.28859#S5.E26)\) applied toSi−1S\_\{i\-1\}\. Conditionally onℋi−1\\mathcal\{H\}\_\{i\-1\}, itemiiis a*fresh*item: its report, label, and randomization seeds are drawn independently of the past, so the conditional law ofYiY\_\{i\}givenℋi−1\\mathcal\{H\}\_\{i\-1\}is exactly the unconditional law of the single\-item incrementZJi,kZ\_\{J\_\{i\},k\}under the \(now fixed\) ruleJiJ\_\{i\}\. This gives, for everyii, that the following inequalities hold almost surely:

𝔼1πk​\[Yi∣ℋi−1\]≥μ0,\|Yi\|≤Binc,Var1⁡\(Yi∣ℋi−1\)≤CV​𝔼1πk​\[Yi∣ℋi−1\],\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\geq\\mu\_\{0\},\\qquad\|Y\_\{i\}\|\\leq B\_\{\\mathrm\{inc\}\},\\qquad\\operatorname\{Var\}\_\{1\}\(Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\)\\leq C\_\{V\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\],\(119\)where the third bound is the conditional form of \([113](https://arxiv.org/html/2609.28859#S10.E113)\):Var1⁡\(Yi∣ℋi−1\)=Var1⁡\(ZJi,k\)≤CV​𝔼1πk​\[ZJi,k\]=CV​𝔼1πk​\[Yi∣ℋi−1\]\\operatorname\{Var\}\_\{1\}\(Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\)=\\operatorname\{Var\}\_\{1\}\(Z\_\{J\_\{i\},k\}\)\\leq C\_\{V\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{J\_\{i\},k\}\]=C\_\{V\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\. SinceYi≤BincY\_\{i\}\\leq B\_\{\\mathrm\{inc\}\}gives𝔼1πk​\[Yi∣ℋi−1\]≤Binc\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\leq B\_\{\\mathrm\{inc\}\}, \([119](https://arxiv.org/html/2609.28859#S10.E119)\) yields the drift\-controlled second moment

𝔼1πk​\[Yi2∣ℋi−1\]=Var1⁡\(Yi∣ℋi−1\)\+\(𝔼1πk​\[Yi∣ℋi−1\]\)2≤\(CV\+Binc\)​𝔼1πk​\[Yi∣ℋi−1\]\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}^\{2\}\\mid\\mathcal\{H\}\_\{i\-1\}\]=\\operatorname\{Var\}\_\{1\}\(Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\)\+\\bigl\(\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\bigr\)^\{2\}\\leq\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\.\(120\)
SinceS0=0≤zkS\_\{0\}=0\\leq z\_\{k\}, letτtrk,k:=min⁡\{i:Si≥ak\}∧Nmain,k≤Nmain,k\\tau\_\{\\mathrm\{trk\},k\}:=\\min\\\{i:S\_\{i\}\\geq a\_\{k\}\\\}\\wedge N\_\{\\mathrm\{main\},k\}\\leq N\_\{\\mathrm\{main\},k\}\. ThenW1,k≤∑i=1τtrk,k𝟏\{Si−1≤zk\}W\_\{1,k\}\\leq\\sum\_\{i=1\}^\{\\tau\_\{\\mathrm\{trk\},k\}\}\\mathbf\{1\}\\\{S\_\{i\-1\}\\leq z\_\{k\}\\\}, so it suffices to bound∑i=1τtrk,k𝟏\{Si−1≤zk\}\\sum\_\{i=1\}^\{\\tau\_\{\\mathrm\{trk\},k\}\}\\mathbf\{1\}\\\{S\_\{i\-1\}\\leq z\_\{k\}\\\}\.

To obtain this bound, we introduce a smooth potentialΦ\\Phiwhose conditional one\-step change is uniformly negative wheneverSi−1≤zkS\_\{i\-1\}\\leq z\_\{k\}and nonpositive otherwise\. Summing this drift inequality up toτtrk,k\\tau\_\{\\mathrm\{trk\},k\}and telescoping the stopped process will then convert the total decrease ofΦ⁡\(Si\)\\Phi\(S\_\{i\}\)into an upper bound on the expected number of indices for whichSi−1≤zkS\_\{i\-1\}\\leq z\_\{k\}\.

*A smooth potential\.*Setθ:=1e⁡\(CV\+Binc\)\>0\\theta:=\\dfrac\{1\}\{e\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)\}\>0, so thatθ⁡\(CV\+Binc\)=1/e\\theta\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)=1/eandθ​Binc≤1/e\\theta B\_\{\\mathrm\{inc\}\}\\leq 1/e, and define

Φ⁡\(s\):=\{zk−s,s≤zk,−1θ​\(1−e−θ⁡\(s−zk\)\),s≥zk\.\\Phi\(s\):=\\begin\{cases\}z\_\{k\}\-s,&s\\leq z\_\{k\},\\\\\[2\.84526pt\] \-\\dfrac\{1\}\{\\theta\}\\bigl\(1\-e^\{\-\\theta\(s\-z\_\{k\}\)\}\\bigr\),&s\\geq z\_\{k\}\.\\end\{cases\}\(121\)Φ\\Phiis convex, non\-increasing, and11\-Lipschitz\. On the linear branchs<zks<z\_\{k\}it hasΦ′​\(s\)=−1\\Phi^\{\\prime\}\(s\)=\-1andΦ′′​\(s\)=0\\Phi^\{\\prime\\prime\}\(s\)=0; on the curved branchs\>zks\>z\_\{k\},

Φ′​\(s\)=−e−θ⁡\(s−zk\),Φ′′​\(s\)=θ​e−θ⁡\(s−zk\)∈\(0,θ\]\.\\Phi^\{\\prime\}\(s\)=\-e^\{\-\\theta\(s\-z\_\{k\}\)\},\\qquad\\Phi^\{\\prime\\prime\}\(s\)=\\theta e^\{\-\\theta\(s\-z\_\{k\}\)\}\\in\(0,\\theta\]\.The two one\-sided values agree atzkz\_\{k\}\(Φ⁡\(zk±\)=0\\Phi\(z\_\{k\}^\{\\pm\}\)=0andΦ′​\(zk±\)=−1\\Phi^\{\\prime\}\(z\_\{k\}^\{\\pm\}\)=\-1\), soΦ∈C1\\Phi\\in C^\{1\}, and the curvature is uniformly bounded:

sups∈ℝΦ′′​\(s\)=θ=1e⁡\(CV\+Binc\),\\sup\_\{s\\in\\mathbb\{R\}\}\\Phi^\{\\prime\\prime\}\(s\)=\\theta=\\frac\{1\}\{e\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)\},\(122\)the supremum being attained ass↓zks\\downarrow z\_\{k\}\. FinallyΦ⁡\(s\)=zk−s→\+∞\\Phi\(s\)=z\_\{k\}\-s\\to\+\\inftyass→−∞s\\to\-\\infty, whileΦ\(s\)→−1/θ\\Phi\(s\)\\to\-1/\\thetaass→\+∞s\\to\+\\infty, soΦ\\Phiis*unbounded above but bounded below*, with

Φ⁡\(S0\)=Φ⁡\(0\)=zk,infs∈ℝΦ⁡\(s\)=−1θ=−e⁡\(CV\+Binc\)\.\\Phi\(S\_\{0\}\)=\\Phi\(0\)=z\_\{k\},\\qquad\\inf\_\{s\\in\\mathbb\{R\}\}\\Phi\(s\)=\-\\tfrac\{1\}\{\\theta\}=\-e\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)\.\(123\)
*One\-step drift inequality\.*We claim that for everyii,

𝔼1πk\[Φ\(Si\)∣ℋi−1\]−Φ\(Si−1\)≤−μ021\{Si−1≤zk\}\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\Phi\(S\_\{i\}\)\\mid\\mathcal\{H\}\_\{i\-1\}\]\-\\Phi\(S\_\{i\-1\}\)\\leq\-\\tfrac\{\\mu\_\{0\}\}\{2\}\\,\\mathbf\{1\}\\\{S\_\{i\-1\}\\leq z\_\{k\}\\\}\.\(124\)SinceΦ∈C1\\Phi\\in C^\{1\}withΦ′′≤M\\Phi^\{\\prime\\prime\}\\leq Meverywhere, the second\-order Taylor inequality

Φ⁡\(y\)≤Φ⁡\(x\)\+Φ′​\(x\)​\(y−x\)\+12​Mx,y​\(y−x\)2,Mx,y:=sup\[x∧y,x∨y\]Φ′′≤θ,\\Phi\(y\)\\leq\\Phi\(x\)\+\\Phi^\{\\prime\}\(x\)\(y\-x\)\+\\tfrac\{1\}\{2\}\\,M\_\{x,y\}\\,\(y\-x\)^\{2\},\\qquad M\_\{x,y\}:=\\sup\_\{\[x\\wedge y,\\,x\\vee y\]\}\\Phi^\{\\prime\\prime\}\\leq\\theta,\(125\)holds for allx,yx,y\(this extends the usual Taylor bound to our piecewise\-C2C^\{2\}Φ\\PhibecauseΦ′\\Phi^\{\\prime\}is absolutely continuous, being continuous with matching one\-sided derivatives at the single kinkzkz\_\{k\}\)\. Applying it withx=Si−1x=S\_\{i\-1\},y=Siy=S\_\{i\}: note thatMx,yM\_\{x,y\}as written here depends on the randomSiS\_\{i\}; since\|Yi\|≤Binc\|Y\_\{i\}\|\\leq B\_\{\\mathrm\{inc\}\}, the interval\[x∧y,x∨y\]\[x\\wedge y,x\\vee y\]is always contained in\[Si−1−Binc,Si−1\+Binc\]\[S\_\{i\-1\}\-B\_\{\\mathrm\{inc\}\},S\_\{i\-1\}\+B\_\{\\mathrm\{inc\}\}\], soMx,y≤θM\_\{x,y\}\\leq\\thetaalmost surely, for every possible realization ofYiY\_\{i\}\. Withy−x=Yiy\-x=Y\_\{i\}, taking𝔼1πk\[⋅∣ℋi−1\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\cdot\\mid\\mathcal\{H\}\_\{i\-1\}\], and using the second\-moment bound \([120](https://arxiv.org/html/2609.28859#S10.E120)\),

𝔼1πk​\[Φ⁡\(Si\)∣ℋi−1\]−Φ⁡\(Si−1\)≤Φ′​\(Si−1\)​𝔼1πk​\[Yi∣ℋi−1\]\+12​𝔼1πk​\[Mx,y​Yi2∣ℋi−1\]\.\\begin\{split\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\Phi\(S\_\{i\}\)\\mid\\mathcal\{H\}\_\{i\-1\}\]\-\\Phi\(S\_\{i\-1\}\)&\\leq\\Phi^\{\\prime\}\(S\_\{i\-1\}\)\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\+\\tfrac\{1\}\{2\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[M\_\{x,y\}Y\_\{i\}^\{2\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\.\\end\{split\}\(126\)
*Low steps*\(Si−1≤zkS\_\{i\-1\}\\leq z\_\{k\}, soΦ′​\(Si−1\)=−1\\Phi^\{\\prime\}\(S\_\{i\-1\}\)=\-1\)\. Using \([120](https://arxiv.org/html/2609.28859#S10.E120)\),Mx,y≤θM\_\{x,y\}\\leq\\thetain \([126](https://arxiv.org/html/2609.28859#S10.E126)\) andθ⁡\(CV\+Binc\)=1/e\\theta\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)=1/e,

𝔼1πk​\[Φ⁡\(Si\)∣ℋi−1\]−Φ⁡\(Si−1\)\\displaystyle\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\Phi\(S\_\{i\}\)\\mid\\mathcal\{H\}\_\{i\-1\}\]\-\\Phi\(S\_\{i\-1\}\)≤\[−1\+12​θ​\(CV\+Binc\)\]​𝔼1πk​\[Yi∣ℋi−1\]\\displaystyle\\leq\\Bigl\[\-1\+\\tfrac\{1\}\{2\}\\theta\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)\\Bigr\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\[\([120](https://arxiv.org/html/2609.28859#S10.E120)\),\([126](https://arxiv.org/html/2609.28859#S10.E126)\),Φ′​\(Si−1\)=−1,Mx,y≤θ\]\\displaystyle\[\\eqref\{eq:dt\-secmom\},\\,\\eqref\{eq:dt\-onestep\},\\ \\Phi^\{\\prime\}\(S\_\{i\-1\}\)=\-1,\\ M\_\{x,y\}\\leq\\theta\\,\]=\[−1\+12​e\]​𝔼1πk​\[Yi∣ℋi−1\]\\displaystyle=\\Bigl\[\-1\+\\tfrac\{1\}\{2e\}\\Bigr\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\[θ\(CV\+Binc\)=1/e\]\\displaystyle\[\\,\\theta\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)=1/e\\,\]≤−12​𝔼1πk​\[Yi∣ℋi−1\]\\displaystyle\\leq\-\\tfrac\{1\}\{2\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\[1−12​e≥12\]\\displaystyle\[\\,1\-\\tfrac\{1\}\{2e\}\\geq\\tfrac\{1\}\{2\}\\,\]≤−μ02\\displaystyle\\leq\-\\tfrac\{\\mu\_\{0\}\}\{2\}\[\([119](https://arxiv.org/html/2609.28859#S10.E119)\)\]\.\\displaystyle\[\\,\\eqref\{eq:dt\-hyp\}\\,\]\.
*High steps*\(Si−1\>zkS\_\{i\-1\}\>z\_\{k\}, soΦ′​\(Si−1\)=−α\\Phi^\{\\prime\}\(S\_\{i\-1\}\)=\-\\alphawithα:=e−θ⁡\(Si−1−zk\)∈\(0,1\)\\alpha:=e^\{\-\\theta\(S\_\{i\-1\}\-z\_\{k\}\)\}\\in\(0,1\)\)\. The step interval\[Si−1∧Si,Si−1∨Si\]\[S\_\{i\-1\}\\wedge S\_\{i\},\\,S\_\{i\-1\}\\vee S\_\{i\}\]lies in\[Si−1−Binc,∞\)\[S\_\{i\-1\}\-B\_\{\\mathrm\{inc\}\},\\infty\); sinceΦ′′\\Phi^\{\\prime\\prime\}is non\-increasing on the curved branch and vanishes belowzkz\_\{k\},

Mx,y≤sups≥Si−1−BincΦ′′​\(s\)≤θ​e−θ⁡\(Si−1−Binc−zk\)=θ​α​eθ​Binc\.M\_\{x,y\}\\leq\\sup\_\{s\\geq S\_\{i\-1\}\-B\_\{\\mathrm\{inc\}\}\}\\Phi^\{\\prime\\prime\}\(s\)\\leq\\theta e^\{\-\\theta\(S\_\{i\-1\}\-B\_\{\\mathrm\{inc\}\}\-z\_\{k\}\)\}=\\theta\\,\\alpha\\,e^\{\\theta B\_\{\\mathrm\{inc\}\}\}\.Substituting into \([126](https://arxiv.org/html/2609.28859#S10.E126)\),

𝔼1πk​\[Φ⁡\(Si\)∣ℋi−1\]−Φ⁡\(Si−1\)\\displaystyle\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\Phi\(S\_\{i\}\)\\mid\\mathcal\{H\}\_\{i\-1\}\]\-\\Phi\(S\_\{i\-1\}\)≤α⁡\[−1\+12​θ​eθ​Binc​\(CV\+Binc\)\]​𝔼1πk​\[Yi∣ℋi−1\]\\displaystyle\\leq\\alpha\\Bigl\[\-1\+\\tfrac\{1\}\{2\}\\theta e^\{\\theta B\_\{\\mathrm\{inc\}\}\}\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)\\Bigr\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\[\([120](https://arxiv.org/html/2609.28859#S10.E120)\),\([126](https://arxiv.org/html/2609.28859#S10.E126)\),Φ′​\(Si−1\)=−α\]\\displaystyle\[\\eqref\{eq:dt\-secmom\},\\,\\eqref\{eq:dt\-onestep\},\\ \\Phi^\{\\prime\}\(S\_\{i\-1\}\)=\-\\alpha\\,\]=α⁡\[−1\+eθ​Binc2​e\]​𝔼1πk​\[Yi∣ℋi−1\]\\displaystyle=\\alpha\\Bigl\[\-1\+\\tfrac\{e^\{\\theta B\_\{\\mathrm\{inc\}\}\}\}\{2e\}\\Bigr\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\[θ\(CV\+Binc\)=1/e\]\\displaystyle\[\\,\\theta\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)=1/e\\,\]≤−α2​𝔼1πk​\[Yi∣ℋi−1\]≤0\\displaystyle\\leq\-\\tfrac\{\\alpha\}\{2\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\ \\leq\\ 0\[eθ​Binc≤e1/e≤e\]\.\\displaystyle\[\\,e^\{\\theta B\_\{\\mathrm\{inc\}\}\}\\leq e^\{1/e\}\\leq e\\,\]\.The extra factor1/e1/ein the definition ofθ\\thetais exactly what keeps this coefficient negative: with the alternative choiceθ=1/\(CV\+Binc\)\\theta=1/\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)the same computation gives−1\+12​eθ​Binc\-1\+\\tfrac\{1\}\{2\}e^\{\\theta B\_\{\\mathrm\{inc\}\}\}, which can be positive \(up to−1\+e2\>0\-1\+\\tfrac\{e\}\{2\}\>0whenθ​Binc\\theta B\_\{\\mathrm\{inc\}\}is close to11\), so the descent property would fail on high steps\. Note also that the linear gain−α​𝔼1πk​\[Yi∣ℋi−1\]\-\\alpha\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]and the curvature cost both carry the factorα\\alpha, so the gain dominates uniformly in the heightSi−1−zk≥0S\_\{i\-1\}\-z\_\{k\}\\geq 0: a down\-crossing ofzkz\_\{k\}raisesΦ\\Phi, but the next\-step positive drift lowers𝔼1πk​\[Φ⁡\(Si\)∣ℋi−1\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\Phi\(S\_\{i\}\)\\mid\\mathcal\{H\}\_\{i\-1\}\]by at least as much, leaving no recharge term\. The two cases together establish \([124](https://arxiv.org/html/2609.28859#S10.E124)\)\.

*Stopping\-time telescoping\.*Becauseτtrk,k≤Nmain,k\\tau\_\{\\mathrm\{trk\},k\}\\leq N\_\{\\mathrm\{main\},k\}is a bounded stopping time,𝟏\{i≤τtrk,k\}\\mathbf\{1\}\\\{i\\leq\\tau\_\{\\mathrm\{trk\},k\}\\\}isℋi−1\\mathcal\{H\}\_\{i\-1\}\-measurable\. Hence the tower property and \([124](https://arxiv.org/html/2609.28859#S10.E124)\) give

𝔼1πk​\[Φ⁡\(Sτtrk,k\)\]−Φ⁡\(S0\)\\displaystyle\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\Phi\(S\_\{\\tau\_\{\\mathrm\{trk\},k\}\}\)\]\-\\Phi\(S\_\{0\}\)=∑i=1Nmain,k𝔼1πk\[𝟏\{i≤τtrk,k\}\(Φ\(Si\)−Φ\(Si−1\)\)\]\\displaystyle=\\sum\_\{i=1\}^\{N\_\{\\mathrm\{main\},k\}\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\\\!\\left\[\\mathbf\{1\}\\\{i\\leq\\tau\_\{\\mathrm\{trk\},k\}\\\}\\bigl\(\\Phi\(S\_\{i\}\)\-\\Phi\(S\_\{i\-1\}\)\\bigr\)\\right\]=∑i=1Nmain,k𝔼1πk\[𝟏\{i≤τtrk,k\}\(𝔼1πk\[Φ\(Si\)∣ℋi−1\]−Φ\(Si−1\)\)\]\\displaystyle=\\sum\_\{i=1\}^\{N\_\{\\mathrm\{main\},k\}\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\\\!\\left\[\\mathbf\{1\}\\\{i\\leq\\tau\_\{\\mathrm\{trk\},k\}\\\}\\left\(\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\Phi\(S\_\{i\}\)\\mid\\mathcal\{H\}\_\{i\-1\}\]\-\\Phi\(S\_\{i\-1\}\)\\right\)\\right\]≤−μ02𝔼1πk\[∑i=1τtrk,k𝟏\{Si−1≤zk\}\]≤−μ02𝔼1πk\[W1,k\]\.\\displaystyle\\leq\-\\tfrac\{\\mu\_\{0\}\}\{2\}\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\\Bigl\[\\sum\_\{i=1\}^\{\\tau\_\{\\mathrm\{trk\},k\}\}\\mathbf\{1\}\\\{S\_\{i\-1\}\\leq z\_\{k\}\\\}\\Bigr\]\\leq\-\\tfrac\{\\mu\_\{0\}\}\{2\}\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[W\_\{1,k\}\]\.Additionally, we haveΦ⁡\(Sτtrk,k\)≥infΦ=−e⁡\(CV\+Binc\)\\Phi\(S\_\{\\tau\_\{\\mathrm\{trk\},k\}\}\)\\geq\\inf\\Phi=\-e\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)from \([123](https://arxiv.org/html/2609.28859#S10.E123)\)\. Combining withΦ⁡\(S0\)=zk\\Phi\(S\_\{0\}\)=z\_\{k\}gives

−e⁡\(CV\+Binc\)−zk≤𝔼1πk​\[Φ⁡\(Sτtrk,k\)\]−Φ⁡\(S0\)≤−μ02​𝔼1πk​\[W1,k\]\.\-e\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)\-z\_\{k\}\\ \\leq\\ \\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[\\Phi\(S\_\{\\tau\_\{\\mathrm\{trk\},k\}\}\)\]\-\\Phi\(S\_\{0\}\)\\ \\leq\\ \-\\tfrac\{\\mu\_\{0\}\}\{2\}\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[W\_\{1,k\}\]\.Rearranging the outer inequality,

𝔼1πk​\[W1,k\]≤2​\(zk\+e⁡\(CV\+Binc\)\)μ0≤2​\(e⁡\(CV\+Binc\)\+1\)​\(zk\+1\)μ0,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[W\_\{1,k\}\]\\leq\\frac\{2\\bigl\(z\_\{k\}\+e\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)\\bigr\)\}\{\\mu\_\{0\}\}\\leq\\frac\{2\\bigl\(e\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)\+1\\bigr\)\(z\_\{k\}\+1\)\}\{\\mu\_\{0\}\},which is theH1H\_\{1\}part of \([84](https://arxiv.org/html/2609.28859#S10.E84)\) withCW:=4​\(e⁡\(CV\+Binc\)\+1\)/μ0C\_\{W\}:=4\\bigl\(e\(C\_\{V\}\+B\_\{\\mathrm\{inc\}\}\)\+1\\bigr\)/\\mu\_\{0\}\. TheH0H\_\{0\}part is identical withSiS\_\{i\}replaced by−Si\-S\_\{i\}, giving the same constant by the symmetric application of Lemma[10\.16](https://arxiv.org/html/2609.28859#S10.Thmtheorem16)\(iii\) noted at the start of the proof, so this singleCWC\_\{W\}covers both terms in \([84](https://arxiv.org/html/2609.28859#S10.E84)\)\.

### 10\.4Proof of Lemma[10\.5](https://arxiv.org/html/2609.28859#S10.Thmtheorem5)

###### Proof 10\.19

Proof of Lemma[10\.5](https://arxiv.org/html/2609.28859#S10.Thmtheorem5)Fix a sufficiently largekk\. We prove the result underH1H\_\{1\}, for which the correct rule is the direction\-11rule and the relevant boundary isaka\_\{k\}\. TheH0H\_\{0\}result follows by applying the same argument to−S\-S, withak,T1,ka\_\{k\},T\_\{1,k\}, and the direction\-11rule replaced bybk,T0,kb\_\{k\},T\_\{0,k\}, and the direction\-00rule\.

*Cost decomposition\.*Fori=1,…,Nmain,ki=1,\\ldots,N\_\{\\mathrm\{main\},k\}, letIiI\_\{i\}indicate that the main stage starts itemii, and define

τmain,k:=∑i=1Nmain,kIi\.\\tau\_\{\\mathrm\{main\},k\}:=\\sum\_\{i=1\}^\{N\_\{\\mathrm\{main\},k\}\}I\_\{i\}\.ThusIi=𝟏\{i≤τmain,k\}I\_\{i\}=\\mathbf\{1\}\\\{i\\leq\\tau\_\{\\mathrm\{main\},k\}\\\}isℋi−1\\mathcal\{H\}\_\{i\-1\}\-measurable\. On a boundary\-crossing path, itemτmain,k\\tau\_\{\\mathrm\{main\},k\}is the terminal item and may be only partially processed; onEfbE\_\{\\mathrm\{fb\}\}, all items are completed andτmain,k=Nmain,k\\tau\_\{\\mathrm\{main\},k\}=N\_\{\\mathrm\{main\},k\}\. Recall that\(Yi,Si\)\(Y\_\{i\},S\_\{i\}\)is the completed\-item continuation defined at the beginning of this subsection\. Classify the started items by the predictable sign test

𝒞:=\{i:Ii=1,Si−1\>zk\},𝒲:=\{i:Ii=1,Si−1≤zk\},\\mathcal\{C\}:=\\\{i:I\_\{i\}=1,\\ S\_\{i\-1\}\>z\_\{k\}\\\},\\qquad\\mathcal\{W\}:=\\\{i:I\_\{i\}=1,\\ S\_\{i\-1\}\\leq z\_\{k\}\\\},and writeNc:=\|𝒞\|N\_\{c\}:=\|\\mathcal\{C\}\|, so that\|𝒲\|=W1,k\|\\mathcal\{W\}\|=W\_\{1,k\}\. Define the sensing cost of fully completing itemiion this continuation by

Ciitem:=\\displaystyle C\_\{i\}^\{\\mathrm\{item\}\}:=\{\}𝟏\{Ui≤ηHJi,k\}cH\+𝟏\{Ui\>ηHJi,k\}\(cAI\+𝟏\{Vi≤ηescJi,k\(Ri\)\}cH\)\.\\displaystyle\\mathbf\{1\}\\\{U\_\{i\}\\leq\\eta^\{\\mathrm\{H\}\}\_\{J\_\{i\},k\}\\\}\\,c\_\{\\mathrm\{H\}\}\+\\mathbf\{1\}\\\{U\_\{i\}\>\\eta^\{\\mathrm\{H\}\}\_\{J\_\{i\},k\}\\\}\\Bigl\(c\_\{\\mathrm\{AI\}\}\+\\mathbf\{1\}\\\{V\_\{i\}\\leq\\eta^\{\\mathrm\{esc\}\}\_\{J\_\{i\},k\}\(R\_\{i\}\)\\\}\\,c\_\{\\mathrm\{H\}\}\\Bigr\)\.LetCk𝒞:=∑i∈𝒞CiitemC^\{\\mathcal\{C\}\}\_\{k\}:=\\sum\_\{i\\in\\mathcal\{C\}\}C\_\{i\}^\{\\mathrm\{item\}\},Ck𝒲:=∑i∈𝒲CiitemC^\{\\mathcal\{W\}\}\_\{k\}:=\\sum\_\{i\\in\\mathcal\{W\}\}C\_\{i\}^\{\\mathrm\{item\}\}, and we haveCkmain≤Ck𝒞\+Ck𝒲C^\{\\mathrm\{main\}\}\_\{k\}\\leq C^\{\\mathcal\{C\}\}\_\{k\}\+C^\{\\mathcal\{W\}\}\_\{k\}\.

Setcmax:=cH\+cAI\+cHc\_\{\\max\}:=c\_\{\\mathrm\{H\}\}\+c\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}, a primitive upper bound on the sensing cost of a completed item\. Taking expectations gives

𝔼1πk​\[Ckmain\]\\displaystyle\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[C^\{\\mathrm\{main\}\}\_\{k\}\]≤𝔼1πk​\[Ck𝒲\]⏟Cost 1: wrong/dead\-zone\+𝔼1πk​\[Ck𝒞\]⏟Cost 2: correct\.\\displaystyle\\leq\\underbrace\{\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[C^\{\\mathcal\{W\}\}\_\{k\}\]\}\_\{\\textnormal\{Cost 1: wrong/dead\-zone\}\}\+\\underbrace\{\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[C^\{\\mathcal\{C\}\}\_\{k\}\]\}\_\{\\textnormal\{Cost 2: correct\}\}\.\(127\)We bound these two terms in turn\.

*Cost 1: wrong and dead\-zone items\.*Each completed\-item sensing cost is at mostcmaxc\_\{\\max\}\. Hence Lemma[10\.4](https://arxiv.org/html/2609.28859#S10.Thmtheorem4)gives

𝔼1πk​\[Ck𝒲\]≤cmax​𝔼1πk​\[W1,k\]≤cmax​CW​\(zk\+1\)\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[C^\{\\mathcal\{W\}\}\_\{k\}\]\\leq c\_\{\\max\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[W\_\{1,k\}\]\\leq c\_\{\\max\}C\_\{W\}\(z\_\{k\}\+1\)\.\(128\)
*Cost 2: correct items\.*LetN1,kplan:=⌈nH,1,k∗⌉\+⌈nAI,1,k∗⌉N^\{\\mathrm\{plan\}\}\_\{1,k\}:=\\lceil n^\{\*\}\_\{\\mathrm\{H\},1,k\}\\rceil\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},1,k\}\\rceil\. If the firstN1,kplanN^\{\\mathrm\{plan\}\}\_\{1,k\}sensed items were all processed with the direction\-11rule, their expected sensing cost would be⌈nH,1,k∗⌉​cH\+⌈nAI,1,k∗⌉​cAI\+⌈nAI,1,k∗⌉​s1,k​cH\\lceil n^\{\*\}\_\{\\mathrm\{H\},1,k\}\\rceil c\_\{\\mathrm\{H\}\}\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},1,k\}\\rceil c\_\{\\mathrm\{AI\}\}\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},1,k\}\\rceil s\_\{1,k\}c\_\{\\mathrm\{H\}\}\. Rounding changes this by at most a primitive constant, so the planned block cost is at mostΓ1​\(T1,k,N¯main,k\)\+O⁡\(1\)\\Gamma\_\{1\}\(T\_\{1,k\},\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\+O\(1\)\. A direction\-11item has planned per\-item cost

cplanned:=𝔼1πk​\[Ciitem\]=η1,kH​cH\+\(1−η1,kH\)​\(cAI\+s1,k​cH\)≤Γ1​\(T1,k,N¯main,k\)\+O⁡\(1\)N1,kplan\.\\displaystyle c\_\{\\mathrm\{planned\}\}:=\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[C^\{\\text\{item\}\}\_\{i\}\]=\\eta^\{\\mathrm\{H\}\}\_\{1,k\}c\_\{\\mathrm\{H\}\}\+\(1\-\\eta^\{\\mathrm\{H\}\}\_\{1,k\}\)\\bigl\(c\_\{\\mathrm\{AI\}\}\+s\_\{1,k\}c\_\{\\mathrm\{H\}\}\\bigr\)\\leq\\frac\{\\Gamma\_\{1\}\(T\_\{1,k\},\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\+O\(1\)\}\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}\.\(129\)
Since all items in𝒞\\mathcal\{C\}apply the direction\-11rule, the tower property gives

𝔼1πk​\[Ck𝒞\]\\displaystyle\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[C^\{\\mathcal\{C\}\}\_\{k\}\]=∑i=1Nmain,k𝔼πk1\[Ii𝟏\{Si−1\>zk\}Ciitem\]=cplanned∑i=1Nmain,k𝔼πk1\[Ii𝟏\{Si−1\>zk\}\]=cplanned𝔼πk1\[Nc\]\.\\displaystyle=\\sum\_\{i=1\}^\{N\_\{\\mathrm\{main\},k\}\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[I\_\{i\}\\mathbf\{1\}\\\{S\_\{i\-1\}\>z\_\{k\}\\\}C\_\{i\}^\{\\mathrm\{item\}\}\]=c\_\{\\mathrm\{planned\}\}\\sum\_\{i=1\}^\{N\_\{\\mathrm\{main\},k\}\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[I\_\{i\}\\mathbf\{1\}\\\{S\_\{i\-1\}\>z\_\{k\}\\\}\]=c\_\{\\mathrm\{planned\}\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[N\_\{c\}\]\.Likewise,

𝔼1πk​\[Sτmain,k\]=𝔼1πk​\[∑i=1Nmain,kIi​𝔼1πk​\[Yi∣ℋi−1\]\],\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[S\_\{\\tau\_\{\\mathrm\{main\},k\}\}\]=\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\\\!\\left\[\\sum\_\{i=1\}^\{N\_\{\\mathrm\{main\},k\}\}I\_\{i\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\right\],and splitting the sum over𝒞\\mathcal\{C\}and𝒲\\mathcal\{W\}yields

𝔼1πk​\[Z1,k\]​𝔼1πk​\[Nc\]=𝔼1πk​\[Sτmain,k\]−𝔼1πk​\[∑i∈𝒲𝔼1πk​\[Yi∣ℋi−1\]\]\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[N\_\{c\}\]=\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[S\_\{\\tau\_\{\\mathrm\{main\},k\}\}\]\-\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\\\!\\left\[\\sum\_\{i\\in\\mathcal\{W\}\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\right\]\.Every deployed rule has conditional drift at leastμ0\>0\\mu\_\{0\}\>0by Lemma[10\.16](https://arxiv.org/html/2609.28859#S10.Thmtheorem16)\(iii\), so the second term on the right is nonnegative\. Moreover,Sτmain,k≤ak\+BincS\_\{\\tau\_\{\\mathrm\{main\},k\}\}\\leq a\_\{k\}\+B\_\{\\mathrm\{inc\}\}pathwise: before a terminal boundary\-crossing item is started,Sτmain,k−1<akS\_\{\\tau\_\{\\mathrm\{main\},k\}\-1\}<a\_\{k\}, and its completed continuation satisfiesYτmain,k≤BincY\_\{\\tau\_\{\\mathrm\{main\},k\}\}\\leq B\_\{\\mathrm\{inc\}\}; on pool exhaustion,Sτmain,k<akS\_\{\\tau\_\{\\mathrm\{main\},k\}\}<a\_\{k\}\. Thus

𝔼1πk​\[Z1,k\]​𝔼1πk​\[Nc\]≤ak\+Binc\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[N\_\{c\}\]\\leq a\_\{k\}\+B\_\{\\mathrm\{inc\}\}\.Combining this bound with the cost identity, \([129](https://arxiv.org/html/2609.28859#S10.E129)\), andN1,kplan​𝔼1πk​\[Z1,k\]≥ak\+ΔkN^\{\\mathrm\{plan\}\}\_\{1,k\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]\\geq a\_\{k\}\+\\Delta\_\{k\}from \([117](https://arxiv.org/html/2609.28859#S10.E117)\), we obtain

𝔼1πk​\[Ck𝒞\]\\displaystyle\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[C^\{\\mathcal\{C\}\}\_\{k\}\]≤\(Γ1​\(T1,k,N¯main,k\)\+O⁡\(1\)\)​ak\+Bincak\+Δk\\displaystyle\\leq\\bigl\(\\Gamma\_\{1\}\(T\_\{1,k\},\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\+O\(1\)\\bigr\)\\frac\{a\_\{k\}\+B\_\{\\mathrm\{inc\}\}\}\{a\_\{k\}\+\\Delta\_\{k\}\}≤Γ1​\(T1,k,N¯main,k\)\+O⁡\(1\),\\displaystyle\\leq\\Gamma\_\{1\}\(T\_\{1,k\},\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\+O\(1\),where the last inequality holds for all largekk, sinceΔk≥Binc\\Delta\_\{k\}\\geq B\_\{\\mathrm\{inc\}\}\.

*Combine\.*Substituting the bounds from Costs 1 and 2 into the decomposition \([127](https://arxiv.org/html/2609.28859#S10.E127)\) gives

𝔼1πk​\[Ckmain\]\\displaystyle\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[C^\{\\mathrm\{main\}\}\_\{k\}\]≤Γ1​\(T1,k,N¯main,k\)\+CM​\(zk\+1\)\.\\displaystyle\\leq\\Gamma\_\{1\}\(T\_\{1,k\},\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\+C\_\{M\}\(z\_\{k\}\+1\)\.\(130\)This is \([85](https://arxiv.org/html/2609.28859#S10.E85)\) underH1H\_\{1\}; the reflected argument at the beginning gives the result underH0H\_\{0\}\.

