Decision Sufficiency of Learned Representation Interfaces for Safe Generative Control

# Decision Sufficiency of Learned Representation Interfaces for Safe Generative Control
Source: [https://arxiv.org/html/2610.27446](https://arxiv.org/html/2610.27446)
###### Abstract

A controller can guide a language model toward safe and useful responses by choosing when to assist, ask for clarification, or refuse\. When the controller uses selected internal features, poor performance alone cannot distinguish missing information from limitations of its decision rules or training\. We study*decision sufficiency*: whether the best controller using a representation can match the minimum average loss attainable with full state information, under the same strategies and evaluation criteria\. For interfaces with finitely many outputs, we compute the optimal loss exactly and separate the three sources of excess loss\. Pairwise checks can miss information loss: states with the same controller input may share an optimal strategy in every pair but not across the whole group\. We bound the number of states needed to reveal a conflict and characterize the fewest outputs an unrestricted interface needs to avoid all conflicts\. We audit selected features from sparse autoencoders, which encode model activations using relatively few active features, in Qwen2\.5, SmolLM2, and Phi\. On SmolLM2 responses scored by two human raters, restricting the controller to two selected on/off features increases its minimum average loss by0\.03450\.0345–0\.11660\.1166relative to full state information, even with the governing policy supplied\. Both independently trained autoencoders show the increase on both evaluation sets\. The audit identifies decision failures that require changing the controller’s input, not merely its architecture or training\.

## 1Introduction

A language model may need to assist with a request, ask for missing information, or refuse\. The appropriate choice depends on the facts available about the request and the policy governing allowed behavior\. Internal activations, the numerical vectors computed inside a model, can inform monitoring and steering\([Li et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib50);[Zou et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib51);[Karnik and Bansal, 2025](https://arxiv.org/html/2610.27446#bib.bib52)\)\. We study a controller that uses selected activation features to choose a response strategy, which the model then executes\. A*representation interface*specifies the controller’s input\. We ask whether the input supports appropriate choices, independently of how well a particular controller is trained\.

Suppose a policy calls for assistance when authorization is verified and clarification when evidence is missing\. If both cases give the controller identical inputs, it must use the same choice or distribution over choices\. Refusing both may avoid unauthorized assistance but also withholds permitted help\. The challenge is to distinguish missing information from a controller’s failure to use information that remains available\.

A sparse autoencoder \(SAE\) learns to reconstruct an activation using relatively few active features\([Cunningham et al\., 2024](https://arxiv.org/html/2610.27446#bib.bib57);[Gao et al\., 2025](https://arxiv.org/html/2610.27446#bib.bib59);[Lieberum et al\., 2024](https://arxiv.org/html/2610.27446#bib.bib60)\)\. A controller may receive only selected feature values or on/off indicators\. Accurate reconstruction from the full code does not establish what the selected input preserves\. A*probe*, a predictor fitted to the selected features, can fail because information is missing, its allowed rules are too restrictive, or its fitting is imperfect\.

We compare the best controller using the interface with the best controller using the*full audited state*: the situation, governing policy, and recorded evidence\. Both select from a fixed library of*continuations*, meaning response strategies that may react to later observations\. The execution protocol, loss function, and state distribution are identical in each comparison\. The loss evaluates safety and response quality;*risk*is its average under the fixed state distribution\. An interface is*decision\-sufficient*when its minimum risk equals the minimum with full state information\. The comparison builds on statistical decision theory and state abstraction\([Blackwell, 1953](https://arxiv.org/html/2610.27446#bib.bib2);[Le Cam, 1964](https://arxiv.org/html/2610.27446#bib.bib3);[Abel et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib18)\)\.

LetUUdenote the full audited state andVVthe complete controller input:

U⏟full audited state⟶V⏟complete controller input⟶choice of continuation\.\\underbrace\{U\}\_\{\\text\{full audited state\}\}\\longrightarrow\\underbrace\{V\}\_\{\\text\{complete controller input\}\}\\longrightarrow\\text\{choice of continuation\}\.LetZZdenote the selected representation features andSSthe governing policy text\. The complete input isV=ZV=Zfor features alone andV=\(Z,S\)V=\(Z,S\)when the policy is also supplied\. For a deterministic interface, states with identical complete inputsVVmust share a choice or distribution over choices\. Supplying policy textSScan distinguish states that selected featuresZZalone merge\. Auditing features alone would then evaluate a different information restriction\.

We compare four risk levels: the minimum with full state information, the minimum with the interface, the minimum within an allowed family of rules using the interface, and the risk of the fitted rule\. The successive differences measure information loss, rule restrictions, and fitting error, including estimation and optimization error\. All three terms can be positive simultaneously\. For a finite deterministic interface, states with identical input form a*cell*\. Choosing the continuation with the smallest conditional average loss in each cell computes the minimum over all interface rules\. The*information gap*is the best interface risk minus the minimum using the full state\.

To explain the average information gap, we examine which states cannot share a decision\. Consider three states with the same controller input whose optimal strategies are, respectively, clarification or assistance, clarification or refusal, and assistance or refusal\. Every pair shares an optimum, but the group does not\. Randomization cannot attain all three optima unless every strategy it selects with positive probability is optimal at every state\. More generally,*regret*is loss above a state’s minimum\. A group is*compatible*when one distribution over continuations keeps regret within a specified nonnegative tolerance at every state\. A*minimal conflict*is an incompatible group that becomes compatible whenever one state is removed\. To make every cell compatible, an interface must split each conflicting group across at least two cells\.

We audit selected SAE features in Qwen2\.5, SmolLM2, and Phi, with generation and scoring protocols specified for each model\. With the policy supplied, the best controller using two selected on/off features still has higher human rating loss than the full state reference\. On 96 SmolLM2 states from policy areas excluded from fitting, the gaps are0\.11660\.1166and0\.10820\.1082for two independently trained SAEs\. The original 80\-state sample gives gaps of0\.03450\.0345and0\.03530\.0353but includes fitting areas\. Qwen2\.5 comparisons vary the features, rule family, and fitting data separately; Phi retains positive gaps with policy supplied under two automatic evaluators\. A controlled example realizes a conflict missed by all pairwise checks, whereas the searched natural language example does not replicate\. The empirical optima concern the recorded responses and scores, not unknown losses on future responses\.

#### Contributions\.

1. 1\.Exact diagnosis of decision loss\.We compute the minimum risk for finite interfaces and separate missing information from limitations of the allowed rules and their fitting\.
2. 2\.Group conflicts and interface requirements\.A hypergraph records states as vertices and minimal incompatible groups as edges\. For a library ofmmcontinuations with randomization allowed, every minimal conflict has at mostmmstates, and the bound is tight\. We connect group incompatibility to risk and characterize the fewest outputs needed by an unrestricted deterministic interface, with global validation of every proposed cell\.
3. 3\.Evidence across inputs and outcome evaluations\.SmolLM2 human ratings and Phi automatic scores show that providing the policy need not remove the selected interface’s information gap\. Qwen2\.5 comparisons separately examine rule restrictions and fitting\.

Sections[2](https://arxiv.org/html/2610.27446#S2)and[3](https://arxiv.org/html/2610.27446#S3)develop the risk and conflict analyses\. Sections[4](https://arxiv.org/html/2610.27446#S4)and[5](https://arxiv.org/html/2610.27446#S5)present the experiments, followed by related work, discussion, and conclusions\. Appendices give conditions for estimated losses, conflict selection after search, and continuous observations\.

Better rules or fitting can reduce loss above the interface’s optimum\. Closing a remaining information gap requires changing what the controller can observe\.

## 2Decision Sufficiency and Exact Risk Analysis

We compare optimal risks under the same losses and continuation library, changing only the information available for selection\. For a finite interface, the comparison reduces to separate optimizations at each output\.

### 2\.1States, continuations, and permitted information

Fix a contextxx, a policy specificationσ\\sigma, and a probability distributionμ\\muon the state space𝒰:=𝒰x\\mathcal\{U\}:=\\mathcal\{U\}\_\{x\}\. LetU∼μU\\sim\\mube the full audited state\. The continuation library isΠ:=Πx=\{π1,…,πm\}\\Pi:=\\Pi\_\{x\}=\\\{\\pi\_\{1\},\\ldots,\\pi\_\{m\}\\\}, withm≥2m\\geq 2and\[m\]:=\{1,…,m\}\[m\]:=\\\{1,\\ldots,m\\\}\. Each continuationπj\\pi\_\{j\}is a response policy that may react to later observations\. The governing specificationσ\\sigmadefines allowed behavior; it is not itself a continuation\.

The expected loss of continuationπj\\pi\_\{j\}at stateuuisℓj​\(u\):=ℓσ,x​\(u,πj\)∈\[0,1\]\\ell\_\{j\}\(u\):=\\ell\_\{\\sigma,x\}\(u,\\pi\_\{j\}\)\\in\[0,1\]\. The expectation includes randomness in its execution\. Define the minimum loss at that state and the regret of continuationπj\\pi\_\{j\}by

ℓ⋆​\(u\):=minj∈\[m\]⁡ℓj​\(u\),ej​\(u\):=ℓj​\(u\)−ℓ⋆​\(u\)\.\\ell^\{\\star\}\(u\):=\\min\_\{j\\in\[m\]\}\\ell\_\{j\}\(u\),\\qquad e\_\{j\}\(u\):=\\ell\_\{j\}\(u\)\-\\ell^\{\\star\}\(u\)\.Regret is the nonnegative excess above the best available continuation\. Zero regret need not mean zero loss\.

LetΔm:=\{q∈\[0,1\]m:∑j=1mqj=1\}\\Delta\_\{m\}:=\\\{q\\in\[0,1\]^\{m\}:\\sum\_\{j=1\}^\{m\}q\_\{j\}=1\\\}be the probability simplex over continuations\. A randomized choiceq∈Δmq\\in\\Delta\_\{m\}selectsπj\\pi\_\{j\}with probabilityqjq\_\{j\}, then executes that policy\. Its loss and regret are

ℓ⁡\(u,q\):=∑jqj​ℓj​\(u\),e⁡\(u,q\):=ℓ⁡\(u,q\)−ℓ⋆​\(u\)=∑jqj​ej​\(u\)\.\\ell\(u,q\):=\\sum\_\{j\}q\_\{j\}\\ell\_\{j\}\(u\),\\qquad e\(u,q\):=\\ell\(u,q\)\-\\ell^\{\\star\}\(u\)=\\sum\_\{j\}q\_\{j\}e\_\{j\}\(u\)\.Randomization mixes complete policies, not their internal actions\. Once the lossesℓj​\(u\)\\ell\_\{j\}\(u\)are specified, the analysis applies to any finite library; no additional temporal assumption is used in the risk formulas\.

Let𝒵\\mathcal\{Z\}be the observation space\. An interface is a Markov kernelK⁡\(d​z∣u\)K\(dz\\mid u\), the conditional distribution of observationZZgiven stateuu\. A controller observesZ∼K\(⋅∣U\)Z\\sim K\(\\cdot\\mid U\)and uses a measurable ruleδ:𝒵→Δm\\delta:\\mathcal\{Z\}\\to\\Delta\_\{m\}\. The componentδj​\(z\)\\delta\_\{j\}\(z\)is the probability of selecting continuationπj\\pi\_\{j\}at inputzz\. Write𝔇Z\\mathfrak\{D\}\_\{Z\}for all such rules,𝒟⊆𝔇Z\\mathcal\{D\}\\subseteq\\mathfrak\{D\}\_\{Z\}for a nonempty allowed family, andδ^∈𝒟\\widehat\{\\delta\}\\in\\mathcal\{D\}for the fitted rule\.

###### Assumption 2\.1\(Audit setting\)\.

The state space𝒰\\mathcal\{U\}and observation space𝒵\\mathcal\{Z\}are standard Borel spaces\. Themmcontinuation losses are measurable and bounded in\[0,1\]\[0,1\], andKKis a Markov kernel\. The nonempty family𝒟⊆𝔇Z\\mathcal\{D\}\\subseteq\\mathfrak\{D\}\_\{Z\}contains the fitted ruleδ^\\widehat\{\\delta\}on the entire declared observation domain\.

Conditional on stateuuand selection ofπj\\pi\_\{j\}, the expected execution loss isℓj​\(u\)\\ell\_\{j\}\(u\); it has no further dependence on the selector or its observation randomness\. All risks use the joint lawμ⁡\(d​u\)​K​\(d​z∣u\)\\mu\(du\)K\(dz\\mid u\)and the same execution protocol for each continuation\. Evaluation ofδ^\\widehat\{\\delta\}holds its training data and randomness fixed\. The policy specificationσ\\sigmais initially fixed; Section[2\.4](https://arxiv.org/html/2610.27446#S2.SS4)handles varying policy text that is supplied to the controller\.

### 2\.2Four risks and three excess losses

The four risks compare access to the full state, unrestricted use of the interface, a restricted family of rules, and the fitted rule:

Rfull\\displaystyle R\_\{\\mathrm\{full\}\}:=𝔼​\[ℓ⋆​\(U\)\],\\displaystyle:=\\mathbb\{E\}\[\\ell^\{\\star\}\(U\)\],\(1\)RZ⋆\\displaystyle R\_\{Z\}^\{\\star\}:=infδ∈𝔇Z𝔼⁡\[ℓ⁡\(U,δ⁡\(Z\)\)\],\\displaystyle:=\\inf\_\{\\delta\\in\\mathfrak\{D\}\_\{Z\}\}\\mathbb\{E\}\[\\ell\(U,\\delta\(Z\)\)\],\(2\)R𝒟\\displaystyle R\_\{\\mathcal\{D\}\}:=infδ∈𝒟𝔼⁡\[ℓ⁡\(U,δ⁡\(Z\)\)\],\\displaystyle:=\\inf\_\{\\delta\\in\\mathcal\{D\}\}\\mathbb\{E\}\[\\ell\(U,\\delta\(Z\)\)\],\(3\)Rδ^\\displaystyle R\_\{\\widehat\{\\delta\}\}:=𝔼​\[ℓ​\(U,δ^​\(Z\)\)\]\.\\displaystyle:=\\mathbb\{E\}\[\\ell\(U,\\widehat\{\\delta\}\(Z\)\)\]\.\(4\)Define the information gapΔinfo:=RZ⋆−Rfull\\Delta\_\{\\mathrm\{info\}\}:=R\_\{Z\}^\{\\star\}\-R\_\{\\mathrm\{full\}\}, the accessibility gapΔaccess:=R𝒟−RZ⋆\\Delta\_\{\\mathrm\{access\}\}:=R\_\{\\mathcal\{D\}\}\-R\_\{Z\}^\{\\star\}, and the fitting gapΔtrain:=Rδ^−R𝒟\\Delta\_\{\\mathrm\{train\}\}:=R\_\{\\widehat\{\\delta\}\}\-R\_\{\\mathcal\{D\}\}\. When the interface and distribution need to be explicit, write

𝔇σinfo​\(K,μ\):=Δinfo=infδ∈𝔇Z𝔼⁡\[e⁡\(U,δ⁡\(Z\)\)\]\.\\mathfrak\{D\}^\{\\mathrm\{info\}\}\_\{\\sigma\}\(K;\\mu\):=\\Delta\_\{\\mathrm\{info\}\}=\\inf\_\{\\delta\\in\\mathfrak\{D\}\_\{Z\}\}\\mathbb\{E\}\[e\(U,\\delta\(Z\)\)\]\.The interface has*decision sufficiency underμ\\mu*when its information gap𝔇σinfo​\(K,μ\)\\mathfrak\{D\}^\{\\mathrm\{info\}\}\_\{\\sigma\}\(K;\\mu\)is zero\.

###### Theorem 2\.2\(Risk ordering and decomposition\)\.

Under Assumption[2\.1](https://arxiv.org/html/2610.27446#S2.Thmassumption1),

Rfull≤RZ⋆≤R𝒟≤Rδ^,R\_\{\\mathrm\{full\}\}\\leq R\_\{Z\}^\{\\star\}\\leq R\_\{\\mathcal\{D\}\}\\leq R\_\{\\widehat\{\\delta\}\},and

Rδ^−Rfull=RZ⋆−Rfull⏟Δinfo\+R𝒟−RZ⋆⏟Δaccess\+Rδ^−R𝒟⏟Δtrain\.R\_\{\\widehat\{\\delta\}\}\-R\_\{\\mathrm\{full\}\}=\\underbrace\{R\_\{Z\}^\{\\star\}\-R\_\{\\mathrm\{full\}\}\}\_\{\\Delta\_\{\\mathrm\{info\}\}\}\+\\underbrace\{R\_\{\\mathcal\{D\}\}\-R\_\{Z\}^\{\\star\}\}\_\{\\Delta\_\{\\mathrm\{access\}\}\}\+\\underbrace\{R\_\{\\widehat\{\\delta\}\}\-R\_\{\\mathcal\{D\}\}\}\_\{\\Delta\_\{\\mathrm\{train\}\}\}\.\(5\)

The information gap remains even for the best rule on the interface\. The accessibility gap measures the extra cost of restricting that rule; the fitting gap measures the remaining error within the allowed family\. The last gap can include both estimation and optimization error\. The identity separates these comparisons, but does not by itself predict how more training data will change them\. All comparisons fix the recorded state, continuation library, and loss\.

### 2\.3Exact optimization for a finite interface

Let𝒰aud=\{u1,…,un\}\\mathcal\{U\}\_\{\\mathrm\{aud\}\}=\\\{u\_\{1\},\\ldots,u\_\{n\}\\\}be a finite audit domain, withn≥1n\\geq 1and state weightsμi≥0\\mu\_\{i\}\\geq 0summing to one\. Let𝒵f=\{z1,…,zL\}\\mathcal\{Z\}\_\{\\mathrm\{f\}\}=\\\{z\_\{1\},\\ldots,z\_\{L\}\\\}, withL≥1L\\geq 1, be the complete observation alphabet\. Define the observation probabilities, loss entries, and joint costs by

Pi​r:=K⁡\(\{zr\}∣ui\),Li​j:=ℓj​\(ui\),Cr​j:=∑i=1nμi​Pi​r​Li​j\.P\_\{ir\}:=K\(\\\{z\_\{r\}\\\}\\mid u\_\{i\}\),\\qquad L\_\{ij\}:=\\ell\_\{j\}\(u\_\{i\}\),\\qquad C\_\{rj\}:=\\sum\_\{i=1\}^\{n\}\\mu\_\{i\}P\_\{ir\}L\_\{ij\}\.Each row ofPPsums to one\. The loss matrix𝐋=\(Li​j\)\\mathbf\{L\}=\(L\_\{ij\}\)contains one entry per state and continuation\. Ifpr:=∑iμi​Pi​r\>0p\_\{r\}:=\\sum\_\{i\}\\mu\_\{i\}P\_\{ir\}\>0, thenCr​j=pr​𝔼​\[ℓj​\(U\)∣Z=zr\]C\_\{rj\}=p\_\{r\}\\mathbb\{E\}\[\\ell\_\{j\}\(U\)\\mid Z=z\_\{r\}\]\. ThusCr​jC\_\{rj\}already includes the probability of outputzrz\_\{r\}\. A rule assigningqr=δ⁡\(zr\)q\_\{r\}=\\delta\(z\_\{r\}\)has risk∑r,jCr​j​qr​j\\sum\_\{r,j\}C\_\{rj\}q\_\{rj\}\.

###### Theorem 2\.3\(Optimum for a finite interface\)\.

For the finite audit distribution and observation channel,

Rfull=∑i=1nμi​minj∈\[m\]​Li​j,RZ⋆=∑r=1Lminj∈\[m\]⁡Cr​j\.R\_\{\\mathrm\{full\}\}=\\sum\_\{i=1\}^\{n\}\\mu\_\{i\}\\min\_\{j\\in\[m\]\}L\_\{ij\},\\qquad R\_\{Z\}^\{\\star\}=\\sum\_\{r=1\}^\{L\}\\min\_\{j\\in\[m\]\}C\_\{rj\}\.At each outputzrz\_\{r\}, every distribution supported onarg​minj⁡Crj\\argmin\_\{j\}C\_\{rj\}is optimal\. In particular, selecting one minimizing continuation suffices\. Outputs of probability zero can be assigned arbitrarily\.

The choices at different outputs have separate simplex constraints, and the objective is linear in each choice\. Forming the dense cost table takesO⁡\(n​L​m\)O\(nLm\)operations; selecting the row minima takesO⁡\(L​m\)O\(Lm\)comparisons \(Algorithm[1](https://arxiv.org/html/2610.27446#alg1)\)\. For a deterministic mapf:𝒰aud→𝒵ff:\\mathcal\{U\}\_\{\\mathrm\{aud\}\}\\to\\mathcal\{Z\}\_\{\\mathrm\{f\}\}, the nonempty setsCr:=\{ui:f⁡\(ui\)=zr\}C\_\{r\}:=\\\{u\_\{i\}:f\(u\_\{i\}\)=z\_\{r\}\\\}are its*cells*\. Every state in a cell receives the same continuation distribution\. With strictly positive state weights, the gap is zero exactly when each cell has a common optimal continuation \(Corollary[A\.4](https://arxiv.org/html/2610.27446#A1.Thmassumption4)\)\. Appendix[A](https://arxiv.org/html/2610.27446#A1)gives the proofs and the corresponding posterior characterization for a general channel\.

### 2\.4Policy information and an observed collision

When policies vary across states, letS=s⁡\(U\)S=s\(U\)denote the governing policy text\. Each lossℓj​\(u\)\\ell\_\{j\}\(u\)is evaluated under the policy in stateuu; the continuation library and execution protocols remain fixed\. LetVVbe the complete controller input:V=ZV=Zfor features alone andV=\(Z,S\)V=\(Z,S\)when policy is also supplied\. Define

RV⋆:=infδ𝔼⁡\[ℓ⁡\(U,δ⁡\(V\)\)\],ΔV:=RV⋆−Rfull,R\_\{V\}^\{\\star\}:=\\inf\_\{\\delta\}\\mathbb\{E\}\[\\ell\(U,\\delta\(V\)\)\],\\qquad\\Delta\_\{V\}:=R\_\{V\}^\{\\star\}\-R\_\{\\mathrm\{full\}\},where the infimum ranges over all measurable rules usingVV\. For the finite deterministic audit, the cells with policy supplied areCz,s:=\{u:f\(u\)=z,s\(u\)=s\}C\_\{z,s\}:=\\\{u:f\(u\)=z,\\ s\(u\)=s\\\}\. Let𝟏​\{A\}\\mathbf\{1\}\\\{A\\\}denote the indicator of eventAA\. Applying the finite optimum gives

RZ,S⋆\\displaystyle R\_\{Z,S\}^\{\\star\}=∑z,sminj⁡𝔼⁡\[𝟏​\{Z=z,S=s\}​ℓj​\(U\)\],\\displaystyle=\\sum\_\{z,s\}\\min\_\{j\}\\mathbb\{E\}\[\\mathbf\{1\}\\\{Z=z,S=s\\\}\\ell\_\{j\}\(U\)\],\(6\)0\\displaystyle 0≤ΔZ,S≤ΔZ\.\\displaystyle\\leq\\Delta\_\{Z,S\}\\leq\\Delta\_\{Z\}\.The sum runs over observed code and policy pairs\. A rule that receives policy textSScan ignore it, which proves the inequality\. A positive gap using featuresZZalone need not remain when policy is supplied\.

#### An example from the human ratings\.

Table[1](https://arxiv.org/html/2610.27446#S2.T1)gives two SmolLM2 states with the same policy and the same two SAE activation indicators,00\. The three continuations ask for clarification, refuse, or assist\. Entries are rating losses in\[0,1\]\[0,1\], with lower values preferred; the scoring rule is defined in Section[4\.3](https://arxiv.org/html/2610.27446#S4.SS3)\.

Table 1:An observed policy\-aware collision under the SmolLM2 seed\-314159, two\-bit interface\. Both requests have the same policy and code00\. Entries are final human composite losses; smaller is better\. The pair is a post hoc illustration, not an independent or higher\-order test\. Exact text and state IDs are given in Appendix[K\.4](https://arxiv.org/html/2610.27446#A11.SS4)\.With equal state weights, access to the full state gives mean loss\(0\+0\.100\)/2=0\.050\(0\+0\.100\)/2=0\.050\. A common choice has column means\(0\.425,0\.3125,0\.400\)\(0\.425,0\.3125,0\.400\), so refusal is optimal with loss0\.31250\.3125\. The two\-state gap is0\.26250\.2625\. The pair was selected after inspecting the data to illustrate an existing collision, not as an independent test or a conflict missed by pairwise checks\. Its full record is in Appendix[K\.4](https://arxiv.org/html/2610.27446#A11.SS4)\.

### 2\.5Restricted rules and estimated losses

The optimum within a restricted rule family requires a separate calculation\. Our affine family maps visible features to continuation probabilities, with simplex constraints on the entire declared observation alphabet\. Its optimum is a linear program; fitting changes the costs, not the feasible domain \(Appendix[A\.3](https://arxiv.org/html/2610.27446#A1.SS3)\)\.

An exact optimum for a supplied loss table need not equal the optimum for unknown expected losses\. Taking a minimum of noisy estimates can favor negative estimation errors\. Proposition[A\.6](https://arxiv.org/html/2610.27446#A1.Thmassumption6)controls the resulting risk changes under a uniform bound on loss errors\. For continuous observations, a finite set of distinct vectors can be memorized by an unrestricted rule, so probe failure alone does not identify missing information\. Quantization also changes the observed input\. Appendix[L](https://arxiv.org/html/2610.27446#A12)gives the conditions needed to transfer a calculation on quantized data to the original observation and to obtain bounds from estimated losses\.

## 3Group Conflicts and Interface Requirements

The average risk gap does not identify which states cannot share a decision\. We describe those groups and determine how many outputs are needed to keep every group compatible\. The grouping results concern deterministic interfaces; the risk analysis in Section[2](https://arxiv.org/html/2610.27446#S2)also permits stochastic channels\.

### 3\.1Shared choices and average risk

Fix a nonempty finite domain𝒰aud=\{u1,…,un\}\\mathcal\{U\}\_\{\\mathrm\{aud\}\}=\\\{u\_\{1\},\\ldots,u\_\{n\}\\\}, them≥2m\\geq 2continuations from Section[2](https://arxiv.org/html/2610.27446#S2), and their regretsej​\(u\)=ℓj​\(u\)−ℓ⋆​\(u\)e\_\{j\}\(u\)=\\ell\_\{j\}\(u\)\-\\ell^\{\\star\}\(u\)\. A complete observation mapf:𝒰aud→\[M\]f:\\mathcal\{U\}\_\{\\mathrm\{aud\}\}\\to\[M\], whereM≥1M\\geq 1and\[M\]:=\{1,…,M\}\[M\]:=\\\{1,\\ldots,M\\\}, has nonempty cellsCc=f−1​\(c\)C\_\{c\}=f^\{\-1\}\(c\)forc∈range⁡\(f\)c\\in\\range\(f\)\. Every state in a cell receives the same continuation distributionq∈Δmq\\in\\Delta\_\{m\}\. If policy text is supplied, it is part offf; states with different policy text cannot belong to the same cell\.

For an allowed regretε≥0\\varepsilon\\geq 0, the acceptable choices at stateuuform the set

𝒢ε​\(u\):=\{q∈Δm:e⁡\(u,q\)≤ε\}\.\\mathcal\{G\}^\{\\varepsilon\}\(u\):=\\\{q\\in\\Delta\_\{m\}:e\(u,q\)\\leq\\varepsilon\\\}\.Each set is nonempty, closed, and convex: a pure optimum is feasible, and regret is linear in the choice probabilities\. For a nonempty groupW⊆𝒰audW\\subseteq\\mathcal\{U\}\_\{\\mathrm\{aud\}\}, define its*margin*as the smallest possible maximum regret across its states:

Γ⁡\(W\):=minq∈Δm⁡maxu∈W⁡e⁡\(u,q\)\.\\Gamma\(W\):=\\min\_\{q\\in\\Delta\_\{m\}\}\\max\_\{u\\in W\}e\(u,q\)\.\(7\)The minimum is attained by compactness ofΔm\\Delta\_\{m\}and continuity of the objective\. A group isε\\varepsilon\-compatible exactly when

Γ\(W\)≤ε⟺⋂u∈W𝒢ε\(u\)≠∅\.\\Gamma\(W\)\\leq\\varepsilon\\quad\\Longleftrightarrow\\quad\\bigcap\_\{u\\in W\}\\mathcal\{G\}^\{\\varepsilon\}\(u\)\\neq\\varnothing\.\(8\)Otherwise the group is incompatible\. Removing states cannot increase the margin:Γ⁡\(V\)≤Γ⁡\(W\)\\Gamma\(V\)\\leq\\Gamma\(W\)for nonemptyV⊆WV\\subseteq W\.

LetΔ\(W\):=\{w=\(wu\)u∈W:wu≥0,∑u∈Wwu=1\}\\Delta\(W\):=\\\{w=\(w\_\{u\}\)\_\{u\\in W\}:w\_\{u\}\\geq 0,\\ \\sum\_\{u\\in W\}w\_\{u\}=1\\\}be the distributions over states inWW\. For fixed state weightsw∈Δ⁡\(W\)w\\in\\Delta\(W\), define

κ⁡\(W,w\):=min⁡∑u∈Wq∈Δm⁡wu​e​\(u,q\)=min⁡∑u∈Wj∈\[m\]⁡wu​ej​\(u\)\.\\kappa\(W,w\):=\\min\_\{q\\in\\Delta\_\{m\}\}\\sum\_\{u\\in W\}w\_\{u\}e\(u,q\)=\\min\_\{j\\in\[m\]\}\\sum\_\{u\\in W\}w\_\{u\}e\_\{j\}\(u\)\.The quantityκ⁡\(W,w\)\\kappa\(W,w\)minimizes average regret, whereasΓ⁡\(W\)\\Gamma\(W\)minimizes the largest regret\. The weightswwdescribe states; the probabilitiesqqdescribe choices of continuation\.

###### Theorem 3\.1\(Relation between average and maximum regret\)\.

For every nonempty finite groupWWandw∈Δ⁡\(W\)w\\in\\Delta\(W\), letβ⁡\(W,w\):=minu∈W⁡wu\\beta\(W,w\):=\\min\_\{u\\in W\}w\_\{u\}\. Then

β⁡\(W,w\)​Γ​\(W\)≤κ⁡\(W,w\)≤Γ⁡\(W\)\.\\beta\(W,w\)\\Gamma\(W\)\\leq\\kappa\(W,w\)\\leq\\Gamma\(W\)\.

Letμ\\mube a probability distribution on the finite audit domain\. The induced channel isKf\(\{c\}∣u\)=𝟏\{f\(u\)=c\}K\_\{f\}\(\\\{c\\\}\\mid u\)=\\mathbf\{1\}\\\{f\(u\)=c\\\}\. For each cell of positive masspc:=∑u∈Ccμ⁡\(u\)p\_\{c\}:=\\sum\_\{u\\in C\_\{c\}\}\\mu\(u\), setwu\(c\):=μ⁡\(u\)/pcw\_\{u\}^\{\(c\)\}:=\\mu\(u\)/p\_\{c\}\. The information gap satisfies

𝔇σinfo\(Kf;μ\)=∑c:pc\>0pcκ\(Cc,w\(c\)\)≤∑c:pc\>0pcΓ\(Cc\)\.\\mathfrak\{D\}^\{\\mathrm\{info\}\}\_\{\\sigma\}\(K\_\{f\};\\mu\)=\\sum\_\{c:p\_\{c\}\>0\}p\_\{c\}\\kappa\(C\_\{c\},w^\{\(c\)\}\)\\leq\\sum\_\{c:p\_\{c\}\>0\}p\_\{c\}\\Gamma\(C\_\{c\}\)\.\(9\)Hence a gap exceedingε\\varepsilonimplies that some cell has margin exceedingε\\varepsilon\. Positivity alone does not imply failure at an arbitrary positive tolerance\. With positive state weights, zero average regret and zero maximum regret are equivalent\. At positive tolerance, a rare state can prevent a uniform guarantee while contributing little to average loss\.

