LODESTAR: Trustworthy Entropy Is Navigated, Not Merely Measured -- Reinforced Polarizer Keeps a Frozen LLM from Being Confidently Misled by the Wrong Evidence

# Trustworthy Entropy Is Navigated,Not Merely Measured Reinforced Polarizer Keeps a Frozen LLMfrom Being Confidently Misled by the Wrong Evidence
Source: [https://arxiv.org/html/2608.11922](https://arxiv.org/html/2608.11922)
Che\-Cheng WuHung\-Chun HsuThanks:Project lead; proposed the methodology, conducted the experiments, and wrote the manuscript\.R10946017@citi\.sinica\.edu\.tw$ˆ‡$Corresponding author:cjwang@citi\.sinica\.edu\.tw\[3pt\]Li\-Yang ChangChuan\-Ju Wang\[7pt\] Research Center for Information Technology InnovationAcademia SinicaTaiwan

###### Abstract

Predictive\-distribution entropy makes a strong selection rule in existing retrieval\-augmented question answering work\. We first verify it on our own benchmarks\. Across five QA benchmarks, keeping the candidate answer that a*frozen*respondent LLM produces with the lowest answer\-token entropy lifts mean answerF1F\_\{1\}from0\.47690\.4769to0\.51480\.5148over the retriever’s top\-ranked passage without any gold answer\. Yet in this paper we show that this lowest\-entropy selection rule, which prior entropy\-based selectors adopt, fails in a specific and consequential way\. A misleading passage makes the respondent*confidently wrong*, driving its entropy down precisely where the signal looks most trustworthy\. Existing work either scopes its claims away from this failure mode or documents it without repairing it; we instead show that the failure comes from the passage that the respondent reads, and the context that passage is read in is an input we can intervene on\. We therefore introduce![[Uncaptioned image]](https://arxiv.org/html/2608.11922v1/figures/assets/noto_2734.png)Lodestar, to the best of our knowledge the first method to score a text intervention by the uncertainty it induces in a third\-party frozen respondent, compared across one question’s candidates, which extends what the respondent reads instead of retraining what it is\.Lodestaruses reinforcement learning to train, once and offline, a*polarizer*—a short fixed natural\-language string inserted into the respondent’s prompt and never into its weights\. Its training labels are built offline from gold answers and two LLM judges; inference reads neither\. We then evaluateLodestarcomprehensively, running every competing selector under the same*frozen*respondent and the same retrieved candidate pool\. On5,0085\{,\}008questions from those five benchmarks,Lodestarattains the highest meanF1F\_\{1\}of any inference\-ready selector, the highest exact match \(0\.41360\.4136\), and the highest GPT\-4o judge score of the frozen\-respondent configurations judged \(0\.64350\.6435\); its three\-seed mean wins all7070method\-by\-dataset cells onF1F\_\{1\}against fourteen published configurations, and it takes the highest macro exact match of any of them,0\.41360\.4136against the best baseline’s0\.40390\.4039, while remaining paired\-significant onF1F\_\{1\}against every one\. It raises the same five\-benchmarkF1F\_\{1\}mean from0\.51480\.5148to0\.53390\.5339, a further\+3\.71%\+3\.71\\%\. The gain holds on both sides of the domain split:0\.46430\.4643to0\.47890\.4789in\-domain on NQ\-Open, and0\.52740\.5274to0\.54760\.5476averaged over SQuAD, TriviaQA, EntityQuestions and WebQuestions\. We further ablate the polarizer string and find that it is what makes the respondent read a misleading passage less often,26\.0%26\.0\\%of the time against30\.3%30\.3\\%\.

## 1Introduction

Retrieval\-augmented generation \(RAG\) answers a question from a retrieved corpus rather than from what the model memorized in training\. In open\-domain question answering the answer therefore turns on which passage the model was handed\. A typical workflow has two steps: \(1\) a retriever returns the corpus passages most similar to the question, and \(2\) a respondent LLM, typically frozen, reads those candidates alongside the question and produces the answer\. The trouble is that not all retrieved passages support the correct answer\. Some are topically relevant but factually misleading, and nothing at inference time marks which is which\. A misleading passage is worse than a useless one, because it supplies a false fact and the respondent’s answer then disagrees with the gold answer\. In Table[1](https://arxiv.org/html/2608.11922#S1.T1)we report how often that happens, on about a thousand questions from each of five common QA datasets, where, depending on the retrieval system, between20\.3%20\.3\\%and35\.0%35\.0\\%of the retrieved candidates mislead the frozen respondent\. What is even worse is that using a stronger retriever or adding a reranker drives that misleading\-passage rate up rather than down\. As a result, deciding which candidate’s answer to trust without gold answers is an active research question in the RAG literature\.

Table 1:Judged\-misleading rate of the retrieved candidate set\.Percentage of a question’s top\-1010passages the frozen judge \(Section[3](https://arxiv.org/html/2608.11922#S3)\) labelsmisleading;macroaverages the five columns\.Entropy\-based selection is an effective way to choose among the candidateswithoutgold answers, taking the entropy of the answer tokens produced under each candidate as that candidate’s score\. The rule is then to keep the answer the respondent produces with the least uncertainty \(i\.e\., the lowest entropy\)\. Across the same five QA datasets, even the simplest entropy signal lifts mean answerF1F\_\{1\}from0\.47690\.4769to0\.51480\.5148over simply taking the retriever’s top\-ranked passage\. Prior work computes this confidence in several forms, all from the respondent’s own distribution: entropy over sampled generations clustered by meaning\([Farquhar et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib7)\), over the generation trajectory\([Moslonka et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib19);[Qiu et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib23)\), and over a short probe rollout\([Song et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib29)\)\. Yet the*confidently\-wrong problem*is where that effectiveness ends\. The passage this rule selects still yields an answer with zero exact match59\.6%59\.6\\%of the time\. That low uncertainty does not imply correctness is documented\([Taparia et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib33);[Ma et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib17)\), and so is the finding that stronger retrieval systematically lowers predictive entropy\([Soudani et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib30)\)\. Neither result says which of one question’s own candidates to trust; that is where the direction reverses, and within a single question a misleading passage lowers the respondent’s entropy \(Figure[1](https://arxiv.org/html/2608.11922#S1.F1)a\)\. The failure is more often sidestepped than met \(Section[2](https://arxiv.org/html/2608.11922#S2)\), leaving the*confidently\-wrong problem*unsolved\.

To proactively address the*confidently\-wrong problem*we introduceLodestar\(Learned Orientation of Directed Entropy, Steering Trustworthy Answer Retrieval\), which leaves the respondent frozen and learns instead one short natural\-language*polarizer*ψ⋆\\psi^\{\\star\}, inserted after the passageppand before the questionqq\. The prompt then reads\[p;ψ⋆;q\]\[\\,p\\,;\\,\\psi^\{\\star\}\\,;\\,q\\,\]\. Across the same five datasets,Lodestarfurther lifts mean answerF1F\_\{1\}from entropy selection’s0\.51480\.5148to0\.53390\.5339\(Section[5](https://arxiv.org/html/2608.11922#S5)\)\. Trained once and offline by reinforcement learning,ψ⋆\\psi^\{\\star\}is optimized against a reward that is not answer correctness but the*within\-question*separation of the frozen respondent’s entropy, raised on misleading passages and kept low on supporting ones \(Figure[1](https://arxiv.org/html/2608.11922#S1.F1)b\)\. We call the resulting signal*directed*entropy, because the polarizer steers the respondent’s entropy in a chosen direction rather than leaving it to be passively measured\.Lodestarruns at inference with no extra model, no sampling and no supervision; the only thing it keeps from training is the polarizerψ⋆\\psi^\{\\star\}, one fixed string, enough by itself to outperform every published method the paper tests \(Section[5](https://arxiv.org/html/2608.11922#S5)\)\. Our contributions can be summarized as follows:

1. 1\.We show that ranking a question’s candidates by entropy alone is unreliable, and repair it with one learned string\.Within a single question, the respondent’s entropy does not reliably separate misleading passages from supporting ones \(Section[3](https://arxiv.org/html/2608.11922#S3)\); one learned string is enough to make it separate them\. Ranking by entropy alone reads a misleading passage more often on average than drawing one of the ten at random,30\.3%30\.3\\%against the pool’s own28\.9%28\.9\\%\. One learned string puts the selection below that floor on all five benchmarks,26\.0%26\.0\\%macro\-averaged \(Tables[1](https://arxiv.org/html/2608.11922#S1.T1)and[3](https://arxiv.org/html/2608.11922#S4.T3)\)\.
2. 2\.We benchmark fourteen published methods re\-purposed as selectors;Lodestarleads them on every metric it is measured on\.Fourteen published configurations, each run on identical pools with an identical frozen respondent under the strongest setting its released code or paper supports \(Section[4](https://arxiv.org/html/2608.11922#S4); per\-row deviations in Appendix[J](https://arxiv.org/html/2608.11922#A10)\), spanning prompt\-search optimizers, uncertainty signals scored from the respondent’s own output, and trained rerankers that never see it\.Lodestarattains the highest mean answerF1F\_\{1\}\(0\.53390\.5339\) and exact match \(0\.41360\.4136\) of any of them, and the highest GPT\-4o judge score of the frozen\-respondent configurations judged \(0\.64350\.6435\); itsF1F\_\{1\}lead is paired\-significant against every configuration tested \(Appendix[D](https://arxiv.org/html/2608.11922#A4)\)\.
3. 3\.We map the polarizer over all nine train–inference pairs of three frozen respondents, Llama\-3\.1\-8B, Qwen2\.5\-7B and Qwen3\.5\-9B; every diagonal helps, though the diagonal is not always where it helps most\.The better entropy signal tracks the model family: first\-tokenH1H\_\{1\}on Llama, all\-token on both Qwen models\. What holds across all three is the polarizer’s effect\. Section[6](https://arxiv.org/html/2608.11922#S6)reports the cross\-respondent comparison\.

![Refer to caption](https://arxiv.org/html/2608.11922v1/fig_overview_tikz.png)Figure 1:Why minimum\-entropy selection is misled, and whatLodestarchanges\.\(a\) The misleading passage makes the frozen respondent become*confidently*wrong: the respondent’s entropy is lowest on that passage \(0\.00130\.0013, the minimum over all ten, against the supporting passage’s0\.00210\.0021\), and the lowest\-entropy rule keeps the answer that passage induces, which is wrong\. \(b\)Lodestarinserts one learned polarizerψ⋆\\psi^\{\\star\}between each passage and the question, optimized to raise the respondent’s entropy on misleading passages while leaving supporting ones near the entropy floor\. The misleading passage rises9\.5×9\.5\\timeson the unrounded probes, to0\.01210\.0121, while the supporting one rises only to0\.00390\.0039, so the same lowest\-entropy rule now selects the supporting passage and the answer is correct\. No gold answers are used at selection time\.
## 2Related Work

#### The confidently\-wrong problem\.

That low uncertainty does not imply correctness is documented in the literature\([Taparia et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib33);[Ma et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib17)\), and[Soudani et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib30)establish that stronger retrieval systematically lowers predictive entropy\. As we show in Table[1](https://arxiv.org/html/2608.11922#S1.T1), retrieving with bge\-m3 puts a higher average share of misleading passages in the top\-1010than retrieving with Contriever,28\.9%28\.9\\%against20\.3%20\.3\\%\. Moreover, on our five pools the respondent’s mean first\-token entropy is1\.221\.22nats111The unit of entropy taken with the natural logarithm\.on judged\-misleading candidates against1\.331\.33on the rest of the candidate set\. A selector that keeps the lowest\-entropy candidate is therefore drawn toward misleading passages and the incorrect answers they induce\. Yet what a selector needs is directional\. Reading a misleading passage should leave the respondent less certain, its answer\-token entropy the higher one\. Neither prior result meets this need\. Both describe how entropy behaves across questions, while a selector only compares the candidates of one question, a gapLodestarcloses by training its polarizer to make misleading passages carry the higher entropy inside each question \(Section[3](https://arxiv.org/html/2608.11922#S3)\)\.

#### Uncertainty estimation for generation\.

Predictive entropy and its refinements are the standard estimators for measuring how uncertain a language model is about its answer\. Semantic entropy\([Farquhar et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib7)\)clusters sampled generations by entailment before computing entropy; EigenScore\([Chen et al\. 2024a](https://arxiv.org/html/2608.11922#bib.bib4)\)and SeaKR\([Yao et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib39)\)read the dispersion \(e\.g\., covariance eigenvalues, Gram determinant\) of internal states across samples; EPR\([Moslonka et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib19)\)andCLeHe\([Qiu et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib23)\)score the generation trajectory, the step\-by\-step entropy of the answer as the respondent decodes it\. Three further signals reach this task from outside hallucination detection\. Min\-K% Prob\([Shi et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib28)\)and its vocabulary\-normalized successor Min\-K%\+\+\([Zhang et al\. 2025a](https://arxiv.org/html/2608.11922#bib.bib41)\)were introduced to test whether a text was part of a model’s training data\. A text the model has seen keeps even its least\-likely tokens probable\. Re\-purposed as selectors, they score the answer conditioned on each passage and keep the passage whose answer scores highest\. SPS\([Hu et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib11)\)scores a passage from its representation alone, without seeing the question\. None of these signals changes the input that produced it\. Yet with a frozen respondent the input is the only part of the system a selector is free to change;Lodestaruses exactly that freedom, learning one string that moves the respondent’s entropy instead of merely reading it\.

#### Entropy\-based answer selection\.

Uncertainty also governs the upstream decision of*whether*to retrieve at all\([Moskvoretskii et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib18)\); our question is which candidate’s answer to keep for the frozen respondent once retrieval has happened\. The closest prior method that does select is IGP\([Song et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib29)\)\. It ranks the retrieved candidatesp∈P⁡\(q\)p\\in P\(q\)by the information gainIG⁡\(p,q\)=H¯L​\(q\)−H¯L​\(q,p\)\\mathrm\{IG\}\(p,q\)=\\bar\{H\}\_\{L\}\(q\)\-\\bar\{H\}\_\{L\}\(q,p\)each delivers to a frozen respondent, withH¯L​\(q,p\)\\bar\{H\}\_\{L\}\(q,p\)the mean normalized entropy of theLLtokens that compose a short probe answer andH¯L​\(q\)\\bar\{H\}\_\{L\}\(q\)the same entropy with no passage in context\. BecauseH¯L​\(q\)\\bar\{H\}\_\{L\}\(q\)is constant once the question is fixed, it cancels within a question, and the rule reduces to a minimum\-entropy selector over the answersa⁡\(q,p\)a\(q,p\)the candidates induce:

a^IGP⏟selected answer=arg⁡maxa⁡\(q,p\),p∈P⁡\(q\)​IG⁡\(p,q\)⏟information gain=arg⁡mina⁡\(q,p\),p∈P⁡\(q\)​H¯L​\(q,p\)⏟measured entropy\.\\underbrace\{\\hat\{a\}^\{\\mathrm\{IGP\}\}\}\_\{\\mathclap\{\\text\{selected answer\}\}\}\\;=\\;\\arg\\max\_\{a\(q,p\),\\ p\\in P\(q\)\}\\underbrace\{\\mathrm\{IG\}\(p,q\)\}\_\{\\text\{information gain\}\}\\;=\\;\\arg\\min\_\{a\(q,p\),\\ p\\in P\(q\)\}\\underbrace\{\\bar\{H\}\_\{L\}\(q,p\)\}\_\{\\mathclap\{\\text\{measured entropy\}\}\}\.\(1\)A second strategy learns the decision rather than scoring it, and this does not by itself escape the confidently\-wrong problem either\. MBA\-RAG\([Tang et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib32)\)learns a bandit policy conditioned on the question alone; a policy that never reads the candidates can decide how to retrieve for a question, but not which of that question’s candidates to trust \(Section[5](https://arxiv.org/html/2608.11922#S5)\)\.

#### Learned guidance for retrieval\-augmented answering\.

A line of work trains a model to emit text that guides how a retrieval\-augmented system answers\. Self\-RAG\([Asai et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib2)\)trains a model to generate reflection tokens; GainRAG\([Jiang et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib13)\)aligns retriever and respondent through a perplexity\-based gain signal whose teacher reads the gold answer; CTRL\-RAG\([Tan et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib31)\)uses reinforcement learning to make generation more context\-faithful; and CRITIC\-R1\([Xiao et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib36)\)leaves the generator frozen and trains a critic instead, rewarding it for aligning its verdicts with a teacher’s judgement of the answer\. What they share is where the training target is read: from the text itself, whether that is an answer’s correctness, its groundedness in the passage, the passage’s relevance, or the quality of a critique of the trajectory\. We stress that Self\-RAG and CTRL\-RAG train the answering model itself\. Their respondent is not frozen, so they solve a different problem from ours, improving the model that answers rather than the evidence a fixed model answers from\.Lodestarchanges only the training signal\. To the best of our knowledge, no prior work scores a text intervention by the uncertainty it induces in a third\-party frozen respondent, compared across the candidates of one question\.