### 10\.5Proof of Lemma[10\.6](https://arxiv.org/html/2609.28859#S10.Thmtheorem6)

###### Proof 10\.20

Proof of Lemma[10\.6](https://arxiv.org/html/2609.28859#S10.Thmtheorem6)Fix a sufficiently largekk\. Forh∈\{0,1\}h\\in\\\{0,1\\\}, let

Nh,kplan:=⌈nH,h,k∗⌉\+⌈nAI,h,k∗⌉N^\{\\mathrm\{plan\}\}\_\{h,k\}:=\\lceil n^\{\*\}\_\{\\mathrm\{H\},h,k\}\\rceil\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},h,k\}\\rceilbe the rounded direction\-hhcount\. We prove the claim underH1H\_\{1\}; theH0H\_\{0\}proof is identical withSSreplaced by−S\-S, the lower boundary−bk\-b\_\{k\}, andN0,kplan,bk,T0,kN^\{\\mathrm\{plan\}\}\_\{0,k\},b\_\{k\},T\_\{0,k\}in place ofN1,kplan,ak,T1,kN^\{\\mathrm\{plan\}\}\_\{1,k\},a\_\{k\},T\_\{1,k\}\. Throughout, use the item\-level process\(Yi,ℋi,Si\)\(Y\_\{i\},\\mathcal\{H\}\_\{i\},S\_\{i\}\)defined at the beginning of the proof of Lemma[10\.4](https://arxiv.org/html/2609.28859#S10.Thmtheorem4)\. By the proof of Lemma[10\.4](https://arxiv.org/html/2609.28859#S10.Thmtheorem4), for everyiithe following inequalities hold almost surely:

𝔼1πk​\[Yi∣ℋi−1\]≥μ0,\|Yi\|≤Binc,Var1⁡\(Yi∣ℋi−1\)≤CV​𝔼1πk​\[Yi∣ℋi−1\]≤CV​Binc\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\geq\\mu\_\{0\},\\qquad\|Y\_\{i\}\|\\leq B\_\{\\mathrm\{inc\}\},\\qquad\\operatorname\{Var\}\_\{1\}\(Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\)\\leq C\_\{V\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\leq C\_\{V\}B\_\{\\mathrm\{inc\}\}\.\(131\)By \([37](https://arxiv.org/html/2609.28859#S5.E37)\),N1,kplan≤Nmain,kN^\{\\mathrm\{plan\}\}\_\{1,k\}\\leq N\_\{\\mathrm\{main\},k\}, and by the proof of Lemma[10\.16](https://arxiv.org/html/2609.28859#S10.Thmtheorem16)\(iii\) \(the inequalityN1,kplan​𝔼1πk​\[Z1,k\]≥T1,kN^\{\\mathrm\{plan\}\}\_\{1,k\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]\\geq T\_\{1,k\}together withN1,kplan=O⁡\(Lk\)N^\{\\mathrm\{plan\}\}\_\{1,k\}=O\(L\_\{k\}\)\),

N1,kplan​𝔼1πk​\[Z1,k\]≥T1,k=ak\+Δk,N1,kplan≤C1​LkN^\{\\mathrm\{plan\}\}\_\{1,k\}\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]\\geq T\_\{1,k\}=a\_\{k\}\+\\Delta\_\{k\},\\qquad N^\{\\mathrm\{plan\}\}\_\{1,k\}\\leq C\_\{1\}L\_\{k\}\(132\)for a primitive constantC1C\_\{1\}, where𝔼1πk​\[Z1,k\]=𝔼1πk​\[Yi∣ℋi−1\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]=\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]on a direction\-11item is the per\-item direction\-11drift\.

*Step 1: onEfbE\_\{\\mathrm\{fb\}\}the statistic stays belowaka\_\{k\}at itemN1,kplanN^\{\\mathrm\{plan\}\}\_\{1,k\}\.*OnEfbE\_\{\\mathrm\{fb\}\}no boundary is crossed in the whole pool, so in particular the upper boundary is not crossed by itemN1,kplan≤Nmain,kN^\{\\mathrm\{plan\}\}\_\{1,k\}\\leq N\_\{\\mathrm\{main\},k\}; hence

Efb⊆\{SN1,kplan<ak\}\.E\_\{\\mathrm\{fb\}\}\\subseteq\\\{S\_\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}<a\_\{k\}\\\}\.\(133\)
*Step 2: drift accumulated by itemN1,kplanN^\{\\mathrm\{plan\}\}\_\{1,k\}on the small\-occupation event\.*OnEfbE\_\{\\mathrm\{fb\}\}, the policy processes at least the firstN1,kplanN^\{\\mathrm\{plan\}\}\_\{1,k\}items\. Letℐk:=\{1≤i≤N1,kplan:Si−1≤zk\}\\mathcal\{I\}\_\{k\}:=\\\{1\\leq i\\leq N^\{\\mathrm\{plan\}\}\_\{1,k\}:S\_\{i\-1\}\\leq z\_\{k\}\\\}be the set of items in this planned block that do not use the direction\-11rule\. SinceW1,kW\_\{1,k\}counts all such items processed before pool exhaustion,\|ℐk\|≤W1,k\|\\mathcal\{I\}\_\{k\}\|\\leq W\_\{1,k\}onEfbE\_\{\\mathrm\{fb\}\}\. Fori∉ℐki\\notin\\mathcal\{I\}\_\{k\}, the policy uses the direction\-11rule and hence𝔼1πk​\[Yi∣ℋi−1\]=𝔼1πk​\[Z1,k\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]=\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]; fori∈ℐki\\in\\mathcal\{I\}\_\{k\}, we have𝔼1πk​\[Yi∣ℋi−1\]≥0\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\geq 0\. Therefore, onEfbE\_\{\\mathrm\{fb\}\},

∑i=1N1,kplan𝔼1πk​\[Yi∣ℋi−1\]=\\displaystyle\\sum\_\{i=1\}^\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]=\(N1,kplan−\|ℐk\|\)​𝔼1πk​\[Z1,k\]\+∑i∈ℐk𝔼1πk​\[Yi∣ℋi−1\]\\displaystyle\\bigl\(N^\{\\mathrm\{plan\}\}\_\{1,k\}\-\|\\mathcal\{I\}\_\{k\}\|\\bigr\)\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]\+\\sum\_\{i\\in\\mathcal\{I\}\_\{k\}\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]≥\\displaystyle\\geqN1,kplan​𝔼1πk​\[Z1,k\]−\|ℐk\|​𝔼1πk​\[Z1,k\]\\displaystyle N^\{\\mathrm\{plan\}\}\_\{1,k\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]\-\|\\mathcal\{I\}\_\{k\}\|\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]≥\\displaystyle\\geqN1,kplan​𝔼1πk​\[Z1,k\]−Binc​W1,k\\displaystyle N^\{\\mathrm\{plan\}\}\_\{1,k\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]\-B\_\{\\mathrm\{inc\}\}\\,W\_\{1,k\}≥\\displaystyle\\geq\(ak\+Δk\)−Binc​W1,k,\\displaystyle\(a\_\{k\}\+\\Delta\_\{k\}\)\-B\_\{\\mathrm\{inc\}\}\\,W\_\{1,k\},where the second inequality uses\|ℐk\|≤W1,k\|\\mathcal\{I\}\_\{k\}\|\\leq W\_\{1,k\}and𝔼1πk​\[Z1,k\]≤Binc\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]\\leq B\_\{\\mathrm\{inc\}\}, and the last uses \([132](https://arxiv.org/html/2609.28859#S10.E132)\)\. Consequently, onEfbE\_\{\\mathrm\{fb\}\}and the small\-occupation event

\{W1,k≤Δk4​Binc\},\\Bigl\\\{W\_\{1,k\}\\leq\\tfrac\{\\Delta\_\{k\}\}\{4B\_\{\\mathrm\{inc\}\}\}\\Bigr\\\},\(134\)the accumulated drift satisfies

∑i=1N1,kplan𝔼1πk​\[Yi∣ℋi−1\]≥ak\+Δk−Binc⋅Δk4​Binc=ak\+3​Δk4\.\\sum\_\{i=1\}^\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\geq a\_\{k\}\+\\Delta\_\{k\}\-B\_\{\\mathrm\{inc\}\}\\cdot\\tfrac\{\\Delta\_\{k\}\}\{4B\_\{\\mathrm\{inc\}\}\}=a\_\{k\}\+\\tfrac\{3\\Delta\_\{k\}\}\{4\}\.\(135\)
*Step 3: martingale tail bound \([138](https://arxiv.org/html/2609.28859#S10.E138)\)\.*Decompose the statistic at itemN1,kplanN^\{\\mathrm\{plan\}\}\_\{1,k\}into its predictable \(drift\) part and a fluctuation part,

SN1,kplan=∑i=1N1,kplan𝔼1πk​\[Yi∣ℋi−1\]\+MN1,kplan,Mn:=∑i=1n\(Yi−𝔼1πk​\[Yi∣ℋi−1\]\)\.S\_\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}=\\sum\_\{i=1\}^\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\+M\_\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\},\\qquad M\_\{n\}:=\\sum\_\{i=1\}^\{n\}\\Bigl\(Y\_\{i\}\-\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\Bigr\)\.
*\(i\)MnM\_\{n\}is a martingale\.*Each𝔼1πk​\[Yi∣ℋi−1\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]isℋi−1\\mathcal\{H\}\_\{i\-1\}\-measurable, and the increments are bounded by \([131](https://arxiv.org/html/2609.28859#S10.E131)\), hence integrable\. The centred increment has zero conditional mean,

𝔼1πk\[Yi−𝔼1πk\[Yi∣ℋi−1\]\|ℋi−1\]=0,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\\\!\\left\[Y\_\{i\}\-\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\mathrel\{\\Big\|\}\\mathcal\{H\}\_\{i\-1\}\\right\]=0,so𝔼1πk​\[Mn∣ℋn−1\]=Mn−1\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[M\_\{n\}\\mid\\mathcal\{H\}\_\{n\-1\}\]=M\_\{n\-1\}; thus\(Mn\)n≥0\(M\_\{n\}\)\_\{n\\geq 0\}is a martingale withM0=0M\_\{0\}=0with respect to\(ℋn\)\(\\mathcal\{H\}\_\{n\}\)\. Its increments are bounded:

\|Yi−𝔼1πk​\[Yi∣ℋi−1\]\|≤\|Yi\|\+𝔼1πk​\[Yi∣ℋi−1\]≤2​Binc\.\\left\|Y\_\{i\}\-\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\right\|\\leq\|Y\_\{i\}\|\+\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\leq 2B\_\{\\mathrm\{inc\}\}\.\(136\)
*\(ii\) Accumulated predictable variance\.*Since the increments ofMnM\_\{n\}are martingale differences, their conditional second moment equals the conditional variance ofYiY\_\{i\}\. We write the accumulated predictable variance as

Vn:=∑i=1n𝔼1πk\[\(Yi−𝔼1πk\[Yi∣ℋi−1\]\)2\|ℋi−1\]=∑i=1nVar1\(Yi∣ℋi−1\)\.V\_\{n\}:=\\sum\_\{i=1\}^\{n\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\\\!\\left\[\\left\(Y\_\{i\}\-\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\right\)^\{2\}\\mathrel\{\\Big\|\}\\mathcal\{H\}\_\{i\-1\}\\right\]=\\sum\_\{i=1\}^\{n\}\\operatorname\{Var\}\_\{1\}\(Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\)\.By the variance bound in \([131](https://arxiv.org/html/2609.28859#S10.E131)\) and then \([132](https://arxiv.org/html/2609.28859#S10.E132)\),Var1⁡\(Yi∣ℋi−1\)≤CV​Binc\\operatorname\{Var\}\_\{1\}\(Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\)\\leq C\_\{V\}B\_\{\\mathrm\{inc\}\}holds a\.s\., and thus:

VN1,kplan=∑i=1N1,kplanVar1⁡\(Yi∣ℋi−1\)≤∑i=1N1,kplanCV​Binc=CV​Binc​N1,kplan≤C1​CV​Binc​Lk\.V\_\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}=\\sum\_\{i=1\}^\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}\\operatorname\{Var\}\_\{1\}\(Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\)\\leq\\sum\_\{i=1\}^\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}C\_\{V\}B\_\{\\mathrm\{inc\}\}=C\_\{V\}B\_\{\\mathrm\{inc\}\}\\,N^\{\\mathrm\{plan\}\}\_\{1,k\}\\leq C\_\{1\}C\_\{V\}B\_\{\\mathrm\{inc\}\}\\,L\_\{k\}\.\(137\)
*\(iii\) Reduction to a martingale lower tail\.*On the eventEfb∩\{W1,k≤Δk4​Binc\}E\_\{\\mathrm\{fb\}\}\\cap\\left\\\{W\_\{1,k\}\\leq\\tfrac\{\\Delta\_\{k\}\}\{4B\_\{\\mathrm\{inc\}\}\}\\right\\\}we haveSN1,kplan<akS\_\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}<a\_\{k\}by \([133](https://arxiv.org/html/2609.28859#S10.E133)\) and∑i≤N1,kplan𝔼1πk​\[Yi∣ℋi−1\]≥ak\+3​Δk4\\sum\_\{i\\leq N^\{\\mathrm\{plan\}\}\_\{1,k\}\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\geq a\_\{k\}\+\\tfrac\{3\\Delta\_\{k\}\}\{4\}by \([135](https://arxiv.org/html/2609.28859#S10.E135)\), hence

MN1,kplan=SN1,kplan−∑i=1N1,kplan𝔼1πk​\[Yi∣ℋi−1\]<ak−\(ak\+3​Δk4\)=−3​Δk4\.M\_\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}=S\_\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}\-\\sum\_\{i=1\}^\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]<a\_\{k\}\-\\Bigl\(a\_\{k\}\+\\tfrac\{3\\Delta\_\{k\}\}\{4\}\\Bigr\)=\-\\tfrac\{3\\Delta\_\{k\}\}\{4\}\.\(OnEfbE\_\{\\mathrm\{fb\}\}the pool is exhausted without stopping, soN1,kplan≤Nmain,kN^\{\\mathrm\{plan\}\}\_\{1,k\}\\leq N\_\{\\mathrm\{main\},k\}and the martingale is defined through itemN1,kplanN^\{\\mathrm\{plan\}\}\_\{1,k\}\.\) ThereforeEfb∩\{W1,k≤Δk4​Binc\}⊆\{MN1,kplan≤−3​Δk4\}E\_\{\\mathrm\{fb\}\}\\cap\\left\\\{W\_\{1,k\}\\leq\\tfrac\{\\Delta\_\{k\}\}\{4B\_\{\\mathrm\{inc\}\}\}\\right\\\}\\subseteq\\\{M\_\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}\\leq\-\\tfrac\{3\\Delta\_\{k\}\}\{4\}\\\}\.

*\(iv\) Freedman’s inequality\.*We use the following form\([Freedman 1975](https://arxiv.org/html/2609.28859#bib.bib15)\): if\(Mn\)\(M\_\{n\}\)is a martingale withM0=0M\_\{0\}=0whose increments satisfyMi−Mi−1≤RM\_\{i\}\-M\_\{i\-1\}\\leq Rfor allii, andVn:=∑i=1n𝔼⁡\[\(Mi−Mi−1\)2∣ℋi−1\]V\_\{n\}:=\\sum\_\{i=1\}^\{n\}\\mathbb\{E\}\[\(M\_\{i\}\-M\_\{i\-1\}\)^\{2\}\\mid\\mathcal\{H\}\_\{i\-1\}\]is its accumulated predictable variance, then for allλ\>0\\lambda\>0andv\>0v\>0,

ℙ\(∃n:Mn≥λandVn≤v\)≤exp\{−λ22​\(v\+R​λ\)\}\.\\mathbb\{P\}\\bigl\(\\exists\\,n:\\ M\_\{n\}\\geq\\lambda\\ \\text\{and\}\\ V\_\{n\}\\leq v\\bigr\)\\leq\\exp\\\!\\left\\\{\-\\frac\{\\lambda^\{2\}\}\{2\(v\+R\\lambda\)\}\\right\\\}\.Apply this to the martingale−Mn\-M\_\{n\}, whose increments𝔼1πk​\[Yi∣ℋi−1\]−Yi≤2​Binc\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\-Y\_\{i\}\\leq 2B\_\{\\mathrm\{inc\}\}are bounded above by2​Binc2B\_\{\\mathrm\{inc\}\}\(by \([131](https://arxiv.org/html/2609.28859#S10.E131)\)\)\. Its accumulated predictable variance also equalsVnV\_\{n\}; sinceVN1,kplan≤C1​CV​Binc​LkV\_\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}\\leq C\_\{1\}C\_\{V\}B\_\{\\mathrm\{inc\}\}\\,L\_\{k\}holds surely by \([137](https://arxiv.org/html/2609.28859#S10.E137)\) \(not merely with high probability\), the event\{VN1,kplan≤C1CVBincLk\}\\\{V\_\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}\\leq C\_\{1\}C\_\{V\}B\_\{\\mathrm\{inc\}\}\\,L\_\{k\}\\\}has probability11, so intersecting with it does not change any probability below, and the single time pointn=N1,kplann=N^\{\\mathrm\{plan\}\}\_\{1,k\}is in particular one instance of the ”∃n\\exists n” event\. Withλ=3​Δk4\\lambda=\\tfrac\{3\\Delta\_\{k\}\}\{4\}andv=CV​Binc​Nmain,kv=C\_\{V\}B\_\{\\mathrm\{inc\}\}N\_\{\\mathrm\{main\},k\},

ℙ1πk\(Efb∩\{W1,k≤Δk4​Binc\}\)≤ℙ1πk\(MN1,kplan≤−3​Δk4\)≤exp\{−\(3​Δk/4\)22​\(C1​CV​Binc​Lk\+2​Binc⋅3​Δk4\)\}\.\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\\bigl\(E\_\{\\mathrm\{fb\}\}\\cap\\\{W\_\{1,k\}\\leq\\tfrac\{\\Delta\_\{k\}\}\{4B\_\{\\mathrm\{inc\}\}\}\\\}\\bigr\)\\leq\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\\Bigl\(M\_\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}\\leq\-\\tfrac\{3\\Delta\_\{k\}\}\{4\}\\Bigr\)\\leq\\exp\\\!\\left\\\{\-\\frac\{\(3\\Delta\_\{k\}/4\)^\{2\}\}\{2\\bigl\(C\_\{1\}C\_\{V\}B\_\{\\mathrm\{inc\}\}\\,L\_\{k\}\+2B\_\{\\mathrm\{inc\}\}\\cdot\\tfrac\{3\\Delta\_\{k\}\}\{4\}\\bigr\)\}\\right\\\}\.\(138\)The denominator equals2​C1​CV​Binc​Lk\+3​Binc​Δk≤2​C1​CV​Binc​Lk\+3​Binc​Lk2C\_\{1\}C\_\{V\}B\_\{\\mathrm\{inc\}\}L\_\{k\}\+3B\_\{\\mathrm\{inc\}\}\\Delta\_\{k\}\\leq 2C\_\{1\}C\_\{V\}B\_\{\\mathrm\{inc\}\}L\_\{k\}\+3B\_\{\\mathrm\{inc\}\}L\_\{k\}\(usingΔk≤Lk\\Delta\_\{k\}\\leq L\_\{k\}\), which is at mostC2​\(CV​Binc\+Binc2\)​LkC\_\{2\}\(C\_\{V\}B\_\{\\mathrm\{inc\}\}\+B\_\{\\mathrm\{inc\}\}^\{2\}\)L\_\{k\}for a primitive constantC2C\_\{2\}; the numerator is916​Δk2\\tfrac\{9\}\{16\}\\Delta\_\{k\}^\{2\}\. Hence \([138](https://arxiv.org/html/2609.28859#S10.E138)\) has the stated formCexp\{−cfbΔk2/Lk\}C\\exp\\\{\-c\_\{\\mathrm\{fb\}\}\\Delta\_\{k\}^\{2\}/L\_\{k\}\\\}withcfb=c/\(Binc2\+CV​Binc\)c\_\{\\mathrm\{fb\}\}=c/\(B\_\{\\mathrm\{inc\}\}^\{2\}\+C\_\{V\}B\_\{\\mathrm\{inc\}\}\)\.

*Step 4: the complementary event\.*By Markov’s inequality and the occupation bound of Lemma[10\.4](https://arxiv.org/html/2609.28859#S10.Thmtheorem4),

ℙ1πk​\(W1,k\>Δk4​Binc\)≤4​Binc​𝔼1πk​\[W1,k\]Δk≤4​BincΔk⋅CW​\(zk\+1\)=C​zk\+1Δk\.\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\\Bigl\(W\_\{1,k\}\>\\tfrac\{\\Delta\_\{k\}\}\{4B\_\{\\mathrm\{inc\}\}\}\\Bigr\)\\leq\\frac\{4B\_\{\\mathrm\{inc\}\}\\,\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[W\_\{1,k\}\]\}\{\\Delta\_\{k\}\}\\leq\\frac\{4B\_\{\\mathrm\{inc\}\}\}\{\\Delta\_\{k\}\}\\cdot C\_\{W\}\(z\_\{k\}\+1\)=C\\,\\frac\{z\_\{k\}\+1\}\{\\Delta\_\{k\}\}\.
*Conclusion\.*SplittingEfbE\_\{\\mathrm\{fb\}\}according to \([134](https://arxiv.org/html/2609.28859#S10.E134)\),

ℙ1πk\(Efb\)≤ℙ1πk\(Efb∩\{W1,k≤Δk4​Binc\}\)\+ℙ1πk\(W1,k\>Δk4​Binc\),\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(E\_\{\\mathrm\{fb\}\}\)\\leq\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\\bigl\(E\_\{\\mathrm\{fb\}\}\\cap\\\{W\_\{1,k\}\\leq\\tfrac\{\\Delta\_\{k\}\}\{4B\_\{\\mathrm\{inc\}\}\}\\\}\\bigr\)\+\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\\Bigl\(W\_\{1,k\}\>\\tfrac\{\\Delta\_\{k\}\}\{4B\_\{\\mathrm\{inc\}\}\}\\Bigr\),and Steps 3–4 bound the two terms by the two terms of \([86](https://arxiv.org/html/2609.28859#S10.E86)\)\. This proves the bound underH1H\_\{1\}, and theH0H\_\{0\}case follows symmetrically\.

### 10\.6Proof of Lemma[10\.7](https://arxiv.org/html/2609.28859#S10.Thmtheorem7)

###### Proof 10\.21

Proof of Lemma[10\.7](https://arxiv.org/html/2609.28859#S10.Thmtheorem7)Fix a sufficiently largekk\. The human fallback completion cost is at mostcH​Nfixed,H​\(α2,k,β2,k\)=O⁡\(Lk\)c\_\{\\mathrm\{H\}\}N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)=O\(L\_\{k\}\)from Lemma[10\.9](https://arxiv.org/html/2609.28859#S10.Thmtheorem9)\. If the AI fallback is unavailable, it is never selected\. If it is selected, thenNfixed,AI​\(α2,k,β2,k\)≤Nmain,k=O⁡\(Lk\)N\_\{\\mathrm\{fixed\},\\mathrm\{AI\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\\leq N\_\{\\mathrm\{main\},k\}=O\(L\_\{k\}\)by the definition of availability in the pre\-commitment rule, so its completion cost is alsoO⁡\(Lk\)O\(L\_\{k\}\)\. Hence the pre\-committed fallback cost is deterministically at mostC​LkCL\_\{k\}for a primitive constantCC, on both the human and AI branches\. SinceCkfb=0C^\{\\mathrm\{fb\}\}\_\{k\}=0onEfbcE\_\{\\mathrm\{fb\}\}^\{c\}andCkfb≤C​LkC^\{\\mathrm\{fb\}\}\_\{k\}\\leq CL\_\{k\}onEfbE\_\{\\mathrm\{fb\}\}, we have the pointwise boundCkfb≤C​Lk​1EfbC^\{\\mathrm\{fb\}\}\_\{k\}\\leq CL\_\{k\}\\,\\mathbf\{1\}\_\{E\_\{\\mathrm\{fb\}\}\}; taking𝔼hπk​\[⋅\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[\\cdot\]of both sides,

𝔼hπk​\[Ckfb\]≤C​Lk​ℙhπk​\(Efb\),\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[C^\{\\mathrm\{fb\}\}\_\{k\}\]\\leq CL\_\{k\}\\,\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{h\}\(E\_\{\\mathrm\{fb\}\}\),and substituting the probability estimate of Lemma[10\.6](https://arxiv.org/html/2609.28859#S10.Thmtheorem6)proves \([87](https://arxiv.org/html/2609.28859#S10.E87)\) withCF:=C​CfbC\_\{F\}:=C\\,C\_\{\\mathrm\{fb\}\}\.

### 10\.7Proof of Corollary[5\.4](https://arxiv.org/html/2609.28859#S5.Thmtheorem4)

###### Proof 10\.22

Proof of Corollary[5\.4](https://arxiv.org/html/2609.28859#S5.Thmtheorem4)To prove Corollary[5\.4](https://arxiv.org/html/2609.28859#S5.Thmtheorem4)we first show that, for everyΔk\\Delta\_\{k\}withΔk→∞\\Delta\_\{k\}\\to\\inftyandΔk=o⁡\(Lk\)\\Delta\_\{k\}=o\(L\_\{k\}\), there are primitive constantsC0<∞C\_\{0\}<\\infty,c0\>0c\_\{0\}\>0, andcG\>0c\_\{G\}\>0such that, for all largekk, the fallback probability satisfies

ℙhπk​\(Efb\)≤C0​exp⁡\{−c0​\(Δk\+cG​Gk\+\)2Lk\},\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{h\}\(E\_\{\\mathrm\{fb\}\}\)\\leq C\_\{0\}\\exp\\left\\\{\-c\_\{0\}\\,\\frac\{\(\\Delta\_\{k\}\+c\_\{G\}G\_\{k\}^\{\+\}\)^\{2\}\}\{L\_\{k\}\}\\right\\\},\(139\)and consequently

maxh⁡𝔼hπk​\[Cπk\]LBk≤1\+O⁡\(ΔkLk\+exp⁡\{−c0​\(Δk\+cG​Gk\+\)2Lk\}\+log⁡LkLk\)\.\\frac\{\\max\_\{h\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[C^\{\\pi\_\{k\}\}\]\}\{\\mathrm\{LB\}\_\{k\}\}\\leq 1\+O\\left\(\\frac\{\\Delta\_\{k\}\}\{L\_\{k\}\}\+\\exp\\left\\\{\-c\_\{0\}\\frac\{\(\\Delta\_\{k\}\+c\_\{G\}G\_\{k\}^\{\+\}\)^\{2\}\}\{L\_\{k\}\}\\right\\\}\+\\frac\{\\log L\_\{k\}\}\{L\_\{k\}\}\\right\)\.\(140\)Then, parts \(i\) and \(ii\) can be obtained by directly plugging in the valuesΔk\\Delta\_\{k\}\.

Forh∈\{0,1\}h\\in\\\{0,1\\\}, letρh∗\\rho\_\{h\}^\{\*\}be the value of

ρh∗:=mina,b,e≥0\\displaystyle\\rho\_\{h\}^\{\*\}:=\\min\_\{a,b,e\\geq 0\}cH​a\+cAI​b\+cH​e\\displaystyle c\_\{\\mathrm\{H\}\}a\+c\_\{\\mathrm\{AI\}\}b\+c\_\{\\mathrm\{H\}\}e\(141\)s\.t\.\\displaystyle\\text\{s\.t\. \}a​JX\(h\)\+b​IR\(h\)\+b​Ψh​\(e/b\)≥1,\\displaystyle aJ\_\{X\}^\{\(h\)\}\+bI\_\{R\}^\{\(h\)\}\+b\\Psi\_\{h\}\(e/b\)\\geq 1,0≤e≤b,\\displaystyle 0\\leq e\\leq b,whereΨh​\(e/b=0\)\\Psi\_\{h\}\(e/b=0\)whenb=e=0b=e=0\. Recallℳh\\mathcal\{M\}\_\{h\}is the set of minimizers of \([141](https://arxiv.org/html/2609.28859#S10.E141)\)\. Define the unbuffered sensing item scale

Nsense,k:=max⁡\{ν1∗​ak,ν0∗​bk\}\.N\_\{\\mathrm\{sense\},k\}:=\\max\\\{\\nu\_\{1\}^\{\*\}a\_\{k\},\\nu\_\{0\}^\{\*\}b\_\{k\}\\\}\.\(142\)
We also use the notations assembled in the proof of Lemma[10\.6](https://arxiv.org/html/2609.28859#S10.Thmtheorem6): the item\-level incrementsYiY\_\{i\}with conditional expectations𝔼1πk​\[Yi∣ℋi−1\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]for everyi≤Nmain,ki\\leq N\_\{\\mathrm\{main\},k\}; thus \([131](https://arxiv.org/html/2609.28859#S10.E131)\) holds\. The rounded direction\-11block size

N1,kplan:=⌈nH,1,k∗⌉\+⌈nAI,1,k∗⌉N^\{\\mathrm\{plan\}\}\_\{1,k\}:=\\lceil n^\{\*\}\_\{\\mathrm\{H\},1,k\}\\rceil\+\\lceil n^\{\*\}\_\{\\mathrm\{AI\},1,k\}\\rceilsatisfiesN1,kplan≤Nmain,kN^\{\\mathrm\{plan\}\}\_\{1,k\}\\leq N\_\{\\mathrm\{main\},k\}andN1,kplan​𝔼1πk​\[Z1,k\]≥T1,k=ak\+ΔkN^\{\\mathrm\{plan\}\}\_\{1,k\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]\\geq T\_\{1,k\}=a\_\{k\}\+\\Delta\_\{k\}\(\([132](https://arxiv.org/html/2609.28859#S10.E132)\) and \([37](https://arxiv.org/html/2609.28859#S5.E37)\)\); the occupation countW1,kW\_\{1,k\}of Lemma[10\.4](https://arxiv.org/html/2609.28859#S10.Thmtheorem4); and the martingaleMn:=∑i=1n\(Yi−𝔼1πk​\[Yi∣ℋi−1\]\)M\_\{n\}:=\\sum\_\{i=1\}^\{n\}\\bigl\(Y\_\{i\}\-\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\bigr\), whose increments are bounded by2​Binc2B\_\{\\mathrm\{inc\}\}and whose accumulated predictable variance satisfiesVn≤CV​Binc​n≤CV​Binc​Nmain,kV\_\{n\}\\leq C\_\{V\}B\_\{\\mathrm\{inc\}\}\\,n\\leq C\_\{V\}B\_\{\\mathrm\{inc\}\}\\,N\_\{\\mathrm\{main\},k\}surely\. Finally, Lemma[10\.16](https://arxiv.org/html/2609.28859#S10.Thmtheorem16)\(iii\) gives a primitive constantμ0\>0\\mu\_\{0\}\>0such that

𝔼1πk​\[Yi∣ℋi−1\]≥μ0for every​i≤Nmain,k​and all large​k\.\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\ \\geq\\ \\mu\_\{0\}\\qquad\\text\{for every \}i\\leq N\_\{\\mathrm\{main\},k\}\\text\{ and all large \}k\.\(143\)
*Step 1: reserve inventory\.*We claim that, beyond the planned block, the pool contains at least

Rk:=Nmain,k−N1,kplan≥\[Gk\+−ν1∗​Δk−2\]\+R\_\{k\}:=N\_\{\\mathrm\{main\},k\}\-N^\{\\mathrm\{plan\}\}\_\{1,k\}\\ \\geq\\ \\left\[G\_\{k\}^\{\+\}\-\\nu\_\{1\}^\{\*\}\\Delta\_\{k\}\-2\\right\]\_\{\+\}\(144\)items\. First, the programΓ1​\(T,N\)\\Gamma\_\{1\}\(T,N\)is positively homogeneous in\(T,nH,nAI,nesc\)\(T;n\_\{\\mathrm\{H\}\},n\_\{\\mathrm\{AI\}\},n\_\{\\mathrm\{esc\}\}\)\. Consequently, whenN¯main,k≥ν1∗​T1,k\\overline\{N\}\_\{\\mathrm\{main\},k\}\\geq\\nu\_\{1\}^\{\*\}T\_\{1,k\}, the minimizers inℳh\\mathcal\{M\}\_\{h\}scaled byT1,kT\_\{1,k\}is feasible forΓ1​\(T1,k,N¯main,k\)\\Gamma\_\{1\}\(T\_\{1,k\},\\overline\{N\}\_\{\\mathrm\{main\},k\}\), attains the unconstrained optimal costρ1∗​T1,k\\rho\_\{1\}^\{\*\}T\_\{1,k\}, and consumesν1∗​T1,k\\nu\_\{1\}^\{\*\}T\_\{1,k\}active items, so the least\-consumption selection satisfiesnH,1,k∗\+nAI,1,k∗≤ν1∗​T1,kn^\{\*\}\_\{\\mathrm\{H\},1,k\}\+n^\{\*\}\_\{\\mathrm\{AI\},1,k\}\\leq\\nu\_\{1\}^\{\*\}T\_\{1,k\}; whenN¯main,k<ν1∗​T1,k\\overline\{N\}\_\{\\mathrm\{main\},k\}<\\nu\_\{1\}^\{\*\}T\_\{1,k\}, the same inequality holds trivially because the capacity constraint forcesnH,1,k∗\+nAI,1,k∗≤N¯main,kn^\{\*\}\_\{\\mathrm\{H\},1,k\}\+n^\{\*\}\_\{\\mathrm\{AI\},1,k\}\\leq\\overline\{N\}\_\{\\mathrm\{main\},k\}\. Hence

N1,kplan≤nH,1,k∗\+nAI,1,k∗\+2≤ν1∗​\(ak\+Δk\)\+2\.N^\{\\mathrm\{plan\}\}\_\{1,k\}\\leq n^\{\*\}\_\{\\mathrm\{H\},1,k\}\+n^\{\*\}\_\{\\mathrm\{AI\},1,k\}\+2\\leq\\nu\_\{1\}^\{\*\}\(a\_\{k\}\+\\Delta\_\{k\}\)\+2\.Second, by \([34](https://arxiv.org/html/2609.28859#S5.E34)\)–\([35](https://arxiv.org/html/2609.28859#S5.E35)\) the pool size dominates the fallback floor, which dominates the unsplit floor becauseNfixed,HN\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}is nonincreasing in both budgets \(Lemma[10\.9](https://arxiv.org/html/2609.28859#S10.Thmtheorem9)\); by \([45](https://arxiv.org/html/2609.28859#S5.E45)\) and \([142](https://arxiv.org/html/2609.28859#S10.E142)\),

Nmain,k≥Nfixed,H​\(α2,k,β2,k\)≥Nfixed,H​\(αk,βk\)=Nsense,k\+Gk≥ν1∗​ak\+Gk\.N\_\{\\mathrm\{main\},k\}\\ \\geq\\ N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\\ \\geq\\ N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{k\},\\beta\_\{k\}\)\\ =\\ N\_\{\\mathrm\{sense\},k\}\+G\_\{k\}\\ \\geq\\ \\nu\_\{1\}^\{\*\}a\_\{k\}\+G\_\{k\}\.Subtracting the two displays givesRk≥Gk−ν1∗​Δk−2R\_\{k\}\\geq G\_\{k\}\-\\nu\_\{1\}^\{\*\}\\Delta\_\{k\}\-2; since alsoRk≥0R\_\{k\}\\geq 0andGk≤Gk\+G\_\{k\}\\leq G\_\{k\}^\{\+\}, the claim \([144](https://arxiv.org/html/2609.28859#S10.E144)\) follows\.