### 3\.2Conflicts missed by pairwise checks

###### Example 3\.2\(Every pair shares an optimum, but the group does not\)\.

Consider three states with losses

askdeclineassistu1010u2001u3100\\begin\{array\}\[\]\{c\|ccc\}&\\textsc\{ask\}&\\textsc\{decline\}&\\textsc\{assist\}\\\\ \\hline\\cr u\_\{1\}&0&1&0\\\\ u\_\{2\}&0&0&1\\\\ u\_\{3\}&1&0&0\\end\{array\}Each state has minimum loss zero, so losses equal regrets\. The pairs\{u1,u2\}\\\{u\_\{1\},u\_\{2\}\\\},\{u1,u3\}\\\{u\_\{1\},u\_\{3\}\\\}, and\{u2,u3\}\\\{u\_\{2\},u\_\{3\}\\\}can chooseask,assist, anddecline, respectively, with zero regret\. For a common choiceq=\(qask,qdecline,qassist\)q=\(q\_\{\\rm ask\},q\_\{\\rm decline\},q\_\{\\rm assist\}\), the three regrets areqdeclineq\_\{\\rm decline\},qassistq\_\{\\rm assist\}, andqaskq\_\{\\rm ask\}\. Their maximum is at least1/31/3; the uniform distribution attains1/31/3\. The triple is therefore incompatible for0≤ε<1/30\\leq\\varepsilon<1/3, while every pair is compatible\.

A*minimal conflict*is an incompatible group for which every proper subset is compatible\. Minimality refers to inclusion, not the smallest cardinality among all conflicts\.

###### Definition 3\.3\(Conflict hypergraph\)\.

Letℋε=\(𝒰aud,ℰε\)\\mathcal\{H\}\_\{\\varepsilon\}=\(\\mathcal\{U\}\_\{\\mathrm\{aud\}\},\\mathcal\{E\}\_\{\\varepsilon\}\), where a nonempty groupWWbelongs toℰε\\mathcal\{E\}\_\{\\varepsilon\}if

⋂u∈W𝒢ε​\(u\)=∅,⋂u∈V𝒢ε​\(u\)≠∅for every proper subset​V⊊W\.\\bigcap\_\{u\\in W\}\\mathcal\{G\}^\{\\varepsilon\}\(u\)=\\varnothing,\\qquad\\bigcap\_\{u\\in V\}\\mathcal\{G\}^\{\\varepsilon\}\(u\)\\neq\\varnothing\\quad\\text\{for every proper subset \}V\\subsetneq W\.The intersection over no states is defined asΔm\\Delta\_\{m\}\. Vertices are audit states, and edges are minimal incompatible groups\.

The global edge set depends on the losses, library, states, and tolerance, not on the current interface\. Its subset

ℰε​\(f\):=\{W∈ℰε:W⊆Cc​for some​c\}\\mathcal\{E\}\_\{\\varepsilon\}\(f\):=\\\{W\\in\\mathcal\{E\}\_\{\\varepsilon\}:W\\subseteq C\_\{c\}\\text\{ for some \}c\\\}contains conflicts hidden within the cells offf\. A new interface may merge states from different old cells, so redesign must consider global conflicts rather than onlyℰε​\(f\)\\mathcal\{E\}\_\{\\varepsilon\}\(f\)\.

###### Theorem 3\.4\(Maximum size of a minimal conflict\)\.

Every minimal conflictW∈ℰεW\\in\\mathcal\{E\}\_\{\\varepsilon\}satisfies2≤\|W\|≤m2\\leq\|W\|\\leq m\. For eachm≥2m\\geq 2, some bounded loss table and tolerance have a minimal conflict of exactlymmstates\.

The upper bound follows from Helly’s theorem in the\(m−1\)\(m\-1\)\-dimensional simplex of continuation probabilities\([Helly, 1923](https://arxiv.org/html/2610.27446#bib.bib38)\)\. With three continuations, every incompatible group thus contains a conflicting pair or triple\. Appendix[B](https://arxiv.org/html/2610.27446#A2)proves tightness and explains when unique optimal choices reduce all minimal conflicts to pairs\.

### 3\.3The risk interpretation of a conflict margin

Allow state weightsλ∈Δ⁡\(W\)\\lambda\\in\\Delta\(W\)to vary whileq∈Δmq\\in\\Delta\_\{m\}remains the controller’s choice\.

###### Theorem 3\.5\(Minimax interpretation of the group margin\)\.

For every nonempty finite groupWW,

Γ⁡\(W\)=maxλ∈Δ⁡\(W\)⁡min⁡∑u∈Wj∈\[m\]⁡λu​ej​\(u\)\.\\Gamma\(W\)=\\max\_\{\\lambda\\in\\Delta\(W\)\}\\min\_\{j\\in\[m\]\}\\sum\_\{u\\in W\}\\lambda\_\{u\}e\_\{j\}\(u\)\.If the complete observation mapffis constant onWW, then

Γ⁡\(W\)=maxλ∈Δ⁡\(W\)⁡𝔇σinfo​\(Kf,λ\),\\Gamma\(W\)=\\max\_\{\\lambda\\in\\Delta\(W\)\}\\mathfrak\{D\}^\{\\mathrm\{info\}\}\_\{\\sigma\}\(K\_\{f\};\\lambda\),where the state distributionλ\\lambdais supported onWW\.

The margin equals the*largest*information gap over distributions on the group, not the gap under any fixed audit distributionμ\\mu\. Finite minimax duality justifies the exchange of optimizations\([Sion, 1958](https://arxiv.org/html/2610.27446#bib.bib39)\)\. Introducing a scalarttgives the linear program

Γ\(W\)=minq,t\{t:q∈Δm,∑jqjej\(u\)≤tfor everyu∈W\}\.\\Gamma\(W\)=\\min\_\{q,t\}\\left\\\{t:q\\in\\Delta\_\{m\},\\quad\\sum\_\{j\}q\_\{j\}e\_\{j\}\(u\)\\leq t\\text\{ for every \}u\\in W\\right\\\}\.The primal solution gives a best common choice; the dual gives state weights attaining the maximum\. For fixedmm, enumerating all candidates requiresO⁡\(nm\)O\(n^\{m\}\)margin problems \(Algorithm[4](https://arxiv.org/html/2610.27446#alg4)\)\.

### 3\.4Output requirements and global validation

A redesigned interface is a mapg:𝒰aud→\[n\]g:\\mathcal\{U\}\_\{\\mathrm\{aud\}\}\\to\[n\]assigning new labels to states\. It is uniformlyε\\varepsilon\-sufficient when every nonempty cell has margin at mostε\\varepsilon\. Define its minimum number of distinct outputs by

𝖬⋆\(ε\):=ming:𝒰aud→\[n\]\{\|range\(g\)\|:Γ\(g−1\(c\)\)≤εfor everyc∈range\(g\)\}\.\\mathsf\{M\}^\{\\star\}\(\\varepsilon\):=\\min\_\{g:\\mathcal\{U\}\_\{\\mathrm\{aud\}\}\\to\[n\]\}\\left\\\{\|\\range\(g\)\|:\\Gamma\(g^\{\-1\}\(c\)\)\\leq\\varepsilon\\text\{ for every \}c\\in\\range\(g\)\\right\\\}\.An edge is monochromatic if all its states have the same label\. The weak chromatic numberχw​\(ℋε\)\\chi\_\{\\mathrm\{w\}\}\(\\mathcal\{H\}\_\{\\varepsilon\}\)is the smallest number of labels with no monochromatic edge\.

###### Theorem 3\.6\(Capacity as a coloring problem\)\.

For unrestricted deterministic interfaces,

𝖬⋆​\(ε\)=χw​\(ℋε\)\.\\mathsf\{M\}^\{\\star\}\(\\varepsilon\)=\\chi\_\{\\mathrm\{w\}\}\(\\mathcal\{H\}\_\{\\varepsilon\}\)\.

A cell is incompatible exactly when it contains a minimal conflict; avoiding monochromatic edges is therefore equivalent to compatibility\([Berge, 1989](https://arxiv.org/html/2610.27446#bib.bib40)\)\. Assigning every state an optimal continuation index uses at mostmmlabels\. Three such labels fit in two bits, although two selected SAE bits need not implement that assignment\. Appendix[C\.2](https://arxiv.org/html/2610.27446#A3.SS2)treats restricted encoders, zero regret, and policy text supplied without charging it to the code alphabet\.

To avoid enumerating every edge, start with anyℱ0⊆ℰε\\mathcal\{F\}\_\{0\}\\subseteq\\mathcal\{E\}\_\{\\varepsilon\}\. At iterationtt, a coloring mixed\-integer linear program \(MILP\) usingℱt\\mathcal\{F\}\_\{t\}returns the minimum label countMtM\_\{t\}and a labelinggtg\_\{t\}\.

###### Proposition 3\.7\(Capacity by constraint generation\)\.

With exact solves,Mt≤𝖬⋆​\(ε\)M\_\{t\}\\leq\\mathsf\{M\}^\{\\star\}\(\\varepsilon\)\. If every nonempty cell ofgtg\_\{t\}has margin at mostε\\varepsilon, thenMt=𝖬⋆​\(ε\)M\_\{t\}=\\mathsf\{M\}^\{\\star\}\(\\varepsilon\)andgtg\_\{t\}is optimal\. Otherwise extract a minimal conflictWtW\_\{t\}from a violating cell, setℱt\+1=ℱt∪\{Wt\}\\mathcal\{F\}\_\{t\+1\}=\\mathcal\{F\}\_\{t\}\\cup\\\{W\_\{t\}\\\}, and repeat\. Exact margin comparisons and MILP solves yield finite termination at the global capacity\.

The retained constraints give a lower bound\. A validated labeling gives a matching upper bound\. A failed validation adds an edge not yet retained; there are finitely many edges\. Algorithms[5](https://arxiv.org/html/2610.27446#alg5)and[6](https://arxiv.org/html/2610.27446#alg6)give the procedures, including the handling of incomplete solves\. Numerical output is reported with solver status and tolerances, not as an exact arithmetic proof\.

### 3\.5Guarantees after searching for a conflict

Letℓ^j​\(u\)∈\[0,1\]\\widehat\{\\ell\}\_\{j\}\(u\)\\in\[0,1\]estimate lossℓj​\(u\)\\ell\_\{j\}\(u\)\. Suppose, with probability at least1−α1\-\\alpha, all errors on the finite audit satisfy\|ℓ^j​\(u\)−ℓj​\(u\)\|≤rℓ\|\\widehat\{\\ell\}\_\{j\}\(u\)\-\\ell\_\{j\}\(u\)\|\\leq r\_\{\\ell\}, where0<α<10<\\alpha<1andrℓ≥0r\_\{\\ell\}\\geq 0\. Definee^j​\(u\):=ℓ^j​\(u\)−mink⁡ℓ^k​\(u\)\\widehat\{e\}\_\{j\}\(u\):=\\widehat\{\\ell\}\_\{j\}\(u\)\-\\min\_\{k\}\\widehat\{\\ell\}\_\{k\}\(u\)and letΓ^​\(W\)\\widehat\{\\Gamma\}\(W\)be the group margin using these estimated regrets\. Every regret and group margin changes by at most2​rℓ2r\_\{\\ell\}on that same event\. For any groupW^\\widehat\{W\}selected using the estimates,

Γ⁡\(W^\)≥Γ^​\(W^\)−2​rℓwith probability at least​1−α\.\\Gamma\(\\widehat\{W\}\)\\geq\\widehat\{\\Gamma\}\(\\widehat\{W\}\)\-2r\_\{\\ell\}\\quad\\text\{with probability at least \}1\-\\alpha\.A lower bound exceedingε\\varepsiloncertifies incompatibility after selection\. Minimality also requires upper bounds for every group formed by deleting one state\. Appendix[C\.3](https://arxiv.org/html/2610.27446#A3.SS3)gives the full results, interval constructions, and conditions for a larger continuation library\. Resampling the existing policy areas does not supply the simultaneous loss bounds assumed here\.

## 4Experimental Design

The experiments compare exact risks under selected SAE inputs, then vary the input, allowed rules, fitting data, or evaluator\. Human ratings are available for SmolLM2\. Appendix[M](https://arxiv.org/html/2610.27446#A13)gives complete model and rating protocols\.

### 4\.1States, response choices, and controller inputs

Each state records a synthetic situation, its governing policy, and relevant evidence\. The original collection has 240 states from 20 policy areas, with three required actions and four paraphrases per area\. Twelve areas \(144 states\) are used for setting selection and fitting; eight \(96 states\) form the original confirmation collection\. SmolLM2 adds 96 states from eight other areas, fixed before generation and scoring\. Phi generates new responses on both collections; the states themselves are reused\.

The available strategies are𝒜:=\{ask,decline,assist\}\\mathcal\{A\}:=\\\{\\textsc\{ask\},\\textsc\{decline\},\\textsc\{assist\}\\\}: request clarification, refuse, or provide assistance\. Every state is evaluated under all three choices\. The restricted input belongs to the controller selecting a strategy, not to every operation of the generator\. The experiments concern one response turn, not arbitrary conversation policies\.

LetFBF\_\{B\}be a set ofBBSAE coordinates andak​\(u\)≥0a\_\{k\}\(u\)\\geq 0the activation of selected coordinatekkat stateuu\. The controller observes the activation indicators

zk\(u\):=𝟏\{ak\(u\)\>0\},Z=\(zk\)k∈FB\.z\_\{k\}\(u\):=\\mathbf\{1\}\\\{a\_\{k\}\(u\)\>0\\\},\\qquad Z=\(z\_\{k\}\)\_\{k\\in F\_\{B\}\}\.Its complete input isV=ZV=ZorV=\(Z,S\)V=\(Z,S\), whereSSis policy text\. When policy is supplied, a cell requires equality of both code and policy string\. The required action label is never a controller input\. SmolLM2 and Phi useB∈\{2,4,8,16,32,64\}B\\in\\\{2,4,8,16,32,64\\\}, withB=2B=2primary\. Features are ranked individually by the reduction in fitting risk relative to a constant input, followed by activation balance and index\. The firstBBcoordinates are retained\. The ranking is not a joint optimization of all subsets, and evaluation conditions do not reselect features\.

### 4\.2Model protocols and automatic scoring

Qwen2\.5\-1\.5B uses public SAEs after layers 17, 18, 19, and 21, with a second Qwen scale as a supporting check\. SmolLM2\-1\.7B and Phi\-4\-mini\-instruct use layers 17 and 23, respectively\. Each has two TopK SAEs trained with different initializations and training orders\. Within a pair, the model, layer, architecture, and training activation corpus are shared\. Layer indices are zero based; full settings appear in Appendix[M](https://arxiv.org/html/2610.27446#A13)\.

The Qwen generator receives the public request, fact name, and selected action, but not the private condition or complete policy\. Its responses are reused across states sharing the public request\. The Qwen evaluator receives the full context and required action label\. SmolLM2 and Phi supply policy and situation to their generators but do not supply the required action to their evaluators\. The comparison across families thus changes generation and scoring protocols as well as the model; it is not a controlled comparison of model identity alone\.

Qwen scores safety, utility, fact handling, action adherence, and overall quality from zero to four with weights\(0\.40,0\.25,0\.15,0\.10,0\.10\)\(0\.40,0\.25,0\.15,0\.10,0\.10\)\. Its score is an expectation under label probabilities corrected for the evaluator’s label prior\. Loss is one minus the weighted score divided by four\. SmolLM2 and Phi instead request integer JSON scores from zero to three on the five dimensions below\. Each generates five responses per state and action, totaling 5,040 responses per model\. Automatic risk uses the mean loss of the five responses\. Mistral is their primary evaluator; Phi is also evaluated with Qwen\-7B and the mean of the two losses\. Normalization to\[0,1\]\[0,1\]does not make the scoring rubrics equivalent\.

Table 2:Models, controller inputs, and evaluations\. Human ratings cover SmolLM2 only\. Generator inputs and scoring rules differ across families\.†Qwen2\.5\-1\.5B results with policy supplied are recovered from rounded summaries for individual policy areas\.
### 4\.3Human ratings and matched observations

Human evaluation covers 80 original SmolLM2 states, including fitting areas, and all 96 locked states\. A SHA\-256 ordering independent of scores fixes one response per state and action, giving 528 items\. After a separate 36\-item pilot, two raters independently score every item\. They see the policy, situation, selected action, and response, but not automatic scores, model or SAE identity, required action, split, or the other rater’s labels\. The pilot is excluded from reported statistics\. Predefined disagreement rules select 100 items for joint adjudication\.

Letssafety,spolicy,suse,sfact,soverall∈\{0,1,2,3\}s\_\{\\rm safety\},s\_\{\\rm policy\},s\_\{\\rm use\},s\_\{\\rm fact\},s\_\{\\rm overall\}\\in\\\{0,1,2,3\\\}denote ratings for safety, policy adherence, usefulness, factuality, and overall quality, respectively\. The human loss is

Lhuman=1−0\.30​ssafety\+0\.30​spolicy\+0\.15​suse\+0\.10​sfact\+0\.15​soverall3\.L^\{\\rm human\}=1\-\\frac\{0\.30s\_\{\\rm safety\}\+0\.30s\_\{\\rm policy\}\+0\.15s\_\{\\rm use\}\+0\.10s\_\{\\rm fact\}\+0\.15s\_\{\\rm overall\}\}\{3\}\.The 100 adjudicated items use resolved scores; the other 428 use the mean of the independent losses\. The composite rating is not an unsafe event probability\. Automatic and human calibration compares scores for the same fixed response, not a human rating with a mean over five responses\. Appendix[M\.4](https://arxiv.org/html/2610.27446#A13.SS4)gives the rubric and procedure\.

ComparingZZwith\(Z,S\)\(Z,S\)holds features, states, responses, and losses fixed\. The SmolLM2 comparison was added after the original study; both Phi observations were specified before generation\. Qwen2\.5\-1\.5B results with policy supplied are recovered from rounded summaries with a stated error bound, not full precision arrays\. Qwen2\.5\-0\.5B uses code alone\.

### 4\.4Risk calculation and resampling

For actiona∈𝒜a\\in\\mathcal\{A\}, letL⁡\(a,u\)∈\[0,1\]L\(a,u\)\\in\[0,1\]be the measured loss at stateuu\. Letp^​\(v\)\\widehat\{p\}\(v\)be the evaluation frequency of inputvv, and𝔼^​\[L​\(a,U\)∣V=v\]\\widehat\{\\mathbb\{E\}\}\[L\(a,U\)\\mid V=v\]the mean loss in its cell\. The empirical optima are

R^V⋆=∑vp^​\(v\)​mina∈𝒜​𝔼^​\[L⁡\(a,U\)∣V=v\],R^full=𝔼^​\[mina∈𝒜⁡L⁡\(a,U\)\]\.\\widehat\{R\}\_\{V\}^\{\\star\}=\\sum\_\{v\}\\widehat\{p\}\(v\)\\min\_\{a\\in\\mathcal\{A\}\}\\widehat\{\\mathbb\{E\}\}\[L\(a,U\)\\mid V=v\],\\qquad\\widehat\{R\}\_\{\\rm full\}=\\widehat\{\\mathbb\{E\}\}\\bigl\[\\min\_\{a\\in\\mathcal\{A\}\}L\(a,U\)\\bigr\]\.\(10\)Their difference is the information gap on the scored table\. Both are reference optimizations over the observed scores, not trained policies tested on future responses\. Comparisons of rule families restrict the allowed mapping; the fitting experiment varies samples from a fixed finite population\.

We resample whole policy areas and recompute both minima\. Tables report individual two\-sided 95% bootstrap intervals and one\-sided 95% lower confidence bounds \(LCBs\)\. Phi uses 2,000 draws and eight areas per collection\. With policy supplied, resampling reweights fixed gaps within policy areas\. A positive LCB describes stability to that reweighting; it does not control error in the underlying response losses\. The intervals are not simultaneous across settings, and a zero fraction of nonpositive draws is not a zeropp\-value\. Formal guarantees require the conditions in Appendices[L](https://arxiv.org/html/2610.27446#A12)and[C\.3](https://arxiv.org/html/2610.27446#A3.SS3)\.

## 5Empirical Results

All optima below use the scored tables from Section[4](https://arxiv.org/html/2610.27446#S4)\. We first report human results with policy supplied, then examine rule and fitting restrictions, other model protocols, and group conflicts\.

### 5\.1Human ratings retain a gap with policy supplied

On the 96 SmolLM2 states held out by policy area, the minimum losses using policy and two feature indicators exceed the minima using the full state by0\.11660\.1166and0\.10820\.1082for the two SAEs\. Their individual one\-sided bootstrap lower bounds are0\.09350\.0935and0\.09010\.0901\(Table[3](https://arxiv.org/html/2610.27446#S5.T3)\)\. No alternative mapping from the same observed inputs to actions can remove these differences on the scored table\. The comparison rules out omission of policy text as the sole reason for the gap; it does not locate the loss of information within the encoding pipeline\.

Table 3:SmolLM2 human\-rated information gaps with two visible features \(B=2B=2\)\.ΔZ\\Delta\_\{Z\}uses the code alone;ΔZ,S\\Delta\_\{Z,S\}also supplies the policy\. Responses, features, and ratings are unchanged\. Intervals and individual LCBs concernΔZ,S\\Delta\_\{Z,S\}\. The original 80\-state sample includes fitting areas; the locked 96\-state collection is held out by policy area\.The original 80\-state sample gives gaps of0\.03450\.0345and0\.03530\.0353but includes fitting areas\. Using either rater alone or their mean before adjudication also gives positive lower bounds \(Table[44](https://arxiv.org/html/2610.27446#A10.T44)\)\. Positive gaps are therefore present before joint adjudication, not only in the final averaged scores\. The observed collision in Table[1](https://arxiv.org/html/2610.27446#S2.T1)illustrates how two different optimal choices can share the same complete input\.

#### Feature budget\.

The feature sets are nested, so a controller with more indicators can ignore the additions\. The optimal risk on a fixed table cannot increase; the curves in Figure[1](https://arxiv.org/html/2610.27446#S5.F1)show the size of the remaining gap, not an independent test of that monotonicity\. Some lower bounds reach zero and many gaps vanish at larger budgets\. These zeros concern evaluated states, not performance on unseen inputs\.

\(a\)SmolLM2: final human loss, locked 96\.\(b\)Phi: Mistral loss, reused locked 96\.
Figure 1:Information gap as nested feature sets expand, with policy supplied\. Lines show the gap on the scored tableΔZ,S\\Delta\_\{Z,S\}and shaded bands show individual 95% intervals from resampling policy areas\. The two panels use different models and losses and are not a model ranking\. For each budget, comparing the two observations keeps features and losses fixed\.
#### Score and action agreement\.

The automatic and human losses have Spearman correlation0\.3790\.379and mean absolute difference0\.3110\.311\. Automatic scores have tied optimal actions in 90 of 176 states\. Choosing the first minimum in the fixed orderask,decline,assistgives51\.1%51\.1\\%agreement; the minimizing sets intersect in80\.7%80\.7\\%of states and coincide in30\.7%30\.7\\%\. Set intersection is weaker than equality, especially with ties\. Risk values do not depend on which tied minimizer is selected\. Appendices[F](https://arxiv.org/html/2610.27446#A6)and[K\.2](https://arxiv.org/html/2610.27446#A11.SS2)give complete statistics\. The separate safety dimension also gives positive locked\-set point gaps,0\.05210\.0521and0\.03120\.0312, but these are descriptive rating differences, not unsafe event probabilities or new confidence bounds \(Appendix[K\.3](https://arxiv.org/html/2610.27446#A11.SS3)\)\.

### 5\.2Information, rule restrictions, and fitting error

At Qwen layer 17, with two indicators and no separate policy input, the minimum risk using the full state is0\.3380\.338and the minimum using the code is0\.4940\.494\. The information gap is0\.1560\.156, with bootstrap interval\[0\.125,0\.180\]\[0\.125,0\.180\]and individual lower bound0\.1290\.129\. All four evaluation codes occur in fitting\. Thus the positive information gap is not a failure to train a response for an unseen code; it remains after optimizing every code assignment\.

Table 4:Qwen2\.5 code\-only risks and their three sources of excess loss on 96 confirmation states\. The four risks compare full\-state access, unrestricted code access, a restricted rule class, and a fitted rule\. Values and gaps are independently rounded; complete intervals appear in Table[24](https://arxiv.org/html/2610.27446#A7.T24)\.At layer 19 withB=16B=16, the information, accessibility, and fitting gaps are0\.0600\.060,0\.0470\.047, and0\.0220\.022, respectively \(Table[4](https://arxiv.org/html/2610.27446#S5.T4)\)\. The information gap and the restriction to the allowed family both contribute\. The 49 observed codes have collision fraction0\.7600\.760and unseen\-code fraction0\.1980\.198\(Table[24](https://arxiv.org/html/2610.27446#A7.T24)\)\.

Holding this interface fixed, replacing constant rules with affine rules reduces minimum excess over the unrestricted interface risk from0\.1430\.143to0\.0470\.047\. Adding pairwise interactions reduces it to0\.0070\.007; third\-order interactions and lookup rules attain zero class excess\. Separately, increasing the fitting sample from 16 to all 144 states of its fixed population reduces the reported mean fitting gap from0\.1300\.130to zero \(Figure[2](https://arxiv.org/html/2610.27446#S5.F2)\)\. Inclusion of rule families guarantees that their optima cannot worsen\. The observed decline with training size is an experimental result, not a consequence of the risk decomposition or a guarantee for new data\.

\(a\)Only the rule class changes\.\(b\)Only the fitting sample size changes\.
Figure 2:Separate comparisons of rule families and fitting samples\. Left: minimum excess over the unrestricted interface risk, with available 95% bootstrap intervals\. Right: the fitted rule’s mean excess over its family optimum\. The panels use reported rounded summaries; full plots and intervals appear in Figure[10](https://arxiv.org/html/2610.27446#A7.F10)\.
### 5\.3Other models, evaluators, and observation choices

For Phi seed 314159 on the original confirmation set, supplying policy text reduces the Mistral loss gap from0\.11100\.1110to0\.05570\.0557, with responses, scores, and features unchanged\. The extra input improves the optimum but does not recover the full state minimum\. Table[5](https://arxiv.org/html/2610.27446#S5.T5)reports the gapΔZ,S\\Delta\_\{Z,S\}using codeZZand policySSfor each model and scoring protocol\.

Table 5:Information loss with policy and SAE code jointly observed\. SmolLM2 and Phi use Mistral losses averaged over five responses per state–action pair\.†Qwen uses Qwen2\.5\-7B scores and rounded per\-policy summaries; its interval endpoints include the5×10−55\\times 10^\{\-5\}rounding correction\. LCBs are individual bootstrap lower bounds\. Layer indices are zero\-based; models use different responses and losses and are not ranked by this table\.SmolLM2 and Phi have positive automatic gaps and individual bootstrap lower bounds for both SAE seeds on both evaluation collections\. Qwen2\.5\-1\.5B also gives positive estimates, recovered from summaries with absolute rounding error at most5×10−55\\times 10^\{\-5\}; interval endpoints include the same correction \(Appendix[J](https://arxiv.org/html/2610.27446#A10)\)\. Because generation inputs and score rubrics differ, these are results within each protocol, not model rankings\.

Replacing Phi’s Mistral scores with Qwen scores or their mean preserves positive primary lower bounds \(Table[26](https://arxiv.org/html/2610.27446#A9.T26)\)\. Deleting one policy area at a time leaves positiveB=2B=2point gaps in all 192 recorded checks, with minimum0\.02360\.0236\. Neither comparison adds independent response data\. The two evaluators have rank correlation0\.35520\.3552, so the shared sign does not imply close agreement of their scores\.

#### Supporting controls\.

Qwen also gives positive gaps with a different evaluator, variance and random feature selection, controls matched on code occupancy, and a second model scale \(Appendix[G](https://arxiv.org/html/2610.27446#A7)\)\. SmolLM2 and Phi diagnostics appear in Appendices[F](https://arxiv.org/html/2610.27446#A6)and[I](https://arxiv.org/html/2610.27446#A9)\. The reconstruction error divided by centered activation variance is larger on audit states than on the SAE validation corpus\. Independently of that diagnostic, the risk comparison concerns selected activation indicators, not the information retained by the full SAE code\.

### 5\.4Group conflicts and required outputs

The controlled benchmark uses the loss pattern in Example[3\.2](https://arxiv.org/html/2610.27446#S3.Thmassumption2)\. Its three states share a layer\-17,B=2B=2code\. Each pair has a common action with zero regret, whereas the smallest maximum regret for the triple is1/31/3, including randomized choices\. The benchmark is a constructed example, not a discovered failure under natural language outcome loss\. The searched natural language candidate has zero positive separation in independent and fresh checks using two paraphrase replicas, five generation replicas, and three judge prompts \(Appendix[G](https://arxiv.org/html/2610.27446#A7)\)\.

\(a\)Compatible pairs and an incompatible triple\.\(b\)Output requirements for 96 states with human ratings\.
Figure 3:Group conflict and required output count\. Left: a controlled three\-state loss matrix has pair margins zero and triple margin1/31/3\. Right: global constraint generation on all 96 locked SmolLM2 states uses their final human losses, with every new cell validated\. The capacity counts unrestricted labels without a separate policy input, not SAE features; the calculation and summary results are described in Appendix[K\.5](https://arxiv.org/html/2610.27446#A11.SS5)\. Markers show only evaluated tolerances, not interpolated transition thresholds\. The two panels use different loss tables\.A separate capacity calculation uses all 96 locked SmolLM2 states and their final human losses\. Unrestricted label assignments require three outputs at tested tolerances00,0\.020\.02,0\.100\.10, and0\.200\.20; two at0\.300\.30and0\.350\.35; and one at0\.400\.40\. The margin with one output is0\.36950\.3695\. Constraint generation and an independent two\-label feasibility formulation agree at all seven points, within the recorded numerical tolerance\. Appendix[K\.5](https://arxiv.org/html/2610.27446#A11.SS5)describes the calculation; the matrix, labelings, and solver records are retained in the author working archive\. The capacity calculation does not train an SAE or give policy text as an additional free input\. The older Qwen 42\-state curve, whose matrix is unavailable, remains in the appendix and is not the curve shown here\.

## 6Related Work

#### Decision sufficiency and compression for a task\.

Blackwell comparison orders experiments by attainable Bayes risks across decision problems\([Blackwell, 1951](https://arxiv.org/html/2610.27446#bib.bib1);[Blackwell, 1953](https://arxiv.org/html/2610.27446#bib.bib2)\), while Le Cam deficiency quantifies approximate comparison and sufficiency\([Le Cam, 1964](https://arxiv.org/html/2610.27446#bib.bib3);[Torgersen, 1991](https://arxiv.org/html/2610.27446#bib.bib4)\)\. Representation\-learning extensions compare feature quality and optimize deficiency\-based objectives\([van Rooyen and Williamson, 2014](https://arxiv.org/html/2610.27446#bib.bib5);[Banerjee and Montúfar, 2018](https://arxiv.org/html/2610.27446#bib.bib6)\)\. Recent formulations study Bayes sufficiency for a chosen loss, compression that preserves optimal actions, and summaries for linear optimization\([Sevetlidis, 2026](https://arxiv.org/html/2610.27446#bib.bib7);[Walsh, 2026](https://arxiv.org/html/2610.27446#bib.bib8);[Ye et al\., 2026](https://arxiv.org/html/2610.27446#bib.bib9)\)\. The information bottleneck trades compression against target information\([Tishby et al\., 1999](https://arxiv.org/html/2610.27446#bib.bib10);[Alemi et al\., 2017](https://arxiv.org/html/2610.27446#bib.bib11)\); usable information additionally restricts the decoder family\([Xu et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib12)\)\.