## 3Methodology

#### The problem: answer selection for a frozen respondent\.

A frozen respondentRRis only as good as the single passage \(possibly misleading, possibly supporting\) it reads\. That choice decides whether its answer is correct\. Given a questionqqand a retrieved candidate setP⁡\(q\)=\{p1,…,pK\}P\(q\)=\\\{p\_\{1\},\\dots,p\_\{K\}\\\}, each candidate read alone leadsRRto one candidate answer, so the pool inducesKKanswers to choose among\.

#### The signal: within\-question entropy separation\.

Within a single question, a misleading passage*lowers*the respondent’s entropy; selection by answer\-token entropy is therefore drawn to the wrong answersRRproduces confidently from misleading passages, over the less confident but correct ones\. We denoteH¯L​\(q,p\)\\bar\{H\}\_\{L\}\(q,p\)as the mean normalized entropy of theLLtokens of the answerRRgenerates from passageppalone, anda⁡\(q,p\)a\(q,p\)as that answer\. The answer selector keepsarg⁡mina⁡\(q,p\),p∈P⁡\(q\)​H¯L​\(q,p\)\\arg\\min\_\{a\(q,p\),\\ p\\in P\(q\)\}\\bar\{H\}\_\{L\}\(q,p\), which is the rule Equation[1](https://arxiv.org/html/2608.11922#S2.E1)showed IGP reduces to\. LetMis⁡\(q\)\\mathrm\{Mis\}\(q\)andSup⁡\(q\)\\mathrm\{Sup\}\(q\)be the passages of questionqqlabelled misleading and supporting\. The quantityLodestaracts on is the within\-question separation between the entropiesRRproduces onMis⁡\(q\)\\mathrm\{Mis\}\(q\)and onSup⁡\(q\)\\mathrm\{Sup\}\(q\), which the polarizer is trained to drive positive inside each question:

ΔMis−Sup​H¯L​\(q\)=meanp∈Mis⁡\(q\)H¯L​\(q,p\)−meanp∈Sup⁡\(q\)H¯L​\(q,p\)\.\\Delta\_\{\\mathrm\{Mis\}\-\\mathrm\{Sup\}\}\\bar\{H\}\_\{L\}\(q\)\\;=\\;\\operatorname\*\{mean\}\_\{p\\in\\mathrm\{Mis\}\(q\)\}\\bar\{H\}\_\{L\}\(q,p\)\\;\-\\;\\operatorname\*\{mean\}\_\{p\\in\\mathrm\{Sup\}\(q\)\}\\bar\{H\}\_\{L\}\(q,p\)\.\(2\)

#### The intervention: directed entropy\.

Lodestar’s methodological contribution is the*RL\-learned polarizer*, one fixed natural\-language stringψ⋆\\psi^\{\\star\}that turns the respondent’s entropy from a quantity we passively measure into one we intervene on\.LodestarleavesRRfrozen and insertsψ⋆\\psi^\{\\star\}after the passageppand before the questionqq\(the prompt is given in Appendix[B](https://arxiv.org/html/2608.11922#A2)\)\. We call the result,H¯L​\(q,p,ψ⋆\)\\bar\{H\}\_\{L\}\(q,p,\\psi^\{\\star\}\), the*directed entropy*\. The polarizerψ⋆\\psi^\{\\star\}is optimized to move it with an intended sign, upward on misleading passages, unchanged on supporting ones\. Under directed entropy the answer selector differs from Equation[1](https://arxiv.org/html/2608.11922#S2.E1)by one symbol, the entropy it ranks now computed underψ⋆\\psi^\{\\star\}:

a⏟^selected answer=arg⁡mina⁡\(q,p,ψ⋆\),p∈P⁡\(q\)⁡H¯L​\(q,p,ψ⋆\)⏟directed entropy\.\\underbrace\{\\hat\{a\}\}\_\{\\mathclap\{\\text\{selected answer\}\}\}\\;=\\;\\arg\\min\_\{a\(q,p;\\psi^\{\\star\}\),\\ p\\in P\(q\)\}\\underbrace\{\\bar\{H\}\_\{L\}\(q,p,\\psi^\{\\star\}\)\}\_\{\\text\{directed entropy\}\}\.\(3\)

#### Training objective\.

The optimal polarizerψ⋆\\psi^\{\\star\}is the string that maximizes the within\-question separation its insertion induces\. A generator policy proposes candidate polarizers from a single fixed prompt that contains neither labels nor gold answers,222The converged strings are reported in Appendix[H](https://arxiv.org/html/2608.11922#A8), where independent seeds are shown to agree on what to say even though the reward never scores wording\.and is trained on their reward by GRPO\([Shao et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib27)\)\. Each candidate polarizerψ\\psiis credited with the separation of Equation[2](https://arxiv.org/html/2608.11922#S3.E2)evaluated underψ\\psi, baseline\-corrected against the same question’s natural separation:

𝒮⁡\(ψ\)=𝔼^q∈B​\[ΔMis−Sup​H¯L​\(q,ψ\)⏟separation with polarizer​ψ​inserted−ΔMis−Sup​H¯L​\(q\)⏟the natural separation, Equation[2](https://arxiv.org/html/2608.11922#S3.E2)\],withΔMis−Sup​H¯L​\(q,ψ\)=meanp∈Mis⁡\(q\)H¯L​\(q,p,ψ\)−meanp∈Sup⁡\(q\)H¯L​\(q,p,ψ\),\\begin\{gathered\}\\mathcal\{S\}\(\\psi\)\\;=\\;\\widehat\{\\mathbb\{E\}\}\_\{q\\in B\}\\Big\[\\,\\underbrace\{\\Delta\_\{\\mathrm\{Mis\}\-\\mathrm\{Sup\}\}\\bar\{H\}\_\{L\}\(q;\\psi\)\}\_\{\\text\{separation with polarizer \}\\psi\\text\{ inserted\}\}\\;\-\\;\\underbrace\{\\Delta\_\{\\mathrm\{Mis\}\-\\mathrm\{Sup\}\}\\bar\{H\}\_\{L\}\(q\)\}\_\{\\text\{the natural separation, Equation~\\ref\{eq:sep\}\}\}\\,\\Big\],\\\\ \\text\{with\}\\quad\\Delta\_\{\\mathrm\{Mis\}\-\\mathrm\{Sup\}\}\\bar\{H\}\_\{L\}\(q;\\psi\)\\;=\\;\\operatorname\*\{mean\}\_\{p\\in\\mathrm\{Mis\}\(q\)\}\\bar\{H\}\_\{L\}\(q,p,\\psi\)\\;\-\\;\\operatorname\*\{mean\}\_\{p\\in\\mathrm\{Sup\}\(q\)\}\\bar\{H\}\_\{L\}\(q,p,\\psi\),\\end\{gathered\}\(4\)where𝔼^q∈B\\widehat\{\\mathbb\{E\}\}\_\{q\\in B\}averages over the question setBBscored at that step, whose labelled passages are sampled from the top\-KKretrieved\.𝒮⁡\(ψ\)\\mathcal\{S\}\(\\psi\)is the polarizer’s reward, zero wheneverψ\\psichanges nothing and larger being better\. The reward functional never reads an answer, gold or generated\. The partitionMis⁡\(q\)/Sup⁡\(q\)\\mathrm\{Mis\}\(q\)/\\mathrm\{Sup\}\(q\)that the reward contrasts is nonetheless built from gold\- and judge\-derived labels, so gold\-freedom is a property of inference, not of training\-data construction\. Appendix[B](https://arxiv.org/html/2608.11922#A2)states the GRPO objective and its advantage in full, together with every setting\.

#### Passage labels\.

Two LLM judges from different model families\([Zheng et al\. 2023](https://arxiv.org/html/2608.11922#bib.bib43);[Verga et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib35);[Thomas et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib34)\), gpt\-oss\-120b\([OpenAI 2025](https://arxiv.org/html/2608.11922#bib.bib21)\)and Qwen2\.5\-72B\-Instruct\([Yang et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib37)\), read each passage independently\. A passage counts asmisleadingonly when both agree that it is, assupportingwhen the respondent’s answer from it exactly matches a gold answer after normalization, and asneutralotherwise; the rare passage meeting both criteria is routed toneutral\. On the5,0005\{,\}000NQ\-Open questions from which the training pool is drawn they agree with Cohen’sκ=0\.675\\kappa=0\.675, and on a held\-out audit set an independent third\-party judge returns their consensus label96%96\\%of the time \(Appendix[G](https://arxiv.org/html/2608.11922#A7)\)\.

#### Inference\.

Inference adds no model, no sampling, no supervision and no passage label\. The respondent is run once per candidate to obtainH¯L​\(q,p,ψ⋆\)\\bar\{H\}\_\{L\}\(q,p,\\psi^\{\\star\}\), and Equation[3](https://arxiv.org/html/2608.11922#S3.E3)returns the argmin\.ψ⋆\\psi^\{\\star\}stays in the prompt while that answer is decoded, which is why Equation[3](https://arxiv.org/html/2608.11922#S3.E3)selects amonga⁡\(q,p,ψ⋆\)a\(q,p;\\psi^\{\\star\}\)rather thana⁡\(q,p\)a\(q,p\)\. The only artifact training carries forward isψ⋆\\psi^\{\\star\}, one category\-level string, and selection reads nothing but the respondent’s own entropy\.

## 4Experimental Settings

### 4\.1Benchmark

#### Datasets and evaluation pools\.

We evaluateLodestarand all baselines on five open\-domain QA datasets:*in\-domain*, Natural Questions\([Kwiatkowski et al\. 2019](https://arxiv.org/html/2608.11922#bib.bib15)\);*out\-of\-domain*, SQuAD, TriviaQA, EntityQuestions and WebQuestions\([Rajpurkar et al\. 2016](https://arxiv.org/html/2608.11922#bib.bib25);[Joshi et al\. 2017](https://arxiv.org/html/2608.11922#bib.bib14);[Sciavolino et al\. 2021](https://arxiv.org/html/2608.11922#bib.bib26);[Berant et al\. 2013](https://arxiv.org/html/2608.11922#bib.bib3)\), none of whose training data is ever seen\. Evaluation pools are fixed in advance and shared by every method:n=1,000n\\\!=\\\!1\{,\}000each except EntityQuestions atn=1,008n\\\!=\\\!1\{,\}008, each with the top ten bge\-m3\([Chen et al\. 2024b](https://arxiv.org/html/2608.11922#bib.bib5)\)candidates from one shared Wikipedia index\.

#### Frozen respondent and answer evaluation protocol\.

The frozen respondent is Llama\-3\.1\-8B\-Instruct\([Grattafiori et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib9)\), its parameters never updated\.333We additionally run the cross\-respondent comparison across two model generations, Qwen2\.5\-7B and Qwen3\.5\-9B, training a polarizer for each respondent and evaluating every pairing both in\-domain and out\-of\-domain; every respondent improves under its own polarizer on both sides of the split \(Section[6](https://arxiv.org/html/2608.11922#S6)\)\.It answers greedily from the selected passage alone, under one shared answering template that carriesψ⋆\\psi^\{\\star\}forLodestarand nothing extra for every other configuration\. EveryF1F\_\{1\}in this paper scores that generated answer against the gold answer\(s\); exact match \(EM\) does the same after normalization\. An LLM judge, GPT\-4o\([OpenAI 2024](https://arxiv.org/html/2608.11922#bib.bib20)\), additionally rates whether the generated answer is equivalent to any gold answer, under a judging prompt adopted verbatim from[Farquhar et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib7)\. Appendix[F](https://arxiv.org/html/2608.11922#A6)carries the exact\-match table and full judge protocol, and Appendix[G](https://arxiv.org/html/2608.11922#A7)reports the judge’s numbers per dataset\.

### 4\.2Polarizer training

#### Policy, data and budget\.

We adopt Qwen3\-4B\-Instruct\([Yang et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib38)\)as the polarizer policy, trading generation quality against training cost at a scale we can afford to run three times, and train it with GRPO \(Appendix[B](https://arxiv.org/html/2608.11922#A2)\)\. The polarizer’s training questions are the full1,9391\{,\}939\-question NQ\-Open training pool, disjoint by construction from every evaluation pool\. The reportedψ⋆\\psi^\{\\star\}, and every polarizer reported in this paper, is the*final*checkpoint of a completed, fixed300300\-step budget\.

#### Entropy estimator\.

Reproducing IGP, whose gain term averages entropy over allLLanswer tokens, we found that scoring the first answer token alone is no worse on the macro mean under a shared answering prompt \(0\.51480\.5148against0\.50010\.5001mean answerF1F\_\{1\}; Appendix[C](https://arxiv.org/html/2608.11922#A3)\)\.L=1L\{=\}1is therefore the settingLodestaruses, in its training reward and in its selector alike\. The choice has precedent on both counts: reading a selection score from the first decoded position alone is established in neural reranking, where[Gangi Reddy et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib8)take a full listwise ordering from the logits of the first generated identifier; and approximating a multi\-token entropy by its first token is a standing convention, one[Clark et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib6)test directly and caution against\. The measurement above asks their question of this task, and answers it\. Each baseline keeps the estimator its own paper specifies\.H¯L\\bar\{H\}\_\{L\}becomes the respondent’s first\-token entropyH1H\_\{1\}, and the decoding a candidate needs to be scored collapses fromLLtokens to one, anLL\-fold saving on every candidate of every question\.

### 4\.3Baselines

Table 2:Passage\-selectionF1F\_\{1\}with a frozen Llama\-3\.1\-8B\-Instruct respondent\.All configurations select one of the same ten bge\-m3 candidates and answer with the same frozen respondent, so the rows differ only in how that passage is chosen\.*Mean*,E​MEMand the LLM\-judge column are five\-pool macro means ofF1F\_\{1\}, of exact match \(Appendix[F](https://arxiv.org/html/2608.11922#A6)\) and of a GPT\-4o answer\-equivalence verdict \(Appendix[G](https://arxiv.org/html/2608.11922#A7)\)\. TheLodestarrow is the mean of the three runs listed beneath it in all three\.¶Min\-K%\+\+\([Zhang et al\. 2025a](https://arxiv.org/html/2608.11922#bib.bib41)\)is the successor of Min\-K% Prob\([Shi et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib28)\), printed for completeness and not among the fourteen ranked configurations\. Our reimplementation matches the official release term for term, and it still scores belowrandom\(Appendix[J](https://arxiv.org/html/2608.11922#A10)\)\.

Every baseline is re\-implemented as a passage selector under the*strongest*configuration its published settings or released code supports, and evaluated on the same pools with the same frozen respondent and the same answering template, withψ⋆\\psi^\{\\star\}the only additional contentLodestarplaces in it\. Fourteen ranked configurations fall into three families, Table[2](https://arxiv.org/html/2608.11922#S4.T2)’s three baseline group headers, differing in where the selection signal comes from: \(i\) a measurement from models as published, most often of uncertainty; \(ii\) a selector model we train; or \(iii\) a polarizer string we search for\. We also report two system\-level comparisons in Appendix[I](https://arxiv.org/html/2608.11922#A9)\(Table[12](https://arxiv.org/html/2608.11922#A9.T12)\): Self\-RAG’s own trained\-respondent configuration and our reproduction of the concurrent CTRL\-RAG preprint\([Tan et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib31)\)\. Both train the answering model itself, so they compare inference classes rather than selection quality and are not ranked\. Table[2](https://arxiv.org/html/2608.11922#S4.T2)’s three reference lines \(random,rank1,oracle\) calibrate rather than compete\. Per\-row configurations, reference\-line definitions, the load\-bearing disclosures about how rows should be read, and every residual deviation from the official scripts are in Appendix[J](https://arxiv.org/html/2608.11922#A10)\.

#### Published signals re\-purposed as selectors\.

Semantic entropy, Self\-RAG \(reflection\), SeaKR, CLeHe, EPR, Min\-K% Prob, IGP, EigenScore and SPS fill nine rows, each run exactly as published, with no training stage in our pipeline\. Seven of them score each candidate from the frozen respondent’s own response, SPS scores the passage representation alone, and Self\-RAG \(reflection\) selects with the method’s officially released critic, which is what keeps it under the frozen respondent\.

#### Search\-based guidance\.

We also ask whether the polarizer can simply be searched for\. GEPA and MIPROv2 answer that underLodestar’s exact reward function, training draws and frozen respondent, so only the optimizer differs\. Both configurations start from the same hand\-written caution seed and select identically at inference \(arg⁡minp​H1​\(q,p,ψ\)\\arg\\min\_\{p\}H\_\{1\}\(q,p,\\psi\)\)\. Each row reports its strongest seed againstLodestar’s three\-seed mean, a comparison deliberately set in the baselines’ favour\.

#### Trained rerankers\.

GainRAG, MBA\-RAG and CRITIC\-R1 replace the signal with a model trained on the same1,9391\{,\}939NQ\-Open questions asLodestar, so NQ is in\-domain and the other four out\-of\-domain\. Each scores the same ten candidates: GainRAG a278278M cross\-encoder distilled from a gold\-conditioned gain teacher, MBA\-RAG a DistilBERT bandit over candidate ranks, and CRITIC\-R1 a GRPO\-trained critic reading each cached draft answer\. Re\-run from stored artifacts on different hardware, all three reproduce their tabulated values \(Δ​F1=0\.0000\\Delta F\_\{1\}\{=\}0\.0000on WebQuestions\)\.