*Step 2: drift accumulated over the whole pool\.*OnEfbE\_\{\\mathrm\{fb\}\}no boundary is crossed, so allNmain,kN\_\{\\mathrm\{main\},k\}items are sensed andSNmain,k<akS\_\{N\_\{\\mathrm\{main\},k\}\}<a\_\{k\}\. Split the predictable sum atN1,kplanN^\{\\mathrm\{plan\}\}\_\{1,k\}\. Among the firstN1,kplanN^\{\\mathrm\{plan\}\}\_\{1,k\}items, those withSi−1\>zkS\_\{i\-1\}\>z\_\{k\}use the direction\-11rule and contribute𝔼1πk​\[Yi∣ℋi−1\]=𝔼1πk​\[Z1,k\]\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]=\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]; the remaining ones number at mostW1,kW\_\{1,k\}and contribute𝔼1πk​\[Yi∣ℋi−1\]≥0\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\geq 0, a loss of at most𝔼1πk​\[Z1,k\]≤Binc\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]\\leq B\_\{\\mathrm\{inc\}\}each\. By \([132](https://arxiv.org/html/2609.28859#S10.E132)\),

∑i=1N1,kplan𝔼1πk​\[Yi∣ℋi−1\]≥N1,kplan​𝔼1πk​\[Z1,k\]−Binc​W1,k≥ak\+Δk−Binc​W1,k\.\\sum\_\{i=1\}^\{N^\{\\mathrm\{plan\}\}\_\{1,k\}\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\ \\geq\\ N^\{\\mathrm\{plan\}\}\_\{1,k\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Z\_\{1,k\}\]\-B\_\{\\mathrm\{inc\}\}\\,W\_\{1,k\}\\ \\geq\\ a\_\{k\}\+\\Delta\_\{k\}\-B\_\{\\mathrm\{inc\}\}\\,W\_\{1,k\}\.Every item beyond the block contributes𝔼1πk​\[Yi∣ℋi−1\]≥μ0\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\geq\\mu\_\{0\}by \([143](https://arxiv.org/html/2609.28859#S10.E143)\), regardless of which rule the sign test selects\. Define the enlarged bufferΔ~k:=Δk\+μ0​Rk\\widetilde\{\\Delta\}\_\{k\}:=\\Delta\_\{k\}\+\\mu\_\{0\}R\_\{k\}\. Therefore

∑i=1Nmain,k𝔼1πk​\[Yi∣ℋi−1\]≥ak\+Δk\+μ0​Rk−Binc​W1,k=ak\+Δ~k−Binc​W1,k\.\\sum\_\{i=1\}^\{N\_\{\\mathrm\{main\},k\}\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\ \\geq\\ a\_\{k\}\+\\Delta\_\{k\}\+\\mu\_\{0\}R\_\{k\}\-B\_\{\\mathrm\{inc\}\}\\,W\_\{1,k\}\\ =\\ a\_\{k\}\+\\widetilde\{\\Delta\}\_\{k\}\-B\_\{\\mathrm\{inc\}\}\\,W\_\{1,k\}\.\(145\)
*Step 3: two martingale tails\.*Settk:=Δ~k/\(4​Binc\)t\_\{k\}:=\\widetilde\{\\Delta\}\_\{k\}/\(4B\_\{\\mathrm\{inc\}\}\)and splitEfbE\_\{\\mathrm\{fb\}\}along\{W1,k≤tk\}\\\{W\_\{1,k\}\\leq t\_\{k\}\\\}\.

\(a\) OnEfb∩\{W1,k≤tk\}E\_\{\\mathrm\{fb\}\}\\cap\\\{W\_\{1,k\}\\leq t\_\{k\}\\\}, combiningSNmain,k<akS\_\{N\_\{\\mathrm\{main\},k\}\}<a\_\{k\}with \([145](https://arxiv.org/html/2609.28859#S10.E145)\),

MNmain,k=SNmain,k−∑i=1Nmain,k𝔼1πk​\[Yi∣ℋi−1\]<ak−\(ak\+Δ~k−Binc​tk\)=−34​Δ~k\.M\_\{N\_\{\\mathrm\{main\},k\}\}=S\_\{N\_\{\\mathrm\{main\},k\}\}\-\\sum\_\{i=1\}^\{N\_\{\\mathrm\{main\},k\}\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\ <\\ a\_\{k\}\-\\bigl\(a\_\{k\}\+\\widetilde\{\\Delta\}\_\{k\}\-B\_\{\\mathrm\{inc\}\}\\,t\_\{k\}\\bigr\)\\ =\\ \-\\tfrac\{3\}\{4\}\\widetilde\{\\Delta\}\_\{k\}\.Freedman’s inequality\([Freedman 1975](https://arxiv.org/html/2609.28859#bib.bib15)\), in the form quoted before \([138](https://arxiv.org/html/2609.28859#S10.E138)\) and applied to−MNmain,k\-M\_\{N\_\{\\mathrm\{main\},k\}\}with increment bound2​Binc2B\_\{\\mathrm\{inc\}\}and predictable variance at mostCV​Binc​Nmain,kC\_\{V\}B\_\{\\mathrm\{inc\}\}N\_\{\\mathrm\{main\},k\}surely, gives

ℙ1πk\(Efb∩\{W1,k≤tk\}\)≤exp\{−\(3​Δ~k/4\)22​\(CV​Binc​Nmain,k\+2​Binc⋅34​Δ~k\)\}≤exp\{−caΔ~k2Lk\},\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\\bigl\(E\_\{\\mathrm\{fb\}\}\\cap\\\{W\_\{1,k\}\\leq t\_\{k\}\\\}\\bigr\)\\leq\\exp\\left\\\{\-\\frac\{\(3\\widetilde\{\\Delta\}\_\{k\}/4\)^\{2\}\}\{2\\bigl\(C\_\{V\}B\_\{\\mathrm\{inc\}\}\\,N\_\{\\mathrm\{main\},k\}\+2B\_\{\\mathrm\{inc\}\}\\cdot\\tfrac\{3\}\{4\}\\widetilde\{\\Delta\}\_\{k\}\\bigr\)\}\\right\\\}\\leq\\exp\\left\\\{\-c\_\{a\}\\frac\{\\widetilde\{\\Delta\}\_\{k\}^\{2\}\}\{L\_\{k\}\}\\right\\\},where the last step usesΔ~k≤Δk\+μ0​Nmain,k=O⁡\(Lk\)\\widetilde\{\\Delta\}\_\{k\}\\leq\\Delta\_\{k\}\+\\mu\_\{0\}N\_\{\\mathrm\{main\},k\}=O\(L\_\{k\}\)\(recallμ0≤Binc\\mu\_\{0\}\\leq B\_\{\\mathrm\{inc\}\}\), so that the denominator is at most a primitive multiple ofLkL\_\{k\}\.

\(b\) For the occupation event we replace the Markov step of Lemma[10\.6](https://arxiv.org/html/2609.28859#S10.Thmtheorem6)by a second application of the same tail bound\. SupposeW1,k\>tkW\_\{1,k\}\>t\_\{k\}and letnnbe the index of the⌈tk⌉\\lceil t\_\{k\}\\rceil\-th sensed item withSi−1≤zkS\_\{i\-1\}\\leq z\_\{k\}; thenn≥⌈tk⌉n\\geq\\lceil t\_\{k\}\\rceiland,

Mn−1=Sn−1−∑i=1n−1𝔼1πk​\[Yi∣ℋi−1\]≤zk−μ0​\(⌈tk⌉−1\)≤−μ02​tkM\_\{n\-1\}=S\_\{n\-1\}\-\\sum\_\{i=1\}^\{n\-1\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{1\}\[Y\_\{i\}\\mid\\mathcal\{H\}\_\{i\-1\}\]\\ \\leq\\ z\_\{k\}\-\\mu\_\{0\}\(\\lceil t\_\{k\}\\rceil\-1\)\\ \\leq\\ \-\\tfrac\{\\mu\_\{0\}\}\{2\}\\,t\_\{k\}for all largekk, becausetk≥Δk/\(4​Binc\)→∞t\_\{k\}\\geq\\Delta\_\{k\}/\(4B\_\{\\mathrm\{inc\}\}\)\\to\\inftyandzk=1z\_\{k\}=1\. Hence writingn′:=n−1n^\{\\prime\}:=n\-1gives

\{W1,k\>tk\}⊆\{∃n′≤Nmain,k−1:−Mn′≥μ0tk/2\}\.\\\{W\_\{1,k\}\>t\_\{k\}\\\}\\subseteq\\\{\\exists\\,n^\{\\prime\}\\leq N\_\{\\mathrm\{main\},k\}\-1:\\,\-M\_\{n^\{\\prime\}\}\\geq\\mu\_\{0\}t\_\{k\}/2\\\}\.SinceVn′≤CV​Binc​Nmain,kV\_\{n^\{\\prime\}\}\\leq C\_\{V\}B\_\{\\mathrm\{inc\}\}\\,N\_\{\\mathrm\{main\},k\}surely for everyn′≤Nmain,kn^\{\\prime\}\\leq N\_\{\\mathrm\{main\},k\}, the same maximal Freedman inequality yields

ℙ1πk​\(W1,k\>tk\)≤exp⁡\{−\(μ0​tk/2\)22​\(CV​Binc​Nmain,k\+2​Binc⋅μ0​tk/2\)\}≤exp⁡\{−cb​Δ~k2Lk\},\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\\bigl\(W\_\{1,k\}\>t\_\{k\}\\bigr\)\\leq\\exp\\left\\\{\-\\frac\{\(\\mu\_\{0\}t\_\{k\}/2\)^\{2\}\}\{2\\bigl\(C\_\{V\}B\_\{\\mathrm\{inc\}\}N\_\{\\mathrm\{main\},k\}\+2B\_\{\\mathrm\{inc\}\}\\cdot\\mu\_\{0\}t\_\{k\}/2\\bigr\)\}\\right\\\}\\leq\\exp\\left\\\{\-c\_\{b\}\\frac\{\\widetilde\{\\Delta\}\_\{k\}^\{2\}\}\{L\_\{k\}\}\\right\\\},usingtk=Δ~k/\(4​Binc\)t\_\{k\}=\\widetilde\{\\Delta\}\_\{k\}/\(4B\_\{\\mathrm\{inc\}\}\)andCV​Binc​Nmain,k≤C1​LkC\_\{V\}B\_\{\\mathrm\{inc\}\}N\_\{\\mathrm\{main\},k\}\\leq C\_\{1\}L\_\{k\}again\. Adding \(a\) and \(b\),ℙ1πk\(Efb\)≤2exp\{−c′Δ~k2/Lk\}\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{1\}\(E\_\{\\mathrm\{fb\}\}\)\\leq 2\\exp\\\{\-c^\{\\prime\}\\widetilde\{\\Delta\}\_\{k\}^\{2\}/L\_\{k\}\\\}withc′:=min⁡\{ca,cb\}c^\{\\prime\}:=\\min\\\{c\_\{a\},c\_\{b\}\\\}\.

It remains to replaceΔ~k=Δk\+μ0​Rk\\widetilde\{\\Delta\}\_\{k\}=\\Delta\_\{k\}\+\\mu\_\{0\}R\_\{k\}byΔk\+cG​Gk\+\\Delta\_\{k\}\+c\_\{G\}G\_\{k\}^\{\+\}\. Setν¯:=max⁡\{ν1∗,ν0∗\}\\bar\{\\nu\}:=\\max\\\{\\nu\_\{1\}^\{\*\},\\nu\_\{0\}^\{\*\}\\\}andcG:=min⁡\{μ0/2,1/\(2​ν¯\),1/2\}c\_\{G\}:=\\min\\\{\\mu\_\{0\}/2,\\,1/\(2\\bar\{\\nu\}\),\\,1/2\\\}\. IfGk\+≥2​\(ν¯​Δk\+2\)G\_\{k\}^\{\+\}\\geq 2\(\\bar\{\\nu\}\\Delta\_\{k\}\+2\), then \([144](https://arxiv.org/html/2609.28859#S10.E144)\) givesRk≥Gk\+/2R\_\{k\}\\geq G\_\{k\}^\{\+\}/2, soΔ~k≥Δk\+\(μ0/2\)​Gk\+≥Δk\+cG​Gk\+\\widetilde\{\\Delta\}\_\{k\}\\geq\\Delta\_\{k\}\+\(\\mu\_\{0\}/2\)G\_\{k\}^\{\+\}\\geq\\Delta\_\{k\}\+c\_\{G\}G\_\{k\}^\{\+\}\. OtherwiseGk\+<2​ν¯​Δk\+4G\_\{k\}^\{\+\}<2\\bar\{\\nu\}\\Delta\_\{k\}\+4, soΔk\+cG​Gk\+≤\(1\+2​cG​ν¯\)​Δk\+4​cG≤2​Δk\+2≤2​Δ~k\+2\\Delta\_\{k\}\+c\_\{G\}G\_\{k\}^\{\+\}\\leq\(1\+2c\_\{G\}\\bar\{\\nu\}\)\\Delta\_\{k\}\+4c\_\{G\}\\leq 2\\Delta\_\{k\}\+2\\leq 2\\widetilde\{\\Delta\}\_\{k\}\+2\. In both casesΔ~k≥12​\(Δk\+cG​Gk\+\)−1\\widetilde\{\\Delta\}\_\{k\}\\geq\\tfrac\{1\}\{2\}\(\\Delta\_\{k\}\+c\_\{G\}G\_\{k\}^\{\+\}\)\-1, and sinceΔk→∞\\Delta\_\{k\}\\to\\infty, for all largekkwe haveΔ~k2≥18​\(Δk\+cG​Gk\+\)2\\widetilde\{\\Delta\}\_\{k\}^\{2\}\\geq\\tfrac\{1\}\{8\}\(\\Delta\_\{k\}\+c\_\{G\}G\_\{k\}^\{\+\}\)^\{2\}\. This proves \([139](https://arxiv.org/html/2609.28859#S10.E139)\) withc0:=c′/8c\_\{0\}:=c^\{\\prime\}/8andC0:=2C\_\{0\}:=2\.

*Step 4: assembly, and parts \(i\)–\(ii\)\.*WriteCπk=cdata​Nmain,k\+Ckmain\+CkfbC^\{\\pi\_\{k\}\}=c\_\{\\mathrm\{data\}\}N\_\{\\mathrm\{main\},k\}\+C^\{\\mathrm\{main\}\}\_\{k\}\+C^\{\\mathrm\{fb\}\}\_\{k\}as in the proof of Theorem[5\.3](https://arxiv.org/html/2609.28859#S5.Thmtheorem3)\. Costs 1 and 2 in the proof of Lemma[10\.5](https://arxiv.org/html/2609.28859#S10.Thmtheorem5)bound the boundary\-crossing runs byΓh​\(Th,k,N¯main,k\)\+O⁡\(1\)\+cmax​CW​\(zk\+1\)\\Gamma\_\{h\}\(T\_\{h,k\},\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\+O\(1\)\+c\_\{\\max\}C\_\{W\}\(z\_\{k\}\+1\), where the occupation term isO⁡\(1\)O\(1\)becausezk=1z\_\{k\}=1\. The proof of Lemma[10\.7](https://arxiv.org/html/2609.28859#S10.Thmtheorem7)gives𝔼hπk​\[Ckfb\]≤C​Lk​ℙhπk​\(Efb\)\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[C^\{\\mathrm\{fb\}\}\_\{k\}\]\\leq CL\_\{k\}\\mathbb\{P\}^\{\\pi\_\{k\}\}\_\{h\}\(E\_\{\\mathrm\{fb\}\}\)\. SinceNmain,k=O⁡\(Lk\)N\_\{\\mathrm\{main\},k\}=O\(L\_\{k\}\), collecting terms, taking the maximum overhh, and applying Lemma[10\.8](https://arxiv.org/html/2609.28859#S10.Thmtheorem8)withffb,k=1/2f\_\{\\mathrm\{fb\},k\}=1/2and \([139](https://arxiv.org/html/2609.28859#S10.E139)\),

maxh⁡𝔼hπk​\[Cπk\]≤LBk\+CP​Δk\+O⁡\(log⁡Lk\)\+C​Lk​exp⁡\{−c0​\(Δk\+cG​Gk\+\)2Lk\}\.\\max\_\{h\}\\mathbb\{E\}^\{\\pi\_\{k\}\}\_\{h\}\[C^\{\\pi\_\{k\}\}\]\\leq\\mathrm\{LB\}\_\{k\}\+C\_\{P\}\\Delta\_\{k\}\+O\(\\log L\_\{k\}\)\+CL\_\{k\}\\exp\\left\\\{\-c\_\{0\}\\frac\{\(\\Delta\_\{k\}\+c\_\{G\}G\_\{k\}^\{\+\}\)^\{2\}\}\{L\_\{k\}\}\\right\\\}\.Dividing byLBk=Θ⁡\(Lk\)\\mathrm\{LB\}\_\{k\}=\\Theta\(L\_\{k\}\)yields \([140](https://arxiv.org/html/2609.28859#S10.E140)\)\.

For part \(i\), takeΔk=κ​Lk​log⁡Lk\\Delta\_\{k\}=\\kappa\\sqrt\{L\_\{k\}\\log L\_\{k\}\}\. Using only\(Δk\+cG​Gk\+\)2≥Δk2=κ2​Lk​log⁡Lk\(\\Delta\_\{k\}\+c\_\{G\}G\_\{k\}^\{\+\}\)^\{2\}\\geq\\Delta\_\{k\}^\{2\}=\\kappa^\{2\}L\_\{k\}\\log L\_\{k\}, the exponential term in \([140](https://arxiv.org/html/2609.28859#S10.E140)\) is at mostLk−c0​κ2=o\(Lk−1/2\)L\_\{k\}^\{\-c\_\{0\}\\kappa^\{2\}\}=o\(L\_\{k\}^\{\-1/2\}\), becausec0​κ2\>1/2c\_\{0\}\\kappa^\{2\}\>1/2by the choiceκ\>1/2​c0\\kappa\>1/\\sqrt\{2c\_\{0\}\}, whileΔk/Lk=κlog⁡Lk/Lk=O~\(Lk−1/2\)\\Delta\_\{k\}/L\_\{k\}=\\kappa\\sqrt\{\\log L\_\{k\}/L\_\{k\}\}=\\widetilde\{O\}\(L\_\{k\}^\{\-1/2\}\)dominateslog⁡Lk/Lk\\log L\_\{k\}/L\_\{k\}\. The ratio is therefore1\+O~\(Lk−1/2\)1\+\\widetilde\{O\}\(L\_\{k\}^\{\-1/2\}\), with no condition onGkG\_\{k\}\.

For part \(ii\), supposeGk\+≥C​Lk​log⁡LkG\_\{k\}^\{\+\}\\geq C\\sqrt\{L\_\{k\}\\log L\_\{k\}\}and takeΔk\\Delta\_\{k\}polylogarithmic withΔk→∞\\Delta\_\{k\}\\to\\infty\. Then\(Δk\+cG​Gk\+\)2≥cG2​C2​Lk​log⁡Lk\(\\Delta\_\{k\}\+c\_\{G\}G\_\{k\}^\{\+\}\)^\{2\}\\geq c\_\{G\}^\{2\}C^\{2\}L\_\{k\}\\log L\_\{k\}, so the exponential term in \([140](https://arxiv.org/html/2609.28859#S10.E140)\) is at mostLk−c0​cG2​C2L\_\{k\}^\{\-c\_\{0\}c\_\{G\}^\{2\}C^\{2\}\}, which isO⁡\(Lk−x\)O\(L\_\{k\}^\{\-x\}\)for any preassigned fixedxxonce the primitive constantCCis large enough, whileΔk/Lk\\Delta\_\{k\}/L\_\{k\}andlog⁡Lk/Lk\\log L\_\{k\}/L\_\{k\}areO~​\(1/Lk\)\\widetilde\{O\}\(1/L\_\{k\}\)\. The ratio is therefore1\+O~​\(1/Lk\)1\+\\widetilde\{O\}\(1/L\_\{k\}\), sharpening part \(i\)\.

## 11Additional Materials for Section[6\.1](https://arxiv.org/html/2609.28859#S6.SS1)

### 11\.1Pilot Concentration

###### Lemma 11\.1\(Pilot concentration\)

Under Assumption[6\.1](https://arxiv.org/html/2609.28859#S6.SS1),ℙh​\(ℰpilot,m\)≥1−δm\\mathbb\{P\}\_\{h\}\(\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}\)\\geq 1\-\\delta\_\{m\}forh∈\{0,1\}h\\in\\\{0,1\\\}\.

###### Proof 11\.2

Proof of Lemma[11\.1](https://arxiv.org/html/2609.28859#S11.Thmtheorem1)The proof proceeds in three steps: a Hoeffding bound on the empirical frequenciesMx​\(r\)/MxM\_\{x\}\(r\)/M\_\{x\}, a deterministic bound on\|f^x​\(r\)−Mx​\(r\)/Mx\|\|\\hat\{f\}\_\{x\}\(r\)\-M\_\{x\}\(r\)/M\_\{x\}\|, and a combination of the two that also yields the rate\.

*Step 1: concentration of the empirical frequencies\.*Fixh∈\{0,1\}h\\in\\\{0,1\\\},x∈\{0,1\}x\\in\\\{0,1\\\}andr∈ℛr\\in\\mathcal\{R\}\. Under Assumption[6\.1](https://arxiv.org/html/2609.28859#S6.SS1), conditional onMxM\_\{x\}the indicators𝟏\{Rx,jpilot=r\}\\mathbf\{1\}\\\{R^\{\\mathrm\{pilot\}\}\_\{x,j\}=r\\\},1≤j≤Mx1\\leq j\\leq M\_\{x\}, are i\.i\.d\.Bernoulli⁡\(fx​\(r\)\)\\operatorname\{Bernoulli\}\(f\_\{x\}\(r\)\), andMx​\(r\)M\_\{x\}\(r\)is their sum\. Then𝔼⁡\[Mx​\(r\)/Mx\|Mx\]=fx​\(r\)\\mathbb\{E\}\[M\_\{x\}\(r\)/M\_\{x\}\|M\_\{x\}\]=f\_\{x\}\(r\)\. Hoeffding’s inequality for sums of independent\[0,1\]\[0,1\]\-valued random variables gives, for everyt\>0t\>0,

ℙh​\(\|Mx​\(r\)Mx−fx​\(r\)\|\>t\|Mx\)≤2​e−2​Mx​t2≤2​e−2​m​t2​almost surely\.\\mathbb\{P\}\_\{h\}\\\!\\left\(\\left\|\\frac\{M\_\{x\}\(r\)\}\{M\_\{x\}\}\-f\_\{x\}\(r\)\\right\|\>t\\Bigg\|M\_\{x\}\\right\)\\leq 2e^\{\-2M\_\{x\}t^\{2\}\}\\leq 2e^\{\-2mt^\{2\}\}\\text\{ almost surely\}\.Then using the law of iterated expectation we have

ℙh​\(\|Mx​\(r\)Mx−fx​\(r\)\|\>t\)≤2​e−2​m​t2\.\\mathbb\{P\}\_\{h\}\\\!\\left\(\\left\|\\frac\{M\_\{x\}\(r\)\}\{M\_\{x\}\}\-f\_\{x\}\(r\)\\right\|\>t\\right\)\\leq 2e^\{\-2mt^\{2\}\}\.Evaluating the bound att=tmt=t\_\{m\}, we obtain

ℙh​\(\|Mx​\(r\)Mx−fx​\(r\)\|\>tm\)≤2​e−2​m​tm2=2​exp⁡\(−log⁡4​\|ℛ\|δm\)=δm2​\|ℛ\|\.\\mathbb\{P\}\_\{h\}\\\!\\left\(\\left\|\\frac\{M\_\{x\}\(r\)\}\{M\_\{x\}\}\-f\_\{x\}\(r\)\\right\|\>t\_\{m\}\\right\)\\leq 2e^\{\-2mt\_\{m\}^\{2\}\}=2\\exp\\\!\\left\(\-\\log\\frac\{4\|\\mathcal\{R\}\|\}\{\\delta\_\{m\}\}\\right\)=\\frac\{\\delta\_\{m\}\}\{2\|\\mathcal\{R\}\|\}\.Since there are exactly2​\|ℛ\|2\|\\mathcal\{R\}\|pairs\(x,r\)\(x,r\), a union bound gives

ℙh\(ℰemp\)≥1−2\|ℛ\|⋅δm2​\|ℛ\|=1−δm,whereℰemp:=\{maxx∈\{0,1\},r∈ℛ\|Mx​\(r\)Mx−fx\(r\)\|≤tm\}\.\\mathbb\{P\}\_\{h\}\(\\mathcal\{E\}\_\{\\mathrm\{emp\}\}\)\\geq 1\-2\|\\mathcal\{R\}\|\\cdot\\frac\{\\delta\_\{m\}\}\{2\|\\mathcal\{R\}\|\}=1\-\\delta\_\{m\},\\qquad\\text\{where\}\\quad\\mathcal\{E\}\_\{\\mathrm\{emp\}\}:=\\left\\\{\\max\_\{x\\in\\\{0,1\\\},\\,r\\in\\mathcal\{R\}\}\\left\|\\frac\{M\_\{x\}\(r\)\}\{M\_\{x\}\}\-f\_\{x\}\(r\)\\right\|\\leq t\_\{m\}\\right\\\}\.
*Step 2: upper bound on\|f^x​\(r\)−Mx​\(r\)/Mx\|\|\\hat\{f\}\_\{x\}\(r\)\-M\_\{x\}\(r\)/M\_\{x\}\|\.*Notice that

f^x​\(r\)=Mx​\(r\)\+λmMx\+\|ℛ\|​λm=\(1−wx\)​Mx​\(r\)Mx\+wx⋅1\|ℛ\|,wx:=\|ℛ\|​λmMx\+\|ℛ\|​λm∈\(0,1\),\\hat\{f\}\_\{x\}\(r\)=\\frac\{M\_\{x\}\(r\)\+\\lambda\_\{m\}\}\{M\_\{x\}\+\|\\mathcal\{R\}\|\\lambda\_\{m\}\}=\(1\-w\_\{x\}\)\\,\\frac\{M\_\{x\}\(r\)\}\{M\_\{x\}\}\+w\_\{x\}\\cdot\\frac\{1\}\{\|\\mathcal\{R\}\|\},\\qquad w\_\{x\}:=\\frac\{\|\\mathcal\{R\}\|\\lambda\_\{m\}\}\{M\_\{x\}\+\|\\mathcal\{R\}\|\\lambda\_\{m\}\}\\in\(0,1\),so the smoothed estimate \([47](https://arxiv.org/html/2609.28859#S6.E47)\) is a convex combination of the empirical frequency and the uniform mass function\. Consequently, for everyxxandrr,

\|f^x​\(r\)−Mx​\(r\)Mx\|=wx​\|1\|ℛ\|−Mx​\(r\)Mx\|≤wx≤\|ℛ\|​λmMx≤\|ℛ\|​λmm,\\left\|\\hat\{f\}\_\{x\}\(r\)\-\\frac\{M\_\{x\}\(r\)\}\{M\_\{x\}\}\\right\|=w\_\{x\}\\left\|\\frac\{1\}\{\|\\mathcal\{R\}\|\}\-\\frac\{M\_\{x\}\(r\)\}\{M\_\{x\}\}\\right\|\\leq w\_\{x\}\\leq\\frac\{\|\\mathcal\{R\}\|\\lambda\_\{m\}\}\{M\_\{x\}\}\\leq\\frac\{\|\\mathcal\{R\}\|\\lambda\_\{m\}\}\{m\},where the first inequality holds because1/\|ℛ\|1/\|\\mathcal\{R\}\|andMx​\(r\)/MxM\_\{x\}\(r\)/M\_\{x\}both lie in\[0,1\]\[0,1\], and the last two inequalities useMx\+\|ℛ\|​λm≥Mx≥mM\_\{x\}\+\|\\mathcal\{R\}\|\\lambda\_\{m\}\\geq M\_\{x\}\\geq m\.

*Step 3: combination and rate\.*Onℰemp\\mathcal\{E\}\_\{\\mathrm\{emp\}\}the triangle inequality yields, for allxxandrr,

\|f^x​\(r\)−fx​\(r\)\|≤\|f^x​\(r\)−Mx​\(r\)Mx\|\+\|Mx​\(r\)Mx−fx​\(r\)\|≤tm\+\|ℛ\|​λmm\.\|\\hat\{f\}\_\{x\}\(r\)\-f\_\{x\}\(r\)\|\\leq\\left\|\\hat\{f\}\_\{x\}\(r\)\-\\frac\{M\_\{x\}\(r\)\}\{M\_\{x\}\}\\right\|\+\\left\|\\frac\{M\_\{x\}\(r\)\}\{M\_\{x\}\}\-f\_\{x\}\(r\)\\right\|\\leq t\_\{m\}\+\\frac\{\|\\mathcal\{R\}\|\\lambda\_\{m\}\}\{m\}\.Becausef^x​\(r\)\\hat\{f\}\_\{x\}\(r\)andfx​\(r\)f\_\{x\}\(r\)are both probabilities in\[0,1\]\[0,1\], we always have\|f^x​\(r\)−fx​\(r\)\|≤1\|\\hat\{f\}\_\{x\}\(r\)\-f\_\{x\}\(r\)\|\\leq 1, so onℰemp\\mathcal\{E\}\_\{\\mathrm\{emp\}\},

maxx∈\{0,1\},r∈ℛ⁡\|f^x​\(r\)−fx​\(r\)\|≤min⁡\{1,tm\+\|ℛ\|​λmm\}=rm\.\\max\_\{x\\in\\\{0,1\\\},\\,r\\in\\mathcal\{R\}\}\|\\hat\{f\}\_\{x\}\(r\)\-f\_\{x\}\(r\)\|\\leq\\min\\left\\\{1,\\;t\_\{m\}\+\\frac\{\|\\mathcal\{R\}\|\\lambda\_\{m\}\}\{m\}\\right\\\}=r\_\{m\}\.Henceℰemp⊆ℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{emp\}\}\\subseteq\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}andℙh​\(ℰpilot,m\)≥ℙh​\(ℰemp\)≥1−δm\\mathbb\{P\}\_\{h\}\(\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}\)\\geq\\mathbb\{P\}\_\{h\}\(\\mathcal\{E\}\_\{\\mathrm\{emp\}\}\)\\geq 1\-\\delta\_\{m\}\.

## 12Additional Materials for Section[6\.2](https://arxiv.org/html/2609.28859#S6.SS2)on Feasibility

Throughout this section, channel\-derived quantities without hats are evaluated at the true channelf=\(f0,f1\)f=\(f\_\{0\},f\_\{1\}\), whereas hatted quantities are evaluated at the plug\-in channelf^=\(f^0,f^1\)\\hat\{f\}=\(\\hat\{f\}\_\{0\},\\hat\{f\}\_\{1\}\)\. Conditional on𝒟m\\mathcal\{D\}\_\{m\}, let𝔼hπ^m,k\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}denote expectation underHhH\_\{h\}for the process induced by the plug\-in policyπ^m,k\\hat\{\\pi\}\_\{m,k\}and the true report channelff, and let𝔼h,f^π^m,k\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h,\\hat\{f\}\}denote the corresponding expectation whenffis replaced byf^\\hat\{f\}, with the label lawX∼Bernoulli⁡\(ph\)X\\sim\\operatorname\{Bernoulli\}\(p\_\{h\}\)unchanged\.

We index the main\-stage likelihood\-ratio updates byt=1,2,…t=1,2,\\ldotsand recallℋt:=σ⁡\(A1,O1,…,At,Ot\)\\mathcal\{H\}\_\{t\}:=\\sigma\\bigl\(A\_\{1\},O\_\{1\},\\ldots,A\_\{t\},O\_\{t\}\\bigr\)\(withℋ0\\mathcal\{H\}\_\{0\}the trivialσ\\sigma\-algebra\{∅,Ω\}\\\{\\emptyset,\\Omega\\\}\) is the filtration generated by the actions and observations up to and including updatett\. Letℋtm:=σ⁡\(𝒟m\)∨ℋt\\mathcal\{H\}^\{m\}\_\{t\}:=\\sigma\(\\mathcal\{D\}\_\{m\}\)\\vee\\mathcal\{H\}\_\{t\}be its pilot\-augmented version\. Also letξt\\xi\_\{t\}be the exact log\-likelihood increment \([77](https://arxiv.org/html/2609.28859#S9.E77)\) of the observation revealed byπ^m,k\\hat\{\\pi\}\_\{m,k\}at updatett, andSt:=∑s≤tξsS\_\{t\}:=\\sum\_\{s\\leq t\}\\xi\_\{s\}\. Sincef0,f1f\_\{0\},f\_\{1\}are unknown,StS\_\{t\}is also unobservable by the policyπ^m,k\\hat\{\\pi\}\_\{m,k\}\. Let

ξ^t:=\{ℓX​\(Xi\),At=H⁡\(i\)​with iteminot previously AI\-queried,ℓ^R​\(Ri\),At=AI⁡\(i\),ℓ^H​\(Xi,Ri\),At=H⁡\(i\)​with reportRialready observed\.\\hat\{\\xi\}\_\{t\}:=\\begin\{cases\}\\ell\_\{X\}\(X\_\{i\}\),&A\_\{t\}=\\mathrm\{H\}\(i\)\\text\{ with item $i$ not previously AI\-queried\},\\\\ \\hat\{\\ell\}\_\{R\}\(R\_\{i\}\),&A\_\{t\}=\\mathrm\{AI\}\(i\),\\\\ \\hat\{\\ell\}\_\{H\}\(X\_\{i\},R\_\{i\}\),&A\_\{t\}=\\mathrm\{H\}\(i\)\\text\{ with report $R\_\{i\}$ already observed\}\.\\end\{cases\}\(146\)ThenS^t:=∑s≤tξ^s\\hat\{S\}\_\{t\}:=\\sum\_\{s\\leq t\}\\hat\{\\xi\}\_\{s\}is the statistic used inπ^m,k\\hat\{\\pi\}\_\{m,k\}\.

We first present an auxiliary Lemma[12\.1](https://arxiv.org/html/2609.28859#S12.Thmtheorem1)in Appendix[12\.1](https://arxiv.org/html/2609.28859#S12.SS1), and then in Appendix[12\.2](https://arxiv.org/html/2609.28859#S12.SS2)we present the proofs of Theorem[6\.1](https://arxiv.org/html/2609.28859#S6.Thmtheorem1)and Corollary[6\.2](https://arxiv.org/html/2609.28859#S6.Thmtheorem2)\.

### 12\.1Likelihood Ratio Bound

Define

w¯m​\(r\):=f¯1,m​\(r\)f¯1,m​\(r\)\+f¯0,m​\(r\),w¯m​\(r\):=f¯1,m​\(r\)f¯1,m​\(r\)\+f¯0,m​\(r\),\\underline\{w\}\_\{m\}\(r\):=\\frac\{\\underline\{f\}\_\{1,m\}\(r\)\}\{\\underline\{f\}\_\{1,m\}\(r\)\+\\overline\{f\}\_\{0,m\}\(r\)\},\\qquad\\overline\{w\}\_\{m\}\(r\):=\\frac\{\\overline\{f\}\_\{1,m\}\(r\)\}\{\\overline\{f\}\_\{1,m\}\(r\)\+\\underline\{f\}\_\{0,m\}\(r\)\},\(147\)wheref¯x,m​\(r\):=\[f^x​\(r\)−rm\]\+\\underline\{f\}\_\{x,m\}\(r\):=\[\\hat\{f\}\_\{x\}\(r\)\-r\_\{m\}\]\_\{\+\}andf¯x,m​\(r\):=min⁡\{1,f^x​\(r\)\+rm\}\\overline\{f\}\_\{x,m\}\(r\):=\\min\\\{1,\\hat\{f\}\_\{x\}\(r\)\+r\_\{m\}\\\}are the lower and upper bound onfx​\(r\)f\_\{x\}\(r\)on the pilot good eventℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}\. Let

ϕrep​\(w\):=log⁡\(1−p1\)​\(1−w\)\+p1​w\(1−p0\)​\(1−w\)\+p0​w,0≤w≤1\.\\phi\_\{\\mathrm\{rep\}\}\(w\):=\\log\\frac\{\(1\-p\_\{1\}\)\(1\-w\)\+p\_\{1\}w\}\{\(1\-p\_\{0\}\)\(1\-w\)\+p\_\{0\}w\},\\qquad 0\\leq w\\leq 1\.\(148\)Then we have thatϕrep​\(w⁡\(r\)\)=ℓR​\(r\)\\phi\_\{\\mathrm\{rep\}\}\(w\(r\)\)=\\ell\_\{R\}\(r\)withw⁡\(r\):=f1​\(r\)f0​\(r\)\+f1​\(r\)w\(r\):=\\frac\{f\_\{1\}\(r\)\}\{f\_\{0\}\(r\)\+f\_\{1\}\(r\)\}, andϕrep​\(w^​\(r\)\)=ℓ^R​\(r\)\\phi\_\{\\mathrm\{rep\}\}\(\\hat\{w\}\(r\)\)=\\hat\{\\ell\}\_\{R\}\(r\)withw^​\(r\):=f^1​\(r\)f^0​\(r\)\+f^1​\(r\)\\hat\{w\}\(r\):=\\frac\{\\hat\{f\}\_\{1\}\(r\)\}\{\\hat\{f\}\_\{0\}\(r\)\+\\hat\{f\}\_\{1\}\(r\)\}for allr∈ℛr\\in\\mathcal\{R\}\. Finally we letεmLR\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}be an upper bound on\|ℓ^R​\(r\)−ℓR​\(r\)\|\|\\hat\{\\ell\}\_\{R\}\(r\)\-\\ell\_\{R\}\(r\)\|onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}:

εmLR:=maxr∈ℛ⁡max⁡\{ℓ^R​\(r\)−ϕrep​\(w¯m​\(r\)\),ϕrep​\(w¯m​\(r\)\)−ℓ^R​\(r\)\}\.\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}:=\\max\_\{r\\in\\mathcal\{R\}\}\\max\\left\\\{\\hat\{\\ell\}\_\{R\}\(r\)\-\\phi\_\{\\mathrm\{rep\}\}\(\\underline\{w\}\_\{m\}\(r\)\),\\phi\_\{\\mathrm\{rep\}\}\(\\overline\{w\}\_\{m\}\(r\)\)\-\\hat\{\\ell\}\_\{R\}\(r\)\\right\\\}\.\(149\)
###### Lemma 12\.1\(Likelihood ratio bound\)

1. \(i\)For everyx∈\{0,1\}x\\in\\\{0,1\\\}andr∈ℛr\\in\\mathcal\{R\}, ℓR​\(r\)\+ℓH​\(x,r\)=ℓX​\(x\),ℓ^R​\(r\)\+ℓ^H​\(x,r\)=ℓX​\(x\)\.\\ell\_\{R\}\(r\)\+\\ell\_\{H\}\(x,r\)=\\ell\_\{X\}\(x\),\\qquad\\hat\{\\ell\}\_\{R\}\(r\)\+\\hat\{\\ell\}\_\{H\}\(x,r\)=\\ell\_\{X\}\(x\)\.\(150\)In particular,\|ℓX\|,\|ℓR\|,\|ℓ^R\|≤Bℓ\|\\ell\_\{X\}\|,\|\\ell\_\{R\}\|,\|\\hat\{\\ell\}\_\{R\}\|\\leq B\_\{\\ell\}and\|ℓH\|,\|ℓ^H\|≤2​Bℓ\|\\ell\_\{H\}\|,\|\\hat\{\\ell\}\_\{H\}\|\\leq 2B\_\{\\ell\}\.
2. \(ii\)Onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}, \|ℓ^R​\(r\)−ℓR​\(r\)\|≤εmLR\(r∈ℛ\),\|\\hat\{\\ell\}\_\{R\}\(r\)\-\\ell\_\{R\}\(r\)\|\\leq\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}\\quad\(r\\in\\mathcal\{R\}\),\(151\)and, on every realized transcript involving at mostNNitems,\|S^t−St\|≤N​εmLR\|\\hat\{S\}\_\{t\}\-S\_\{t\}\|\\leq N\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}at every update time\.