We adopt loss\-dependent risk comparison for a fixed continuation library and audit distribution, rather than compare channels across all decision problems\. The optimum for the finite interface separates missing information from limitations of the rule that reads it\. Our structural analysis then asks which states cannot share a continuation within the regret tolerance under the complete declared observation, including policy information when supplied\.

#### State abstraction and closed\-loop information states\.

State abstraction preserves rewards, transitions, values, or near\-optimal behavior in Markov decision processes\([Givan et al\., 2003](https://arxiv.org/html/2610.27446#bib.bib13);[Li et al\., 2006](https://arxiv.org/html/2610.27446#bib.bib14);[Ferns et al\., 2004](https://arxiv.org/html/2610.27446#bib.bib15);[Ferns et al\., 2011](https://arxiv.org/html/2610.27446#bib.bib16);[Abel et al\., 2016](https://arxiv.org/html/2610.27446#bib.bib17);[Abel et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib18)\)\. Learned control representations seek to preserve latent dynamics, invariant task\-relevant structure, or exploration information\([Gelada et al\., 2019](https://arxiv.org/html/2610.27446#bib.bib19);[Zhang et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib20);[Zhang et al\., 2021](https://arxiv.org/html/2610.27446#bib.bib21);[Misra et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib23)\); control sufficiency has also been analyzed for mutual\-information objectives\([Rakelly et al\., 2021](https://arxiv.org/html/2610.27446#bib.bib22)\)\. Recent vision\-language\-action work addresses aliasing between similar observations requiring different actions\([Lee et al\., 2026](https://arxiv.org/html/2610.27446#bib.bib24)\)\.

Belief states and predictive state representations summarize histories for recursive planning\([Smallwood and Sondik, 1973](https://arxiv.org/html/2610.27446#bib.bib25);[Kaelbling et al\., 1998](https://arxiv.org/html/2610.27446#bib.bib26);[Littman et al\., 2001](https://arxiv.org/html/2610.27446#bib.bib27);[Singh et al\., 2004](https://arxiv.org/html/2610.27446#bib.bib28);[Boots et al\., 2011](https://arxiv.org/html/2610.27446#bib.bib29)\); approximate information states quantify the cost of compressed histories\([Subramanian et al\., 2022](https://arxiv.org/html/2610.27446#bib.bib30)\)\. Options formalize temporally extended closed\-loop policies\([Sutton et al\., 1999](https://arxiv.org/html/2610.27446#bib.bib31);[Abel et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib18)\)\. Our continuation library follows the established closed\-loop view\. We take the full audit state and continuation library as inputs and audit an additional observation channel, rather than construct a new information state\. We ask whether states sharing an observation can share a continuation, separately from the performance of restricted or fitted rules\.

#### Set\-valued decisions and higher\-order compatibility\.

Reject\-option classification permits abstention\([Chow, 1970](https://arxiv.org/html/2610.27446#bib.bib32);[Bartlett and Wegkamp, 2008](https://arxiv.org/html/2610.27446#bib.bib33);[Charoenphakdee et al\., 2021](https://arxiv.org/html/2610.27446#bib.bib34)\), while set\-valued prediction controls error, coverage, or output\-set size\([Denis and Hebiri, 2017](https://arxiv.org/html/2610.27446#bib.bib35);[Sadinle et al\., 2019](https://arxiv.org/html/2610.27446#bib.bib36);[Bates et al\., 2021](https://arxiv.org/html/2610.27446#bib.bib68);[Vovk et al\., 2005](https://arxiv.org/html/2610.27446#bib.bib37)\)\. These methods constrain predictions for each input\. Our uniform compatibility criterion instead requires one continuation distribution to satisfy a regret tolerance at every state in a deterministic interface cell\. Pairwise intersections can all be nonempty while the joint intersection is empty\.

Helly’s theorem bounds the size of a minimal convex witness\([Helly, 1923](https://arxiv.org/html/2610.27446#bib.bib38)\), and minimax duality relates its margin to a worst\-case state mixture\([Sion, 1958](https://arxiv.org/html/2610.27446#bib.bib39)\)\. Weak hypergraph coloring characterizes how many labels separate all minimal conflicts\([Berge, 1989](https://arxiv.org/html/2610.27446#bib.bib40)\)\. We connect these classical results to exact risk for the finite interface and the capacity of unrestricted deterministic interfaces, using conflicts over the entire audit domain, not only within existing cells\. On a finite audit domain, grouping by optimal action is sufficient when each state has a unique optimal pure continuation and the nonnegative tolerance is below half the minimum second\-best regret margin \(Proposition[B\.2](https://arxiv.org/html/2610.27446#A2.Thmassumption2)\)\. Outside that regime, a pairwise conflict graph need not describe joint compatibility\.

#### SAE interfaces and latent safety control\.

Sparse coding and sparse autoencoders learn dictionaries with sparse codes\([Olshausen and Field, 1996](https://arxiv.org/html/2610.27446#bib.bib53);[Makhzani and Frey, 2013](https://arxiv.org/html/2610.27446#bib.bib54)\)\. Language\-model studies examine superposition\([Elhage et al\., 2022](https://arxiv.org/html/2610.27446#bib.bib55)\), interpretable dictionary features\([Bricken et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib56);[Cunningham et al\., 2024](https://arxiv.org/html/2610.27446#bib.bib57)\), SAE scaling\([Gao et al\., 2025](https://arxiv.org/html/2610.27446#bib.bib59);[Templeton et al\., 2024](https://arxiv.org/html/2610.27446#bib.bib58)\), public dictionaries\([Lieberum et al\., 2024](https://arxiv.org/html/2610.27446#bib.bib60)\), and concept geometry\([Bhalla et al\., 2026](https://arxiv.org/html/2610.27446#bib.bib61)\)\. Our target is decision information in a selected support code or coordinate set, not reconstruction from the full SAE dictionary\.

Safe reinforcement learning develops constrained policies\([García and Fernández, 2015](https://arxiv.org/html/2610.27446#bib.bib41);[Achiam et al\., 2017](https://arxiv.org/html/2610.27446#bib.bib42);[Chow et al\., 2018](https://arxiv.org/html/2610.27446#bib.bib43)\), shields\([Alshiekh et al\., 2018](https://arxiv.org/html/2610.27446#bib.bib44);[Carr et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib45)\), and reachability or predictive safety filters\([Fisac et al\., 2019](https://arxiv.org/html/2610.27446#bib.bib46);[Wabersich and Zeilinger, 2021](https://arxiv.org/html/2610.27446#bib.bib47);[Brunke et al\., 2022](https://arxiv.org/html/2610.27446#bib.bib48)\)\. Related activation studies extract latent knowledge\([Burns et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib49)\)or steer generation\([Li et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib50);[Zou et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib51);[Karnik and Bansal, 2025](https://arxiv.org/html/2610.27446#bib.bib52)\)\. We audit the specified interface before selecting a downstream rule: a positive gap on the same finite loss table cannot be removed by changing that rule alone\. The result is complementary to designing or evaluating a particular monitor or steering method\.

#### Evaluation and generative safety\.

Language\-model evaluators score open\-ended outputs and compare candidate responses\([Liu et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib63);[Zheng et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib62);[Kim et al\., 2024](https://arxiv.org/html/2610.27446#bib.bib64)\); reward\-model benchmarks assess preference judgments\([Lambert et al\., 2025](https://arxiv.org/html/2610.27446#bib.bib65)\)\. Judging studies identify biases\([Wang et al\., 2024](https://arxiv.org/html/2610.27446#bib.bib66)\)and develop calibrated tests with imperfect evaluators\([Feng et al\., 2026](https://arxiv.org/html/2610.27446#bib.bib67)\)\. Risk\-control procedures calibrate predictions from finite data\([Angelopoulos et al\., 2025](https://arxiv.org/html/2610.27446#bib.bib69)\), while concentration inequalities provide bounds for bounded loss estimates\([Hoeffding, 1963](https://arxiv.org/html/2610.27446#bib.bib70)\)\. Our certificate concerns a witness selected after searching state groups\. Valid simultaneous loss bounds or independent certification data preserve the guarantee after selection\.

Toxicity benchmarks, red teaming, principle\-based training, and safeguard classifiers evaluate or modify generative behavior\([Gehman et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib71);[Perez et al\., 2022](https://arxiv.org/html/2610.27446#bib.bib72);[Bai et al\., 2022](https://arxiv.org/html/2610.27446#bib.bib73);[Inan et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib74)\)\. Our question is which safe and useful continuations the declared observation can support, not only whether an individual output is safe\. The human study tests the finite\-interface finding under a separate loss construction; it does not establish interchangeability of automatic and human scores\. Appendix[H](https://arxiv.org/html/2610.27446#A8)provides the extended comparison\.

## 7Discussion

#### What the risk comparison identifies\.

For fixed losses and continuations, a positive information gapΔinfo\\Delta\_\{\\mathrm\{info\}\}rules out matching the full state optimum by changing only the controller\. The accessibility gapΔaccess\\Delta\_\{\\mathrm\{access\}\}can decrease when the allowed rule family expands\. The fitting gapΔtrain\\Delta\_\{\\mathrm\{train\}\}may decrease with better estimation or optimization, but more data alone need not reduce it in every run\. These statements compare specified optima; they do not identify every cause of poor deployed performance\.

#### Encoding matters, not only the number of bits\.

With three continuations, an unrestricted encoder can transmit an optimal action index using three labels, which fit in two bits\. The positive gaps for selected SAE indicators do not prove that every pair of features fails, or that two bits are inherently insufficient\. They identify a failure of the selected partition\. Comparing the full state directly with the final indicators does not locate whether a distinction was lost in the model activation, SAE encoding, coordinate selection, or thresholding\. Those stages require separate comparisons\.

#### What the experiments support\.

Supplying policy text does not eliminate the observed gaps\. Human and automatic losses give the same sign in the tested SmolLM2 settings but often differ on individual scores and optimal action sets\. The losses measure rated safety and quality, not unsafe event probabilities\. The controlled group example demonstrates a possible failure of pairwise checks; the unsuccessful natural language replication gives no estimate of its prevalence\.

The model families use different generation and scoring protocols\. Within each SmolLM2 or Phi SAE pair, only initialization and training order vary; the activation corpus is shared\. Human ratings cover SmolLM2 only, and Phi reuses the policy collections\. The original human sample includes fitting areas, whereas the locked collection is held out by policy area\. These comparisons support the specific configurations, not all models, encoders, or policy distributions\.

#### Limits of inference from a fixed table\.

The empirical minima use recorded response scores, not unknown losses on future responses\. Minimizing noisy scores can change the apparent gap even when individual scores are unbiased\. Resampling policy areas measures variation under different weights on the recorded tables; it does not control generation or judge error\. Likewise, a zero gap on distinct evaluation codes can result from unrestricted lookup and gives no guarantee on unseen inputs\. The additional assumptions for statistical certification appear in Appendices[L](https://arxiv.org/html/2610.27446#A12)and[C\.3](https://arxiv.org/html/2610.27446#A3.SS3)\.

## 8Conclusion

Decision sufficiency compares what an interface permits with what the full state permits under the same loss and continuations\. Exact optimization for finite interfaces separates that information gap from restrictions on the controller and imperfect fitting\. Minimal conflicts identify groups that cannot share an acceptable choice; their complete set determines the output count for an unrestricted deterministic interface\.

The tested SAE indicators retain positive gaps with policy supplied, including under SmolLM2 human ratings\. A controlled example shows why pairwise checks can miss a group conflict, but the natural language candidate does not replicate\. The conclusions concern the specified responses and loss tables\. Within that scope, they distinguish limitations of the chosen input from errors a different controller could avoid\.

## Reproducibility Statement

The arXiv source package contains the manuscript, bibliography, figures, and LaTeX tables\. Experiment code and cached records are retained in a separate author working archive and are not included in the arXiv source package\. The working archive contains SmolLM2 ratings and adjudication inputs, cached SmolLM2 and Phi records, and CPU analysis scripts\. Its 96\-state capacity records include the matrix, labelings, margins, and independent feasibility checks\. Qwen results with policy supplied use rounded notebook summaries; its full precision historical arrays and 42\-state matrix are unavailable\. Large SAE checkpoints and activation shards are not included\. Appendix[M\.6](https://arxiv.org/html/2610.27446#A13.SS6)lists the inputs and commands\. Analysis scripts do not create human labels or retrain models\.

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## Appendix AAdditional Risk Results and Proofs

The results below establish the risk comparisons, their posterior interpretation, and the calculations for affine rules and uncertain losses\. The audit setting and all decision permissions are those of Section[2](https://arxiv.org/html/2610.27446#S2)\.

### A\.1Proofs of the main risk comparisons

###### Proof of Theorem[2\.2](https://arxiv.org/html/2610.27446#S2.Thmassumption2)\.

For every stateuuand choiceq∈Δmq\\in\\Delta\_\{m\},ℓ⋆​\(u\)≤ℓ⁡\(u,q\)\\ell^\{\\star\}\(u\)\\leq\\ell\(u,q\)\. Taking expectations and then an infimum provesRfull≤RZ⋆R\_\{\\mathrm\{full\}\}\\leq R\_\{Z\}^\{\\star\}\. The inclusionsδ^∈𝒟⊆𝔇Z\\widehat\{\\delta\}\\in\\mathcal\{D\}\\subseteq\\mathfrak\{D\}\_\{Z\}giveRZ⋆≤R𝒟≤Rδ^R\_\{Z\}^\{\\star\}\\leq R\_\{\\mathcal\{D\}\}\\leq R\_\{\\widehat\{\\delta\}\}\. Adding and subtractingRZ⋆R\_\{Z\}^\{\\star\}andR𝒟R\_\{\\mathcal\{D\}\}proves the decomposition\. Every step uses the same loss and audit distribution\. ∎

###### Proof of Theorem[2\.3](https://arxiv.org/html/2610.27446#S2.Thmassumption3)\.

A selector observing stateuiu\_\{i\}can choose a minimizing column ofLi​jL\_\{ij\}, givingRfull=∑iμi​minj​Li​jR\_\{\\mathrm\{full\}\}=\\sum\_\{i\}\\mu\_\{i\}\\min\_\{j\}L\_\{ij\}\. A rule assigningqr∈Δmq\_\{r\}\\in\\Delta\_\{m\}to outputzrz\_\{r\}has risk

∑i,r,jμi​Pi​r​Li​j​qr​j=∑r,jCr​j​qr​j\.\\sum\_\{i,r,j\}\\mu\_\{i\}P\_\{ir\}L\_\{ij\}q\_\{rj\}=\\sum\_\{r,j\}C\_\{rj\}q\_\{rj\}\.Eachqrq\_\{r\}has its own simplex constraint\. Its linear objective has minimumminj⁡Cr​j\\min\_\{j\}C\_\{rj\}, attained by any distribution supported on the minimizing columns\. Summing over outputs proves the formula\. A row with zero observation probability has zero cost for every column\. ∎

###### Justification of Equation \([6](https://arxiv.org/html/2610.27446#S2.E6)\)\.

On the finite audit, the complete observation\(Z,S\)\(Z,S\)has a finite alphabet\. Apply Theorem[2\.3](https://arxiv.org/html/2610.27446#S2.Thmassumption3)to its joint observation cellsCz,sC\_\{z,s\}\. Their unnormalized costs are𝔼⁡\[𝟏​\{Z=z,S=s\}​ℓj​\(U\)\]\\mathbb\{E\}\[\\mathbf\{1\}\\\{Z=z,S=s\\\}\\ell\_\{j\}\(U\)\], giving the displayed sum of minima\. The full\-state oracle cannot be outperformed by either observation rule\. Moreover, any code\-only rule can be implemented by a policy\-aware rule that ignoresSS, soRfull≤RZ,S⋆≤RZ⋆R\_\{\\mathrm\{full\}\}\\leq R\_\{Z,S\}^\{\\star\}\\leq R\_\{Z\}^\{\\star\}\. Subtracting the same full\-state risk gives0≤ΔZ,S≤ΔZ0\\leq\\Delta\_\{Z,S\}\\leq\\Delta\_\{Z\}\. ∎

### A\.2Posterior conflict and zero information loss

#### Residual uncertainty after observing the interface\.

LetPZP\_\{Z\}be the marginal law ofZZ, and letμz:=P\(U∈⋅∣Z=z\)\\mu\_\{z\}:=P\(U\\in\\cdot\\mid Z=z\)be a regular conditional state distribution\. Assumption[2\.1](https://arxiv.org/html/2610.27446#S2.Thmassumption1)ensures that such a conditional distribution exists\. The posterior expected regret of continuationπj\\pi\_\{j\}and the smallest posterior expected regret are

e¯j​\(z\):=∫ej​\(u\)​μz​\(𝑑u\),κ⁡\(z\):=min⁡∑j=1mq∈Δm⁡qj​e¯j​\(z\)=minj∈\[m\]⁡e¯j​\(z\)\.\\bar\{e\}\_\{j\}\(z\):=\\int e\_\{j\}\(u\)\\,\\mu\_\{z\}\(du\),\\qquad\\kappa\(z\):=\\min\_\{q\\in\\Delta\_\{m\}\}\\sum\_\{j=1\}^\{m\}q\_\{j\}\\bar\{e\}\_\{j\}\(z\)=\\min\_\{j\\in\[m\]\}\\bar\{e\}\_\{j\}\(z\)\.The selector can choose a different continuation for each observationzz, but it must use one distribution for the full states remaining possible underμz\\mu\_\{z\}\. The posterior conflictκ⁡\(z\)\\kappa\(z\)measures the resulting minimum average regret\.

###### Theorem A\.1\(Posterior\-conflict decomposition\)\.

Under Assumption[2\.1](https://arxiv.org/html/2610.27446#S2.Thmassumption1),

𝔇σinfo​\(K,μ\)=𝔼Z​\[κ⁡\(Z\)\]\.\\mathfrak\{D\}^\{\\mathrm\{info\}\}\_\{\\sigma\}\(K;\\mu\)=\\mathbb\{E\}\_\{Z\}\[\\kappa\(Z\)\]\.The information gap is zero if and only if the nonnegative posterior conflictκ⁡\(Z\)\\kappa\(Z\)is zeroPZP\_\{Z\}\-almost surely\.

###### Proposition A\.2\(Zero\-gap characterization\)\.

Under Assumption[2\.1](https://arxiv.org/html/2610.27446#S2.Thmassumption1), the following are equivalent:

1. 1\.The interface information deficiency𝔇σinfo​\(K,μ\)\\mathfrak\{D\}^\{\\mathrm\{info\}\}\_\{\\sigma\}\(K;\\mu\)is zero\.
2. 2\.ForPZP\_\{Z\}\-almost every observationzz, some continuation indexj∈\[m\]j\\in\[m\]satisfiesμz​\(\{u:ej​\(u\)=0\}\)=1\\mu\_\{z\}\(\\\{u:e\_\{j\}\(u\)=0\\\}\)=1\.
3. 3\.There exists a measurable pure selectorj⋆:𝒵→\[m\]j^\{\\star\}:\\mathcal\{Z\}\\to\[m\]such thatej⋆​\(Z\)​\(U\)=0e\_\{j^\{\\star\}\(Z\)\}\(U\)=0almost surely\.

Zero average information loss therefore requires a common optimal continuation under almost every posterior distribution, not merely a potentially different optimum for each hidden state\. The later uniform criterion uses a nonnegative toleranceε\\varepsilonand requires one continuation distribution to have regret at mostε\\varepsilonat*every*listed state in a cell\. Average sufficiency and uniform compatibility are not interchangeable at positive tolerance\.

###### Proposition A\.3\(Data processing for representation interfaces\)\.

LetG⁡\(d​y∣z\)G\(dy\\mid z\)be a Markov kernel into a standard Borel observation space, and obtainYYfromZZwithout additional information aboutUU, so thatU→Z→YU\\to Z\\to Yis a Markov chain\. Under the same audit distribution, continuation library, and loss,

𝔇σinfo​\(G∘K,μ\)≥𝔇σinfo​\(K,μ\)\.\\mathfrak\{D\}^\{\\mathrm\{info\}\}\_\{\\sigma\}\(G\\circ K;\\mu\)\\geq\\mathfrak\{D\}^\{\\mathrm\{info\}\}\_\{\\sigma\}\(K;\\mu\)\.

Post\-processing can increase the best attainable loss but cannot reduce it\. In particular, a positive gap after analysis\-only quantization does not by itself establish a positive gap for the original unquantized observation\.

###### Proof of Theorem[A\.1](https://arxiv.org/html/2610.27446#A1.Thmassumption1)\.

For any measurable ruleδ\\delta, conditional expectation gives

𝔼⁡\[e⁡\(U,δ⁡\(Z\)\)\]=𝔼Z​\[∑jδj​\(Z\)​e¯j​\(Z\)\]≥𝔼Z​\[κ⁡\(Z\)\]\.\\mathbb\{E\}\[e\(U,\\delta\(Z\)\)\]=\\mathbb\{E\}\_\{Z\}\\\!\\left\[\\sum\_\{j\}\\delta\_\{j\}\(Z\)\\bar\{e\}\_\{j\}\(Z\)\\right\]\\geq\\mathbb\{E\}\_\{Z\}\[\\kappa\(Z\)\]\.The finitely many conditional regretse¯j\\bar\{e\}\_\{j\}have measurable versions\. Choosing the smallest minimizing index at each observation defines a measurable pure rule and attains equality\. Nonnegativity ofκ\\kappathen makes its expectation zero exactly whenκ⁡\(Z\)=0\\kappa\(Z\)=0almost surely\. ∎

###### Proof of Proposition[A\.2](https://arxiv.org/html/2610.27446#A1.Thmassumption2)\.

The posterior formula makes zero deficiency equivalent tominj⁡e¯j​\(z\)=0\\min\_\{j\}\\bar\{e\}\_\{j\}\(z\)=0for almost every observationzz\. Each regret is nonnegative, soe¯j​\(z\)=∫ej​\(u\)​μz​\(𝑑u\)=0\\bar\{e\}\_\{j\}\(z\)=\\int e\_\{j\}\(u\)\\,\\mu\_\{z\}\(du\)=0exactly whenej​\(u\)=0e\_\{j\}\(u\)=0forμz\\mu\_\{z\}\-almost every state\. The first two statements are therefore equivalent\. Choosing the smallest minimizing index as in the posterior proof gives a measurable selector with zero regret almost surely\. Conversely, any such selector has expected regret zero, and no rule can have negative regret\. ∎

###### Proof of Proposition[A\.3](https://arxiv.org/html/2610.27446#A1.Thmassumption3)\.

For any measurable ruleη\\etausingYY, define

δj​\(z\):=∫ηj​\(y\)​G​\(𝑑y∣z\),j∈\[m\]\.\\delta\_\{j\}\(z\):=\\int\\eta\_\{j\}\(y\)G\(dy\\mid z\),\\qquad j\\in\[m\]\.These components are measurable, nonnegative, and sum to one, soδ∈𝔇Z\\delta\\in\\mathfrak\{D\}\_\{Z\}\. The Markov relationU→Z→YU\\to Z\\to Yand linearity of loss give𝔼⁡\[ℓ⁡\(U,δ⁡\(Z\)\)\]=𝔼⁡\[ℓ⁡\(U,η⁡\(Y\)\)\]\\mathbb\{E\}\[\\ell\(U,\\delta\(Z\)\)\]=\\mathbb\{E\}\[\\ell\(U,\\eta\(Y\)\)\]\. Every risk available fromYYis therefore available fromZZ\. Taking infima and subtracting the same full state risk proves the claim\. ∎

#### Deterministic cells and uniform compatibility\.

For a deterministic interfacef:𝒰aud→𝒵ff:\\mathcal\{U\}\_\{\\mathrm\{aud\}\}\\to\\mathcal\{Z\}\_\{\\mathrm\{f\}\}, define the occupied cellCr:=\{ui:f⁡\(ui\)=zr\}C\_\{r\}:=\\\{u\_\{i\}:f\(u\_\{i\}\)=z\_\{r\}\\\}for each nonempty cell\. Its robust regret of a shared continuation is

Γ⁡\(Cr\):=minq∈Δm⁡maxu∈Cr⁡e⁡\(u,q\)\.\\Gamma\(C\_\{r\}\):=\\min\_\{q\\in\\Delta\_\{m\}\}\\max\_\{u\\in C\_\{r\}\}e\(u,q\)\.Unlike posterior conflict, which averages over states,Γ⁡\(Cr\)\\Gamma\(C\_\{r\}\)requires one continuation distribution to control the worst regret in the cell\.

###### Corollary A\.4\(Finite\-cell zero\-gap criterion\)\.

If the deterministic finite audit has strictly positive state weights,𝔇σinfo​\(K,μ\)=0\\mathfrak\{D\}^\{\\mathrm\{info\}\}\_\{\\sigma\}\(K;\\mu\)=0if and only if every occupied cellCrC\_\{r\}has a pure continuation optimal at every state inCrC\_\{r\}\. Equivalently,Γ⁡\(Cr\)=0\\Gamma\(C\_\{r\}\)=0for every occupied cell\.

Positive weights make the criterion apply to every listed state, rather than only to states of positive probability\. At positive regret tolerance, randomization may improve a shared decision\.

###### Proof of Corollary[A\.4](https://arxiv.org/html/2610.27446#A1.Thmassumption4)\.

Positive state weights give every listed state positive conditional probability in its occupied cell\. The zero\-gap characterization therefore requires a common optimal pure continuation in each cell, which givesΓ⁡\(Cr\)=0\\Gamma\(C\_\{r\}\)=0\. Conversely, a minimizing distribution forΓ⁡\(Cr\)=0\\Gamma\(C\_\{r\}\)=0exists by compactness\. At each state,∑jqj​ej​\(u\)=0\\sum\_\{j\}q\_\{j\}e\_\{j\}\(u\)=0; nonnegative regrets force every index withqj\>0q\_\{j\}\>0to be optimal throughout the cell\. At least one such index exists\. Choosing a common optimum in each cell gives zero information gap\. ∎

### A\.3Exact restricted affine rules

#### Affine probabilities\.

Return to a declared finite observation alphabet\. Letφ:𝒵f→ℝdφ\\varphi:\\mathcal\{Z\}\_\{\\mathrm\{f\}\}\\to\\mathbb\{R\}^\{d\_\{\\varphi\}\}be a fixed visible feature map, wheredφd\_\{\\varphi\}is its dimension, and setφr:=φ⁡\(zr\)\\varphi\_\{r\}:=\\varphi\(z\_\{r\}\)\. An affine randomized rule assignsqr=A​φr\+bq\_\{r\}=A\\varphi\_\{r\}\+b, withA∈ℝm×dφA\\in\\mathbb\{R\}^\{m\\times d\_\{\\varphi\}\}andb∈ℝmb\\in\\mathbb\{R\}^\{m\}\. Its parameter bound satisfies1/m≤Bpar<∞1/m\\leq B\_\{\\mathrm\{par\}\}<\\infty;BparB\_\{\\mathrm\{par\}\}is distinct from the SAE feature budgetBB\.

###### Proposition A\.5\(Restricted affine\-randomized risk is a linear program\)\.

For the affine class with simplex constraints on the entire declared alphabet,

R𝒟=minA,b,\{qr\}\\displaystyle R\_\{\\mathcal\{D\}\}=\\min\_\{A,b,\\\{q\_\{r\}\\\}\}∑r=1L∑j=1mCr​j​qr​j\\displaystyle\\sum\_\{r=1\}^\{L\}\\sum\_\{j=1\}^\{m\}C\_\{rj\}q\_\{rj\}subject to\\displaystyle\\text\{subject to\}qr=A​φr\+b,\\displaystyle q\_\{r\}=A\\varphi\_\{r\}\+b,r∈\[L\],\\displaystyle r\\in\[L\],qr∈Δm,\\displaystyle q\_\{r\}\\in\\Delta\_\{m\},r∈\[L\],\\displaystyle r\\in\[L\],\|Aj​k\|≤Bpar,\|bj\|≤Bpar,\\displaystyle\|A\_\{jk\}\|\\leq B\_\{\\mathrm\{par\}\},\\quad\|b\_\{j\}\|\\leq B\_\{\\mathrm\{par\}\},j∈\[m\],k∈\[dφ\]\.\\displaystyle j\\in\[m\],\\ k\\in\[d\_\{\\varphi\}\]\.The uniform ruleA=0A=0,bj=1/mb\_\{j\}=1/mis feasible, and the minimum is attained\.

Fitting replacesCr​jC\_\{rj\}by training\-sample costs, changing the objective but not the feasible observation domain\. Simplex constraints must still hold on declared evaluation codes absent from training\. Observations outside that declared domain require a separately specified feasible extension\. For a policy\-aware audit, the cost table and visible features must use\(Z,S\)\(Z,S\)rather thanZZalone\. The linear program uses standard convex\-optimization tools\([Boyd and Vandenberghe, 2004](https://arxiv.org/html/2610.27446#bib.bib75)\); Algorithm[2](https://arxiv.org/html/2610.27446#alg2)separates its rule\-class and fitting gaps\.

###### Proof of Proposition[A\.5](https://arxiv.org/html/2610.27446#A1.Thmassumption5)\.

Substitutingqr=A​φr\+bq\_\{r\}=A\\varphi\_\{r\}\+binto the finite risk gives the stated linear objective and linear constraints\. The choiceA=0A=0,bj=1/mb\_\{j\}=1/mis feasible becauseBpar≥1/mB\_\{\\rm par\}\\geq 1/m\. Parameter and simplex bounds make the feasible set compact, so the continuous objective attains its minimum\. Replacing evaluation costs by training costs leaves the same feasible family of rules\. ∎

### A\.4Effect of errors in the loss table

#### Group margin\.

For any nonempty state groupW⊆𝒰audW\\subseteq\\mathcal\{U\}\_\{\\mathrm\{aud\}\}, defineΓ⁡\(W\):=minq∈Δm⁡maxu∈W⁡e⁡\(u,q\)\\Gamma\(W\):=\\min\_\{q\\in\\Delta\_\{m\}\}\\max\_\{u\\in W\}e\(u,q\)using the regrets of the specified loss table\.

###### Proposition A\.6\(Loss\-table perturbation\)\.

Fix the finite audit weights and observation channel\. If two loss matrices in\[0,1\]n×m\[0,1\]^\{n\\times m\}differ by at mostrℓ≥0r\_\{\\ell\}\\geq 0entrywise, their full\-state risks and optimal interface risks each differ by at mostrℓr\_\{\\ell\}\. Their information gaps differ by at most2​rℓ2r\_\{\\ell\}, and their robust marginsΓ⁡\(W\)\\Gamma\(W\)differ by at most2​rℓ2r\_\{\\ell\}for every nonempty state groupWW\.

An exact finite\-table optimum and an accurate estimate of expected outcome loss are separate requirements\. The perturbation radiusrℓr\_\{\\ell\}transfers accuracy of the loss entries to accuracy of the resulting risks; a positive plug\-in gap without such error control does not certify a positive gap in expected loss\.

###### Proof of Proposition[A\.6](https://arxiv.org/html/2610.27446#A1.Thmassumption6)\.

Let the two loss tables be𝐋\\mathbf\{L\}and𝐋~\\widetilde\{\\mathbf\{L\}\}\. At every state, a fixed continuation mixture has losses differing by at mostrℓr\_\{\\ell\}\. The same holds after averaging over the fixed observation channel and state weights\. For two objectives whose pointwise difference is at mostrℓr\_\{\\ell\}on the same feasible set, their infima differ by at mostrℓr\_\{\\ell\}\. Apply this fact to the full state and interface optima; their difference changes by at most2​rℓ2r\_\{\\ell\}\.

