Nous: Learning and Certifying Memory Decisions Before Source Calibration

# Learning and CertifyingMemory Decisions Before Source Calibration
Source: [https://arxiv.org/html/2610.00094](https://arxiv.org/html/2610.00094)
## Nous: Learning and Certifying Memory Decisions Before Source Calibration

Pranav SinghAffiliation:Department of Mathematics, Indian Institute of Technology RoparAffiliation:Student ID: 2023MCB1308Email:[2023mcb1308@iitrpr\.ac\.in](mailto:)

September 2026

###### Abstract

Belief\-based agent memory needs reliable decisions about current state, yet its evidence may be noisy, copied, or stale\. Must a memory calibrate its sources before it can improve its decisions? We separate learning, calibration, and revision certification\. On one four\-model hidden Markov family, learning an unknown Bayes decision requiresΘ⁡\(ℓ−2\)\\Theta\(\\ell^\{\-2\}\)records and certifying its improvement over an informative incumbent takesO⁡\(ℓ−2\)O\(\\ell^\{\-2\}\)fresh records from the same observation law, while fixed\-precision source estimation requiresΘ⁡\(ℓ−4\)\\Theta\(\\ell^\{\-4\}\)as persistenceℓ\\ellvanishes\. Thus learning and certifying useful decisions can require quadratically fewer records than source calibration\. A broader model class retains the decision rate and source lower bound\. Under an unknown identity\-plus\-background report channel, we then characterize the sharp identified interval for a policy’s improvement and derive a finite\-sample certificate using observable witness regions, without pure\-class anchors\. Unlike an unrestricted\-channel certificate, it can authorize one\-direction corrections\. A robustness extension tolerates bounded history\-dependent misspecification and conditional copying; split\-trained witnesses apply to arbitrary history spaces with explicit power conditions\. We integrate policy\-bound receipts with Nous Dimensions and test 45,000 held\-out mutable\-state histories and 9,000 episodes in three external MiniGrid memory environments with an introduced noisy\-report interface\. The new certificate accepts 9/9 improvements over a constant incumbent and 4/9 over last\-write\-wins, versus none for the earlier certificate in MiniGrid\. A complete strength/bias study distinguishes unidentifiable improvements from finite\-sample rejections\. Strong established inference baselines remain competitive or better; the result is a statistical account of when memory decisions can be learned and justified without recovering source reliability, not a universally superior memory algorithm\.

## 1Introduction

An agent can accumulate evidence faster than it learns how much to trust it\. A report can describe a superseded state; ten reports can repeat one mistake; and a confident posterior can reflect assumed reliability rather than measured reliability\. Nous represents entity attributes as categorical belief*Dimensions*and observational updates as*Deltas*\[[1](https://arxiv.org/html/2610.00094#bib.bib1)\]\. This makes uncertainty and revision explicit, but leaves a fundamental question:*which parts of a memory’s behavior actually require source calibration?*

We distinguish three tasks\.*Calibration*estimates numerical source informativeness\.*Decision learning*learns which current state to report\.*Certification*decides whether replacing an incumbent policy improves expected accuracy\. These tasks share evidence but need not share statistical difficulty\. Our central result constructs one four\-world HMM family where*learning and certifying useful decisions can require quadratically fewer records than source calibration*\. The Bayes decision is unknown, the incumbent is informative, and learning, fresh auditing, and source estimation all receive the same full observation law\. First\-order signal reveals the decision orientation and its improvement; source scale appears only at second order\. This is a finite\-family minimax separation, not pointwise hardness at every model or a universal characterization of memory environments\.

Certification adds a different challenge\. A later biased report can favor a candidate that predicts the preferred label more frequently, even when that candidate is less accurate about truth\. Requiring improvement against every positively oriented channel is valid but can be unnecessarily conservative: many such channels cannot have produced the observed report distribution\. We instead compare policies over the channels*compatible with the data*\. This yields an attainable identified gain interval, a sharp population sign condition, and a finite\-sample witness certificate\. It resolves an important structural failure of the unrestricted\-channel rule: inability to authorize any entirely one\-direction correction\.

The contribution is theory\-led, with an executable connection to Nous rather than a claim that combining existing components creates a new inference algorithm\. We organize the contributions around three claims:\(1\) Shared\-observation separation\(Theorem[1](https://arxiv.org/html/2610.00094#Thmtheorem1)\): learning and powered certification useO⁡\(ℓ−2\)O\(\\ell^\{\-2\}\)records each, whereas source estimation needsΘ⁡\(ℓ−4\)\\Theta\(\\ell^\{\-4\}\), even with the orientation given\.\(2\) Sharp policy\-gain identification\(Theorem[3](https://arxiv.org/html/2610.00094#Thmtheorem3)\): categorical shared\-background reporting yields an attainable clean\-gain interval without pure\-class anchors\.\(3\) Finite\-sample certification\(Section[5](https://arxiv.org/html/2610.00094#S5)\): witness certificates extend to bounded history dependence, conditional copying, split\-trained witnesses with power guarantees, and fresh repeated releases\. These are theorem\-level claims, not claims to have invented their statistical ingredients\. A mutable\-state ledger study tests temporal inference and policy receipts; an external simulator study tests memory\-dependent action outcomes\. We retain negative results and stronger baselines, and separate what cannot be identified from what our finite\-sample test merely fails to detect\.

## 2Memory decisions and the evidence contract

LetZ∈\{1,…,K\}Z\\in\\\{1,\\ldots,K\\\}be an exclusive state,HHthe information available when memory is queried, andYYa later report\. The incumbentb⁡\(H\)b\(H\)and candidateg⁡\(H\)g\(H\)are fixed before calibration\. Define clean and observable gains

Δ\\displaystyle\\Delta=𝔼\[𝟏\{g\(H\)=Z\}−𝟏\{b\(H\)=Z\}\],\\displaystyle=\\mathbb\{E\}\[\\mathbf\{1\}\\\{g\(H\)=Z\\\}\-\\mathbf\{1\}\\\{b\(H\)=Z\\\}\],\(1\)ΔY\\displaystyle\\Delta\_\{Y\}=𝔼\[𝟏\{g\(H\)=Y\}−𝟏\{b\(H\)=Y\}\]\.\\displaystyle=\\mathbb\{E\}\[\\mathbf\{1\}\\\{g\(H\)=Y\\\}\-\\mathbf\{1\}\\\{b\(H\)=Y\\\}\]\.\(2\)The target isΔ\>0\\Delta\>0on a specified deployment population\. This does not certify each individual revision, numerical posterior calibration, or safety under arbitrary adversaries\. Within\-history sources may be dependent; records used for calibration must be iid from the deployment law\.

#### A Nous integration, not an unrelated replacement\.

A Dimension retains the categorical belief state\. The native BayesianUpdater remains a candidate\-generating baseline\. A reference adapter snapshots the incumbent Dimension, binds a candidate distribution and policy identifier to that snapshot, and applies or declines the proposal according to a calibration receipt\. It rejects stale snapshots or mismatched policy identifiers\. A separatePolicyDeltarecords this operation, since an observational Delta’s numerical reliability field cannot honestly be filled by an accuracy certificate\. The adapter leaves the production updater and persistence format unchanged\. Hashes bind research artifacts; they are not adversarial authentication\.

#### Information boundaries\.

Training/development may fit and select policies, but calibration is reserved for the locked comparison\. Test truth is evaluator\-only\. A later report can concern a future state, as in the learning theorem, or audit the query’s pinned version, as in the revision experiments\. These are distinct channels\. A pinned audit is generated independently of the policy’s answer with constant noise parameters; version pinning by itself would not enforce this assumption in a real system\. Test policies never receive their future audit or clean truth\.

## 3Learning and certification before source calibration

Fix0<ℓ≤1/80<\\ell\\leq 1/8\. Each observation is the full recordO=\(X0,X1,Y\)O=\(X\_\{0\},X\_\{1\},Y\), whereXt∈\{−1,\+1\}3X\_\{t\}\\in\\\{\-1,\+1\\\}^\{3\}andY=X2,1Y=X\_\{2,1\}\. The target isZ1∈\{−1,\+1\}Z\_\{1\}\\in\\\{\-1,\+1\\\}\. Records are independent three\-step realizations, not overlapping windows from one trajectory\. Letℳℓ\\mathcal\{M\}\_\{\\ell\}contain stationary balanced binary HMMs with

ℙ⁡\(Zt\+1=z′∣Zt=z\)=\(1\+λ​z​z′\)/2,λ∈\[4​ℓ/9,ℓ\]\.\\mathbb\{P\}\(Z\_\{t\+1\}=z^\{\\prime\}\\mid Z\_\{t\}=z\)=\(1\+\\lambda zz^\{\\prime\}\)/2,\\qquad\\lambda\\in\[4\\ell/9,\\ell\]\.Vector emissions are independent across time conditional on the latent path, have strictly positive support, and satisfyf−​\(x\)=f\+​\(−x\)f\_\{\-\}\(x\)=f\_\{\+\}\(\-x\)\. Dependence among the three simultaneous sources is unrestricted\. Anchor informativenessa=𝔼⁡\[Xt,1∣Zt=\+1\]a=\\mathbb\{E\}\[X\_\{t,1\}\\mid Z\_\{t\}=\+1\]lies in\[1/4,4/5\]\[1/4,4/5\]; its known positive orientation fixes latent\-label ambiguity\. Both learners know this class and receive the sameOO, without knowing its parameters\.

#### One observation mechanism for all three tasks\.

The central construction uses the same future report for learning and auditing, without an additional pinned\-audit sensor\. Lets∈\{−1,1\}s\\in\\\{\-1,1\\\},c∈\{1,3/2\}c\\in\\\{1,3/2\\\},χ⁡\(x\)=x1​x2​x3\\chi\(x\)=x\_\{1\}x\_\{2\}x\_\{3\}, and

ηs​\(x\)=14​x1\+38​s​χ​\(x\),fz​\(x\)=18​\[1\+z​c​ηs​\(x\)\],λ=ℓ/c2\.\\eta\_\{s\}\(x\)=\\tfrac\{1\}\{4\}x\_\{1\}\+\\tfrac\{3\}\{8\}s\\chi\(x\),\\quad f\_\{z\}\(x\)=\\tfrac\{1\}\{8\}\[1\+zc\\eta\_\{s\}\(x\)\],\\quad\\lambda=\\ell/c^\{2\}\.These four worlds belong toℳℓ\\mathcal\{M\}\_\{\\ell\}; their source informativeness isa=c/4a=c/4\. All procedures observe the fullO=\(X0,X1,X2,1\)O=\(X\_\{0\},X\_\{1\},X\_\{2,1\}\)\.

###### Theorem 1\(Learning and certification precede source calibration\)\.

For fixed failure probabilities below1/41/4and excess\-risk tolerance below3/1283/128, Bayes decision learning on this family requiresΘ⁡\(ℓ−2\)\\Theta\(\\ell^\{\-2\}\)records\. Learninggs=s​χ​\(X1\)g\_\{s\}=s\\chi\(X\_\{1\}\)and certifying its improvement over the informative incumbentb=X1,1b=X\_\{1,1\}using the witness certificate on a fresh split of the same records requiresO⁡\(ℓ−2\)O\(\\ell^\{\-2\}\)records per split\. Uniform source estimation to error below1/161/16requiresΘ⁡\(ℓ−4\)\\Theta\(\\ell^\{\-4\}\)total records, even ifssis given and all splits are pooled\.

HereΔ=c/16\\Delta=c/16,m=0m=0, andΔY=ℓ/64\\Delta\_\{Y\}=\\ell/64\. Both witness regions may be the whole history space\. For each decision orientation the two scales remain calibration\-hard; for each scale the orientation must be learned\. Appendix[A](https://arxiv.org/html/2610.00094#A1)gives explicit learning/acceptance probabilities, constants and both source bounds\. This is still a family\-wise rate statement, with large sufficient constants\. It demonstrates compatibility of the tasks, not indispensability of every component: simpler symmetric\-noise comparisons also work on this balanced subfamily\. The quartic source difficulty exploits jointly unknown informativeness and transition persistence; it does not apply with persistence known\. Indeed, on this subfamily knownλ\\lambdaandℓ\\elldeterminec=ℓ/λc=\\sqrt\{\\ell/\\lambda\}\.