![Refer to caption](https://arxiv.org/html/2608.11922v1/fig11_cross_respondent.png)Figure 2:Cross\-respondent transfer of the polarizer under a mean\-normalized\-entropy \(H¯L\\bar\{H\}\_\{L\}\) selector\.Rows are the respondent the polarizerψ\\psiwas trained against, columns the frozen respondent that selects and answers; each of the eighteen facets prints answerF1F\_\{1\}withψ\\psiinserted \(top\) andΔ\\Deltaagainst the same\-pass configuration without it \(bottom\)\. Every diagonal is positive on both panels; all four negative facets are off\-diagonal\. Cell colour gives the sign ofΔ\\Delta, its depth the magnitude; largest gain\+0\.0500\+0\.0500, largest harm−0\.0188\-0\.0188\.Table 3:The polarizer ablation under two metrics\.Both blocks compareLodestaragainstLodestarwithψ⋆\\psi^\{\\star\}removed, i\.e\., plain first\-token entropy selection\. Rates areP⁡\(misleading∣selected\)P\(\\text\{\{misleading\}\}\\mid\\text\{selected\}\)atK=10K\{=\}10on the bge\-m3 pool, whose own rate is28\.928\.9;macroaverages the five\.

## 5Main Results

#### Head\-to\-head against published signals\.

Lodestarattains the highest meanF1F\_\{1\}of any frozen\-respondent method,0\.53390\.5339against0\.51480\.5148for plain first\-token entropy selection and0\.47690\.4769for the retriever’s own top\-1 passage\. It takes the highest exact match \(0\.41360\.4136, Table[9](https://arxiv.org/html/2608.11922#A6.T9)\) on the same pools, and the highest GPT\-4o judge score of the frozen\-respondent configurations judged \(0\.64350\.6435, Appendix[G](https://arxiv.org/html/2608.11922#A7)\)\. Throughout this section*meanF1F\_\{1\}*is the macro mean over the five pools, the*Mean*column of Table[2](https://arxiv.org/html/2608.11922#S4.T2)\. The step from the entropy selector holds on both sides of the split that mean conceals:0\.46430\.4643to0\.47890\.4789in\-domain on NQ, and0\.52740\.5274to0\.54760\.5476averaged over SQuAD, TriviaQA, EntityQuestions and WebQuestions\. TheLodestarrow of Table[2](https://arxiv.org/html/2608.11922#S4.T2), the mean of its three training seeds, also wins all7070method\-by\-dataset cells onF1F\_\{1\}, and takes the highest macro exact match of any ranked row,0\.41360\.4136against semantic entropy’s0\.40390\.4039\(Table[9](https://arxiv.org/html/2608.11922#A6.T9)\)\. The sweep is a property of that row rather than of each run behind it: seed4343loses the EntityQuestionsF1F\_\{1\}cell to GainRAG \(0\.46610\.4661against0\.46750\.4675\), while every run does win all five datasets against both entropy selection andrank1\(Appendix[E](https://arxiv.org/html/2608.11922#A5)\)\. Those cells span the fourteen frozen\-respondent\-comparable baseline configurations we rank, meaning every configuration that picks one of those ten candidates and leaves the respondent frozen\. The three runs beneath theLodestarrow differ in the training\-data seed alone\. It reshuffles the grouping and ordering of the1,9391\{,\}939training questions, while which questions exist, the train/held\-out split and the trainer’s own seed are held fixed, so the three converged polarizers are scored on one held\-out set \(Appendix[E](https://arxiv.org/html/2608.11922#A5)\)\. EveryF1F\_\{1\}advantage ofLodestarover a baseline configuration is paired\-significant under both att\-test and Wilcoxon \(Appendix[D](https://arxiv.org/html/2608.11922#A4)\)\.

#### Reporting conventions\.

Table[2](https://arxiv.org/html/2608.11922#S4.T2)reports the mean of three training runs and lists them individually; the three differ only in their training\-data seed, each a300300\-step run over the same1,9391\{,\}939questions in a different grouping and order, with their spread reported in Appendix[E](https://arxiv.org/html/2608.11922#A5)\. A table that instead reports a single run says so in its caption\. Every headline number uses the reward of Equation[4](https://arxiv.org/html/2608.11922#S3.E4)\.

## 6The Polarizer Ablation

#### The ablation\.

We ablateLodestarby removing the polarizerψ⋆\\psi^\{\\star\}and changing nothing else\. The frozen respondent, the pools and thearg⁡min\\arg\\minselection rule are identical, so the two configurations differ only in whether the string is inserted\. That configuration is also the strongest untrained selector in our benchmark \(Appendix[J](https://arxiv.org/html/2608.11922#A10)\)\. Table[3](https://arxiv.org/html/2608.11922#S4.T3)gives it per dataset under two metrics: answerF1F\_\{1\}, the outcome, and how often the selected passage is judgedmisleading, the quantity the reward acts on\. On the latter the string helps on5/55/5datasets, from2\.02\.0points \(TriviaQA\) to6\.46\.4\(EntityQuestions\), and carries the rule below the pool’s own28\.9%28\.9\\%, which entropy selection alone does not reach\. Averaged over the three training seeds \(Appendix[E](https://arxiv.org/html/2608.11922#A5)\) the gain over the ablated configuration is\+0\.0191±0\.0054\+0\.0191\\pm 0\.0054, with per\-dataset gains from\+0\.005\+0\.005\(TriviaQA\) to\+0\.031\+0\.031\(SQuAD\) and5/55/5datasets positive under every seed\. That lead is paired\-significant under*every*seed individually, the three seeds givingpw=2\.4×10−8p\_\{w\}=2\.4\\times 10^\{\-8\},1\.8×10−41\.8\\times 10^\{\-4\}and2\.9×10−112\.9\\times 10^\{\-11\}, under the protocol of Appendix[D](https://arxiv.org/html/2608.11922#A4), whose table carries the seed\-4242run alone\. Appendix[H](https://arxiv.org/html/2608.11922#A8)decomposes the entropy shift the string produces, by passage class\.

#### Cross\-respondent transfer\.

Figure[2](https://arxiv.org/html/2608.11922#S4.F2)repeats the comparison between inserting the polarizerψ⋆\\psi^\{\\star\}and leaving it out, under a second selection signal \(i\.e\.,H¯L\\bar\{H\}\_\{L\}\) and across three frozen respondents from two model families, Llama\-3\.1\-8B, Qwen2\.5\-7B and Qwen3\.5\-9B\. All four negative facets are transfers to a respondent the string was never trained against\. In the Qwen3\.5\-9B column the answer sequence ends at<\|endoftext\|\>rather than<\|im\_end\|\>, because that is where its generation actually stops\. Restricting it to questions where both answers finished leaves the signs unchanged\. Pooled across the figure’s two panels \(5,0085\{,\}008questions\) the diagonal stays at\+0\.0395\+0\.0395against\+0\.0477\+0\.0477unrestricted, and the Llama\-3\.1\-8B transfer grows rather than shrinks\.

## 7Conclusion

We first show that the frozen respondent’s first\-token entropy chooses passages well enough to beat the retriever’s own ranking with no gold answer,0\.51480\.5148mean answerF1F\_\{1\}against its0\.47690\.4769, but that it fails in a way invisible from where prior work measures entropy\. The standard entropy selector keeps the lowest\-entropy candidate \(Equation[1](https://arxiv.org/html/2608.11922#S2.E1)\) and is drawn toward the passages it should avoid: pooled over all questions, judged\-misleading candidates carry lower entropy than the rest \(1\.221\.22nats against1\.331\.33; Section[2](https://arxiv.org/html/2608.11922#S2)\)\. Across the five pools it reads a judged\-misleading passage more often on average than picking one of the ten at random \(Table[3](https://arxiv.org/html/2608.11922#S4.T3)\)\. We propose![[Uncaptioned image]](https://arxiv.org/html/2608.11922v1/figures/assets/noto_2734.png)Lodestar, which repairs this without touching the respondent: a GRPO\-trained policy writes the polarizerψ⋆\\psi^\{\\star\}, one short natural\-language string whose only job is to navigate the respondent’s entropy in the intended direction\. Across5,0085\{,\}008questions from five QA benchmarks that one string is enough to giveLodestarthe highest meanF1F\_\{1\}of any inference\-ready selector,0\.53390\.5339, its three\-seed mean winning every method\-by\-datasetF1F\_\{1\}cell against fourteen published configurations \(Table[2](https://arxiv.org/html/2608.11922#S4.T2)\)\. Of that lead, the ablation removingψ⋆\\psi^\{\\star\}with the respondent, pools and selection rule unchanged attributes\+0\.0191±0\.0054\+0\.0191\\pm 0\.0054F1F\_\{1\}to the string itself, positive on5/55/5datasets under all three training seeds \(Section[6](https://arxiv.org/html/2608.11922#S6)\)\. All of it costs one forward pass per candidate and no gold answers\. Overall, entropy is the right signal here but only once directed rather than passively measured: one learned string turns within\-question entropy from a signal that misleads into one that selects, with the respondent left frozen\.