Each regret subtracts a row minimum from a loss entry, so regret entries differ by at most2​rℓ2r\_\{\\ell\}\. Convex combinations, a maximum over a fixed groupWW, and a minimum overΔm\\Delta\_\{m\}preserve this uniform bound\. Hence the group margins differ by at most2​rℓ2r\_\{\\ell\}\. ∎

## Appendix BProofs for Group Conflicts and Unique Optimal Choices

### B\.1Proofs for shared decisions and minimal conflicts

#### Existence, compatibility, and monotonicity\.

For a nonempty finite state groupWW, the mapq↦max⁡∑ju∈W⁡ej​\(u\)​qjq\\mapsto\\max\_\{u\\in W\}\\sum\_\{j\}e\_\{j\}\(u\)q\_\{j\}is continuous on the compact simplexΔm\\Delta\_\{m\}\. A minimizing shared selection therefore exists\. If the acceptable sets have a common memberqq, every state regret is at mostε\\varepsilon, soΓ⁡\(W\)≤ε\\Gamma\(W\)\\leq\\varepsilon\. Conversely, if the minimum margin is at mostε\\varepsilon, a minimizingqqbelongs to all acceptable sets\. For nonemptyV⊆WV\\subseteq W, the maximum overVVis no larger than the maximum overWWfor everyqq\. Taking minima givesΓ⁡\(V\)≤Γ⁡\(W\)\\Gamma\(V\)\\leq\\Gamma\(W\)\. A finite incompatible set contains an inclusion\-minimal incompatible subset: delete a state whenever the remaining set is still incompatible, and stop when no deletion is possible\. Monotonicity makes all smaller proper subsets compatible at termination\.

###### Proof of Theorem[3\.1](https://arxiv.org/html/2610.27446#S3.Thmassumption1)\.

For every continuation distributionqq, letuq∈arg​maxu∈W⁡e​\(u,q\)u\_\{q\}\\in\\argmax\_\{u\\in W\}e\(u,q\)\. Nonnegative regrets and the smallest state weightβ=minu⁡wu\\beta=\\min\_\{u\}w\_\{u\}give

β​maxu∈W⁡e⁡\(u,q\)≤wuq​e​\(uq,q\)≤∑u∈Wwu​e​\(u,q\)≤maxu∈W⁡e⁡\(u,q\)\.\\beta\\max\_\{u\\in W\}e\(u,q\)\\leq w\_\{u\_\{q\}\}e\(u\_\{q\},q\)\\leq\\sum\_\{u\\in W\}w\_\{u\}e\(u,q\)\\leq\\max\_\{u\\in W\}e\(u,q\)\.Taking the minimum overqqproves both bounds, including the caseβ=0\\beta=0\. For a deterministic interface, the finite optimum also gives

𝔇σinfo\(Kf;μ\)=∑c:pc\>0minj∑u∈Ccμ\(u\)ej\(u\)=∑c:pc\>0pcκ\(Cc,w\(c\)\)\.\\mathfrak\{D\}^\{\\mathrm\{info\}\}\_\{\\sigma\}\(K\_\{f\};\\mu\)=\\sum\_\{c:p\_\{c\}\>0\}\\min\_\{j\}\\sum\_\{u\\in C\_\{c\}\}\\mu\(u\)e\_\{j\}\(u\)=\\sum\_\{c:p\_\{c\}\>0\}p\_\{c\}\\kappa\(C\_\{c\},w^\{\(c\)\}\)\.If every cell of positive mass has margin at mostε\\varepsilon, the upper bound and∑cpc=1\\sum\_\{c\}p\_\{c\}=1make this sum at mostε\\varepsilon\. ∎

###### Proof of Theorem[3\.4](https://arxiv.org/html/2610.27446#S3.Thmassumption4)\.

Every singleton has an optimal pure continuation, so a conflict has at least two states\. Each acceptable set is convex in the affine hull ofΔm\\Delta\_\{m\}, whose dimension ism−1m\-1\. Helly’s theorem\([Helly, 1923](https://arxiv.org/html/2610.27446#bib.bib38)\)gives an empty intersection among at mostmmsets whenever the full finite intersection is empty\. An inclusion\-minimal conflict must itself have at mostmmstates\.

For tightness at everym≥2m\\geq 2, takemmstates and lossesℓj\(ui\)=𝟏\{j=i\}\\ell\_\{j\}\(u\_\{i\}\)=\\mathbf\{1\}\\\{j=i\\\}\. Each row has minimum zero\. Atε=0\\varepsilon=0, stateuiu\_\{i\}requiresqi=0q\_\{i\}=0\. Allmmrequirements contradict∑iqi=1\\sum\_\{i\}q\_\{i\}=1\. After any stateuku\_\{k\}is deleted, selecting continuationkkgives zero regret at every remaining state\. The full group is therefore a minimal conflict of sizemm\. ∎

###### Proof of Theorem[3\.5](https://arxiv.org/html/2610.27446#S3.Thmassumption5)\.

For a fixed groupWW, the payoff∑u,jλu​qj​ej​\(u\)\\sum\_\{u,j\}\\lambda\_\{u\}q\_\{j\}e\_\{j\}\(u\)is bilinear on the compact convex setsΔ⁡\(W\)\\Delta\(W\)andΔm\\Delta\_\{m\}\. Finite minimax duality\([Sion, 1958](https://arxiv.org/html/2610.27446#bib.bib39)\)gives

Γ⁡\(W\)\\displaystyle\\Gamma\(W\)=minq∈Δm⁡max⁡∑u,jλ∈Δ⁡\(W\)⁡λu​qj​ej​\(u\)\\displaystyle=\\min\_\{q\\in\\Delta\_\{m\}\}\\max\_\{\\lambda\\in\\Delta\(W\)\}\\sum\_\{u,j\}\\lambda\_\{u\}q\_\{j\}e\_\{j\}\(u\)=maxλ∈Δ⁡\(W\)⁡min⁡∑u,jq∈Δm⁡λu​qj​ej​\(u\)\\displaystyle=\\max\_\{\\lambda\\in\\Delta\(W\)\}\\min\_\{q\\in\\Delta\_\{m\}\}\\sum\_\{u,j\}\\lambda\_\{u\}q\_\{j\}e\_\{j\}\(u\)=maxλ∈Δ⁡\(W\)⁡min⁡∑u∈Wj⁡λu​ej​\(u\)\.\\displaystyle=\\max\_\{\\lambda\\in\\Delta\(W\)\}\\min\_\{j\}\\sum\_\{u\\in W\}\\lambda\_\{u\}e\_\{j\}\(u\)\.The first equality expresses a maximum over states as a maximum over state distributions; the last uses linearity inqq\. The final maximum is attained because a finite minimum of linear functions is continuous\. Ifffis constant onWW, the finite interface formula gives𝔇σinfo​\(Kf,λ\)=min⁡∑uj⁡λu​ej​\(u\)\\mathfrak\{D\}^\{\\mathrm\{info\}\}\_\{\\sigma\}\(K\_\{f\};\\lambda\)=\\min\_\{j\}\\sum\_\{u\}\\lambda\_\{u\}e\_\{j\}\(u\)\. Maximizing overλ\\lambdaproves the risk interpretation\. ∎

### B\.2Unique optimal choices and positive tolerances

The main example has two optimal pure choices at every state\. The example below instead has one optimal choice per state and shows why positive tolerance can still create a higher\-order conflict\.

###### Example B\.1\(Pairwise compatibility without joint compatibility\)\.

Consider three statesu1,u2,u3u\_\{1\},u\_\{2\},u\_\{3\}and three continuationsa,b,ca,b,c, with losses

abcu1011u2101u3110\\begin\{array\}\[\]\{c\|ccc\}&a&b&c\\\\ \\hline\\cr u\_\{1\}&0&1&1\\\\ u\_\{2\}&1&0&1\\\\ u\_\{3\}&1&1&0\\end\{array\}and toleranceε=1/2\\varepsilon=1/2\. Every state has oracle loss zero, so the loss entries are also regrets\. A shared selectionq=\(qa,qb,qc\)∈Δ3q=\(q\_\{a\},q\_\{b\},q\_\{c\}\)\\in\\Delta\_\{3\}must satisfy

𝒢1/2​\(u1\)\\displaystyle\\mathcal\{G\}^\{1/2\}\(u\_\{1\}\)=\{q∈Δ3:qa≥1/2\},\\displaystyle=\\\{q\\in\\Delta\_\{3\}:q\_\{a\}\\geq 1/2\\\},𝒢1/2​\(u2\)\\displaystyle\\mathcal\{G\}^\{1/2\}\(u\_\{2\}\)=\{q∈Δ3:qb≥1/2\},\\displaystyle=\\\{q\\in\\Delta\_\{3\}:q\_\{b\}\\geq 1/2\\\},𝒢1/2​\(u3\)\\displaystyle\\mathcal\{G\}^\{1/2\}\(u\_\{3\}\)=\{q∈Δ3:qc≥1/2\}\.\\displaystyle=\\\{q\\in\\Delta\_\{3\}:q\_\{c\}\\geq 1/2\\\}\.For the pair\{u1,u2\}\\\{u\_\{1\},u\_\{2\}\\\}, regrets are1−qa1\-q\_\{a\}and1−qb1\-q\_\{b\}\. Sinceqa\+qb≤1q\_\{a\}\+q\_\{b\}\\leq 1, their maximum is at least1/21/2, attained by\(1/2,1/2,0\)\(1/2,1/2,0\)\. The same argument applies to every pair\. For the triple,

maxi⁡e⁡\(ui,q\)=1−min⁡\{qa,qb,qc\}≥2/3,\\max\_\{i\}e\(u\_\{i\},q\)=1\-\\min\\\{q\_\{a\},q\_\{b\},q\_\{c\}\\\}\\geq 2/3,with equality at the uniform distribution\. Every pair is therefore compatible at tolerance1/21/2, but the triple is not\.

#### When the pairwise view is sufficient\.

A unique optimum does not remove all higher\-order conflicts at arbitrary positive tolerance, because randomized selections can become acceptable\. A uniform separation from the second\-best choice gives a sufficient small\-tolerance regime\.

###### Proposition B\.2\(Unique\-optimum reduction at small tolerance\)\.

Suppose every state has a unique optimal pure continuation with indexj⋆​\(u\)∈\[m\]j^\{\\star\}\(u\)\\in\[m\]\. Define the smallest second\-best regret by

ρ:=minu∈𝒰aud⁡minj≠j⋆​\(u\)​ej​\(u\)\.\\rho:=\\min\_\{u\\in\\mathcal\{U\}\_\{\\mathrm\{aud\}\}\}\\min\_\{j\\neq j^\{\\star\}\(u\)\}e\_\{j\}\(u\)\.Finiteness and uniqueness implyρ\>0\\rho\>0\. For0≤ε<ρ/20\\leq\\varepsilon<\\rho/2, two states are incompatible if and only if their optimal continuation indices differ\. Every minimal conflict is then a pair, and the minimum number of unrestricted deterministic interface outputs with compatible cells equals the number of distinct optimal continuation indices\.

Example[B\.1](https://arxiv.org/html/2610.27446#A2.Thmassumption1)hasρ=1\\rho=1andε=1/2\\varepsilon=1/2, so it lies outside the proposition’s strict range\.

###### Proof of Proposition[B\.2](https://arxiv.org/html/2610.27446#A2.Thmassumption2)\.

There are finitely many strictly positive nonoptimal regrets, so their minimumρ\\rhois positive\. States with the same optimal index share a pure optimum\. For two states with distinct optimal indicesaaandbb,

e⁡\(u,q\)≥ρ⁡\(1−qa\),e⁡\(v,q\)≥ρ⁡\(1−qb\)\.e\(u,q\)\\geq\\rho\(1\-q\_\{a\}\),\\qquad e\(v,q\)\\geq\\rho\(1\-q\_\{b\}\)\.A common choice with regret at mostε<ρ/2\\varepsilon<\\rho/2would requireqa\+qb≥2​\(1−ε/ρ\)\>1q\_\{a\}\+q\_\{b\}\\geq 2\(1\-\\varepsilon/\\rho\)\>1, which is impossible\. Every group containing different optimal indices therefore contains an incompatible pair; a group with only one optimal index is compatible\. Minimal conflicts are exactly the pairs with different indices\. Distinct indices require distinct outputs, and assigning each state its optimal index attains that output count\. ∎

## Appendix CCapacity Extensions and Statistical Certification

### C\.1Proofs for capacity and global validation

###### Proof of Theorem[3\.6](https://arxiv.org/html/2610.27446#S3.Thmassumption6)\.

A monochromatic conflict lies entirely in one cell and makes that cell incompatible\. Conversely, every incompatible finite cell contains an inclusion\-minimal incompatible subset, which is a monochromatic edge of the global hypergraph\. A labeling has compatible cells exactly when it has no monochromatic edge\. Minimizing its number of outputs is therefore the weak coloring problem\. ∎

###### Proof of Proposition[3\.7](https://arxiv.org/html/2610.27446#S3.Thmassumption7)\.

Every compatible labeling satisfies all retained conflict constraints, soMt≤𝖬⋆​\(ε\)M\_\{t\}\\leq\\mathsf\{M\}^\{\\star\}\(\\varepsilon\)\. If the candidate labelinggtg\_\{t\}passes every cell check, it is feasible for the full problem withMtM\_\{t\}outputs; hence𝖬⋆​\(ε\)≤Mt\\mathsf\{M\}^\{\\star\}\(\\varepsilon\)\\leq M\_\{t\}\.

Otherwise a violating cell contains a minimal conflictWtW\_\{t\}\. The edge cannot already be retained, becausegtg\_\{t\}would violate its constraint\. Each failed iteration adds a new edge\. At most

∑s=2min⁡\{n,m\}\(ns\)−\|ℱ0\|\\sum\_\{s=2\}^\{\\min\\\{n,m\\\}\}\\binom\{n\}\{s\}\-\|\\mathcal\{F\}\_\{0\}\|new edges can be added\. The sum is zero ifmin⁡\{n,m\}<2\\min\\\{n,m\\\}<2\. Assigning each state a separate label is always feasible, so no iteration has an infeasible full coloring problem\. Once no missing edge remains, the candidate passes validation and the bounds coincide\. Deleting a state whenever the remaining margin exceedsε\\varepsilonfinds a minimal conflict: every proper subset lies inside a compatible single deletion at termination\. ∎

#### Why local conflicts are not enough\.

Use the three\-state loss table in Example[B\.1](https://arxiv.org/html/2610.27446#A2.Thmassumption1), now at toleranceε=0\\varepsilon=0\. Suppose the old interface cells are\{u1,u2\}\\\{u\_\{1\},u\_\{2\}\\\}and\{u3\}\\\{u\_\{3\}\\\}\. Local search finds only the conflict\{u1,u2\}\\\{u\_\{1\},u\_\{2\}\\\}\. The single retained constraint allows a two\-label redesign that mergesu3u\_\{3\}with one of the other states\. Every pair actually requires different optimal continuations at zero tolerance, so the true capacity is three\. Checking all*new*cells detects the missing constraints\. Algorithms[5](https://arxiv.org/html/2610.27446#alg5)and[6](https://arxiv.org/html/2610.27446#alg6)implement the coloring and validation steps; solver tolerances and optimality status must accompany numerical outputs\.

### C\.2Restricted encoders, zero regret, and observable policy

Let𝔎\\mathfrak\{K\}be a restricted family of deterministic finite\-output maps on the same audit domain\. Its minimum feasible output count is

𝖬𝔎⋆\(ε\):=min\{\|range\(g\)\|:g∈𝔎,gis uniformlyε\-sufficient\},\\mathsf\{M\}^\{\\star\}\_\{\\mathfrak\{K\}\}\(\\varepsilon\):=\\min\\\{\|\\range\(g\)\|:g\\in\\mathfrak\{K\},\\ g\\text\{ is uniformly \}\\varepsilon\\text\{\-sufficient\}\\\},with value\+∞\+\\inftyif no map in the family is feasible\. Restricting the maps cannot improve the optimum:𝖬𝔎⋆​\(ε\)≥𝖬⋆​\(ε\)\\mathsf\{M\}^\{\\star\}\_\{\\mathfrak\{K\}\}\(\\varepsilon\)\\geq\\mathsf\{M\}^\{\\star\}\(\\varepsilon\)\. An SAE support map may fail to realize an optimal abstract labeling even when it has enough possible outputs\. For an allowed output countM≥1M\\geq 1, the inverse curve is

ε⋆​\(M\):=inf\{ε≥0:𝖬⋆​\(ε\)≤M\}\.\\varepsilon^\{\\star\}\(M\):=\\inf\\\{\\varepsilon\\geq 0:\\mathsf\{M\}^\{\\star\}\(\\varepsilon\)\\leq M\\\}\.The valueε⋆​\(M\)\\varepsilon^\{\\star\}\(M\)is the smallest uniform regret tolerance achievable with at mostMMunrestricted outputs\.

#### A useful special case: zero regret\.

For each continuationπj\\pi\_\{j\}, letOj:=\{u∈𝒰aud:ej​\(u\)=0\}O\_\{j\}:=\\\{u\\in\\mathcal\{U\}\_\{\\mathrm\{aud\}\}:e\_\{j\}\(u\)=0\\\}be the states where it is optimal\. The family\{Oj\}j=1m\\\{O\_\{j\}\\\}\_\{j=1\}^\{m\}covers the audit domain because every state has an optimal continuation\.

###### Proposition C\.1\(Capacity bounds and zero\-regret set cover\)\.

For everyε≥0\\varepsilon\\geq 0,

1≤𝖬⋆​\(ε\)≤min⁡\{n,m\}\.1\\leq\\mathsf\{M\}^\{\\star\}\(\\varepsilon\)\\leq\\min\\\{n,m\\\}\.At zero tolerance,

𝖬⋆​\(0\)=minJ⊆\[m\]⁡\{\|J\|:⋃j∈JOj=𝒰aud\}\.\\mathsf\{M\}^\{\\star\}\(0\)=\\min\_\{J\\subseteq\[m\]\}\\left\\\{\|J\|:\\bigcup\_\{j\\in J\}O\_\{j\}=\\mathcal\{U\}\_\{\\mathrm\{aud\}\}\\right\\\}\.Computing the capacity is NP\-hard in general when the continuation countmmis part of the input\.

The zero\-regret formula selects the smallest collection of continuations that supplies an optimum for every state\. Each selected continuation can serve one label cell\. The at\-most\-mmoutput bound does not imply that a restricted SAE can implement the optimal\-action labeling with a small feature budget\.

#### Capacity when policy is supplied separately\.

For a pooled\-policy audit, letUUdenote the full audited state and evaluate each lossℓj​\(u\)\\ell\_\{j\}\(u\)under the policy carried by stateuu\. LetS=s⁡\(U\)S=s\(U\)be the finite\-valued policy observation supplied at no code cost, and let𝒮:=s⁡\(𝒰aud\)\\mathcal\{S\}:=s\(\\mathcal\{U\}\_\{\\mathrm\{aud\}\}\)be its values on the audit domain\. For each policy values∈𝒮s\\in\\mathcal\{S\}, define

𝒰s:=\{u∈𝒰aud:s⁡\(u\)=s\},ℋε,s:=\(𝒰s,\{W∈ℰε:W⊆𝒰s\}\)\.\\mathcal\{U\}\_\{s\}:=\\\{u\\in\\mathcal\{U\}\_\{\\mathrm\{aud\}\}:s\(u\)=s\\\},\\qquad\\mathcal\{H\}\_\{\\varepsilon,s\}:=\\left\(\\mathcal\{U\}\_\{s\},\\\{W\\in\\mathcal\{E\}\_\{\\varepsilon\}:W\\subseteq\\mathcal\{U\}\_\{s\}\\\}\\right\)\.DefineMS⋆​\(ε\)M\_\{S\}^\{\\star\}\(\\varepsilon\)as the smallest code alphabet for a mapggsuch that every nonempty complete\-observation cell\{u:g\(u\)=c,s\(u\)=s\}\\\{u:g\(u\)=c,\\ s\(u\)=s\\\}is compatible\.

###### Corollary C\.2\(Capacity with free finite policy information\)\.

Over unrestricted deterministic code maps on the full audit state,

MS⋆​\(ε\)=maxs∈𝒮⁡χw​\(ℋε,s\)\.M\_\{S\}^\{\\star\}\(\\varepsilon\)=\\max\_\{s\\in\\mathcal\{S\}\}\\chi\_\{\\mathrm\{w\}\}\(\\mathcal\{H\}\_\{\\varepsilon,s\}\)\.

Each policy requires its own compatible partition, but code labels can be reused across policies because the selector also observesSS\. The maximum counts code values, not the sum of code–policy pairs\. The encoder is unrestricted in this corollary; a restricted encoder requires its own realizability analysis\.

###### Proof of Proposition[C\.1](https://arxiv.org/html/2610.27446#A3.Thmassumption1)\.

A nonempty domain requires a label\. Assigning each state an optimal continuation index gives compatible cells at regret zero, using at mostmin⁡\{n,m\}\\min\\\{n,m\\\}labels\.

For a cell with zero shared regret, a feasible distributionqqsatisfies∑jqj​ej​\(u\)=0\\sum\_\{j\}q\_\{j\}e\_\{j\}\(u\)=0at every state\. Nonnegativity makes any index withqj\>0q\_\{j\}\>0optimal throughout the cell\. One such index per cell yields a cover by optimal\-state sets\. Conversely, a coverJ⊆\[m\]J\\subseteq\[m\]yields a compatible labeling by assigning each state a covering index\. Taking minima proves the set\-cover formula\.

To show NP\-hardness, take a set\-cover instance whose sets cover its finite universe\. Use elements as states and sets as continuations, with loss zero for membership and one otherwise\. Every row has minimum zero, and the zero\-regret capacity equals the minimum cover size\. The matrix has polynomial size, proving the claim whenmmis part of the input\. ∎

###### Proof of Corollary[C\.2](https://arxiv.org/html/2610.27446#A3.Thmassumption2)\.

On each policy group𝒰s\\mathcal\{U\}\_\{s\}, a valid code must use at leastχw​\(ℋε,s\)\\chi\_\{\\mathrm\{w\}\}\(\\mathcal\{H\}\_\{\\varepsilon,s\}\)labels\. Thus the maximum of these numbers is a lower bound\. LetMMequal that maximum and color each policy group using a common palette\[M\]\[M\]\. Define the unrestricted encoder by the coloring associated with the state’s policy\. Policy textSSdistinguishes groups even when their code labels agree, so all cells of the complete observation are compatible\. The combined encoder attains the lower bound\. ∎

### C\.3Loss uncertainty, larger libraries, and search

#### From loss uncertainty to margin uncertainty\.

Return to the fixed finite audit domain and continuation library\. Letℓ^j​\(u\)∈\[0,1\]\\widehat\{\\ell\}\_\{j\}\(u\)\\in\[0,1\]estimate the true expected lossℓj​\(u\)\\ell\_\{j\}\(u\)\. Define the estimated oracle loss, regret, and group margin by

ℓ^⋆​\(u\)\\displaystyle\\widehat\{\\ell\}^\{\\star\}\(u\):=minj∈\[m\]⁡ℓ^j​\(u\),\\displaystyle:=\\min\_\{j\\in\[m\]\}\\widehat\{\\ell\}\_\{j\}\(u\),e^j​\(u\)\\displaystyle\\widehat\{e\}\_\{j\}\(u\):=ℓ^j​\(u\)−ℓ^⋆​\(u\),\\displaystyle:=\\widehat\{\\ell\}\_\{j\}\(u\)\-\\widehat\{\\ell\}^\{\\star\}\(u\),Γ^​\(W\)\\displaystyle\\widehat\{\\Gamma\}\(W\):=minq∈Δm⁡max⁡∑j=1mu∈W⁡qj​e^j​\(u\)\.\\displaystyle:=\\min\_\{q\\in\\Delta\_\{m\}\}\\max\_\{u\\in W\}\\sum\_\{j=1\}^\{m\}q\_\{j\}\\widehat\{e\}\_\{j\}\(u\)\.For a loss error radiusrℓ≥0r\_\{\\ell\}\\geq 0, define the simultaneous event

𝒜ℓ:=\{maxu∈𝒰aud,j∈\[m\]\|ℓ^j\(u\)−ℓj\(u\)\|≤rℓ\}\.\\mathcal\{A\}\_\{\\ell\}:=\\left\\\{\\max\_\{u\\in\\mathcal\{U\}\_\{\\mathrm\{aud\}\},\\ j\\in\[m\]\}\|\\widehat\{\\ell\}\_\{j\}\(u\)\-\\ell\_\{j\}\(u\)\|\\leq r\_\{\\ell\}\\right\\\}\.Assumeℙ⁡\(𝒜ℓ\)≥1−α\\mathbb\{P\}\(\\mathcal\{A\}\_\{\\ell\}\)\\geq 1\-\\alpha, where0<α<10<\\alpha<1\. On𝒜ℓ\\mathcal\{A\}\_\{\\ell\}, the oracle minimum also changes by at mostrℓr\_\{\\ell\}, so regrets change by at most2​rℓ2r\_\{\\ell\}\. Consequently,

\|Γ^​\(W\)−Γ⁡\(W\)\|≤2​rℓfor every nonempty​W⊆𝒰aud\|\\widehat\{\\Gamma\}\(W\)\-\\Gamma\(W\)\|\\leq 2r\_\{\\ell\}\\quad\\text\{for every nonempty \}W\\subseteq\\mathcal\{U\}\_\{\\mathrm\{aud\}\}\(11\)holds on the same event\. The uniform bound covers all candidate groups without an additional union bound over the groups searched\.

#### Declaring the continuation class being certified\.

For the declared library, writeΓΠ​\(W\):=Γ​\(W\)\\Gamma\_\{\\Pi\}\(W\):=\\Gamma\(W\)\. If a conclusion is instead intended for a larger classΠtarget⊇Π\\Pi\_\{\\mathrm\{target\}\}\\supseteq\\Pi, the target margin needs a separate definition\. Letℓ⁡\(u,π\)∈\[0,1\]\\ell\(u,\\pi\)\\in\[0,1\]extend the same expected loss to the target class, withℓ⁡\(u,πj\)=ℓj​\(u\)\\ell\(u,\\pi\_\{j\}\)=\\ell\_\{j\}\(u\)for every declared continuation\. Let𝒬target\\mathcal\{Q\}\_\{\\mathrm\{target\}\}be all finitely supported probability distributions over the target class\. Define

ℓtarget⋆​\(u\)\\displaystyle\\ell\_\{\\mathrm\{target\}\}^\{\\star\}\(u\):=infπ∈Πtargetℓ⁡\(u,π\),\\displaystyle:=\\inf\_\{\\pi\\in\\Pi\_\{\\mathrm\{target\}\}\}\\ell\(u,\\pi\),Γtarget​\(W\)\\displaystyle\\Gamma\_\{\\mathrm\{target\}\}\(W\):=infQ∈𝒬targetmaxu∈W⁡\(𝔼π∼Q​ℓ​\(u,π\)−ℓtarget⋆​\(u\)\)\.\\displaystyle:=\\inf\_\{Q\\in\\mathcal\{Q\}\_\{\\mathrm\{target\}\}\}\\max\_\{u\\in W\}\\left\(\\mathbb\{E\}\_\{\\pi\\sim Q\}\\ell\(u,\\pi\)\-\\ell\_\{\\mathrm\{target\}\}^\{\\star\}\(u\)\\right\)\.For such a larger target, assume a justified fixed nonnegative library\-error boundηΠ​\(W\)\\eta\_\{\\Pi\}\(W\)satisfying

Γtarget​\(W\)≥ΓΠ​\(W\)−ηΠ​\(W\)for every nonempty​W⊆𝒰aud\.\\Gamma\_\{\\mathrm\{target\}\}\(W\)\\geq\\Gamma\_\{\\Pi\}\(W\)\-\\eta\_\{\\Pi\}\(W\)\\quad\\text\{for every nonempty \}W\\subseteq\\mathcal\{U\}\_\{\\mathrm\{aud\}\}\.\(12\)For the declared\-library target, setΓtarget=ΓΠ\\Gamma\_\{\\mathrm\{target\}\}=\\Gamma\_\{\\Pi\}andηΠ=0\\eta\_\{\\Pi\}=0\. Repeated scoring does not supplyηΠ\\eta\_\{\\Pi\}, the error bound for library approximation\. A sufficient condition for the bound is given below in Equation \([13](https://arxiv.org/html/2610.27446#A3.E13)\)\.

###### Theorem C\.3\(Search\-valid finite\-witness certificate\)\.

Supposeℙ⁡\(𝒜ℓ\)≥1−α\\mathbb\{P\}\(\\mathcal\{A\}\_\{\\ell\}\)\\geq 1\-\\alphaand the library\-error bound in Equation \([12](https://arxiv.org/html/2610.27446#A3.E12)\) holds\. LetW^\\widehat\{W\}be any nonempty group selected from the audit domain using the estimated losses\. With probability at least1−α1\-\\alpha,

Γtarget​\(W^\)≥Γ^​\(W^\)−2​rℓ−ηΠ​\(W^\)\.\\Gamma\_\{\\mathrm\{target\}\}\(\\widehat\{W\}\)\\geq\\widehat\{\\Gamma\}\(\\widehat\{W\}\)\-2r\_\{\\ell\}\-\\eta\_\{\\Pi\}\(\\widehat\{W\}\)\.If the right\-hand side exceedsε\\varepsilon, the selected group isε\\varepsilon\-incompatible for the specified target class\. For the declared\-library target, if the complete interfaceffis constant onW^\\widehat\{W\}, then also

maxλ∈Δ⁡\(W^\)⁡𝔇σinfo​\(Kf,λ\)≥Γ^​\(W^\)−2​rℓ\.\\max\_\{\\lambda\\in\\Delta\(\\widehat\{W\}\)\}\\mathfrak\{D\}^\{\\mathrm\{info\}\}\_\{\\sigma\}\(K\_\{f\};\\lambda\)\\geq\\widehat\{\\Gamma\}\(\\widehat\{W\}\)\-2r\_\{\\ell\}\.

The probability guarantee survives data\-dependent witness selection because Equation \([11](https://arxiv.org/html/2610.27446#A3.E11)\) holds for every group at once\. If only a fixed\-witness interval is available, selecting a witness and then applying that interval to the same data is not justified\. Independent certification data provide one valid alternative: freeze the selected group before estimating its certification losses\.

#### Incompatibility is not the same as minimality\.

A lower margin bound for a group proves only that the group cannot share an acceptable selection\. To certify a*minimal*conflict, all proper subgroups must be compatible\. By monotonicity ofΓ\\Gamma, it suffices to check every one\-state deletion\.

###### Corollary C\.4\(Certifying minimality in the declared library\)\.

On the event𝒜ℓ\\mathcal\{A\}\_\{\\ell\}, keep the declared library as the target\. A groupWWwith\|W\|≥2\|W\|\\geq 2is a minimalε\\varepsilon\-incompatible group if

Γ^​\(W\)−2​rℓ\\displaystyle\\widehat\{\\Gamma\}\(W\)\-2r\_\{\\ell\}\>ε,\\displaystyle\>\\varepsilon,Γ^​\(W∖\{u\}\)\+2​rℓ\\displaystyle\\widehat\{\\Gamma\}\(W\\setminus\\\{u\\\}\)\+2r\_\{\\ell\}≤ε\\displaystyle\\leq\\varepsilonfor every​u∈W\.\\displaystyle\\text\{for every \}u\\in W\.The conclusion also holds for a data\-selected groupWWunder the same simultaneous event\.