Table 1:Theorem[1](https://arxiv.org/html/2610.00094#Thmtheorem1): first\-order decision and improvement signals, second\-order source\-scale information\. All order\-ℓ\\ellterms in the full observation law are independent ofcc\. Certification uses fresh records; all rates are family\-wise with fixed accuracy and failure tolerances\.
#### Beyond the four\-world family\.

The broader class admits a uniform decision guarantee, but its calibration lower bound need not be attained by the decision\-hard subfamily\.

###### Theorem 2\(Matched decision–calibration separation\)\.

For fixed excess\-risk tolerance0<ϵ<\.050<\\epsilon<\.05and fixed failure probability0<δ<1/40<\\delta<1/4, the minimax sample complexity of learning a current\-state decision overℳℓ\\mathcal\{M\}\_\{\\ell\}isΘ⁡\(ℓ−2\)\\Theta\(\\ell^\{\-2\}\)\. Uniformly estimatingaawith error strictly below1/81/8and failure probability below1/41/4requiresΩ⁡\(ℓ−4\)\\Omega\(\\ell^\{\-4\}\)records on the same class and experiment\. The decision comparator is the Bayes rule based on both snapshots\.

The future anchor satisfies𝔼\[Y∣Z1,H\]=λaZ1\\mathbb\{E\}\[Y\\mid Z\_\{1\},H\]=\\lambda aZ\_\{1\}\. Thus future\-label excess risk is exactlyλ​a\\lambda atimes clean\-state excess risk\. There are 64 possible histories; empirical future\-label majority in each cell gives

RZ​\(g^\)−RZ​\(g∗\)≤18ℓ​log⁡\(2⋅264/δ\)2​n\.R\_\{Z\}\(\\hat\{g\}\)\-R\_\{Z\}\(g^\{\*\}\)\\leq\\frac\{18\}\{\\ell\}\\sqrt\{\\frac\{\\log\(2\\cdot 2^\{64\}/\\delta\)\}\{2n\}\}\.\(3\)This does not estimateaaorλ\\lambda\. Decision\-hard full\-record laws haveO⁡\(ℓ2\)O\(\\ell^\{2\}\)KL, whereas parameter\-hard laws match the lag\-one coefficient and differ only at orderℓ2\\ell^\{2\}in the lag\-two coefficient, givingO⁡\(ℓ4\)O\(\\ell^\{4\}\)KL\. Appendix[B](https://arxiv.org/html/2610.00094#A2)gives proofs and simulations\. Unlike Theorem[1](https://arxiv.org/html/2610.00094#Thmtheorem1), this result supplies no matching parameter upper bound\. Neither result claims pointwise dominance over HMM inference; quartic parameter difficulty and prediction/parameter stability contrasts have substantial precedents \(Section[7](https://arxiv.org/html/2610.00094#S7)\)\.

## 4Which policy improvements are identifiable?

Deployment reports need not have the learning theorem’s symmetric noise\. Assume the categorical report channel has the form

C=ρ​I\+u​𝟏𝖳,ρ\>0,u≥0,𝟏𝖳​u=1−ρ\.C=\\rho I\+u\\mathbf\{1\}^\{\\mathsf\{T\}\},\\qquad\\rho\>0,\\quad u\\geq 0,\\quad\\mathbf\{1\}^\{\\mathsf\{T\}\}u=1\-\\rho\.\(4\)With probabilityρ\\rhothe report preserves the state; otherwise it draws from an unknown shared background distribution\. This covers every positively oriented binary class\-conditional channel, but not every categorical confusion matrix\. Setqk​\(h\)=ℙ⁡\(Y=k∣H=h\)q\_\{k\}\(h\)=\\mathbb\{P\}\(Y=k\\mid H=h\)andmk=ℙ⁡\(g=k\)−ℙ⁡\(b=k\)m\_\{k\}=\\mathbb\{P\}\(g=k\)\-\\mathbb\{P\}\(b=k\)\. Then

q⁡\(h\)=ρ​p​\(h\)\+u,ΔY=ρ​Δ\+m𝖳​u,q\(h\)=\\rho p\(h\)\+u,\\qquad\\Delta\_\{Y\}=\\rho\\Delta\+m^\{\\mathsf\{T\}\}u,\(5\)wherepk​\(h\)=ℙ⁡\(Z=k∣H=h\)p\_\{k\}\(h\)=\\mathbb\{P\}\(Z=k\\mid H=h\)\. The mass\-change term creates a bias trap\. Ignoring observable compatibility gives the sufficient ruleΔY\>maxk⁡mk\\Delta\_\{Y\}\>\\max\_\{k\}m\_\{k\}\. In binary labels0,10,1, this isΔY\>\|𝔼⁡\[g−b\]\|\\Delta\_\{Y\}\>\|\\mathbb\{E\}\[g\-b\]\|\. It is impossible for that rule to accept an entirely one\-direction change\.

###### Theorem 3\(Sharp compatible\-channel gain interval\)\.

SupposeHHhas finite positive\-mass support andq⁡\(h\)q\(h\)is nonconstant\. WriteℓkY=minh⁡qk​\(h\)\\ell\_\{k\}^\{Y\}=\\min\_\{h\}q\_\{k\}\(h\), so∑kℓkY<1\\sum\_\{k\}\\ell\_\{k\}^\{Y\}<1\. The full compatible family is

0≤uk≤ℓkY,ρ=1−∑kuk,p⁡\(h\)=q⁡\(h\)−u1−∑kuk\.0\\leq u\_\{k\}\\leq\\ell\_\{k\}^\{Y\},\\quad\\rho=1\-\\sum\_\{k\}u\_\{k\},\\quad p\(h\)=\\frac\{q\(h\)\-u\}\{1\-\\sum\_\{k\}u\_\{k\}\}\.\(6\)The attainable clean gains form the interval between the minimum and maximum of

F⁡\(u\)=ΔY−m𝖳​u1−∑kukover the box in \([6](https://arxiv.org/html/2610.00094#S4.E6)\)\.F\(u\)=\\frac\{\\Delta\_\{Y\}\-m^\{\\mathsf\{T\}\}u\}\{1\-\\sum\_\{k\}u\_\{k\}\}\\quad\\text\{over the box in \\eqref\{eq:box\}\}\.\(7\)In particular, every compatible law has strictly positive gain if and only if

ΔY\>∑k\(mk\)\+​ℓkY\.\\Delta\_\{Y\}\>\\sum\_\{k\}\(m\_\{k\}\)\_\{\+\}\\ell\_\{k\}^\{Y\}\.\(8\)

#### Proof\.

Nonnegativity ofpprequires the box constraints\. Conversely every point in the box gives a positiveρ\\rhoand normalized nonnegativep⁡\(h\)p\(h\), and hence a joint lawℙ⁡\(H=h\)​pZ​\(h\)​CY,Z\\mathbb\{P\}\(H=h\)p\_\{Z\}\(h\)C\_\{Y,Z\}with the observed distribution\. Equation \([5](https://arxiv.org/html/2610.00094#S4.E5)\) yieldsFF\. The box is compact and connected and its denominator stays positive, giving attainable endpoints and all intermediate values\. Minimizing the numerator independently sets positive\-mmcoordinates toℓkY\\ell\_\{k\}^\{Y\}and negative\-mmcoordinates to zero, proving \([8](https://arxiv.org/html/2610.00094#S4.E8)\)\. Linear\-fractional optimization obtains the endpoints \(Appendix[C](https://arxiv.org/html/2610.00094#A3)\)\.□\\square

No pure\-clean\-class history is assumed, and the channel is generally not identified\. Binary noise compatibility itself is established in prior work; here the compatible set is used for the policy\-gain functional and its categorical shared\-background counterpart\. If the lower endpoint is nonpositive, there exists an observationally identical non\-improving world: additional data from this law alone cannot establish uniformly positive improvement\.

## 5Finite\-sample certification and repeated releases

Estimating every conditional report probability can be expensive\. Choose, before calibration, one history\-only witness regionAkA\_\{k\}per class\. Regions need not be pure, disjoint, or exhaustive\. Sinceuk≤ℙ⁡\(Y=k∣H∈Ak\)u\_\{k\}\\leq\\mathbb\{P\}\(Y=k\\mid H\\in A\_\{k\}\), an upper bound on a regional report frequency is also an upper bound on the unknown background mass\.

Fornniid records compute an upper Clopper–Pearson boundUkU\_\{k\}for that regional frequency with one\-sided errorα/\(2​K\)\\alpha/\(2K\)\. SetUk=1U\_\{k\}=1for an empty region\. LetΔ^Y\\widehat\{\\Delta\}\_\{Y\}andm^\\hat\{m\}be the empirical quantities in \([2](https://arxiv.org/html/2610.00094#S2.E2)\) and \([5](https://arxiv.org/html/2610.00094#S4.E5)\), and set

L=Δ^Y−∑k\(m^k\)\+​Uk−8​log⁡\(2/α\)n\.L=\\widehat\{\\Delta\}\_\{Y\}\-\\sum\_\{k\}\(\\hat\{m\}\_\{k\}\)\_\{\+\}U\_\{k\}\-\\sqrt\{\\frac\{8\\log\(2/\\alpha\)\}\{n\}\}\.\(9\)
###### Theorem 4\(Witness certificate\)\.

Under \([4](https://arxiv.org/html/2610.00094#S4.E4)\), fixed policies and history\-only regions, and iid calibration from the deployment law,ℙ\{L\>0\}≤α\\mathbb\{P\}\\\{L\>0\\\}\\leq\\alphawheneverΔ≤0\\Delta\\leq 0\. More generallyL≤ρ​ΔL\\leq\\rho\\Deltawith probability at least1−α1\-\\alpha; whenL\>0L\>0, it is also a positive lower bound onΔ\\Delta\. The same records may be used for witness bounds and gain estimation\.

#### Proof\.

The simultaneous eventuk≤Uku\_\{k\}\\leq U\_\{k\}has probability at least1−α/21\-\\alpha/2\. On it, the empirical worst numerator over\[0,U\]\[0,U\]is no larger than the empirical numerator at the fixed trueuu\. Its iid summands

Ti\(u\)=𝟏\{gi=Yi\}−𝟏\{bi=Yi\}−ugi\+ubiT\_\{i\}\(u\)=\\mathbf\{1\}\\\{g\_\{i\}=Y\_\{i\}\\\}\-\\mathbf\{1\}\\\{b\_\{i\}=Y\_\{i\}\\\}\-u\_\{g\_\{i\}\}\+u\_\{b\_\{i\}\}lie in\[−2,2\]\[\-2,2\]and have meanρ​Δ\\rho\\Delta\. One\-sided Hoeffding with errorα/2\\alpha/2gives the stated radius\. A union bound completes the proof without conditioning on data\-derivedUU, or taking a union over fitted vertices\.□\\square

The finite\-sample witness box can be larger than the population box\. Thus the certificate is valid but not sharp and need not dominate an exact binary test at everynn\. For a finite history alphabet, independent training can select minimum\-frequency cells; increasing training and calibration counts recovers the sharp condition whenever its margin is strictly positive\. A failed test is not a claim that the candidate is harmful\. A truncated valuemax⁡\(0,L\)\\max\(0,L\)is not an unconditional clean\-gain lower bound when the test fails\.

#### Repeated release control\.

At releasettspendαt=α/\[t⁡\(t\+1\)\]\\alpha\_\{t\}=\\alpha/\[t\(t\+1\)\]\. Policies and witnesses may depend on past releases but must precede that release’s fresh iid calibration\. Conditional validity and∑tαt=α\\sum\_\{t\}\\alpha\_\{t\}=\\alphabound the probability of ever accepting a non\-improving release byα\\alpha\. This is ordinary error spending, not a new sequential\-testing result\. Reusing calibration or repeatedly inspecting a fixed\-α\\alphatest is not justified by this statement\.