###### Proof 12\.2

Proof of Lemma[12\.1](https://arxiv.org/html/2609.28859#S12.Thmtheorem1)Throughout, recall thatfx​\(r\)\>0f\_\{x\}\(r\)\>0for allx,rx,r\(Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1); Section[6](https://arxiv.org/html/2609.28859#S6)relaxes only the knowledge off0,f1f\_\{0\},f\_\{1\}, not their positivity\), and thatf^x​\(r\)\>0\\hat\{f\}\_\{x\}\(r\)\>0and∑rf^x​\(r\)=1\\sum\_\{r\}\\hat\{f\}\_\{x\}\(r\)=1by the smoothing in \([47](https://arxiv.org/html/2609.28859#S6.E47)\)\. Hence all objects below are well\-defined for both the true channelf=\(f0,f1\)f=\(f\_\{0\},f\_\{1\}\)and the plug\-in channelf^=\(f^0,f^1\)\\hat\{f\}=\(\\hat\{f\}\_\{0\},\\hat\{f\}\_\{1\}\)\.

*Part \(i\)\.*We prove the two identities in \([150](https://arxiv.org/html/2609.28859#S12.E150)\) in turn\.

For the first identity, fixx∈\{0,1\}x\\in\\\{0,1\\\}andr∈ℛr\\in\\mathcal\{R\}\. By \([3](https://arxiv.org/html/2609.28859#S3.E3)\),qh​\(r\)=ph​f1​\(r\)/gh​\(r\)q\_\{h\}\(r\)=p\_\{h\}f\_\{1\}\(r\)/g\_\{h\}\(r\), hence1−qh​\(r\)=\(gh​\(r\)−ph​f1​\(r\)\)/gh​\(r\)=\(1−ph\)​f0​\(r\)/gh​\(r\)1\-q\_\{h\}\(r\)=\(g\_\{h\}\(r\)\-p\_\{h\}f\_\{1\}\(r\)\)/g\_\{h\}\(r\)=\(1\-p\_\{h\}\)f\_\{0\}\(r\)/g\_\{h\}\(r\), and the conditional\-label mass function \([4](https://arxiv.org/html/2609.28859#S3.E4)\) can be written as

ρh​\(x∣r\)=phx​\(1−ph\)1−x​fx​\(r\)gh​\(r\),h∈\{0,1\}\.\\rho\_\{h\}\(x\\mid r\)=\\frac\{p\_\{h\}^\{x\}\(1\-p\_\{h\}\)^\{1\-x\}f\_\{x\}\(r\)\}\{g\_\{h\}\(r\)\},\\qquad h\\in\\\{0,1\\\}\.In the ratio of the two hypotheses the factorfx​\(r\)f\_\{x\}\(r\)is common to the numerator and the denominator and cancels:

ρ1​\(x∣r\)ρ0​\(x∣r\)=p1x​\(1−p1\)1−xp0x​\(1−p0\)1−x⋅g0​\(r\)g1​\(r\)=eℓX​\(x\)​g0​\(r\)g1​\(r\)\.\\frac\{\\rho\_\{1\}\(x\\mid r\)\}\{\\rho\_\{0\}\(x\\mid r\)\}=\\frac\{p\_\{1\}^\{x\}\(1\-p\_\{1\}\)^\{1\-x\}\}\{p\_\{0\}^\{x\}\(1\-p\_\{0\}\)^\{1\-x\}\}\\cdot\\frac\{g\_\{0\}\(r\)\}\{g\_\{1\}\(r\)\}=e^\{\\ell\_\{X\}\(x\)\}\\,\\frac\{g\_\{0\}\(r\)\}\{g\_\{1\}\(r\)\}\.Taking logarithms and addingℓR​\(r\)=log⁡\(g1​\(r\)/g0​\(r\)\)\\ell\_\{R\}\(r\)=\\log\(g\_\{1\}\(r\)/g\_\{0\}\(r\)\)on both sides givesℓR​\(r\)\+ℓH​\(x,r\)=ℓX​\(x\)\\ell\_\{R\}\(r\)\+\\ell\_\{H\}\(x,r\)=\\ell\_\{X\}\(x\)\.

For the second identity, the plug\-in objects obey the same algebra: by \([49](https://arxiv.org/html/2609.28859#S6.E49)\),q^h​\(r\)=ph​f^1​\(r\)/g^h​\(r\)\\hat\{q\}\_\{h\}\(r\)=p\_\{h\}\\hat\{f\}\_\{1\}\(r\)/\\hat\{g\}\_\{h\}\(r\), hence1−q^h​\(r\)=\(g^h​\(r\)−ph​f^1​\(r\)\)/g^h​\(r\)=\(1−ph\)​f^0​\(r\)/g^h​\(r\)1\-\\hat\{q\}\_\{h\}\(r\)=\(\\hat\{g\}\_\{h\}\(r\)\-p\_\{h\}\\hat\{f\}\_\{1\}\(r\)\)/\\hat\{g\}\_\{h\}\(r\)=\(1\-p\_\{h\}\)\\hat\{f\}\_\{0\}\(r\)/\\hat\{g\}\_\{h\}\(r\), and \([49](https://arxiv.org/html/2609.28859#S6.E49)\) gives

ρ^h​\(x∣r\)=phx​\(1−ph\)1−x​f^x​\(r\)g^h​\(r\)\.\\hat\{\\rho\}\_\{h\}\(x\\mid r\)=\\frac\{p\_\{h\}^\{x\}\(1\-p\_\{h\}\)^\{1\-x\}\\hat\{f\}\_\{x\}\(r\)\}\{\\hat\{g\}\_\{h\}\(r\)\}\.Repeating the computation above withf^x,g^h,ρ^h\\hat\{f\}\_\{x\},\\hat\{g\}\_\{h\},\\hat\{\\rho\}\_\{h\}in place offx,gh,ρhf\_\{x\},g\_\{h\},\\rho\_\{h\}\(the factorf^x​\(r\)\\hat\{f\}\_\{x\}\(r\)now cancels in the ratio\) yieldsℓ^R​\(r\)\+ℓ^H​\(x,r\)=ℓX​\(x\)\\hat\{\\ell\}\_\{R\}\(r\)\+\\hat\{\\ell\}\_\{H\}\(x,r\)=\\ell\_\{X\}\(x\)\. The right\-hand side is the*true*ℓX\\ell\_\{X\}becausep0,p1p\_\{0\},p\_\{1\}are known and enterg^h,q^h\\hat\{g\}\_\{h\},\\hat\{q\}\_\{h\}unchanged\.

For the increment bounds, use the normalized weights defined above\. Since

\(1−ph\)​\(1−w⁡\(r\)\)\+ph​w​\(r\)=\(1−ph\)​f0​\(r\)\+ph​f1​\(r\)f0​\(r\)\+f1​\(r\)=gh​\(r\)f0​\(r\)\+f1​\(r\),\(1\-p\_\{h\}\)\(1\-w\(r\)\)\+p\_\{h\}w\(r\)=\\frac\{\(1\-p\_\{h\}\)f\_\{0\}\(r\)\+p\_\{h\}f\_\{1\}\(r\)\}\{f\_\{0\}\(r\)\+f\_\{1\}\(r\)\}=\\frac\{g\_\{h\}\(r\)\}\{f\_\{0\}\(r\)\+f\_\{1\}\(r\)\},the normalizerf0​\(r\)\+f1​\(r\)f\_\{0\}\(r\)\+f\_\{1\}\(r\)cancels in the ratio definingϕrep\\phi\_\{\\mathrm\{rep\}\}in \([148](https://arxiv.org/html/2609.28859#S12.E148)\), andℓR​\(r\)=ϕrep​\(w⁡\(r\)\)\\ell\_\{R\}\(r\)=\\phi\_\{\\mathrm\{rep\}\}\(w\(r\)\); the same computation withf^\\hat\{f\}in place offfgivesℓ^R​\(r\)=ϕrep​\(w^​\(r\)\)\\hat\{\\ell\}\_\{R\}\(r\)=\\phi\_\{\\mathrm\{rep\}\}\(\\hat\{w\}\(r\)\)\. Becauseϕrep\\phi\_\{\\mathrm\{rep\}\}is increasing withϕrep​\(0\)=log⁡1−p11−p0=ℓX​\(0\)\\phi\_\{\\mathrm\{rep\}\}\(0\)=\\log\\frac\{1\-p\_\{1\}\}\{1\-p\_\{0\}\}=\\ell\_\{X\}\(0\)andϕrep​\(1\)=log⁡p1p0=ℓX​\(1\)\\phi\_\{\\mathrm\{rep\}\}\(1\)=\\log\\frac\{p\_\{1\}\}\{p\_\{0\}\}=\\ell\_\{X\}\(1\), bothℓR\\ell\_\{R\}andℓ^R\\hat\{\\ell\}\_\{R\}take values in\[ℓX​\(0\),ℓX​\(1\)\]⊆\[−Bℓ,Bℓ\]\[\\ell\_\{X\}\(0\),\\ell\_\{X\}\(1\)\]\\subseteq\[\-B\_\{\\ell\},B\_\{\\ell\}\]\. Finally, \([150](https://arxiv.org/html/2609.28859#S12.E150)\) and the triangle inequality give\|ℓH\|=\|ℓX−ℓR\|≤2​Bℓ\|\\ell\_\{H\}\|=\|\\ell\_\{X\}\-\\ell\_\{R\}\|\\leq 2B\_\{\\ell\}and likewise\|ℓ^H\|≤2​Bℓ\|\\hat\{\\ell\}\_\{H\}\|\\leq 2B\_\{\\ell\}\.

*Part \(ii\)\.*First, direct differentiation gives

ϕrep′​\(w\)=p1−p0\{\(1−p1\)​\(1−w\)\+p1​w\}​\{\(1−p0\)​\(1−w\)\+p0​w\}\>0,\\phi\_\{\\mathrm\{rep\}\}^\{\\prime\}\(w\)=\\frac\{p\_\{1\}\-p\_\{0\}\}\{\\\{\(1\-p\_\{1\}\)\(1\-w\)\+p\_\{1\}w\\\}\\\{\(1\-p\_\{0\}\)\(1\-w\)\+p\_\{0\}w\\\}\}\>0,soϕrep\\phi\_\{\\mathrm\{rep\}\}is increasing\. Onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}we have\|f^x​\(r\)−fx​\(r\)\|≤rm\|\\hat\{f\}\_\{x\}\(r\)\-f\_\{x\}\(r\)\|\\leq r\_\{m\}, sofx​\(r\)≥max⁡\{f^x​\(r\)−rm,0\}=f¯x,m​\(r\)f\_\{x\}\(r\)\\geq\\max\\\{\\hat\{f\}\_\{x\}\(r\)\-r\_\{m\},0\\\}=\\underline\{f\}\_\{x,m\}\(r\)andfx​\(r\)≤min⁡\{1,f^x​\(r\)\+rm\}=f¯x,m​\(r\)f\_\{x\}\(r\)\\leq\\min\\\{1,\\hat\{f\}\_\{x\}\(r\)\+r\_\{m\}\\\}=\\overline\{f\}\_\{x,m\}\(r\); deterministically alsof^x​\(r\)∈\[f¯x,m​\(r\),f¯x,m​\(r\)\]\\hat\{f\}\_\{x\}\(r\)\\in\[\\underline\{f\}\_\{x,m\}\(r\),\\overline\{f\}\_\{x,m\}\(r\)\]\. The denominators in \([147](https://arxiv.org/html/2609.28859#S12.E147)\) are positive becausef¯1,m​\(r\)\+f¯0,m​\(r\)≥f¯0,m​\(r\)≥f^0​\(r\)\>0\\underline\{f\}\_\{1,m\}\(r\)\+\\overline\{f\}\_\{0,m\}\(r\)\\geq\\overline\{f\}\_\{0,m\}\(r\)\\geq\\hat\{f\}\_\{0\}\(r\)\>0andf¯1,m​\(r\)\+f¯0,m​\(r\)≥f¯1,m​\(r\)≥f^1​\(r\)\>0\\overline\{f\}\_\{1,m\}\(r\)\+\\underline\{f\}\_\{0,m\}\(r\)\\geq\\overline\{f\}\_\{1,m\}\(r\)\\geq\\hat\{f\}\_\{1\}\(r\)\>0\. The map\(a,b\)↦a/\(a\+b\)\(a,b\)\\mapsto a/\(a\+b\)is nondecreasing ina≥0a\\geq 0and nonincreasing inb≥0b\\geq 0on\{a\+b\>0\}\\\{a\+b\>0\\\}, so bothw⁡\(r\)w\(r\)andw^​\(r\)\\hat\{w\}\(r\)defined above lie in\[w¯m​\(r\),w¯m​\(r\)\]\[\\underline\{w\}\_\{m\}\(r\),\\overline\{w\}\_\{m\}\(r\)\]\. By the representation established in part \(i\),ℓR​\(r\)=ϕrep​\(w⁡\(r\)\)\\ell\_\{R\}\(r\)=\\phi\_\{\\mathrm\{rep\}\}\(w\(r\)\)andℓ^R​\(r\)=ϕrep​\(w^​\(r\)\)\\hat\{\\ell\}\_\{R\}\(r\)=\\phi\_\{\\mathrm\{rep\}\}\(\\hat\{w\}\(r\)\), andϕrep\\phi\_\{\\mathrm\{rep\}\}is increasing, so both lie in the interval\[ϕrep​\(w¯m​\(r\)\),ϕrep​\(w¯m​\(r\)\)\]\[\\phi\_\{\\mathrm\{rep\}\}\(\\underline\{w\}\_\{m\}\(r\)\),\\phi\_\{\\mathrm\{rep\}\}\(\\overline\{w\}\_\{m\}\(r\)\)\]\. Two numbers in a common interval differ by at most the distance from either of them to the farther endpoint; hence

\|ℓ^R​\(r\)−ℓR​\(r\)\|≤max⁡\{ℓ^R​\(r\)−ϕrep​\(w¯m​\(r\)\),ϕrep​\(w¯m​\(r\)\)−ℓ^R​\(r\)\}≤εmLR,\|\\hat\{\\ell\}\_\{R\}\(r\)\-\\ell\_\{R\}\(r\)\|\\leq\\max\\left\\\{\\hat\{\\ell\}\_\{R\}\(r\)\-\\phi\_\{\\mathrm\{rep\}\}\(\\underline\{w\}\_\{m\}\(r\)\),\\;\\phi\_\{\\mathrm\{rep\}\}\(\\overline\{w\}\_\{m\}\(r\)\)\-\\hat\{\\ell\}\_\{R\}\(r\)\\right\\\}\\leq\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\},proving \([151](https://arxiv.org/html/2609.28859#S12.E151)\)\.

Moreover,S^t−St=∑i:AI\-scored, label not yet revealed\(ℓ^R\(Ri\)−ℓR\(Ri\)\)\\hat\{S\}\_\{t\}\-S\_\{t\}=\\sum\_\{i:\\;\\text\{AI\-scored, label not yet revealed\}\}\\bigl\(\\hat\{\\ell\}\_\{R\}\(R\_\{i\}\)\-\\ell\_\{R\}\(R\_\{i\}\)\\bigr\)according to part \(i\)\. Therefore we have\|S^t−St\|≤N​εmLR\|\\hat\{S\}\_\{t\}\-S\_\{t\}\|\\leq N\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}\.

### 12\.2Proofs of Theorem[6\.1](https://arxiv.org/html/2609.28859#S6.Thmtheorem1)and Corollary[6\.2](https://arxiv.org/html/2609.28859#S6.Thmtheorem2)

###### Proof 12\.3

Proof of Theorem[6\.1](https://arxiv.org/html/2609.28859#S6.Thmtheorem1)Fix a realized pilot inℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}and condition on𝒟m\\mathcal\{D\}\_\{m\}throughout; all plug\-in quantities and design coefficients \(f^x\\hat\{f\}\_\{x\},ℓ^R\\hat\{\\ell\}\_\{R\},ℓ^H\\hat\{\\ell\}\_\{H\},N^main,m,k\\widehat\{N\}\_\{\\mathrm\{main\},m,k\},η^J,m,kH\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{J,m,k\},η^J,m,kesc\\hat\{\\eta\}^\{\\mathrm\{esc\}\}\_\{J,m,k\}, andωm,k\\omega\_\{m,k\}\) are then deterministic\.

If the guarded outer problem \([55](https://arxiv.org/html/2609.28859#S6.E55)\) is infeasible, the safe\-default policy queries a human on every item\. Hence every observed increment equals the exact direct\-label incrementℓX\\ell\_\{X\}, soξ^t=ξt\\hat\{\\xi\}\_\{t\}=\\xi\_\{t\}and, pathwise,S^t=St\\hat\{S\}\_\{t\}=S\_\{t\}for everytt\. Therefore, the boundary\-route argument below applies directly withωm,k=0\\omega\_\{m,k\}=0\. It remains to consider the guarded branch, in whichωm,k=N^main,m,k​εmLR\\omega\_\{m,k\}=\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}\.

*Step 1: conditional martingale property of the true likelihood process\.*The policy selects each sensing action using the plug\-in statisticS^t−1\\hat\{S\}\_\{t\-1\}, and that action depends only on the fixed pilot, the revealed history, and a fresh hypothesis\-independent randomization seed; it does not depend on the not\-yet\-revealed observationOtO\_\{t\}\. Moreover, the main sample and the policy randomization are independent of𝒟m\\mathcal\{D\}\_\{m\}by Assumption[6\.1](https://arxiv.org/html/2609.28859#S6.SS1)\. Thus, conditionally on\(ℋt−1m,At\)\(\\mathcal\{H\}^\{m\}\_\{t\-1\},A\_\{t\}\), the observationOtO\_\{t\}retains its true model law, and the one\-step likelihood\-ratio identities \([79](https://arxiv.org/html/2609.28859#S9.E79)\)–\([81](https://arxiv.org/html/2609.28859#S9.E81)\), followed by averaging overAtA\_\{t\}, give

𝔼0π^m,k​\[eξt∣ℋt−1m\]=1,𝔼1π^m,k​\[e−ξt∣ℋt−1m\]=1\.\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{0\}\[e^\{\\xi\_\{t\}\}\\mid\\mathcal\{H\}^\{m\}\_\{t\-1\}\]=1,\\qquad\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[e^\{\-\\xi\_\{t\}\}\\mid\\mathcal\{H\}^\{m\}\_\{t\-1\}\]=1\.Consequently,eSte^\{S\_\{t\}\}is a nonnegative\(ℋtm\)\(\\mathcal\{H\}^\{m\}\_\{t\}\)\-martingale underℙ0π^m,k\(⋅∣𝒟m\)\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{0\}\(\\cdot\\mid\\mathcal\{D\}\_\{m\}\), ande−Ste^\{\-S\_\{t\}\}is one underℙ1π^m,k\(⋅∣𝒟m\)\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\(\\cdot\\mid\\mathcal\{D\}\_\{m\}\), both with initial value11\.

*Step 2: boundary routes\.*Letτb\\tau\_\{\\mathrm\{b\}\}be the index of the last paid update in the main stage: the update at which a boundary is first crossed, or, if no crossing occurs, the last paid update before entering fallback\. LetE^\+:=\{S^τb≥ak\+ωm,kand policy stops before fallback\}\\hat\{E\}\_\{\+\}:=\\bigl\\\{\\hat\{S\}\_\{\\tau\_\{\\mathrm\{b\}\}\}\\geq a\_\{k\}\+\\omega\_\{m,k\}\\text\{ and policy stops before fallback\}\\bigr\\\}andE^−:=\{S^τb≤−\(bk\+ωm,k\)and policy stops before fallback\}\\hat\{E\}\_\{\-\}:=\\bigl\\\{\\hat\{S\}\_\{\\tau\_\{\\mathrm\{b\}\}\}\\leq\-\(b\_\{k\}\+\\omega\_\{m,k\}\)\\text\{ and policy stops before fallback\}\\bigr\\\}\.

Given𝒟m\\mathcal\{D\}\_\{m\}, the main\-stage horizon is fixed and each item contributes at most two updates; henceτb\\tau\_\{\\mathrm\{b\}\}is an\(ℋtm\)\(\\mathcal\{H\}^\{m\}\_\{t\}\)\-stopping time bounded by2​N^main,m,k2\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\. Step 1 and bounded optional stopping therefore give

𝔼0π^m,k​\[eSτb∣𝒟m\]=1,𝔼1π^m,k​\[e−Sτb∣𝒟m\]=1\.\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{0\}\[e^\{S\_\{\\tau\_\{\\mathrm\{b\}\}\}\}\\mid\\mathcal\{D\}\_\{m\}\]=1,\\qquad\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[e^\{\-S\_\{\\tau\_\{\\mathrm\{b\}\}\}\}\\mid\\mathcal\{D\}\_\{m\}\]=1\.OnE^\+\\hat\{E\}\_\{\+\}, Lemma[12\.1](https://arxiv.org/html/2609.28859#S12.Thmtheorem1)\(ii\) and the guard definition ofωm,k\\omega\_\{m,k\}implySτb≥S^τb−N^main,m,k​εmLR=S^τb−ωm,k≥akS\_\{\\tau\_\{\\mathrm\{b\}\}\}\\ \\geq\\ \\hat\{S\}\_\{\\tau\_\{\\mathrm\{b\}\}\}\-\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}\\ =\\ \\hat\{S\}\_\{\\tau\_\{\\mathrm\{b\}\}\}\-\\omega\_\{m,k\}\\ \\geq\\ a\_\{k\}, and hence

ℙ0π^m,k​\(E^\+∣𝒟m\)≤α1,k​𝔼0π^m,k​\[eSτb​𝟏E^\+∣𝒟m\]≤α1,k​𝔼0π^m,k​\[eSτb∣𝒟m\]=α1,k\.\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{0\}\(\\hat\{E\}\_\{\+\}\\mid\\mathcal\{D\}\_\{m\}\)\\leq\\alpha\_\{1,k\}\\,\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{0\}\\bigl\[e^\{S\_\{\\tau\_\{\\mathrm\{b\}\}\}\}\\mathbf\{1\}\_\{\\hat\{E\}\_\{\+\}\}\\mid\\mathcal\{D\}\_\{m\}\\bigr\]\\leq\\alpha\_\{1,k\}\\,\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{0\}\\bigl\[e^\{S\_\{\\tau\_\{\\mathrm\{b\}\}\}\}\\mid\\mathcal\{D\}\_\{m\}\\bigr\]=\\alpha\_\{1,k\}\.Similarly, onE^−\\hat\{E\}\_\{\-\},Sτb≤S^τb\+ωm,k≤−bkS\_\{\\tau\_\{\\mathrm\{b\}\}\}\\leq\\hat\{S\}\_\{\\tau\_\{\\mathrm\{b\}\}\}\+\\omega\_\{m,k\}\\leq\-b\_\{k\}, soℙ1π^m,k​\(E^−∣𝒟m\)≤β1,k\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\(\\hat\{E\}\_\{\-\}\\mid\\mathcal\{D\}\_\{m\}\)\\leq\\beta\_\{1,k\}\.

*Step 3: fallback route\.*On the complementE^fb:=\(E^\+∪E^−\)c\\hat\{E\}\_\{\\mathrm\{fb\}\}:=\(\\hat\{E\}\_\{\+\}\\cup\\hat\{E\}\_\{\-\}\)^\{c\}, the policy reveals all labels in the fixed set𝒥^H,k\\hat\{\\mathcal\{J\}\}\_\{\\mathrm\{H\},k\}and applies the exact randomized full\-label Neyman–Pearson test at levels\(α2,k,β2,k\)\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\. The set is available becauseN^main,m,k≥Nfixed,H​\(α2,k,β2,k\)\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)in both branches of the policy\. Its labels and the test randomization are independent of𝒟m\\mathcal\{D\}\_\{m\}, so the test retains these levels conditionally on𝒟m\\mathcal\{D\}\_\{m\}\. Finally, intersecting a test\-error event withE^fb\\hat\{E\}\_\{\\mathrm\{fb\}\}can only reduce its probability \(equivalently,𝟏E^fb≤1\\mathbf\{1\}\_\{\\hat\{E\}\_\{\\mathrm\{fb\}\}\}\\leq 1\)\. Thus Lemma[9\.3](https://arxiv.org/html/2609.28859#S9.Thmtheorem3)gives

ℙ0π^m,k​\(E^fb∩\{fallback rejects\}∣𝒟m\)≤α2,k,ℙ1π^m,k​\(E^fb∩\{fallback accepts\}∣𝒟m\)≤β2,k\.\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{0\}\\bigl\(\\hat\{E\}\_\{\\mathrm\{fb\}\}\\cap\\\{\\text\{fallback rejects\}\\\}\\mid\\mathcal\{D\}\_\{m\}\\bigr\)\\leq\\alpha\_\{2,k\},\\qquad\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\\bigl\(\\hat\{E\}\_\{\\mathrm\{fb\}\}\\cap\\\{\\text\{fallback accepts\}\\\}\\mid\\mathcal\{D\}\_\{m\}\\bigr\)\\leq\\beta\_\{2,k\}\.
*Step 4: route decomposition\.*The eventsE^\+,E^−,E^fb\\hat\{E\}\_\{\+\},\\hat\{E\}\_\{\-\},\\hat\{E\}\_\{\\mathrm\{fb\}\}are disjoint and exhaustive \(at the first exit timeS^τb\\hat\{S\}\_\{\\tau\_\{\\mathrm\{b\}\}\}cannot be simultaneously≥ak\+ωm,k\>0\\geq a\_\{k\}\+\\omega\_\{m,k\}\>0and≤−\(bk\+ωm,k\)<0\\leq\-\(b\_\{k\}\+\\omega\_\{m,k\}\)<0\), and the policy rejects through exactly one of two disjoint routes:

\{δπ^m,k=1\}=E^\+∪\(E^fb∩\{fallback rejects\}\),\\\{\\delta^\{\\hat\{\\pi\}\_\{m,k\}\}=1\\\}=\\hat\{E\}\_\{\+\}\\ \\cup\\ \\bigl\(\\hat\{E\}\_\{\\mathrm\{fb\}\}\\cap\\\{\\text\{fallback rejects\}\\\}\\bigr\),a disjoint union\. Steps 2 and 3 and the budget splitα1,k\+α2,k=αk\\alpha\_\{1,k\}\+\\alpha\_\{2,k\}=\\alpha\_\{k\}of \([23](https://arxiv.org/html/2609.28859#S5.E23)\) give

ℙ0π^m,k​\(δπ^m,k=1∣𝒟m\)≤α1,k\+α2,k=αk\.\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{0\}\(\\delta^\{\\hat\{\\pi\}\_\{m,k\}\}=1\\mid\\mathcal\{D\}\_\{m\}\)\\leq\\alpha\_\{1,k\}\+\\alpha\_\{2,k\}=\\alpha\_\{k\}\.Similarly we haveℙ1π^m,k​\(δπ^m,k=0∣𝒟m\)≤β1,k\+β2,k=βk\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\(\\delta^\{\\hat\{\\pi\}\_\{m,k\}\}=0\\mid\\mathcal\{D\}\_\{m\}\)\\leq\\beta\_\{1,k\}\+\\beta\_\{2,k\}=\\beta\_\{k\}\. This proves Theorem[6\.1](https://arxiv.org/html/2609.28859#S6.Thmtheorem1)\.

###### Proof 12\.4

Proof of Corollary[6\.2](https://arxiv.org/html/2609.28859#S6.Thmtheorem2)The eventℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}isσ⁡\(𝒟m\)\\sigma\(\\mathcal\{D\}\_\{m\}\)\-measurable: in \([48](https://arxiv.org/html/2609.28859#S6.E48)\),f^x\\hat\{f\}\_\{x\}is computed from𝒟m\\mathcal\{D\}\_\{m\}, whilefxf\_\{x\}andrmr\_\{m\}are deterministic\. By the tower property,

ℙ0π^m,k​\(δπ^m,k=1\)=𝔼0​\[ℙ0π^m,k​\(δπ^m,k=1∣𝒟m\)\]≤αk​ℙ0​\(ℰpilot,m\)\+ℙ0​\(ℰpilot,mc\)≤αk\+δm,\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{0\}\(\\delta^\{\\hat\{\\pi\}\_\{m,k\}\}=1\)=\\mathbb\{E\}\_\{0\}\\bigl\[\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{0\}\(\\delta^\{\\hat\{\\pi\}\_\{m,k\}\}=1\\mid\\mathcal\{D\}\_\{m\}\)\\bigr\]\\leq\\alpha\_\{k\}\\,\\mathbb\{P\}\_\{0\}\(\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}\)\+\\mathbb\{P\}\_\{0\}\(\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}^\{\\,c\}\)\\leq\\alpha\_\{k\}\+\\delta\_\{m\},where the expectation𝔼0\\mathbb\{E\}\_\{0\}is with respect to the pilot data𝒟m\\mathcal\{D\}\_\{m\}underH0H\_\{0\}, the first inequality applies Theorem[6\.1](https://arxiv.org/html/2609.28859#S6.Thmtheorem1)onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}and bounds the conditional probability by11on the complement, and the second usesℙ0​\(ℰpilot,mc\)≤δm\\mathbb\{P\}\_\{0\}\(\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}^\{\\,c\}\)\\leq\\delta\_\{m\}from Lemma[11\.1](https://arxiv.org/html/2609.28859#S11.Thmtheorem1)\. The type\-II bound follows in the same way underℙ1π^m,k\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}applied to\{δπ^m,k=0\}\\\{\\delta^\{\\hat\{\\pi\}\_\{m,k\}\}=0\\\}withβk\\beta\_\{k\}in place ofαk\\alpha\_\{k\}, proving Corollary[6\.2](https://arxiv.org/html/2609.28859#S6.Thmtheorem2)\.

## 13Additional Materials for Section[6\.2](https://arxiv.org/html/2609.28859#S6.SS2)on First\-order Cost Optimality

Conditional on𝒟m\\mathcal\{D\}\_\{m\}, let𝔼hπ^m,k\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}denote expectation underHhH\_\{h\}for the process induced by the plug\-in policyπ^m,k\\hat\{\\pi\}\_\{m,k\}and the true report channelff, and let𝔼h,f^π^m,k\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h,\\hat\{f\}\}denote the corresponding expectation whenffis replaced byf^\\hat\{f\}, with the label lawX∼Bernoulli⁡\(ph\)X\\sim\\operatorname\{Bernoulli\}\(p\_\{h\}\)unchanged\.

Throughout this section, we index the main\-stage items byi=1,…,N^main,m,ki=1,\\ldots,\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}, letℋi−1m\\mathcal\{H\}^\{m\}\_\{i\-1\}be the pilot\-augmented filtration generated by𝒟m\\mathcal\{D\}\_\{m\}and the firsti−1i\-1processed items, letJi∈\{0,1,∗\}J\_\{i\}\\in\\\{0,1,\*\\\}be the regime of itemii\(determined fromS^i−1\\hat\{S\}\_\{i\-1\}and the dead zone±zk\\pm z\_\{k\}in Algorithm[2](https://arxiv.org/html/2609.28859#alg2), henceℋi−1m\\mathcal\{H\}^\{m\}\_\{i\-1\}\-measurable\)\. WriteUi,ViU\_\{i\},V\_\{i\}for the fresh randomization seeds of itemii, and define its plug\-in\-statistic increment and sensing cost by

Z^iitem​\(η\)\\displaystyle\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\eta\):=𝟏\{Ui≤ηH\}ℓX\(Xi\)\+𝟏\{Ui\>ηH\}\(ℓ^R\(Ri\)\+𝟏\{Vi≤ηesc\(Ri\)\}ℓ^H\(Xi,Ri\)\),\\displaystyle:=\\mathbf\{1\}\\\{U\_\{i\}\\leq\\eta^\{\\mathrm\{H\}\}\\\}\\,\\ell\_\{X\}\(X\_\{i\}\)\+\\mathbf\{1\}\\\{U\_\{i\}\>\\eta^\{\\mathrm\{H\}\}\\\}\\Bigl\(\\hat\{\\ell\}\_\{R\}\(R\_\{i\}\)\+\\mathbf\{1\}\\\{V\_\{i\}\\leq\\eta^\{\\mathrm\{esc\}\}\(R\_\{i\}\)\\\}\\,\\hat\{\\ell\}\_\{H\}\(X\_\{i\},R\_\{i\}\)\\Bigr\),\(152\)Ciitem​\(η\)\\displaystyle C^\{\\mathrm\{item\}\}\_\{i\}\(\\eta\):=𝟏\{Ui≤ηH\}cH\+𝟏\{Ui\>ηH\}\(cAI\+𝟏\{Vi≤ηesc\(Ri\)\}cH\)\.\\displaystyle:=\\mathbf\{1\}\\\{U\_\{i\}\\leq\\eta^\{\\mathrm\{H\}\}\\\}\\,c\_\{\\mathrm\{H\}\}\+\\mathbf\{1\}\\\{U\_\{i\}\>\\eta^\{\\mathrm\{H\}\}\\\}\\Bigl\(c\_\{\\mathrm\{AI\}\}\+\\mathbf\{1\}\\\{V\_\{i\}\\leq\\eta^\{\\mathrm\{esc\}\}\(R\_\{i\}\)\\\}\\,c\_\{\\mathrm\{H\}\}\\Bigr\)\.Then, forj∈\{0,1,∗\}j\\in\\\{0,1,\*\\\},Z^iitem​\(η^j,m,k\)\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)is the one\-item plug\-in\-statistic increment produced by the ruleη^j,m,k:=\(η^j,m,kH,η^j,m,kesc\)\\hat\{\\eta\}\_\{j,m,k\}:=\(\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{j,m,k\},\\hat\{\\eta\}^\{\\mathrm\{esc\}\}\_\{j,m,k\}\)on a fresh item\. LetS^i=∑ℓ≤iZ^ℓitem​\(η^Jℓ,m,k\)\\hat\{S\}\_\{i\}=\\sum\_\{\\ell\\leq i\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{\\ell\}\(\\hat\{\\eta\}\_\{J\_\{\\ell\},m,k\}\)denote the cumulative plug\-in log\-likelihood statistic after completing the firstiiitems, whereJℓJ\_\{\\ell\}is the sensing rule used by itemℓ\\ell\. For the item\-level analysis below, we use a completed\-item continuation\. We generate\(Xi,Ri,Ui,Vi\)\(X\_\{i\},R\_\{i\},U\_\{i\},V\_\{i\}\)for every acquired item and recursively defineJiJ\_\{i\}fromS^i−1\\hat\{S\}\_\{i\-1\}andS^i=S^i−1\+Z^iitem​\(η^Ji,m,k\)\\hat\{S\}\_\{i\}=\\hat\{S\}\_\{i\-1\}\+\\hat\{Z\}\_\{i\}^\{\\mathrm\{item\}\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\. If the actual policy stops during or before itemii, we counterfactually complete that item and continue the recursion for the remaining items\. This continuation is used only for the proof and does not represent additional queries or costs incurred by the actual policy\.