For a three\-state witness, the first inequality certifies failure of the triple and the three upper bounds certify success of every pair\. A positive triple lower bound alone cannot establish a pairwise\-pass/triple\-fail result\.

#### A rollout\-based simultaneous radius\.

For each of then​mnmstate–continuation pairs, letYu​j​k∈\[0,1\]Y\_\{ujk\}\\in\[0,1\],k=1,…,Nk=1,\\ldots,N, beN≥1N\\geq 1independent scores with common meanνj​\(u\)\\nu\_\{j\}\(u\)under a fixed scoring protocol\. Setℓ^j​\(u\):=N−1​∑k=1NYu​j​k\\widehat\{\\ell\}\_\{j\}\(u\):=N^\{\-1\}\\sum\_\{k=1\}^\{N\}Y\_\{ujk\}\. Suppose a known deterministic judge\-bias boundbjudge≥0b\_\{\\mathrm\{judge\}\}\\geq 0satisfies\|νj​\(u\)−ℓj​\(u\)\|≤bjudge\|\\nu\_\{j\}\(u\)\-\\ell\_\{j\}\(u\)\|\\leq b\_\{\\mathrm\{judge\}\}for every audited pair\. Independence between different pairs is not required\.

###### Corollary C\.5\(One simple rollout radius\)\.

Under the stated bounded\-score, within\-pair independence, and uniform\-bias conditions, a simultaneous loss radius with error probability at mostα∈\(0,1\)\\alpha\\in\(0,1\)is

rℓ=bjudge\+log⁡\(2​n​m/α\)2​N\.r\_\{\\ell\}=b\_\{\\mathrm\{judge\}\}\+\\sqrt\{\\frac\{\\log\(2nm/\\alpha\)\}\{2N\}\}\.

The square\-root term controls score sampling through Hoeffding’s inequality and a union bound over then​mnmpairs\([Hoeffding, 1963](https://arxiv.org/html/2610.27446#bib.bib70)\);bjudgeb\_\{\\mathrm\{judge\}\}controls systematic error relative to the target loss\. If the bias bound or library approximation holds only on a separate probability event, its failure probability must also enter the guarantee\. The human\-study packet maximum does not establish a uniform judge\-bias bound\. Algorithm[7](https://arxiv.org/html/2610.27446#alg7)keeps incompatibility certification and minimality certification separate\.

### C\.4Proofs of the statistical certification results

#### A sufficient condition for the library approximation bound\.

The inequality in Equation \([12](https://arxiv.org/html/2610.27446#A3.E12)\) is an assumption for transfer to a larger continuation class, not a consequence of loss\-estimation accuracy\. Enlarging the library also changes the statewise oracle baseline, so library inclusion alone does not specify the size of the regret\-margin change\. One sufficient condition can be stated in terms of expected losses\. Fix a nonempty state groupWWand a numberη≥0\\eta\\geq 0\. Suppose every target mixtureQ∈𝒬targetQ\\in\\mathcal\{Q\}\_\{\\mathrm\{target\}\}has a declared\-library selectionq∈Δmq\\in\\Delta\_\{m\}with

∑j=1mqj​ℓj​\(u\)≤𝔼π∼Q​ℓ​\(u,π\)\+ηfor every​u∈W\.\\sum\_\{j=1\}^\{m\}q\_\{j\}\\ell\_\{j\}\(u\)\\leq\\mathbb\{E\}\_\{\\pi\\sim Q\}\\ell\(u,\\pi\)\+\\eta\\quad\\text\{for every \}u\\in W\.\(13\)The selectionqqmay depend onQQandWW, but must work for all states inWWsimultaneously\. Because the declared library is contained in the target class,ℓ⋆​\(u\)≥ℓtarget⋆​\(u\)\\ell^\{\\star\}\(u\)\\geq\\ell\_\{\\mathrm\{target\}\}^\{\\star\}\(u\)\. Hence a covering selectionqqsatisfies

e⁡\(u,q\)\\displaystyle e\(u,q\)=∑jqj​ℓj​\(u\)−ℓ⋆​\(u\)\\displaystyle=\\sum\_\{j\}q\_\{j\}\\ell\_\{j\}\(u\)\-\\ell^\{\\star\}\(u\)≤𝔼π∼Q​ℓ​\(u,π\)−ℓtarget⋆​\(u\)\+η\.\\displaystyle\\leq\\mathbb\{E\}\_\{\\pi\\sim Q\}\\ell\(u,\\pi\)\-\\ell\_\{\\mathrm\{target\}\}^\{\\star\}\(u\)\+\\eta\.Taking the maximum over states, then the infimum over target mixtures, givesΓΠ​\(W\)≤Γtarget​\(W\)\+η\\Gamma\_\{\\Pi\}\(W\)\\leq\\Gamma\_\{\\mathrm\{target\}\}\(W\)\+\\eta\. Thus Equation \([13](https://arxiv.org/html/2610.27446#A3.E13)\) justifies usingηΠ​\(W\)=η\\eta\_\{\\Pi\}\(W\)=\\eta\. The infimum argument does not require an optimal target mixture to exist\. Uniform validity across searched groups requires the coverage condition for every candidate group\. No such coverage condition is inferred merely from the number of scored rollouts\.

###### Proof of Theorem[C\.3](https://arxiv.org/html/2610.27446#A3.Thmassumption3)\.

On𝒜ℓ\\mathcal\{A\}\_\{\\ell\}, the minimum loss in each row changes by at mostrℓr\_\{\\ell\}\. Each regret therefore changes by at most2​rℓ2r\_\{\\ell\}\. Convex averaging, maximization over states, and minimization over choices preserve the uniform bound, giving\|Γ^​\(W\)−ΓΠ​\(W\)\|≤2​rℓ\|\\widehat\{\\Gamma\}\(W\)\-\\Gamma\_\{\\Pi\}\(W\)\|\\leq 2r\_\{\\ell\}for every nonemptyWW\. The library assumption then yields

Γtarget​\(W\)≥ΓΠ​\(W\)−ηΠ​\(W\)≥Γ^​\(W\)−2​rℓ−ηΠ​\(W\)\.\\Gamma\_\{\\rm target\}\(W\)\\geq\\Gamma\_\{\\Pi\}\(W\)\-\\eta\_\{\\Pi\}\(W\)\\geq\\widehat\{\\Gamma\}\(W\)\-2r\_\{\\ell\}\-\\eta\_\{\\Pi\}\(W\)\.Since the same event covers every group, the inequality holds for the selected groupW^\\widehat\{W\}with probability at least1−α1\-\\alpha\. A lower bound aboveε\\varepsilonexcludes every target choice with regret at mostε\\varepsilonthroughout the group\. For the declared library and a constant observation onW^\\widehat\{W\}, Theorem[3\.5](https://arxiv.org/html/2610.27446#S3.Thmassumption5)gives the stated maximum risk gap over state distributions\. ∎

###### Proof of Corollary[C\.4](https://arxiv.org/html/2610.27446#A3.Thmassumption4)\.

On𝒜ℓ\\mathcal\{A\}\_\{\\ell\}, the full\-group lower bound givesΓ⁡\(W\)\>ε\\Gamma\(W\)\>\\varepsilon, while each deletion upper bound givesΓ⁡\(W∖\{u\}\)≤ε\\Gamma\(W\\setminus\\\{u\\\}\)\\leq\\varepsilon\. Every proper nonempty subset lies in a deletion and has no larger margin\. The empty group is compatible by definition\. ThusWWis a minimal conflict\. Simultaneous coverage permits choosingWWusing the same estimates\. ∎

###### Proof of Corollary[C\.5](https://arxiv.org/html/2610.27446#A3.Thmassumption5)\.

For each pair\(u,j\)\(u,j\), Hoeffding’s inequality gives

ℙ⁡\(\|ℓ^j​\(u\)−νj​\(u\)\|\>a\)≤2​e−2​N​a2\.\\mathbb\{P\}\\bigl\(\|\\widehat\{\\ell\}\_\{j\}\(u\)\-\\nu\_\{j\}\(u\)\|\>a\\bigr\)\\leq 2e^\{\-2Na^\{2\}\}\.Takinga=log⁡\(2​n​m/α\)/\(2​N\)a=\\sqrt\{\\log\(2nm/\\alpha\)/\(2N\)\}and a union bound over then​mnmpairs bounds all sampling errors byaawith probability at least1−α1\-\\alpha\. Adding\|νj​\(u\)−ℓj​\(u\)\|≤bjudge\|\\nu\_\{j\}\(u\)\-\\ell\_\{j\}\(u\)\|\\leq b\_\{\\rm judge\}proves the radius\. Independence is needed within each pair, not between pairs\. ∎

## Appendix DAlgorithms, Complexity, and Failure Modes

The procedures below specify their inputs, search domains, and stopping conditions\. Mathematical claims assume exact optimization; numerical implementations must also report solver status and tolerance\.

Algorithm 1Exact finite\-interface diagnosis1:State weights

\{μi\}i=1n\\\{\\mu\_\{i\}\\\}\_\{i=1\}^\{n\}, channel matrix

P∈\[0,1\]n×LP\\in\[0,1\]^\{n\\times L\}, loss matrix

𝐋=\(Li​j\)∈\[0,1\]n×m\\mathbf\{L\}=\(L\_\{ij\}\)\\in\[0,1\]^\{n\\times m\}, with

n,L≥1n,L\\geq 1and

m≥2m\\geq 2\.

2:

RfullR\_\{\\mathrm\{full\}\},

RZ⋆R\_\{Z\}^\{\\star\}, an optimal unrestricted rule

δZ⋆\\delta\_\{Z\}^\{\\star\}, and the interface information gap\.

3:Check

μi≥0\\mu\_\{i\}\\geq 0,

∑iμi=1\\sum\_\{i\}\\mu\_\{i\}=1, and

∑rPi​r=1\\sum\_\{r\}P\_\{ir\}=1for each

ii\.

4:Compute

Rfull←∑i=1nμi​minj∈\[m\]​Li​jR\_\{\\mathrm\{full\}\}\\leftarrow\\sum\_\{i=1\}^\{n\}\\mu\_\{i\}\\min\_\{j\\in\[m\]\}L\_\{ij\}\.

5:foreach interface output

r∈\[L\]r\\in\[L\]do

6:foreach continuation

j∈\[m\]j\\in\[m\]do

7:

Cr​j←∑i=1nμi​Pi​r​Li​jC\_\{rj\}\\leftarrow\\sum\_\{i=1\}^\{n\}\\mu\_\{i\}P\_\{ir\}L\_\{ij\}\.

8:endfor

9:Choose

jr⋆∈arg​minj∈\[m\]⁡Crjj\_\{r\}^\{\\star\}\\in\\argmin\_\{j\\in\[m\]\}C\_\{rj\}\.

10:Set

δZ,j⋆\(zr\)←𝟏\{j=jr⋆\}\\delta\_\{Z,j\}^\{\\star\}\(z\_\{r\}\)\\leftarrow\\mathbf\{1\}\\\{j=j\_\{r\}^\{\\star\}\\\}for every

j∈\[m\]j\\in\[m\]\.

11:endfor

12:

RZ⋆←∑r=1LCr​jr⋆R\_\{Z\}^\{\\star\}\\leftarrow\\sum\_\{r=1\}^\{L\}C\_\{rj\_\{r\}^\{\\star\}\}\.

13:

Δinfo←RZ⋆−Rfull\\Delta\_\{\\mathrm\{info\}\}\\leftarrow R\_\{Z\}^\{\\star\}\-R\_\{\\mathrm\{full\}\}\.

14:return

\(Rfull,RZ⋆,δZ⋆,Δinfo\)\(R\_\{\\mathrm\{full\}\},R\_\{Z\}^\{\\star\},\\delta\_\{Z\}^\{\\star\},\\Delta\_\{\\mathrm\{info\}\}\)\.

Algorithm 2Restricted\-rule and finite\-training audit1:Evaluation joint costs

Cr​jC\_\{rj\}and training joint costs

C^r​jtr\\widehat\{C\}^\{\\mathrm\{tr\}\}\_\{rj\}on the same

LLobservation codes and

mmchoices; visible features

\{φr\}r=1L\\\{\\varphi\_\{r\}\\\}\_\{r=1\}^\{L\}and finite parameter bound

Bpar≥1/mB\_\{\\mathrm\{par\}\}\\geq 1/m\. Costs include observation probabilities\.

2:

R𝒟R\_\{\\mathcal\{D\}\},

Rδ^R\_\{\\widehat\{\\delta\}\}, accessibility gap, and fitting gap\.

3:Compute

RZ⋆←∑r=1Lminj∈\[m\]⁡Cr​jR\_\{Z\}^\{\\star\}\\leftarrow\\sum\_\{r=1\}^\{L\}\\min\_\{j\\in\[m\]\}C\_\{rj\}\.

4:Solve Proposition[A\.5](https://arxiv.org/html/2610.27446#A1.Thmassumption5)with

CCto obtain

q𝒟⋆q^\{\\star\}\_\{\\mathcal\{D\}\}and

R𝒟R\_\{\\mathcal\{D\}\}\.

5:Solve the same linear program with

C^tr\\widehat\{C\}^\{\\mathrm\{tr\}\}to obtain the learned rule

q^\\widehat\{q\}, keeping all simplex constraints unchanged\.

6:Evaluate

q^\\widehat\{q\}on

CCto obtain

Rδ^R\_\{\\widehat\{\\delta\}\}\.

7:

Δaccess←R𝒟−RZ⋆\\Delta\_\{\\mathrm\{access\}\}\\leftarrow R\_\{\\mathcal\{D\}\}\-R\_\{Z\}^\{\\star\}\.

8:

Δtrain←Rδ^−R𝒟\\Delta\_\{\\mathrm\{train\}\}\\leftarrow R\_\{\\widehat\{\\delta\}\}\-R\_\{\\mathcal\{D\}\}\.

9:return

\(R𝒟,Rδ^,Δaccess,Δtrain\)\(R\_\{\\mathcal\{D\}\},R\_\{\\widehat\{\\delta\}\},\\Delta\_\{\\mathrm\{access\}\},\\Delta\_\{\\mathrm\{train\}\}\)\.

Algorithm 3Continuous\-interface certificate with loss uncertainty1:

n≥1n\\geq 1independent identically distributed audit pairs

\(Ut,Zt\)\(U\_\{t\},Z\_\{t\}\); a fixed library of

m≥2m\\geq 2choices; simultaneous loss intervals

\[L¯t​j,L¯t​j\]⊆\[0,1\]\[\\underline\{L\}\_\{tj\},\\overline\{L\}\_\{tj\}\]\\subseteq\[0,1\]with failure probability

αℓ\\alpha\_\{\\ell\}; a partition

QhQ\_\{h\}with

L≥1L\\geq 1cells and a valid modulus

ω\\omega, both fixed independently of the sample; error budget

0<αs<10<\\alpha\_\{s\}<1with

0≤αℓ<1−αs0\\leq\\alpha\_\{\\ell\}<1\-\\alpha\_\{s\}\.

2:A lower bound valid with probability at least

1−αs−αℓ1\-\\alpha\_\{s\}\-\\alpha\_\{\\ell\}\.

3:Reject the certificate request if interval coverage, the modulus, or partition independence is not justified\.

4:Compute

C^r​j−\\widehat\{C\}^\{\-\}\_\{rj\}and

R^full\+\\widehat\{R\}^\{\+\}\_\{\\mathrm\{full\}\}from Theorem[L\.5](https://arxiv.org/html/2610.27446#A12.Thmassumption5)\.

5:

rn←log⁡\(4​L​m/αs\)/\(2​n\)r\_\{n\}\\leftarrow\\sqrt\{\\log\(4Lm/\\alpha\_\{s\}\)/\(2n\)\};

sn←log⁡\(4/αs\)/\(2​n\)s\_\{n\}\\leftarrow\\sqrt\{\\log\(4/\\alpha\_\{s\}\)/\(2n\)\}\.

6:

LBQ←∑r=1Lmax⁡\{0,minj⁡C^r​j−−rn\}\\mathrm\{LB\}\_\{Q\}\\leftarrow\\sum\_\{r=1\}^\{L\}\\max\\\{0,\\min\_\{j\}\\widehat\{C\}^\{\-\}\_\{rj\}\-r\_\{n\}\\\}\.

7:

UBfull←min⁡\{1,R^full\+\+sn\}\\mathrm\{UB\}\_\{\\mathrm\{full\}\}\\leftarrow\\min\\\{1,\\widehat\{R\}^\{\+\}\_\{\\mathrm\{full\}\}\+s\_\{n\}\\\}\.

8:return

max⁡\{0,LBQ−UBfull−2​ω​\(h\)\}\\max\\\{0,\\mathrm\{LB\}\_\{Q\}\-\\mathrm\{UB\}\_\{\\mathrm\{full\}\}\-2\\omega\(h\)\\\}\.

Algorithm 4Minimal conflict extraction with an explicit search scope1:Finite regret matrix

𝐄\\mathbf\{E\}, tolerance

ε≥0\\varepsilon\\geq 0, continuation count

mm, and a family of disjoint search groups

𝒞\\mathcal\{C\}\. Use current observation cells for local diagnosis; use

𝒞=\{𝒰aud\}\\mathcal\{C\}=\\\{\\mathcal\{U\}\_\{\\mathrm\{aud\}\}\\\}for the complete global conflict system\.

2:All minimal conflicts contained in at least one declared search group\.

3:Initialize

ℰout←∅\\mathcal\{E\}\_\{\\mathrm\{out\}\}\\leftarrow\\varnothing\.

4:foreach group

C∈𝒞C\\in\\mathcal\{C\}do

5:for

s=2,…,min⁡\{m,\|C\|\}s=2,\\ldots,\\min\\\{m,\|C\|\\\}do

6:foreach subset

W⊆CW\\subseteq Cwith

\|W\|=s\|W\|=sdo

7:ifno recorded edge is contained in

WWthen

8:Solve the margin LP for

Γ⁡\(W\)\\Gamma\(W\)and its dual state weights\.

9:if

Γ⁡\(W\)\>ε\\Gamma\(W\)\>\\varepsilonthen

10:Add

WWand its margin and dual weights to

ℰout\\mathcal\{E\}\_\{\\mathrm\{out\}\}\.

11:endif

12:endif

13:endfor

14:endfor

15:endfor

16:return

ℰout\\mathcal\{E\}\_\{\\mathrm\{out\}\}and the declared search scope\.

Algorithm 5Coloring MILP for a declared global constraint subset1:Audit states

u1,…,unu\_\{1\},\\ldots,u\_\{n\}and retained valid conflicts

ℱt⊆ℰε\\mathcal\{F\}\_\{t\}\\subseteq\\mathcal\{E\}\_\{\\varepsilon\}\.

2:The relaxed optimum

MtM\_\{t\}and labeling

gtg\_\{t\}if optimality is proved; otherwise a lower bound, any feasible incumbent, and solver status\. An optimum is the global capacity only with complete constraints or successful validation of every candidate cell\.

3:Create binary variables

xi​cx\_\{ic\}and

ycy\_\{c\}for

i,c∈\[n\]i,c\\in\[n\]\.

4:Minimize

∑c=1nyc\\sum\_\{c=1\}^\{n\}y\_\{c\}\.

5:Add

∑cxi​c=1\\sum\_\{c\}x\_\{ic\}=1for every state

iiand

xi​c≤ycx\_\{ic\}\\leq y\_\{c\}for every

\(i,c\)\(i,c\)\.

6:foreach retained conflict

W∈ℱtW\\in\\mathcal\{F\}\_\{t\}and label

c∈\[n\]c\\in\[n\]do

7:Add

∑ui∈Wxi​c≤\|W\|−1\\sum\_\{u\_\{i\}\\in W\}x\_\{ic\}\\leq\|W\|\-1\.

8:endfor

9:Add

yc≥yc\+1y\_\{c\}\\geq y\_\{c\+1\}for

c<nc<n\.

10:Solve the MILP and record its status, lower bound, and any feasible incumbent\.

11:ifoptimality is provedthen

12:return

\(Mt,gt,optimal\)\(M\_\{t\},g\_\{t\},\\text\{optimal\}\)\.

13:else

14:returnthe lower bound, any incumbent, and the stopping status; do not report exact capacity\.

15:endif

Algorithm 6Global capacity by validation and conflict generation1:Full finite regret table, tolerance

ε\\varepsilon, and any valid initial edge subset

ℱ0\\mathcal\{F\}\_\{0\}\(possibly empty or obtained from local diagnosis\)\.

2:Global capacity and a valid labeling under exact solves/comparisons\.

3:Set

t←0t\\leftarrow 0\.

4:loop

5:Call Algorithm[5](https://arxiv.org/html/2610.27446#alg5)with

ℱt\\mathcal\{F\}\_\{t\}\.

6:ifthe relaxed optimum is not provedthen

7:Validate any incumbent to obtain an upper bound if possible\.

8:returnavailable lower and upper bounds and solver status; global optimality is unresolved\.

9:endif

10:Let

\(Mt,gt\)\(M\_\{t\},g\_\{t\}\)be the proved relaxed optimum and its labeling\.

11:Compute

Γ⁡\(C\)\\Gamma\(C\)for every nonempty new cell

C=gt−1​\(c\)C=g\_\{t\}^\{\-1\}\(c\)\.

12:ifany margin comparison is unresolved at the numerical tolerancethen

13:returnavailable bounds and the unresolved cell; do not claim exact capacity\.

14:endif

15:ifevery new cell satisfies

Γ⁡\(C\)≤ε\\Gamma\(C\)\\leq\\varepsilonthen

16:return

\(𝖬⋆​\(ε\)=Mt,gt\)\(\\mathsf\{M\}^\{\\star\}\(\\varepsilon\)=M\_\{t\},g\_\{t\}\)with matching lower and upper bounds\.

17:endif

18:For each violating cell, repeatedly remove a state while the remaining group still has margin greater than

ε\\varepsilon\.

19:Add each resulting minimal conflict to obtain

ℱt\+1\\mathcal\{F\}\_\{t\+1\}; set

t←t\+1t\\leftarrow t\+1\.

20:endloop

Algorithm 7Search and certify a higher\-order conflict1:Search loss estimates on a finite audit domain and declared library; complete deterministic observation cells; tolerance

ε≥0\\varepsilon\\geq 0; error budget

α∈\(0,1\)\\alpha\\in\(0,1\); a specified target class and justified library bound

ηΠ\\eta\_\{\\Pi\}; either a valid domain\-wide loss bound or independent certification measurements\.

2:Incompatibility status for the target and, when justified, minimality status for the declared library\.

3:Run Algorithm[4](https://arxiv.org/html/2610.27446#alg4)with the search estimates and the declared local cells to obtain candidate groups\.

4:ifno candidate is returnedthen

5:return“no candidate selected\.”

6:endif

7:Select a nonempty candidate

W^\\widehat\{W\}by the declared search criterion\.

8:ifa valid simultaneous loss bound covers the entire searched matrixthen

9:Use that estimated matrix and its domain\-wide radius

rℓr\_\{\\ell\}\.

10:else

11:Freeze

W^\\widehat\{W\}before collecting independent certification data\.

12:Estimate losses on

W^×Π\\widehat\{W\}\\times\\Piand construct a valid conditional simultaneous radius

rℓr\_\{\\ell\}with error probability at most

α\\alpha\.

13:endif

14:Compute

Γ^​\(W^\)\\widehat\{\\Gamma\}\(\\widehat\{W\}\)from the chosen certification loss matrix\.

15:

LCBW←Γ^​\(W^\)−2​rℓ−ηΠ​\(W^\)\\mathrm\{LCB\}\_\{W\}\\leftarrow\\widehat\{\\Gamma\}\(\\widehat\{W\}\)\-2r\_\{\\ell\}\-\\eta\_\{\\Pi\}\(\\widehat\{W\}\)\.

16:Initialize incompatibility and minimality statuses to “not certified\.”

17:if

LCBW\>ε\\mathrm\{LCB\}\_\{W\}\>\\varepsilonthen

18:Mark incompatibility for the specified target as certified\.

19:ifthe target is the declared library and every single deletion upper bound in Corollary[C\.4](https://arxiv.org/html/2610.27446#A3.Thmassumption4)is at most

ε\\varepsilonthen

20:Mark declared\-library minimality as certified\.

21:endif

22:endif

23:return

\(W^,LCBW,incompatibility status,minimality status\)\(\\widehat\{W\},\\mathrm\{LCB\}\_\{W\},\\text\{incompatibility status\},\\text\{minimality status\}\)\.

Algorithm 8Decision\-sufficiency audit with separated diagnostic and design branches1:Full audit states, complete observation channel

VV\(including any policy side information\), finite continuation library, loss estimates and valid error bounds when available, restricted rule class, and tolerance\.

2:Finite\-table risks, local conflicts, and only those population or capacity certificates whose assumptions have been checked\.

3:Freeze the observation contract, continuation library, loss definition, and audit scope\.

4:ifthe complete observation is finitethen

5:Compute finite\-table full\-state and optimal interface risks by Algorithm[1](https://arxiv.org/html/2610.27446#alg1)\.

6:Label plug\-in results as finite\-table results unless loss estimation error is controlled\.

7:else

8:Apply Algorithm[3](https://arxiv.org/html/2610.27446#alg3)only with valid loss intervals and modulus; otherwise report that this procedure provides no population bound\.

9:endif

10:Audit the restricted and fitted rules over the same feasible observation domain\.

11:ifthe complete observation defines deterministic finite cellsthen

12:Run Algorithm[4](https://arxiv.org/html/2610.27446#alg4)on those cells for local diagnosis at the declared tolerance\.

13:For unrestricted code\-only design, run Algorithm[6](https://arxiv.org/html/2610.27446#alg6)on the full audit domain, or enumerate the full global hypergraph first\.

14:With free finite policy side information, solve the global problem separately within each policy and use Corollary[C\.2](https://arxiv.org/html/2610.27446#A3.Thmassumption2)\.

15:Certify selected local conflicts using Algorithm[7](https://arxiv.org/html/2610.27446#alg7)only when the required loss\-error event is valid\.

16:else

17:Do not apply deterministic coloring to posterior supports of a stochastic channel without a separate justified construction\.

18:endif

19:Return risks, uncertainty assumptions, search scope, solver status, and claim boundaries\.

### D\.1Computational cost and assumptions

#### Reference implementation and numerical decisions\.

The CPU implementation retained in the author working archive uses the at\-most\-mmfeasible label bound from Proposition[C\.1](https://arxiv.org/html/2610.27446#A3.Thmassumption1)\. It records solver optimality, new\-cell margins, added conflicts, and an explicit numerical tolerance\. The theorems assume exact arithmetic; numerical LP/MILP outputs must be interpreted with the recorded tolerance, and cases at a decision boundary must not be presented as proofs in exact arithmetic\.

## Appendix EComplete Empirical Claim Ledger

Table[6](https://arxiv.org/html/2610.27446#A5.T6)links each claim to its evidence and limits\. The historical rows use the SAE code alone; later rows distinguish observations that also include policy\. Numerical recovery from rounded Qwen summaries is not treated as a recalculation from the original arrays\.

Table 6:Evidence for each claim for the complete audit\.IDCandidate claimStatusEvidence boundaryC1A frozen low\-budget finite SAE interface has positive information loss on untouched Qwen2\.5 policy areas\.SupportedLayer 17,B=2B=2: gap0\.1560\.156with one\-sided 95% lower bound0\.1290\.129\.C2A restricted rule class adds positive accessibility loss after the interface is fixed\.SupportedLayer 19,B=16B=16: gap0\.0470\.047with one\-sided lower bound0\.0310\.031; nested classes drive the gap to zero\.C3The primary frozen\-interface conclusion is stable across two automatic outcome evaluators\.SupportedQwen2\.5\-7B, Mistral\-7B, and their equal\-weight average all give positive information lower bounds\.C4The low\-budget information result appears at a second Qwen2\.5 model scale with a separate public SAE release\.SupportedQwen2\.5\-0\.5B layer 17,B=2B=2: gap0\.2100\.210with one\-sided lower bound0\.1920\.192\.C5Positive information loss is not tied to the supervised feature selector\.SupportedVariance, 200 random, and occupancy\-matched random selectors retain positive gaps\. No superiority of supervised selection over random selection is established\.C6Pairwise compatibility can miss a group\-level conflict in real SAE support patterns\.Supported in the controlled benchmarkExact pair margin00, triple margin1/31/3, separation1/31/3\.C7A searched language example passes all pair checks but fails for the triple in new data\.Not replicatedIndependent and fresh semantic/generation/judge replications give zero positive separation\.C8The low\-budget result transfers to a non\-Qwen model family\.Supported on one non\-Qwen familySmolLM2\-1\.7B layer 17 gives positiveB=2B=2automatic gaps on the old confirmation set and the fresh locked set for both SAE seeds\.C9The SmolLM2 sign is stable across independently trained SAE seeds\.Supported for two SmolLM2 seedsTwo width\-16,38416\{,\}384,k=64k=64TopK SAEs trained on eight million activation tokens each pass QC and give positive low\-budget gaps\. The seeds share the model, layer, architecture, and cached corpus activations\.C10The sign transfers to a policy collection locked before model generation\.SupportedThe fresh 96\-state collection gives automatic gaps0\.0710\.071and0\.0770\.077atB=2B=2, with positive one\-sided lower bounds\.C11Blinded human outcome loss retains a positive low\-budget information gap\.SupportedFinal adjudicated SmolLM2ZZ\-onlyB=2B=2gaps are0\.0780\.078and0\.0990\.099on the old 80\-state sample and0\.1480\.148and0\.1330\.133on the fresh 96\-state set; all one\-sided lower bounds are positive\.C12Two human raters apply the ordinal rubric with measured agreement\.Supported with dimension\-specific variationWeightedκ\\kapparanges from0\.4720\.472to0\.8110\.811; composite human\-loss ICC\(2,1\) is0\.7460\.746with 95% interval\[0\.686,0\.798\]\[0\.686,0\.798\]\. One hundred triggered items are jointly resolved\.C13Automatic outcome loss is closely calibrated to final human judgments\.Not supportedSpearman correlation0\.3790\.379, MAE0\.3110\.311, fixed\-order action\-label agreement51\.1%51\.1\\%\(set intersection80\.7%80\.7\\%, set equality30\.7%30\.7\\%\), and strong action\-specific signed error\.C14A packet maximum error provides a uniform population judge\-error bound\.Not supportedThe observed maximum absolute error is0\.9000\.900, but the packet maximum is not a population\-level uniform certificate\.C15The combined studies prove the result for all SAEs or all model families\.Not supportedThe audit covers three language\-model families under different observation contracts, with two SAE training seeds in each of SmolLM2 and Phi\. Human evidence is SmolLM2\-only\.C16Phi has positive primary information loss when active policy is supplied\.Supported on the saved finite collectionsAtB=2B=2, Mistral gaps0\.03310\.0331–0\.05570\.0557; both seeds and both collections have positive individual LCBs\. Qwen\-loss and mean\-loss checks agree on the sign\.C17One policy area drives the primary Phi sign\.Not supported by the deletion checkAll 192 single area deletion point estimates across two observations and three loss constructions remain positive; minimum0\.02360\.0236\.C18The two Phi evaluators closely agree numerically\.Not supportedSpearman0\.35520\.3552, mean absolute difference0\.32200\.3220across 5,040 shared responses\.C19The earlier human study validates Phi responses\.Not supportedThe Phi human packets are blank\. No SmolLM2 human label is transferred to a Phi response\.C20A positive gap using code alone implies a positive gap with policy supplied\.Not validThe implication is false in general\. Separate paired reanalysis now supplies the evidence reported in C22 and C23\.C21Phi adds a previously unseen policy collection\.Not supportedThe original and locked collections are reused, although fixed before generating the new Phi responses\.C22SmolLM2’s policy\-aware information gap remains positive under automatic and human loss\.Supported on cached finite tablesFour automatic and four final humanB=2B=2gaps have positive individual bootstrap lower bounds\. Human gaps are0\.03450\.0345–0\.11660\.1166\.C23Primary Qwen policy\-aware gaps remain positive\.Supported with a rounding boundPer\-policy notebook summaries give0\.13695±0\.000050\.13695\\pm 0\.00005and0\.03875±0\.000050\.03875\\pm 0\.00005\. No full\-precision state\-array rerun or new 0\.5B policy\-aware result is claimed\.C24Final SmolLM2 human losses are reproducible from completed ratings\.RecomputedBoth 528\-row files and all 100 resolved rows match the packet\. Final losses agree with the earlier item\-level archive to numerical precision; inputs and the CPU script are retained in the author working archive\.C25Local conflicts alone determine global capacity\.Not valid in generalComplete global constraints or validation of every new cell are required\. Proposition[3\.7](https://arxiv.org/html/2610.27446#S3.Thmassumption7)and the cross\-cell regression test support the global constraint\-generation algorithm\.C26Unbiased single\-rollout scores suffice for the exact\-vector population certificate\.Not valid in generalTheorem[L\.5](https://arxiv.org/html/2610.27446#A12.Thmassumption5)uses simultaneous expected\-loss intervals\. The noisy\-minimum counterexample shows why unbiased single\-rollout scores alone do not establish the certificate\.
## Appendix FComplete SmolLM2 Results and Human Evaluation

This appendix contains the SmolLM2 results and human evaluation\. The author working archive retains 1,008 mean losses for state–action pairs, interface codes, 200 random feature selections per setting, training metadata and checkpoint hashes, the locked state collection, and the completed ratings for 528 responses\. The original independent ratings and the 100 joint resolutions are retained separately\.