#### A channel assumption cannot be certified from agreement alone\.

LetHHbe a fair bit,g=Hg=H,b=1−Hb=1\-H\. Two worlds haveℙ⁡\(Z=H∣H\)=\.8\\mathbb\{P\}\(Z=H\\mid H\)=\.8and\.2\.2, respectively\. At a history withℙ⁡\(Z=1∣H\)=\.8\\mathbb\{P\}\(Z=1\\mid H\)=\.8, set\(q0,q1\)=\(\.1,\.6\)\(q\_\{0\},q\_\{1\}\)=\(\.1,\.6\); if it equals\.2\.2, set\(q0,q1\)=\(\.4,\.9\)\(q\_\{0\},q\_\{1\}\)=\(\.4,\.9\)\. Both worlds haveℙ⁡\(Y=1∣H\)=\.5\\mathbb\{P\}\(Y=1\\mid H\)=\.5and positive channel orientation, but their gains are\+\.6\+\.6and−\.6\-\.6\. Arbitrary history\-dependent reporting therefore defeats observable\-only comparison\. Likewise, an independent fractionβ\\betaof reports copying the candidate addsβ​ℙ​\(g≠b\)\\beta\\mathbb\{P\}\(g\\neq b\)to the uncontaminated paired gain, potentially rewarding a harmful revision\. Known source roots help audit dependence but do not by themselves validate the assumed channel\.

#### Bounded misspecification rather than an exact channel\.

Let the actual channel satisfyess​suph,zTV\(Ch\(⋅∣z\),ρez\+u\)≤ϵ\\mathop\{\\mathrm\{ess\\,sup\}\}\_\{h,z\}\\mathrm\{TV\}\(C\_\{h\}\(\\cdot\\mid z\),\\rho e\_\{z\}\+u\)\\leq\\epsilonfor oneρ\>0\\rho\>0,u≥0u\\geq 0,∑kuk=1−ρ\\sum\_\{k\}u\_\{k\}=1\-\\rho\. SetVk=min⁡\(1,Uk\+ϵ\)V\_\{k\}=\\min\(1,U\_\{k\}\+\\epsilon\)andd^=n−1∑i𝟏\{gi≠bi\}\\hat\{d\}=n^\{\-1\}\\sum\_\{i\}\\mathbf\{1\}\\\{g\_\{i\}\\neq b\_\{i\}\\\}\.

###### Theorem 5\(Bounded\-misspecification certificate\)\.

For policies and history\-only witnesses fixed independently of iid calibration,

Lϵ=Δ^Y−∑k\(m^k\)\+​Vk−2​ϵ​d^−\(4\+2​ϵ\)​log⁡\(2/α\)2​nL\_\{\\epsilon\}=\\hat\{\\Delta\}\_\{Y\}\-\\sum\_\{k\}\(\\hat\{m\}\_\{k\}\)\_\{\+\}V\_\{k\}\-2\\epsilon\\hat\{d\}\-\(4\+2\\epsilon\)\\sqrt\{\\frac\{\\log\(2/\\alpha\)\}\{2n\}\}satisfiesℙ⁡\(Lϵ≤ρ​Δ\)≥1−α\\mathbb\{P\}\(L\_\{\\epsilon\}\\leq\\rho\\Delta\)\\geq 1\-\\alpha\. Accepting only whenLϵ\>0L\_\{\\epsilon\}\>0has level\-α\\alphaerror forΔ≤0\\Delta\\leq 0; on the same event an acceptedLϵL\_\{\\epsilon\}lower\-boundsΔ\\Delta\.

The proof and local tightness examples are in Appendix[H](https://arxiv.org/html/2610.00094#A8)\. A conditional arbitrary\-copy budgetβ\\betaon top of base deviationγ\\gammagivesϵ=γ\+β−γ​β\\epsilon=\\gamma\+\\beta\-\\gamma\\beta\. These are externally justified assumptions or sensitivity parameters, not quantities identified from the same reports\. Fixed history\-only witnesses must be enforced by the caller: the mask API cannot verify their construction\. Section[6\.4](https://arxiv.org/html/2610.00094#S6.SS4)adds sensitivity and learned\-witness experiments\. Appendix[I](https://arxiv.org/html/2610.00094#A9)also gives a quantitative acceptance condition for split\-trained score\-threshold witnesses on arbitrary histories, separating region rarity, approximation error and sample size\. It composes candidate learning, independent auditing and Nous receipts without channel identification\. The region\-mass and approximation conditions concern sufficient power only: type\-I validity does not require knowingπ\\piorξ\\xi, or finding good regions\.

## 6Experiments

We separate inference, certification, and action outcomes\. All studies retain protocols, split/episode seeds, code hashes, calibration records and test predictions\. No human annotation or model\-generated ground truth is used\. The new gate is not fitted to previously exposed occupancy/EEG test labels\.

### 6\.1Mutable\-state Nous audit harness

Nine frozen runs cover independent, copied, and delayed\-copy reporting\. Each uses 4,000 training, 1,000 development, 5,000 calibration and 5,000 test histories, totaling 45,000 held\-out queries\. A versioned entity ledger executes compare\-and\-swap commands, rejects stale versions and treats duplicate command identifiers idempotently\. Withheld current truth is checked by ledger replay\. The reports themselves are generated from a controlled HMM; domain display names are not separate real\-world distributions\.

All methods receive four five\-source snapshots with times and recorded roots\. Sources 2–4 copy source 1 with probability\.9\.9in the copy conditions; delayed copy additionally delivers one first\-snapshot report last\. Native Nous executes its actual updater in arrival order with declared reliability\.75\.75\. Root\-capped Nous uses each recorded root once per snapshot\. Strong baselines include a current anchor and independent/joint\-emission HMMs, with 25 EM iterations and three restarts\. Selection uses development next\-anchor accuracy and is locked before the independent pinned audit with\(q0,q1\)=\(\.45,\.95\)\(q\_\{0\},q\_\{1\}\)=\(\.45,\.95\)\. Clean truth never enters fitting or selection\.

The earlier unrestricted\-channel certificate accepts the selected candidate in all nine runs: selected accuracy is 93\.97%, 84\.59% and 84\.38% across the three regimes, versus arrival LWW’s 74\.82%, 68\.41% and 58\.34%\. Joint HMM reaches 93\.70%, 85\.40% and 85\.28%, beating the selected policy in both copy regimes\. The gain is principally temporal inference, not gating\. Separately, ordinary paired report agreement accepts a harmful constant\-one proposal in all nine runs; the unrestricted certificate rejects all nine\. Appendix[D](https://arxiv.org/html/2610.00094#A4)reports all baselines and paired intervals\. Every ledger and Nous revision receipt was reconstructed; this frozen study is not relabelled as a new evaluation of the compatible\-channel gate\.

### 6\.2External memory\-dependent agent tasks

We use MiniGrid Memory S7, S13 and S17Random\[[16](https://arxiv.org/html/2610.00094#bib.bib16)\], with an introduced noisy\-memory interface\. The native task requires remembering an initial key/ball cue, crossing a corridor and choosing the matching branch\. The cue is static: these experiments complement, not replace, the mutable\-state study\. An observation\-only routing controller obtains the cue and branch\-object observations without reading hidden grid cells or success/failure metadata\. The memory interface receives only five noisy reports of the cue; the controller’s clean cue is evaluator/wrapper\-only after report generation\.

Source accuracies are\(\.85,\.60,\.70,\.65,\.70\)\(\.85,\.60,\.70,\.65,\.70\); sources 2–4 copy source 1 with probability\.9\.9\. An independent calibration audit has\(q0,q1\)=\(\.1,\.9\)\(q\_\{0\},q\_\{1\}\)=\(\.1,\.9\), unknown to the certifier\. The fixed candidate is root\-capped Nous, with arrival LWW and constant zero as separately analyzed incumbents\. WitnessAkA\_\{k\}is the event that source 0 reports a value other thankk\. Each environment has three runs, each with 5,000 calibration and 1,000 test episodes\. These are separate level\-\.05 comparisons, not a joint level\-\.05 guarantee for choosing between incumbents\.

Calibration stages the simulator camera at the cue to avoid replaying the corridor; this declared observer intervention supplies only noisy reports to calibration\. All 9,000 test episodes execute the full observation\-based route\. At the junction, both terminal choices are executed in cloned simulator states; every method shares the route and environment seed\. Actual terminal rewards, actions and success are saved and replayed\. All 54,000 calibration/test seeds are distinct\. This is not an unmodified MiniGrid leaderboard or an LLM\-agent comparison: it isolates the memory\-dependent decision\.

Table 2:External simulator task success \(%\), mean±\\pmsample SD over three independent 1,000\-episode runs per environment\. All methods use the same introduced report interface and shared routing controller\.Table[2](https://arxiv.org/html/2610.00094#S6.T2)shows that the old gate accepts no revision in either comparison\. The compatible gate accepts 9/9 candidates against constant zero and 4/9 against LWW\. All accepted candidates improve held\-out success against their specified incumbent\. It still rejects five beneficial LWW replacements; the source\-0 baseline beats the candidate in every run\. These outcomes show the practical removal of a certification obstruction, not superior fusion or an empirical proof of the error guarantee\. Paired intervals, override counts and actual rewards are retained \(Appendix[E](https://arxiv.org/html/2610.00094#A5)\)\.

### 6\.3Categorical, delayed\-probe and repeated\-release controls

A frozen synthetic study crossesK∈\{2,3,5\}K\\in\\\{2,3,5\\\}, probe lagd∈\{1,3,6\}d\\in\\\{1,3,6\\\},n∈\{2000,8000\}n\\in\\\{2000,8000\\\}, and beneficial, null and harmful fixed policies, with 100 repetitions per cell\. Report signal isρ=\.8​\(\.8\)d\\rho=\.8\(\.8\)^\{d\}and background class\-zero mass is\.75\.75\. The compatible certificate accepts none of the 1,800 beneficial proposals and none of the 3,600 null/harmful proposals\. This is a substantive zero\-power result, not evidence that no beneficial policies exist\. In 200 null series with ten fresh releases and the stated error spending, no series accepts a false improvement; the 95% binomial upper bound on that event frequency is 1\.83%\.

To diagnose the failure, a separately frozen follow\-up evaluates the complete gridK∈\{2,3,5\}K\\in\\\{2,3,5\\\},ρ∈\{\.2,\.4,\.6,\.8,\.95\}\\rho\\in\\\{\.2,\.4,\.6,\.8,\.95\\\}and background class\-zero massv0∈\{\.25,\.5,\.75\}v\_\{0\}\\in\\\{\.25,\.5,\.75\\\}, atn=32000n=32000and 100 repetitions\. The candidate corrects only last\-class\-to\-zero incumbent errors\. Figure[1](https://arxiv.org/html/2610.00094#S6.F1)shows all cells, alongside whether the exact compatible interval excludes zero\. The old gate accepts zero throughout\. The new gate has 100/100 acceptances in 23 cells, partial power in two, and zero power in 20\. Sixteen cells have nonpositive sharp lower endpoints; their failure is an identification issue\. The remaining four zero\-power cells have positive population margins but insufficient finite\-sample power\. This follow\-up is diagnostic, not a hidden retuning of the first protocol\.

![Refer to caption](https://arxiv.org/html/2610.00094v1/compatible_phase.png)Figure 1:Compatible\-gate acceptance percentages over the complete fresh strength/bias grid, 100 repetitions per cell andn=32000n=32000\. A star denotes a nonpositive sharp population gain lower bound\. All true gains are positive; some cannot be certified from the observed law\. The old gate accepts none\.
### 6\.4Sensitivity and learned witnesses

For every frozen MiniGrid comparison we computeϵmax=sup\{ϵ∈\[0,1\]:Lϵ\>0\}\\epsilon\_\{\\max\}=\\sup\\\{\\epsilon\\in\[0,1\]:L\_\{\\epsilon\}\>0\\\}from calibration only\. The bound is decreasing; acceptance is strict below the root, not at it\. For the nine accepted constant\-incumbent comparisons the roots range from3\.03%3\.03\\%to5\.07%5\.07\\%TV; the four accepted LWW comparisons tolerate only0\.18%0\.18\\%–1\.37%1\.37\\%\. The other five comparisons do not accept even at zero\. These post\-hoc sensitivities expose fragility, not estimated channel errors or joint guarantees across comparisons \(Appendix[J](https://arxiv.org/html/2610.00094#A10)\)\. Source calibration estimates latent numerical reliabilities; a TV budget is an external structural contract that may itself require domain knowledge or independent auditing\. The small LWW limits expose cases where deployment is not robustly justified unless that demanding contract can be supported\.