In Appendix[13\.1](https://arxiv.org/html/2609.28859#S13.SS1), we present auxiliary Lemma[13\.1](https://arxiv.org/html/2609.28859#S13.Thmtheorem1)to prove Theorem[6\.3](https://arxiv.org/html/2609.28859#S6.Thmtheorem3)\. Then in Appendices[13\.2](https://arxiv.org/html/2609.28859#S13.SS2)–[13\.5](https://arxiv.org/html/2609.28859#S13.SS5), we present the proof of Theorem[6\.3](https://arxiv.org/html/2609.28859#S6.Thmtheorem3)and other lemmas\.

### 13\.1Uniform Pilot Perturbation

Let

f⋆:=minx∈\{0,1\},r∈ℛ⁡fx​\(r\)f\_\{\\star\}:=\\min\_\{x\\in\\\{0,1\\\},\\,r\\in\\mathcal\{R\}\}f\_\{x\}\(r\)\(153\)denote the smallest true channel mass, which is strictly positive by Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)\. RecallBℓ=max⁡\{log⁡p1p0,log⁡1−p01−p1\}B\_\{\\ell\}=\\max\\left\\\{\\log\\frac\{p\_\{1\}\}\{p\_\{0\}\},\\;\\log\\frac\{1\-p\_\{0\}\}\{1\-p\_\{1\}\}\\right\\\}\. We also letεmcost:=cH​\|ℛ\|​rm\\varepsilon\_\{m\}^\{\\mathrm\{cost\}\}:=c\_\{\\mathrm\{H\}\}\|\\mathcal\{R\}\|r\_\{m\}be an upper bound \(conditional onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}\) on the deviation between the true\-channel and plug\-in\-channel expected one\-item sensing cost\.

For a count triple\(nH,nAI,nesc\)\(n\_\{\\mathrm\{H\}\},n\_\{\\mathrm\{AI\}\},n\_\{\\mathrm\{esc\}\}\)withnH,nAI≥0n\_\{\\mathrm\{H\}\},n\_\{\\mathrm\{AI\}\}\\geq 0and0≤nesc≤nAI0\\leq n\_\{\\mathrm\{esc\}\}\\leq n\_\{\\mathrm\{AI\}\}, define its true and plug\-in information values by

Iplan\(h\)\\displaystyle I^\{\(h\)\}\_\{\\mathrm\{plan\}\}:=nH​JX\(h\)\+nAI​IR\(h\)\+nAI​Ψh​\(nesc/nAI\),\\displaystyle:=n\_\{\\mathrm\{H\}\}J\_\{X\}^\{\(h\)\}\+n\_\{\\mathrm\{AI\}\}I\_\{R\}^\{\(h\)\}\+n\_\{\\mathrm\{AI\}\}\\Psi\_\{h\}\(n\_\{\\mathrm\{esc\}\}/n\_\{\\mathrm\{AI\}\}\),\(154\)I^plan\(h\)\\displaystyle\\hat\{I\}^\{\(h\)\}\_\{\\mathrm\{plan\}\}:=nH​JX\(h\)\+nAI​I^R\(h\)\+nAI​Ψ^h​\(nesc/nAI\),\\displaystyle:=n\_\{\\mathrm\{H\}\}J\_\{X\}^\{\(h\)\}\+n\_\{\\mathrm\{AI\}\}\\hat\{I\}\_\{R\}^\{\(h\)\}\+n\_\{\\mathrm\{AI\}\}\\hat\{\\Psi\}\_\{h\}\(n\_\{\\mathrm\{esc\}\}/n\_\{\\mathrm\{AI\}\}\),whereΨ^h​\(nesc/nAI\)\\hat\{\\Psi\}\_\{h\}\(n\_\{\\mathrm\{esc\}\}/n\_\{\\mathrm\{AI\}\}\)whennAI=nesc=0n\_\{\\mathrm\{AI\}\}=n\_\{\\mathrm\{esc\}\}=0\.

###### Lemma 13\.1\(Uniform pilot perturbation\)

1. \(i\)Onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}, for everyh∈\{0,1\}h\\in\\\{0,1\\\}and every one\-item randomized sensing rule, \|𝔼hπ^m,k​\[Z^iitem​\(η\)∣𝒟m\]−𝔼h,f^π^m,k​\[Z^iitem​\(η\)∣𝒟m\]\|≤εmdr,\\displaystyle\\left\|\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m,k\}\}\[\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\eta\)\\mid\\mathcal\{D\}\_\{m\}\]\-\\mathbb\{E\}\_\{h,\\hat\{f\}\}^\{\\hat\{\\pi\}\_\{m,k\}\}\[\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\eta\)\\mid\\mathcal\{D\}\_\{m\}\]\\right\|\\leq\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\},\(155\)\|𝔼hπ^m,k​\[Ciitem​\(η\)∣𝒟m\]−𝔼h,f^π^m,k​\[Ciitem​\(η\)∣𝒟m\]\|≤εmcost\.\\displaystyle\\left\|\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m,k\}\}\[C^\{\\mathrm\{item\}\}\_\{i\}\(\\eta\)\\mid\\mathcal\{D\}\_\{m\}\]\-\\mathbb\{E\}\_\{h,\\hat\{f\}\}^\{\\hat\{\\pi\}\_\{m,k\}\}\[C^\{\\mathrm\{item\}\}\_\{i\}\(\\eta\)\\mid\\mathcal\{D\}\_\{m\}\]\\right\|\\leq\\varepsilon\_\{m\}^\{\\mathrm\{cost\}\}\.
2. \(ii\)There is a primitive constantLloc<∞L\_\{\\mathrm\{loc\}\}<\\inftysuch that, onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}and wheneverrm≤f⋆/4r\_\{m\}\\leq f\_\{\\star\}/4,q^h\\hat\{q\}\_\{h\},1−q^h1\-\\hat\{q\}\_\{h\}andg^h\\hat\{g\}\_\{h\}are lower bounded by a constant that only depends on the primitives, and moreover, εmLR\\displaystyle\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}\+maxh\|I^R\(h\)−IR\(h\)\|\+maxh,r⁡\|d^\(h\)​\(r\)−d\(h\)​\(r\)\|\+max⁡sups∈\[0,1\]h⁡\|Ψ^h​\(s\)−Ψh​\(s\)\|≤Lloc​rm\.\\displaystyle\+\\max\_\{h\}\|\\hat\{I\}\_\{R\}^\{\(h\)\}\-I\_\{R\}^\{\(h\)\}\|\+\\max\_\{h,r\}\|\\hat\{d\}^\{\(h\)\}\(r\)\-d^\{\(h\)\}\(r\)\|\+\\max\_\{h\}\\sup\_\{s\\in\[0,1\]\}\|\\hat\{\\Psi\}\_\{h\}\(s\)\-\\Psi\_\{h\}\(s\)\|\\leq L\_\{\\mathrm\{loc\}\}r\_\{m\}\.\(156\)
3. \(iii\)Onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}and wheneverrm≤f⋆/4r\_\{m\}\\leq f\_\{\\star\}/4, for everyN≥1N\\geq 1and every count triple\(nH,nAI,nesc\)\(n\_\{\\mathrm\{H\}\},n\_\{\\mathrm\{AI\}\},n\_\{\\mathrm\{esc\}\}\)as in \([154](https://arxiv.org/html/2609.28859#S13.E154)\) withnH\+nAI≤Nn\_\{\\mathrm\{H\}\}\+n\_\{\\mathrm\{AI\}\}\\leq N, maxh∈\{0,1\}⁡\|Iplan\(h\)−I^plan\(h\)\|≤Lloc​N​rm\.\\max\_\{h\\in\\\{0,1\\\}\}\\bigl\|I^\{\(h\)\}\_\{\\mathrm\{plan\}\}\-\\hat\{I\}^\{\(h\)\}\_\{\\mathrm\{plan\}\}\\bigr\|\\leq L\_\{\\mathrm\{loc\}\}Nr\_\{m\}\.\(157\)

###### Proof 13\.2

Proof of Lemma[13\.1](https://arxiv.org/html/2609.28859#S13.Thmtheorem1)Throughout, recall thatfx​\(r\)\>0f\_\{x\}\(r\)\>0for allx,rx,r\(Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1)\), and thatf^x​\(r\)\>0\\hat\{f\}\_\{x\}\(r\)\>0and∑rf^x​\(r\)=1\\sum\_\{r\}\\hat\{f\}\_\{x\}\(r\)=1by the smoothing in \([47](https://arxiv.org/html/2609.28859#S6.E47)\)\. Hence all objects below are well\-defined for both the true channelf=\(f0,f1\)f=\(f\_\{0\},f\_\{1\}\)and the plug\-in channelf^=\(f^0,f^1\)\\hat\{f\}=\(\\hat\{f\}\_\{0\},\\hat\{f\}\_\{1\}\)\.

*Part \(i\)\.*Fixh∈\{0,1\}h\\in\\\{0,1\\\}, fix a one\-item randomized sensing rule\(ηH,ηesc\)\(\\eta^\{\\mathrm\{H\}\},\\eta^\{\\mathrm\{esc\}\}\), and work onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}; the bounds obtained below are uniform inhhand in the rule\. WritePh​\(x,r\):=phx​\(1−ph\)1−x​fx​\(r\)P\_\{h\}\(x,r\):=p\_\{h\}^\{x\}\(1\-p\_\{h\}\)^\{1\-x\}f\_\{x\}\(r\)andP^h​\(x,r\):=phx​\(1−ph\)1−x​f^x​\(r\)\\hat\{P\}\_\{h\}\(x,r\):=p\_\{h\}^\{x\}\(1\-p\_\{h\}\)^\{1\-x\}\\hat\{f\}\_\{x\}\(r\)for the joint mass functions of\(X,R\)\(X,R\)under the true and plug\-in report channels, respectively\. For everyrr,

\|g^h​\(r\)−gh​\(r\)\|≤ph​\|f^1​\(r\)−f1​\(r\)\|\+\(1−ph\)​\|f^0​\(r\)−f0​\(r\)\|≤rm,\|\\hat\{g\}\_\{h\}\(r\)\-g\_\{h\}\(r\)\|\\leq p\_\{h\}\|\\hat\{f\}\_\{1\}\(r\)\-f\_\{1\}\(r\)\|\+\(1\-p\_\{h\}\)\|\\hat\{f\}\_\{0\}\(r\)\-f\_\{0\}\(r\)\|\\leq r\_\{m\},so‖gh−g^h‖1≤\|ℛ\|​rm\\\|g\_\{h\}\-\\hat\{g\}\_\{h\}\\\|\_\{1\}\\leq\|\\mathcal\{R\}\|r\_\{m\}\. The same computation for the joint lawsPhP\_\{h\}andP^h\\hat\{P\}\_\{h\}gives

∑x,r\|Ph−P^h\|\(x,r\)=∑r\[ph​\|f^1−f1\|​\(r\)\+\(1−ph\)​\|f^0−f0\|​\(r\)\]≤\|ℛ\|​rm\.\\sum\_\{x,r\}\|P\_\{h\}\-\\hat\{P\}\_\{h\}\|\(x,r\)=\\sum\_\{r\}\\bigl\[p\_\{h\}\|\\hat\{f\}\_\{1\}\-f\_\{1\}\|\(r\)\+\(1\-p\_\{h\}\)\|\\hat\{f\}\_\{0\}\-f\_\{0\}\|\(r\)\\bigr\]\\leq\|\\mathcal\{R\}\|r\_\{m\}\.
*Drift\.*WriteF⁡\(x,r\):=ℓ^R​\(r\)\+ηesc​\(r\)​ℓ^H​\(x,r\)F\(x,r\):=\\hat\{\\ell\}\_\{R\}\(r\)\+\\eta^\{\\mathrm\{esc\}\}\(r\)\\hat\{\\ell\}\_\{H\}\(x,r\)\. The direct\-human contributions to the two expectations are identical, so

𝔼hπ^m,k​\[Z^iitem​\(η\)∣𝒟m\]−𝔼h,f^π^m,k​\[Z^iitem​\(η\)∣𝒟m\]=\(1−ηH\)​∑x,r\(Ph−P^h\)​\(x,r\)​F​\(x,r\)\.\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m,k\}\}\[\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\eta\)\\mid\\mathcal\{D\}\_\{m\}\]\-\\mathbb\{E\}\_\{h,\\hat\{f\}\}^\{\\hat\{\\pi\}\_\{m,k\}\}\[\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\eta\)\\mid\\mathcal\{D\}\_\{m\}\]=\(1\-\\eta^\{\\mathrm\{H\}\}\)\\sum\_\{x,r\}\\bigl\(P\_\{h\}\-\\hat\{P\}\_\{h\}\\bigr\)\(x,r\)\\,F\(x,r\)\.Lemma[12\.1](https://arxiv.org/html/2609.28859#S12.Thmtheorem1)\(i\) gives\|F\|≤Bℓ\+2​Bℓ=3​Bℓ\|F\|\\leq B\_\{\\ell\}\+2B\_\{\\ell\}=3B\_\{\\ell\}, so

\|𝔼hπ^m,k\[Z^iitem\(η\)∣𝒟m\]−𝔼h,f^π^m,k\[Z^iitem\(η\)∣𝒟m\]\|≤\(1−ηH\)⋅3Bℓ∑x,r\|Ph−P^h\|\(x,r\)≤3\|ℛ\|Bℓrm=εmdr\.\\left\|\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m,k\}\}\[\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\eta\)\\mid\\mathcal\{D\}\_\{m\}\]\-\\mathbb\{E\}\_\{h,\\hat\{f\}\}^\{\\hat\{\\pi\}\_\{m,k\}\}\[\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\eta\)\\mid\\mathcal\{D\}\_\{m\}\]\\right\|\\leq\(1\-\\eta^\{\\mathrm\{H\}\}\)\\cdot 3B\_\{\\ell\}\\,\\sum\_\{x,r\}\|P\_\{h\}\-\\hat\{P\}\_\{h\}\|\(x,r\)\\leq 3\|\\mathcal\{R\}\|B\_\{\\ell\}r\_\{m\}=\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}\.
*Cost\.*The termsηH​cH\+\(1−ηH\)​cAI\\eta^\{\\mathrm\{H\}\}c\_\{\\mathrm\{H\}\}\+\(1\-\\eta^\{\\mathrm\{H\}\}\)c\_\{\\mathrm\{AI\}\}are common to the two expected costs, so they cancel in the difference, leaving

\|𝔼hπ^m,k​\[Ciitem​\(η\)∣𝒟m\]−𝔼h,f^π^m,k​\[Ciitem​\(η\)∣𝒟m\]\|\\displaystyle\\left\|\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m,k\}\}\[C^\{\\mathrm\{item\}\}\_\{i\}\(\\eta\)\\mid\\mathcal\{D\}\_\{m\}\]\-\\mathbb\{E\}\_\{h,\\hat\{f\}\}^\{\\hat\{\\pi\}\_\{m,k\}\}\[C^\{\\mathrm\{item\}\}\_\{i\}\(\\eta\)\\mid\\mathcal\{D\}\_\{m\}\]\\right\|=\(1−ηH\)​cH​\|∑r\(gh​\(r\)−g^h​\(r\)\)​ηesc​\(r\)\|≤cH​‖gh−g^h‖1≤cH​\|ℛ\|​rm=εmcost\.\\displaystyle\\quad=\(1\-\\eta^\{\\mathrm\{H\}\}\)\\,c\_\{\\mathrm\{H\}\}\\left\|\\sum\_\{r\}\\bigl\(g\_\{h\}\(r\)\-\\hat\{g\}\_\{h\}\(r\)\\bigr\)\\eta^\{\\mathrm\{esc\}\}\(r\)\\right\|\\leq c\_\{\\mathrm\{H\}\}\\\|g\_\{h\}\-\\hat\{g\}\_\{h\}\\\|\_\{1\}\\leq c\_\{\\mathrm\{H\}\}\|\\mathcal\{R\}\|r\_\{m\}=\\varepsilon\_\{m\}^\{\\mathrm\{cost\}\}\.This and the drift bound prove \([155](https://arxiv.org/html/2609.28859#S13.E155)\)\.

*Part \(ii\)\.*Work onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}and assumerm≤f⋆/4r\_\{m\}\\leq f\_\{\\star\}/4\. For allx,rx,r,f^x​\(r\)≥fx​\(r\)−rm≥f⋆−f⋆/4=3​f⋆/4\\hat\{f\}\_\{x\}\(r\)\\geq f\_\{x\}\(r\)\-r\_\{m\}\\geq f\_\{\\star\}\-f\_\{\\star\}/4=3f\_\{\\star\}/4andf¯x,m​\(r\)≥f^x​\(r\)−rm≥f⋆/2\\underline\{f\}\_\{x,m\}\(r\)\\geq\\hat\{f\}\_\{x\}\(r\)\-r\_\{m\}\\geq f\_\{\\star\}/2, whilef^x​\(r\)≤f¯x,m​\(r\)≤1\\hat\{f\}\_\{x\}\(r\)\\leq\\overline\{f\}\_\{x,m\}\(r\)\\leq 1\. Thus all offx​\(r\),f^x​\(r\),f¯x,m​\(r\),f¯x,m​\(r\)f\_\{x\}\(r\),\\hat\{f\}\_\{x\}\(r\),\\underline\{f\}\_\{x,m\}\(r\),\\overline\{f\}\_\{x,m\}\(r\)lie in\[f⋆/2,1\]\[f\_\{\\star\}/2,1\]\.

*\(a\) Show thatmaxh,r⁡\|d^\(h\)​\(r\)−d\(h\)​\(r\)\|\+maxh⁡\|I^R\(h\)−IR\(h\)\|≤C1​rm=O⁡\(rm\)\\max\_\{h,r\}\|\\hat\{d\}^\{\(h\)\}\(r\)\-d^\{\(h\)\}\(r\)\|\+\\max\_\{h\}\|\\hat\{I\}\_\{R\}^\{\(h\)\}\-I\_\{R\}^\{\(h\)\}\|\\leq C\_\{1\}r\_\{m\}=O\(r\_\{m\}\)\.*Sincegh​\(r\)g\_\{h\}\(r\)andg^h​\(r\)\\hat\{g\}\_\{h\}\(r\)are convex combinations off0​\(r\),f1​\(r\)f\_\{0\}\(r\),f\_\{1\}\(r\)and off^0​\(r\),f^1​\(r\)\\hat\{f\}\_\{0\}\(r\),\\hat\{f\}\_\{1\}\(r\)respectively,

1≥gh​\(r\)≥min⁡\{f0​\(r\),f1​\(r\)\}≥f⋆,1≥g^h​\(r\)≥min⁡\{f^0​\(r\),f^1​\(r\)\}≥3​f⋆4,1\\geq g\_\{h\}\(r\)\\geq\\min\\\{f\_\{0\}\(r\),f\_\{1\}\(r\)\\\}\\geq f\_\{\\star\},\\qquad 1\\geq\\hat\{g\}\_\{h\}\(r\)\\geq\\min\\\{\\hat\{f\}\_\{0\}\(r\),\\hat\{f\}\_\{1\}\(r\)\\\}\\geq\\frac\{3f\_\{\\star\}\}\{4\},and the posteriors satisfyqh​\(r\)=ph​f1​\(r\)/gh​\(r\)≥p0​f⋆q\_\{h\}\(r\)=p\_\{h\}f\_\{1\}\(r\)/g\_\{h\}\(r\)\\geq p\_\{0\}f\_\{\\star\},1−qh​\(r\)=\(1−ph\)​f0​\(r\)/gh​\(r\)≥\(1−p1\)​f⋆1\-q\_\{h\}\(r\)=\(1\-p\_\{h\}\)f\_\{0\}\(r\)/g\_\{h\}\(r\)\\geq\(1\-p\_\{1\}\)f\_\{\\star\}, and likewiseq^h​\(r\)≥p0⋅3​f⋆/4\\hat\{q\}\_\{h\}\(r\)\\geq p\_\{0\}\\cdot 3f\_\{\\star\}/4and1−q^h​\(r\)≥\(1−p1\)⋅3​f⋆/41\-\\hat\{q\}\_\{h\}\(r\)\\geq\(1\-p\_\{1\}\)\\cdot 3f\_\{\\star\}/4\. Hence all ofqh​\(r\),q^h​\(r\)q\_\{h\}\(r\),\\hat\{q\}\_\{h\}\(r\)lie in\[cpost,1−cpost\]\[c\_\{\\mathrm\{post\}\},1\-c\_\{\\mathrm\{post\}\}\], wherecpost:=12​min⁡\{p0,1−p1\}​f⋆c\_\{\\mathrm\{post\}\}:=\\tfrac\{1\}\{2\}\\min\\\{p\_\{0\},1\-p\_\{1\}\\\}f\_\{\\star\}\. Next, writing

f^1​\(r\)g^h​\(r\)−f1​\(r\)gh​\(r\)=\(f^1​\(r\)−f1​\(r\)\)​gh​\(r\)\+f1​\(r\)​\(gh​\(r\)−g^h​\(r\)\)gh​\(r\)​g^h​\(r\)\\frac\{\\hat\{f\}\_\{1\}\(r\)\}\{\\hat\{g\}\_\{h\}\(r\)\}\-\\frac\{f\_\{1\}\(r\)\}\{g\_\{h\}\(r\)\}=\\frac\{\(\\hat\{f\}\_\{1\}\(r\)\-f\_\{1\}\(r\)\)\\,g\_\{h\}\(r\)\+f\_\{1\}\(r\)\\,\(g\_\{h\}\(r\)\-\\hat\{g\}\_\{h\}\(r\)\)\}\{g\_\{h\}\(r\)\\hat\{g\}\_\{h\}\(r\)\}and using\|f^x​\(r\)−fx​\(r\)\|≤rm\|\\hat\{f\}\_\{x\}\(r\)\-f\_\{x\}\(r\)\|\\leq r\_\{m\}together with\|g^h​\(r\)−gh​\(r\)\|≤rm\|\\hat\{g\}\_\{h\}\(r\)\-g\_\{h\}\(r\)\|\\leq r\_\{m\}\(proof of part \(i\)\),

\|q^h​\(r\)−qh​\(r\)\|≤ph​rm⋅1\+1⋅rmf⋆⋅\(3​f⋆/4\)=8​ph​rm3​f⋆2≤8​rm3​f⋆2\.\|\\hat\{q\}\_\{h\}\(r\)\-q\_\{h\}\(r\)\|\\leq p\_\{h\}\\,\\frac\{r\_\{m\}\\cdot 1\+1\\cdot r\_\{m\}\}\{f\_\{\\star\}\\cdot\(3f\_\{\\star\}/4\)\}=\\frac\{8\\,p\_\{h\}\\,r\_\{m\}\}\{3f\_\{\\star\}^\{2\}\}\\leq\\frac\{8r\_\{m\}\}\{3f\_\{\\star\}^\{2\}\}\.The mapkl\(⋅∥⋅\)\\operatorname\{kl\}\(\\cdot\\\|\\cdot\)is continuously differentiable, hence Lipschitz, on the compact square\[cpost,1−cpost\]2\[c\_\{\\mathrm\{post\}\},1\-c\_\{\\mathrm\{post\}\}\]^\{2\}, with constant depending only oncpostc\_\{\\mathrm\{post\}\}; since both\(qh​\(r\),q1−h​\(r\)\)\(q\_\{h\}\(r\),q\_\{1\-h\}\(r\)\)and\(q^h​\(r\),q^1−h​\(r\)\)\(\\hat\{q\}\_\{h\}\(r\),\\hat\{q\}\_\{1\-h\}\(r\)\)lie in this square,

\|d^\(h\)​\(r\)−d\(h\)​\(r\)\|≤C⁡\(\|q^h​\(r\)−qh​\(r\)\|\+\|q^1−h​\(r\)−q1−h​\(r\)\|\)=O⁡\(rm\)\.\|\\hat\{d\}^\{\(h\)\}\(r\)\-d^\{\(h\)\}\(r\)\|\\leq C\\bigl\(\|\\hat\{q\}\_\{h\}\(r\)\-q\_\{h\}\(r\)\|\+\|\\hat\{q\}\_\{1\-h\}\(r\)\-q\_\{1\-h\}\(r\)\|\\bigr\)=O\(r\_\{m\}\)\.Similarly,log\\logis Lipschitz on\[3​f⋆/4,1\]\[3f\_\{\\star\}/4,1\]with constant4/\(3​f⋆\)4/\(3f\_\{\\star\}\), and\|log⁡\(g^h​\(r\)/g^1−h​\(r\)\)\|≤log⁡\(4/\(3​f⋆\)\)\|\\log\(\\hat\{g\}\_\{h\}\(r\)/\\hat\{g\}\_\{1\-h\}\(r\)\)\|\\leq\\log\(4/\(3f\_\{\\star\}\)\), so

\|I^R\(h\)−IR\(h\)\|\\displaystyle\|\\hat\{I\}\_\{R\}^\{\(h\)\}\-I\_\{R\}^\{\(h\)\}\|≤∑r\|g^h​\(r\)−gh​\(r\)​\|\|log⁡g^h​\(r\)g^1−h​\(r\)\|\+∑rgh​\(r\)\|​log⁡g^h​\(r\)−log⁡gh​\(r\)\|\\displaystyle\\leq\\sum\_\{r\}\|\\hat\{g\}\_\{h\}\(r\)\-g\_\{h\}\(r\)\|\\,\\left\|\\log\\frac\{\\hat\{g\}\_\{h\}\(r\)\}\{\\hat\{g\}\_\{1\-h\}\(r\)\}\\right\|\+\\sum\_\{r\}g\_\{h\}\(r\)\\,\\bigl\|\\log\\hat\{g\}\_\{h\}\(r\)\-\\log g\_\{h\}\(r\)\\bigr\|\+∑rgh\(r\)\|logg^1−h\(r\)−logg1−h\(r\)\|≤\|ℛ\|rmlog43​f⋆\+8​rm3​f⋆\.\\displaystyle\\qquad\+\\sum\_\{r\}g\_\{h\}\(r\)\\,\\bigl\|\\log\\hat\{g\}\_\{1\-h\}\(r\)\-\\log g\_\{1\-h\}\(r\)\\bigr\|\\;\\leq\\;\|\\mathcal\{R\}\|r\_\{m\}\\log\\frac\{4\}\{3f\_\{\\star\}\}\+\\frac\{8r\_\{m\}\}\{3f\_\{\\star\}\}\.Collecting terms yields a primitive constantC1C\_\{1\}, depending only onp0,p1,f⋆,\|ℛ\|p\_\{0\},p\_\{1\},f\_\{\\star\},\|\\mathcal\{R\}\|, with

maxh,r⁡\|d^\(h\)​\(r\)−d\(h\)​\(r\)\|\+maxh⁡\|I^R\(h\)−IR\(h\)\|≤C1​rm\.\\max\_\{h,r\}\|\\hat\{d\}^\{\(h\)\}\(r\)\-d^\{\(h\)\}\(r\)\|\+\\max\_\{h\}\|\\hat\{I\}\_\{R\}^\{\(h\)\}\-I\_\{R\}^\{\(h\)\}\|\\leq C\_\{1\}r\_\{m\}\.
*\(b\) Show thatεmLR=O⁡\(rm\)\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}=O\(r\_\{m\}\)\.*Fixr∈ℛr\\in\\mathcal\{R\}\. Recall by construction \([148](https://arxiv.org/html/2609.28859#S12.E148)\) ofϕrep\\phi\_\{\\mathrm\{rep\}\},ℓ^R​\(r\)=ϕrep​\(w^​\(r\)\)\\hat\{\\ell\}\_\{R\}\(r\)=\\phi\_\{\\mathrm\{rep\}\}\(\\hat\{w\}\(r\)\)withw^​\(r\)=f^1​\(r\)f^0​\(r\)\+f^1​\(r\)∈\[w¯m​\(r\),w¯m​\(r\)\]\\hat\{w\}\(r\)=\\frac\{\\hat\{f\}\_\{1\}\(r\)\}\{\\hat\{f\}\_\{0\}\(r\)\+\\hat\{f\}\_\{1\}\(r\)\}\\in\[\\underline\{w\}\_\{m\}\(r\),\\overline\{w\}\_\{m\}\(r\)\]\.ϕrep\\phi\_\{\\mathrm\{rep\}\}is increasing, so

ϕrep​\(w¯m​\(r\)\)≤ℓ^R​\(r\)≤ϕrep​\(w¯m​\(r\)\)\.\\phi\_\{\\mathrm\{rep\}\}\(\\underline\{w\}\_\{m\}\(r\)\)\\leq\\hat\{\\ell\}\_\{R\}\(r\)\\leq\\phi\_\{\\mathrm\{rep\}\}\(\\overline\{w\}\_\{m\}\(r\)\)\.Consequently both terms inside the inner maximum in the definition \([149](https://arxiv.org/html/2609.28859#S12.E149)\) ofεmLR\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}are at mostϕrep​\(w¯m​\(r\)\)−ϕrep​\(w¯m​\(r\)\)\\phi\_\{\\mathrm\{rep\}\}\(\\overline\{w\}\_\{m\}\(r\)\)\-\\phi\_\{\\mathrm\{rep\}\}\(\\underline\{w\}\_\{m\}\(r\)\), whence

εmLR≤maxr∈ℛ⁡\{ϕrep​\(w¯m​\(r\)\)−ϕrep​\(w¯m​\(r\)\)\}\.\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}\\leq\\max\_\{r\\in\\mathcal\{R\}\}\\bigl\\\{\\phi\_\{\\mathrm\{rep\}\}\(\\overline\{w\}\_\{m\}\(r\)\)\-\\phi\_\{\\mathrm\{rep\}\}\(\\underline\{w\}\_\{m\}\(r\)\)\\bigr\\\}\.We bound the right\-hand side in two steps: a uniform bound onϕrep′\\phi\_\{\\mathrm\{rep\}\}^\{\\prime\}, and a bound on the widthw¯m​\(r\)−w¯m​\(r\)\\overline\{w\}\_\{m\}\(r\)\-\\underline\{w\}\_\{m\}\(r\)\.

First, we have that

supw∈\[0,1\]ϕrep′​\(w\)≤Cϕ:=p1−p0min⁡\{p0,1−p0\}​min​\{p1,1−p1\},\\sup\_\{w\\in\[0,1\]\}\\phi\_\{\\mathrm\{rep\}\}^\{\\prime\}\(w\)\\leq C\_\{\\phi\}:=\\frac\{p\_\{1\}\-p\_\{0\}\}\{\\min\\\{p\_\{0\},1\-p\_\{0\}\\\}\\,\\min\\\{p\_\{1\},1\-p\_\{1\}\\\}\},and the mean\-value theorem givesϕrep​\(w¯m​\(r\)\)−ϕrep​\(w¯m​\(r\)\)≤Cϕ​\(w¯m​\(r\)−w¯m​\(r\)\)\\phi\_\{\\mathrm\{rep\}\}\(\\overline\{w\}\_\{m\}\(r\)\)\-\\phi\_\{\\mathrm\{rep\}\}\(\\underline\{w\}\_\{m\}\(r\)\)\\leq C\_\{\\phi\}\\bigl\(\\overline\{w\}\_\{m\}\(r\)\-\\underline\{w\}\_\{m\}\(r\)\\bigr\)\.

Second, writeW⁡\(a,b\):=a/\(a\+b\)W\(a,b\):=a/\(a\+b\), so that, by \([147](https://arxiv.org/html/2609.28859#S12.E147)\),

w¯m​\(r\)=W⁡\(f¯1,m​\(r\),f¯0,m​\(r\)\),w¯m​\(r\)=W⁡\(f¯1,m​\(r\),f¯0,m​\(r\)\),w^​\(r\)=W⁡\(f^1​\(r\),f^0​\(r\)\),\\underline\{w\}\_\{m\}\(r\)=W\(\\underline\{f\}\_\{1,m\}\(r\),\\overline\{f\}\_\{0,m\}\(r\)\),\\qquad\\overline\{w\}\_\{m\}\(r\)=W\(\\overline\{f\}\_\{1,m\}\(r\),\\underline\{f\}\_\{0,m\}\(r\)\),\\qquad\\hat\{w\}\(r\)=W\(\\hat\{f\}\_\{1\}\(r\),\\hat\{f\}\_\{0\}\(r\)\),where all three argument pairs lie in the box\[f⋆/2,1\]2\[f\_\{\\star\}/2,1\]^\{2\}, on which the partial derivatives

∂W∂a=b\(a\+b\)2,∂W∂b=−a\(a\+b\)2\\frac\{\\partial W\}\{\\partial a\}=\\frac\{b\}\{\(a\+b\)^\{2\}\},\\qquad\\frac\{\\partial W\}\{\\partial b\}=\-\\frac\{a\}\{\(a\+b\)^\{2\}\}are bounded in absolute value by1/\(a\+b\)≤1/f⋆1/\(a\+b\)\\leq 1/f\_\{\\star\}\. Moreover,rm≤f⋆/4<f^x​\(r\)r\_\{m\}\\leq f\_\{\\star\}/4<\\hat\{f\}\_\{x\}\(r\)makes the positive\-part clip inf¯x,m​\(r\)=\[f^x​\(r\)−rm\]\+\\underline\{f\}\_\{x,m\}\(r\)=\[\\hat\{f\}\_\{x\}\(r\)\-r\_\{m\}\]\_\{\+\}inactive, so

\|f^1​\(r\)−f¯1,m​\(r\)\|=rm,\|f¯0,m​\(r\)−f^0​\(r\)\|≤rm,\|\\hat\{f\}\_\{1\}\(r\)\-\\underline\{f\}\_\{1,m\}\(r\)\|=r\_\{m\},\\qquad\|\\overline\{f\}\_\{0,m\}\(r\)\-\\hat\{f\}\_\{0\}\(r\)\|\\leq r\_\{m\},and likewise\|f¯1,m​\(r\)−f^1​\(r\)\|≤rm\|\\overline\{f\}\_\{1,m\}\(r\)\-\\hat\{f\}\_\{1\}\(r\)\|\\leq r\_\{m\}and\|f^0​\(r\)−f¯0,m​\(r\)\|=rm\|\\hat\{f\}\_\{0\}\(r\)\-\\underline\{f\}\_\{0,m\}\(r\)\|=r\_\{m\}\. The box is convex, so the segments joining\(f^1​\(r\),f^0​\(r\)\)\(\\hat\{f\}\_\{1\}\(r\),\\hat\{f\}\_\{0\}\(r\)\)to\(f¯1,m​\(r\),f¯0,m​\(r\)\)\(\\underline\{f\}\_\{1,m\}\(r\),\\overline\{f\}\_\{0,m\}\(r\)\)and to\(f¯1,m​\(r\),f¯0,m​\(r\)\)\(\\overline\{f\}\_\{1,m\}\(r\),\\underline\{f\}\_\{0,m\}\(r\)\)stay inside it, and the mean\-value theorem yields

w^​\(r\)−w¯m​\(r\)≤\|f^1​\(r\)−f¯1,m​\(r\)\|\+\|f¯0,m​\(r\)−f^0​\(r\)\|f⋆≤2​rmf⋆,w¯m​\(r\)−w^​\(r\)≤2​rmf⋆,\\hat\{w\}\(r\)\-\\underline\{w\}\_\{m\}\(r\)\\leq\\frac\{\|\\hat\{f\}\_\{1\}\(r\)\-\\underline\{f\}\_\{1,m\}\(r\)\|\+\|\\overline\{f\}\_\{0,m\}\(r\)\-\\hat\{f\}\_\{0\}\(r\)\|\}\{f\_\{\\star\}\}\\leq\\frac\{2r\_\{m\}\}\{f\_\{\\star\}\},\\qquad\\overline\{w\}\_\{m\}\(r\)\-\\hat\{w\}\(r\)\\leq\\frac\{2r\_\{m\}\}\{f\_\{\\star\}\},sow¯m​\(r\)−w¯m​\(r\)≤4​rm/f⋆\\overline\{w\}\_\{m\}\(r\)\-\\underline\{w\}\_\{m\}\(r\)\\leq 4r\_\{m\}/f\_\{\\star\}\. Combining the two steps,

εmLR≤Cϕ⋅4​rmf⋆=4​Cϕ​rmf⋆\.\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}\\leq C\_\{\\phi\}\\cdot\\frac\{4r\_\{m\}\}\{f\_\{\\star\}\}=\\frac\{4C\_\{\\phi\}r\_\{m\}\}\{f\_\{\\star\}\}\.
*\(c\) Show thatmax⁡sups∈\[0,1\]h⁡\|Ψ^h​\(s\)−Ψh​\(s\)\|=O⁡\(rm\)\\max\_\{h\}\\sup\_\{s\\in\[0,1\]\}\|\\hat\{\\Psi\}\_\{h\}\(s\)\-\\Psi\_\{h\}\(s\)\|=O\(r\_\{m\}\)\.*The true frontierΨh\\Psi\_\{h\}is defined by the inf\-representation \([15](https://arxiv.org/html/2609.28859#S4.E15)\), and the dual\-representation part of the proof of Lemma[4\.7](https://arxiv.org/html/2609.28859#S4.Thmtheorem7)\(a finite linear programming duality argument\) uses only that the report marginal is a strictly positive mass function and that the follow\-up gains are nonnegative and finite, whichg^h\>0\\hat\{g\}\_\{h\}\>0and0≤d^\(h\)<∞0\\leq\\hat\{d\}^\{\(h\)\}<\\inftysatisfy\. Hence the plug\-in frontier \([50](https://arxiv.org/html/2609.28859#S6.E50)\) admits the same representation\. Writing

Φh​\(λ,s\):=λ​s\+∑r∈ℛgh​\(r\)​\[d\(h\)​\(r\)−λ\]\+,Φ^h​\(λ,s\):=λ​s\+∑r∈ℛg^h​\(r\)​\[d^\(h\)​\(r\)−λ\]\+,\\Phi\_\{h\}\(\\lambda,s\):=\\lambda s\+\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{h\}\(r\)\\left\[d^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\},\\qquad\\hat\{\\Phi\}\_\{h\}\(\\lambda,s\):=\\lambda s\+\\sum\_\{r\\in\\mathcal\{R\}\}\\hat\{g\}\_\{h\}\(r\)\\left\[\\hat\{d\}^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\},we therefore haveΨh​\(s\)=infλ≥0Φh​\(λ,s\)\\Psi\_\{h\}\(s\)=\\inf\_\{\\lambda\\geq 0\}\\Phi\_\{h\}\(\\lambda,s\)andΨ^h​\(s\)=infλ≥0Φ^h​\(λ,s\)\\hat\{\\Psi\}\_\{h\}\(s\)=\\inf\_\{\\lambda\\geq 0\}\\hat\{\\Phi\}\_\{h\}\(\\lambda,s\)\.