## References

- Agrawal et al\. \(2026\)Lakshya A\. Agrawal, Shangyin Tan, Dilara Soylu, Noah Ziems, Rishi Khare, Krista Opsahl\-Ong, Arnav Singhvi, Herumb Shandilya, Michael J\. Ryan, Meng Jiang, Christopher Potts, Koushik Sen, Alexandros G\. Dimakis, Ion Stoica, Dan Klein, Matei Zaharia, and Omar Khattab\.GEPA: Reflective prompt evolution can outperform reinforcement learning\.In*International Conference on Learning Representations \(ICLR\)*, 2026\.arXiv:2507\.19457\.
- Asai et al\. \(2024\)Akari Asai, Zeqiu Wu, Yizhong Wang, Avirup Sil, and Hannaneh Hajishirzi\.Self\-RAG: Learning to retrieve, generate, and critique through self\-reflection\.In*International Conference on Learning Representations \(ICLR\)*, 2024\.arXiv:2310\.11511\.
- Berant et al\. \(2013\)Jonathan Berant, Andrew Chou, Roy Frostig, and Percy Liang\.Semantic parsing on Freebase from question\-answer pairs\.In*Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing \(EMNLP\)*, pp\. 1533–1544, 2013\.
- Chen et al\. \(2024a\)Chao Chen, Kai Liu, Ze Chen, Yi Gu, Yue Wu, Mingyuan Tao, Zhihang Fu, and Jieping Ye\.INSIDE: LLMs’ internal states retain the power of hallucination detection\.In*International Conference on Learning Representations \(ICLR\)*, 2024a\.arXiv:2402\.03744\.
- Chen et al\. \(2024b\)Jianlyu Chen, Shitao Xiao, Peitian Zhang, Kun Luo, Defu Lian, and Zheng Liu\.M3\-embedding: Multi\-linguality, multi\-functionality, multi\-granularity text embeddings through self\-knowledge distillation\.In*Findings of the Association for Computational Linguistics: ACL*, pp\. 2318–2335, 2024b\.doi:10\.18653/v1/2024\.findings\-acl\.137\.arXiv:2402\.03216\.
- Clark et al\. \(2025\)Christian Clark, Byung\-Doh Oh, and William Schuler\.How well does first\-token entropy approximate word entropy as a psycholinguistic predictor?In*Proceedings of the 14th International Joint Conference on Natural Language Processing and the 4th Conference of the Asia\-Pacific Chapter of the Association for Computational Linguistics \(IJCNLP\-AACL\)*, pp\. 47–57, 2025\.doi:10\.18653/v1/2025\.ijcnlp\-short\.4\.arXiv:2507\.22209\.
- Farquhar et al\. \(2024\)Sebastian Farquhar, Jannik Kossen, Lorenz Kuhn, and Yarin Gal\.Detecting hallucinations in large language models using semantic entropy\.*Nature*, 630:625–630, 2024\.doi:10\.1038/s41586\-024\-07421\-0\.
- Gangi Reddy et al\. \(2024\)Revanth Gangi Reddy, JaeHyeok Doo, Yifei Xu, Md Arafat Sultan, Deevya Swain, Avirup Sil, and Heng Ji\.FIRST: Faster improved listwise reranking with single token decoding\.In*Proceedings of the 2024 Conference on Empirical Methods in Natural Language Processing \(EMNLP\)*, pp\. 8642–8652, 2024\.doi:10\.18653/v1/2024\.emnlp\-main\.491\.
- Grattafiori et al\. \(2024\)Aaron Grattafiori, Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al\-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Alex Vaughan, et al\.The Llama 3 herd of models\.*arXiv preprint arXiv:2407\.21783*, 2024\.
- Hu et al\. \(2022\)Edward J\. Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen\-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen\.LoRA: Low\-rank adaptation of large language models\.In*International Conference on Learning Representations \(ICLR\)*, 2022\.arXiv:2106\.09685\.
- Hu et al\. \(2026\)Zhanghao Hu, Qinglin Zhu, Siya Qi, Yulan He, Hanqi Yan, and Lin Gui\.Beyond perplexity: Let the reader select retrieval summaries via spectrum projection score\.In*AAAI Conference on Artificial Intelligence*, pp\. 31104–31112, 2026\.doi:10\.1609/aaai\.v40i37\.40371\.arXiv:2508\.05909\.
- Izacard et al\. \(2022\)Gautier Izacard, Mathilde Caron, Lucas Hosseini, Sebastian Riedel, Piotr Bojanowski, Armand Joulin, and Edouard Grave\.Unsupervised dense information retrieval with contrastive learning\.*Transactions on Machine Learning Research*, 2022\.arXiv:2112\.09118\.
- Jiang et al\. \(2025\)Yi Jiang, Sendong Zhao, Jianbo Li, Haochun Wang, and Bing Qin\.GainRAG: Preference alignment in retrieval\-augmented generation through gain signal synthesis\.In*Annual Meeting of the Association for Computational Linguistics \(ACL\)*, 2025\.arXiv:2505\.18710\.
- Joshi et al\. \(2017\)Mandar Joshi, Eunsol Choi, Daniel S\. Weld, and Luke Zettlemoyer\.TriviaQA: A large scale distantly supervised challenge dataset for reading comprehension\.In*Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics \(ACL\)*, pp\. 1601–1611, 2017\.
- Kwiatkowski et al\. \(2019\)Tom Kwiatkowski, Jennimaria Palomaki, Olivia Redfield, Michael Collins, Ankur Parikh, Chris Alberti, Danielle Epstein, Illia Polosukhin, Jacob Devlin, Kenton Lee, Kristina Toutanova, Llion Jones, Matthew Kelcey, Ming\-Wei Chang, Andrew M\. Dai, Jakob Uszkoreit, Quoc Le, and Slav Petrov\.Natural questions: A benchmark for question answering research\.*Transactions of the Association for Computational Linguistics*, 7:452–466, 2019\.
- Liu et al\. \(2025\)Zichen Liu, Changyu Chen, Wenjun Li, Penghui Qi, Tianyu Pang, Chao Du, Wee Sun Lee, and Min Lin\.Understanding R1\-zero\-like training: A critical perspective\.In*Conference on Language Modeling \(COLM\)*, 2025\.arXiv:2503\.20783\.
- Ma et al\. \(2025\)Huan Ma, Jiadong Pan, Jing Liu, Yan Chen, Joey Tianyi Zhou, Guangyu Wang, Qinghua Hu, Hua Wu, Changqing Zhang, and Haifeng Wang\.Semantic energy: Detecting LLM hallucination beyond entropy\.*arXiv preprint arXiv:2508\.14496*, 2025\.
- Moskvoretskii et al\. \(2025\)Viktor Moskvoretskii, Maria Marina, Mikhail Salnikov, Nikolay Ivanov, Sergey Pletenev, Daria Galimzianova, Nikita Krayko, Vasily Konovalov, Irina Nikishina, and Alexander Panchenko\.Adaptive retrieval without self\-knowledge? bringing uncertainty back home\.In*Annual Meeting of the Association for Computational Linguistics \(ACL\)*, 2025\.ACL Anthology 2025\.acl\-long\.319\.
- Moslonka et al\. \(2026\)Charles Moslonka, Hicham Randrianarivo, Arthur Garnier, and Emmanuel Malherbe\.Learned hallucination detection in black\-box LLMs using token\-level entropy production rate\.In*European Conference on Information Retrieval \(ECIR\)*, volume 16483 of*Lecture Notes in Computer Science*, pp\. 115–130\. Springer, 2026\.doi:10\.1007/978\-3\-032\-21289\-4˙8\.arXiv:2509\.04492\.
- OpenAI \(2024\)OpenAI\.GPT\-4o system card\.*arXiv preprint arXiv:2410\.21276*, 2024\.
- OpenAI \(2025\)OpenAI\.gpt\-oss\-120b & gpt\-oss\-20b Model Card\.*arXiv preprint arXiv:2508\.10925*, 2025\.
- Opsahl\-Ong et al\. \(2024\)Krista Opsahl\-Ong, Michael J\. Ryan, Josh Purtell, David Broman, Christopher Potts, Matei Zaharia, and Omar Khattab\.Optimizing instructions and demonstrations for multi\-stage language model programs\.In*Proceedings of EMNLP*, 2024\.
- Qiu et al\. \(2025\)Zexuan Qiu, Zijing Ou, Bin Wu, Jingjing Li, Aiwei Liu, and Irwin King\.Entropy\-based decoding for retrieval\-augmented large language models\.In*North American Chapter of the Association for Computational Linguistics \(NAACL\)*, 2025\.ACL Anthology 2025\.naacl\-long\.236\.
- Qwen Team \(2026\)Qwen Team\.Qwen3\.5: Towards Native Multimodal Agents, 2026\.[https://qwen\.ai/blog?id=qwen3\.5](https://qwen.ai/blog?id=qwen3.5)\.
- Rajpurkar et al\. \(2016\)Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang\.SQuAD: 100,000\+ questions for machine comprehension of text\.In*Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing \(EMNLP\)*, pp\. 2383–2392, 2016\.
- Sciavolino et al\. \(2021\)Christopher Sciavolino, Zexuan Zhong, Jinhyuk Lee, and Danqi Chen\.Simple entity\-centric questions challenge dense retrievers\.In*Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing \(EMNLP\)*, 2021\.arXiv:2109\.08535\.
- Shao et al\. \(2024\)Zhihong Shao, Peiyi Wang, Qihao Zhu, Runxin Xu, Junxiao Song, Xiao Bi, Haowei Zhang, Mingchuan Zhang, Y\. K\. Li, Y\. Wu, and Daya Guo\.DeepSeekMath: Pushing the limits of mathematical reasoning in open language models\.*arXiv preprint arXiv:2402\.03300*, 2024\.
- Shi et al\. \(2024\)Weijia Shi, Anirudh Ajith, Mengzhou Xia, Yangsibo Huang, Daogao Liu, Terra Blevins, Danqi Chen, and Luke Zettlemoyer\.Detecting pretraining data from large language models\.In*International Conference on Learning Representations \(ICLR\)*, 2024\.arXiv:2310\.16789\.
- Song et al\. \(2026\)Zhipeng Song, Yizhi Zhou, Xiangyu Kong, Jiulong Jiao, Xinrui Bao, Xu You, Xueqing Shi, Yuhang Zhou, and Heng Qi\.Less is more for RAG: Information gain pruning for generator\-aligned reranking and evidence selection\.*arXiv preprint arXiv:2601\.17532*, 2026\.
- Soudani et al\. \(2025\)Heydar Soudani, Evangelos Kanoulas, and Faegheh Hasibi\.Why uncertainty estimation methods fall short in RAG: An axiomatic analysis\.In*Findings of the Association for Computational Linguistics: ACL*, 2025\.ACL Anthology 2025\.findings\-acl\.852\.
- Tan et al\. \(2026\)Zhehao Tan, Yihan Jiao, Dan Yang, Junjie Wang, Duolin Sun, Jie Feng, Xidong Wang, Lei Liu, Yue Shen, Jian Wang, and Jinjie Gu\.CTRL\-RAG: Contrastive likelihood reward based reinforcement learning for context\-faithful RAG models\.*arXiv preprint arXiv:2603\.04406*, 2026\.
- Tang et al\. \(2025\)Xiaqiang Tang, Qiang Gao, Jian Li, Nan Du, Qi Li, and Sihong Xie\.MBA\-RAG: a bandit approach for adaptive retrieval\-augmented generation through question complexity\.In*International Conference on Computational Linguistics \(COLING\)*, 2025\.arXiv:2412\.01572\.
- Taparia et al\. \(2026\)Aditya Taparia, Ransalu Senanayake, Kowshik Thopalli, and Vivek Narayanaswamy\.The anatomy of uncertainty in LLMs\.*arXiv preprint arXiv:2603\.24967*, 2026\.
- Thomas et al\. \(2024\)Paul Thomas, Seth Spielman, Nick Craswell, and Bhaskar Mitra\.Large language models can accurately predict searcher preferences\.In*Proceedings of the 47th International ACM SIGIR Conference on Research and Development in Information Retrieval \(SIGIR\)*, pp\. 1930–1940, 2024\.doi:10\.1145/3626772\.3657707\.arXiv:2309\.10621\.
- Verga et al\. \(2024\)Pat Verga, Sebastian Hofstätter, Sophia Althammer, Yixuan Su, Aleksandra Piktus, Arkady Arkhangorodsky, Minjie Xu, Naomi White, and Patrick Lewis\.Replacing judges with juries: Evaluating LLM generations with a panel of diverse models\.*arXiv preprint arXiv:2404\.18796*, 2024\.
- Xiao et al\. \(2026\)Wenhan Xiao, Ziwei Zhang, Chuanyue Yu, Xingcheng Fu, Qingyun Sun, Runhua Xu, and Jianxin Li\.CRITIC\-R1: Learning structured critics for retrieval\-augmented generation\.*arXiv preprint arXiv:2605\.29886*, 2026\.
- Yang et al\. \(2024\)An Yang, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chengyuan Li, Dayiheng Liu, Fei Huang, Haoran Wei, et al\.Qwen2\.5 technical report\.*arXiv preprint arXiv:2412\.15115*, 2024\.
- Yang et al\. \(2025\)An Yang, Anfeng Li, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chengyuan Gao, Chengen Huang, Chenxu Lv, et al\.Qwen3 technical report\.*arXiv preprint arXiv:2505\.09388*, 2025\.
- Yao et al\. \(2025\)Zijun Yao, Weijian Qi, Liangming Pan, Shulin Cao, Linmei Hu, Weichuan Liu, Lei Hou, and Juanzi Li\.SeaKR: Self\-aware knowledge retrieval for adaptive retrieval augmented generation\.In*Annual Meeting of the Association for Computational Linguistics \(ACL\)*, pp\. 27022–27043, 2025\.doi:10\.18653/v1/2025\.acl\-long\.1312\.arXiv:2406\.19215\.
- Yu et al\. \(2025\)Qiying Yu, Zheng Zhang, Ruofei Zhu, Yufeng Yuan, Xiaochen Zuo, Yu Yue, Weinan Dai, Tiantian Fan, Gaohong Liu, Juncai Liu, Lingjun Liu, Xin Liu, Haibin Lin, Zhiqi Lin, Bole Ma, Guangming Sheng, Yuxuan Tong, Chi Zhang, Mofan Zhang, Ru Zhang, Wang Zhang, Hang Zhu, Jinhua Zhu, Jiaze Chen, Jiangjie Chen, Chengyi Wang, Hongli Yu, Yuxuan Song, Xiangpeng Wei, Hao Zhou, Jingjing Liu, Wei\-Ying Ma, Ya\-Qin Zhang, Lin Yan, Yonghui Wu, and Mingxuan Wang\.DAPO: An open\-source LLM reinforcement learning system at scale\.In*Advances in Neural Information Processing Systems \(NeurIPS\)*, 2025\.arXiv:2503\.14476\.
- Zhang et al\. \(2025a\)Jingyang Zhang, Jingwei Sun, Eric Yeats, Yang Ouyang, Martin Kuo, Jianyi Zhang, Hao Frank Yang, and Hai Li\.Min\-k%\+\+: Improved baseline for detecting pre\-training data from large language models\.In*International Conference on Learning Representations \(ICLR\)*, 2025a\.arXiv:2404\.02936\.
- Zhang et al\. \(2025b\)Yanzhao Zhang, Mingxin Li, Dingkun Long, Xin Zhang, Huan Lin, Baosong Yang, Pengjun Xie, An Yang, Dayiheng Liu, Junyang Lin, Fei Huang, and Jingren Zhou\.Qwen3 embedding: Advancing text embedding and reranking through foundation models\.*arXiv preprint arXiv:2506\.05176*, 2025b\.
- Zheng et al\. \(2023\)Lianmin Zheng, Wei\-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin, Zhuohan Li, Dacheng Li, Eric P\. Xing, Hao Zhang, Joseph E\. Gonzalez, and Ion Stoica\.Judging LLM\-as\-a\-judge with MT\-Bench and chatbot arena\.In*Advances in Neural Information Processing Systems \(NeurIPS\), Datasets and Benchmarks Track*, 2023\.arXiv:2306\.05685\.

## Appendix contents

[A](https://arxiv.org/html/2608.11922#A1)

Notation\.Every symbol used in the paper, collected\.

[B](https://arxiv.org/html/2608.11922#A2)

GRPO objective and hyper\-parameters\.The objective of Equation[5](https://arxiv.org/html/2608.11922#A2.E5)in full, and every hyper\-parameter\.

[C](https://arxiv.org/html/2608.11922#A3)

Aligned\-probe ablation\.What IGP’s gain term scores once its probe reads the answering prompt, against first\-token entropy on the same five pools\.

[D](https://arxiv.org/html/2608.11922#A4)

Paired significance tests\.Lodestaragainst each method onF1F\_\{1\}\.

[E](https://arxiv.org/html/2608.11922#A5)

Run and seed stability\.What separates the three training runs, and where the tested run sits among them\.

[F](https://arxiv.org/html/2608.11922#A6)

Exact\-match results and paired tests\.The exact\-match counterpart of Table[2](https://arxiv.org/html/2608.11922#S4.T2), and the same paired tests on per\-question EM differences\.

[G](https://arxiv.org/html/2608.11922#A7)

LLM\-judge scores per dataset\.Table[2](https://arxiv.org/html/2608.11922#S4.T2)’s judge column opened up pool by pool, including the one pool where the judge andF1F\_\{1\}disagree\.

[H](https://arxiv.org/html/2608.11922#A8)

Learned polarizer text and its effect on the entropy\.The converged strings themselves, the two framings independent seeds reach under a reward that never scores wording, and how the string shifts the entropy of supporting and non\-supporting passages\.

[I](https://arxiv.org/html/2608.11922#A9)

CTRL\-RAG reproduction\.Its strongest checkpoint, why no single checkpoint of that run can stand in for the method, what its reward was maximising while the format decayed, and the scope limits on what we do and do not claim at its published scale\.

[J](https://arxiv.org/html/2608.11922#A10)

Baseline configurations and disclosures\.Official settings, the disclosures about how individual rows should be read, and every residual deviation from the published scripts\.

## Appendix ANotation

Table[4](https://arxiv.org/html/2608.11922#A1.T4)collects the paper’s symbols for reference\.

Table 4:Symbols used in the paper\.Defined at first use in Sections[3](https://arxiv.org/html/2608.11922#S3)and[4](https://arxiv.org/html/2608.11922#S4); collected here for reference\.
## Appendix BGRPO objective and hyper\-parameters

This section states the GRPO objective in full:

𝒥\(θ\)=𝔼\[min\(ρA^,clip\(ρ,1−εlow,1\+εhigh\)A^\)\]−β𝔻KL\[πθ∥πref\],\\mathcal\{J\}\(\\theta\)\\;=\\;\\mathbb\{E\}\\Big\[\\min\\\!\\big\(\\rho\\,\\hat\{A\},\\;\\operatorname\{clip\}\(\\rho,\\,1\-\\varepsilon\_\{\\text\{low\}\},\\,1\+\\varepsilon\_\{\\text\{high\}\}\)\\,\\hat\{A\}\\big\)\\Big\]\\;\-\\;\\beta\\,\\mathbb\{D\}\_\{\\mathrm\{KL\}\}\\big\[\\pi\_\{\\theta\}\\,\\\|\\,\\pi\_\{\\text\{ref\}\}\\big\],\(5\)whereρ\\rhois the token\-level importance ratio againstπθold\\pi\_\{\\theta\_\{\\text\{old\}\}\},ρi,t​\(θ\)=πθ​\(ψi,t∣x,ψi,<t\)/πθold​\(ψi,t∣x,ψi,<t\)\\rho\_\{i,t\}\(\\theta\)=\\pi\_\{\\theta\}\(\\psi\_\{i,t\}\\mid x,\\psi\_\{i,<t\}\)\\,/\\,\\pi\_\{\\theta\_\{\\text\{old\}\}\}\(\\psi\_\{i,t\}\\mid x,\\psi\_\{i,<t\}\), andπref\\pi\_\{\\text\{ref\}\}is the frozen initial policy\. At each training stepπθ\\pi\_\{\\theta\}samples a group ofGGpolarizers from the single fixed promptxx, and allGGcandidates are scored by Equation[4](https://arxiv.org/html/2608.11922#S3.E4)on the same batch of labelled questions; a step draws1616such groups, the128128sampled strings of Table[5](https://arxiv.org/html/2608.11922#A2.T5)\. GRPO\([Shao et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib27)\)replaces a learned value baseline with the statistics of the sampled group: the advantage of polarizerψi\\psi\_\{i\}is its reward standardized over theGGrewards of that group,

A^i=𝒮⁡\(ψi\)−mean⁡\{𝒮⁡\(ψj\)\}j=1Gstd⁡\{𝒮⁡\(ψj\)\}j=1G,\\hat\{A\}\_\{i\}\\;=\\;\\frac\{\\mathcal\{S\}\(\\psi\_\{i\}\)\\;\-\\;\\operatorname\{mean\}\\big\\\{\\mathcal\{S\}\(\\psi\_\{j\}\)\\big\\\}\_\{j=1\}^\{G\}\}\{\\operatorname\{std\}\\big\\\{\\mathcal\{S\}\(\\psi\_\{j\}\)\\big\\\}\_\{j=1\}^\{G\}\},\(6\)broadcast to every token ofψi\\psi\_\{i\}\. Following[Liu et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib16)and[Yu et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib40), we aggregate token\-mean over the group, so the sequence\-level reward is not silently re\-weighted by polarizer length\. The clip range is asymmetric,\(εlow,εhigh\)=\(0\.2,0\.28\)\(\\varepsilon\_\{\\text\{low\}\},\\varepsilon\_\{\\text\{high\}\}\)=\(0\.2,0\.28\), and the KL term usesβ=10−3\\beta=10^\{\-3\}with the low\-variancek3k\_\{3\}estimator, kept in the loss rather than in the reward so that Equation[4](https://arxiv.org/html/2608.11922#S3.E4)remains the*only*quantity the advantage ranks\. Because allGGpolarizers of a group are scored on the same questions under the same frozen respondent, this advantage ranks nothing but phrasings of the polarizer\.

Table[5](https://arxiv.org/html/2608.11922#A2.T5)lists the configuration behind Equations[5](https://arxiv.org/html/2608.11922#A2.E5)and[6](https://arxiv.org/html/2608.11922#A2.E6)\. Every polarizer reported in this paper is read at the final step of the300300\-step budget, with*no*per\-run checkpoint selection of any kind\. The polarizer policy emits its polarizer wrapped in<critique\>\.\.\.</critique\>tags, a name inherited from the training harness whose contents are the polarizerψ\\psiof Section[3](https://arxiv.org/html/2608.11922#S3)and nothing else, and the string is parsed out before scoring\. The policy prompt contains only the fixed category\-level instruction and unlabelled exemplar question–passage pairs, never a label, a gold answer, or respondent feedback\.ψ⋆\\psi^\{\\star\}itself is inserted into the respondent’s prompt rather than the policy’s, after the passageppand before the questionqq: it readsPassages:pp/An expert’s analysis of the passage above:ψ⋆\\psi^\{\\star\}/Question:qq\.

Table 5:GRPO configurationused for every training run reported in this paper\.
## Appendix CAligned\-probe ablation

Appendix[J](https://arxiv.org/html/2608.11922#A10)’s IGP entry attributes part of its position to a prompt mismatch rather than to its gain formulation: its published implementation probes with its own short prompt, whereas the polarizer ablation’sH1​\(q,p\)H\_\{1\}\(q,p\)\(Section[6](https://arxiv.org/html/2608.11922#S6)\) is read from the very prompt that produces the scored answer\. We isolate that factor by re\-scoring IGP’s per\-passage all\-token mean entropyH¯L​\(q,p\)\\bar\{H\}\_\{L\}\(q,p\)under the shared answering prompt, with the same frozen respondent on the same five pools \(Table[6](https://arxiv.org/html/2608.11922#A3.T6)\)\. The two macro means are close,0\.51480\.5148forH1H\_\{1\}against0\.50010\.5001for the aligned all\-token mean, but that closeness holds only on average: per pool the two disagree in both directions,H1H\_\{1\}ahead by\+0\.0339\+0\.0339on TriviaQA and the aligned mean ahead by0\.00700\.0070on SQuAD\.