A separately specified synthetic study uses 16\-dimensional continuous histories, design\-trained ridge report scores, five design\-quantile regions per class, independent region selection, calibration and held\-out truth evaluation\. The candidate is a learned policy for Nous integration, not its native updater\. Table[3](https://arxiv.org/html/2610.00094#S6.T3)reports all 12 cells: learned witnesses help at moderate deviation but all rules decline atϵ=\.10\\epsilon=\.10despite beneficial candidates\. This tests learned regions, not natural\-language extraction or LLM agents\.

Table 3:Continuous\-history study: acceptance counts across four replicates per externally controlled TV budget; mean paired clean test gain over constant zero\. Comparisons are separately fixed tests, not joint\-α\\alphaselection\.

## 7Related work and scope of contribution

[Abraham et al\. \[2\]](https://arxiv.org/html/2610.00094#bib.bib2), Theorem 1 and Corollary 1, already establish quartic dependence on proximity to rank deficiency for HMM emission estimation\.[Gassiat et al\. \[3\]](https://arxiv.org/html/2610.00094#bib.bib3), Theorem 10 and Section 5, give clustering guarantees and explicitly contrast stable prediction with unstable parameter recovery near independence\. Theorem[2](https://arxiv.org/html/2610.00094#Thmtheorem2)supplies matching decision rates and a parameter obstruction on one specified independent\-record class; it does not claim either prior insight as new\.

[Scott et al\. \[5\]](https://arxiv.org/html/2610.00094#bib.bib5), Proposition 3, characterize the entire feasible binary contamination family, not only its maximally denoised member\. Binary compatibility geometry is therefore a foundation, not a first discovery here\.[Katz\-Samuels et al\. \[13\]](https://arxiv.org/html/2610.00094#bib.bib13)study multiclass mutual contamination, including a common\-background example, and recover base distributions under joint irreducibility and mixing assumptions\. We do not claim the channel family as new: Theorems[3](https://arxiv.org/html/2610.00094#Thmtheorem3)–[4](https://arxiv.org/html/2610.00094#Thmtheorem4)instead retain nonidentification and target a fixed policy gain under a restricted forward channel, without joint irreducibility\.[Menon et al\. \[4\]](https://arxiv.org/html/2610.00094#bib.bib4), Propositions 5–10, connect noisy posteriors, decision thresholds and extremal probabilities\.[Reeve and Kabán \[15\]](https://arxiv.org/html/2610.00094#bib.bib15)obtain classification rates using quantitative approach to clean posterior extrema\. Our gain bounds retain nonidentification without pure\-class assumptions and furnish a finite\-sample policy test; confidence bounds and linear\-fractional optimization are standard\.[Fogliato et al\. \[14\]](https://arxiv.org/html/2610.00094#bib.bib14)study sensitivity of evaluation to biased proxy labels\.[Guerdan et al\. \[6\]](https://arxiv.org/html/2610.00094#bib.bib6)already exploit cancellation on policy\-agreement regions and discuss measurement\-error extensions\. We do not claim cancellation or partial\-identification\-based comparison as new in general\. Applying a fixed gain functional to known compatibility geometry is not by itself the principal novelty claim\. Theorem[1](https://arxiv.org/html/2610.00094#Thmtheorem1)establishes the shared\-observation separation: one full observation law supports both learning and valid, powered certification before source estimation, without an extra audit sensor\.

Dependent truth discovery\[[7](https://arxiv.org/html/2610.00094#bib.bib7)\]predates memory\-agent applications\. Nous supplies the belief/Delta abstraction\[[1](https://arxiv.org/html/2610.00094#bib.bib1)\]; transactional memory commits are explicit in MemTX and MemTxn\[[8](https://arxiv.org/html/2610.00094#bib.bib8),[9](https://arxiv.org/html/2610.00094#bib.bib9)\]\. StateMemBench uses executable event programs for evolving\-state evaluation\[[10](https://arxiv.org/html/2610.00094#bib.bib10)\]\. Our ledger is an audit harness, not the first executable memory benchmark\. The principal contribution is the shared\-observation task separation, supported by the broader compatible\-channel memory\-policy theory\.

## 8Limitations and conclusion

The learning theorem assumes a fixed finite alphabet, complementary emissions, a balanced chain and positive anchor orientation\. Its asymptotic separation does not make its constants small or imply a universally better learning algorithm\. Certification requires a positively oriented reference channel and fresh iid calibration from deployment\. The extension permits bounded history dependence and conditional copying with externally justified budgets; unbounded feedback and deployment shift remain outside its guarantee\. The categorical channel is restricted and the finite\-sample bound conservative\.

The external simulator adds genuine action outcomes, but its noise is controlled, its cue static and its routing shared\. Neither it nor the ledger evaluates natural\-language extraction or establishes real\-world source independence\. Previously exposed UCI diagnostics remain negative evidence: a nominal gate accepted an EEG revision reducing accuracy from 70\.69% to 37\.07%; the old asymmetric gate rejected that revision but also occupancy’s improvement from 55\.36% to 99\.49%\[[12](https://arxiv.org/html/2610.00094#bib.bib12),[11](https://arxiv.org/html/2610.00094#bib.bib11)\]\. Those dependent time\-series data do not validate the theorem’s assumptions\. No posterior\-calibration, inherited benchmark\-gain or poisoning\-defense claim follows from the new results\.

Learning and certifying useful decisions can require quadratically fewer records than source calibration \(Theorem[1](https://arxiv.org/html/2610.00094#Thmtheorem1)\)\. The lesson for Nous is constructive: keep beliefs and source calibration, but do not make every useful state decision wait for numerical source identification\. Learn decisions where the observations support them; compare replacements over compatible channels; and record the resulting policy\-level evidence separately from individual observational updates\. The separation theorem and the measured certification boundary identify precisely where this distinction helps and where additional information is needed\.

## AI use statement

Generative AI assisted theory and proof development, experimental design, synthetic\-data generation, implementation and testing, literature analysis, result interpretation, and manuscript preparation\. The author reviewed the AI\-assisted work and takes responsibility for the final content\. Automated checks and replay are not independent proof review or replication\.

## Ethics statement

The new experiments involve no human participants or private dialogue disclosure\. Misusing an expected\-accuracy certificate as an individual safety guarantee would be harmful; the interface records its population\-level scope\. Known synthetic copying roots are not evidence that real provenance can be recovered reliably\. No deployment in a consequential decision system is evaluated\.

## Reproducibility statement

Appendices provide proof details, protocol specifications and paired results\. The accompanying local reproducibility package contains the reference implementation, frozen inputs, policy locks, simulator traces and audit scripts\. No public upload or independent replication is asserted\. A separate conference\-formatted version is maintained; this preprint identifies its author and does not by itself indicate a conference submission\.

## References

- \[1\]P\. Singh\. When Does Belief\-Based Agent Memory Help? Reliability\-Conditional Updating and Provenance\-Capped Poisoning Defense\. arXiv:2606\.22030, 2026\.
- \[2\]K\. Abraham, E\. Gassiat, and Z\. Naulet\. Fundamental limits for learning hidden Markov model parameters\.*IEEE Transactions on Information Theory*, 2023\. arXiv:2106\.12936\.
- \[3\]E\. Gassiat, I\. Kaddouri, and Z\. Naulet\. Clustering Risk in Nonparametric Hidden Markov and I\.I\.D\. Models\.*Annals of Statistics*, 53\(6\):2409–2429, 2025\.
- \[4\]A\. Menon, B\. van Rooyen, C\. S\. Ong, and B\. Williamson\. Learning from Corrupted Binary Labels via Class\-Probability Estimation\.*ICML*, PMLR 37:125–134, 2015\.
- \[5\]C\. Scott, G\. Blanchard, and G\. Handy\. Classification with Asymmetric Label Noise: Consistency and Maximal Denoising\.*COLT*, PMLR 30:489–511, 2013\.
- \[6\]L\. Guerdan, A\. Coston, K\. Holstein, and Z\. S\. Wu\. Predictive Performance Comparison of Decision Policies Under Confounding\.*ICML*, PMLR 235:16673–16705, 2024\.
- \[7\]X\. L\. Dong, L\. Berti\-Equille, and D\. Srivastava\. Truth Discovery and Copying Detection in a Dynamic World\.*Proceedings of the VLDB Endowment*, 2\(1\):562–573, 2009\.
- \[8\]X\. Li et al\. MemTX: Transactional Belief Commit for Stateful Agent Memory\. arXiv:2607\.23929, 2026\.
- \[9\]H\. Cui et al\. MemTxn: A Transaction Boundary for Source\-Supported Updates and Complete\-State Recovery in Agent Memory\. arXiv:2607\.27834, 2026\.
- \[10\]X\. Fan, M\. Liu, R\. Yang, S\. Ouyang, and J\. Han\. Can Agent Memory Systems Track Evolving State? arXiv:2608\.19652, 2026\.
- \[11\]L\. Candanedo\. Occupancy Detection \[Dataset\]\. UCI Machine Learning Repository, 2016\.[https://doi\.org/10\.24432/C5X01N](https://doi.org/10.24432/C5X01N)\.
- \[12\]O\. Roesler\. EEG Eye State \[Dataset\]\. UCI Machine Learning Repository, 2013\.[https://doi\.org/10\.24432/C57G7J](https://doi.org/10.24432/C57G7J)\.
- \[13\]J\. Katz\-Samuels, G\. Blanchard, and C\. Scott\. Decontamination of Mutual Contamination Models\.*Journal of Machine Learning Research*, 20\(41\):1–57, 2019\.
- \[14\]R\. Fogliato, A\. Chouldechova, and M\. G’Sell\. Fairness Evaluation in Presence of Biased Noisy Labels\.*AISTATS*, PMLR 108:2325–2336, 2020\.
- \[15\]H\. W\. J\. Reeve and A\. Kabán\. Classification with unknown class\-conditional label noise on non\-compact feature spaces\.*COLT*, PMLR 99:2624–2651, 2019\. arXiv:1902\.05627\.
- \[16\]Farama Foundation\. MiniGrid Memory environment documentation and implementation, version 3\.0\.0\.[https://minigrid\.farama\.org/environments/minigrid/MemoryEnv/](https://minigrid.farama.org/environments/minigrid/MemoryEnv/)\. Accessed September 7, 2026\.

## Appendix AProof of the shared\-experiment theorem

WriteA=1/4A=1/4,B=3/8B=3/8,ηs​\(x\)=A​x1\+s​B​χ​\(x\)\\eta\_\{s\}\(x\)=Ax\_\{1\}\+sB\\chi\(x\),c∈\{1,3/2\}c\\in\\\{1,3/2\\\}, andλ=ℓ/c2\\lambda=\\ell/c^\{2\}\. Sincec⁡\(A\+B\)≤15/16c\(A\+B\)\\leq 15/16, all emissions are positive\. The anchor parametera=c​Aa=cAand transition parameter both lie in the original model class\. Hidden\-path summation gives

Ps,c​\(x0,x1,y\)=1\+ℓ​e0​e1\+ℓ​A​y​e1\+\(ℓ2/c2\)​A​y​e0128,et=ηs​\(xt\)\.P\_\{s,c\}\(x\_\{0\},x\_\{1\},y\)=\\frac\{1\+\\ell e\_\{0\}e\_\{1\}\+\\ell Aye\_\{1\}\+\(\\ell^\{2\}/c^\{2\}\)Aye\_\{0\}\}\{128\},\\quad e\_\{t\}=\\eta\_\{s\}\(x\_\{t\}\)\.Every cell is at least1/2561/256\. MarginallyX1X\_\{1\}is uniform, so bothgs=s​χ​\(X1\)g\_\{s\}=s\\chi\(X\_\{1\}\)andb=X1,1b=X\_\{1,1\}predict either label with probability1/21/2\.