### F\.1Saved configuration and provenance

The SmolLM2 run uses SmolLM2\-1\.7B\-Instruct revision31b70e2e869a7173562077fd711b654946d38674and Mistral\-7B\-Instruct\-v0\.3 revisionc170c708c41dac9275d15a8fff4eca08d52bab71\. Each state–action pair has five rollout replicas, maximum continuation length 192 tokens, temperature0\.70\.7, and top\-ppvalue0\.90\.9\. The evaluator output is limited to 160 new tokens and permits four parse retries\.

SAE training uses FineWeb\-Edusample\-10BTrevision87f09149ef4734204d70ed1d046ddc9ca3f2b8f9\. Corpus contexts have 128 tokens\. The activation cache uses 32,768\-token shards\. Each SAE is optimized with AdamW, learning rate2×10−42\\times 10^\{\-4\}, 500 warm\-up steps, zero weight decay, gradient\-norm clip1\.01\.0, auxiliary dead\-feature coefficient0\.031250\.03125, and a 1,000\-step dead\-feature window\. The fresh collection seed is 602214, the human packet seed is 8675309, the bootstrap seed is 99173, and the two SAE seeds are 314159 and 271828\.

The run used Python 3\.13\.15, PyTorch 2\.11\.0 with CUDA 12\.8, Transformers 4\.57\.6, Datasets 5\.0\.1, Accelerate 1\.14\.0, NumPy 2\.1\.3, Pandas 2\.2\.3, SciPy 1\.16\.3, and an NVIDIA A100\-SXM4\-40GB GPU\. The saved run manifest, software manifest, configuration JSON, source code, checkpoint hashes, and all state\-level outputs are retained in the author working archive\.

Table 7:SmolLM2 audit and SAE training settings\.
### F\.2SmolLM2 results using the code alone and reconstruction diagnostics

The historical SmolLM2 tables in this subsection use cells defined by the SAE code alone\. Section[5\.3](https://arxiv.org/html/2610.27446#S5.SS3)and Appendix[J](https://arxiv.org/html/2610.27446#A10)separately recompute policy\-aware risk from the same frozen codes and losses\.

#### Independent SAE quality control\.

Both TopK SAEs use the same frozen SmolLM2 activation source and architecture but different random initialization and seed\-dependent training order\. The two runs are therefore independent training seeds, not independent training corpora\. Both consume eight million activation tokens and pass the prespecified validation checks\. Validation FVU is0\.00890\.0089and0\.00880\.0088, mean reconstruction cosine is0\.9220\.922for both seeds, meanL0L\_\{0\}is exactly6464, all16,38416\{,\}384features are active on validation, and the observed dead\-feature fraction is zero\.

The audit\-domain centered FVU is much larger:35\.2235\.22and35\.8835\.88on the old audit states, and7\.777\.77and8\.128\.12on the fresh states\. Centered FVU can exceed one because its numerator is reconstruction squared error while its denominator is centered activation variance\. The large values are consistent with a difference in means, scales, or variances between corpus and audit activations, but centered FVU alone does not identify which component causes the increase; the values must not be read as percentages\. Reconstruction cosine on the audit states remains between0\.8770\.877and0\.8830\.883\. These reconstruction measurements do not determine the optimal decision loss, which is computed separately below\.

Table 8:Quality\-control diagnostics for the two independently initialized and trained TopK SAEs\. Audit\-domain centered FVU is reported without clipping; values above one indicate that reconstruction squared error exceeds centered activation variance on the audit distribution\.
#### Exact risks for the selected support code\.

Fornnequally weighted states, letLi​aL\_\{ia\}be the mean automatic loss for stateiiand actiona∈𝒜a\\in\\mathcal\{A\}, where𝒜=\{ask,decline,assist\}\\mathcal\{A\}=\\\{\\textsc\{ask\},\\textsc\{decline\},\\textsc\{assist\}\\\}\. LetCzC\_\{z\}contain the states with codezz; the sum below uses nonempty cells\. The full state and interface risks are

Rfull=1n​∑i=1nmina∈𝒜⁡Li​a,RZ⋆=∑z\|Cz\|n​mina∈𝒜​1\|Cz\|​∑i∈CzLi​a\.R\_\{\\mathrm\{full\}\}=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\min\_\{a\\in\\mathcal\{A\}\}L\_\{ia\},\\qquad R\_\{Z\}^\{\\star\}=\\sum\_\{z\}\\frac\{\|C\_\{z\}\|\}\{n\}\\min\_\{a\\in\\mathcal\{A\}\}\\frac\{1\}\{\|C\_\{z\}\|\}\\sum\_\{i\\in C\_\{z\}\}L\_\{ia\}\.The information gap isΔinfo=RZ⋆−Rfull\\Delta\_\{\\mathrm\{info\}\}=R\_\{Z\}^\{\\star\}\-R\_\{\\mathrm\{full\}\}\.

At the primary budgetB=2B=2, both SAE seeds give a positive gap on both collections\. The old confirmation gaps are0\.1360\.136and0\.1420\.142; the fresh locked gaps are0\.0710\.071and0\.0770\.077\. Every one\-sided 95% lower bound is positive, and none of the 2,000 cluster\-bootstrap replicates has a nonpositive gap\.

Table 9:Primary automatic SmolLM2ZZ\-only results atB=2B=2\. Active policy is not a separate lookup key\.The supervised\-selector gap decreases with feature budget\. On the old confirmation set, the seed\-314159 curve is0\.1360\.136,0\.1180\.118,0\.0500\.050,0\.0040\.004, and approximately zero atB=2,4,8,16,32B=2,4,8,16,32, respectively; seed 271828 follows the same pattern\. On the fresh set, seed 314159 gives0\.0710\.071,0\.0680\.068,0\.0400\.040,0\.0210\.021,0\.0020\.002, and zero, while seed 271828 reaches zero byB=16B=16\. The larger feature sets refine the earlier partitions, so their optimal risk cannot increase\. A zero gap means that each occupied cell has a common optimal action; it does not require every state to have a distinct code\.

Figure file not found at compile time\.figures/figure\_30\_cross\_family\_old\_confirmation\_budget\.pdf Automatic information gap by feature budget on the old confirmation collection for the two independently trained SAE seeds\.

\(a\)Old confirmation collection\.
Figure file not found at compile time\.figures/figure\_31\_cross\_family\_fresh\_locked\_budget\.pdf Automatic information gap by feature budget on the fresh locked policy collection for the two independently trained SAE seeds\.

\(b\)Fresh locked collection\.

Figure 4:SmolLM2ZZ\-only automatic information\-gap curves\. Error bars are cluster bootstrap over policy areas 95% intervals\.
#### Selector and occupancy checks\.

The variance selector also gives positiveB=2B=2gaps for every seed and collection:0\.1760\.176and0\.1490\.149on the old confirmation set, and0\.0470\.047and0\.0510\.051on the fresh set\. The 200 random selectors have meanB=2B=2gaps0\.1950\.195on the old set and about0\.0770\.077on the fresh set\. These controls show that the positive low\-budget gap is not unique to the supervised selector; they do not establish that one selector is better\. Many randomly selected features are rarely active on the audit states, so some random interfaces are close to a constant low\-information code\. The occupancy\-matched controls in the Qwen2\.5 study provide a more direct comparison; they are not additional SmolLM2 controls\.

AtB=2B=2, the supervised interface has four fitting codes and four evaluation codes\. The evaluation collision\-state fraction is0\.9900\.990or1\.0001\.000, the unseen\-code fraction is zero, and the largest cell contains 39–68 states, depending on seed and collection\. AsBBincreases, collision decreases and unseen evaluation codes increase\. For example, atB=64B=64the fresh set has 96 unique singleton evaluation codes but every evaluation code is unseen in the 144\-state fitting split\. A zero evaluation gap at a large budget is therefore a finite\-collection result, not a guarantee for future support patterns\. Complete selector and occupancy tables are reported in Appendix[F](https://arxiv.org/html/2610.27446#A6)\.

### F\.3Automatic cross\-family selector results

Table 10:Complete supervised\-selector results for the SmolLM2ZZ\-only automatic audit\. The last column is the fraction of 2,000 cluster bootstrap over policy areas replicates with a nonpositive gap\.CollectionSAE seedBBRfullR\_\{\\mathrm\{full\}\}RZ⋆R\_\{Z\}^\{\\star\}Δinfo\\Delta\_\{\\mathrm\{info\}\}95% CI95% LCBPrboot⁡\(Δ≤0\)\\Pr\_\{\\rm boot\}\(\\Delta\\leq 0\)Fresh locked 9627182820\.2010\.2780\.077\[0\.042, 0\.088\]0\.0470\.000Fresh locked 9627182840\.2010\.2580\.057\[0\.023, 0\.069\]0\.0280\.000Fresh locked 9627182880\.2010\.2400\.038\[0\.014, 0\.044\]0\.0160\.000Fresh locked 96271828160\.2010\.2010\.000\[0\.000, 0\.000\]0\.0000\.846Fresh locked 96271828320\.2010\.2010\.000\[0\.000, 0\.000\]0\.0000\.850Fresh locked 96271828640\.2010\.2010\.000\[0\.000, 0\.000\]0\.0000\.859Fresh locked 9631415920\.2010\.2730\.071\[0\.037, 0\.098\]0\.0420\.000Fresh locked 9631415940\.2010\.2690\.068\[0\.034, 0\.075\]0\.0390\.000Fresh locked 9631415980\.2010\.2420\.040\[0\.017, 0\.056\]0\.0210\.001Fresh locked 96314159160\.2010\.2220\.021\[0\.002, 0\.030\]0\.0040\.006Fresh locked 96314159320\.2010\.2030\.002\[0\.000, 0\.005\]0\.0000\.294Fresh locked 96314159640\.2010\.2010\.000\[0\.000, 0\.000\]0\.0000\.847Old confirmation 9627182820\.1420\.2840\.142\[0\.084, 0\.191\]0\.0910\.000Old confirmation 9627182840\.1420\.2480\.106\[0\.061, 0\.134\]0\.0650\.000Old confirmation 9627182880\.1420\.2120\.070\[0\.033, 0\.086\]0\.0370\.000Old confirmation 96271828160\.1420\.1590\.017\[0\.000, 0\.025\]0\.0010\.017Old confirmation 96271828320\.1420\.1420\.000\[0\.000, 0\.000\]0\.0000\.650Old confirmation 96271828640\.1420\.1420\.000\[0\.000, 0\.000\]0\.0000\.663Old confirmation 9631415920\.1420\.2780\.136\[0\.100, 0\.164\]0\.1050\.000Old confirmation 9631415940\.1420\.2600\.118\[0\.073, 0\.132\]0\.0760\.000Old confirmation 9631415980\.1420\.1920\.050\[0\.016, 0\.059\]0\.0180\.001Old confirmation 96314159160\.1420\.1460\.004\[0\.000, 0\.009\]0\.0000\.119Old confirmation 96314159320\.1420\.1420\.000\[0\.000, 0\.000\]0\.0000\.643Old confirmation 96314159640\.1420\.1420\.000\[0\.000, 0\.000\]0\.0000\.649Table 11:Complete variance\-selector results for the SmolLM2ZZ\-only automatic audit\.CollectionSAE seedBBRfullR\_\{\\mathrm\{full\}\}RZ⋆R\_\{Z\}^\{\\star\}Δinfo\\Delta\_\{\\mathrm\{info\}\}95% CI95% LCBFresh locked 9627182820\.2010\.2520\.051\[0\.026, 0\.070\]0\.030Fresh locked 9627182840\.2010\.2510\.050\[0\.025, 0\.066\]0\.029Fresh locked 9627182880\.2010\.2390\.038\[0\.009, 0\.047\]0\.012Fresh locked 96271828160\.2010\.2040\.003\[0\.000, 0\.007\]0\.000Fresh locked 96271828320\.2010\.2010\.000\[0\.000, 0\.000\]0\.000Fresh locked 96271828640\.2010\.2010\.000\[0\.000, 0\.000\]0\.000Fresh locked 9631415920\.2010\.2490\.047\[0\.028, 0\.064\]0\.031Fresh locked 9631415940\.2010\.2480\.047\[0\.027, 0\.061\]0\.030Fresh locked 9631415980\.2010\.2410\.039\[0\.015, 0\.052\]0\.019Fresh locked 96314159160\.2010\.2150\.013\[0\.001, 0\.020\]0\.002Fresh locked 96314159320\.2010\.2020\.001\[0\.000, 0\.001\]0\.000Fresh locked 96314159640\.2010\.2010\.000\[0\.000, 0\.000\]0\.000Old confirmation 9627182820\.1420\.2910\.149\[0\.113, 0\.188\]0\.118Old confirmation 9627182840\.1420\.2870\.145\[0\.087, 0\.168\]0\.092Old confirmation 9627182880\.1420\.2100\.068\[0\.028, 0\.076\]0\.032Old confirmation 96271828160\.1420\.1590\.017\[0\.000, 0\.029\]0\.000Old confirmation 96271828320\.1420\.1420\.000\[0\.000, 0\.000\]0\.000Old confirmation 96271828640\.1420\.1420\.000\[0\.000, 0\.000\]0\.000Old confirmation 9631415920\.1420\.3180\.176\[0\.118, 0\.211\]0\.126Old confirmation 9631415940\.1420\.2820\.140\[0\.092, 0\.174\]0\.098Old confirmation 9631415980\.1420\.2230\.081\[0\.039, 0\.087\]0\.044Old confirmation 96314159160\.1420\.1420\.000\[0\.000, 0\.000\]0\.000Old confirmation 96314159320\.1420\.1420\.000\[0\.000, 0\.000\]0\.000Old confirmation 96314159640\.1420\.1420\.000\[0\.000, 0\.000\]0\.000Table 12:Complete random\-selector information\-gap summary for the SmolLM2ZZ\-only audit\. Each row summarizes 200 random selectors\.CollectionSAE seedBBNNMeanSDMinMedianMaxFresh locked 9627182822000\.0770\.0010\.0620\.0770\.077Fresh locked 9627182842000\.0770\.0010\.0670\.0770\.077Fresh locked 9627182882000\.0770\.0020\.0580\.0770\.077Fresh locked 96271828162000\.0770\.0020\.0600\.0770\.077Fresh locked 96271828322000\.0760\.0030\.0490\.0770\.077Fresh locked 96271828642000\.0750\.0050\.0490\.0770\.077Fresh locked 9631415922000\.0770\.0010\.0670\.0770\.077Fresh locked 9631415942000\.0770\.0010\.0610\.0770\.077Fresh locked 9631415982000\.0770\.0010\.0660\.0770\.077Fresh locked 96314159162000\.0770\.0020\.0630\.0770\.077Fresh locked 96314159322000\.0760\.0030\.0540\.0770\.077Fresh locked 96314159642000\.0750\.0040\.0560\.0770\.077Old confirmation 9627182822000\.1950\.0000\.1950\.1950\.195Old confirmation 9627182842000\.1950\.0000\.1950\.1950\.195Old confirmation 9627182882000\.1950\.0000\.1880\.1950\.195Old confirmation 96271828162000\.1950\.0010\.1880\.1950\.195Old confirmation 96271828322000\.1940\.0020\.1720\.1950\.195Old confirmation 96271828642000\.1940\.0030\.1670\.1950\.195Old confirmation 9631415922000\.1950\.0000\.1950\.1950\.195Old confirmation 9631415942000\.1950\.0000\.1920\.1950\.195Old confirmation 9631415982000\.1950\.0020\.1720\.1950\.195Old confirmation 96314159162000\.1940\.0030\.1650\.1950\.195Old confirmation 96314159322000\.1940\.0050\.1520\.1950\.195Old confirmation 96314159642000\.1930\.0060\.1350\.1950\.195
### F\.4Finite\-code occupancy

Table 13:Complete supervised\-selector occupancy diagnostics for the SmolLM2ZZ\-only audit\. Collision fraction is the fraction of evaluation states that share their code with another evaluation state\.CollectionSAE seedBBFit codesEval codesCollision frac\.Unseen frac\.Largest cellFresh locked 962718282441\.0000\.00048Fresh locked 96271828416120\.9790\.00031Fresh locked 96271828858380\.7710\.60412Fresh locked 9627182816125800\.2810\.9904Fresh locked 9627182832143920\.0731\.0003Fresh locked 9627182864144960\.0001\.0001Fresh locked 963141592441\.0000\.00068Fresh locked 96314159414100\.9790\.02137Fresh locked 96314159868170\.9580\.34425Fresh locked 9631415916125540\.6250\.9908Fresh locked 9631415932141910\.1041\.0002Fresh locked 9631415964144960\.0001\.0001Old confirmation 962718282441\.0000\.00046Old confirmation 96271828416140\.9580\.00034Old confirmation 96271828858380\.8020\.0839Old confirmation 9627182816125760\.3440\.5944Old confirmation 9627182832143950\.0210\.9382Old confirmation 9627182864144960\.0000\.9901Old confirmation 963141592440\.9900\.00039Old confirmation 96314159414120\.9900\.02122Old confirmation 96314159868480\.6670\.27114Old confirmation 9631415916125870\.1770\.8333Old confirmation 9631415932141960\.0000\.9901Old confirmation 9631415964144960\.0001\.0001
### F\.5Complete SmolLM2 human agreement and code\-only results

The human analysis evaluates fixed SmolLM2 packet continuations\. The historical code\-only results below and the policy\-aware reanalysis in Section[5\.3](https://arxiv.org/html/2610.27446#S5.SS3)use the same final labels\. No human evaluation of Qwen2\.5 or Phi responses is reported\.

#### Pre\-adjudication agreement\.

Agreement is computed before any joint discussion\. Quadratic\-weighted Cohen’sκ\\kappais0\.7480\.748for safety,0\.6460\.646for policy adherence,0\.8110\.811for usefulness,0\.4720\.472for factuality, and0\.6320\.632for overall quality\. Ordinal Krippendorff’sα\\alpharanges from0\.4820\.482to0\.7910\.791\. Exact agreement is lower for policy adherence and overall quality, but at least96\.4%96\.4\\%of all ordinal ratings differ by at most one point\. Factuality has90\.7%90\.7\\%exact agreement, and most scores are 3\. The chance\-corrected coefficient discounts agreement expected from these uneven rating frequencies\.

The composite human\-loss ICC\(2,1\) is

0\.746with 95% cluster\-bootstrap interval​\[0\.686,0\.798\]\.0\.746\\quad\\text\{with 95\\% cluster\-bootstrap interval \}\[0\.686,0\.798\]\.The two composite losses have Pearson correlation0\.7900\.790, Spearman correlation0\.7720\.772, mean absolute difference0\.1190\.119, and RMSE0\.1640\.164\. Rater B is stricter on average: mean loss is0\.4530\.453for Rater B and0\.3780\.378for Rater A\. Agreement on the recommended action is nearly exact, with Cohen’sκ=0\.991\\kappa=0\.991and99\.4%99\.4\\%exact agreement\. Agreement is lower for the binary acceptable label, withκ=0\.563\\kappa=0\.563and83\.0%83\.0\\%exact agreement\. Critical\-error labels haveκ=0\.745\\kappa=0\.745and79\.5%79\.5\\%exact agreement\.

Table 14:Pre\-adjudication inter\-rater agreement\. Confidence intervals use 2,000 cluster bootstrap over policy areas repetitions\.Table 15:Pre\-adjudication agreement for categorical fields\.\(a\)Independent rater losses\.\(b\)Automatic and final human loss\.
Figure 5:Two different agreement questions on the same SmolLM2 packet\. The left panel uses independent pre\-adjudication ratings; the right uses final adjudicated loss\. Neither panel supplies a population\-uniform judge\-error bound\.
#### Adjudication\.

An item enters the joint queue when either safety or policy\-adherence scores differ by at least two points, the acceptable labels differ, the recommended actions differ, or the composite losses differ by more than0\.250\.25\. Exactly 100 of 528 items, or18\.9%18\.9\\%, trigger at least one condition\. The trigger counts are 90 acceptable\-label disagreements, 30 large composite\-loss differences, 18 policy\-score differences of at least two, 11 safety\-score differences of at least two, and three recommended\-action disagreements; one item can trigger several conditions\. Sixty\-seven items enter only because the acceptable labels differ\.

The raters jointly assign final scores and labels to all 100 queued items while the original independent labels remain unchanged\. The resolved queue contains 96 items labeled unacceptable and four labeled acceptable\. Final recommended actions are 50assist, 33decline, and 17ask\. Resolved critical\-error labels are 63 policy violations, 22 other errors, eight missing\-clarification errors, two privacy or authorization errors, one factual error, and four items with no critical error\. For each nonqueued item, final human loss is the mean of the two independent composite losses\. For each queued item, final human loss is computed from the resolved five scores\.

#### Comparison of automatic and human scores after adjudication\.

Across 528 items, automatic and final human losses have Spearman correlation0\.3790\.379and Pearson correlation0\.3880\.388\. Mean signed error, defined as automatic loss minus human loss, is−0\.063\-0\.063, with cluster bootstrap over policy areas 95% interval\[−0\.111,−0\.018\]\[\-0\.111,\-0\.018\]\. The automatic evaluator is therefore optimistic on average\. MAE is0\.3110\.311with 95% interval\[0\.290,0\.331\]\[0\.290,0\.331\]and a one\-sided 95% upper bound of0\.3270\.327\. RMSE is0\.3670\.367\. The largest observed absolute error in the packet is0\.9000\.900; this packet maximum is not a population\-level uniform error bound\.

Using the first minimizing action in the fixed orderask,decline,assist, state\-level automatic–human label agreement is51\.1%51\.1\\%\. Automatic scores have tied minima in 90 of 176 states; Appendix[K\.2](https://arxiv.org/html/2610.27446#A11.SS2)separately reports minimizing\-set intersection \(80\.7%80\.7\\%\) and equality \(30\.7%30\.7\\%\)\. The saved pairwise action\-preference accuracy is55\.8%55\.8\\%; its pair ordering and tie convention are also documented there\. Agreement is61\.3%61\.3\\%on the old 80\-state sample and42\.7%42\.7\\%on the fresh 96\-state collection\. The final human best action matches the hidden required action for86\.3%86\.3\\%of old states and92\.7%92\.7\\%of fresh states, while the automatic best action matches for55\.0%55\.0\\%and36\.5%36\.5\\%, respectively\.

Calibration is strongly action dependent\. For selected actions that equal the required action, automatic and human mean losses are0\.1110\.111and0\.1100\.110forask,0\.5190\.519and0\.1960\.196fordecline, and0\.0040\.004and0\.2880\.288forassist\. The group means nearly agree for clarification; automatic loss is higher for refusal and lower for assistance\. Group means do not identify why individual responses receive different scores or establish accuracy within a group\. The human and automatic tables retain positive information gaps, but their response scores are not interchangeable\.

\(a\)Action\-dependent mean losses\.\(b\)First\-minimum label agreement by collection\.

Figure 6:Action\-specific calibration and state\-level agreement\. Numerical records and counts appear in Tables[16](https://arxiv.org/html/2610.27446#A6.T16)and[17](https://arxiv.org/html/2610.27446#A6.T17)\. Matching group means do not imply close item\-level agreement\.Table 16:Action\-specific calibration split by whether the selected action equals the required action\.Table 17:First\-minimum action\-label agreement after adjudication, using the fixed orderask,decline,assistto break ties\. Set\-based comparisons appear in Appendix[K\.2](https://arxiv.org/html/2610.27446#A11.SS2)\.
#### Final human\-loss information gaps\.

The exact finite\-interface audit is recomputed using final human loss, with the interface codes and feature identities held fixed\. The primary budget is againB=2B=2\. On the old 80\-state human sample, seed 314159 gives

Rfullhuman=0\.216,RZ⋆,human=0\.293,Δinfohuman=0\.078,R\_\{\\mathrm\{full\}\}^\{\\mathrm\{human\}\}=0\.216,\\qquad R\_\{Z\}^\{\\star,\\mathrm\{human\}\}=0\.293,\\qquad\\Delta\_\{\\mathrm\{info\}\}^\{\\mathrm\{human\}\}=0\.078,with 95% interval\[0\.047,0\.111\]\[0\.047,0\.111\]and one\-sided lower bound0\.0520\.052\. Seed 271828 gives gap0\.0990\.099, interval\[0\.062,0\.119\]\[0\.062,0\.119\], and lower bound0\.0680\.068\. On the fresh locked 96\-state collection, seed 314159 gives gap0\.1480\.148, interval\[0\.118,0\.161\]\[0\.118,0\.161\], and lower bound0\.1240\.124; seed 271828 gives gap0\.1330\.133, interval\[0\.095,0\.155\]\[0\.095,0\.155\], and lower bound0\.0990\.099\. Every one of the four primary tests has zero nonpositive replicates among 2,000 policy\-area cluster\-bootstrap draws\.

Table 18:Final adjudicated SmolLM2 human\-loss results for theZZ\-only observation atB=2B=2\. Active policy is not a separate action\-selector input\.The human gap decreases with feature budget\. It is positive for both seeds and both collections atB=2,4,8B=2,4,8\. AtB=16B=16, the old\-set gaps are0\.00060\.0006for both seeds, the fresh seed\-271828 gap is0\.00220\.0022, and the fresh seed\-314159 gap is0\.0280\.028with one\-sided lower bound0\.0100\.010\. AtB=32B=32andB=64B=64, the gaps are zero or numerically negligible\. The nested feature sets cannot increase optimal risk\. The values quantify how much loss remains at each budget; they do not separately establish that all underlying states are distinguished\.

Figure file not found at compile time\.figures/figure\_34\_human\_gap\_curve\_old\.pdf Final human\-loss information gap by feature budget on the old 80\-state human sample\.

\(a\)Old 80\-state human sample\.
Figure file not found at compile time\.figures/figure\_35\_human\_gap\_curve\_fresh\.pdf Final human\-loss information gap by feature budget on the fresh locked 96\-state collection\.

\(b\)Fresh locked 96\-state collection\.

Figure 7:Final adjudicated human\-loss information\-gap curves for the two independently trained SAE seeds\.
#### Single\-rater sensitivity\.

Before adjudication, each rater’s labels are also analyzed separately atB=2B=2\. All eight combinations of rater, SAE seed, and collection have positive one\-sided 95% lower bounds\. For example, old\-sample lower bounds range from0\.0430\.043to0\.0950\.095, and fresh\-sample lower bounds range from0\.0880\.088to0\.1250\.125\. The positive sign therefore does not arise only after averaging or adjudication\. The exact rater\-specific values are given in Appendix[F](https://arxiv.org/html/2610.27446#A6)\.

### F\.6Final human finite\-interface results across all budgets

Table 19:Complete final adjudicated SmolLM2 human\-lossZZ\-only finite\-interface results\. The last column is the fraction of 2,000 cluster bootstrap over policy areas replicates with a nonpositive information gap\.CollectionSAE seedBBRfullhumanR\_\{\\mathrm\{full\}\}^\{\\rm human\}RZ⋆,humanR\_\{Z\}^\{\\star,\\rm human\}Δinfohuman\\Delta\_\{\\rm info\}^\{\\rm human\}95% CI95% LCBPrboot⁡\(Δ≤0\)\\Pr\_\{\\rm boot\}\(\\Delta\\leq 0\)Fresh locked 9627182820\.1860\.3200\.133\[0\.095, 0\.155\]0\.0990\.000Fresh locked 9627182840\.1860\.2830\.097\[0\.055, 0\.123\]0\.0590\.000Fresh locked 9627182880\.1860\.2370\.051\[0\.019, 0\.056\]0\.0230\.000Fresh locked 96271828160\.1860\.1890\.002\[0\.000, 0\.004\]0\.0000\.079Fresh locked 96271828320\.1860\.1870\.000\[0\.000, 0\.001\]0\.0000\.186Fresh locked 96271828640\.1860\.1860\.000\[0\.000, 0\.000\]0\.0000\.313Fresh locked 9631415920\.1860\.3350\.148\[0\.118, 0\.161\]0\.1240\.000Fresh locked 9631415940\.1860\.3170\.130\[0\.096, 0\.145\]0\.1010\.000Fresh locked 9631415980\.1860\.2790\.093\[0\.055, 0\.118\]0\.0610\.000Fresh locked 96314159160\.1860\.2140\.028\[0\.009, 0\.042\]0\.0100\.000Fresh locked 96314159320\.1860\.1860\.000\[0\.000, 0\.000\]0\.0000\.328Fresh locked 96314159640\.1860\.1860\.000\[0\.000, 0\.000\]0\.0000\.309Old human sample 8027182820\.2160\.3140\.099\[0\.062, 0\.119\]0\.0680\.000Old human sample 8027182840\.2160\.2840\.069\[0\.035, 0\.081\]0\.0380\.000Old human sample 8027182880\.2160\.2630\.047\[0\.018, 0\.056\]0\.0200\.000Old human sample 80271828160\.2160\.2160\.001\[0\.000, 0\.002\]0\.0000\.308Old human sample 80271828320\.2160\.2160\.000\[0\.000, 0\.000\]0\.0000\.522Old human sample 80271828640\.2160\.2160\.000\[0\.000, 0\.000\]0\.0000\.526Old human sample 8031415920\.2160\.2930\.078\[0\.047, 0\.111\]0\.0520\.000Old human sample 8031415940\.2160\.2880\.073\[0\.040, 0\.094\]0\.0440\.000Old human sample 8031415980\.2160\.2440\.028\[0\.007, 0\.039\]0\.0090\.000Old human sample 80314159160\.2160\.2160\.001\[0\.000, 0\.001\]0\.0000\.239Old human sample 80314159320\.2160\.2160\.000\[0\.000, 0\.000\]0\.0000\.500Old human sample 80314159640\.2160\.2160\.000\[0\.000, 0\.000\]0\.0000\.526Table 20:Single\-rater sensitivity analysis at the primary budgetB=2B=2\. Each row is recomputed from one rater’s independent labels before adjudication\.
### F\.7Human agreement, adjudication, and evaluator calibration

Adjudication rates differ across policy areas\. The counts below describe that variation but do not identify its cause\.