Table 6:First\-token entropy against the same\-prompt all\-token mean that IGP’s gain term uses, on the five canonical pools, scored with the frozen respondent under the shared answering prompt \(as in Table[2](https://arxiv.org/html/2608.11922#S4.T2)\)\. The two macro means are close \(0\.51480\.5148vs\.0\.50010\.5001\), but per pool the two variants disagree in both directions:H1H\_\{1\}leads by its largest margin on TriviaQA \(\+0\.0339\+0\.0339\), while the aligned all\-token mean leads by its largest margin on SQuAD \(−0\.0070\-0\.0070\); the small mean gap is not per\-pool equivalence\.The narrowing does not move IGP’s ranked row\. Every baseline is ranked under the strongest configuration its published settings or released code supports \(Section[4\.3](https://arxiv.org/html/2608.11922#S4.SS3)\)\. Probing with the answering prompt is a change we make to IGP’s gain term, not an option those settings offer\. Table[2](https://arxiv.org/html/2608.11922#S4.T2)therefore keeps IGP at its published configuration’s macro mean of0\.48440\.4844, and0\.50010\.5001stands as an ablation result\. The aligned probe recovers about half the distance from that row toH1H\_\{1\}’s0\.51480\.5148, and leavesH1H\_\{1\}ahead on the macro mean\.

## Appendix DPaired significance tests

Because all methods select from the same ten candidates of the same question, we test per\-questionF1F\_\{1\}differences with paired bootstrap \(10,00010\{,\}000resamples\), a pairedtt\-test and Wilcoxon signed\-rank \(Table[7](https://arxiv.org/html/2608.11922#A4.T7)\), every row of it scored on all five pools and all5,0085\{,\}008questions\. Every test in this appendix is paired on the seed\-4242run’s per\-question selections, that being the fixed\-budget run whose polarizer is used at inference; Table[2](https://arxiv.org/html/2608.11922#S4.T2)’sLodestarrow is instead the mean of three training runs, and is characterized by its±0\.0054\\pm 0\.0054spread rather than by app\-value\. Three rows carry a disclosure with their marker\. The§§configuration’s original run predates the per\-question logging these tests need and was re\-run with it, greedy CLeHe reproducing its tabulated macroF1F\_\{1\}to−0\.0007\-0\.0007, so the test is of the re\-run\. The∗configuration is reconstructed under first\-wins tie\-breaking, which*favours*official semantic entropy: its tie rate is6262–79%79\\%with∼2\\sim\\\!2distinct scores per1010candidates, so first\-wins imports a rank\-1 prior\. Removing it \(random\_tiebreak\) puts that method at meanF1F\_\{1\}0\.47740\.4774, indistinguishable fromrank1’s0\.47690\.4769\. The∥row’s paired vector comes from EPR’s official\-budget configuration \(Appendix[J](https://arxiv.org/html/2608.11922#A10)\), macro0\.49550\.4955, rather than from the ranked2424\-token row, macro0\.49680\.4968, so itsΔ​F1\\Delta F\_\{1\}reads\+0\.0389\+0\.0389where the ranked row’s macro difference is\+0\.0375\+0\.0375\.

Table 7:Paired significance ofLodestaragainst each methodon per\-questionF1F\_\{1\}differences\. Every row is tested against the sameLodestarrun on the same questions, so the rows differ only in the method being compared\.Δ​F1\\Delta F\_\{1\}is that difference in answerF1F\_\{1\}as defined in Section[4](https://arxiv.org/html/2608.11922#S4), reported with its95%95\\%bootstrap interval and two paired tests, and bold marksp<0\.05p<0\.05on both\. Rows are banded by method family, in Table[2](https://arxiv.org/html/2608.11922#S4.T2)’s order;§§marks a configuration re\-run for per\-question logging and∗a configuration reconstructed under first\-wins tie\-breaking, and∥one whose paired vector is its official\-budget run; all three are stated in full in the text\.MethodReferenceΔ​F1\\Delta F\_\{1\}95%95\\%CIptp\_\{t\}pWilcoxonp\_\{\\text\{Wilcoxon\}\}Published signals re\-purposed as selectorsSemantic entropy∗\([Farquhar et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib7)\)\+0\.0246\\mathbf\{\+0\.0246\}\[\+0\.0149,\+0\.0342\]\[\+0\.0149,\\,\+0\.0342\]4\.0×10−74\.0\{\\times\}10^\{\-7\}4\.4×10−74\.4\{\\times\}10^\{\-7\}Self\-RAG \(reflection\)\([Asai et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib2)\)\+0\.0272\\mathbf\{\+0\.0272\}\[\+0\.0171,\+0\.0373\]\[\+0\.0171,\\,\+0\.0373\]1\.8×10−71\.8\{\\times\}10^\{\-7\}4\.1×10−74\.1\{\\times\}10^\{\-7\}SeaKR\([Yao et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib39)\)\+0\.0295\\mathbf\{\+0\.0295\}\[\+0\.0207,\+0\.0383\]\[\+0\.0207,\\,\+0\.0383\]7\.3×10−117\.3\{\\times\}10^\{\-11\}7\.9×10−117\.9\{\\times\}10^\{\-11\}CLeHe§§\([Qiu et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib23)\)\+0\.0348\\mathbf\{\+0\.0348\}\[\+0\.0258,\+0\.0438\]\[\+0\.0258,\\,\+0\.0438\]2\.3×10−142\.3\{\\times\}10^\{\-14\}5\.1×10−145\.1\{\\times\}10^\{\-14\}EPR∥\([Moslonka et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib19)\)\+0\.0389\\mathbf\{\+0\.0389\}\[\+0\.0299,\+0\.0480\]\[\+0\.0299,\\,\+0\.0480\]3\.5×10−173\.5\{\\times\}10^\{\-17\}1\.1×10−161\.1\{\\times\}10^\{\-16\}Min\-K% Prob\([Shi et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib28)\)\+0\.0420\\mathbf\{\+0\.0420\}\[\+0\.0330,\+0\.0512\]\[\+0\.0330,\\,\+0\.0512\]1\.3×10−191\.3\{\\times\}10^\{\-19\}4\.1×10−194\.1\{\\times\}10^\{\-19\}IGP\([Song et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib29)\)\+0\.0500\\mathbf\{\+0\.0500\}\[\+0\.0398,\+0\.0602\]\[\+0\.0398,\\,\+0\.0602\]2\.6×10−212\.6\{\\times\}10^\{\-21\}1\.2×10−201\.2\{\\times\}10^\{\-20\}EigenScore\([Chen et al\. 2024a](https://arxiv.org/html/2608.11922#bib.bib4)\)\+0\.0526\\mathbf\{\+0\.0526\}\[\+0\.0427,\+0\.0622\]\[\+0\.0427,\\,\+0\.0622\]5\.1×10−265\.1\{\\times\}10^\{\-26\}4\.4×10−254\.4\{\\times\}10^\{\-25\}SPS\([Hu et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib11)\)\+0\.1965\\mathbf\{\+0\.1965\}\[\+0\.1840,\+0\.2091\]\[\+0\.1840,\\,\+0\.2091\]8\.4×10−2068\.4\{\\times\}10^\{\-206\}3\.8×10−1693\.8\{\\times\}10^\{\-169\}Trained reranker \(respondent\-free\)GainRAG\([Jiang et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib13)\)\+0\.0393\\mathbf\{\+0\.0393\}\[\+0\.0289,\+0\.0496\]\[\+0\.0289,\\,\+0\.0496\]6\.3×10−146\.3\{\\times\}10^\{\-14\}2\.2×10−132\.2\{\\times\}10^\{\-13\}MBA\-RAG\([Tang et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib32)\)\+0\.0579\\mathbf\{\+0\.0579\}\[\+0\.0468,\+0\.0689\]\[\+0\.0468,\\,\+0\.0689\]4\.3×10−254\.3\{\\times\}10^\{\-25\}4\.4×10−244\.4\{\\times\}10^\{\-24\}CRITIC\-R1\([Xiao et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib36)\)\+0\.0719\\mathbf\{\+0\.0719\}\[\+0\.0613,\+0\.0826\]\[\+0\.0613,\\,\+0\.0826\]4\.6×10−394\.6\{\\times\}10^\{\-39\}1\.3×10−361\.3\{\\times\}10^\{\-36\}Search\-based guidanceGEPA\([Agrawal et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib1)\)\+0\.0183\\mathbf\{\+0\.0183\}\[\+0\.0105,\+0\.0262\]\[\+0\.0105,\\,\+0\.0262\]4\.1×10−64\.1\{\\times\}10^\{\-6\}5\.7×10−65\.7\{\\times\}10^\{\-6\}MIPROv2\([Opsahl\-Ong et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib22)\)\+0\.0212\\mathbf\{\+0\.0212\}\[\+0\.0134,\+0\.0290\]\[\+0\.0134,\\,\+0\.0290\]1\.4×10−71\.4\{\\times\}10^\{\-7\}1\.8×10−71\.8\{\\times\}10^\{\-7\}Reference lines*rank1*\+0\.0575\\mathbf\{\+0\.0575\}\[\+0\.0464,\+0\.0685\]\[\+0\.0464,\\,\+0\.0685\]9\.1×10−259\.1\{\\times\}10^\{\-25\}9\.0×10−249\.0\{\\times\}10^\{\-24\}*oracle**−0\.1943\-0\.1943**\[−0\.2035,−0\.1853\]\[\-0\.2035,\\,\-0\.1853\]*≈0\\approx 05\.6×10−2875\.6\{\\times\}10^\{\-287\}Polarizer ablationLodestarw/o polarizer\+0\.0197\\mathbf\{\+0\.0197\}\[\+0\.0128,\+0\.0267\]\[\+0\.0128,\\,\+0\.0267\]2\.1×10−82\.1\{\\times\}10^\{\-8\}2\.4×10−82\.4\{\\times\}10^\{\-8\}

EveryΔ​F1\\Delta F\_\{1\},95%95\\%confidence interval andpp\-value behind these comparisons is in Table[7](https://arxiv.org/html/2608.11922#A4.T7), and the exact\-match tests are in the last two columns of Table[9](https://arxiv.org/html/2608.11922#A6.T9)\(Appendix[F](https://arxiv.org/html/2608.11922#A6)\)\. “Winning all7070cells” is a statement about cell\-wise values rather than about per\-dataset statistical significance; the statistically grounded form of the claim is “highest meanF1F\_\{1\}, paired\-significant on both tests against all fourteen baseline configurations ranked”\.

#### Plain entropy against the retrieval prior, respondent by respondent\.

The polarizer’s value is easiest to read where the signal it steers has none\. On NQ, and within a single pass that recomputes both configurations question by question \(n=1,000n=1\{,\}000\), selection by first\-tokenH1H\_\{1\}alone is statistically indistinguishable from taking the retriever’s top\-ranked passage on the two Qwen respondents\. Each figure comes from that respondent’s own pass, which recomputes both configurations together, so the contrast is exactly paired within it\. On Qwen2\.5\-7B the difference is\+0\.0139\+0\.0139\[−0\.0113,\+0\.0391\]\[\-0\.0113,\\,\+0\.0391\],pw=0\.42p\_\{w\}=0\.42; on Qwen3\.5\-9B it is\+0\.003\+0\.003,pw=0\.64p\_\{w\}=0\.64\. Llama\-3\.1\-8B is the one respondent of the three on which the untrained signal separates from the retrieval prior by itself—0\.46430\.4643againstrank1’s0\.42110\.4211on its own NQ pass, and0\.51480\.5148against0\.47690\.4769over the five pools \(Table[2](https://arxiv.org/html/2608.11922#S4.T2)\)\. Each respondent’s polarizer is trained on that same first\-token statistic, so the comparison is matched, and inserting it carries those two pastrank1, to0\.37010\.3701over0\.34520\.3452and0\.40860\.4086over0\.35170\.3517\. Two qualifications belong beside that\. The all\-token estimator, which we do not train on,*does*separate fromrank1on Qwen2\.5\-7B \(\+0\.0395\+0\.0395\[\+0\.0143,\+0\.0649\]\[\+0\.0143,\\,\+0\.0649\],pw=2\.8×10−3p\_\{w\}=2\.8\\times 10^\{\-3\}\), so the failure is first\-tokenH1H\_\{1\}’s rather than every untrained signal’s\. And a signal’s standalone value does not predict what the reward can extract from it, Qwen3\.5\-9B having the weakest standalone signal of the three and the largest gain over its ownH1H\_\{1\}configuration,\+0\.0544\+0\.0544,pw=8\.3×10−10p\_\{w\}=8\.3\\times 10^\{\-10\}\.

## Appendix ERun and seed stability

Table[2](https://arxiv.org/html/2608.11922#S4.T2)reportsLodestaras the mean of three training runs and lists those runs beneath itsLodestarrow\. This appendix says what separates them, which is the*training\-data seed*and nothing else\. Each of the three is its own300300\-step run under the inference\-time configuration, evaluated at its final checkpoint, and each consumes its own training file built by the same builder under seed4242,4343or4444\. That seed draws4,8004\{,\}800rows of1616question groups per step from the same1,9391\{,\}939\-question training pool, so300300steps is one epoch over the file and what the seed varies is the grouping and ordering of the training questions, not which questions exist\. Nothing else differs: the train/held\-out split seed is pinned at4242in all three, so they share one held\-out split byte for byte, and the trainer’s own seed is never overridden\. They land at meanF1F\_\{1\}0\.53430\.5343\(seed4242\),0\.52830\.5283\(seed4343\) and0\.53900\.5390\(seed4444\), that is0\.5339±0\.00540\.5339\\pm 0\.0054, with a gain over entropy of\+0\.0191±0\.0054\+0\.0191\\pm 0\.0054and every run winning all five datasets against both entropy selection andrank1\. Two properties of that spread matter for how the rest of the paper should be read\. No run involves a checkpoint choice of any kind, each being the final step of the fixed budget \(Appendix[B](https://arxiv.org/html/2608.11922#A2)\)\. And the seed\-4242run, which carries every paired test in this paper, is the*median*of the three rather than the best of them, so the tested run is a representative draw and not a selected peak\.

#### An execution floor for that spread\.

A fourth run places the±0\.0054\\pm 0\.0054against something\. The seed\-4242configuration was trained a second time on the identical training file, again for the full300300\-step budget and read at its own final checkpoint, and it lands at0\.53330\.5333against that seed’s0\.53430\.5343\. What separates the two is execution alone, meaning floating\-point accumulation order, rollout sampling, and where the reward readers were placed\. The variability therefore has three levels of its own\. A same\-seed re\-run moves the five\-dataset mean by0\.00100\.0010, a change of training\-data seed moves it by0\.00540\.0054\(sd\), and the method effect across the three seeds runs from\+0\.0135\+0\.0135to\+0\.0243\+0\.0243, so the quantity this paper reports is more than an order of magnitude above the noisiest level beneath it\.

## Appendix FExact\-match results and paired tests

Table[9](https://arxiv.org/html/2608.11922#A6.T9)repeats the comparison of Table[2](https://arxiv.org/html/2608.11922#S4.T2)under exact match, with the same three training seeds averaged into theLodestarrow and listed beneath it\. The ordering is largely preserved and every claim made onF1F\_\{1\}holds under EM\. Against the fourteen ranked configurationsLodestartakes the highest macro exact match,0\.41360\.4136against official semantic entropy’s0\.40390\.4039, and it does so without winning every cell: on WebQuestions the three\-seed mean trails that method,0\.22830\.2283against0\.23000\.2300, a deficit belonging to the mean more than to every run, since seed4444takes the cell on its own at0\.23100\.2310while seeds4242and4343land at0\.22800\.2280and0\.22600\.2260\. The system\-level Self\-RAG row, marked§in the table, is excluded from that count, because it answers with its own trained generator in an extraction configuration and is not frozen\-respondent\-comparable; that disclosure is given under*Trained generators*in Appendix[J](https://arxiv.org/html/2608.11922#A10), with every other per\-row configuration\. Thepolarizer ablationrow carries no seed breakdown, that configuration having no polarizer and being identical under all three training seeds\. Our CTRL\-RAG reproduction is not a row here either, as in Table[2](https://arxiv.org/html/2608.11922#S4.T2); it is reported in the case study of Appendix[I](https://arxiv.org/html/2608.11922#A9), where its exact\-match numbers are tabulated\.

The last two columns of Table[9](https://arxiv.org/html/2608.11922#A6.T9)repeat the paired tests of Table[7](https://arxiv.org/html/2608.11922#A4.T7)on the per\-question EM differences of the same5,0085\{,\}008aligned questions\. Their values are aligned on the×\\timesso that the exponents can be compared down the column, and every one is printed rather than marked, so how close a comparison runs to0\.050\.05can be read off its own row\. Every conclusion of theF1F\_\{1\}tests survives under EM, the narrowest margin being the lead over official semantic entropy, which clears0\.050\.05on both tests but by less than any other row \(pt=2\.3×10−2p\_\{t\}=2\.3\\times 10^\{\-2\},pWilcoxon=4\.8×10−2p\_\{\\text\{Wilcoxon\}\}=4\.8\\times 10^\{\-2\}\), which is unsurprising for a coarser integer\-valued score on what was already the closest comparison underF1F\_\{1\}\. The polarizer ablation, the cleanest test of the contribution, remains significant under EM on both tests\.