#### Learning and a matching lower bound\.

The orientation moment satisfies𝔼⁡\[χ⁡\(X1\)​Y\]=s​ℓ​A​B=3​s​ℓ/32\\mathbb\{E\}\[\\chi\(X\_\{1\}\)Y\]=s\\ell AB=3s\\ell/32, independent ofcc\. Its empirical sign, with fixed tie breaking, learnssswith failure at mostδ\\deltafor

nlearn≥\(2048/9\)​ℓ−2​log⁡\(1/δ\)\.n\_\{\\rm learn\}\\geq\(2048/9\)\\ell^\{\-2\}\\log\(1/\\delta\)\.The full\-history posterior mean has numeratorc⁡\[ηs​\(X1\)\+λ​ηs​\(X0\)\]c\[\\eta\_\{s\}\(X\_\{1\}\)\+\\lambda\\eta\_\{s\}\(X\_\{0\}\)\]and denominator1\+λ​c2​ηs​\(X0\)​ηs​\(X1\)\>01\+\\lambda c^\{2\}\\eta\_\{s\}\(X\_\{0\}\)\\eta\_\{s\}\(X\_\{1\}\)\>0\. Its sign isgsg\_\{s\}, becauseB−A−λ⁡\(A\+B\)≥3/64B\-A\-\\lambda\(A\+B\)\\geq 3/64\. Thus the learner finds the full\-history Bayes rule, despite using only a particular moment of the full available records\.

For fixedcc, the twosslaws differ in four orthogonal Walsh characters, givingKL⁡\(P\+,c,P−,c\)≤\(9/32\)​ℓ2\\mathrm\{KL\}\(P\_\{\+,c\},P\_\{\-,c\}\)\\leq\(9/32\)\\ell^\{2\}\. Forn≤16/\(9​ℓ2\)n\\leq 16/\(9\\ell^\{2\}\), product KL and Pinsker give TV at most1/21/2\. Decodessfrom the sign of𝔼U​\[g⁡\(H\)​χ​\(X1\)\]\\mathbb\{E\}\_\{U\}\[g\(H\)\\chi\(X\_\{1\}\)\], whereUUis uniform on all 64 histories\. Clean regret is

c2​𝔼U​\[\(s​χ​\(X1\)−g⁡\(H\)\)​\(ηs​\(X1\)\+λ​ηs​\(X0\)\)\]\.\\frac\{c\}\{2\}\\mathbb\{E\}\_\{U\}\[\(s\\chi\(X\_\{1\}\)\-g\(H\)\)\(\\eta\_\{s\}\(X\_\{1\}\)\+\\lambda\\eta\_\{s\}\(X\_\{0\}\)\)\]\.If decoding is wrong, the parity contribution is at leastBBand the remaining contributions have total magnitude at mostA\+λ⁡\(A\+B\)A\+\\lambda\(A\+B\)\. Hence regret is at leastc⁡\[B−A−λ⁡\(A\+B\)\]/2≥3/128c\[B\-A\-\\lambda\(A\+B\)\]/2\\geq 3/128\. Le Cam implies worst\-case failure at least1/41/4for smaller tolerances\. Randomized rules use their conditional mean prediction in the decoder\.

#### Certifying the same learned candidate with the same future report\.

GivenZ1,HZ\_\{1\},H, the future report has meanλ​a​Z1\\lambda aZ\_\{1\}\. It is therefore already a symmetric shared\-background audit of current state, withρ=ℓ​A/c\\rho=\\ell A/candu0=u1=\(1−ρ\)/2u\_\{0\}=u\_\{1\}=\(1\-\\rho\)/2\. No second sensor or clean\-label sample is supplied\. Candidate and incumbent accuracies are\(1\+c​B\)/2\(1\+cB\)/2and\(1\+c​A\)/2\(1\+cA\)/2, so

Δ=c/16,m0=m1=0,ΔY=ρ​Δ=ℓ/64\.\\Delta=c/16,\\qquad m\_\{0\}=m\_\{1\}=0,\\qquad\\Delta\_\{Y\}=\\rho\\Delta=\\ell/64\.Use the certificate of Theorem[4](https://arxiv.org/html/2610.00094#Thmtheorem4)on a fresh iid split with both witness regions equal to the whole history space\. Conditional validity also covers the incorrectly learned orientation, whose gain is−c\(A\+B\)/2<0\-c\(A\+B\)/2<0\. Conditional on correct learning, putt=2​log⁡\(4/η\)/nauditt=\\sqrt\{2\\log\(4/\\eta\)/n\_\{\\rm audit\}\}\. One\-sided Hoeffding for the proxy gain and two\-sided Hoeffding for the binary mass change yieldΔ^Y≥ℓ/64−t\\hat\{\\Delta\}\_\{Y\}\\geq\\ell/64\-tand\|m^1\|≤t\|\\hat\{m\}\_\{1\}\|\\leq texcept with probability at most3​η/43\\eta/4\. SinceUk≤1U\_\{k\}\\leq 1andm^0=−m^1\\hat\{m\}\_\{0\}=\-\\hat\{m\}\_\{1\},

L≥ℓ/64−2​t−8​log⁡\(2/α\)/naudit\.L\\geq\\ell/64\-2t\-\\sqrt\{8\\log\(2/\\alpha\)/n\_\{\\rm audit\}\}\.Thus

naudit\>4096​\[2​2​log⁡\(4/η\)\+8​log⁡\(2/α\)\]2​ℓ−2n\_\{\\rm audit\}\>4096\\bigl\[2\\sqrt\{2\\log\(4/\\eta\)\}\+\\sqrt\{8\\log\(2/\\alpha\)\}\\bigr\]^\{2\}\\ell^\{\-2\}suffices for acceptance with probability at least1−η1\-\\etaconditional on correct learning\. Joint correct learning and acceptance has probability at least1−δ−η1\-\\delta\-\\eta, while non\-improving acceptance is bounded byα\\alpha\. These sufficient constants are large; the claim is a rate comparison, not practical small\-sample efficiency or superiority of this gate on this family\.

#### Source estimation remains quartically hard, with a matching upper bound\.

For each fixedss, switchingccpreserves all order\-ℓ\\ellterms of the full law\. The difference isℓ2​\(5/9\)​A​y​ηs​\(x0\)/128\\ell^\{2\}\(5/9\)Ay\\eta\_\{s\}\(x\_\{0\}\)/128\. Using𝔼U​\[ηs​\(X0\)2\]=13/64\\mathbb\{E\}\_\{U\}\[\\eta\_\{s\}\(X\_\{0\}\)^\{2\}\]=13/64and the cell floor gives, in either direction,

KL⁡\(Ps,1,Ps,3/2\)≤χ2​\(Ps,1,Ps,3/2\)≤\(325/41472\)​ℓ4\.\\mathrm\{KL\}\(P\_\{s,1\},P\_\{s,3/2\}\)\\leq\\chi^\{2\}\(P\_\{s,1\},P\_\{s,3/2\}\)\\leq\(325/41472\)\\ell^\{4\}\.Forn≤20736/\(325​ℓ4\)n\\leq 20736/\(325\\ell^\{4\}\), TV is at most1/21/2\. The anchor parameters1/41/4and3/83/8differ by1/81/8; testing at midpoint5/165/16yieldsmaxc⁡ℙ⁡\(\|a^−a\|≥1/16\)≥1/4\\max\_\{c\}\\mathbb\{P\}\(\|\\hat\{a\}\-a\|\\geq 1/16\)\\geq 1/4\. This bound applies to all observed records pooled across learning and certification splits, even ifssis known\. Receipts cannot add information beyond those records\.

Conversely𝔼⁡\[X0,1​Y\]=ℓ2​A2/c2\\mathbb\{E\}\[X\_\{0,1\}Y\]=\\ell^\{2\}A^\{2\}/c^\{2\}equalsℓ2/16\\ell^\{2\}/16orℓ2/36\\ell^\{2\}/36, independent ofss\. Threshold its empirical mean at the midpoint and output the corresponding anchor parameter\. The moment gap is5​ℓ2/1445\\ell^\{2\}/144, and Hoeffding bounds source\-error probability byexp\[−n\(5ℓ2/144\)2/8\]\\exp\[\-n\(5\\ell^\{2\}/144\)^\{2\}/8\]\. This proves the matching upper rate on this finite family, not on the entire original model class\.

Exact rational checks enumerate all observable cells and hidden paths at three persistence values, checking Bayes signs, clean/proxy gains, balanced prediction masses and both chi\-squared bounds\. These support the algebra; the inequalities above establish the interval\-wide result\. The construction is deliberately restricted and still minimax, not a pointwise statement\.

## Appendix BProof of the common\-experiment separation

### B\.1Uniform decision upper bound

Conditional emission independence and the symmetric transition imply𝔼\[Y∣Z1,H\]=λaZ1\\mathbb\{E\}\[Y\\mid Z\_\{1\},H\]=\\lambda aZ\_\{1\}\. Hence conditional report error is symmetric and independent ofHHgivenZ1Z\_\{1\}\. For the finite class𝒢\\mathcal\{G\}of all2642^\{64\}deterministic policies, Hoeffding and a union bound give simultaneous risk\-estimation error at mostϵn=log⁡\(2​\|𝒢\|/δ\)/\(2​n\)\\epsilon\_\{n\}=\\sqrt\{\\log\(2\|\\mathcal\{G\}\|/\\delta\)/\(2n\)\}\. An empirical minimizer has excess future risk at most2​ϵn2\\epsilon\_\{n\}\. The positive symmetric bridge preserves the conditional Bayes rule and divides this excess byλ​a≥ℓ/9\\lambda a\\geq\\ell/9, giving \([3](https://arxiv.org/html/2610.00094#S3.E3)\)\.

### B\.2Nontrivial decision lower bound

Seta=\.3a=\.3,b=\.5b=\.5,λ=ℓ\\lambda=\\ell, and letχ⁡\(x\)=x1​x2​x3\\chi\(x\)=x\_\{1\}x\_\{2\}x\_\{3\},ηs​\(x\)=a​x1\+s​b​χ​\(x\)\\eta\_\{s\}\(x\)=ax\_\{1\}\+sb\\chi\(x\)fors∈\{−1,\+1\}s\\in\\\{\-1,\+1\\\}\. Definefs​\(x∣z\)=\[1\+z​ηs​\(x\)\]/8f\_\{s\}\(x\\mid z\)=\[1\+z\\eta\_\{s\}\(x\)\]/8\. These are positive complementary emissions with anchor informativenessaa\. Summing the three latent states gives the full observed law

Ps​\(x0,x1,y\)=1\+λ​ηs​\(x0\)​ηs​\(x1\)\+λ​a​y​ηs​\(x1\)\+λ2​a​y​ηs​\(x0\)128\.P\_\{s\}\(x\_\{0\},x\_\{1\},y\)=\\frac\{1\+\\lambda\\eta\_\{s\}\(x\_\{0\}\)\\eta\_\{s\}\(x\_\{1\}\)\+\\lambda ay\\eta\_\{s\}\(x\_\{1\}\)\+\\lambda^\{2\}ay\\eta\_\{s\}\(x\_\{0\}\)\}\{128\}\.\(10\)Every cell exceeds1/2561/256, since\|ηs\|≤\.8\|\\eta\_\{s\}\|\\leq\.8andλ≤1/8\\lambda\\leq 1/8\. Writingχt=χ⁡\(xt\)\\chi\_\{t\}=\\chi\(x\_\{t\}\), the difference is

P\+−P−=2​a​b128​\[λ⁡\(x0,1​χ1\+χ0​x1,1\+y​χ1\)\+λ2​y​χ0\]\.P\_\{\+\}\-P\_\{\-\}=\\frac\{2ab\}\{128\}\\left\[\\lambda\(x\_\{0,1\}\\chi\_\{1\}\+\\chi\_\{0\}x\_\{1,1\}\+y\\chi\_\{1\}\)\+\\lambda^\{2\}y\\chi\_\{0\}\\right\]\.The four Walsh characters in brackets are orthogonal under the uniform 128\-cell law\. Therefore

KL\(P\+∥P−\)\\displaystyle\\operatorname\{KL\}\(P\_\{\+\}\\\|P\_\{\-\}\)≤∑o\(P\+​\(o\)−P−​\(o\)\)2P−​\(o\)\\displaystyle\\leq\\sum\_\{o\}\\frac\{\(P\_\{\+\}\(o\)\-P\_\{\-\}\(o\)\)^\{2\}\}\{P\_\{\-\}\(o\)\}\(11\)≤8​a2​b2​\(3​λ2\+λ4\)≤32​a2​b2​λ2\.\\displaystyle\\leq 8a^\{2\}b^\{2\}\(3\\lambda^\{2\}\+\\lambda^\{4\}\)\\leq 32a^\{2\}b^\{2\}\\lambda^\{2\}\.\(12\)Pinsker and product additivity giveTV⁡\(P\+n,P−n\)≤4​a​b​λ​n≤1/2\\operatorname\{TV\}\(P\_\{\+\}^\{n\},P\_\{\-\}^\{n\}\)\\leq 4ab\\lambda\\sqrt\{n\}\\leq 1/2forn≤1/\(64​a2​b2​ℓ2\)n\\leq 1/\(64a^\{2\}b^\{2\}\\ell^\{2\}\)\.