First we restrict both infima to a common compact interval\. Setdmax:=maxh,r⁡max⁡\{d\(h\)​\(r\),d^\(h\)​\(r\)\}d\_\{\\max\}:=\\max\_\{h,r\}\\max\\\{d^\{\(h\)\}\(r\),\\hat\{d\}^\{\(h\)\}\(r\)\\\}; since by step \(a\) all posteriorsqh​\(r\),q^h​\(r\)q\_\{h\}\(r\),\\hat\{q\}\_\{h\}\(r\)lie in\[cpost,1−cpost\]\[c\_\{\\mathrm\{post\}\},1\-c\_\{\\mathrm\{post\}\}\], andkl\(⋅∥⋅\)\\operatorname\{kl\}\(\\cdot\\\|\\cdot\)is bounded on the compact square\[cpost,1−cpost\]2\[c\_\{\\mathrm\{post\}\},1\-c\_\{\\mathrm\{post\}\}\]^\{2\}by a constant depending only oncpostc\_\{\\mathrm\{post\}\}, we havedmax≤C2d\_\{\\max\}\\leq C\_\{2\}for a primitive constantC2C\_\{2\}on the present event\. Forλ≥dmax\\lambda\\geq d\_\{\\max\}every positive part vanishes, soΦh​\(λ,s\)=λ​s≥dmax​s=Φh​\(dmax,s\)\\Phi\_\{h\}\(\\lambda,s\)=\\lambda s\\geq d\_\{\\max\}s=\\Phi\_\{h\}\(d\_\{\\max\},s\), and likewise forΦ^h\\hat\{\\Phi\}\_\{h\}; the infima overλ≥dmax\\lambda\\geq d\_\{\\max\}are thus attained atλ=dmax\\lambda=d\_\{\\max\}, and

Ψh​\(s\)=inf0≤λ≤dmaxΦh​\(λ,s\),Ψ^h​\(s\)=inf0≤λ≤dmaxΦ^h​\(λ,s\)\.\\Psi\_\{h\}\(s\)=\\inf\_\{0\\leq\\lambda\\leq d\_\{\\max\}\}\\Phi\_\{h\}\(\\lambda,s\),\\qquad\\hat\{\\Psi\}\_\{h\}\(s\)=\\inf\_\{0\\leq\\lambda\\leq d\_\{\\max\}\}\\hat\{\\Phi\}\_\{h\}\(\\lambda,s\)\.
Next we show that the two objectives are uniformly close on this common domain\. Fixλ∈\[0,dmax\]\\lambda\\in\[0,d\_\{\\max\}\]ands∈\[0,1\]s\\in\[0,1\]\. Theλ​s\\lambda sterms coincide, and adding and subtracting∑rg^h​\(r\)​\[d\(h\)​\(r\)−λ\]\+\\sum\_\{r\}\\hat\{g\}\_\{h\}\(r\)\\left\[d^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}splits the remainder into two sums:

Φh​\(λ,s\)−Φ^h​\(λ,s\)=∑r\(gh​\(r\)−g^h​\(r\)\)​\[d\(h\)​\(r\)−λ\]\+\+∑rg^h​\(r\)​\(\[d\(h\)​\(r\)−λ\]\+−\[d^\(h\)​\(r\)−λ\]\+\)\.\\Phi\_\{h\}\(\\lambda,s\)\-\\hat\{\\Phi\}\_\{h\}\(\\lambda,s\)=\\sum\_\{r\}\\bigl\(g\_\{h\}\(r\)\-\\hat\{g\}\_\{h\}\(r\)\\bigr\)\\left\[d^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}\+\\sum\_\{r\}\\hat\{g\}\_\{h\}\(r\)\\bigl\(\\left\[d^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}\-\\left\[\\hat\{d\}^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}\\bigr\)\.In the first sum,0≤\[d\(h\)​\(r\)−λ\]\+≤d\(h\)​\(r\)≤dmax≤C20\\leq\\left\[d^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}\\leq d^\{\(h\)\}\(r\)\\leq d\_\{\\max\}\\leq C\_\{2\}for everyrr, so its absolute value is at mostC2​‖gh−g^h‖1≤C2​\|ℛ\|​rmC\_\{2\}\\\|g\_\{h\}\-\\hat\{g\}\_\{h\}\\\|\_\{1\}\\leq C\_\{2\}\|\\mathcal\{R\}\|r\_\{m\}, using‖gh−g^h‖1≤\|ℛ\|​rm\\\|g\_\{h\}\-\\hat\{g\}\_\{h\}\\\|\_\{1\}\\leq\|\\mathcal\{R\}\|r\_\{m\}from the proof of part \(i\)\. In the second sum, the Lipschitz property\|\[a\]\+−\[b\]\+\|≤\|a−b\|\|\\left\[a\\right\]\_\{\+\}\-\\left\[b\\right\]\_\{\+\}\|\\leq\|a\-b\|gives\|\[d\(h\)​\(r\)−λ\]\+−\[d^\(h\)​\(r\)−λ\]\+\|≤\|d^\(h\)​\(r\)−d\(h\)​\(r\)\|\|\\left\[d^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}\-\\left\[\\hat\{d\}^\{\(h\)\}\(r\)\-\\lambda\\right\]\_\{\+\}\|\\leq\|\\hat\{d\}^\{\(h\)\}\(r\)\-d^\{\(h\)\}\(r\)\|, so, since∑rg^h​\(r\)=1\\sum\_\{r\}\\hat\{g\}\_\{h\}\(r\)=1, its absolute value is at mostmaxr⁡\|d^\(h\)​\(r\)−d\(h\)​\(r\)\|≤C1​rm\\max\_\{r\}\|\\hat\{d\}^\{\(h\)\}\(r\)\-d^\{\(h\)\}\(r\)\|\\leq C\_\{1\}r\_\{m\}by step \(a\)\. Hence, settingδΨ:=\(C2​\|ℛ\|\+C1\)​rm\\delta\_\{\\Psi\}:=\(C\_\{2\}\|\\mathcal\{R\}\|\+C\_\{1\}\)r\_\{m\},

\|Φh​\(λ,s\)−Φ^h​\(λ,s\)\|≤δΨfor all​λ∈\[0,dmax\],s∈\[0,1\],h∈\{0,1\}\.\\bigl\|\\Phi\_\{h\}\(\\lambda,s\)\-\\hat\{\\Phi\}\_\{h\}\(\\lambda,s\)\\bigr\|\\leq\\delta\_\{\\Psi\}\\qquad\\text\{for all \}\\lambda\\in\[0,d\_\{\\max\}\],\\ s\\in\[0,1\],\\ h\\in\\\{0,1\\\}\.
Finally we pass from the objectives to their infima\. Fixs∈\[0,1\]s\\in\[0,1\]\. For everyλ∈\[0,dmax\]\\lambda\\in\[0,d\_\{\\max\}\]the previous display givesΦh​\(λ,s\)≤Φ^h​\(λ,s\)\+δΨ\\Phi\_\{h\}\(\\lambda,s\)\\leq\\hat\{\\Phi\}\_\{h\}\(\\lambda,s\)\+\\delta\_\{\\Psi\}; taking the infimum overλ∈\[0,dmax\]\\lambda\\in\[0,d\_\{\\max\}\]on both sides and using the truncated representations above,

Ψh​\(s\)=inf0≤λ≤dmaxΦh​\(λ,s\)≤inf0≤λ≤dmax\{Φ^h​\(λ,s\)\+δΨ\}=Ψ^h​\(s\)\+δΨ\.\\Psi\_\{h\}\(s\)=\\inf\_\{0\\leq\\lambda\\leq d\_\{\\max\}\}\\Phi\_\{h\}\(\\lambda,s\)\\leq\\inf\_\{0\\leq\\lambda\\leq d\_\{\\max\}\}\\bigl\\\{\\hat\{\\Phi\}\_\{h\}\(\\lambda,s\)\+\\delta\_\{\\Psi\}\\bigr\\\}=\\hat\{\\Psi\}\_\{h\}\(s\)\+\\delta\_\{\\Psi\}\.Exchanging the roles ofΦh\\Phi\_\{h\}andΦ^h\\hat\{\\Phi\}\_\{h\}givesΨ^h​\(s\)≤Ψh​\(s\)\+δΨ\\hat\{\\Psi\}\_\{h\}\(s\)\\leq\\Psi\_\{h\}\(s\)\+\\delta\_\{\\Psi\}, whence\|Ψ^h​\(s\)−Ψh​\(s\)\|≤δΨ\|\\hat\{\\Psi\}\_\{h\}\(s\)\-\\Psi\_\{h\}\(s\)\|\\leq\\delta\_\{\\Psi\}\. SinceδΨ\\delta\_\{\\Psi\}depends on neitherssnorhh,

max⁡sups∈\[0,1\]h⁡\|Ψ^h​\(s\)−Ψh​\(s\)\|≤δΨ=\(C2​\|ℛ\|\+C1\)​rm\.\\max\_\{h\}\\sup\_\{s\\in\[0,1\]\}\|\\hat\{\\Psi\}\_\{h\}\(s\)\-\\Psi\_\{h\}\(s\)\|\\leq\\delta\_\{\\Psi\}=\\bigl\(C\_\{2\}\|\\mathcal\{R\}\|\+C\_\{1\}\\bigr\)r\_\{m\}\.Summing the bounds of \(a\), \(b\), and \(c\) proves \([156](https://arxiv.org/html/2609.28859#S13.E156)\) withLloc:=4​Cϕ/f⋆\+2​C1\+C2​\|ℛ\|L\_\{\\mathrm\{loc\}\}:=4C\_\{\\phi\}/f\_\{\\star\}\+2C\_\{1\}\+C\_\{2\}\|\\mathcal\{R\}\|, a primitive constant depending only onp0,p1,f⋆,\|ℛ\|p\_\{0\},p\_\{1\},f\_\{\\star\},\|\\mathcal\{R\}\|\.

*Part \(iii\)\.*IfnAI=0n\_\{\\mathrm\{AI\}\}=0, both perspective terms vanish andIplan\(h\)=I^plan\(h\)I^\{\(h\)\}\_\{\\mathrm\{plan\}\}=\\hat\{I\}^\{\(h\)\}\_\{\\mathrm\{plan\}\}, so the bound is trivial\. Otherwise, thenH​JX\(h\)n\_\{\\mathrm\{H\}\}J\_\{X\}^\{\(h\)\}terms in \([154](https://arxiv.org/html/2609.28859#S13.E154)\) coincide becausep0,p1p\_\{0\},p\_\{1\}are known, so, for eachhh,

\|Iplan\(h\)−I^plan\(h\)\|≤nAI​\(\|I^R\(h\)−IR\(h\)\|\+sups∈\[0,1\]\|Ψ^h​\(s\)−Ψh​\(s\)\|\)≤N​Lloc​rm,\\bigl\|I^\{\(h\)\}\_\{\\mathrm\{plan\}\}\-\\hat\{I\}^\{\(h\)\}\_\{\\mathrm\{plan\}\}\\bigr\|\\leq n\_\{\\mathrm\{AI\}\}\\left\(\|\\hat\{I\}\_\{R\}^\{\(h\)\}\-I\_\{R\}^\{\(h\)\}\|\+\\sup\_\{s\\in\[0,1\]\}\|\\hat\{\\Psi\}\_\{h\}\(s\)\-\\Psi\_\{h\}\(s\)\|\\right\)\\leq NL\_\{\\mathrm\{loc\}\}r\_\{m\},using \([156](https://arxiv.org/html/2609.28859#S13.E156)\) andnAI≤Nn\_\{\\mathrm\{AI\}\}\\leq N\. This proves \([157](https://arxiv.org/html/2609.28859#S13.E157)\)\.

### 13\.2Proof of Theorem[6\.3](https://arxiv.org/html/2609.28859#S6.Thmtheorem3)

###### Proof 13\.3

Proof of Theorem[6\.3](https://arxiv.org/html/2609.28859#S6.Thmtheorem3)To prove Theorem[6\.3](https://arxiv.org/html/2609.28859#S6.Thmtheorem3), we show that onℰpilot,mpilot,k\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\_\{\\mathrm\{pilot\},k\}\}and for all sufficiently largekk, there exist primitive constantsc\>0c\>0andC1,…,C5<∞C\_\{1\},\\dots,C\_\{5\}<\\inftysuch that

maxh⁡𝔼hπ^mpilot,k,k​\[Cπ^mpilot,k,k∣𝒟mpilot,k\]LBk\\displaystyle\\frac\{\\max\_\{h\}\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\[C^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\\mid\\mathcal\{D\}\_\{m\_\{\\mathrm\{pilot\},k\}\}\]\}\{\\mathrm\{LB\}\_\{k\}\}≤1\+C1​ΔkLk\+C2​rmpilot,k\+C3​zk\+1Δmpilot,k,keff\+C4​exp⁡\{−c​\(Δmpilot,k,keff\)2Lk\}\+C5​log⁡LkLk,\\displaystyle\\quad\\leq 1\+C\_\{1\}\\frac\{\\Delta\_\{k\}\}\{L\_\{k\}\}\+C\_\{2\}r\_\{m\_\{\\mathrm\{pilot\},k\}\}\+C\_\{3\}\\frac\{z\_\{k\}\+1\}\{\\Delta^\{\\mathrm\{eff\}\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\+C\_\{4\}\\exp\\left\\\{\-c\\frac\{\(\\Delta^\{\\mathrm\{eff\}\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\)^\{2\}\}\{L\_\{k\}\}\\right\\\}\+C\_\{5\}\\frac\{\\log L\_\{k\}\}\{L\_\{k\}\},\(158\)where

Δm,keff:=Δk−ωm,k\.\\Delta^\{\\mathrm\{eff\}\}\_\{m,k\}:=\\Delta\_\{k\}\-\\omega\_\{m,k\}\.\(159\)
In the proof of Theorem[6\.3](https://arxiv.org/html/2609.28859#S6.Thmtheorem3), we always assume that the conditions of Theorem[6\.3](https://arxiv.org/html/2609.28859#S6.Thmtheorem3)hold\. That is, Assumption[5\.3](https://arxiv.org/html/2609.28859#S5.SS3)holds,ffb,kf\_\{\\mathrm\{fb\},k\}is bounded away from zero and one,\(Δk,zk\)\(\\Delta\_\{k\},z\_\{k\}\)satisfy \([41](https://arxiv.org/html/2609.28859#S5.E41)\) and \([42](https://arxiv.org/html/2609.28859#S5.E42)\), andmpilot,k→∞m\_\{\\mathrm\{pilot\},k\}\\to\\inftyandLk​rmpilot,k=o⁡\(Δk\)L\_\{k\}r\_\{m\_\{\\mathrm\{pilot\},k\}\}=o\(\\Delta\_\{k\}\)\. For brevity, we state the auxiliary lemmas without explicitly rewriting those assumptions\.

The proof of Theorem[6\.3](https://arxiv.org/html/2609.28859#S6.Thmtheorem3)has an analogous roadmap to the proof of Theorem[5\.3](https://arxiv.org/html/2609.28859#S5.Thmtheorem3)\. The conditional expected cost of policyπ^m,k\\hat\{\\pi\}\_\{m,k\}conditional on𝒟m\\mathcal\{D\}\_\{m\},𝔼hπ^m,k​\[Cπ^m,k\|𝒟m\]\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[C^\{\\hat\{\\pi\}\_\{m,k\}\}\|\\mathcal\{D\}\_\{m\}\], has three parts: the data acquisition costcdata​N^main,m,kc\_\{\\mathrm\{data\}\}\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}, the conditional expected main\-stage sensing cost𝔼hπ^m,k​\[C^main\|𝒟m\]\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\widehat\{C\}^\{\\mathrm\{main\}\}\|\\mathcal\{D\}\_\{m\}\]before fallback, and the conditional expected fallback test cost𝔼hπ^m,k​\[C^fb\|𝒟m\]\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\widehat\{C\}^\{\\mathrm\{fb\}\}\|\\mathcal\{D\}\_\{m\}\]:𝔼hπ^m,k​\[Cπ^m,k\|𝒟m\]=cdata​N^main,m,k\+𝔼hπ^m,k​\[C^main\|𝒟m\]\+𝔼hπ^m,k​\[C^fb\|𝒟m\]\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[C^\{\\hat\{\\pi\}\_\{m,k\}\}\|\\mathcal\{D\}\_\{m\}\]=c\_\{\\mathrm\{data\}\}\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\+\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\widehat\{C\}^\{\\mathrm\{main\}\}\|\\mathcal\{D\}\_\{m\}\]\+\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\widehat\{C\}^\{\\mathrm\{fb\}\}\|\\mathcal\{D\}\_\{m\}\]\.

Lemma[13\.4](https://arxiv.org/html/2609.28859#S13.Thmtheorem4)below bounds the main\-stage cost\. Proof of Lemma[13\.4](https://arxiv.org/html/2609.28859#S13.Thmtheorem4)is in Appendix[13\.5](https://arxiv.org/html/2609.28859#S13.SS5)\.

###### Lemma 13\.4\(Main\-stage cost\)

There exists a constantC<∞C<\\inftysuch that for everyh∈\{0,1\}h\\in\\\{0,1\\\}and for all sufficiently largem,km,k, onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\},

𝔼hπ^m,k​\[C^main∣𝒟m\]≤Γ^h​\(T^h,m,kpilot​\(N¯^main,m,k\),N¯^main,m,k\)\+C​Lk​rm\+C⁡\(zk\+1\)\.\\displaystyle\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\widehat\{C\}^\{\\mathrm\{main\}\}\\mid\\mathcal\{D\}\_\{m\}\]\\leq\\hat\{\\Gamma\}\_\{h\}\(\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{h,m,k\}\(\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\),\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\)\+CL\_\{k\}r\_\{m\}\+C\(z\_\{k\}\+1\)\.

LetE^fb\\hat\{E\}\_\{\\mathrm\{fb\}\}be the guarded fallback event, as in the proof of Theorem[6\.1](https://arxiv.org/html/2609.28859#S6.Thmtheorem1)\. Lemma[13\.5](https://arxiv.org/html/2609.28859#S13.Thmtheorem5)below bounds the fallback probabilityℙhπ^m,k​\(E^fb∣𝒟m\)\\mathbb\{P\}\_\{h\}^\{\\hat\{\\pi\}\_\{m,k\}\}\(\\hat\{E\}\_\{\\mathrm\{fb\}\}\\mid\\mathcal\{D\}\_\{m\}\)\. Given thatN^main,m,k=O⁡\(Lk\)\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}=O\(L\_\{k\}\)\(see Lemma[13\.8](https://arxiv.org/html/2609.28859#S13.Thmtheorem8)in Appendix[13\.3](https://arxiv.org/html/2609.28859#S13.SS3)\), the expected fallback cost is thereforeO\(Lk\[\(zk\+1\)/Δm,keff\+exp\{−c\(Δm,keff\)2/Lk\}\]\)O\(L\_\{k\}\[\(z\_\{k\}\+1\)/\\Delta^\{\\mathrm\{eff\}\}\_\{m,k\}\+\\exp\\\{\-c\(\\Delta^\{\\mathrm\{eff\}\}\_\{m,k\}\)^\{2\}/L\_\{k\}\\\}\]\)\. Proof of Lemma[13\.5](https://arxiv.org/html/2609.28859#S13.Thmtheorem5)is in Appendix[13\.4](https://arxiv.org/html/2609.28859#S13.SS4)\.

###### Lemma 13\.5\(Fallback probability\)

AssumeΔm,keff≥4​Bℓ\\Delta^\{\\mathrm\{eff\}\}\_\{m,k\}\\geq 4B\_\{\\ell\}\. There are primitive constantsC,c\>0C,c\>0such that for everyh∈\{0,1\}h\\in\\\{0,1\\\}and for sufficiently largem,km,k, onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\},

ℙhπ^m,k​\(E^fb∣𝒟m\)≤C​zk\+1Δm,keff\+C​exp⁡\{−c​\(Δm,keff\)2Lk\}\.\\mathbb\{P\}\_\{h\}^\{\\hat\{\\pi\}\_\{m,k\}\}\(\\hat\{E\}\_\{\\mathrm\{fb\}\}\\mid\\mathcal\{D\}\_\{m\}\)\\leq C\\frac\{z\_\{k\}\+1\}\{\\Delta^\{\\mathrm\{eff\}\}\_\{m,k\}\}\+C\\exp\\left\\\{\-c\\frac\{\(\\Delta^\{\\mathrm\{eff\}\}\_\{m,k\}\)^\{2\}\}\{L\_\{k\}\}\\right\\\}\.\(160\)

Combining Lemmas[13\.4](https://arxiv.org/html/2609.28859#S13.Thmtheorem4)and[13\.5](https://arxiv.org/html/2609.28859#S13.Thmtheorem5)gives

𝔼hπ^m,k​\[Cπ^m,k∣𝒟m\]\\displaystyle\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m,k\}\}\[C^\{\\hat\{\\pi\}\_\{m,k\}\}\\mid\\mathcal\{D\}\_\{m\}\]≤N^main,m,k​cdata\+Γ^h​\(T^h,m,kpilot​\(N¯^main,m,k\),N¯^main,m,k\)\+C​Lk​rm\+C⁡\(zk\+1\)\\displaystyle\\leq\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}c\_\{\\mathrm\{data\}\}\+\\hat\{\\Gamma\}\_\{h\}\\\!\\left\(\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{h,m,k\}\(\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\),\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\\right\)\+CL\_\{k\}r\_\{m\}\+C\(z\_\{k\}\+1\)\+C​Lk​\[zk\+1Δm,keff\+exp⁡\{−c​\(Δm,keff\)2Lk\}\]\.\\displaystyle\\quad\+CL\_\{k\}\\left\[\\frac\{z\_\{k\}\+1\}\{\\Delta^\{\\mathrm\{eff\}\}\_\{m,k\}\}\+\\exp\\left\\\{\-c\\frac\{\(\\Delta^\{\\mathrm\{eff\}\}\_\{m,k\}\)^\{2\}\}\{L\_\{k\}\}\\right\\\}\\right\]\.\(161\)The final lemma connects𝔼hπ^m,k​\[Cπ^m,k\]\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[C^\{\\hat\{\\pi\}\_\{m,k\}\}\]toLBk\\mathrm\{LB\}\_\{k\}:

###### Lemma 13\.6\(Pilot outer stability\)

Onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}, for all sufficiently largem,km,k, the guarded outer problem \([55](https://arxiv.org/html/2609.28859#S6.E55)\) is feasible and

N^main,m,k​cdata\+maxh⁡Γ^h​\(T^h,m,kpilot​\(N¯^main,m,k\),N¯^main,m,k\)\\displaystyle\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}c\_\{\\mathrm\{data\}\}\+\\max\_\{h\}\\hat\{\\Gamma\}\_\{h\}\\\!\\left\(\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{h,m,k\}\(\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\),\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\\right\)≤minN∈ℤ\+,N≥Nfixed,H​\(α2,k,β2,k\)⁡\{N​cdata\+maxh⁡Γh​\(Th,k,N\)\}\+O⁡\(Lk​rm\+1\)\.\\displaystyle\\qquad\\leq\\min\_\{N\\in\\mathbb\{Z\}\_\{\+\},N\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\}\\left\\\{Nc\_\{\\mathrm\{data\}\}\+\\max\_\{h\}\\Gamma\_\{h\}\(T\_\{h,k\},N\)\\right\\\}\+O\(L\_\{k\}r\_\{m\}\+1\)\.\(162\)

Proof of Lemma[13\.6](https://arxiv.org/html/2609.28859#S13.Thmtheorem6)is in Appendix[13\.3](https://arxiv.org/html/2609.28859#S13.SS3)\. Finally, Lemma[13\.8](https://arxiv.org/html/2609.28859#S13.Thmtheorem8)in Appendix[13\.3](https://arxiv.org/html/2609.28859#S13.SS3)givesωmpilot,k,k=O⁡\(Lk​rmpilot,k\)=o⁡\(Δk\)\\omega\_\{m\_\{\\mathrm\{pilot\},k\},k\}=O\(L\_\{k\}r\_\{m\_\{\\mathrm\{pilot\},k\}\}\)=o\(\\Delta\_\{k\}\), soΔmpilot,k,keff=Δk​\{1−o⁡\(1\)\}\\Delta^\{\\mathrm\{eff\}\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}=\\Delta\_\{k\}\\\{1\-o\(1\)\\\}\. Combine \([162](https://arxiv.org/html/2609.28859#S13.E162)\) and \([161](https://arxiv.org/html/2609.28859#S13.E161)\), and then applying Lemma[10\.8](https://arxiv.org/html/2609.28859#S10.Thmtheorem8)gives the desired argument\.

### 13\.3Proof of Lemma[13\.6](https://arxiv.org/html/2609.28859#S13.Thmtheorem6)

###### Proof 13\.7

Proof of Lemma[13\.6](https://arxiv.org/html/2609.28859#S13.Thmtheorem6)Sincerm→0r\_\{m\}\\to 0asm→∞m\\to\\infty, we haverm≤f⋆/4r\_\{m\}\\leq f\_\{\\star\}/4for all sufficiently largemm, so Lemma[13\.1](https://arxiv.org/html/2609.28859#S13.Thmtheorem1)\(ii\) applies; moreoverεmdr=3​\|ℛ\|​Bℓ​rm→0\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}=3\|\\mathcal\{R\}\|B\_\{\\ell\}r\_\{m\}\\to 0and, by the first term of \([156](https://arxiv.org/html/2609.28859#S13.E156)\),εmLR≤Lloc​rm\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}\\leq L\_\{\\mathrm\{loc\}\}r\_\{m\}\. Asymptotic statements are understood along any pilot\-size sequencem=mk→∞m=m\_\{k\}\\to\\inftysatisfying \([57](https://arxiv.org/html/2609.28859#S6.E57)\), and “for all sufficiently largem,km,k” refers to such sequences\.

In order to prove Lemma[13\.6](https://arxiv.org/html/2609.28859#S13.Thmtheorem6), we constructN′∈ℤ\+N^\{\\prime\}\\in\\mathbb\{Z\}\_\{\+\}such that

F^m,kpilot​\(N′\)≤minN∈ℤ\+,N≥Nfixed,H​\(α2,k,β2,k\)⁡\{N​cdata\+maxh⁡Γh​\(Th,k,N\)\}\+O⁡\(Lk​rm\+1\)\.\\hat\{F\}^\{\\mathrm\{pilot\}\}\_\{m,k\}\(N^\{\\prime\}\)\\leq\\min\_\{N\\in\\mathbb\{Z\}\_\{\+\},N\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\}\\left\\\{Nc\_\{\\mathrm\{data\}\}\+\\max\_\{h\}\\Gamma\_\{h\}\(T\_\{h,k\},N\)\\right\\\}\+O\(L\_\{k\}r\_\{m\}\+1\)\.Then, the argument follows given thatF^m,kpilot​\(N¯^main,m,k\)≤F^m,kpilot​\(N′\)\\hat\{F\}^\{\\mathrm\{pilot\}\}\_\{m,k\}\(\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\)\\leq\\hat\{F\}^\{\\mathrm\{pilot\}\}\_\{m,k\}\(N^\{\\prime\}\)\. Recall from \([34](https://arxiv.org/html/2609.28859#S5.E34)\) thatN¯main,k∈arg​minN≥Nfixed,H​\(α2,k,β2,k\)⁡FkΔ​\(N\)\\overline\{N\}\_\{\\mathrm\{main\},k\}\\in\\operatorname\*\{arg\\,min\}\_\{N\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)\}F\_\{k\}^\{\\Delta\}\(N\)and that according to Lemma[10\.11](https://arxiv.org/html/2609.28859#S10.Thmtheorem11),N¯main,k=O⁡\(Lk\)\\overline\{N\}\_\{\\mathrm\{main\},k\}=O\(L\_\{k\}\)\.

For eachhh, let\(nH,h∗,nAI,h∗,nesc,h∗\)\(n^\{\*\}\_\{\\mathrm\{H\},h\},n^\{\*\}\_\{\\mathrm\{AI\},h\},n^\{\*\}\_\{\\mathrm\{esc\},h\}\)be aΓh\\Gamma\_\{h\}\-optimizer at\(Th,k,N¯main,k\)\(T\_\{h,k\},\\overline\{N\}\_\{\\mathrm\{main\},k\}\); its true information valueIplan\(h\)I^\{\(h\)\}\_\{\\mathrm\{plan\}\}is at leastTh,kT\_\{h,k\}by feasibility\. By \([157](https://arxiv.org/html/2609.28859#S13.E157)\) withN=N¯main,kN=\\overline\{N\}\_\{\\mathrm\{main\},k\}, the plug\-in value of\(nH,h∗,nAI,h∗,nesc,h∗\)\(n^\{\*\}\_\{\\mathrm\{H\},h\},n^\{\*\}\_\{\\mathrm\{AI\},h\},n^\{\*\}\_\{\\mathrm\{esc\},h\}\)satisfiesI^plan\(h\)≥Iplan\(h\)−Lloc​N¯main,k​rm≥Th,k−Lloc​N¯main,k​rm\\hat\{I\}^\{\(h\)\}\_\{\\mathrm\{plan\}\}\\geq I^\{\(h\)\}\_\{\\mathrm\{plan\}\}\-L\_\{\\mathrm\{loc\}\}\\overline\{N\}\_\{\\mathrm\{main\},k\}r\_\{m\}\\geq T\_\{h,k\}\-L\_\{\\mathrm\{loc\}\}\\overline\{N\}\_\{\\mathrm\{main\},k\}r\_\{m\}\. Append

Rk:=⌈Lloc​N¯main,k​rm\+\(N¯main,k\+1\)​εmdrJmin−εmdr⌉,Jmin:=minh⁡JX\(h\),R\_\{k\}:=\\left\\lceil\\frac\{L\_\{\\mathrm\{loc\}\}\\overline\{N\}\_\{\\mathrm\{main\},k\}r\_\{m\}\+\(\\overline\{N\}\_\{\\mathrm\{main\},k\}\+1\)\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}\}\{J\_\{\\min\}\-\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}\}\\right\\rceil,\\qquad J\_\{\\min\}:=\\min\_\{h\}J\_\{X\}^\{\(h\)\},direct\-human items; the denominator is positive for all largemmbecauseεmdr→0\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}\\to 0whileJmin\>0J\_\{\\min\}\>0is a primitive constant\. LetN′:=N¯main,k\+RkN^\{\\prime\}:=\\overline\{N\}\_\{\\mathrm\{main\},k\}\+R\_\{k\}\. We now verify that\(nH,h∗\+Rk,nAI,h∗,nesc,h∗\)\(n^\{\*\}\_\{\\mathrm\{H\},h\}\+R\_\{k\},\\,n^\{\*\}\_\{\\mathrm\{AI\},h\},\\,n^\{\*\}\_\{\\mathrm\{esc\},h\}\)is feasible to \([51](https://arxiv.org/html/2609.28859#S6.E51)\) atT=T^h,m,kpilot​\(N′\)T=\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{h,m,k\}\(N^\{\\prime\}\)andN=N′N=N^\{\\prime\}\. First, by constructionnesc,h∗≤nAI,h∗n^\{\*\}\_\{\\mathrm\{esc\},h\}\\leq n^\{\*\}\_\{\\mathrm\{AI\},h\}and

nH,h∗\+Rk\+nAI,h∗≤N¯main,k\+Rk=N′\.n^\{\*\}\_\{\\mathrm\{H\},h\}\+R\_\{k\}\+n^\{\*\}\_\{\\mathrm\{AI\},h\}\\leq\\overline\{N\}\_\{\\mathrm\{main\},k\}\+R\_\{k\}=N^\{\\prime\}\.Additionally, the plug\-in information value provided by the triple underHhH\_\{h\}satisfies

\(nH,h∗\+Rk\)​JX\(h\)\+nAI,h∗​I^R\(h\)\+nAI,h∗​Ψ^h​\(nesc,h∗/nAI,h∗\)≥nH,h∗​JX\(h\)\+Rk​Jmin\+nAI,h∗​I^R\(h\)\+nAI,h∗​Ψ^h​\(nesc,h∗/nAI,h∗\)≥I^plan\(h\)\+Rk​Jmin≥Th,k−Lloc​N¯main,k​rm\+Rk​\(Jmin−εmdr\)\+Rk​εmdr≥Th,k\+\(N′\+1\)​εmdr=T^h,m,kpilot​\(N′\)\.\\begin\{split\}&\(n^\{\*\}\_\{\\mathrm\{H\},h\}\+R\_\{k\}\)J^\{\(h\)\}\_\{X\}\+n^\{\*\}\_\{\\mathrm\{AI\},h\}\\hat\{I\}^\{\(h\)\}\_\{R\}\+n^\{\*\}\_\{\\mathrm\{AI\},h\}\\hat\{\\Psi\}\_\{h\}\(n^\{\*\}\_\{\\mathrm\{esc\},h\}/n^\{\*\}\_\{\\mathrm\{AI\},h\}\)\\\\ \\geq&n^\{\*\}\_\{\\mathrm\{H\},h\}J^\{\(h\)\}\_\{X\}\+R\_\{k\}J\_\{\\min\}\+n^\{\*\}\_\{\\mathrm\{AI\},h\}\\hat\{I\}^\{\(h\)\}\_\{R\}\+n^\{\*\}\_\{\\mathrm\{AI\},h\}\\hat\{\\Psi\}\_\{h\}\(n^\{\*\}\_\{\\mathrm\{esc\},h\}/n^\{\*\}\_\{\\mathrm\{AI\},h\}\)\\\\ \\geq&\\hat\{I\}^\{\(h\)\}\_\{\\mathrm\{plan\}\}\+R\_\{k\}J\_\{\\min\}\\\\ \\geq&T\_\{h,k\}\-L\_\{\\mathrm\{loc\}\}\\overline\{N\}\_\{\\mathrm\{main\},k\}r\_\{m\}\+R\_\{k\}\\bigl\(J\_\{\\min\}\-\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}\\bigr\)\+R\_\{k\}\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}\\\\ \\geq&T\_\{h,k\}\+\(N^\{\\prime\}\+1\)\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}\\\\ =&\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{h,m,k\}\(N^\{\\prime\}\)\.\\end\{split\}
SinceN′≥N¯main,k≥Nfixed,H​\(α2,k,β2,k\)N^\{\\prime\}\\geq\\overline\{N\}\_\{\\mathrm\{main\},k\}\\geq N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\),N′N^\{\\prime\}is feasible to \([55](https://arxiv.org/html/2609.28859#S6.E55)\)\. Moreover, since the objectives ofΓh\\Gamma\_\{h\}andΓ^h\\hat\{\\Gamma\}\_\{h\}share the channel\-free cost coefficients\(cH,cAI,cH\)\(c\_\{\\mathrm\{H\}\},c\_\{\\mathrm\{AI\}\},c\_\{\\mathrm\{H\}\}\), the repaired triple costsΓh​\(Th,k,N¯main,k\)\+Rk​cH\\Gamma\_\{h\}\(T\_\{h,k\},\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\+R\_\{k\}c\_\{\\mathrm\{H\}\}in the plug\-in program, so

F^m,kpilot​\(N′\)≤N′​cdata\+maxh⁡Γh​\(Th,k,N¯main,k\)\+Rk​cH=FkΔ​\(N¯main,k\)\+Rk​\(cdata\+cH\)\.\\widehat\{F\}^\{\\mathrm\{pilot\}\}\_\{m,k\}\(N^\{\\prime\}\)\\leq N^\{\\prime\}c\_\{\\mathrm\{data\}\}\+\\max\_\{h\}\\Gamma\_\{h\}\(T\_\{h,k\},\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\+R\_\{k\}c\_\{\\mathrm\{H\}\}=F^\{\\Delta\}\_\{k\}\(\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\+R\_\{k\}\(c\_\{\\mathrm\{data\}\}\+c\_\{\\mathrm\{H\}\}\)\.The left\-hand side of \([162](https://arxiv.org/html/2609.28859#S13.E162)\) equalsF^m,kpilot​\(N¯^main,m,k\)\+cdata≤F^m,kpilot​\(N′\)\+cdata\\widehat\{F\}^\{\\mathrm\{pilot\}\}\_\{m,k\}\(\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\)\+c\_\{\\mathrm\{data\}\}\\leq\\widehat\{F\}^\{\\mathrm\{pilot\}\}\_\{m,k\}\(N^\{\\prime\}\)\+c\_\{\\mathrm\{data\}\}by \([55](https://arxiv.org/html/2609.28859#S6.E55)\)\. SinceN¯main,k=O⁡\(Lk\)\\overline\{N\}\_\{\\mathrm\{main\},k\}=O\(L\_\{k\}\)andεmdr=O⁡\(rm\)\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}=O\(r\_\{m\}\), the numerator ofRkR\_\{k\}isO⁡\(Lk​rm\)O\(L\_\{k\}r\_\{m\}\), soRk=O⁡\(Lk​rm\+1\)R\_\{k\}=O\(L\_\{k\}r\_\{m\}\+1\), and \([162](https://arxiv.org/html/2609.28859#S13.E162)\) follows\.