Table 21:Adjudication\-trigger rate by policy area\.CollectionPolicy areaItemsTriggeredFractionFresh lockedlaboratory\_sample\_chain\_of\_custody36140\.389Fresh lockedadditive\_manufacturing\_ventilation36110\.306Fresh lockedfood\_cold\_chain\_handling36110\.306Fresh lockedmuseum\_artifact\_conservation36100\.278Fresh lockedindustrial\_robot\_lockout3660\.167Fresh lockedassistive\_accommodation\_communications3650\.139Fresh lockedmultilingual\_intake\_interpretation3640\.111Fresh lockedopen\_source\_license\_compliance3620\.056Old human samplefield\_equipment\_checkout1260\.500Old human sampleconfidential\_document\_sharing1240\.333Old human samplepurchase\_approval1240\.333Old human samplewine\_tasting\_registration1240\.333Old human samplearchive\_room\_visit1230\.250Old human samplecampus\_lab\_access1220\.167Old human sampledonor\_contact\_list\_export1220\.167Old human samplemedical\_record\_release1220\.167Old human samplepayroll\_bank\_change1220\.167Old human sampleproject\_dashboard\_settings1220\.167Old human samplestudent\_record\_access1220\.167Old human samplecampus\_drone\_flight\_booking1210\.083Old human sampleprivate\_calendar\_sharing1210\.083Old human samplerental\_vehicle\_booking1210\.083Old human sampleresearch\_data\_export1210\.083Old human sampleafter\_hours\_building\_access1200\.000Old human samplelicensed\_software\_download1200\.000Old human sampleminor\_photo\_release1200\.000Old human sampletraining\_simulator\_booking1200\.000Old human samplevolunteer\_background\_clearance1200\.000Table 22:Final adjudicated comparison of automatic and human scores by selected action, collection, and required\-action match\. Signed error is automatic loss minus human loss\.GroupingGroupNNAuto meanHuman meanBiasMAERMSESpearmanρ\\rhoselected\_actiondecline1760\.4790\.4160\.0630\.2780\.344\-0\.069selected\_actionask1760\.2860\.338\-0\.0530\.2980\.3440\.504selected\_actionassist1760\.3080\.507\-0\.2000\.3560\.4090\.581collectionold\_2402400\.2990\.418\-0\.1190\.2920\.3290\.362collectionfresh\_locked\_962880\.4060\.423\-0\.0160\.3270\.3960\.395matches\_required\_actionTrue1760\.1960\.204\-0\.0080\.2650\.339\-0\.195matches\_required\_actionFalse3520\.4380\.529\-0\.0900\.3340\.3800\.350

## Appendix GAdditional Experiments and Diagnostic Figures

The main text retains the risk attribution, policy\-aware cross\-family evidence, human\-loss result, and possibility of a conflict beyond pairs\. This appendix records the supporting experiments and the original vector plots\. Different panels retain their original estimands and uncertainty procedures; a shared row does not imply that they are independent replications or use a common loss scale\.

### G\.1Qwen2\.5 settings and full frozen risk profiles

Table 23:Primary Qwen2\.5 audit\. Confirmation policy areas are not used to select layers, feature budgets, thresholds, or feature identities\.Table 24:HistoricalZZ\-only Qwen2\.5 risk decomposition on untouched policy areas\. Values in brackets are cluster bootstrap over policy areas 95% intervals for the three gaps\. Lower risk is better\.Figure file not found at compile time\.figures/figure\_18\_certified\_positive\_four\_risks\.pdf Frozen four\-risk result for layer 17 and budget 2\.

\(a\)Primary positive\-information setting\.
Figure file not found at compile time\.figures/figure\_19\_certified\_accessibility\_four\_risks\.pdf Frozen four\-risk result for layer 19 and budget 16\.

\(b\)Mixed information and accessibility setting\.

Figure 8:Four risks on the eight untouched confirmation policy areas\. Error bars are domain\-cluster bootstrap 95% intervals for each risk\. The selected interfaces are frozen before confirmation\.
### G\.2Budget and rule interventions: complete historical diagnostics

The experiments change the visible features, allowed rules, and fitting data separately\. Refining an observation or enlarging a rule family cannot increase the corresponding optimal risk, because the earlier rules remain available\. A decrease from additional training data is an empirical result, not a consequence of the decomposition in Equation \([5](https://arxiv.org/html/2610.27446#S2.E5)\)\.

#### Interface information\.

Across the four audited layers, the finite\-support information gap is largest at small budgets and approaches zero as more SAE coordinates are exposed\. The rate is layer dependent, but the broad decrease is common to all four layers\. The budget sweep is descriptive because it was available during setting selection; the frozen untouched\-domain values in Table[24](https://arxiv.org/html/2610.27446#A7.T24)provide the frozen confirmation results\.

#### Decision\-rule accessibility\.

At the frozen layer\-19,B=16B=16interface, a constant rule has accessibility gap0\.1430\.143with 95% interval\[0\.086,0\.200\]\[0\.086,0\.200\]\. A first\-order affine rule reduces the gap to0\.0470\.047with interval\[0\.016,0\.078\]\[0\.016,0\.078\]\. Adding pairwise interactions reduces the gap to0\.0070\.007with interval\[0,0\.016\]\[0,0\.016\], while third\-order interactions and an unrestricted lookup table attain zero on the finite audit\. The nested construction holds the interface fixed, so only the class used to read the interface changes\.

#### Finite training\.

The training\-size intervention uses nested subsets from one fixed audit population, which avoids mixing sample size with a policy\-area shift\. The mean finite\-training gap decreases from0\.1300\.130at 16 training situations to0\.0720\.072at 36,0\.0200\.020at 72,0\.0090\.009at 108, and zero at all 144 fitting situations\.

\(a\)Selected SAE feature budget\.\(b\)Collision and unseen\-code rates\.
Figure 9:Qwen2\.5 budget and occupancy diagnostics\. Bands summarize the saved domain partitions\. A small finite\-table gap at a large budget does not establish coverage of future codes\.\(a\)Nested rule classes on confirmation areas\.\(b\)Fitting\-set sizes within one audit population\.
Figure 10:Original intervention plots, with their saved uncertainty displays\. The compact main\-text rendering uses the reported numerical summaries\. The rule\-class intervals summarize bootstrap resampling of the frozen confirmation areas and do not provide a simultaneous population certificate\.\(a\)Layer 17, two features\.\(b\)Layer 19, sixteen features\.
Figure 11:Setting\-selection risk profiles\. These descriptive plots are not the untouched confirmation estimates in Table[24](https://arxiv.org/html/2610.27446#A7.T24)\.\(a\)Descriptive nested\-rule sweep\.\(b\)Affine parameter\-bound check\.
Figure 12:Rule\-class and optimizer diagnostics\. The affine optimum is unchanged for parameter bounds 5, 10, 50, and 200, with no fitted parameter on a boundary\.

### G\.3Evaluator, selector, and model\-scale robustness

#### Independent evaluators\.

The primary layer\-17,B=2B=2interface is held fixed while the loss table is replaced\. The information gap is0\.1560\.156under Qwen2\.5\-7B,0\.1340\.134under the independent Mistral\-7B evaluator, and0\.1530\.153under the equal\-weight average of both evaluators\. The Mistral interval is\[0\.100,0\.162\]\[0\.100,0\.162\], and its one\-sided lower bound is0\.1040\.104\. For the layer\-19,B=16B=16interface, the independent evaluator gives information gap0\.0790\.079and accessibility gap0\.0660\.066, with positive one\-sided lower bounds0\.0510\.051and0\.0430\.043\. Across all state–action losses, the two evaluators have Spearman correlation0\.8030\.803and agree on the best action for91\.3%91\.3\\%of situations\. The positive gap persists under both scoring rules on the same responses\. The comparison does not establish the accuracy of either evaluator\.

#### Feature selectors\.

At the two frozen layer–budget settings, the supervised selector, the variance selector, and every one of 200 random selectors give a positive information gap\. For layer 17 andB=2B=2, the supervised gap is0\.1560\.156, the variance gap is0\.2030\.203, and the random\-selector mean is0\.1890\.189\. For layer 19 andB=16B=16, the corresponding values are0\.0600\.060,0\.1310\.131, and0\.0940\.094\. Forty occupancy\-matched random controls per setting also have positive mean gaps\. The result tests whether positive information loss depends on one feature selector\. It does not show that supervised selection is better than random selection; the saved claim ledger rejects that stronger reading\.

#### Second audited model scale\.

The full finite\-interface pipeline is repeated on Qwen2\.5\-0\.5B\-Instruct with the separatesae\-qwen2\.5\-0\.5B\-resrelease\. At layer 17 andB=2B=2, the information gap is0\.2100\.210with 95% interval\[0\.188,0\.223\]\[0\.188,0\.223\]\. At layer 19 andB=16B=16, the information gap is0\.0610\.061with interval\[0\.031,0\.079\]\[0\.031,0\.079\], and the accessibility gap is0\.0840\.084with interval\[0\.055,0\.104\]\[0\.055,0\.104\]\. The cross\-scale budget curve again falls as more selected features are exposed\. The replication reduces dependence on one model size, but both audited models remain in the Qwen2\.5 family\.

\(a\)Frozen interface across evaluators\.\(b\)Frozen settings across selectors\.
Figure 13:Qwen2\.5 evaluator and selector robustness\. Left: bootstrap over policy areas intervals\. Right: 200 random selectors and the named selector results\. These comparisons do not establish selector superiority\.\(a\)Second Qwen2\.5 scale, with its own SAE\.\(b\)Descriptive selector budget sweep\.
Figure 14:Cross\-scale and selector budget diagnostics for code\-only observations\. Neither panel is a new Qwen2\.5\-0\.5B policy\-aware experiment\.

### G\.4Reconstruction, output count, and decision information

Reconstruction loss and decision information answer different questions\. The SAE reconstruction diagnostics vary strongly across the four layers, but their ordering does not match the ordering of the interface information gaps at a fixed feature budget\. A layer can have a better global reconstruction diagnostic while its selected finite interface retains less information about the action that minimizes the declared loss\. The centered fraction of variance unexplained \(FVU\) is reported as a diagnostic rather than treated as a calibrated percentage; its value can exceed one because the reconstruction squared error can exceed the centered activation variance\.

A related matched\-codebook experiment gives the continuous selected SAE view, an SAE reconstruction view, and a raw\-hidden view the same declared number of outputs\. Their information gaps still differ\. Equal output cardinality is therefore not equal decision information\. Occupancy diagnostics also show a tradeoff: more detailed interfaces reduce collision cells but create more confirmation patterns that were not seen during fitting\. The matched codebook plots are shown below\.

\(a\)Layer reconstruction versus decision loss\.\(b\)Matched finite output counts\.
Figure 15:Reconstruction and output count do not substitute for a decision\-loss audit\. The codebook comparison holds the declared number of outputs fixed\.\(a\)Occupancy of the matched codebooks\.\(b\)Alternative splits and selector settings\.
Figure 16:Coverage and split diagnostics\. Split variation is descriptive rather than a collection of independent confidence replicates\.
### G\.5Continuous observations and fitted\-rule diagnostics

#### Specified Gaussian observation channel\.

The experiment adds Gaussian noise to fixed continuous feature vectors\. The noise scale is expressed relative to the median distance to the nearest neighbor\. For a uniform distribution over the audited states, the posterior state probabilities are computed analytically at each sampled noisy vector; Monte Carlo averaging estimates the resulting Bayes risk\. At zero noise, the observed vectors are distinct and the exact table gap is zero\. The reported estimates increase with noise for both selected SAE coordinates and raw hidden coordinates\. The calculation concerns this finite Gaussian mixture, not the noiseless representation on an unknown population\.

#### Fitted continuous rules\.

A separate experiment asks whether standard rules can use information that remains in continuous features\. At layer 19 with 64 selected SAE coordinates, cost\-sensitivekk\-nearest neighbors, a linear rule, and a multilayer perceptron have held\-out practical excess losses0\.0470\.047,0\.0470\.047, and0\.0350\.035\. With 64 raw\-hidden principal components, the corresponding values are0\.0880\.088,0\.2470\.247, and0\.1050\.105\. The comparison concerns the tested representation–rule pairs; it is not an exact lower bound on the continuous raw\-hidden interface and does not imply that every SAE rule beats every raw\-hidden rule\.

\(a\)SAE and raw\-hidden observations\.\(b\)Selected SAE observations across layers\.
Figure 17:Monte Carlo estimates of Bayes risk for the specified Gaussian channel\. Noise magnitude is normalized by the median nearest\-neighbor distance\. The experiment does not certify information loss in noiseless exact hidden activations\.\(a\)Fitted rules on continuous features\.\(b\)Reported held\-out excess losses\.

Figure 18:Continuous representation–rule comparisons, not exact information\-loss lower bounds\. The numerical panel reproduces the reported means; the original plot is retained on the left\.

### G\.6Higher\-order search, capacity, and sanity checks

The controlled higher\-order benchmark constructs three states whose acceptable action sets are \{ask,assist\}, \{ask,decline\}, and \{decline,assist\}\. Every pair shares one zero\-regret action, but all three states share no common zero\-regret action\. At layer 17 andB=2B=2, the SAE support pattern collapses the three states\. The largest pair margin is zero, the triple margin is1/31/3, and the separation is1/31/3\. The witness is exact relative to the declared policy\-validity loss table\. It proves the logical need for group\-level checks without relying on a searched natural\-language example\.

The controlled\-triple sweep repeats the construction across layers and budgets\. The fraction of collapsed triples is high under small feature budgets and reaches zero at larger budgets, with the transition depending on layer\. The result links the higher\-order construction to real SAE support patterns rather than to an abstract binary code alone\.

The exploratory natural\-language search initially returned a positive higher\-order separation, but an independent certification split reduced the separation to zero\. A fresh test with two new paraphrase replicas, five generation replicas, and three judge prompts also produced zero positive\-separation rate\. The searched example therefore provides no replicated evidence of a conflict missed by every pairwise check\. The negative replication plots are retained in Appendix[G](https://arxiv.org/html/2610.27446#A7)\.

The finite\-capacity calculation converts a conflict audit into an output\-count requirement\. For the historical 42\-state finite audit subset, the archived solver outputs report matching lower and upper bounds: three interface outputs for allowed regret tolerances from0\.020\.02through0\.300\.30, and two outputs at tolerances0\.350\.35and0\.400\.40\. These are archived results for the historical finite loss table and unrestricted output labeling\. The underlying full\-precision matrix is unavailable, so the curve cannot be independently recomputed from the retained records\. The original code enumerates all pairs and triples globally; the source audit and separate algorithm regression tests are described in Appendix[J](https://arxiv.org/html/2610.27446#A10)\. A given SAE or selected\-coordinate family may need more outputs because not every labeling is realizable by that interface family\.

#### Sanity controls and evaluator diagnostics\.

Across five saved domain partitions, the mean information gap is0\.1280\.128for the actual low\-budget SAE interface,0\.2080\.208for a constant interface,0\.2000\.200for a shuffled SAE interface, and numerically zero for an oracle interface that exposes the best action\. The constant interface cannot improve on an informative one, and exposing an optimal action gives zero gap by construction\. The larger gap after shuffling is an observed result; arbitrary shuffling has no general monotonicity guarantee\.

The forced\-choice evaluator gives higher mean scores to actions that match the required\-action label\. Mean matching scores range from2\.622\.62to2\.822\.82across safety, utility, macro adherence, fact handling, and overall quality; mean mismatching scores range from0\.720\.72to1\.101\.10\. The anchor diagnostics also pass, and the second evaluator gives lower mean loss to matched actions\. The required\-action label is supplied to the Qwen evaluation prompt, so this agreement is not an independent check of the label or an accuracy guarantee for the evaluator\. Human ratings provide the separate comparison in Section[5\.1](https://arxiv.org/html/2610.27446#S5.SS1)\.

![Refer to caption](https://arxiv.org/html/2610.27446v1/figure_14_controlled_exact_higher_order_witness.png)\(a\)Declared three\-state regret table\.
\(b\)Collapse rate across layers and budgets\.
Figure 19:Original controlled higher\-order evidence\. Every pair has a shared zero\-regret action, while the triple has none\. The collapse sweep uses real SAE support patterns\.\(a\)Saved 42\-state capacity curve\.\(b\)Searched natural\-language witness and replication\.
Figure 20:Different evidence levels: the saved capacity curve is reported from archived solver outputs and cannot be independently recomputed without its missing historical matrix; the natural\-language higher\-order candidate does not replicate\.\(a\)Fresh semantic, generation, and judge replication\.\(b\)Actual, constant, shuffled, and oracle interfaces\.
Figure 21:Falsification and sanity checks\. Fresh natural\-language differences are at floating\-point scale\. The interface controls have the expected risk ordering\.\(a\)Historical evaluator discrimination\.\(b\)Saved control means across five splits\.

Figure 22:Gross measurement and implementation checks\. Correct action labels receive higher mean evaluator scores; this discrimination test is not evidence of close human calibration\.

## Appendix HExtended Literature Map and Detailed Comparisons

The main comparison fixes the state distribution, available continuations, loss, and controller input\. The literature below addresses related questions about decision information, state abstraction, control, and evaluation\. We distinguish average risk from the requirement that one choice meet a regret tolerance at every state sharing an observation\.

### H\.1Comparison of statistical experiments and decision\-theoretic sufficiency

Blackwell’s comparison of experiments orders information channels by the decision performance that they permit across decision problems\([Blackwell, 1951](https://arxiv.org/html/2610.27446#bib.bib1);[Blackwell, 1953](https://arxiv.org/html/2610.27446#bib.bib2)\)\. Le Cam’s theory of deficiency gives a quantitative notion of approximate comparison and sufficiency\([Le Cam, 1964](https://arxiv.org/html/2610.27446#bib.bib3);[Torgersen, 1991](https://arxiv.org/html/2610.27446#bib.bib4)\)\. Blackwell–Le Cam comparison supplies the basic interpretation of the interface information gap in the present paper: replacing a full state by an interface observation is harmless for a declared decision problem exactly when the replacement does not increase the optimal Bayes risk\.

Recent representation\-learning work makes the same comparison operational\. Deficiency has been used to define generic feature quality\([van Rooyen and Williamson, 2014](https://arxiv.org/html/2610.27446#bib.bib5)\)and to derive a variational bottleneck for learning representations that preserve downstream decision performance\([Banerjee and Montúfar, 2018](https://arxiv.org/html/2610.27446#bib.bib6)\)\. Loss\-dependent Bayes sufficiency identifies a coarsest quotient when the Bayes action is almost surely unique\([Sevetlidis, 2026](https://arxiv.org/html/2610.27446#bib.bib7)\)\. Action\-sufficient compression similarly merges support states that require the same optimal action\([Walsh, 2026](https://arxiv.org/html/2610.27446#bib.bib8)\), while decision\-sufficient datasets preserve the information needed to solve a family of linear optimization problems\([Ye et al\., 2026](https://arxiv.org/html/2610.27446#bib.bib9)\)\.

We use risk comparison to audit an existing interface\. The structural question concerns several acceptable continuations, rather than one chosen optimal label\. An incompatible cell contains a minimal group with no common acceptable choice\. Such groups characterize feasible interface partitions\. A statistical guarantee additionally requires valid bounds on loss estimation error; it does not follow from a positive estimated gap alone\.

### H\.2Information bottlenecks and task\-dependent representation learning

The information bottleneck formalizes representation learning as a trade\-off between compression of the input and retention of information about a target\([Tishby et al\., 1999](https://arxiv.org/html/2610.27446#bib.bib10)\)\. Variational formulations make the objective usable with neural encoders\([Alemi et al\., 2017](https://arxiv.org/html/2610.27446#bib.bib11)\)\. The theory of usable information further makes information relative to a computational family, rather than treating all statistically present information as equally available\([Xu et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib12)\)\. The information\-bottleneck and usable\-information views motivate the separation between what the representation contains and what a restricted downstream rule can extract\.

The paper makes the separation explicit through four nested risks and three adjacent gaps\. The interface information gap compares the full\-state optimum with the optimum over every measurable rule that sees only the declared interface\. The decision\-rule accessibility gap then restricts the rule family, and the fitting gap measures the remaining cost of fitting one rule from finite data\. Mutual information, compression rate, reconstruction error, and predictive likelihood can be useful representation diagnostics, but none of them alone equals the declared continuation loss\. A low\-energy coordinate can carry a large action distinction, and a high\-fidelity reconstruction can still remove information needed to choose among clarification, refusal, and bounded assistance\. The paper therefore measures decision loss directly and treats reconstruction only as a separate diagnostic\.

### H\.3State abstraction, bisimulation, and control\-sufficient representations

State abstraction in Markov decision processes studies when states may be merged while preserving planning or control properties\. Exact model reduction and homomorphism\-style equivalence relations preserve rewards, transitions, values, or optimal policies\([Givan et al\., 2003](https://arxiv.org/html/2610.27446#bib.bib13);[Li et al\., 2006](https://arxiv.org/html/2610.27446#bib.bib14)\)\. Bisimulation metrics extend exact equivalence to graded state similarity and provide value\-loss bounds for finite and continuous state spaces\([Ferns et al\., 2004](https://arxiv.org/html/2610.27446#bib.bib15);[Ferns et al\., 2011](https://arxiv.org/html/2610.27446#bib.bib16)\)\. Approximate abstraction work then gives explicit suboptimality guarantees when aggregation is inexact\([Abel et al\., 2016](https://arxiv.org/html/2610.27446#bib.bib17);[Abel et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib18)\)\.

Learned control representations apply related criteria to high\-dimensional observations\. DeepMDP learns a latent model by matching rewards and transition distributions\([Gelada et al\., 2019](https://arxiv.org/html/2610.27446#bib.bib19)\)\. Causal and invariant state representations seek features that retain stable return\-relevant structure across environments\([Zhang et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib20);[Zhang et al\., 2021](https://arxiv.org/html/2610.27446#bib.bib21)\)\. Mutual\-information objectives have been analyzed according to whether they are sufficient for control\([Rakelly et al\., 2021](https://arxiv.org/html/2610.27446#bib.bib22)\), and kinematic abstraction supports efficient exploration in rich\-observation environments\([Misra et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib23)\)\. Recent work on vision\-language\-action models directly diagnoses state aliasing when visually similar states require different low\-level actions\([Lee et al\., 2026](https://arxiv.org/html/2610.27446#bib.bib24)\)\.

Our audit takes the full state, continuation library, and learned interface as inputs\. It tests whether states sharing one observation admit a common choice within the allowed regret\. Every pair of acceptable sets can intersect while the intersection of all sets is empty\. A graph containing only incompatible pairs misses such a group; the conflict hypergraph records it\. Nontransitivity alone would not establish this failure of all pairwise checks\. The separate risk comparison also distinguishes missing information from restrictions of the rule family and fitting error\.

### H\.4Partial observability, predictive states, and closed\-loop continuations

Partially observable control replaces the unobserved physical state with an information state\. In a finite partially observable Markov decision process, the posterior belief is sufficient for finite\-horizon control\([Smallwood and Sondik, 1973](https://arxiv.org/html/2610.27446#bib.bib25);[Kaelbling et al\., 1998](https://arxiv.org/html/2610.27446#bib.bib26)\)\. Predictive state representations instead summarize a history by predictions of future tests\([Littman et al\., 2001](https://arxiv.org/html/2610.27446#bib.bib27);[Singh et al\., 2004](https://arxiv.org/html/2610.27446#bib.bib28)\), and spectral predictive\-state methods connect representation learning with planning\([Boots et al\., 2011](https://arxiv.org/html/2610.27446#bib.bib29)\)\. Approximate information states quantify the loss from using a learned or compressed history summary in partially observed planning and reinforcement learning\([Subramanian et al\., 2022](https://arxiv.org/html/2610.27446#bib.bib30)\)\. Options provide a related formalism for temporally extended closed\-loop actions whose internal policies react to future observations\([Sutton et al\., 1999](https://arxiv.org/html/2610.27446#bib.bib31)\)\.

Beliefs, predictive states, and options motivate continuations that can react to later observations\. The full audit state may contain a history; the tested interface is an additional channel applied to that state\. A conflict is relative to the available continuations\. Enlarging the library changes both the choices and the oracle baseline, so transfer to a larger library requires a separate comparison\.

### H\.5Set\-valued prediction, abstention, and higher\-order compatibility

Classification with rejection permits a predictor to abstain when the cost of a forced label is too high\([Chow, 1970](https://arxiv.org/html/2610.27446#bib.bib32);[Bartlett and Wegkamp, 2008](https://arxiv.org/html/2610.27446#bib.bib33);[Charoenphakdee et al\., 2021](https://arxiv.org/html/2610.27446#bib.bib34)\)\. Confidence\-set and set\-valued classifiers return several plausible labels while controlling set size, error, or coverage\([Denis and Hebiri, 2017](https://arxiv.org/html/2610.27446#bib.bib35);[Sadinle et al\., 2019](https://arxiv.org/html/2610.27446#bib.bib36)\)\. Conformal prediction gives finite\-sample coverage for prediction sets under exchangeability\([Vovk et al\., 2005](https://arxiv.org/html/2610.27446#bib.bib37)\)\. Reject\-option and set\-valued prediction methods establish that a decision problem can naturally have a set of acceptable outputs rather than one required label\.

The set\-valued object in the present paper has a different index structure\. For each full stateuu, the acceptable set𝒢ε​\(u\)\\mathcal\{G\}^\{\\varepsilon\}\(u\)contains continuation distributions whose regret atuuis at mostε\\varepsilon\. Uniform compatibility at toleranceε\\varepsilonrequires the intersection

⋂u∈Cz𝒢ε​\(u\)\\bigcap\_\{u\\in C\_\{z\}\}\\mathcal\{G\}^\{\\varepsilon\}\(u\)to be nonempty for every interface cellCzC\_\{z\}\. Set\-valued classification usually controls which labels may be returned for one input; the present intersection asks whether one downstream decision can serve several full states that the interface makes indistinguishable\.

Convex geometry and game theory then determine the finite witness structure\. Helly’s theorem converts an empty intersection of convex acceptable sets in an\(m−1\)\(m\-1\)\-dimensional continuation simplex into a subfamily of at mostmmsets\([Helly, 1923](https://arxiv.org/html/2610.27446#bib.bib38)\)\. Sion’s minimax theorem gives the dual state\-mixture interpretation of the robust witness margin\([Sion, 1958](https://arxiv.org/html/2610.27446#bib.bib39)\)\. Weak hypergraph coloring characterizes how many interface labels are needed to split every minimal conflict\([Berge, 1989](https://arxiv.org/html/2610.27446#bib.bib40)\)\. These are classical tools\. Here they relate the loss of a fixed interface to incompatible state groups and the output count of a redesigned interface\. A gap exceedingε\\varepsilonimplies a cell with margin exceedingε\\varepsilon; positivity alone does not establish conflict at an arbitrary tolerance\. Certifying a group selected from estimated losses also requires simultaneous error control or independent certification data\.

### H\.6Safe reinforcement learning, shielding, and latent control

Safe reinforcement learning studies reward optimization under safety costs, constraints, or intervention mechanisms\([García and Fernández, 2015](https://arxiv.org/html/2610.27446#bib.bib41);[Achiam et al\., 2017](https://arxiv.org/html/2610.27446#bib.bib42);[Chow et al\., 2018](https://arxiv.org/html/2610.27446#bib.bib43)\)\. Shielding places a verified filter between a learned policy and the environment\([Alshiekh et al\., 2018](https://arxiv.org/html/2610.27446#bib.bib44);[Carr et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib45)\)\. Reachability\-based filters and model\-predictive safety filters override or modify unsafe proposed controls\([Fisac et al\., 2019](https://arxiv.org/html/2610.27446#bib.bib46);[Wabersich and Zeilinger, 2021](https://arxiv.org/html/2610.27446#bib.bib47)\)\. Reviews of safe learning in robotics organize these methods by their assumptions about dynamics, uncertainty, constraints, and certification\([Brunke et al\., 2022](https://arxiv.org/html/2610.27446#bib.bib48)\)\.

A related language\-model line reads or changes internal activations\. Latent knowledge methods search for information represented internally without direct supervision\([Burns et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib49)\)\. Inference\-time intervention and representation engineering learn activation directions used to steer model behavior\([Li et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib50);[Zou et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib51)\)\. Latent reachability has also been proposed for early detection and steering of language\-model misalignment\([Karnik and Bansal, 2025](https://arxiv.org/html/2610.27446#bib.bib52)\)\.

Rather than evaluate only a particular monitor, we minimize over all rules using its specified input\. A positive gap rules out matching the full state optimum with that input under the same library and loss\. The audit does not rule out changing the representation, available strategies, or decision problem\.

### H\.7Sparse autoencoders and activation\-space analysis

Sparse coding represents an observation by a small number of dictionary elements and has long been used to study distributed structure\([Olshausen and Field, 1996](https://arxiv.org/html/2610.27446#bib.bib53)\)\.kk\-sparse autoencoders impose a fixed activation budget in the learned code\([Makhzani and Frey, 2013](https://arxiv.org/html/2610.27446#bib.bib54)\)\. Work on superposition argues that neural networks may represent more features than available neurons by using non\-orthogonal directions\([Elhage et al\., 2022](https://arxiv.org/html/2610.27446#bib.bib55)\)\. Sparse autoencoders \(SAEs\) are now widely used to recover sparse, interpretable activation features from language models\([Bricken et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib56);[Cunningham et al\., 2024](https://arxiv.org/html/2610.27446#bib.bib57)\)\.

Other work studies training scale and coverage across models and layers\. Large SAEs have been trained and evaluated with explicit reconstruction–sparsity controls\([Templeton et al\., 2024](https://arxiv.org/html/2610.27446#bib.bib58);[Gao et al\., 2025](https://arxiv.org/html/2610.27446#bib.bib59)\), and Gemma Scope provides public SAEs across models and layers\([Lieberum et al\., 2024](https://arxiv.org/html/2610.27446#bib.bib60)\)\. Activation geometry can also be distributed across groups of SAE features rather than represented by one isolated feature\([Bhalla et al\., 2026](https://arxiv.org/html/2610.27446#bib.bib61)\)\. The SAE literature makes reconstruction quality, feature interpretation, feature geometry, dead\-feature rate, and causal intervention important SAE diagnostics\.

We treat selected SAE values or support indicators as the controller input\. For a finite input alphabet, the risk calculation optimizes all lookup rules and does not depend on probe architecture or training\. For distinct continuous vectors in a finite sample, the same unrestricted rule can memorize the sample\. Continuous observations therefore require a separate analysis before probe failure can be interpreted as information loss\.

### H\.8Language\-model judges and finite\-sample evaluation

Large language models are frequently used as evaluators for open\-ended model outputs\. Chatbot Arena uses pairwise model comparisons at scale\([Zheng et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib62)\), and G\-Eval prompts a strong model to score generated text along declared dimensions\([Liu et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib63)\)\. Fine\-grained evaluator models such as Prometheus seek reproducible rubric\-conditioned judgments\([Kim et al\., 2024](https://arxiv.org/html/2610.27446#bib.bib64)\)\. RewardBench evaluates reward models and preference judges across several behavioral categories\([Lambert et al\., 2025](https://arxiv.org/html/2610.27446#bib.bib65)\)\. At the same time, position bias can make model\-based pairwise evaluation unreliable\([Wang et al\., 2024](https://arxiv.org/html/2610.27446#bib.bib66)\)\.