Summing outYY, the unnormalized current posterior margin isηs​\(x1\)\+λ​ηs​\(x0\)\\eta\_\{s\}\(x\_\{1\}\)\+\\lambda\\eta\_\{s\}\(x\_\{0\}\)\. Sinceb−a−λ⁡\(a\+b\)≥\.1b\-a\-\\lambda\(a\+b\)\\geq\.1, its sign iss​χ1s\\chi\_\{1\}\. Both Bayes accuracies are\(1\+b\)/2=\.75\(1\+b\)/2=\.75\. If𝔼U\\mathbb\{E\}\_\{U\}denotes uniform averaging over the 64 histories, any realized deterministic policy has excess

Rs​\(g\)−Rs​\(gs∗\)=b2−12​𝔼U​\[g⁡\(ηs​\(x1\)\+λ​ηs​\(x0\)\)\]\.R\_\{s\}\(g\)\-R\_\{s\}\(g\_\{s\}^\{\*\}\)=\\frac\{b\}\{2\}\-\\frac\{1\}\{2\}\\mathbb\{E\}\_\{U\}\[g\(\\eta\_\{s\}\(x\_\{1\}\)\+\\lambda\\eta\_\{s\}\(x\_\{0\}\)\)\]\.\(13\)Decode the model sign usingsign⁡\(𝔼U​\[g​χ1\]\)\\operatorname\{sign\}\(\\mathbb\{E\}\_\{U\}\[g\\chi\_\{1\}\]\), with a fixed tie rule\. When this decoding is wrong, the main parity term cannot be positive; the remaining terms have magnitude at mosta\+λ⁡\(a\+b\)a\+\\lambda\(a\+b\)\. The excess is thus at least\[b−a−λ⁡\(a\+b\)\]/2≥\.05\[b\-a\-\\lambda\(a\+b\)\]/2\\geq\.05\. Le Cam’s two\-point inequality gives maximum decoding error at least\(1−TV\)/2≥1/4\(1\-\\operatorname\{TV\}\)/2\\geq 1/4, and therefore the same lower bound for probability of excess at least\.05\.05\. Randomized learners are covered by adjoining their independent random seed; the same bounded\-term argument applies to a randomized policy’s expected prediction\. This proves the quadratic decision lower bound\.

### B\.3Parameter lower bound without discarding observations

LetUt=Zt​EtU\_\{t\}=Z\_\{t\}E\_\{t\}, where the independent errors have𝔼⁡\[Et\]=a\\mathbb\{E\}\[E\_\{t\}\]=a\. SetXt,1=UtX\_\{t,1\}=U\_\{t\}andXt,j=Ut​Bt,jX\_\{t,j\}=U\_\{t\}B\_\{t,j\}forj=2,3j=2,3, with independentBB’s flipping with known probability\.1\.1\. These emissions have full support and belong to the same class\. Conditional on the observed anchor triple\(U0,U1,U2\)\(U\_\{0\},U\_\{1\},U\_\{2\}\), the four additional source entries are generated by a parameter\-independent channel\. Conversely the full observation contains the anchor triple\. The KL chain rule therefore gives exact equality between full\-record KL and anchor\-triple KL\.

The latter law is

Pa,λ​\(u,v,w\)=1\+λ​a2​\(u​v\+v​w\)\+λ2​a2​u​w8\.P\_\{a,\\lambda\}\(u,v,w\)=\\frac\{1\+\\lambda a^\{2\}\(uv\+vw\)\+\\lambda^\{2\}a^\{2\}uw\}\{8\}\.Choose\(a0,λ0\)=\(\.5,ℓ\)\(a\_\{0\},\\lambda\_\{0\}\)=\(\.5,\\ell\)and\(a1,λ1\)=\(\.75,4​ℓ/9\)\(a\_\{1\},\\lambda\_\{1\}\)=\(\.75,4\\ell/9\)\. Both are inℳℓ\\mathcal\{M\}\_\{\\ell\}\. Their lag\-one coefficients coincide atℓ/4\\ell/4, while lag\-two coefficients differ byd=5​ℓ2/36d=5\\ell^\{2\}/36\. Every scalar cell is at least1/161/16, giving

KL\(P0∥P1\)≤2d2=25ℓ4/648,TV\(P0n,P1n\)≤5ℓ2n/36\.\\operatorname\{KL\}\(P\_\{0\}\\\|P\_\{1\}\)\\leq 2d^\{2\}=25\\ell^\{4\}/648,\\qquad\\operatorname\{TV\}\(P\_\{0\}^\{n\},P\_\{1\}^\{n\}\)\\leq 5\\ell^\{2\}\\sqrt\{n\}/36\.Forn≤324/\(25​ℓ4\)n\\leq 324/\(25\\ell^\{4\}\), total variation is at most1/21/2\. Thresholding an estimate at5/85/8turns it into a test\. An incorrect test implies estimation error at least1/81/8, so

infa^maxj∈\{0,1\}ℙj\{\|a^−aj\|≥1/8\}≥1/4\.\\inf\_\{\\hat\{a\}\}\\max\_\{j\\in\\\{0,1\\\}\}\\mathbb\{P\}\_\{j\}\\\{\|\\hat\{a\}\-a\_\{j\}\|\\geq 1/8\\\}\\geq 1/4\.The two minimax lower bounds use different least\-favorable subfamilies inside one model class\. The parameter result is not obtained by withholding source observations available to the decision learner\.

Figure 2:The broader\-class result \(Theorem[2](https://arxiv.org/html/2610.00094#Thmtheorem2)\): one common experiment, different least\-favorable subfamilies\. Exact 128\-cell laws exhibit quadratic decision\-hard and quartic parameter\-hard KL\. Future\-label ERM risk approximately collapses againstn​ℓ2n\\ell^\{2\}\(500 repetitions per point\)\. These retained simulations illustrate the proof, not establish it; they are not experiments on the four\-world family of Theorem[1](https://arxiv.org/html/2610.00094#Thmtheorem1)\.

## Appendix CDetails of compatible\-channel certification

#### Linear program for sharp endpoints\.

Sett=1/\(1−∑kuk\)t=1/\(1\-\\sum\_\{k\}u\_\{k\}\)andv=t​uv=tu\. Minimize or maximizeΔY​t−m𝖳​v\\Delta\_\{Y\}t\-m^\{\\mathsf\{T\}\}vsubject to

t−∑kvk=1,0≤vk≤ℓkY​t,t≥1\.t\-\\sum\_\{k\}v\_\{k\}=1,\\qquad 0\\leq v\_\{k\}\\leq\\ell\_\{k\}^\{Y\}t,\\qquad t\\geq 1\.The inverse isu=v/tu=v/t\. The population implementation refuses constantqq: then∑kℓkY=1\\sum\_\{k\}\\ell\_\{k\}^\{Y\}=1and the compact positive\-denominator argument no longer applies\. This restriction is unnecessary for the finite\-sample witness test, which remains valid for constant channels withρ\>0\\rho\>0\.

#### Binary sign condition\.

Use labels0,10,1, writer⁡\(h\)=ℙ⁡\(Y=1∣H=h\)r\(h\)=\\mathbb\{P\}\(Y=1\\mid H=h\)andm=ℙ⁡\(g=1\)−ℙ⁡\(b=1\)m=\\mathbb\{P\}\(g=1\)\-\\mathbb\{P\}\(b=1\)\. Since the two mass changes are\(−m,m\)\(\-m,m\), the sharp positive\-gain condition is

ΔY\>\{m​minh​r​\(h\),m≥0,\(−m\)​\(1−maxh⁡r⁡\(h\)\),m<0\.\\Delta\_\{Y\}\>\\begin\{cases\}m\\min\_\{h\}r\(h\),&m\\geq 0,\\\\ \(\-m\)\(1\-\\max\_\{h\}r\(h\)\),&m<0\.\\end\{cases\}For example, letHHbe a fair bit and the clean state equalHHwith probability\.8\.8\. Letg=Hg=H,b=0b=0,ρ=\.8\\rho=\.8and background probabilities\(\.25,\.75\)\(\.25,\.75\)\. Thenr⁡\(0\)=\.31r\(0\)=\.31,r⁡\(1\)=\.79r\(1\)=\.79,m=\.5m=\.5andΔY=\.29\\Delta\_\{Y\}=\.29\. The sharp numerator is\.135\>0\.135\>0, so a one\-direction correction is identifiable\. The unrestricted ruleΔY\>\|m\|\\Delta\_\{Y\}\>\|m\|cannot accept it\.

#### Exact witness bound\.

If classkkhasNkN\_\{k\}records in its fixed region andSkS\_\{k\}of their reports equalkk, set

Uk=\{1,Nk=0​or​Sk=Nk,BetaQuantile1−α/\(2​K\)⁡\(Sk\+1,Nk−Sk\),otherwise\.U\_\{k\}=\\begin\{cases\}1,&N\_\{k\}=0\\text\{ or \}S\_\{k\}=N\_\{k\},\\\\ \\operatorname\{BetaQuantile\}\_\{1\-\\alpha/\(2K\)\}\(S\_\{k\}\+1,N\_\{k\}\-S\_\{k\}\),&\\text\{otherwise\}\.\\end\{cases\}Conditionally on the region count,SkS\_\{k\}is binomial with the regional report probability\. Exact conditional coverage therefore implies unconditional coverage, including random region counts\. Regions may overlap: the argument uses a union bound, not independence\. Learning regions from the same reports used in this calculation is not covered by the theorem\.

#### Finite\-history consistency\.

For each class, select a minimum empirical report\-probability history from an independent training sample, with fixed tie breaking\. If every history has positive probability and both training and calibration sizes diverge, uniform convergence on the finite alphabet ensures selected population frequencies converge toℓkY\\ell\_\{k\}^\{Y\}\. Counts in the selected calibration regions diverge and their binomial upper bounds converge to the same values\. Empirical policy gains/masses converge and the Hoeffding radius vanishes\. Thus every fixed pair with strictly positive sharp numerator is eventually accepted with probability tending to one\. No corresponding claim is made at a zero numerator\.

#### Old comparator implementation\.

For binarydi=gi−bid\_\{i\}=g\_\{i\}\-b\_\{i\}, the earlier test requires both means ofdi​Yid\_\{i\}Y\_\{i\}and−di​\(1−Yi\)\-d\_\{i\}\(1\-Y\_\{i\}\)to be positive\. Each is a\{−1,0,1\}\\\{\-1,0,1\\\}score\. With positive/negative countsW,LW,L, its exact one\-sided sign\-test p\-value isℙ\{Binomial\(W\+L,1/2\)≥W\}\\mathbb\{P\}\\\{\\operatorname\{Binomial\}\(W\+L,1/2\)\\geq W\\\}, or one ifW\+L=0W\+L=0\. Both tests must reject at levelα\\alpha; intersection\-union logic gives a level\-α\\alphaguarantee\. An entirely upward revision makes the second score nonpositive everywhere\. ForK\>2K\>2, the earlier implementation requires every mean ofDi−𝟏\{gi=k\}\+𝟏\{bi=k\}D\_\{i\}\-\\mathbf\{1\}\\\{g\_\{i\}=k\\\}\+\\mathbf\{1\}\\\{b\_\{i\}=k\\\}to exceed8​log⁡\(1/α\)/n\\sqrt\{8\\log\(1/\\alpha\)/n\}\. Its population condition is sufficient, not sharp\.