#### LLM\-as\-judge protocol\.

Because bothF1F\_\{1\}and EM are lexical, we additionally audit answer correctness with an LLM judge, so that no conclusion rests on string overlap alone\. To keep the audit independent of our own design choices, the judging prompt is adopted*verbatim*from the official code release of the semantic\-entropy paper\([Farquhar et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib7)\), in its multi\-reference branch, since every dataset here ships a list of gold answers\. The judge isgpt\-4o\-2024\-08\-06\([OpenAI 2024](https://arxiv.org/html/2608.11922#bib.bib20)\), queried at temperature00with a ten\-token decoding budget and constrained to a yes/no verdict, which we map to11and00; a reply that is neither is re\-queried once and otherwise counted as incorrect\. With the per\-example fields as placeholders, the prompt is reproduced in Table[8](https://arxiv.org/html/2608.11922#A6.T8)\.

Table 8:The LLM\-judge prompt, adopted*verbatim*from the official release of[Farquhar et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib7)in its multi\-reference branch\. Braced tokens are the per\-example fields substituted at query time\.Prompt templateInstruction
We are assessing the quality of answers to the following question: \{question\}
The following are expected answers to this question: \{correct\_answers\}\.
The proposed answer is: \{predicted\_answer\}
Within the context of the question, does the proposed answer mean the same as any of the expected answers? Respond only with yes or no\.Constrained answer cue
Response:Table 9:Exact match on the same pools and methods as Table[2](https://arxiv.org/html/2608.11922#S4.T2)\.Its rows differ from that table’s in two places: Min\-K%\+\+, printed there for completeness, is not repeated, and the system\-level Self\-RAG row \(§\) is added\. Every cell is11where the respondent’s generated answer matches a gold answer after SQuAD normalization and00otherwise, averaged over the pool, so the rows carry that table’s one\-row\-per\-method rule and family grouping under a second metric\. The last two columns are that row’s paired tests againstLodestaron per\-question EM differences, under the protocol of Appendix[D](https://arxiv.org/html/2608.11922#A4); a — there marks a row that admits no per\-question vector, and§a system\-level row, whose test therefore compares inference classes rather than selection quality\. TheLodestarrow is the mean of the three training seeds listed beneath it, as in Table[2](https://arxiv.org/html/2608.11922#S4.T2); reference lines keep their band’s italic, and the*oracle*pair is the one comparison that runs againstLodestarrather than for it\.Answer exact match \(EM\)↑\\uparrowPaired tests vs\.Lodestar\(ours\)MethodReferenceIn\-domainOut\-of\-domainMeanptp\_\{t\}pWilcoxonp\_\{\\text\{Wilcoxon\}\}NQSQuADTriviaQAEntityQWebQRL\-learned polarizer![[Uncaptioned image]](https://arxiv.org/html/2608.11922v1/figures/assets/noto_2734.png)Lodestar\(ours\)✦·✧·✦·⋆\\star★··⋆\\star·✦·✧··⋆\\star·✦0\.36630\.48170\.64230\.34920\.22830\.4136——– seed 420\.36700\.47700\.64200\.36010\.22800\.4148——– seed 430\.36200\.48000\.64700\.33830\.22600\.4107——– seed 440\.37000\.48800\.63800\.34920\.23100\.4152——Published signals re\-purposed as selectorsSemantic entropy\([Farquhar et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib7)\)0\.35900\.47100\.63000\.32940\.23000\.40392\.3×10−24\.8×10−2SeaKR\([Yao et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib39)\)0\.34500\.46300\.62100\.32740\.22300\.39594\.4×10−54\.1×10−4Self\-RAG \(reflection\)\([Asai et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib2)\)0\.34500\.44500\.60700\.31350\.20600\.38334\.3×10−94\.0×10−7Min\-K% Prob\([Shi et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib28)\)0\.32600\.44100\.61500\.30260\.21400\.37972\.1×10−132\.5×10−10CLeHe\([Qiu et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib23)\)0\.32100\.45300\.59600\.30950\.21200\.37832\.3×10−159\.0×10−12EPR\([Moslonka et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib19)\)0\.32100\.44400\.58900\.30750\.21500\.37536\.1×10−181\.1×10−13EigenScore\([Chen et al\. 2024a](https://arxiv.org/html/2608.11922#bib.bib4)\)0\.31200\.44300\.59300\.30260\.20500\.37111\.6×10−172\.4×10−13IGP\([Song et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib29)\)0\.31500\.40500\.60300\.29860\.18200\.36073\.6×10−231\.8×10−17SPS\([Hu et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib11)\)0\.21200\.18000\.49700\.16570\.16500\.24397\.1×10−1476\.0×10−98Trained reranker \(respondent\-free\)GainRAG\([Jiang et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib13)\)0\.32800\.40100\.60400\.31940\.20100\.37079\.0×10−164\.5×10−12MBA\-RAG\([Tang et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib32)\)0\.31400\.41200\.57500\.30560\.18900\.35917\.4×10−221\.6×10−16CRITIC\-R1\([Xiao et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib36)\)0\.30100\.38000\.54800\.26690\.19300\.33782\.6×10−424\.3×10−31Search\-based guidanceMIPROv2\([Opsahl\-Ong et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib22)\)0\.34600\.47400\.62200\.33230\.21900\.39871\.2×10−48\.8×10−4GEPA\([Agrawal et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib1)\)0\.34300\.46800\.62900\.32740\.21800\.39711\.7×10−52\.0×10−4Trained generatorSelf\-RAG§\([Asai et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib2)\)0\.39900\.08000\.29900\.09230\.11200\.19651\.2×10−2011\.3×10−128Reference lines*random**0\.1928**0\.1731**0\.4795**0\.1752**0\.1550**0\.2351*——*rank1**0\.3160**0\.4120**0\.5750**0\.3056**0\.1890**0\.3595**1\.5**×**10−21**2\.6**×**10−16**oracle**0\.5710**0\.6930**0\.7700**0\.5099**0\.3960**0\.5880**2\.4**×**10−229**3\.5**×**10−143*Polarizer ablationLodestarw/o polarizer*ψ⋆\\psi^\{\\star\}removed: plainarg⁡minp⁡H1\\arg\\min\_\{p\}H\_\{1\}*0\.36200\.46000\.64000\.34330\.21600\.40432\.1×10−37\.6×10−3

## Appendix GLLM\-judge scores per dataset

Table[2](https://arxiv.org/html/2608.11922#S4.T2)reports the LLM\-judge column as a five\-pool mean; Table[10](https://arxiv.org/html/2608.11922#A7.T10)opens it up per dataset, in the same layout and row order\. TheF1F\_\{1\}ordering is reproduced pool by pool with one exception worth stating: on WebQuestions the Self\-RAG \(reflection\) configuration leads the seed\-4242run that carries this table’s paired tests,0\.55100\.5510against0\.54300\.5430\(0\.54970\.5497for the three\-seed mean\), where under tokenF1F\_\{1\}Lodestarwins that cell\. The disagreement is confined to one pool and does not move the mean, whereLodestarleads the best baseline by\+0\.0208\+0\.0208on that run and by\+0\.0223\+0\.0223on the three\-seed mean, but it is the kind of per\-pool reversal a five\-pool average hides, which is why we print the breakdown\.

Table 10:LLM\-judge accuracy per dataset\(GPT\-4o answer\-equivalence, five pools\), opening up the five\-pool mean of Table[2](https://arxiv.org/html/2608.11922#S4.T2)\. Same rows and grouping as Table[9](https://arxiv.org/html/2608.11922#A6.T9), withLodestarin bold and reference lines in italic\. The last two columns are that row’s paired tests againstLodestaron per\-question judge verdicts, under the protocol of Appendix[D](https://arxiv.org/html/2608.11922#A4), and the*oracle*pair is the one comparison that runs againstLodestarrather than for it\. TheLodestarrow is the mean of the three training seeds listed beneath it, as in Table[2](https://arxiv.org/html/2608.11922#S4.T2), while the two test columns read the seed\-4242run alone\. A — is never a zero and never a test that fell short: in the first six columns it is a measurement this table does not report, and in the last two a comparison we did not run\.In\-domainOut\-of\-domainPaired tests vs\.Lodestar\(ours\)MethodReferenceNQSQuADTriviaQAEntityQWebQMeanptp\_\{t\}pWilcoxonp\_\{\\text\{Wilcoxon\}\}RL\-learned polarizer![[Uncaptioned image]](https://arxiv.org/html/2608.11922v1/figures/assets/noto_2734.png)Lodestar\(ours\)✦·✧·✦·⋆\\star★··⋆\\star·✦·✧··⋆\\star·✦0\.58030\.72370\.75700\.60680\.54970\.6435——– seed 420\.57800\.72100\.75300\.61510\.54300\.6420——– seed 430\.57200\.72200\.75900\.58330\.54600\.6365——– seed 440\.59100\.72800\.75900\.62200\.56000\.6520——Published signals re\-purposed as selectorsSelf\-RAG \(reflection\)\([Asai et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib2)\)0\.55200\.68700\.72200\.59420\.55100\.62127\.6×10−43\.6×10−3Semantic entropy\([Farquhar et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib7)\)0\.55200\.66100\.73900\.53170\.51700\.60013\.5×10−133\.6×10−10CLeHe\([Qiu et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib23)\)0\.55400\.68400\.73200\.53770\.52400\.60635\.1×10−111\.5×10−8EPR\([Moslonka et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib19)\)0\.55300\.67400\.72700\.53570\.52300\.60258\.0×10−136\.7×10−10SeaKR\([Yao et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib39)\)0\.53900\.68600\.73300\.52180\.52300\.60061\.9×10−144\.2×10−11IGP\([Song et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib29)\)0\.54500\.61700\.72900\.55260\.52400\.59351\.9×10−157\.9×10−12Min\-K% Prob\([Shi et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib28)\)0\.53500\.64700\.72800\.51790\.51500\.58861\.3×10−212\.4×10−16EigenScore\([Chen et al\. 2024a](https://arxiv.org/html/2608.11922#bib.bib4)\)0\.50500\.66000\.70900\.50600\.49700\.57548\.6×10−303\.1×10−22SPS\([Hu et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib11)\)0\.36900\.32600\.59300\.29860\.45100\.40755\.0×10−2123\.9×10−134Trained reranker \(respondent\-free\)GainRAG\([Jiang et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib13)\)0\.55600\.61200\.72400\.60020\.53500\.60543\.9×10−93\.7×10−7MBA\-RAG\([Tang et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib32)\)0\.51200\.62200\.68400\.56250\.49300\.57473\.3×10−243\.2×10−18CRITIC\-R1\([Xiao et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib36)\)0\.52900\.61300\.68700\.50400\.52300\.57121\.6×10−271\.3×10−20Search\-based guidanceGEPA\([Agrawal et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib1)\)0\.56700\.71000\.74900\.54960\.52700\.62057\.0×10−61\.0×10−4MIPROv2\([Opsahl\-Ong et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib22)\)0\.55300\.69000\.74900\.56350\.51500\.61419\.3×10−97\.0×10−7Trained generatorSelf\-RAG§\([Asai et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib2)\)0\.51900\.61100\.68500\.61610\.55700\.59762\.6×10−105\.0×10−8Reference lines*random**0\.3560**0\.3195**0\.5797**0\.3008**0\.4057**0\.3923*——*rank1**0\.5130**0\.6220**0\.6840**0\.5625**0\.4930**0\.5749**4\.4**×**10−24**3\.9**×**10−18**oracle**0\.8240**0\.9220**0\.8710**0\.7778**0\.7600**0\.8310**1\.4**×**10−255**2\.1**×**10−156*Polarizer ablationLodestarw/o polarizer*ψ⋆\\psi^\{\\star\}removed: plainarg⁡minp⁡H1\\arg\\min\_\{p\}H\_\{1\}*0\.56000\.68300\.74900\.55850\.51400\.61293\.4×10−121\.9×10−9

Every mean in Table[10](https://arxiv.org/html/2608.11922#A7.T10)reproduces the corresponding cell of Table[2](https://arxiv.org/html/2608.11922#S4.T2)exactly wherever both tables report one, which is the check that the two columns are the same measurement and not two independent judge runs\. Semantic entropy’s0\.60010\.6001is opened up per pool here as well, having been obtained the same way: the corrected run’s cached selections looked up in that one judging pass, with no query of its own\.

The two paired\-test columns are the one place in this table where the three training seeds do not average\. TheirLodestarconfiguration is the seed\-4242run’s per\-question verdict file, whose five means are the seed\-4242row, so neither test speaks for the three\-seed mean that theLodestarrow and Table[2](https://arxiv.org/html/2608.11922#S4.T2)report\. They add no query to the judging the rest of the table already rests on, that pass having scored all ten candidates of every one of the5,0085\{,\}008questions, so each configuration’s per\-question verdict is the verdict at the rank it selected\. A — in those two columns is therefore a comparison we did not run rather than a test that ran and fell short, and only two kinds of row still carry one\.Lodestarand its seeds are the comparison’s own reference, and the*random*line is an expectation over the ten ranks rather than a per\-question selection, so neither admits a paired vector at all\. The system\-level Self\-RAG row is tested and reads§: that method trains its own answering model, so its test compares inference classes rather than selection quality and the row stays outside the fourteen frozen\-respondent\-comparable configurations of Table[2](https://arxiv.org/html/2608.11922#S4.T2), which it is not a row of\. The values themselves are aligned on the×\\timesso that the exponents can be compared down the column\.

#### Auditing the labelling judge\.

The judge above scores answers; the passage labels of Section[3](https://arxiv.org/html/2608.11922#S3)come from a separate pair of labelling judges, whose inter\-judge agreement is Cohen’sκ=0\.675\\kappa=0\.675\. A separate benchmark audits the same judge pair against an independent 70B bf16 third\-party judge, Llama\-3\.1\-70B\-Instruct\([Grattafiori et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib9)\), on1,3001\{,\}300held\-out rows: agreement with the dual\-judge consensus label, measured on the600600consensus rows, is96%96\\%, with a1\.5%1\.5\\%false\-positive rate on200200no\-injection anchors\. The two figures are not the same measurement and do not confirm each other:κ\\kappais chance\-corrected and the96%96\\%is not, and the two are computed on different sets of rows\.

## Appendix HLearned polarizer text and its effect on the entropy

The converged strings are interpretable, and independent seeds agree on what to say even though the reward never scores wording\. The inference\-timeψ⋆\\psi^\{\\star\}opens by*directing doubt*: “*The passage may address a similar entity or context butnotthe specific one asked for, and it might contain outdated, fictional, or misattributed information*…”\. The second seed converges on a paraphrase of the same instruction: “*a related or similarly named topic but doesnotdirectly or correctly answer the questioned topic; it could be referencing a different entity, event, or timeframe*…”\. Since the reward is defined purely onH1H\_\{1\}separation, this convergence is evidence that the objective has a stable linguistic optimum rather than that any phrasing was supplied\.

The opening clause is not, however, a reliable predictor of quality\. The third seed instead opens by*affirming*: “*The passage maydirectly addressthe surface\-level query but could mislead by providing outdated, incomplete, or contextually adjacent information*…”\. It reaches meanF1F\_\{1\}0\.52830\.5283, below both direct\-doubt runs \(0\.53430\.5343and0\.53900\.5390\) yet above plain entropy selection’s0\.51480\.5148on the same pools\. All three runs win all five datasets against both entropy selection andrank1\(Appendix[E](https://arxiv.org/html/2608.11922#A5)\)\. Affirm\-then\-qualify and direct\-doubt framings are therefore both reachable optima of the same objective, and we report the wording as interpretable rather than as diagnostic\.

Table 11:The polarizer across the300300\-step budget, greedy\-decoded from the five retained checkpoints of the deployed run, one box per checkpoint\. The decode is still moving through the first three; steps240240and300300emit byte\-identical strings, so the four boxes cover all five checkpoints, and the last of them is theψ⋆\\psi^\{\\star\}every number in this paper is computed from\.Step 60 — still moving The passage may directly address a surface\-level fact but could mislead if it refers to a related entity, a different time period, a variant form of the subject, or a fictionalized version rather than the actual entity\. It might also contain outdated statistics, approximate numbers, or misattributed details that appear plausible but are not accurate for the specific query\.

Step 120 — still moving The passage may address a similar entity or context but not the specific one asked for, and it might contain outdated, approximate, or misattributed figures that do not accurately reflect the correct quantity or detail\. It could also misrepresent the scope or boundaries of the subject, leading to a misleading answer\.

Step 180 — still moving The passage may address a similar entity or context but not the specific one asked for, and it might contain outdated, approximate, or misattributed figures that misrepresent the correct answer\.

Steps 240 and 300 — byte\-identical, and the deployedψ⋆\\psi^\{\\star\} The passage may address a similar entity or context but not the specific one asked for, and it might contain outdated, fictional, or misattributed information that misrepresents the true answer\.

#### How the string is reached, and in what sense it is one string\.

Table[11](https://arxiv.org/html/2608.11922#A8.T11)showsψ⋆\\psi^\{\\star\}at five checkpoints spanning the300300\-step budget of the deployed run\. The wording is not stable early: across the ten checkpoints dumped every3030steps the greedy decode takes seven distinct values, and the early strings are longer and carry extra qualifying clauses \(375375characters at step6060against194194at step300300\)\. The deployed string first appears at step240240and is byte\-identical at270270and300300\.