Lemma[13\.8](https://arxiv.org/html/2609.28859#S13.Thmtheorem8)is an immediate consequence of Lemma[13\.6](https://arxiv.org/html/2609.28859#S13.Thmtheorem6)\.

###### Lemma 13\.8

Onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}and for all sufficiently largem,km,k,N^main,m,k=O⁡\(Lk\)\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}=O\(L\_\{k\}\)andωm,k=O⁡\(Lk​rm\)=o⁡\(Δk\)\\omega\_\{m,k\}=O\(L\_\{k\}r\_\{m\}\)=o\(\\Delta\_\{k\}\)\.

###### Proof 13\.9

Proof of Lemma[13\.8](https://arxiv.org/html/2609.28859#S13.Thmtheorem8)According to Lemma[13\.6](https://arxiv.org/html/2609.28859#S13.Thmtheorem6),N^main,m,k​cdata≤FkΔ​\(N¯main,k\)\+O⁡\(Lk​rm\+1\)=O⁡\(Lk\)\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}c\_\{\\mathrm\{data\}\}\\leq F^\{\\Delta\}\_\{k\}\(\\overline\{N\}\_\{\\mathrm\{main\},k\}\)\+O\(L\_\{k\}r\_\{m\}\+1\)=O\(L\_\{k\}\)by Lemma[10\.11](https://arxiv.org/html/2609.28859#S10.Thmtheorem11)andLk​rm=o⁡\(Δk\)=o⁡\(Lk\)L\_\{k\}r\_\{m\}=o\(\\Delta\_\{k\}\)=o\(L\_\{k\}\)\. HenceN^main,m,k=O⁡\(Lk\)\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}=O\(L\_\{k\}\), and, by the first term of \([156](https://arxiv.org/html/2609.28859#S13.E156)\) withLlocL\_\{\\mathrm\{loc\}\}the constant of Lemma[13\.1](https://arxiv.org/html/2609.28859#S13.Thmtheorem1)\(ii\),ωm,k=N^main,m,k​εmLR≤N^main,m,k​Lloc​rm=O⁡\(Lk​rm\)=o⁡\(Δk\)\\omega\_\{m,k\}=\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\\varepsilon\_\{m\}^\{\\mathrm\{LR\}\}\\leq\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}L\_\{\\mathrm\{loc\}\}r\_\{m\}=O\(L\_\{k\}r\_\{m\}\)=o\(\\Delta\_\{k\}\)\.

### 13\.4Proof of Lemma[13\.5](https://arxiv.org/html/2609.28859#S13.Thmtheorem5)

We first provide an auxiliary Lemma[13\.10](https://arxiv.org/html/2609.28859#S13.Thmtheorem10)together with its proof, and then prove Lemma[13\.5](https://arxiv.org/html/2609.28859#S13.Thmtheorem5)at the end of this section\.

Writeσ1:=1\\sigma\_\{1\}:=1andσ0:=−1\\sigma\_\{0\}:=\-1\. The signed one\-item means under the true and plug\-in channels are, respectively,𝔼hπ^m,k​\[σh​Z^iitem​\(η^j,m,k\)∣𝒟m\]\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]and𝔼h,f^π^m,k​\[σh​Z^iitem​\(η^j,m,k\)∣𝒟m\]\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h,\\hat\{f\}\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\.

According to \([155](https://arxiv.org/html/2609.28859#S13.E155)\) applied withη=η^j,m,k\\eta=\\hat\{\\eta\}\_\{j,m,k\}, onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}, we have

\|𝔼hπ^m,k​\[σh​Z^iitem​\(η^j,m,k\)∣𝒟m\]−𝔼h,f^π^m,k​\[σh​Z^iitem​\(η^j,m,k\)∣𝒟m\]\|≤εmdr\.\\left\|\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\-\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h,\\hat\{f\}\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\right\|\\leq\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}\.\(163\)
###### Lemma 13\.10\(Uniform pilot increment conditions\)

LetN^h,m,kplan:=⌈n^H,h,m,k∗⌉\+⌈n^AI,h,m,k∗⌉\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{h,m,k\}:=\\lceil\\hat\{n\}^\{\*\}\_\{\\mathrm\{H\},h,m,k\}\\rceil\+\\lceil\\hat\{n\}^\{\*\}\_\{\\mathrm\{AI\},h,m,k\}\\rceil\. Forh∈\{0,1\}h\\in\\\{0,1\\\}, the correct\-direction rounded block also satisfies

N^h,m,kplan​𝔼hπ^m,k​\[σh​Z^iitem​\(η^h,m,k\)∣𝒟m\]≥Th,k\.\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{h,m,k\}\\,\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{h,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq T\_\{h,k\}\.\(164\)
Moreover, there are primitive constantscμ\>0c\_\{\\mu\}\>0andCV<∞C\_\{V\}<\\inftysuch that, onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}and for everyj∈\{0,1,∗\}j\\in\\\{0,1,\*\\\}and all sufficiently largem,km,k, the following inequalities hold almost surely,

\|Z^itemi\(η^j,m,k\)\|≤2Bℓ,𝔼π^m,kh\[σhZ^itemi\(η^j,m,k\)∣𝒟m\]≥cμ,Varh⁡\(Z^iitem​\(η^j,m,k\)∣𝒟m\)≤CV​𝔼hπ^m,k​\[σh​Z^iitem​\(η^j,m,k\)∣𝒟m\]\.\\begin\{split\}\|\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\|\\leq 2B\_\{\\ell\},\\qquad\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq c\_\{\\mu\},\\\\ \\operatorname\{Var\}\_\{h\}\(\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\)\\leq C\_\{V\}\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\.\\end\{split\}\(165\)

###### Proof 13\.11

Proof of Lemma[13\.10](https://arxiv.org/html/2609.28859#S13.Thmtheorem10)Work onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}for all sufficiently largem,km,k, such that the guarded outer problem \([55](https://arxiv.org/html/2609.28859#S6.E55)\) is feasible,N^main,m,k=O⁡\(Lk\)\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}=O\(L\_\{k\}\),rm≤f⋆/4r\_\{m\}\\leq f\_\{\\star\}/4andεmdr=3​\|ℛ\|​Bℓ​rm→0\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}=3\|\\mathcal\{R\}\|B\_\{\\ell\}r\_\{m\}\\to 0\. We have that

𝔼h,f^π^m,k​\[σh​Z^iitem​\(η^j,m,k\)∣𝒟m\]=η^j,m,kH​JX\(h\)\+\(1−η^j,m,kH\)​\(I^R\(h\)\+∑r∈ℛg^h​\(r\)​η^j,m,kesc​\(r\)​d^\(h\)​\(r\)\)\.\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h,\\hat\{f\}\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]=\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{j,m,k\}J\_\{X\}^\{\(h\)\}\+\(1\-\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{j,m,k\}\)\\Bigl\(\\hat\{I\}\_\{R\}^\{\(h\)\}\+\\sum\_\{r\\in\\mathcal\{R\}\}\\hat\{g\}\_\{h\}\(r\)\\,\\hat\{\\eta\}^\{\\mathrm\{esc\}\}\_\{j,m,k\}\(r\)\\,\\hat\{d\}^\{\(h\)\}\(r\)\\Bigr\)\.*Step 1: show thatN^h,m,kplan​𝔼hπ^m,k​\[σh​Z^iitem​\(η^h,m,k\)∣𝒟m\]≥Th,k\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{h,m,k\}\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{h,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq T\_\{h,k\}\.*Repeating the derivation of \([117](https://arxiv.org/html/2609.28859#S10.E117)\) for the plug\-in program gives

N^h,m,kplan​𝔼h,f^π^m,k​\[σh​Z^iitem​\(η^h,m,k\)∣𝒟m\]≥T^h,m,kpilot​\(N¯^main,m,k\)=Th,k\+N^main,m,k​εmdr,\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{h,m,k\}\\,\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h,\\hat\{f\}\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{h,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{h,m,k\}\(\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\)=T\_\{h,k\}\+\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\},usingN^main,m,k=N¯^main,m,k\+1\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}=\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\+1in \([53](https://arxiv.org/html/2609.28859#S6.E53)\)\. Combining the above display with \([163](https://arxiv.org/html/2609.28859#S13.E163)\) andN^h,m,kplan≤N^main,m,k\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{h,m,k\}\\leq\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\(which follows sinceN^h,m,kplan≤⌈n^H,h,m,k∗\+n^AI,h,m,k∗⌉\+1≤⌈N¯^main,m,k⌉\+1=N^main,m,k\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{h,m,k\}\\leq\\lceil\\hat\{n\}^\{\*\}\_\{\\mathrm\{H\},h,m,k\}\+\\hat\{n\}^\{\*\}\_\{\\mathrm\{AI\},h,m,k\}\\rceil\+1\\leq\\lceil\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\\rceil\+1=\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\) gives

N^h,m,kplan​𝔼hπ^m,k​\[σh​Z^iitem​\(η^h,m,k\)∣𝒟m\]\\displaystyle\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{h,m,k\}\\,\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{h,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]≥N^h,m,kplan​𝔼h,f^π^m,k​\[σh​Z^iitem​\(η^h,m,k\)∣𝒟m\]−N^h,m,kplan​εmdr\\displaystyle\\geq\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{h,m,k\}\\,\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h,\\hat\{f\}\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{h,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\-\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{h,m,k\}\\,\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}≥Th,k\+\(N^main,m,k−N^h,m,kplan\)​εmdr\\displaystyle\\geq T\_\{h,k\}\+\\bigl\(\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\-\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{h,m,k\}\\bigr\)\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}≥Th,k,\\displaystyle\\geq T\_\{h,k\},which is \([164](https://arxiv.org/html/2609.28859#S13.E164)\)\.

*Step 2: show that𝔼hπ^m,k​\[σh​Z^iitem​\(η^h,m,k\)∣𝒟m\]≥cplan\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{h,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq c\_\{\\mathrm\{plan\}\}for a primitivecplan\>0c\_\{\\mathrm\{plan\}\}\>0for thecorrectrule\.*According to Step 1,𝔼hπ^m,k​\[σh​Z^iitem​\(η^h,m,k\)∣𝒟m\]≥Th,k/N^h,m,kplan≥Th,k/N^main,m,k\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{h,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq T\_\{h,k\}/\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{h,m,k\}\\geq T\_\{h,k\}/\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\(note thatN^h,m,kplan≥1\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{h,m,k\}\\geq 1becauseT^h,m,kpilot​\(N¯^main,m,k\)\>0\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{h,m,k\}\(\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\)\>0forces a nonzero optimizer\(n^H,h,m,k∗,n^AI,h,m,k∗,n^esc,h,m,k∗\)≠\(0,0,0\)\(\\hat\{n\}^\{\*\}\_\{\\mathrm\{H\},h,m,k\},\\hat\{n\}^\{\*\}\_\{\\mathrm\{AI\},h,m,k\},\\hat\{n\}^\{\*\}\_\{\\mathrm\{esc\},h,m,k\}\)\\neq\(0,0,0\)\. Using thatTh,k≥min⁡\{ak,bk\}≥min⁡\{log⁡\(1/α1,k\),log⁡\(1/β1,k\)\}≥cT​LkT\_\{h,k\}\\geq\\min\\\{a\_\{k\},b\_\{k\}\\\}\\geq\\min\\\{\\log\(1/\\alpha\_\{1,k\}\),\\log\(1/\\beta\_\{1,k\}\)\\\}\\geq c\_\{T\}L\_\{k\}for a primitivecT\>0c\_\{T\}\>0and all largekkand thatN^main,m,k=O⁡\(Lk\)\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}=O\(L\_\{k\}\), the correct rule satisfies𝔼hπ^m,k​\[σh​Z^iitem​\(η^h,m,k\)∣𝒟m\]≥Th,k/N^main,m,k≥cplan\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{h,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq T\_\{h,k\}/\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\\geq c\_\{\\mathrm\{plan\}\}for a primitivecplan\>0c\_\{\\mathrm\{plan\}\}\>0with the correct sensing ruleη^h,m,k\\hat\{\\eta\}\_\{h,m,k\}underHhH\_\{h\}\.

*Step 3: show that𝔼hπ^m,k​\[σh​Z^iitem​\(η^j,m,k\)∣𝒟m\]≥cμ\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq c\_\{\\mu\}for a primitive constantcμc\_\{\\mu\}for everyh∈\{0,1\}h\\in\\\{0,1\\\}andj∈\{0,1,∗\}j\\in\\\{0,1,\*\\\}\.*

We first show that𝔼h,f^π^m,k​\[σh​Z^iitem​\(η^j,m,k\)∣𝒟m\]≥c×0​𝔼1−h,f^π^m,k​\[σ1−h​Z^iitem​\(η^j,m,k\)∣𝒟m\]\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h,\\hat\{f\}\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq c\_\{\\times\}^\{0\}\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\-h,\\hat\{f\}\}\[\\sigma\_\{1\-h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]for a primitive constantc×0c^\{0\}\_\{\\times\}\. Analogous to the proof of Lemma[10\.16](https://arxiv.org/html/2609.28859#S10.Thmtheorem16)\(ii\), it suffices to show that there exists a constantccmp\>0c\_\{\\mathrm\{cmp\}\}\>0such that for bothhhand everyrr,

JX\(h\)≥ccmp​JX\(1−h\),I^R\(h\)≥ccmp​I^R\(1−h\),d^\(h\)​\(r\)≥ccmp​d^\(1−h\)​\(r\)J\_\{X\}^\{\(h\)\}\\geq c\_\{\\mathrm\{cmp\}\}J\_\{X\}^\{\(1\-h\)\},\\qquad\\hat\{I\}\_\{R\}^\{\(h\)\}\\geq c\_\{\\mathrm\{cmp\}\}\\hat\{I\}\_\{R\}^\{\(1\-h\)\},\\qquad\\hat\{d\}^\{\(h\)\}\(r\)\\geq c\_\{\\mathrm\{cmp\}\}\\hat\{d\}^\{\(1\-h\)\}\(r\)and

mg:=minh,r⁡g^h​\(r\)g^1−h​\(r\)\>0\.m\_\{g\}:=\\min\_\{h,r\}\\frac\{\\hat\{g\}\_\{h\}\(r\)\}\{\\hat\{g\}\_\{1\-h\}\(r\)\}\>0\.To verify these bounds uniformly over the realized pilot, work onℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}and takemmsufficiently large thatrm≤f⋆/4r\_\{m\}\\leq f\_\{\\star\}/4\. By Lemma[13\.1](https://arxiv.org/html/2609.28859#S13.Thmtheorem1)\(ii\), all coordinates of the plug\-in report lawsg^h\\hat\{g\}\_\{h\}and the plug\-in conditional\-label lawsρ^h\(⋅∣r\)\\hat\{\\rho\}\_\{h\}\(\\cdot\\mid r\)are bounded below by some primitive constant, let it beq⋆q\_\{\\star\}\. The same is true of the fixed direct\-label laws because0<p0<p1<10<p\_\{0\}<p\_\{1\}<1\.

If two probability vectorsP,QP,Qon a common finite alphabet have all coordinates at leastq⋆q\_\{\\star\}, then Pinsker’s inequality and the chi\-square upper bound give

D\(P∥Q\)≥12∥P−Q∥12≥12∥P−Q∥22,D\(Q∥P\)≤∑x\(Q⁡\(x\)−P⁡\(x\)\)2P⁡\(x\)≤q⋆−1∥P−Q∥22\.D\(P\\\|Q\)\\geq\\frac\{1\}\{2\}\\lVert P\-Q\\rVert\_\{1\}^\{2\}\\geq\\frac\{1\}\{2\}\\lVert P\-Q\\rVert\_\{2\}^\{2\},\\qquad D\(Q\\\|P\)\\leq\\sum\_\{x\}\\frac\{\(Q\(x\)\-P\(x\)\)^\{2\}\}\{P\(x\)\}\\leq q\_\{\\star\}^\{\-1\}\\lVert P\-Q\\rVert\_\{2\}^\{2\}\.Consequently,

D\(P∥Q\)≥q⋆2D\(Q∥P\)\.D\(P\\\|Q\)\\geq\\frac\{q\_\{\\star\}\}\{2\}D\(Q\\\|P\)\.This inequality also holds whenP=QP=Q, since then both divergences are zero\. Applying it to the direct\-label pair, the plug\-in report pair, and each plug\-in conditional\-label pair proves the three component comparisons above withccmp:=q⋆/2c\_\{\\mathrm\{cmp\}\}:=q\_\{\\star\}/2\. Moreover,g^h​\(r\)≥q⋆\\hat\{g\}\_\{h\}\(r\)\\geq q\_\{\\star\}andg^1−h​\(r\)≤1\\hat\{g\}\_\{1\-h\}\(r\)\\leq 1implymg≥q⋆\>0m\_\{g\}\\geq q\_\{\\star\}\>0\. Thus forj∈\{0,1,∗\}j\\in\\\{0,1,\*\\\}, we have𝔼h,f^π^m,k​\[σh​Z^iitem​\(η^j,m,k\)∣𝒟m\]≥c×0​𝔼1−h,f^π^m,k​\[σ1−h​Z^iitem​\(η^j,m,k\)∣𝒟m\]\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h,\\hat\{f\}\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq c\_\{\\times\}^\{0\}\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\-h,\\hat\{f\}\}\[\\sigma\_\{1\-h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]wherec×0:=ccmp​min⁡\{1,mg\}\>0c^\{0\}\_\{\\times\}:=c\_\{\\mathrm\{cmp\}\}\\min\\\{1,m\_\{g\}\\\}\>0is a primitive constant that does not depend onmmorkk\.

Now, analogous to the proof of Lemma[10\.16](https://arxiv.org/html/2609.28859#S10.Thmtheorem16)\(iii\), we have that𝔼h,f^π^m,k​\[σh​Z^iitem​\(η^h,m,k\)∣𝒟m\]≥Th,k/N^main,m,k≥cplan\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h,\\hat\{f\}\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{h,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq T\_\{h,k\}/\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\\geq c\_\{\\mathrm\{plan\}\}for a primitivecplan\>0c\_\{\\mathrm\{plan\}\}\>0forh∈\{0,1\}h\\in\\\{0,1\\\}\. Hence the preceding comparison gives𝔼h,f^π^m,k​\[σh​Z^iitem​\(η^1−h,m,k\)∣𝒟m\]≥c×0​cplan\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h,\\hat\{f\}\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{1\-h,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq c^\{0\}\_\{\\times\}c\_\{\\mathrm\{plan\}\}for the wrong rule, while the dead\-zone averaging rule gives𝔼h,f^π^m,k​\[σh​Z^iitem​\(η^∗,m,k\)∣𝒟m\]≥cplan/4\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h,\\hat\{f\}\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{\*,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq c\_\{\\mathrm\{plan\}\}/4\. Letcplug:=min⁡\{cplan,c×0​cplan,cplan/4\}\>0c\_\{\\mathrm\{plug\}\}:=\\min\\\{c\_\{\\mathrm\{plan\}\},c\_\{\\times\}^\{0\}c\_\{\\mathrm\{plan\}\},c\_\{\\mathrm\{plan\}\}/4\\\}\>0\. Then𝔼h,f^π^m,k​\[σh​Z^iitem​\(η^j,m,k\)∣𝒟m\]≥cplug\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h,\\hat\{f\}\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq c\_\{\\mathrm\{plug\}\}for everyhhandj∈\{0,1,∗\}j\\in\\\{0,1,\*\\\}\. By \([163](https://arxiv.org/html/2609.28859#S13.E163)\), uniformly over these six choices,

𝔼hπ^m,k​\[σh​Z^iitem​\(η^j,m,k\)∣𝒟m\]≥𝔼h,f^π^m,k​\[σh​Z^iitem​\(η^j,m,k\)∣𝒟m\]−εmdr\.\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h,\\hat\{f\}\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\-\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}\.Sinceεmdr→0\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}\\to 0, it is at mostcplug/2c\_\{\\mathrm\{plug\}\}/2for all sufficiently largemm\. Thus

𝔼hπ^m,k​\[σh​Z^iitem​\(η^j,m,k\)∣𝒟m\]≥cplug/2=:cμ\>0,\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq c\_\{\\mathrm\{plug\}\}/2=:c\_\{\\mu\}\>0,which proves the mean bound in \([165](https://arxiv.org/html/2609.28859#S13.E165)\)\.

*Step 4: show that\|Z^iitem​\(η^j,m,k\)\|≤2​Bℓ\|\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\|\\leq 2B\_\{\\ell\}andVarh⁡\(Z^iitem​\(η^j,m,k\)∣𝒟m\)≤CV​𝔼hπ^m,k​\[σh​Z^iitem​\(η^j,m,k\)∣𝒟m\]\\operatorname\{Var\}\_\{h\}\(\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\)\\leq C\_\{V\}\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\.*The inequality\|Z^iitem​\(η^j,m,k\)\|≤2​Bℓ\|\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\|\\leq 2B\_\{\\ell\}follows straightforwardly from the definition ofZ^iitem​\(η^j,m,k\)\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)and Lemma[12\.1](https://arxiv.org/html/2609.28859#S12.Thmtheorem1)\(i\)\. Additionally

Varh⁡\(Z^iitem​\(η^j,m,k\)∣𝒟m\)≤𝔼hπ^m,k​\[Z^iitem​\(η^j,m,k\)2∣𝒟m\]≤4​Bℓ2≤4​Bℓ2cμ​𝔼hπ^m,k​\[σh​Z^iitem​\(η^j,m,k\)∣𝒟m\]\.\\operatorname\{Var\}\_\{h\}\(\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\)\\leq\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)^\{2\}\\mid\\mathcal\{D\}\_\{m\}\]\\leq 4B\_\{\\ell\}^\{2\}\\leq\\frac\{4B\_\{\\ell\}^\{2\}\}\{c\_\{\\mu\}\}\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{h\}\[\\sigma\_\{h\}\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\.TakingCV:=4​Bℓ2/cμC\_\{V\}:=4B\_\{\\ell\}^\{2\}/c\_\{\\mu\}proves the remaining assertion of \([165](https://arxiv.org/html/2609.28859#S13.E165)\)\.

###### Proof 13\.12

Proof of Lemma[13\.5](https://arxiv.org/html/2609.28859#S13.Thmtheorem5)We prove the claims underH1H\_\{1\}\. TheH0H\_\{0\}proof is identical after applying the argument to the reflected process\(−S^i,−Z^iitem​\(η^j,m,k\)\)\(\-\\hat\{S\}\_\{i\},\-\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{j,m,k\}\)\), interchanging the indices00and11, and replacingaka\_\{k\}bybkb\_\{k\}; the upper guarded boundary then becomes the reflection of the lower guarded boundary, and the direction\-11rule becomes the direction\-00rule\. Work underℙ1π^m,k\(⋅∣𝒟m\)\\mathbb\{P\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\(\\cdot\\mid\\mathcal\{D\}\_\{m\}\)for a realized pilot inℰpilot,m\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\}, for all sufficiently largem,km,kas in Lemma[13\.10](https://arxiv.org/html/2609.28859#S13.Thmtheorem10)\.

Conditionally on𝒟m\\mathcal\{D\}\_\{m\}, itemiiis fresh: its label and report are drawn independently of the previous items by Assumption[3\.1](https://arxiv.org/html/2609.28859#S3.SS1), its randomization seeds are fresh by construction, and all of these are independent of𝒟m\\mathcal\{D\}\_\{m\}by Assumption[6\.1](https://arxiv.org/html/2609.28859#S6.SS1)\. SinceJiJ\_\{i\}isℋi−1m\\mathcal\{H\}^\{m\}\_\{i\-1\}\-measurable,

𝔼1π^m,k\[Z^iitem\(η^Ji,m,k\)∣ℋi−1m\]=𝔼1π^m,k\[Z^iitem\(η^Ji,m,k\)∣𝒟m,Ji\],\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\\mid\\mathcal\{H\}^\{m\}\_\{i\-1\}\]=\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\\mid\\mathcal\{D\}\_\{m\},J\_\{i\}\],and

Var1⁡\(Z^iitem​\(η^Ji,m,k\)∣ℋi−1m\)=Var1⁡\(Z^iitem​\(η^Ji,m,k\)∣𝒟m,Ji\)\.\\operatorname\{Var\}\_\{1\}\(\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\\mid\\mathcal\{H\}^\{m\}\_\{i\-1\}\)=\\operatorname\{Var\}\_\{1\}\(\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\\mid\\mathcal\{D\}\_\{m\},J\_\{i\}\)\.Since the inequalities in \([165](https://arxiv.org/html/2609.28859#S13.E165)\) hold for allj∈\{0,1,∗\}j\\in\\\{0,1,\*\\\}, the following holds almost surely:

𝔼π^m,k1\[Z^itemi\(η^Ji,m,k\)∣ℋmi−1\]≥cμ,\|Z^itemi\(η^Ji,m,k\)\|≤2Bℓ,Var1⁡\(Z^iitem​\(η^Ji,m,k\)∣ℋi−1m\)≤CV​𝔼1π^m,k​\[Z^iitem​\(η^Ji,m,k\)∣ℋi−1m\]\.\\begin\{split\}\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\\mid\\mathcal\{H\}^\{m\}\_\{i\-1\}\]\\geq c\_\{\\mu\},\\qquad\|\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\|\\leq 2B\_\{\\ell\},\\\\ \\operatorname\{Var\}\_\{1\}\(\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\\mid\\mathcal\{H\}^\{m\}\_\{i\-1\}\)\\leq C\_\{V\}\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\\mid\\mathcal\{H\}^\{m\}\_\{i\-1\}\]\.\\end\{split\}
*Step 1: direction tracking\.*The preceding display is exactly the bounded\-increment, positive\-drift, and variance–drift input \([119](https://arxiv.org/html/2609.28859#S10.E119)\) of Lemma[10\.4](https://arxiv.org/html/2609.28859#S10.Thmtheorem4), withcμc\_\{\\mu\}playing the role of the drift lower boundμ0\\mu\_\{0\}there, now verified for\(S^i,Z^iitem​\(η^Ji,m,k\)\)\(\\hat\{S\}\_\{i\},\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\)under the conditional measure\. The proof of that lemma uses only these abstract hypotheses and the predictability of the regime selection, which holds here becauseJiJ\_\{i\}is determined fromS^i−1\\hat\{S\}\_\{i\-1\}\. It therefore shows that the numberW1,m,kW\_\{1,m,k\}of items processed against the true direction underH1H\_\{1\}\(the analogue ofW1,kW\_\{1,k\}there\) satisfies

𝔼1π^m,k​\[W1,m,k∣𝒟m\]≤C⁡\(zk\+1\),\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[W\_\{1,m,k\}\\mid\\mathcal\{D\}\_\{m\}\]\\leq C\(z\_\{k\}\+1\),where the primitive constant absorbs the factor1/cμ1/c\_\{\\mu\}\.

*Step 2: fallback probability\.*The increment conditions above also give the abstract input \([131](https://arxiv.org/html/2609.28859#S10.E131)\) of Lemma[10\.6](https://arxiv.org/html/2609.28859#S10.Thmtheorem6)\. Its planned\-block input \([132](https://arxiv.org/html/2609.28859#S10.E132)\) is supplied by Lemma[13\.10](https://arxiv.org/html/2609.28859#S13.Thmtheorem10): by \([164](https://arxiv.org/html/2609.28859#S13.E164)\),

N^1,m,kplan​𝔼1π^m,k​\[Z^iitem​\(η^1,m,k\)∣𝒟m\]≥T1,k=ak\+Δk,\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{1,m,k\}\\,\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{1,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\ \\geq\\ T\_\{1,k\}=a\_\{k\}\+\\Delta\_\{k\},whileN^1,m,kplan≤N^main,m,k\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{1,m,k\}\\leq\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}andN^main,m,k=O⁡\(Lk\)\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}=O\(L\_\{k\}\)by Lemma[13\.8](https://arxiv.org/html/2609.28859#S13.Thmtheorem8)\. The guarded policy stops at the boundaryak\+ωm,ka\_\{k\}\+\\omega\_\{m,k\}\(resp\.−\(bk\+ωm,k\)\-\(b\_\{k\}\+\\omega\_\{m,k\}\)\), so the margin between the planned block drift and the stopping boundary is

\(ak\+Δk\)−\(ak\+ωm,k\)=Δk−ωm,k=Δm,keff\>0,\(a\_\{k\}\+\\Delta\_\{k\}\)\-\(a\_\{k\}\+\\omega\_\{m,k\}\)=\\Delta\_\{k\}\-\\omega\_\{m,k\}=\\Delta^\{\\mathrm\{eff\}\}\_\{m,k\}\>0,which replacesΔk\\Delta\_\{k\}throughout: the Freedman margin becomes3​Δm,keff/43\\Delta^\{\\mathrm\{eff\}\}\_\{m,k\}/4and the occupation thresholdΔm,keff/\(8​Bℓ\)\\Delta^\{\\mathrm\{eff\}\}\_\{m,k\}/\(8B\_\{\\ell\}\), exactly as in the proof of Lemma[10\.6](https://arxiv.org/html/2609.28859#S10.Thmtheorem6)\. Running the proof of Lemma[10\.6](https://arxiv.org/html/2609.28859#S10.Thmtheorem6)with the boundaryak\+ωm,ka\_\{k\}\+\\omega\_\{m,k\}and marginΔm,keff\\Delta^\{\\mathrm\{eff\}\}\_\{m,k\}in place ofaka\_\{k\}andΔk\\Delta\_\{k\}— the accumulated predictable variance at the end of the block is at most2​CV​Bℓ​N^1,m,kplan=O⁡\(Lk\)2C\_\{V\}B\_\{\\ell\}\\,\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{1,m,k\}=O\(L\_\{k\}\)as in its display \([137](https://arxiv.org/html/2609.28859#S10.E137)\), so the Freedman exponent becomes\(Δm,keff\)2/Lk\(\\Delta^\{\\mathrm\{eff\}\}\_\{m,k\}\)^\{2\}/L\_\{k\}— yields \([160](https://arxiv.org/html/2609.28859#S13.E160)\)\.

### 13\.5Proof of Lemma[13\.4](https://arxiv.org/html/2609.28859#S13.Thmtheorem4)

###### Proof 13\.13

Proof of Lemma[13\.4](https://arxiv.org/html/2609.28859#S13.Thmtheorem4)We prove the claims underH1H\_\{1\}\. TheH0H\_\{0\}proof is identical after applying the argument to the reflected process\(−S^i,−Z^iitem​\(η^Ji,m,k\)\)\(\-\\hat\{S\}\_\{i\},\-\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\), interchanging the indices00and11, and replacingaka\_\{k\}bybkb\_\{k\}; the upper guarded boundary then becomes the reflection of the lower guarded boundary, and the direction\-11rule becomes the direction\-00rule\.

RecallC^main\\widehat\{C\}^\{\\mathrm\{main\}\}is the sensing cost incurred during the main stage, before the fallback completion is executed\. We follow the proof of Lemma[10\.5](https://arxiv.org/html/2609.28859#S10.Thmtheorem5)\. LetIiI\_\{i\}indicate that itemiiis started and writeτ^main,m,k:=∑i=1N^main,m,kIi\\hat\{\\tau\}\_\{\\mathrm\{main\},m,k\}:=\\sum\_\{i=1\}^\{\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}\}I\_\{i\}, soIi=𝟏\{i≤τ^main,m,k\}I\_\{i\}=\\mathbf\{1\}\\\{i\\leq\\hat\{\\tau\}\_\{\\mathrm\{main\},m,k\}\\\}isℋi−1m\\mathcal\{H\}^\{m\}\_\{i\-1\}\-measurable\. Define𝒞^:=\{i:Ii=1,S^i−1\>zk\}\\hat\{\\mathcal\{C\}\}:=\\\{i:I\_\{i\}=1,\\hat\{S\}\_\{i\-1\}\>z\_\{k\}\\\},𝒲^:=\{i:Ii=1,S^i−1≤zk\}\\hat\{\\mathcal\{W\}\}:=\\\{i:I\_\{i\}=1,\\hat\{S\}\_\{i\-1\}\\leq z\_\{k\}\\\}, andN^c:=\|𝒞^\|\\hat\{N\}\_\{c\}:=\|\\hat\{\\mathcal\{C\}\}\|; then\|𝒲^\|=W1,m,k\|\\hat\{\\mathcal\{W\}\}\|=W\_\{1,m,k\}\.