#### Verification\.

Tests enumerate all vertices of random compatible boxes forK=2,3,5K=2,3,5, compare their gain extrema with the LP, reconstruct the attaining clean laws, and verify that they reproduce the observed law\. Fixed\-seed binary/categorical one\-direction examples pass the new certificate; harmful and empty\-sample examples fail\. These checks supplement but do not replace the analytic proof\.

## Appendix DMutable\-state protocol and complete results

At each step a ledger attempts a state change with probability\.12\.12, with failed\-version and duplicate\-command controls\. The display labels asset room, deployment slot and task owner are not independently validated domains\. Four report snapshots have known synthetic event/arrival times and roots\. Policies receive reports, not initial state, commands or withheld ledger truth\. The candidate selection set comprises native/root\-capped Nous, smoothed suffix\-one/two future predictors and independent/joint\-emission HMMs\. Selection uses only development next\-anchor accuracy\. The selected policy is an HMM in seven runs and a suffix\-two predictor in two\. There are 36 distinct split seeds\.

Table 4:All principal baselines on the frozen mutable\-state study\. Percentage accuracy, mean±\\pmsample SD across three 5,000\-history test runs per regime\.Table 5:Selected certified policy versus LWW\. Harmful/beneficial counts concern individual overrides and are distinct from the population\-level gate guarantee\. Intervals are paired normal approximations, not bootstrap or simultaneous CIs\.Total per\-cell generation, fitting, inference and receipt\-writing time summed to 42\.55 seconds on the local host\. The runtime is not hardware\-normalized\. Full\-array audits reconstruct locked policies, predictions and certificates and replay all 45,000 ledgers and Dimension receipts without changing selection\. Where reconstruction reruns HMM fitting, it uses the original training split and frozen seed, never held\-out truth\.

## Appendix EExternal simulator protocol and paired results

MiniGrid 3\.0\.0 and Gymnasium 1\.3\.0 execute the S7, S13 and S17Random Memory environments\. Controller smoke tests use seeds 0–11; final episode seeds are independently derived from the protocol namespace, environment, run, split and record index and checked for collisions after conversion to 32 bits\. Fixed policies precede all final calibration and test generation\.

#### Input and action audit\.

The controller first returns west to the cue room, observes the object while facing north, traverses east to the junction and observes the upper branch object\. It uses canonical partial observations for routing, never the hidden success coordinate\. The noisy wrapper receives the cue to generate reports, then every memory policy receives only the reports and recorded copying roots\. Branch choice uses its predicted cue and the visible branch\-object identity\. The clean cue retained by the runner is used for evaluator checks only\. No history of pristine cue observations is supplied to a memory baseline\. All methods share routing, including its known layout conventions; this is memory\-module evaluation, not learned navigation\.

Calibration resets the same environment family but deliberately stages the camera at the cue; it does not claim a calibration action rollout\. This changes routing cost, not the cue/report population used by the fixed policy and witness\. Test has no camera\-staging shortcut\. The runner deep\-copies the junction state and executes both possible terminal decisions to obtain exact paired action outcomes\. The audit independently replays both saved action sequences from reset for all 9,000 episode seeds, reconstructs 36 old/new certificates and checks 54,000 distinct calibration/test seeds and all hashes\.

Table 6:Compatible\-gate deployment versus its specified incumbent\. A declined candidate leaves the incumbent unchanged\. Paired 95% intervals use the normal approximation for the mean per\-episode accuracy difference\. They are descriptive, not simultaneous across the 18 comparisons\.Saved result files additionally contain harmful and beneficial overrides, override fraction and actual environment reward\. The certificate targets success/accuracy, not MiniGrid’s step\-penalized reward; changing the final turn sequence can change reward even at equal success\. The source\-0 baseline is not oracle\-tuned: its report index and evaluation were fixed in the protocol, although the controlled generator gives it higher accuracy than any other individual source\.

## Appendix FComplete control laws and interpretation

For categorical controlsHHis uniform onKKclasses\. Except for null cases,ℙ⁡\(Z=H∣H\)=\.8\\mathbb\{P\}\(Z=H\\mid H\)=\.8and each alternative state has probability\.2/\(K−1\)\.2/\(K\-1\)\. The one\-direction candidate isg⁡\(H\)=Hg\(H\)=H; its incumbent maps onlyH=K−1H=K\-1to zero\. Thus its true gain is\(\.8−\.2/\(K−1\)\)/K\(\.8\-\.2/\(K\-1\)\)/K\. The harmful candidate is constant zero againstb⁡\(H\)=Hb\(H\)=H, with gain1/K−\.81/K\-\.8\. In null casesZZis uniform independent ofHH,g⁡\(H\)=Hg\(H\)=Handb⁡\(H\)=\(H\+1\)modKb\(H\)=\(H\+1\)\\bmod K\. Reports preserveZZwith probabilityρ\\rhoand otherwise draw from\(v0,\(1−v0\)/\(K−1\),…\)\(v\_\{0\},\(1\-v\_\{0\}\)/\(K\-1\),\\ldots\)\. Fixed witnesses areAk=\{H≠k\}A\_\{k\}=\\\{H\\neq k\\\}\.

The lag controls useρ=\.8​\(\.8\)d\\rho=\.8\(\.8\)^\{d\}\. This is a controlled effective lag\-to\-signal law, not an empirical estimate from natural trajectories\. It matches the decay of the identity component under a symmetric categorical transition channel, with the resulting background encoded in the report law\. Longer lags therefore weaken the report evidence explicitly\.

There are 5,400 independent calibration repetitions in the initial controls, 4,500 in the phase follow\-up, and 2,000 fresh sequential\-release calibrations\. Fixed\-cell repetitions retain seeds, decisions, empirical margins and radii; sequential runs retain decisions with seeds recoverable from the protocol\. Generator hashes allow regeneration\. Exact Clopper–Pearson intervals for acceptance frequencies are stored for every cell\. In a 100\-run cell, 0/100 gives a 95% upper bound of about 3\.62%, and 100/100 a lower bound of about 96\.38%; these do not replace theoretical type\-I control\.

The sharp interval is computed from the known population law only to diagnose identification in the synthetic study\. Neither that law, trueρ\\rho, trueuu, nor clean states are supplied to the finite\-sample certificate\. The phase follow\-up was motivated by the initial zero\-power result and is labelled accordingly; all 45 cells, including its 20 zero\-power cells, are reported\.

## Appendix GAdditional arithmetic and assumption scrutiny

Symbolic summation over hidden paths verifies the matched construction’s observable law; exact integer Walsh inner products and rational endpoint checks verify its orthogonality and positive posterior margin\. These checks support particular proof steps, not automated verification of the full theorem\. For a binary one\-direction comparison, complete conditional\-binomial summation evaluates C2 acceptance across 27 specified laws \(sample sizes 128, 512, 2048; three gains and three report channels\)\. Among the 18 null or harmful laws, the largest computed acceptance probability is1\.16×10−211\.16\\times 10^\{\-21\}atα=\.05\\alpha=\.05\. This is not a supremum over all null laws\. Probabilities use floating\-point arithmetic, not exact rational arithmetic or Monte Carlo\. A separate exhaustive small\-sample enumeration checks the summation against the implemented certificate\.

An intentionally invalid control chooses witness masks using calibration labels instead of histories: it accepts a zero\-population\-gain proposal, whereas fixed history\-only witnesses reject on the same records\. This violates C2’s pre\-calibration witness requirement, not its conclusion\. The low\-level mask API cannot verify that requirement from boolean arrays alone; callers must enforce it\. Neither this audit nor a saved mask hash proves independence\. The reproducibility package retains the checks and the counterexample\.

These are self\-audits\. Bibliographic comparison cannot establish that no equivalent result exists elsewhere; automated verification does not imply independent scientific scrutiny or author approval\.

## Appendix HBounded channel misspecification

LetCh\(⋅∣z\)C\_\{h\}\(\\cdot\\mid z\)be the actual audit channel and let a single reference channel beC0\(⋅∣z\)=ρez\+uC\_\{0\}\(\\cdot\\mid z\)=\\rho e\_\{z\}\+u, withρ\>0\\rho\>0,u≥0u\\geq 0and∑kuk=1−ρ\\sum\_\{k\}u\_\{k\}=1\-\\rho\. Assume

ess​suph,zTV\(Ch\(⋅∣z\),C0\(⋅∣z\)\)≤ϵ,TV\(P,Q\)=12∑k\|Pk−Qk\|\.\\mathop\{\\mathrm\{ess\\,sup\}\}\_\{h,z\}\\mathrm\{TV\}\\bigl\(C\_\{h\}\(\\cdot\\mid z\),C\_\{0\}\(\\cdot\\mid z\)\\bigr\)\\leq\\epsilon,\\qquad\\mathrm\{TV\}\(P,Q\)=\\tfrac\{1\}\{2\}\\sum\_\{k\}\|P\_\{k\}\-Q\_\{k\}\|\.This permits bounded history dependence; the clean deployment law and iid calibration assumptions remain unchanged\. LetUkU\_\{k\}be the existing binomial witness bounds,Vk=min⁡\(1,Uk\+ϵ\)V\_\{k\}=\\min\(1,U\_\{k\}\+\\epsilon\), andd^=n−1∑i𝟏\{gi≠bi\}\\hat\{d\}=n^\{\-1\}\\sum\_\{i\}\\mathbf\{1\}\\\{g\_\{i\}\\neq b\_\{i\}\\\}\. Atϵ=0\\epsilon=0Theorem[5](https://arxiv.org/html/2610.00094#Thmtheorem5)recovers the original witness certificate\.

#### Proof of Theorem[5](https://arxiv.org/html/2610.00094#Thmtheorem5)\.

Convexity givesTV⁡\(q⁡\(h\),ρ​p​\(h\)\+u\)≤ϵ\\mathrm\{TV\}\(q\(h\),\\rho p\(h\)\+u\)\\leq\\epsilon, souk≤qk​\(h\)\+ϵu\_\{k\}\\leq q\_\{k\}\(h\)\+\\epsilon\. Binomial coverage therefore places the fixed trueuuin\[0,V\]\[0,V\]with failure probability at mostα/2\\alpha/2\. The contrastfh\(y\)=𝟏\{g\(h\)=y\}−𝟏\{b\(h\)=y\}f\_\{h\}\(y\)=\\mathbf\{1\}\\\{g\(h\)=y\\\}\-\\mathbf\{1\}\\\{b\(h\)=y\\\}has oscillation2𝟏\{g\(h\)≠b\(h\)\}2\\mathbf\{1\}\\\{g\(h\)\\neq b\(h\)\\\}\. The TV expectation inequality bounds its conditional bias above by2ϵ𝟏\{g\(h\)≠b\(h\)\}2\\epsilon\\mathbf\{1\}\\\{g\(h\)\\neq b\(h\)\\\}\. Consequently the fixed\-uuscores

Si=fHi\(Yi\)−ugi\+ubi−2ϵ𝟏\{gi≠bi\}S\_\{i\}=f\_\{H\_\{i\}\}\(Y\_\{i\}\)\-u\_\{g\_\{i\}\}\+u\_\{b\_\{i\}\}\-2\\epsilon\\mathbf\{1\}\\\{g\_\{i\}\\neq b\_\{i\}\\\}have mean at mostρ​Δ\\rho\\Deltaand lie in\[−2−2​ϵ,2\]\[\-2\-2\\epsilon,2\]\. One\-sided Hoeffding at tailα/2\\alpha/2gives the displayed radius\. Minimization over\[0,V\]\[0,V\]cannot exceed the empirical true\-uuscore\. The union bound proves the result without conditioning on estimated witnesses\.□\\square

#### Conditional copying and tight local constants\.

Suppose a base channel is withinγ\\gammaofC0C\_\{0\}and is replaced by an arbitrary report channel with conditional probabilityt⁡\(h,z\)≤βt\(h,z\)\\leq\\beta\. Convexity bounds the resulting TV by\(1−t\)​γ\+t≤γ\+β−γ​β\(1\-t\)\\gamma\+t\\leq\\gamma\+\\beta\-\\gamma\\beta\. This includes candidate copying but is stronger than a bound on the average copied fraction\. Movingϵ\\epsilonprobability from the incumbent label to the candidate label attains paired bias2​ϵ2\\epsilonon disagreement\. Removingϵ\\epsilonmass from a background coordinate attains the witness inflation constant one\. These local constants are optimal; neither global sharpness of the combined bound nor optimality of its concentration radius is claimed\.