Within a step,ψ⋆\\psi^\{\\star\}is the survivor of a group rather than a single output:πθ\\pi\_\{\\theta\}samplesG=8G=8polarizers at temperature1\.11\.1and GRPO ranks them against one another \(Table[5](https://arxiv.org/html/2608.11922#A2.T5)\)\. Those training rollouts were not written to disk, so the within\-step spread*during*training cannot be recovered from this run\. What can be measured is the converged policy re\-sampled afterwards, and it is close to a point mass: drawing88samples from the final checkpoint at the training temperature returns eight byte\-identical strings, and widening to512512draws at temperature1\.01\.0returns the deployedψ⋆\\psi^\{\\star\}502502times \(98\.0%98\.0\\%\), the ten that differ occurring once each and moving only inside the clause that lists what may be wrong with the passage\. A control at temperature2\.02\.0returns six distinct strings from eight draws, so the sampler is live rather than silently greedy\. This measures the policy after training rather than the group the optimizer ranked at step300300, and it speaks to how the reported string was obtained, not to whether the polarizer helps\.

#### What the string does to the entropy\.

Decomposing the shiftH1​\(q,p,ψ⋆\)−H1​\(q,p\)H\_\{1\}\(q,p,\\psi^\{\\star\}\)\-H\_\{1\}\(q,p\)by passage class over the five pools, a zero\-inference calculation on the cached probes, showsψ⋆\\psi^\{\\star\}acting almost entirely on one side\. Passages the respondent answers correctly from move by\+0\.007\+0\.007on average and passages it fails on by\+0\.258\+0\.258, a difference\-in\-differences of\+0\.251\+0\.251\. The effect concentrates where selection fails\. The per\-question*lure*, the wrong passage with the lowest plain entropy and therefore the one a plain\-entropy selector actually picks, is pushed up by\+0\.377\+0\.377, against\+0\.106\+0\.106for that question’s best supporting passage\. These figures are a descriptive decomposition of cached probe values, not a causal account of what the reward taught the polarizer policy\.

## Appendix ICTRL\-RAG reproduction

CTRL\-RAG\([Tan et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib31)\)is a concurrent preprint that trains the generator for context faithfulness rather than a selector, so it is a paradigm relative rather than a same\-task competitor\. We reproduced it end\-to\-end and report it as a case study rather than as a row of Tables[2](https://arxiv.org/html/2608.11922#S4.T2),[7](https://arxiv.org/html/2608.11922#A4.T7)and[9](https://arxiv.org/html/2608.11922#A6.T9)for one measurable reason:*the run has no stable protocol\-compliant checkpoint*, so every candidate number would misrepresent it, and each would do so differently\. The clean short\-answer format survives to roughly9090–106106steps of the300300\-step protocol; after that the raw output collapses and stays collapsed—macroF1F\_\{1\}0\.13440\.1344at step120120,0\.12760\.1276at step300300\(Table[13](https://arxiv.org/html/2608.11922#A9.T13)\),*every*exact match lost at both—and the extraction rule we pre\-registered against the earlier failure mode fires on only9%9\\%–10%10\\%of questions against the next variant\. Table[12](https://arxiv.org/html/2608.11922#A9.T12)still prints both trained generators besideLodestar, and each of its two rows needs its provenance stated\. The CTRL\-RAG cells are the strongest checkpoint of the run, RL step8080, so Table[13](https://arxiv.org/html/2608.11922#A9.T13)’s trajectory rather than that row is the reproduction’s full record, and its judge column carries the same five\-pool macro, so step8080reads0\.69250\.6925in both tables\. The Self\-RAG cells are that method’s system\-level row, the configuration its§disclosure attaches to in Table[9](https://arxiv.org/html/2608.11922#A6.T9), where its per\-pool exact match is printed\. Self\-RAG still selects one passage, as every ranked row does, while CTRL\-RAG reads all ten in retrieval order and exposes no selection step at all\.

Table 12:Trained generators besideLodestar, in the format of Table[2](https://arxiv.org/html/2608.11922#S4.T2)\.Both comparators train the answering model itself, so these are comparisons of inference class rather than of selection quality, and the two rows sit outside Table[2](https://arxiv.org/html/2608.11922#S4.T2)’s frozen\-respondent ranking\. Therespondentcolumn marks that model as frozen \( \) or fine\-tuned \( \), and the judge column is the five\-pool GPT\-4o macro of Appendix[G](https://arxiv.org/html/2608.11922#A7)’s protocol\.RespondentAnswerF1F\_\{1\}↑\\uparrow\(generated answer vs\. gold answers\)MethodReferenceIn\-domainOut\-of\-domainMeanLLM\-Judge\(GPT\-4o\)NQSQuADTriviaQAEntityQWebQ![[Uncaptioned image]](https://arxiv.org/html/2608.11922v1/figures/assets/noto_2734.png)Lodestar\(ours\)0\.47830\.59160\.71630\.49430\.39080\.53430\.6420Self\-RAG\([Asai et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib2)\)0\.47960\.25870\.46840\.29770\.31470\.36380\.5976CTRL\-RAG\([Tan et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib31)\)0\.54920\.63180\.71980\.58170\.47750\.59200\.6925

#### The strongest checkpoint, stated plainly\.

The reproduction beatsLodestaron every pool, its best checkpoint reaching meanF1F\_\{1\}0\.5920\\mathbf\{0\.5920\}againstLodestar’s0\.53430\.5343\(Table[13](https://arxiv.org/html/2608.11922#A9.T13)\)\. That reference row is the seed\-4242training run in every column, where Table[2](https://arxiv.org/html/2608.11922#S4.T2)reports the three\-seed mean,0\.53390\.5339macroF1F\_\{1\}and0\.41360\.4136EM\. Every row of the table is read under the raw protocol, on which the pre\-registered extraction step changes step2020by less than10−410^\{\-4\}, and the mid RL row is the last checkpoint whose sampled responses never hit the length cap\. Against step2020, on the same5,0085\{,\}008aligned questions, the paired difference runs*against*us,Δ​F1=−0\.0533\\Delta F\_\{1\}=\-0\.0533\[−0\.0646,−0\.0420\]\[\-0\.0646,\-0\.0420\]and−0\.0665\-0\.0665\[−0\.0792,−0\.0538\]\[\-0\.0792,\-0\.0538\]under EM\. Two qualifications belong next to that number, and neither subtracts from it\. The comparison is one of inference class rather than selection quality, since this checkpoint is a gold\-supervised, fully fine\-tuned copy of the very respondent every ranked row keeps frozen and it reads all ten passages per query\. What that class costs is the concrete form of the distinction\. It requires gold answer supervision, per\-domain fine\-tuning, serving of the respondent itself and a ten\-passage prefill per query, none of which is available in the frozen\-respondent settings this paper targets, whereLodestarselects at one forward pass per candidate, needs no gold answers, and serves no model beyond the respondent that answers\. And the margin is bought by*supervised fine\-tuning on in\-benchmark gold answers, not by the reinforcement objective the method is about*: the SFT anchor scores0\.58600\.5860before a single RL step, three hundred steps of RL add at most\+0\.0060\+0\.0060over it, and then remove everything\. We report the anchor row precisely so that this decomposition is visible rather than implied\.

#### What the judge says about the collapse\.

At step300300every string metric is destroyed—F1F\_\{1\}0\.12760\.1276and EM exactly0\.00000\.0000—yet the GPT\-4o judge still credits0\.58450\.5845of answers as equivalent, against0\.67270\.6727–0\.69250\.6925for the three intact checkpoints\. Roughly85%85\\%of the judge score survives anF1F\_\{1\}collapse to a fifth of its value, which is direct evidence that the collapse is a change of surface form and not a loss of content\. Every figure in that column is a five\-pool macro, so it reads both down the rows of Table[13](https://arxiv.org/html/2608.11922#A9.T13)and against the macros quoted elsewhere, whereLodestarstands at0\.64350\.6435over three seeds \(Table[10](https://arxiv.org/html/2608.11922#A7.T10)\) and at0\.64200\.6420on the seed\-4242run this table’s reference row reports\. Figure[3](https://arxiv.org/html/2608.11922#A9.F3)plots what the run was maximising while that surface form decayed\. The reward rises over exactly the steps in which the sampled responses pin to the length cap, and CTRL\-RAG’s objective scores no property of the output format\. The surface form that the string metrics of Table[13](https://arxiv.org/html/2608.11922#A9.T13)measure is therefore not a quantity this run was optimising; what it was optimising rewards length outright, as the next paragraph derives\.

Figure 3:CTRL\-RAG reproduction, the RL stage measured two ways\.Both panels are the SFT\-anchored seed\-4242run whose checkpoints Table[13](https://arxiv.org/html/2608.11922#A9.T13)reports, read from its training log\.*Top:*critic/score/mean, the quantity the objective maximises\.*Bottom:*the fraction of sampled responses truncated at the3232\-token cap\. That cap is not binding on a well\-formed answer: under the policy’s own tokenizer the1,8481\{,\}848gold answer strings of the NQ pool average3\.73\.7tokens and the longest runs to1313, so truncation records a change of output format rather than an answer that did not fit\. The reward rises as the format degrades, from mean0\.15770\.1577over the8484steps at which no response reaches the cap to mean0\.34660\.3466over the165165steps at which every response does\.Table 13:CTRL\-RAG reproduction, stage by stage\.The four CTRL\-RAG rows are its SFT anchor at zero RL steps and three points early, mid and late in the300300\-step budget, all read under the raw protocol, so a column is comparable down the rows\. Those three points are fixed by the run rather than selected from it:2020is the first checkpoint it wrote under its2020\-step save interval,8080is the last whose sampled responses never reach the length cap \(Figure[3](https://arxiv.org/html/2608.11922#A9.F3); the next saved checkpoint,100100, already truncates36%36\\%of them\), and300300is the final step of the budget, the checkpoint Appendix[B](https://arxiv.org/html/2608.11922#A2)’s no\-per\-run\-selection rule reports for every trained configuration\.*Judge*is the GPT\-4o answer\-equivalence protocol of Table[2](https://arxiv.org/html/2608.11922#S4.T2), macro\-averaged over the same five pools as every other judge figure in the paper\. Bold marks the best of the four CTRL\-RAG rows, andLodestaris a reference row that is the seed\-4242training run in every column, where Table[2](https://arxiv.org/html/2608.11922#S4.T2)reports the three\-seed mean\.
#### Why the optimum is format\-degenerate\.

The collapse is not an instability the run fell into; it is the objective being maximised\. Written out, the reward optimised here is

Rhybrid​\(y\)\\displaystyle R\_\{\\mathrm\{hybrid\}\}\(y\)=RCLR′​\(y\)⋅𝟏​\(y∗⊆y\),\\displaystyle=R^\{\\prime\}\_\{\\mathrm\{CLR\}\}\(y\)\\cdot\\mathbf\{1\}\(y^\{\*\}\\subseteq y\),RCLR​\(y\)\\displaystyle\\qquad R\_\{\\mathrm\{CLR\}\}\(y\)=ℰ⁡\(y\)​1​\(ℰ⁡\(y\)\>τ\)/T,\\displaystyle=\\mathcal\{E\}\(y\)\\,\\mathbf\{1\}\\big\(\\mathcal\{E\}\(y\)\>\\tau\\big\)\\big/\\sqrt\{T\},\(7\)ℰ⁡\(y\)\\displaystyle\\mathcal\{E\}\(y\)=S⁡\(y∣D\)−mind∈D\+⁡S⁡\(y∣D∖\{d\}\),\\displaystyle=S\(y\\mid D\)\-\\min\_\{d\\in D^\{\+\}\}S\(y\\mid D\\setminus\\\{d\\\}\),S\(y∣⋅\)\\displaystyle\\qquad S\(y\\mid\\cdot\)=∑t=1TlogP\(yt∣y<t,q,⋅\),\\displaystyle=\\textstyle\\sum\_\{t=1\}^\{T\}\\log P\(y\_\{t\}\\mid y\_\{<t\},q,\\cdot\),withTTthe answer’s token count andτ=1\\tau=1\. Two properties of that form push the same way onTT\. First,ℰ⁡\(y\)\\mathcal\{E\}\(y\)is a difference of two sums over the*same*TTtokens, so it is extensive in length: each additional token contributes one more per\-token contrast, while the normaliser grows only asT\\sqrt\{T\}\. Any positive average per\-token contrastcctherefore leavesRCLR≈c​TR\_\{\\mathrm\{CLR\}\}\\approx c\\sqrt\{T\}, rising without bound inTT—a sub\-linear normaliser does not cancel an extensive numerator, it only slows it, and the threshold pushes the same way because a longeryyclearsℰ⁡\(y\)\>τ\\mathcal\{E\}\(y\)\>\\taumore easily\. Second, the accuracy factor is a*containment*test, and containment is monotone under appending: oncey∗y^\{\*\}occurs inyy, no continuation can remove it\. Padding is thus free on the factor that checks correctness and strictly profitable on the factor that does not, and the in\-group min\-max rescale is order\-preserving, so in every group the longer surviving sibling carries the larger advantage\. The joint optimum is to emit the gold string and then continue to the cap, which is exactly the endpoint Table[13](https://arxiv.org/html/2608.11922#A9.T13)records: exact match0\.00000\.0000while the judge still credits0\.58450\.5845, i\.e\.y∗y^\{\*\}is still there and everything after it is reward being collected\. The analysis fixes the mechanism; Figure[3](https://arxiv.org/html/2608.11922#A9.F3)fixes its sign, the reward rising over precisely the steps in which the responses lengthen\. Note that this argument is a property of the reward’s shape alone, so the frozen\-scorer deviation of Appendix[J](https://arxiv.org/html/2608.11922#A10)does not bear on it\.

#### Scoped conclusion\.

Our claim is bounded by our adaptation, which Appendix[J](https://arxiv.org/html/2608.11922#A10)discloses in full:*under that adaptation*, the contrastive\-likelihood objective’s optimum is format\-degenerate, the SFT anchor delays that optimum in proportion to its strength without preventing it, and pre\-declared cleaning rules are evaded by the next variant\. We explicitly do*not*claim this of CTRL\-RAG at its published scale, where the official7474k SFT corpus, a batch of10241024and a reward that reads likelihoods from the policy itself rather than from a frozen copy are all untested here\. The length incentive we audited exists under any conditional likelihood, but its interaction with a larger anchor and a smaller gradient noise scale is an open question\.

## Appendix JBaseline configurations and disclosures

This appendix gives the per\-row configuration behind the bare method names of Table[2](https://arxiv.org/html/2608.11922#S4.T2), together with the disclosures about how those rows should be read\. Every configuration follows the fairness contract of the*Baselines*paragraph \(Section[4](https://arxiv.org/html/2608.11922#S4)\), and every method\-by\-dataset cell of theentropy,rank1andoraclecolumns is identical across all runs, confirming that every method ran on exactly the same items\. The three reference lines arerandom, the expectedF1F\_\{1\}of a uniform draw;rank1, the retriever’s own top\-ranked passage; andoracle, the best of the ten in hindsight\.

GainRAG, CRITIC\-R1 and MBA\-RAG are the only baselines that need a training artifact of their own, so they are the rows for which the paper’s own numbers cannot answer whether we ran the other method correctly or ran it badly in our own favour\. We therefore re\-ran all three from their stored artifacts on the WebQuestions pool, on different hardware from the original runs, and all three return their tabulated value exactly \(0\.37530\.3753,0\.36740\.3674and0\.34910\.3491;Δ​F1=0\.0000\\Delta F\_\{1\}=0\.0000in each case\)\. MBA\-RAG’s re\-run coincides with therank1line because its bandit selects rank 1 on all1,0001\{,\}000questions of that pool, which is that policy’s collapse onto the retrieval prior reproducing rather than a failed re\-run\.

### J\.1Training\-free selection signals

These rows score each candidate from the frozen respondent’s own response and select by that score, except forSPS, which scores the passage representation alone, and Self\-RAG \(reflection\), which scores with the method’s officially released critic\. None of them is trained for this paper\. Each row’s configuration and the disclosure attaching to it are given below, one method per entry, in Table[2](https://arxiv.org/html/2608.11922#S4.T2)’s order\. Multi\-sample methods use a uniformk=5k\\\!=\\\!5budget unless an official row is given\.