For the completed\-item continuation, recall thatCiitem​\(η^Ji,m,k\)C\_\{i\}^\{\\mathrm\{item\}\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)corresponds to the cost when itemiiuses sensing ruleη^Ji,m,k\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\. Setcmax:=cH\+cAI\+cHc\_\{\\max\}:=c\_\{\\mathrm\{H\}\}\+c\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}\. Analogous to the cost decomposition \([127](https://arxiv.org/html/2609.28859#S10.E127)\),

𝔼1π^m,k​\[C^main∣𝒟m\]≤𝔼1π^m,k​\[∑i∈𝒲^Ciitem​\(η^Ji,m,k\)∣𝒟m\]\+𝔼1π^m,k​\[∑i∈𝒞^Ciitem​\(η^Ji,m,k\)∣𝒟m\]\.\\begin\{split\}\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[\\widehat\{C\}^\{\\mathrm\{main\}\}\\mid\\mathcal\{D\}\_\{m\}\]\\leq\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\\bigg\[\\sum\_\{i\\in\\hat\{\\mathcal\{W\}\}\}C\_\{i\}^\{\\mathrm\{item\}\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\\bigg\]\+\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\\bigg\[\\sum\_\{i\\in\\hat\{\\mathcal\{C\}\}\}C\_\{i\}^\{\\mathrm\{item\}\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\\bigg\]\.\\end\{split\}
*Cost 1: wrong and dead\-zone items\.*Analogous to Cost 1 in the proof of Lemma[10\.5](https://arxiv.org/html/2609.28859#S10.Thmtheorem5),

𝔼1π^m,k​\[∑i∈𝒲^Ciitem​\(η^Ji,m,k\)∣𝒟m\]≤cmax​𝔼1π^m,k​\[W1,m,k∣𝒟m\]≤C⁡\(zk\+1\)\.\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[\\sum\_\{i\\in\\hat\{\\mathcal\{W\}\}\}C\_\{i\}^\{\\mathrm\{item\}\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\leq c\_\{\\max\}\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[W\_\{1,m,k\}\\mid\\mathcal\{D\}\_\{m\}\]\\leq C\(z\_\{k\}\+1\)\.
*Cost 2: correct items\.*Under the correct sensing ruleη^1,m,k\\hat\{\\eta\}\_\{1,m,k\}, the cost of itemiiisCiitem\(η^1,m,k\)=𝟏\{Ui≤η^1,m,kH\}cH\+𝟏\{Ui\>η^1,m,kH\}\(cAI\+𝟏\{Vi≤η^1,m,kesc\(Ri\)\}cH\)C\_\{i\}^\{\\mathrm\{item\}\}\(\\hat\{\\eta\}\_\{1,m,k\}\)=\\mathbf\{1\}\\\{U\_\{i\}\\leq\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{1,m,k\}\\\}c\_\{\\mathrm\{H\}\}\+\\mathbf\{1\}\\\{U\_\{i\}\>\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{1,m,k\}\\\}\\Bigl\(c\_\{\\mathrm\{AI\}\}\+\\mathbf\{1\}\\\{V\_\{i\}\\leq\\hat\{\\eta\}^\{\\mathrm\{esc\}\}\_\{1,m,k\}\(R\_\{i\}\)\\\}c\_\{\\mathrm\{H\}\}\\Bigr\)\. Conditional on the realized pilot, its plug\-in\-channel and true\-channel expected costs are, respectively,

c^1,m,kitem:=𝔼1,f^π^m,k​\[Ciitem​\(η^1,m,k\)∣𝒟m\]=η^1,m,kH​cH\+\(1−η^1,m,kH\)​\(cAI\+cH​∑r∈ℛg^1​\(r\)​η^1,m,kesc​\(r\)\),\\displaystyle\\widehat\{c\}^\{\\mathrm\{item\}\}\_\{1,m,k\}:=\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1,\\hat\{f\}\}\[C\_\{i\}^\{\\mathrm\{item\}\}\(\\hat\{\\eta\}\_\{1,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]=\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{1,m,k\}c\_\{\\mathrm\{H\}\}\+\(1\-\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{1,m,k\}\)\\left\(c\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}\\sum\_\{r\\in\\mathcal\{R\}\}\\hat\{g\}\_\{1\}\(r\)\\hat\{\\eta\}^\{\\mathrm\{esc\}\}\_\{1,m,k\}\(r\)\\right\),\(166\)c1,m,kitem:=𝔼1π^m,k​\[Ciitem​\(η^1,m,k\)∣𝒟m\]=η^1,m,kH​cH\+\(1−η^1,m,kH\)​\(cAI\+cH​∑r∈ℛg1​\(r\)​η^1,m,kesc​\(r\)\)\.\\displaystyle c^\{\\mathrm\{item\}\}\_\{1,m,k\}:=\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[C\_\{i\}^\{\\mathrm\{item\}\}\(\\hat\{\\eta\}\_\{1,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]=\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{1,m,k\}c\_\{\\mathrm\{H\}\}\+\(1\-\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{1,m,k\}\)\\left\(c\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}\\sum\_\{r\\in\\mathcal\{R\}\}g\_\{1\}\(r\)\\hat\{\\eta\}^\{\\mathrm\{esc\}\}\_\{1,m,k\}\(r\)\\right\)\.The rounded\-block calculation gives

c^1,m,kitem≤Γ^1​\(T^1,m,kpilot​\(N¯^main,m,k\),N¯^main,m,k\)\+O⁡\(1\)N^1,m,kplan\.\\widehat\{c\}^\{\\mathrm\{item\}\}\_\{1,m,k\}\\leq\\frac\{\\hat\{\\Gamma\}\_\{1\}\(\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{1,m,k\}\(\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\),\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\)\+O\(1\)\}\{\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{1,m,k\}\}\.Moreover, \([166](https://arxiv.org/html/2609.28859#S13.E166)\) makes the channel perturbation explicit:

\|c1,m,kitem−c^1,m,kitem\|\\displaystyle\\left\|c^\{\\mathrm\{item\}\}\_\{1,m,k\}\-\\widehat\{c\}^\{\\mathrm\{item\}\}\_\{1,m,k\}\\right\|=\(1−η^1,m,kH\)​cH​\|∑r∈ℛ\(g1​\(r\)−g^1​\(r\)\)​η^1,m,kesc​\(r\)\|\\displaystyle=\(1\-\\hat\{\\eta\}^\{\\mathrm\{H\}\}\_\{1,m,k\}\)c\_\{\\mathrm\{H\}\}\\left\|\\sum\_\{r\\in\\mathcal\{R\}\}\\bigl\(g\_\{1\}\(r\)\-\\hat\{g\}\_\{1\}\(r\)\\bigr\)\\hat\{\\eta\}^\{\\mathrm\{esc\}\}\_\{1,m,k\}\(r\)\\right\|≤εmcost=cH​\|ℛ\|​rm\.\\displaystyle\\leq\\varepsilon\_\{m\}^\{\\mathrm\{cost\}\}=c\_\{\\mathrm\{H\}\}\|\\mathcal\{R\}\|r\_\{m\}\.Thusc1,m,kitem≤c^1,m,kitem\+εmcostc^\{\\mathrm\{item\}\}\_\{1,m,k\}\\leq\\widehat\{c\}^\{\\mathrm\{item\}\}\_\{1,m,k\}\+\\varepsilon\_\{m\}^\{\\mathrm\{cost\}\}\. Since membership in𝒞^\\hat\{\\mathcal\{C\}\}is predictable, by the tower property,

𝔼1π^m,k​\[∑i∈𝒞^Ciitem​\(η^Ji,m,k\)∣𝒟m\]=c1,m,kitem​𝔼1π^m,k​\[N^c∣𝒟m\]≤Γ^1​\(T^1,m,kpilot​\(N¯^main,m,k\),N¯^main,m,k\)\+O⁡\(1\)N^1,m,kplan​𝔼1π^m,k​\[N^c∣𝒟m\]\+εmcost​N^main,m,k,\\begin\{split\}&\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[\\sum\_\{i\\in\\hat\{\\mathcal\{C\}\}\}C\_\{i\}^\{\\mathrm\{item\}\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\\\ =&c^\{\\mathrm\{item\}\}\_\{1,m,k\}\\,\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[\\hat\{N\}\_\{c\}\\mid\\mathcal\{D\}\_\{m\}\]\\\\ \\leq&\\frac\{\\hat\{\\Gamma\}\_\{1\}\(\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{1,m,k\}\(\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\),\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\)\+O\(1\)\}\{\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{1,m,k\}\}\\,\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[\\hat\{N\}\_\{c\}\\mid\\mathcal\{D\}\_\{m\}\]\+\\varepsilon\_\{m\}^\{\\mathrm\{cost\}\}\\widehat\{N\}\_\{\\mathrm\{main\},m,k\},\\end\{split\}where the last term isO⁡\(Lk​rm\)O\(L\_\{k\}r\_\{m\}\)becauseN^main,m,k=O⁡\(Lk\)\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}=O\(L\_\{k\}\)\.

The predictable drift calculation in Cost 2 of the proof of Lemma[10\.5](https://arxiv.org/html/2609.28859#S10.Thmtheorem5)applies withSiS\_\{i\}replaced byS^i\\hat\{S\}\_\{i\}\. The complementary items have nonnegative conditional drift, and the completed continuation satisfiesS^τ^main,m,k≤ak\+ωm,k\+2​Bℓ\\hat\{S\}\_\{\\hat\{\\tau\}\_\{\\mathrm\{main\},m,k\}\}\\leq a\_\{k\}\+\\omega\_\{m,k\}\+2B\_\{\\ell\}\. Hence

𝔼1π^m,k​\[Z^iitem​\(η^1,m,k\)∣𝒟m\]​𝔼1π^m,k​\[N^c∣𝒟m\]≤ak\+ωm,k\+2​Bℓ\.\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{1,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\,\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[\\hat\{N\}\_\{c\}\\mid\\mathcal\{D\}\_\{m\}\]\\leq a\_\{k\}\+\\omega\_\{m,k\}\+2B\_\{\\ell\}\.Combining the last two displays withN^1,m,kplan​𝔼1π^m,k​\[Z^iitem​\(η^1,m,k\)∣𝒟m\]≥ak\+Δk\\widehat\{N\}^\{\\mathrm\{plan\}\}\_\{1,m,k\}\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[\\hat\{Z\}^\{\\mathrm\{item\}\}\_\{i\}\(\\hat\{\\eta\}\_\{1,m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]\\geq a\_\{k\}\+\\Delta\_\{k\}from \([164](https://arxiv.org/html/2609.28859#S13.E164)\), analogously to the end of Cost 2 in that proof,

𝔼1π^m,k​\[∑i∈𝒞^Ciitem​\(η^Ji,m,k\)∣𝒟m\]\\displaystyle\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[\\sum\_\{i\\in\\hat\{\\mathcal\{C\}\}\}C\_\{i\}^\{\\mathrm\{item\}\}\(\\hat\{\\eta\}\_\{J\_\{i\},m,k\}\)\\mid\\mathcal\{D\}\_\{m\}\]≤\(Γ^1​\(T^1,m,kpilot​\(N¯^main,m,k\),N¯^main,m,k\)\+O⁡\(1\)\)​ak\+ωm,k\+2​Bℓak\+Δk\+O⁡\(Lk​rm\)\\displaystyle\\leq\(\\hat\{\\Gamma\}\_\{1\}\(\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{1,m,k\}\(\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\),\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\)\+O\(1\)\)\\frac\{a\_\{k\}\+\\omega\_\{m,k\}\+2B\_\{\\ell\}\}\{a\_\{k\}\+\\Delta\_\{k\}\}\+O\(L\_\{k\}r\_\{m\}\)≤Γ^1​\(T^1,m,kpilot​\(N¯^main,m,k\),N¯^main,m,k\)\+O⁡\(1\)\+O⁡\(Lk​rm\),\\displaystyle\\leq\\hat\{\\Gamma\}\_\{1\}\(\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{1,m,k\}\(\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\),\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\)\+O\(1\)\+O\(L\_\{k\}r\_\{m\}\),becauseΔm,keff=Δk−ωm,k≥4​Bℓ\\Delta^\{\\mathrm\{eff\}\}\_\{m,k\}=\\Delta\_\{k\}\-\\omega\_\{m,k\}\\geq 4B\_\{\\ell\}\.

Combining Costs 1–2,

𝔼1π^m,k​\[C^main∣𝒟m\]≤Γ^1​\(T^1,m,kpilot​\(N¯^main,m,k\),N¯^main,m,k\)\+C​Lk​rm\+C⁡\(zk\+1\),\\begin\{split\}\\mathbb\{E\}^\{\\hat\{\\pi\}\_\{m,k\}\}\_\{1\}\[\\widehat\{C\}^\{\\mathrm\{main\}\}\\mid\\mathcal\{D\}\_\{m\}\]\\leq\\hat\{\\Gamma\}\_\{1\}\(\\widehat\{T\}^\{\\mathrm\{pilot\}\}\_\{1,m,k\}\(\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\),\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\)\+CL\_\{k\}r\_\{m\}\+C\(z\_\{k\}\+1\),\\end\{split\}whereO⁡\(1\)O\(1\)is absorbed intoC⁡\(zk\+1\)C\(z\_\{k\}\+1\)\.

## 14Additional Materials for Section[6\.3](https://arxiv.org/html/2609.28859#S6.SS3)

### 14\.1Proof of Corollary[6\.5](https://arxiv.org/html/2609.28859#S6.Thmtheorem5)

###### Proof 14\.1

Proof of Corollary[6\.5](https://arxiv.org/html/2609.28859#S6.Thmtheorem5)To prove Corollary[6\.5](https://arxiv.org/html/2609.28859#S6.Thmtheorem5), we show that there exist primitive constantsc\>0c\>0andC1,⋯,C5,C6,C7<∞C\_\{1\},\\cdots,C\_\{5\},C\_\{6\},C\_\{7\}<\\inftysuch that for all sufficiently largekk,

maxh⁡𝔼hπ^mpilot,k,k​\[Cmpilot,kpilot\+Cπ^mpilot,k,k\]LBk≤1\+C1​ΔkLk\+C2​rmpilot,k\+C3​zk\+1Δk\+C4​exp⁡\{−c​\(Δk\)2Lk\}\+C5​log⁡LkLk\+C6​mpilot,kLk\+C7​mpilot,k−2\.\\begin\{split\}&\\,\\frac\{\\max\_\{h\}\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\[C\_\{m\_\{\\mathrm\{pilot\},k\}\}^\{\\mathrm\{pilot\}\}\+C^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\]\}\{\\mathrm\{LB\}\_\{k\}\}\\leq 1\+C\_\{1\}\\frac\{\\Delta\_\{k\}\}\{L\_\{k\}\}\+C\_\{2\}r\_\{m\_\{\\mathrm\{pilot\},k\}\}\+C\_\{3\}\\frac\{z\_\{k\}\+1\}\{\\Delta\_\{k\}\}\\\\ &\\qquad\\qquad\+C\_\{4\}\\exp\\left\\\{\-c\\frac\{\(\\Delta\_\{k\}\)^\{2\}\}\{L\_\{k\}\}\\right\\\}\+C\_\{5\}\\frac\{\\log L\_\{k\}\}\{L\_\{k\}\}\+C\_\{6\}\\frac\{m\_\{\\mathrm\{pilot\},k\}\}\{L\_\{k\}\}\+C\_\{7\}m^\{\-2\}\_\{\\mathrm\{pilot\},k\}\.\\end\{split\}\(167\)Then the argument follows by plugging in the values ofΔk\\Delta\_\{k\}\.

*Step 1: the expected pilot charge\.*To prove \([167](https://arxiv.org/html/2609.28859#S14.E167)\), we first show that𝔼h​\[Mtot\]=Θ⁡\(mpilot,k\)\\mathbb\{E\}\_\{h\}\[M\_\{\\mathrm\{tot\}\}\]=\\Theta\(m\_\{\\mathrm\{pilot\},k\}\)\. By construction,Mtot=min\{n≥1:minx∈\{0,1\}∑i=1n𝟏\{Xipilot=x\}≥m\}M\_\{\\mathrm\{tot\}\}=\\min\\\{n\\geq 1:\\min\_\{x\\in\\\{0,1\\\}\}\\sum^\{n\}\_\{i=1\}\\mathbf\{1\}\\\{X^\{\\mathrm\{pilot\}\}\_\{i\}=x\\\}\\geq m\\\}\. ThusMtot≥2​m=2​mpilot,kM\_\{\\mathrm\{tot\}\}\\geq 2m=2m\_\{\\mathrm\{pilot\},k\}almost surely\. To calculate the expectation, letBh∼Bin⁡\(2​mpilot,k,ph\)B\_\{h\}\\sim\\operatorname\{Bin\}\(2m\_\{\\mathrm\{pilot\},k\},p\_\{h\}\)be the number of label\-one observations among the first2​mpilot,k2m\_\{\\mathrm\{pilot\},k\}pilot items\. IfBh<mpilot,kB\_\{h\}<m\_\{\\mathrm\{pilot\},k\}, then the label\-zero quota has already been reached andmpilot,k−Bhm\_\{\\mathrm\{pilot\},k\}\-B\_\{h\}additional label\-one observations are required, with conditional expected waiting time\(mpilot,k−Bh\)/ph\(m\_\{\\mathrm\{pilot\},k\}\-B\_\{h\}\)/p\_\{h\}\. Similarly, ifBh\>mpilot,kB\_\{h\}\>m\_\{\\mathrm\{pilot\},k\}, thenBh−mpilot,kB\_\{h\}\-m\_\{\\mathrm\{pilot\},k\}additional label\-zero observations are required, with conditional expected waiting time\(Bh−mpilot,k\)/\(1−ph\)\(B\_\{h\}\-m\_\{\\mathrm\{pilot\},k\}\)/\(1\-p\_\{h\}\)\. Hence

𝔼h​\[Mtot\]=2​mpilot,k\+𝔼h​\[\(mpilot,k−Bh\)\+\]ph\+𝔼h​\[\(Bh−mpilot,k\)\+\]1−ph\.\\mathbb\{E\}\_\{h\}\[M\_\{\\mathrm\{tot\}\}\]=2m\_\{\\mathrm\{pilot\},k\}\+\\frac\{\\mathbb\{E\}\_\{h\}\[\(m\_\{\\mathrm\{pilot\},k\}\-B\_\{h\}\)\_\{\+\}\]\}\{p\_\{h\}\}\+\\frac\{\\mathbb\{E\}\_\{h\}\[\(B\_\{h\}\-m\_\{\\mathrm\{pilot\},k\}\)\_\{\+\}\]\}\{1\-p\_\{h\}\}\.\(168\)
Suppose first thatph<1/2p\_\{h\}<1/2\. Since𝔼h​\[\(Bh−mpilot,k\)\+\]−𝔼h​\[\(mpilot,k−Bh\)\+\]=𝔼h​\[Bh−mpilot,k\]=mpilot,k​\(2​ph−1\)\\mathbb\{E\}\_\{h\}\[\(B\_\{h\}\-m\_\{\\mathrm\{pilot\},k\}\)\_\{\+\}\]\-\\mathbb\{E\}\_\{h\}\[\(m\_\{\\mathrm\{pilot\},k\}\-B\_\{h\}\)\_\{\+\}\]=\\mathbb\{E\}\_\{h\}\[B\_\{h\}\-m\_\{\\mathrm\{pilot\},k\}\]=m\_\{\\mathrm\{pilot\},k\}\(2p\_\{h\}\-1\), equation \([168](https://arxiv.org/html/2609.28859#S14.E168)\) can be rewritten as

𝔼h​\[Mtot\]=mpilot,kph\+𝔼h​\[\(Bh−mpilot,k\)\+\]ph​\(1−ph\)\.\\mathbb\{E\}\_\{h\}\[M\_\{\\mathrm\{tot\}\}\]=\\frac\{m\_\{\\mathrm\{pilot\},k\}\}\{p\_\{h\}\}\+\\frac\{\\mathbb\{E\}\_\{h\}\[\(B\_\{h\}\-m\_\{\\mathrm\{pilot\},k\}\)\_\{\+\}\]\}\{p\_\{h\}\(1\-p\_\{h\}\)\}\.A binomial Chernoff bound givesℙh​\(Bh≥mpilot,k\)≤e−ch​mpilot,k\\mathbb\{P\}\_\{h\}\(B\_\{h\}\\geq m\_\{\\mathrm\{pilot\},k\}\)\\leq e^\{\-c\_\{h\}m\_\{\\mathrm\{pilot\},k\}\}for somech\>0c\_\{h\}\>0\. Since\(Bh−mpilot,k\)\+≤mpilot,k\(B\_\{h\}\-m\_\{\\mathrm\{pilot\},k\}\)\_\{\+\}\\leq m\_\{\\mathrm\{pilot\},k\}, it follows that

𝔼h​\[Mtot\]=mpilot,kph\+O⁡\(mpilot,k​e−ch​mpilot,k\)\.\\mathbb\{E\}\_\{h\}\[M\_\{\\mathrm\{tot\}\}\]=\\frac\{m\_\{\\mathrm\{pilot\},k\}\}\{p\_\{h\}\}\+O\\\!\\left\(m\_\{\\mathrm\{pilot\},k\}e^\{\-c\_\{h\}m\_\{\\mathrm\{pilot\},k\}\}\\right\)\.The caseph\>1/2p\_\{h\}\>1/2is symmetric\. Thus, wheneverph≠1/2p\_\{h\}\\neq 1/2,

𝔼h​\[Mtot\]=mpilot,kmin⁡\{ph,1−ph\}\+O⁡\(mpilot,k​e−ch​mpilot,k\)\.\\mathbb\{E\}\_\{h\}\[M\_\{\\mathrm\{tot\}\}\]=\\frac\{m\_\{\\mathrm\{pilot\},k\}\}\{\\min\\\{p\_\{h\},1\-p\_\{h\}\\\}\}\+O\\\!\\left\(m\_\{\\mathrm\{pilot\},k\}e^\{\-c\_\{h\}m\_\{\\mathrm\{pilot\},k\}\}\\right\)\.\(169\)Ifph=1/2p\_\{h\}=1/2, a direct calculation using the symmetry ofBh∼Bin⁡\(2​mpilot,k,1/2\)B\_\{h\}\\sim\\operatorname\{Bin\}\(2m\_\{\\mathrm\{pilot\},k\},1/2\)gives

𝔼h\[Mtot\]=2mpilot,k\+2mpilot,k\(2​mpilot,kmpilot,k\)4mpilot,k=2mpilot,k\+2mpilot,kπ\+O\(mpilot,k−1/2\)\.\\mathbb\{E\}\_\{h\}\[M\_\{\\mathrm\{tot\}\}\]=2m\_\{\\mathrm\{pilot\},k\}\+2m\_\{\\mathrm\{pilot\},k\}\\frac\{\\binom\{2m\_\{\\mathrm\{pilot\},k\}\}\{m\_\{\\mathrm\{pilot\},k\}\}\}\{4^\{m\_\{\\mathrm\{pilot\},k\}\}\}=2m\_\{\\mathrm\{pilot\},k\}\+2\\sqrt\{\\frac\{m\_\{\\mathrm\{pilot\},k\}\}\{\\pi\}\}\+O\(m\_\{\\mathrm\{pilot\},k\}^\{\-1/2\}\)\.Becausep0<p1p\_\{0\}<p\_\{1\}, we haveminh∈\{0,1\}⁡min⁡\{ph,1−ph\}=min⁡\{p0,1−p1\}=:p⋆\\min\_\{h\\in\\\{0,1\\\}\}\\min\\\{p\_\{h\},1\-p\_\{h\}\\\}=\\min\\\{p\_\{0\},1\-p\_\{1\}\\\}=:p\_\{\\star\}, and the preceding two displays combine into

maxh∈\{0,1\}⁡𝔼h​\[Mtot\]=mpilot,kp⋆​\{1\+o⁡\(1\)\},2​mpilot,k≤maxh∈\{0,1\}⁡𝔼h​\[Mtot\]=O⁡\(mpilot,k\)\.\\max\_\{h\\in\\\{0,1\\\}\}\\mathbb\{E\}\_\{h\}\[M\_\{\\mathrm\{tot\}\}\]=\\frac\{m\_\{\\mathrm\{pilot\},k\}\}\{p\_\{\\star\}\}\\\{1\+o\(1\)\\\},\\qquad 2m\_\{\\mathrm\{pilot\},k\}\\leq\\max\_\{h\\in\\\{0,1\\\}\}\\mathbb\{E\}\_\{h\}\[M\_\{\\mathrm\{tot\}\}\]=O\(m\_\{\\mathrm\{pilot\},k\}\)\.\(170\)Since every pilot item is acquired and labeled by both the AI and the human,Cmpilot,kpilot=\(cdata\+cAI\+cH\)​MtotC^\{\\mathrm\{pilot\}\}\_\{m\_\{\\mathrm\{pilot\},k\}\}=\(c\_\{\\mathrm\{data\}\}\+c\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}\)M\_\{\\mathrm\{tot\}\}, and therefore

maxh∈\{0,1\}⁡𝔼h​\[Cmpilot,kpilot\]=cdata\+cAI\+cHp⋆​mpilot,k​\{1\+o⁡\(1\)\}=Θ⁡\(mpilot,k\)\.\\max\_\{h\\in\\\{0,1\\\}\}\\mathbb\{E\}\_\{h\}\[C^\{\\mathrm\{pilot\}\}\_\{m\_\{\\mathrm\{pilot\},k\}\}\]=\\frac\{c\_\{\\mathrm\{data\}\}\+c\_\{\\mathrm\{AI\}\}\+c\_\{\\mathrm\{H\}\}\}\{p\_\{\\star\}\}\\,m\_\{\\mathrm\{pilot\},k\}\\\{1\+o\(1\)\\\}=\\Theta\(m\_\{\\mathrm\{pilot\},k\}\)\.\(171\)
*Step 2: the unconditional main\-stage cost\.*Letℰk:=ℰpilot,mpilot,k\\mathcal\{E\}\_\{k\}:=\\mathcal\{E\}\_\{\\mathrm\{pilot\},m\_\{\\mathrm\{pilot\},k\}\}\. By the pilot concentration result,ℙh​\(ℰkc\)≤mpilot,k−2\\mathbb\{P\}\_\{h\}\(\\mathcal\{E\}\_\{k\}^\{c\}\)\\leq m\_\{\\mathrm\{pilot\},k\}^\{\-2\}forh∈\{0,1\}h\\in\\\{0,1\\\}\. Onℰk\\mathcal\{E\}\_\{k\}, when \([57](https://arxiv.org/html/2609.28859#S6.E57)\) holds, we have thatΔm,keff=Δk−ωm,k=Δk​\(1−o⁡\(1\)\)\\Delta^\{\\mathrm\{eff\}\}\_\{m,k\}=\\Delta\_\{k\}\-\\omega\_\{m,k\}=\\Delta\_\{k\}\(1\-o\(1\)\)\(see Lemma[13\.8](https://arxiv.org/html/2609.28859#S13.Thmtheorem8)\)\. Additionally, Theorem[6\.3](https://arxiv.org/html/2609.28859#S6.Thmtheorem3)bounds the expected cost conditional on the pilot data𝒟mpilot,k\\mathcal\{D\}\_\{m\_\{\\mathrm\{pilot\},k\}\}; averaging that bound over pilot realizations inℰk\\mathcal\{E\}\_\{k\}gives, for primitive constantsC<∞C<\\inftyandc\>0c\>0,

𝔼hπ^mpilot,k,k​\[Cπ^mpilot,k,k∣ℰk\]≤LBk​\(1\+C​Ξk\),\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\\bigl\[C^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\\mid\\mathcal\{E\}\_\{k\}\\bigr\]\\leq\\mathrm\{LB\}\_\{k\}\\left\(1\+C\\,\\Xi\_\{k\}\\right\),\(172\)where

Ξk:=ΔkLk\+rmpilot,k\+zk\+1Δk\+exp⁡\{−c​\(Δk\)2Lk\}\+log⁡LkLk\.\\Xi\_\{k\}:=\\frac\{\\Delta\_\{k\}\}\{L\_\{k\}\}\+r\_\{m\_\{\\mathrm\{pilot\},k\}\}\+\\frac\{z\_\{k\}\+1\}\{\\Delta\_\{k\}\}\+\\exp\\left\\\{\-c\\frac\{\(\\Delta\_\{k\}\)^\{2\}\}\{L\_\{k\}\}\\right\\\}\+\\frac\{\\log L\_\{k\}\}\{L\_\{k\}\}\.On the complementℰkc\\mathcal\{E\}\_\{k\}^\{c\}, in the worst case we may have to query both the AI and human on allN^main,mpilot,k,k\\widehat\{N\}\_\{\\mathrm\{main\},m\_\{\\mathrm\{pilot\},k\},k\}items\. We now show thatN^main,mpilot,k,k=O⁡\(Lk\)\\widehat\{N\}\_\{\\mathrm\{main\},m\_\{\\mathrm\{pilot\},k\},k\}=O\(L\_\{k\}\)holds uniformly over all pilot realizations\. LetJmin:=minh⁡JX\(h\)\>0J\_\{\\min\}:=\\min\_\{h\}J\_\{X\}^\{\(h\)\}\>0and recall thatεmdr=3​\|ℛ\|​Bℓ​rm\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}=3\|\\mathcal\{R\}\|B\_\{\\ell\}r\_\{m\}is deterministic withεmdr→0\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}\\to 0, soεmdr<Jmin/2\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}<J\_\{\\min\}/2for all largekk\. Consider the human\-only candidate

NH,k:=max⁡\{Nfixed,H​\(α2,k,β2,k\),⌈maxh∈\{0,1\}⁡Th,k\+εmdrJX\(h\)−εmdr⌉\}\.N\_\{H,k\}:=\\max\\left\\\{N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\),\\left\\lceil\\max\_\{h\\in\\\{0,1\\\}\}\\frac\{T\_\{h,k\}\+\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}\}\{J\_\{X\}^\{\(h\)\}\-\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}\}\\right\\rceil\\right\\\}\.For everyhh,NH,k​JX\(h\)≥Th,k\+\(NH,k\+1\)​εmdrN\_\{H,k\}J\_\{X\}^\{\(h\)\}\\geq T\_\{h,k\}\+\(N\_\{H,k\}\+1\)\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}, and this constraint involves only the known information rateJX\(h\)J\_\{X\}^\{\(h\)\}and the deterministic guardεmdr\\varepsilon\_\{m\}^\{\\mathrm\{dr\}\}, so the all\-human plan\(nH,nAI,nesc\)=\(NH,k,0,0\)\(n\_\{\\mathrm\{H\}\},n\_\{\\mathrm\{AI\}\},n\_\{\\mathrm\{esc\}\}\)=\(N\_\{H,k\},0,0\)is feasible for the guarded plug\-in program atN=NH,kN=N\_\{H,k\}regardless of the realized pilot; in particular the guarded outer problem \([55](https://arxiv.org/html/2609.28859#S6.E55)\) is feasible for all largekk\. By optimality ofN¯^main,m,k\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\},

cdata​N¯^main,m,k≤F^m,kpilot​\(N¯^main,m,k\)≤F^m,kpilot​\(NH,k\)≤\(cdata\+cH\)​NH,k\.c\_\{\\mathrm\{data\}\}\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\\leq\\widehat\{F\}^\{\\mathrm\{pilot\}\}\_\{m,k\}\\bigl\(\\widehat\{\\overline\{N\}\}\_\{\\mathrm\{main\},m,k\}\\bigr\)\\leq\\widehat\{F\}^\{\\mathrm\{pilot\}\}\_\{m,k\}\(N\_\{H,k\}\)\\leq\(c\_\{\\mathrm\{data\}\}\+c\_\{\\mathrm\{H\}\}\)N\_\{H,k\}\.SinceTh,k=O⁡\(Lk\)T\_\{h,k\}=O\(L\_\{k\}\)andNfixed,H​\(α2,k,β2,k\)=O⁡\(Lk\)N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)=O\(L\_\{k\}\), we haveNH,k=O⁡\(Lk\)N\_\{H,k\}=O\(L\_\{k\}\)and henceN^main,m,k=O⁡\(Lk\)\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}=O\(L\_\{k\}\); if the guarded outer problem is infeasible, the safe default givesN^main,m,k=Nfixed,H​\(α2,k,β2,k\)=O⁡\(Lk\)\\widehat\{N\}\_\{\\mathrm\{main\},m,k\}=N\_\{\\mathrm\{fixed\},\\mathrm\{H\}\}\(\\alpha\_\{2,k\},\\beta\_\{2,k\}\)=O\(L\_\{k\}\)directly\.

It thus follows from the previous argument that onℰkc\\mathcal\{E\}^\{c\}\_\{k\}the worst\-case cost is at mostO⁡\(Lk\)O\(L\_\{k\}\)\. Combining the two events,

𝔼hπ^mpilot,k,k​\[Cπ^mpilot,k,k\]=ℙh​\(ℰk\)​𝔼hπ^mpilot,k,k​\[Cπ^mpilot,k,k∣ℰk\]\+ℙh​\(ℰkc\)​𝔼hπ^mpilot,k,k​\[Cπ^mpilot,k,k∣ℰkc\]≤LBk​\(1\+C​Ξk\)\+O⁡\(Lk​mpilot,k−2\)\.\\begin\{split\}\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\\bigl\[C^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\\bigr\]&=\\mathbb\{P\}\_\{h\}\(\\mathcal\{E\}\_\{k\}\)\\,\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\\bigl\[C^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\\mid\\mathcal\{E\}\_\{k\}\\bigr\]\+\\mathbb\{P\}\_\{h\}\(\\mathcal\{E\}\_\{k\}^\{c\}\)\\,\\mathbb\{E\}\_\{h\}^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\\bigl\[C^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\\mid\\mathcal\{E\}\_\{k\}^\{c\}\\bigr\]\\\\ &\\leq\\mathrm\{LB\}\_\{k\}\\left\(1\+C\\,\\Xi\_\{k\}\\right\)\+O\\\!\\left\(L\_\{k\}m\_\{\\mathrm\{pilot\},k\}^\{\-2\}\\right\)\.\\end\{split\}\(173\)
Adding \([173](https://arxiv.org/html/2609.28859#S14.E173)\) to \([171](https://arxiv.org/html/2609.28859#S14.E171)\), and dividing byLBk=Θ⁡\(Lk\)\\mathrm\{LB\}\_\{k\}=\\Theta\(L\_\{k\}\)\(Theorem[5\.3](https://arxiv.org/html/2609.28859#S5.Thmtheorem3)\(i\)\) yields \([167](https://arxiv.org/html/2609.28859#S14.E167)\)\.

*Step 3: Plug inΔk=Lk2/3\\Delta\_\{k\}=L\_\{k\}^\{2/3\}\.*For the choice in Corollary[6\.4](https://arxiv.org/html/2609.28859#S6.Thmtheorem4),Δk=Lk2/3\\Delta\_\{k\}=L\_\{k\}^\{2/3\}andmpilot,k=⌈Lk2/3​\(log⁡Lk\)2⌉m\_\{\\mathrm\{pilot\},k\}=\\left\\lceil L\_\{k\}^\{2/3\}\(\\log L\_\{k\}\)^\{2\}\\right\\rceil\. In this case,

rmpilot,k=O\(Lk−1/3\(logLk\)−1/2\),Lkrmpilot,k=o\(Δk\),r\_\{m\_\{\\mathrm\{pilot\},k\}\}=O\\\!\\left\(L\_\{k\}^\{\-1/3\}\(\\log L\_\{k\}\)^\{\-1/2\}\\right\),\\qquad L\_\{k\}r\_\{m\_\{\\mathrm\{pilot\},k\}\}=o\(\\Delta\_\{k\}\),and, forzk=1z\_\{k\}=1, substituting into \([167](https://arxiv.org/html/2609.28859#S14.E167)\) therefore gives

maxh⁡𝔼h​\[Cmpilot,kpilot\+Cπ^mpilot,k,k\]LBk≤1\+O\(Lk−1/3\(logLk\)2\)=1\+O~\(Lk−1/3\)\.\\frac\{\\max\_\{h\}\\mathbb\{E\}\_\{h\}\[C\_\{m\_\{\\mathrm\{pilot\},k\}\}^\{\\mathrm\{pilot\}\}\+C^\{\\hat\{\\pi\}\_\{m\_\{\\mathrm\{pilot\},k\},k\}\}\]\}\{\\mathrm\{LB\}\_\{k\}\}\\leq 1\+O\\\!\\left\(L\_\{k\}^\{\-1/3\}\(\\log L\_\{k\}\)^\{2\}\\right\)=1\+\\widetilde\{O\}\(L\_\{k\}^\{\-1/3\}\)\.