Regression tests check zero\-budget equivalence, categorical population envelopes, attained local constants, monotonicity in the budget, and an explicit null\-gain history\-dependent attack accepted by the original rule but rejected at the correct budget\. A beneficial control also accepts\. These are implementation checks, not a replacement for a frozen power study\. Budgets are not inferred from noisy reports\. Selecting a convenient budget after inspection does not establish its validity\. Fresh\-release error spending continues to apply conditionally when each budget contract holds\.

## Appendix ILearned\-witness power and end\-to\-end composition

Histories need not have finite support\. For fixed regions with probabilities at leastπ\>0\\pi\>0, writerk=ℙ⁡\(Y=k∣H∈Ak\)r\_\{k\}=\\mathbb\{P\}\(Y=k\\mid H\\in A\_\{k\}\)and define

M=ΔY−∑k\(mk\)\+​min⁡\(1,rk\+ϵ\)−2​ϵ​d,d=ℙ⁡\(g≠b\)\.M=\\Delta\_\{Y\}\-\\sum\_\{k\}\(m\_\{k\}\)\_\{\+\}\\min\(1,r\_\{k\}\+\\epsilon\)\-2\\epsilon d,\\qquad d=\\mathbb\{P\}\(g\\neq b\)\.For a desired power failure probabilityη\\eta, set

a\\displaystyle a=2​log⁡\(10​\(K\+1\)/η\)/n,b\\displaystyle=\\sqrt\{2\\log\(10\(K\+1\)/\\eta\)/n\},\\qquad b=log⁡\(10/η\)/\(2​n\),\\displaystyle=\\sqrt\{\\log\(10/\\eta\)/\(2n\)\},c\\displaystyle c=log⁡\(5​K/η\)\+log⁡\(2​K/α\)n​π,\\displaystyle=\\frac\{\\sqrt\{\\log\(5K/\\eta\)\}\+\\sqrt\{\\log\(2K/\\alpha\)\}\}\{\\sqrt\{n\\pi\}\},B\\displaystyle B=\(4\+2​ϵ\)​log⁡\(2/α\)2​n\+\(K\+1\)​a\+2​ϵ​b\+c\.\\displaystyle=\(4\+2\\epsilon\)\\sqrt\{\\frac\{\\log\(2/\\alpha\)\}\{2n\}\}\+\(K\+1\)a\+2\\epsilon b\+c\.
###### Proposition 6\(Fixed\-region acceptance power\)\.

Ifn​π≥8​log⁡\(5​K/η\)n\\pi\\geq 8\\log\(5K/\\eta\)andM\>BM\>B, the bounded\-misspecification certificate accepts with probability at least1−η1\-\\eta\.

#### Proof\.

Chernoff and a union bound giveNk≥n​π/2N\_\{k\}\\geq n\\pi/2for all regions except with probabilityη/5\\eta/5\. Simultaneous Hoeffding bounds controlΔ^Y\\hat\{\\Delta\}\_\{Y\}and allm^k\\hat\{m\}\_\{k\}withinaa, with failureη/5\\eta/5, andd^\\hat\{d\}withinbb, with failureη/5\\eta/5\. Conditional binomial Hoeffding controls each regional frequency above byrk\+log⁡\(5​K/η\)/\(2​Nk\)r\_\{k\}\+\\sqrt\{\\log\(5K/\\eta\)/\(2N\_\{k\}\)\}, with joint failureη/5\\eta/5\. Inverting the lower\-tail binomial Hoeffding bound gives deterministically

Uk≤min⁡\(1,r^k\+log⁡\(2​K/α\)/\(2​Nk\)\)\.U\_\{k\}\\leq\\min\\left\(1,\\hat\{r\}\_\{k\}\+\\sqrt\{\\log\(2K/\\alpha\)/\(2N\_\{k\}\)\}\\right\)\.ThusV^k≤min⁡\(1,rk\+ϵ\)\+c\\hat\{V\}\_\{k\}\\leq\\min\(1,r\_\{k\}\+\\epsilon\)\+con these events\. Clipping and positive parts are 1\-Lipschitz, and∑k\(mk\)\+≤1\\sum\_\{k\}\(m\_\{k\}\)\_\{\+\}\\leq 1\. The empirical numerator is at leastM−\(K\+1\)​a−2​ϵ​b−cM\-\(K\+1\)a\-2\\epsilon b\-c\. Subtracting the radius and union\-bounding the four events proves the claim\.□\\square

#### Learned score regions\.

Use an independent design split to learn arbitrary scoressk​\(h\)s\_\{k\}\(h\)and fixJJthreshold regions per class\. Conditional on design, suppose every region has mass at leastπ\\piand the best per\-class region has report probability at mostℓkY\+ξ\\ell\_\{k\}^\{Y\}\+\\xi, whereℓkY=ess​infh⁡qk​\(h\)\\ell\_\{k\}^\{Y\}=\\mathop\{\\mathrm\{ess\\,inf\}\}\_\{h\}q\_\{k\}\(h\)\. Select the smallest empirical report frequency on a separate selection split of sizeNN, breaking ties deterministically\. IfN​π≥8​log⁡\(2​K​J/ηtr\)N\\pi\\geq 8\\log\(2KJ/\\eta\_\{\\rm tr\}\), all selected regions satisfy

rk≤ℓkY\+ξ\+2​log⁡\(4​K​J/ηtr\)N​πr\_\{k\}\\leq\\ell\_\{k\}^\{Y\}\+\\xi\+2\\sqrt\{\\frac\{\\log\(4KJ/\\eta\_\{\\rm tr\}\)\}\{N\\pi\}\}with probability at least1−ηtr1\-\\eta\_\{\\rm tr\}\. To prove this, union\-bound Chernoff overK​JKJcounts with errorηtr/2\\eta\_\{\\rm tr\}/2, and two\-sided conditional Hoeffding over their frequencies with errorηtr/2\\eta\_\{\\rm tr\}/2\. Empirical minimization costs twice the resulting uniform deviation\. IfSϵS\_\{\\epsilon\}denotesMMwithrkr\_\{k\}replaced byℓkY\\ell\_\{k\}^\{Y\}, then

Sϵ\>ξ\+2​log⁡\(4​K​J/ηtr\)/\(N​π\)\+BS\_\{\\epsilon\}\>\\xi\+2\\sqrt\{\\log\(4KJ/\\eta\_\{\\rm tr\}\)/\(N\\pi\)\}\+Bensures acceptance probability at least1−ηtr−η1\-\\eta\_\{\\rm tr\}\-\\eta\. This does not assert that useful large regions exist for every representation\. All population mass and approximation assumptions must be justified; they are not certified by substituting calibration estimates into this condition\.

#### Nous composition\.

Learn a candidate and region family on design data; select regions on separate selection data; lock the policies, regions and externally justified channel budget; audit on fresh iid records; and issue a policy\-level Nous receipt only ifLϵ\>0L\_\{\\epsilon\}\>0\. Conditional on design and selection, Theorem[5](https://arxiv.org/html/2610.00094#Thmtheorem5)bounds non\-improving acceptance byα\\alpha\. Integrating over those splits preserves this guarantee without identifyingρ\\rho,uuor clean posteriors\. The displayed power condition additionally guarantees useful acceptance\. Theorem[1](https://arxiv.org/html/2610.00094#Thmtheorem1)supplies a concrete same\-report instance of this composition\. More generally, the HMM learning theorem is one possible candidate learner, not a requirement that its future\-state reports be the pinned audit channel\. Repeated fresh releases retain the stated error\-spending guarantee\. The receipt concerns the locked policy population, not correctness of an individual memory value\.

## Appendix JSensitivity and continuous\-witness protocol

Figure 3:All 18 frozen MiniGrid calibration comparisons under increasing assumed TV deviation, with no held\-out labels used\. A line above zero certifies only under the stated channel contract\. Five LWW lines start below zero\. This is post\-hoc sensitivity, not a confidence interval for actual misspecification\.The scalar lower bound is continuous, piecewise linear and strictly decreasing for nonempty samples: clipped witness penalties are nondecreasing, and the concentration radius strictly increases\. Roots are computed numerically to10−1410^\{\-14\}tolerance\. Empty accepting sets are recorded as null, not zero robustness\. The complete per\-comparison values are retained in the package\. Selecting an attractive root cannot establish an unknown channel contract; actual deviation must still be externally bounded\.

#### Fresh learned\-witness design\.

The protocol and runner were written before the first run\. There are four replicates at eachγ∈\{0,\.03,\.10\}\\gamma\\in\\\{0,\.03,\.10\\\}, with 4,000 design, 4,000 selection, 8,000 calibration and 5,000 test records per cell\. All 48 split seeds are distinct and derived from a new namespace\. The study is synthetic:H∼N⁡\(0,I16\)H\\sim N\(0,I\_\{16\}\),ℙ⁡\(Z=1∣H\)=sigmoid⁡\(2​H0\+H1\)\\mathbb\{P\}\(Z=1\\mid H\)=\\mathrm\{sigmoid\}\(2H\_\{0\}\+H\_\{1\}\), and

ℙ⁡\(Y=1∣H,Z\)=\.1\+\.8​Z\+γ​tanh⁡\(H2\)\.\\mathbb\{P\}\(Y=1\\mid H,Z\)=\.1\+\.8Z\+\\gamma\\tanh\(H\_\{2\}\)\.Thus the reference hasρ=\.8\\rho=\.8,u=\(\.1,\.1\)u=\(\.1,\.1\), and the per\-history/class TV deviation is at mostγ\\gammaby construction\. This is not a fitted bound\.

On design data fit ridge least squares toYYfrom an intercept and all 16 coordinates, using unnormalized penalty one including the intercept\. Freeze the candidateg\(h\)=𝟏\{s\(h\)\>\.5\}g\(h\)=\\mathbf\{1\}\\\{s\(h\)\>\.5\\\}against constant zero; no clean labels train it\. The witness scores ares1=ss\_\{1\}=sands0=1−ss\_\{0\}=1\-s\. For each score the five thresholds are design quantiles\.1,\.2,\.3,\.4,\.5\.1,\.2,\.3,\.4,\.5\. Independent selection data choose the threshold with minimum empirical class\-specific audit frequency, with deterministic tie breaking\. The coefficients, thresholds, selected indices and budget are locked before inspecting calibration or test labels\. Only then apply the certificate to fresh calibration\.

Comparators use the entire history space or fixed regionsH0\>0H\_\{0\}\>0for class zero andH0≤0H\_\{0\}\\leq 0for class one\. All three use the same candidate, incumbent, calibration records and separately fixedα=\.05\\alpha=\.05tests\. Saved results include nominal and correct\-budget rules, region counts, all certificate terms, test accuracy and paired gain/standard error for every cell\. The correct\-budget acceptances are in the main table; no method is selected by test accuracy or calibration acceptance\. Conditional validity holds regardless of region quality\. The stronger population mass/approximation assumptions of the power proposition are not claimed to have been empirically certified\.

The audit regenerates every split, refits design coefficients, reconstructs selection, verifies locks and all certificates, and recomputes held\-out metrics from saved data\. This is self\-reproduction, not independent replication\. There is no language model, textual extraction or new external environment in this experiment\. The existing external MiniGrid and mutable\-state results remain unchanged, and their strong baselines and negative outcomes remain\.