- •Semantic entropy\([Farquhar et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib7)\)runs in its official strict\-entailment form\. Its row is inflated by the table’s first\-wins tie\-break and should not be read as second place: under that setting it ties on6262–79%79\\%of questions with only∼2\\sim\\\!2distinct scores per1010candidates, so first\-wins silently imports a retrieval\-rank\-1 prior\. Breaking ties at random gives meanF1F\_\{1\}0\.47740\.4774, statistically indistinguishable fromrank1’s0\.47690\.4769\. Official code at[https://github\.com/jlko/semantic\_uncertainty](https://github.com/jlko/semantic_uncertainty)\.
- •Self\-RAG \(reflection\)\([Asai et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib2)\)selects with the method’s officially released critic\. We score it over the full vocabulary instead of the official top\-50005000log\-prob truncation, a deviation strictly favourable to it\. Official code at[https://github\.com/AkariAsai/self\-rag](https://github.com/AkariAsai/self-rag)\.
- •SeaKR\([Yao et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib39)\)runs at its officialn=20n\\\!=\\\!20\. Our row omits the official newline stop\-strings, fixed seed and empty\-sample filtering; restoring all three moves meanF1F\_\{1\}by\+0\.0006\+0\.0006, mixed in sign per dataset, and leaves every per\-dataset comparison againstLodestarunchanged\. Official code at[https://github\.com/THU\-KEG/SeaKR](https://github.com/THU-KEG/SeaKR)\.
- •CLeHe\([Qiu et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib23)\)is our*adaptation*of that method’s entropy signal into a selector\. The official method is a token\-level soft fusion that never selects a single passage, so this row carries no inferential value about the published method\. Official code at[https://github\.com/zexuanqiu/entropy\-based\-decoding](https://github.com/zexuanqiu/entropy-based-decoding)\.
- •EPR\([Moslonka et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib19)\)runs its official sequence\-mean rate\. Our row generates2424tokens where the official script uses200200; a rerun at the official budget moves meanF1F\_\{1\}by−0\.0012\-0\.0012, so the row we rank against is the marginally*stronger*configuration\. Official code at[https://github\.com/artefactory/artefactual](https://github.com/artefactory/artefactual), release tagECIR2026\.
- •Min\-K% Prob\([Shi et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib28)\)runs at its officialk=20k\\\!=\\\!20, adapted to selection by scoring each passage’s own cached greedy answer under the shared answering prompt\. Official code at[https://github\.com/swj0419/detect\-pretrain\-code](https://github.com/swj0419/detect-pretrain-code)\.
- •IGP\([Song et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib29)\)runs under its*published*configuration, which probes with its own short prompt rather than the answering prompt\. That misalignment, not the gain formulation, is the likely main cause of its position \(Section[2](https://arxiv.org/html/2608.11922#S2)\); the aligned\-probe ablation of Appendix[C](https://arxiv.org/html/2608.11922#A3)recovers about half the gap\.
- •EigenScore\([Chen et al\. 2024a](https://arxiv.org/html/2608.11922#bib.bib4)\)runs the released code atk=5k\\\!=\\\!5\(K=10K\\\!=\\\!10,T=0\.5T\\\!=\\\!0\.5,max\_new\_tokens=256=\\\!256\) without Feature Clipping, which the official repository also omits, so it corresponds to that paper’s “EigenScore \(w/o\)” variant\. The released code also follows top\-k=10k\\\!=\\\!10where the paper states55, and pools at the EOS position\. Official code at[https://github\.com/alibaba/eigenscore](https://github.com/alibaba/eigenscore)\.
- •SPS\([Hu et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib11)\)runs its released scoring function, which reads the candidate text and never the question\. In SPS’s own protocol the candidates are query\-conditioned rewrites of one evidence set, so nothing is lost by this; ours are ten*different*passages, and a score blind to the question cannot prefer the one that answers it\. Its low row is that mismatch, which we state before reporting it, and we keep the row beside therandomreference line so its position is checkable rather than asserted: its correlation with per\-candidate answerF1F\_\{1\}is−0\.005\-0\.005to−0\.082\-0\.082on our pools, its meanF1F\_\{1\}0\.33800\.3380sits at that line’s0\.32710\.3271\(within−0\.5\-0\.5to\+2\.4\+2\.4pairedzzper dataset\), the implementation was verified against the released code element by element \(subspace basis, PCA rule, max\-pooled penultimate representation, bare\-sentence input, residual score, argmin\), and the argmax control that would have exposed a sign error is worse on all five datasets\. Official code at[https://github\.com/HU\-xiaobai/Code\-of\-Beyond\-Perplexity\-Let\-the\-Reader\-Select\-Retrieval\-Summaries\-via\-Spectrum\-Projection\-Score](https://github.com/HU-xiaobai/Code-of-Beyond-Perplexity-Let-the-Reader-Select-Retrieval-Summaries-via-Spectrum-Projection-Score)\.
- •Min\-K%\+\+\([Zhang et al\. 2025a](https://arxiv.org/html/2608.11922#bib.bib41)\)runs the identical transplant, differing only in its vocabulary\-normalized per\-token statistic\. Its paper fixes no defaultkk, so the run also sweepsk=10\.\.100k\{=\}10\.\.100, and the best point \(k=10k\{=\}10, macro0\.29120\.2912\) still sits below every other ranked row\. Its below\-randomrow is a property of the statistic under this transplant, not an implementation defect: our reimplementation matches the official release term for term, and an argmin control does not rescue the score\. Official code at[https://github\.com/zjysteven/mink\-plus\-plus](https://github.com/zjysteven/mink-plus-plus)\.

Every ranked row with tie logging other than semantic entropy has a tie rate≤1\.2%\\leq 1\.2\\%and reproduces its first\-wins numbers under random tie\-breaking\. Plain first\-token entropy, the polarizer ablation ofLodestar, is not ranked as a baseline row of Tables[2](https://arxiv.org/html/2608.11922#S4.T2)and[9](https://arxiv.org/html/2608.11922#A6.T9); being our own method minus one component, it is reported with the ablations \(Section[6](https://arxiv.org/html/2608.11922#S6)\) and paired\-tested in Table[7](https://arxiv.org/html/2608.11922#A4.T7)\. At meanF1F\_\{1\}0\.51480\.5148it is nonetheless the strongest*untrained*selector in the benchmark, behind only the searched GEPA string among inference\-ready configurations, so whatLodestarimproves on is a demanding comparison rather than a favourable one\.

### J\.2Searched guidance

The prompt\-optimization rows ask whetherLodestar’s GRPO stage can be replaced by searching the polarizer directly\. Both configurations consumeLodestar’s exact reward function, training draws and frozen respondent, search under the same9696\-token polarizer cap, and select identically at inference \(arg⁡minp​H1​\(q,p,ψ\)\\arg\\min\_\{p\}H\_\{1\}\(q,p,\\psi\)\); only the optimizer differs\. Both also start from the*same hand\-written caution seed*, a manually written polarizer directing doubt at passages that look relevant but do not answer the exact question, and on the never\-searched pools the searched strings improve on it by at most\+0\.003\+0\.003meanF1F\_\{1\}\. That seed is itself evaluated under the identical protocol and reaches meanF1F\_\{1\}0\.51290\.5129and EM0\.40050\.4005,*below*the no\-polarizer ablation’s0\.51480\.5148: writing the polarizer by hand adds nothing over inserting no polarizer at all, so the ordering across the three ways of obtaining the string is hand\-written0\.51290\.5129, searched0\.51310\.5131–0\.51600\.5160, GRPO\-learned0\.53390\.5339\. Each optimizer’s configuration is given below, one per entry, in Table[2](https://arxiv.org/html/2608.11922#S4.T2)’s order\.

- •GEPA\([Agrawal et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib1)\)runs its official implementation, reflective evolution with Pareto candidate selection and gpt\-oss\-120b as the reflection LM, at two settings\. At its official sample\-efficiency setting, all six runs we launched—three seeds at each of two reflection\-minibatch sizes—returned the hand\-written seed unchanged, and not for want of proposals: at GEPA’s default minibatch of three,2323–2525of the reflection LM’s rewrites per run cleared the minibatch screen and were fully validated without ever displacing the seed on the Pareto frontier, and at a minibatch of ten no rewrite cleared the screen at all\. The second setting runs each seed to saturation under plateau early\-stopping; the table reports the strongest saturated seed\.
- •MIPROv2\([Opsahl\-Ong et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib22)\), a non\-reflective Bayesian instruction search, runs under the same harness at the same sample\-efficiency setting\. Unlike GEPA there, every seed improved on the hand\-written seed internally \(\+0\.005\+0\.005to\+0\.107\+0\.107separation\)—the same harness accepting improvements whenever the optimizer finds one, which rules out a harness fault behind GEPA’s unchanged returns—and the table reports the strongest seed\.

Each row therefore prints its strongest seed againstLodestar’s three\-seed mean, a comparison deliberately set in the baselines’ favour\. On the never\-searched pools, however, both searched strings land within\+0\.003\+0\.003meanF1F\_\{1\}of the untuned hand\-written caution—and significantly belowLodestar\(Table[7](https://arxiv.org/html/2608.11922#A4.T7)\)\. Section[5](https://arxiv.org/html/2608.11922#S5)reports the full comparison\.

### J\.3Trained selectors

These three rows replace the selection signal with a model trained on our data, while the same frozen respondent still answers\.

- •GainRAG\([Jiang et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib13)\)is the only baseline that, likeLodestar,*trains*on our data, so we control its training distribution exactly\. It enters only as its NQ\-trained*inference\-ready*distilled selector: its per\-candidate gain signal reads the gold answer and is therefore not a selector one can run at inference, so we report that signal at the end of this entry rather than as a table row\. Following its official recipe, per\-candidate gains are synthesised as contrastive\-decoding perplexities of the gold answer \(α=0\.5\\alpha\{=\}0\.5\) under the same frozen respondent and the same answering prompt that produced the cached utilities; a respondent\-generated pseudo\-passage joins the ten bge\-m3 candidates \(1111per question\); questions whose highest\-gain candidate fails GainRAG’s lenient containment\-EM filter are dropped; and bge\-reranker\-base is distillation\-finetuned on the soft gain distribution \(KL loss on−log⁡\(PPL\+1\)\-\\log\(\\mathrm\{PPL\}\{\+\}1\)targets, group size1111, lr6×10−56\\times 10^\{\-5\},22epochs, official defaults otherwise, final checkpoint, no model selection\)\. Crucially, its training questions are*exactly*the1,9391\{,\}939NQ\-open train questionsLodestartrains on \(the filter keeps1,8831\{,\}883\), so both methods see the same supervision distribution and share an in\-domain pool\. Two benchmark adaptations from the paper’s setting: candidates come from the shared bge\-m3 top\-1010pools rather than Contriever top\-2020, and reranker input length is capped at the model’s true512512\-token limit rather than the released script’s512\+512512\{\+\}512\. At test time the selector scores the same ten candidates as every other row, one278278M cross\-encoder forward per candidate with no respondent and no gold; its full workflow with the pseudo\-passage action enabled is not cell\-comparable and is excluded from the tables, where it would reach meanF1F\_\{1\}0\.49070\.4907\. The*teacher*signal is likewise kept out of the tables, but we report it because it bounds what the distillation could have delivered—selecting by a gain that scores the gold answer is selecting with the answer in hand\. Reproduced on our pools it reaches meanF1F\_\{1\}0\.62130\.6213over the four datasets where it is available, againstLodestar’s0\.48880\.4888on those same four, while the distilled selector retains0\.44800\.4480on those four \(0\.49510\.4951over all five pools, its Table[2](https://arxiv.org/html/2608.11922#S4.T2)row\)\. That gap is distillation loss rather than a reproduction failure, and it is why GainRAG’s inference\-ready row sits mid\-table\.
- •MBA\-RAG\([Tang et al\. 2025](https://arxiv.org/html/2608.11922#bib.bib32)\)trains a DistilBERT bandit policy with anϵ\\epsilon\-greedy rewardra=A⁡\(y,y^a\)−λ​C​\(a\)r\_\{a\}=A\(y,\\hat\{y\}\_\{a\}\)\-\\lambda C\(a\)\. Its published arms are retrieval*strategies*; our benchmark fixes retrieval, so the arm set is adapted to the ten candidate ranks of the shared pools, and since every arm then costs exactly one respondent call the cost term is degenerate andλ\\lambdais set to00\. Rewards are the same cached per\-passageF1F\_\{1\}utilities that define our labels\.*Result:*the trained policy collapses onto the retrieval prior, reaching meanF1F\_\{1\}0\.47650\.4765; the paired comparison againstLodestaris in Table[7](https://arxiv.org/html/2608.11922#A4.T7)\.
- •CRITIC\-R1\([Xiao et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib36)\)trains a structured critic \(verdict/location/reason/fix schema\) with two\-stage GRPO under consensus teacher supervision\. We adapt it to selection: the trained critic scores each candidate’s cached draft answer by its verdict\-token probabilities \(P⁡\(CORRECT\)−P⁡\(INCORRECT\)P\(\\texttt\{CORRECT\}\)\-P\(\\texttt\{INCORRECT\}\), one forward per candidate\), the best\-scored passage is selected, and the same frozen respondent answers\. Training follows the official recipe at the official scale: Qwen2\.5\-3B\-Instruct critic with LoRA\([Hu et al\. 2022](https://arxiv.org/html/2608.11922#bib.bib10)\)r=16/α=16r\{=\}16/\\alpha\{=\}16, lr10−610^\{\-6\}, KL0\.0030\.003,5,0005\{,\}000critique samples at the official five\-epoch exposure, two stages \(CJA then gated DQA\) with the paper’s conservative verdict matrix, the training triples drawn only fromLodestar’s train questions and consensus supervision \(K=3K\{=\}3\) from a gpt\-oss\-120b teacher over the same cached draft answers\.*Result:*the critic trains healthily by its own reward yet its selection ends at meanF1F\_\{1\}0\.46260\.4626, belowrank1’s0\.47690\.4769; the paired comparison againstLodestaris in Table[7](https://arxiv.org/html/2608.11922#A4.T7)\. An implementation audit accompanies the number, and the residual gap is the method rather than the harness: teacher\-verdict alignment and within\-question ranking are decoupled skills on this benchmark, and the two training stages reward the former\.

### J\.4Trained generators \(system\-level\)

In these two rows the answering model itself is trained, so they are*system\-level*comparisons rather than cell\-level ones and are grouped separately in the tables\.

- •Self\-RAG\([Asai et al\. 2024](https://arxiv.org/html/2608.11922#bib.bib2)\)is the releasedselfrag\-llama2\-7bgenerator running its official reflection\-token inference over the same ten candidates\. Unlike every other row it replaces the frozen respondent with a trained one, a Llama\-2\-7B fine\-tuned on Self\-RAG’s own150150k\-instance corpus, so its row answers the question “does a purpose\-trained retrieval\-augmented generator beat a frozen respondent steered by one learned string?”*Result and metric diagnostic\.*Its reflection\-based passage selection is sound \(frozen\-respondentF1F\_\{1\}at its chosen passages matches plain entropy within±0\.015\\pm 0\.015on every pool\), but its answers are style\-mismatched with token\-levelF1F\_\{1\}: on well\-formed out\-of\-domain questions it answers in full sentences \(mean11\.911\.9words vs\. the frozen respondent’s4\.24\.2\), which tokenF1F\_\{1\}punishes even when the fact is present\. Its table row therefore uses its stronger configuration under our one\-row policy, official inference plus a gold\-agnostic extraction step, which recovers meanF1F\_\{1\}from0\.33460\.3346to0\.36380\.3638\(Table[12](https://arxiv.org/html/2608.11922#A9.T12)\);Lodestarloses only the NQ cell to it, by0\.0010\.001, consistent with NQ\-style data in its training corpus\. Under Self\-RAG’s*own*official metric, containment match, the comparison is a statistical tie, because containment rewards its verbose style; we report both metrics rather than choosing either\.
- •CTRL\-RAG\([Tan et al\. 2026](https://arxiv.org/html/2608.11922#bib.bib31)\)also trains the generator itself rather than a selector: full\-parameter GRPO on the very Llama\-3\.1\-8B\-Instruct checkpoint we otherwise keep frozen, optimizing the paper’s contrastive\-likelihood hybrid rewardRhybrid=RCLR′⋅𝟏​\(y∗⊆y\)R\_\{\\mathrm\{hybrid\}\}=R^\{\\prime\}\_\{\\mathrm\{CLR\}\}\\cdot\\mathbf\{1\}\(y^\{\*\}\\subseteq y\), whose leave\-one\-out likelihood gap is scored online by a frozen\-likelihood service held at the initial policy\. We keep its two\-stage SFT\-then\-RL structure but rebuild the SFT stage inside the benchmark, on1,8391\{,\}839of the1,9391\{,\}939NQ training questionsLodestartrains on, with the first gold answer as the target, in place of the official7474k multi\-hop SFT corpus, which would inject supervision from outside this comparison\. It reads the shared top\-1010concatenation and answers directly, with no selection step, so its cells are not frozen\-respondent\-comparable and are not claimed in the per\-cell win count\. It is kept out of Tables[2](https://arxiv.org/html/2608.11922#S4.T2),[7](https://arxiv.org/html/2608.11922#A4.T7)and[9](https://arxiv.org/html/2608.11922#A6.T9), and appears only beside Self\-RAG in Table[12](https://arxiv.org/html/2608.11922#A9.T12); its scores, formatting pathology and the reason no single checkpoint of the run can stand in for the method are reported together in the reproduction case study of Appendix[I](https://arxiv.org/html/2608.11922#A9)\.