Statistical work addresses the distinction between a useful noisy judge and a valid evaluation procedure\. Imperfect\-judge testing can preserve type\-I error control after a human calibration stage\([Feng et al\., 2026](https://arxiv.org/html/2610.27446#bib.bib67)\)\. Risk\-control methods construct prediction sets or deployment rules with finite\-sample bounds\([Bates et al\., 2021](https://arxiv.org/html/2610.27446#bib.bib68);[Angelopoulos et al\., 2025](https://arxiv.org/html/2610.27446#bib.bib69)\), while simultaneous concentration inequalities support uniform control across many bounded quantities\([Hoeffding, 1963](https://arxiv.org/html/2610.27446#bib.bib70)\)\.

We analyze automatic scores and human ratings as separate loss tables\. Their gaps may agree in sign without their response scores being close\. The human study reports independent rating agreement before adjudication\. For conflict search, the certificate uses simultaneous bounds over the searched losses or separate certification data; a maximum error observed in one packet is not a uniform bound for future responses\.

### H\.9Generative\-model safeguards, red teaming, and path\-level safety

Generative\-model safety work often evaluates a completed output, a prompt– response pair, or a model\-wide failure rate\. RealToxicityPrompts measures toxic text degeneration under naturally occurring prompts\([Gehman et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib71)\)\. Red\-teaming methods search for prompts that elicit harmful behavior\([Perez et al\., 2022](https://arxiv.org/html/2610.27446#bib.bib72)\)\. Constitutional AI uses written principles and model feedback to train a less harmful assistant\([Bai et al\., 2022](https://arxiv.org/html/2610.27446#bib.bib73)\), while Llama Guard provides an input–output safeguard classifier for conversational systems\([Inan et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib74)\)\.

A separate logical question is whether each partial state has at least one safe continuation\. Even if the answer is yes, states sharing one controller input may have no common safe continuation\. Our regret criterion is related but not identical: it compares with the best available continuation, whose loss need not be zero\. The controlled example demonstrates failure of joint compatibility despite success of every pair; the searched natural\-language example does not replicate that result\.

### H\.10Summary of the comparison

Table 25:Relation between the present paper and the closest technical lines of work\.Line of workWhat the prior line establishesTarget in the present paperComparison of experiments and Bayes sufficiencyOrders information channels by attainable risk and characterizes exact or approximate loss\-dependent sufficiency\([Blackwell, 1953](https://arxiv.org/html/2610.27446#bib.bib2);[Le Cam, 1964](https://arxiv.org/html/2610.27446#bib.bib3);[van Rooyen and Williamson, 2014](https://arxiv.org/html/2610.27446#bib.bib5);[Sevetlidis, 2026](https://arxiv.org/html/2610.27446#bib.bib7)\)\.Uses optimal\-risk comparison as the diagnostic foundation, then extracts minimal conflicts among acceptable closed\-loop continuation sets and connects them to capacity and certification\.Action\-sufficient compression and decision\-sufficient dataRetains the distinctions needed to recover one optimal action or an optimal solution to a declared optimization family\([Walsh, 2026](https://arxiv.org/html/2610.27446#bib.bib8);[Ye et al\., 2026](https://arxiv.org/html/2610.27446#bib.bib9)\)\.Allows several acceptable continuations at each state\. Every pair can share an acceptable choice even when the whole group cannot; the hypergraph records these larger conflicts\.Information bottlenecksTrades compression against predictive or computationally usable information\([Tishby et al\., 1999](https://arxiv.org/html/2610.27446#bib.bib10);[Alemi et al\., 2017](https://arxiv.org/html/2610.27446#bib.bib11);[Xu et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib12)\)\.Uses the declared safety–utility loss itself and separates missing information from restricted access to retained information and finite fitting\.MDP abstraction and bisimulationConstructs exact or approximate summaries that preserve models, values, policies, or exploration structure\([Givan et al\., 2003](https://arxiv.org/html/2610.27446#bib.bib13);[Ferns et al\., 2011](https://arxiv.org/html/2610.27446#bib.bib16);[Abel et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib18);[Misra et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib23)\)\.Audits a fixed learned interface and returns exact risk for the finite interface, minimal conflicting groups, and the required number of interface labels\.Partial observability and predictive statesBuilds belief, information, or predictive states sufficient for recursive planning; options define temporally extended closed\-loop actions\([Kaelbling et al\., 1998](https://arxiv.org/html/2610.27446#bib.bib26);[Singh et al\., 2004](https://arxiv.org/html/2610.27446#bib.bib28);[Subramanian et al\., 2022](https://arxiv.org/html/2610.27446#bib.bib30);[Sutton et al\., 1999](https://arxiv.org/html/2610.27446#bib.bib31)\)\.Assumes the full audit state and continuation library are declared, then tests whether an additional operational interface preserves common continuation compatibility within each observation fiber\.Set\-valued prediction and abstentionReturns several plausible labels or rejects a prediction while controlling coverage, ambiguity, or cost\([Bartlett and Wegkamp, 2008](https://arxiv.org/html/2610.27446#bib.bib33);[Denis and Hebiri, 2017](https://arxiv.org/html/2610.27446#bib.bib35);[Sadinle et al\., 2019](https://arxiv.org/html/2610.27446#bib.bib36)\)\.Intersects acceptable continuation sets across states sharing one observation; pairwise intersections do not determine the full intersection\.Safe control, shielding, and latent steeringOptimizes constrained policies or applies a monitor, shield, filter, or latent intervention to a chosen observation\([Achiam et al\., 2017](https://arxiv.org/html/2610.27446#bib.bib42);[Alshiekh et al\., 2018](https://arxiv.org/html/2610.27446#bib.bib44);[Fisac et al\., 2019](https://arxiv.org/html/2610.27446#bib.bib46);[Li et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib50);[Karnik and Bansal, 2025](https://arxiv.org/html/2610.27446#bib.bib52)\)\.Tests whether the chosen observation interface is adequate before the monitoring or steering rule is trusted\.SAE and activation\-space analysisStudies sparse feature recovery, interpretation, reconstruction, scaling, causal effects, and activation geometry\([Cunningham et al\., 2024](https://arxiv.org/html/2610.27446#bib.bib57);[Gao et al\., 2025](https://arxiv.org/html/2610.27446#bib.bib59);[Lieberum et al\., 2024](https://arxiv.org/html/2610.27446#bib.bib60);[Bhalla et al\., 2026](https://arxiv.org/html/2610.27446#bib.bib61)\)\.Treats an SAE\-derived code or coordinate set as an operational control interface and measures the exact decision loss induced by its state collisions\.Model\-based judges and risk controlMeasures open\-ended outputs with learned evaluators and develops calibration or finite\-sample risk procedures for imperfect measurements\([Liu et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib63);[Wang et al\., 2024](https://arxiv.org/html/2610.27446#bib.bib66);[Feng et al\., 2026](https://arxiv.org/html/2610.27446#bib.bib67);[Angelopoulos et al\., 2025](https://arxiv.org/html/2610.27446#bib.bib69)\)\.Uses automatic and human loss constructions separately, preserves validity after witness search, and does not infer close item\-level calibration from a positive aggregate interface gap\.Generative safeguardsDetects harmful prompts or completed outputs and trains safer assistants\([Gehman et al\., 2020](https://arxiv.org/html/2610.27446#bib.bib71);[Perez et al\., 2022](https://arxiv.org/html/2610.27446#bib.bib72);[Bai et al\., 2022](https://arxiv.org/html/2610.27446#bib.bib73);[Inan et al\., 2023](https://arxiv.org/html/2610.27446#bib.bib74)\)\.Tests whether states sharing a controller input have a common acceptable continuation; a conflict can first appear for a triple\.#### Relation to a single optimal label\.

On a finite audit with one optimal pure continuation at each state, letρ\\rhobe the smallest regret of a nonoptimal continuation\. For0≤ε<ρ/20\\leq\\varepsilon<\\rho/2, different optimal labels are already incompatible in pairs \(Proposition[B\.2](https://arxiv.org/html/2610.27446#A2.Thmassumption2)\)\. With larger tolerances or several acceptable choices, a single selected label need not describe all compatible groups\. The hypergraph supplies the complete finite set of incompatibility constraints\. Statistical claims about estimated constraints still require the loss error assumptions in Appendix[C](https://arxiv.org/html/2610.27446#A3)\.

## Appendix IComplete Phi Third\-Family Results

This appendix reports the source\-derived Phi results\. Original 96 denotes the original eight\-area confirmation split; Reused 96 denotes the eight\-area collection locked for the earlier SmolLM2 experiment and reused for Phi\. The fitting split has 144 original states\. The original experiment archive is retained assupplementary/phi/PHI\_THIRD\_FAMILY\_RESULTS\_original\.zipin the author working archive\. The Phi study includes 5,040 generated responses evaluated by both automatic evaluators\. No human ratings were collected for the Phi responses\. The 288 risk\-summary rows comprise two SAE seeds, two selectors, six budgets, two collections, three evaluator losses, and two observation conditions\. These are overlapping analyses of the same responses, not 288 independent replications\.

### I\.1Configuration, provenance, and validation

The active protocol identifies Phi\-4\-mini\-instruct revisioncfbefacb99257ffa30c83adab238a50856ac3083; Mistral\-7B\-Instruct\-v0\.3 revisionc170c708c41dac9275d15a8fff4eca08d52bab71; Qwen2\.5\-7B\-Instruct revisiona09a35458c702b33eeacc393d103063234e8bc28; and FineWeb\-Edusample\-10BTrevision87f09149ef4734204d70ed1d046ddc9ca3f2b8f9\. Block index 23 has width 3,072 and depth fraction0\.750\.75\. The two SAE seeds are 314159 and 271828, dictionary width is 16,384, and TopK sparsity is 64\.

Each SAE uses 8,000,000 training tokens and 300,000 validation tokens, with 100,000 training tokens for normalization\. Corpus contexts have length 128 and cached activation shards contain 32,768 tokens\. SAE batches contain 1,024 activation tokens; AdamW uses learning rate2×10−42\\times 10^\{\-4\}, 500 warm\-up steps, zero weight decay, gradient norm clipping at 1, auxiliary coefficient0\.031250\.03125, and a 1,000\-step dead\-feature window\. Both training records end at step 7,813 and 8,000,000 consumed tokens\. Validation requires FVU at most0\.450\.45, cosine at least0\.750\.75, unobserved\-feature fraction at most0\.50\.5, and mean\-L0L\_\{0\}deviation from 64 at most3\.23\.2\. Both runs pass these saved checks\.

Generation uses five replicas per state–action pair, at most 192 new tokens, temperature0\.70\.7, and top\-p=0\.9p=0\.9\. The two evaluators each score all 5,040 responses, with a 160\-token output limit and up to four format retries\. Mistral scores parse on the first attempt; four Qwen scores require one format retry\. There are no missing or duplicated state–action–replica keys\. The saved execution takes approximately 4\.63 hours on an A100\-SXM4\-40GB\. The recorded environment is Python 3\.13\.15, PyTorch 2\.11\.0\+cu128, Transformers 4\.57\.6, Datasets 4\.4\.1, Hugging Face Hub 0\.36\.2, Accelerate 1\.14\.0, NumPy 2\.1\.3, Pandas 2\.2\.3, and SciPy 1\.16\.3\.

Cached feature arrays and item\-level scores independently reproduce all 288 risk/bootstrap summaries and all 192 deletion\-level estimates to floating\-point precision\. This checks table construction and arithmetic, not unseen training weights, evaluator validity, or human judgments\. Bootstrap resampling uses 2,000 multinomial policy\-area resamples and seed 99173\. An eight\-cluster percentile interval remains an approximate, per\-setting uncertainty summary rather than the formal simultaneous certificate in Section[3](https://arxiv.org/html/2610.27446#S3)\.

### I\.2Phi retains a positive gap when the active policy is observed

The Phi run completes all 5,040 generations and both complete evaluator passes\. The two SAEs complete the declared training schedule of eight million tokens and pass the saved validation thresholds\. No Phi human labels have been collected\.

#### Policy\-aware primary results\.

Table[26](https://arxiv.org/html/2610.27446#A9.T26)reports the primary\(Z,S\)\(Z,S\)observation atB=2B=2\. Under Mistral loss, the information gaps are0\.05570\.0557and0\.04910\.0491on the original confirmation set and0\.04360\.0436and0\.03310\.0331on the reused locked collection, for seeds 314159 and 271828, respectively\. The corresponding one\-sided 95% bootstrap LCBs are0\.03670\.0367,0\.02660\.0266,0\.02250\.0225, and0\.01850\.0185\. Every primary Qwen\-loss and mean\-loss check also has a positive individual LCB\. The tables below include both risk minima, two\-sided intervals, and the complete budget sweeps\.

Table 26:Phi primary policy\-aware results at feature budgetB=2B=2\. Each entry is the information gapΔ^Z,S\\widehat\{\\Delta\}\_\{Z,S\}followed by its individual one\-sided 95% bootstrap over policy areas lower bound in brackets\. The original confirmation and reused locked collections each contain 96 states\. Mean loss averages the two evaluators before risk optimization\.These results retain the information restriction after active policy is supplied: some states under the same policy and SAE code still cannot share the lowest\-loss macro\-continuation under the saved loss table\. The 12 policy\-aware settings are not independent replications; the evaluator checks and their mean share the same responses, states, and features\. All 2,000 bootstrap gaps in each primary setting exceed the saved numerical tolerance, but the zero nonpositive fraction is not app\-value or a simultaneous certificate\.

#### Paired observation comparison\.

Providing the active policy reduces or leaves unchanged the exact information gap in every primary Phi setting, as required by Equation \([6](https://arxiv.org/html/2610.27446#S2.E6)\)\. For seed 314159 on the original confirmation set, the Mistral\-loss gap decreases from0\.11100\.1110forZZalone to0\.05570\.0557for\(Z,S\)\(Z,S\)\. The positive policy\-aware gap therefore cannot be explained solely by hiding the active policy\. Both observation conditions keep all other inputs fixed\. The historical Qwen2\.5 and SmolLM2 tables estimate code\-only risk; the separate policy\-aware analyses in Section[5\.3](https://arxiv.org/html/2610.27446#S5.SS3)have their own stated source precision\.

#### Budget and policy\-area sensitivity\.

Figure[23](https://arxiv.org/html/2610.27446#A9.F23)shows the policy\-aware Mistral\-loss curves\. All supervisedB=2B=2andB=4B=4settings have positive individual LCBs across the two seeds, collections, and three loss constructions\. FromB=8B=8onward some LCBs reach zero; atB=32B=32andB=64B=64many finite\-table gaps vanish\. Zero gap on these states does not establish sufficiency on future states\. In particular, the complete\(Z,S\)\(Z,S\)keys are unseen in the fitting split because the evaluation policy texts differ from the fitting policy texts\. The exact evaluation\-set optimum is not the performance of a learned lookup table on those unseen keys\.

\(a\)Original confirmation collection\.\(b\)Reused locked collection\.
Figure 23:Phi policy\-aware information gaps under Mistral loss\. Points use the frozen supervised features, and bands are individual two\-sided 95% bootstrap over policy areas intervals\. Budget positions are equally spaced for readability\. Both the policy text and selected SAE support code are observed\.AtB=2B=2, deleting any one of the eight policy areas leaves 84 states\. Every deletion\-level gap is positive across both seeds, both collections, all three loss constructions, and both observation conditions: 192 point estimates in total, with minimum0\.02360\.0236\. No single deletion removes the positive gap in these collections\. Table[37](https://arxiv.org/html/2610.27446#A9.T37)summarizes the deletion ranges\.

#### Reconstruction quality and distribution shift\.

Validation centered fraction of variance unexplained \(FVU\) is0\.01620\.0162and0\.01630\.0163; mean reconstruction cosine is0\.92770\.9277for both SAE seeds, meanL0L\_\{0\}is 64, and all 16,384 features appear at least once on validation\. Reconstruction on audit states is weaker\. Centered FVU is17\.312517\.3125and16\.651816\.6518on the original 240 states, and2\.83552\.8355and2\.77052\.7705on the reused 96 states\. Uncentered relative squared reconstruction error is0\.27040\.2704–0\.30620\.3062across these four audit settings\. On the original audit distribution, centered activation energy per vector is232\.49232\.49, compared with uncentered energy13,143\.0413\{,\}143\.04\. The smaller denominator explains why centered FVU is much larger than uncentered relative error, but does not remove the reconstruction error itself\. Passing corpus validation does not establish accurate reconstruction on the audit domain\. Table[34](https://arxiv.org/html/2610.27446#A9.T34)reports the distribution\-specific diagnostics\.

#### Evaluator comparison\.

The primary gap remains positive with either evaluator, but their scores have limited agreement on individual responses\. Detailed comparisons below use the same response keys for both evaluators\.

### I\.3Policy\-aware and code\-only primary results

The complete observation is explicitlyV=\(Z,S\)V=\(Z,S\)orV=ZV=Z\. Equal\-weight losses are averaged before minimization\. Full\-state, best\-interface, and gap values can differ slightly from arithmetic on the displayed rounded numbers because the calculations use unrounded values\.

Table 27:Phi full primary results for observation\(Z,S\)\(Z,S\)atB=2B=2\.RV⋆R\_\{V\}^\{\\star\}denotes the exact finite\-table optimum for the stated observationVV\. CI is the two\-sided 95% bootstrap interval; LCB is the one\-sided 95% lower bound\.f0f\_\{0\}is the fraction of 2,000 bootstrap gaps at or below the saved10−1210^\{\-12\}numerical tolerance, not app\-value\.EvaluatorR^full\\widehat\{R\}\_\{\\mathrm\{full\}\}R^V⋆\\widehat\{R\}\_\{V\}^\{\\star\}Δ^V\\widehat\{\\Delta\}\_\{V\}95% CI95% LCBf0f\_\{0\}Original 96, seed 314159Mistral0\.07420\.12990\.0557\[0\.0332, 0\.0761\]0\.03670\.000Qwen0\.16650\.24940\.0830\[0\.0554, 0\.1151\]0\.05880\.000Mean loss0\.14470\.20150\.0568\[0\.0339, 0\.0801\]0\.03810\.000Original 96, seed 271828Mistral0\.07420\.12320\.0491\[0\.0230, 0\.0777\]0\.02660\.000Qwen0\.16650\.20290\.0365\[0\.0198, 0\.0517\]0\.02290\.000Mean loss0\.14470\.17440\.0296\[0\.0136, 0\.0469\]0\.01570\.000Reused 96, seed 314159Mistral0\.10060\.14430\.0436\[0\.0191, 0\.0718\]0\.02250\.000Qwen0\.19100\.24940\.0584\[0\.0394, 0\.0783\]0\.04240\.000Mean loss0\.17310\.24390\.0708\[0\.0437, 0\.0999\]0\.04770\.000Reused 96, seed 271828Mistral0\.10060\.13380\.0331\[0\.0163, 0\.0515\]0\.01850\.000Qwen0\.19100\.23740\.0465\[0\.0250, 0\.0692\]0\.02830\.000Mean loss0\.17310\.22010\.0470\[0\.0296, 0\.0704\]0\.03200\.000Table 28:Phi full primary results for observationZZonly atB=2B=2\.RV⋆R\_\{V\}^\{\\star\}denotes the exact finite\-table optimum for the stated observationVV\. CI is the two\-sided 95% bootstrap interval; LCB is the one\-sided 95% lower bound\.f0f\_\{0\}is the fraction of 2,000 bootstrap gaps at or below the saved10−1210^\{\-12\}numerical tolerance, not app\-value\.EvaluatorR^full\\widehat\{R\}\_\{\\mathrm\{full\}\}R^V⋆\\widehat\{R\}\_\{V\}^\{\\star\}Δ^V\\widehat\{\\Delta\}\_\{V\}95% CI95% LCBf0f\_\{0\}Original 96, seed 314159Mistral0\.07420\.18520\.1110\[0\.0732, 0\.1296\]0\.07990\.000Qwen0\.16650\.26750\.1010\[0\.0647, 0\.1460\]0\.06900\.000Mean loss0\.14470\.22630\.0816\[0\.0495, 0\.1115\]0\.05400\.000Original 96, seed 271828Mistral0\.07420\.15170\.0775\[0\.0442, 0\.0978\]0\.04820\.000Qwen0\.16650\.21520\.0487\[0\.0254, 0\.0654\]0\.03080\.000Mean loss0\.14470\.19320\.0485\[0\.0262, 0\.0587\]0\.02880\.000Reused 96, seed 314159Mistral0\.10060\.16610\.0655\[0\.0376, 0\.0917\]0\.04140\.000Qwen0\.19100\.29830\.1073\[0\.0445, 0\.1581\]0\.05030\.000Mean loss0\.17310\.26830\.0952\[0\.0664, 0\.1173\]0\.07080\.000Reused 96, seed 271828Mistral0\.10060\.16190\.0612\[0\.0332, 0\.0788\]0\.03680\.000Qwen0\.19100\.29380\.1028\[0\.0419, 0\.1248\]0\.04820\.000Mean loss0\.17310\.24350\.0704\[0\.0436, 0\.0946\]0\.04780\.000Table 29:Paired observation\-condition comparison for the same Phi responses, losses, frozen features, andB=2B=2\. Supplying active policy can refine aZZ\-only cell and never increases the exact optimal interface risk\. Differences are computed before rounding\.
### I\.4Complete supervised\-selector and variance\-selector sweeps

Only fitting data determine the feature identities\. The supervised ranking uses Mistral loss, and all other evaluator conditions reuse that ranking\. No interface is selected using the Phi confirmation losses\. The 288 rows below retain both positive and zero\-gap results; they do not extend theB=2B=2claim to every budget\.

Table 30:Complete Phi supervised\-selector results for observation\(Z,S\)\(Z,S\)\. Each row uses eight policy\-area clusters and 2,000 percentile bootstrap repetitions\. Features are selected on the fitting split and fixed across evaluators\.f0f\_\{0\}uses the saved10−1210^\{\-12\}tolerance; intervals are individual, not simultaneous\.BBEvaluatorR^full\\widehat\{R\}\_\{\\mathrm\{full\}\}R^V⋆\\widehat\{R\}\_\{V\}^\{\\star\}Δ^V\\widehat\{\\Delta\}\_\{V\}95% CI95% LCBf0f\_\{0\}Original 96, seed 3141592Mistral0\.07420\.12990\.0557\[0\.0332, 0\.0761\]0\.03670\.0002Qwen0\.16650\.24940\.0830\[0\.0554, 0\.1151\]0\.05880\.0002Mean loss0\.14470\.20150\.0568\[0\.0339, 0\.0801\]0\.03810\.0004Mistral0\.07420\.11020\.0360\[0\.0201, 0\.0535\]0\.02240\.0004Qwen0\.16650\.23280\.0663\[0\.0367, 0\.1005\]0\.04130\.0004Mean loss0\.14470\.18600\.0413\[0\.0202, 0\.0642\]0\.02290\.0008Mistral0\.07420\.08490\.0107\[0\.0013, 0\.0250\]0\.00240\.0048Qwen0\.16650\.18310\.0167\[0\.0023, 0\.0354\]0\.00230\.0248Mean loss0\.14470\.15320\.0085\[0\.0022, 0\.0152\]0\.00330\.00416Mistral0\.07420\.07790\.0037\[0\.0000, 0\.0112\]0\.00000\.33416Qwen0\.16650\.16790\.0015\[0\.0000, 0\.0044\]0\.00000\.33416Mean loss0\.14470\.14580\.0010\[0\.0000, 0\.0031\]0\.00000\.33432Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Qwen0\.16650\.16650\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.16650\.16650\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.000Original 96, seed 2718282Mistral0\.07420\.12320\.0491\[0\.0230, 0\.0777\]0\.02660\.0002Qwen0\.16650\.20290\.0365\[0\.0198, 0\.0517\]0\.02290\.0002Mean loss0\.14470\.17440\.0296\[0\.0136, 0\.0469\]0\.01570\.0004Mistral0\.07420\.09700\.0228\[0\.0117, 0\.0351\]0\.01310\.0004Qwen0\.16650\.18690\.0204\[0\.0077, 0\.0348\]0\.00960\.0014Mean loss0\.14470\.15870\.0140\[0\.0066, 0\.0217\]0\.00750\.0008Mistral0\.07420\.08550\.0114\[0\.0035, 0\.0204\]0\.00400\.0048Qwen0\.16650\.17460\.0081\[0\.0000, 0\.0198\]0\.00000\.0888Mean loss0\.14470\.14990\.0052\[0\.0011, 0\.0103\]0\.00150\.00416Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00016Qwen0\.16650\.16650\.0000\[0\.0000, 0\.0000\]0\.00001\.00016Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Qwen0\.16650\.16650\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.16650\.16650\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.000Reused 96, seed 3141592Mistral0\.10060\.14430\.0436\[0\.0191, 0\.0718\]0\.02250\.0002Qwen0\.19100\.24940\.0584\[0\.0394, 0\.0783\]0\.04240\.0002Mean loss0\.17310\.24390\.0708\[0\.0437, 0\.0999\]0\.04770\.0004Mistral0\.10060\.12800\.0274\[0\.0135, 0\.0418\]0\.01550\.0004Qwen0\.19100\.23810\.0471\[0\.0286, 0\.0673\]0\.03160\.0004Mean loss0\.17310\.22230\.0493\[0\.0335, 0\.0684\]0\.03560\.0008Mistral0\.10060\.11460\.0140\[0\.0036, 0\.0280\]0\.00490\.0008Qwen0\.19100\.21170\.0207\[0\.0112, 0\.0315\]0\.01260\.0008Mean loss0\.17310\.19210\.0190\[0\.0102, 0\.0292\]0\.01140\.00016Mistral0\.10060\.10420\.0035\[0\.0012, 0\.0065\]0\.00150\.00416Qwen0\.19100\.19950\.0085\[0\.0035, 0\.0135\]0\.00420\.00016Mean loss0\.17310\.17800\.0049\[0\.0005, 0\.0109\]0\.00050\.02232Mistral0\.10060\.10210\.0015\[0\.0000, 0\.0044\]0\.00000\.33932Qwen0\.19100\.19100\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Mean loss0\.17310\.17330\.0002\[0\.0000, 0\.0006\]0\.00000\.33964Mistral0\.10060\.10060\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.19100\.19100\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.17310\.17310\.0000\[0\.0000, 0\.0000\]0\.00001\.000Reused 96, seed 2718282Mistral0\.10060\.13380\.0331\[0\.0163, 0\.0515\]0\.01850\.0002Qwen0\.19100\.23740\.0465\[0\.0250, 0\.0692\]0\.02830\.0002Mean loss0\.17310\.22010\.0470\[0\.0296, 0\.0704\]0\.03200\.0004Mistral0\.10060\.12660\.0259\[0\.0100, 0\.0455\]0\.01170\.0004Qwen0\.19100\.23740\.0465\[0\.0250, 0\.0692\]0\.02830\.0004Mean loss0\.17310\.20870\.0356\[0\.0234, 0\.0495\]0\.02540\.0008Mistral0\.10060\.11780\.0172\[0\.0034, 0\.0351\]0\.00460\.0058Qwen0\.19100\.20440\.0134\[0\.0067, 0\.0220\]0\.00750\.0008Mean loss0\.17310\.18840\.0153\[0\.0079, 0\.0229\]0\.00920\.00016Mistral0\.10060\.10660\.0059\[0\.0000, 0\.0178\]0\.00000\.36016Qwen0\.19100\.19340\.0024\[0\.0000, 0\.0054\]0\.00000\.10716Mean loss0\.17310\.17690\.0038\[0\.0002, 0\.0105\]0\.00020\.02532Mistral0\.10060\.10060\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Qwen0\.19100\.19100\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Mean loss0\.17310\.17310\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mistral0\.10060\.10060\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.19100\.19100\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.17310\.17310\.0000\[0\.0000, 0\.0000\]0\.00001\.000Table 31:Complete Phi variance\-selector results for observation\(Z,S\)\(Z,S\)\. Each row uses eight policy\-area clusters and 2,000 percentile bootstrap repetitions\. Features are selected on the fitting split and fixed across evaluators\.f0f\_\{0\}uses the saved10−1210^\{\-12\}tolerance; intervals are individual, not simultaneous\.BBEvaluatorR^full\\widehat\{R\}\_\{\\mathrm\{full\}\}R^V⋆\\widehat\{R\}\_\{V\}^\{\\star\}Δ^V\\widehat\{\\Delta\}\_\{V\}95% CI95% LCBf0f\_\{0\}Original 96, seed 3141592Mistral0\.07420\.17030\.0961\[0\.0646, 0\.1288\]0\.06890\.0002Qwen0\.16650\.31330\.1468\[0\.1208, 0\.1716\]0\.12480\.0002Mean loss0\.14470\.24950\.1048\[0\.0911, 0\.1164\]0\.09350\.0004Mistral0\.07420\.12770\.0535\[0\.0323, 0\.0810\]0\.03400\.0004Qwen0\.16650\.24250\.0761\[0\.0539, 0\.0981\]0\.05770\.0004Mean loss0\.14470\.19540\.0507\[0\.0361, 0\.0662\]0\.03820\.0008Mistral0\.07420\.08490\.0107\[0\.0028, 0\.0191\]0\.00420\.0048Qwen0\.16650\.18730\.0208\[0\.0110, 0\.0298\]0\.01250\.0008Mean loss0\.14470\.16080\.0160\[0\.0091, 0\.0224\]0\.01010\.00016Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00016Qwen0\.16650\.16710\.0006\[0\.0000, 0\.0019\]0\.00000\.33416Mean loss0\.14470\.14490\.0002\[0\.0000, 0\.0006\]0\.00000\.33432Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Qwen0\.16650\.16650\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.16650\.16650\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.000Original 96, seed 2718282Mistral0\.07420\.21190\.1377\[0\.0984, 0\.1753\]0\.10400\.0002Qwen0\.16650\.32090\.1544\[0\.1227, 0\.1889\]0\.12730\.0002Mean loss0\.14470\.26640\.1217\[0\.0998, 0\.1427\]0\.10290\.0004Mistral0\.07420\.16180\.0876\[0\.0625, 0\.1166\]0\.06610\.0004Qwen0\.16650\.24700\.0805\[0\.0574, 0\.0992\]0\.06150\.0004Mean loss0\.14470\.21030\.0656\[0\.0466, 0\.0843\]0\.04970\.0008Mistral0\.07420\.09390\.0197\[0\.0059, 0\.0362\]0\.00750\.0008Qwen0\.16650\.18350\.0170\[0\.0049, 0\.0306\]0\.00680\.0008Mean loss0\.14470\.15760\.0129\[0\.0038, 0\.0239\]0\.00500\.00016Mistral0\.07420\.07800\.0039\[0\.0004, 0\.0084\]0\.00040\.02216Qwen0\.16650\.16940\.0029\[0\.0000, 0\.0075\]0\.00000\.08816Mean loss0\.14470\.14680\.0021\[0\.0002, 0\.0050\]0\.00020\.02532Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Qwen0\.16650\.16650\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.16650\.16650\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.000Reused 96, seed 3141592Mistral0\.10060\.15030\.0497\[0\.0240, 0\.0777\]0\.02740\.0002Qwen0\.19100\.24130\.0503\[0\.0344, 0\.0669\]0\.03730\.0002Mean loss0\.17310\.24470\.0716\[0\.0433, 0\.1021\]0\.04780\.0004Mistral0\.10060\.13520\.0346\[0\.0129, 0\.0642\]0\.01480\.0004Qwen0\.19100\.22460\.0336\[0\.0224, 0\.0465\]0\.02390\.0004Mean loss0\.17310\.20770\.0346\[0\.0244, 0\.0443\]0\.02590\.0008Mistral0\.10060\.11920\.0185\[0\.0053, 0\.0376\]0\.00620\.0008Qwen0\.19100\.21140\.0204\[0\.0105, 0\.0325\]0\.01170\.0008Mean loss0\.17310\.18920\.0161\[0\.0087, 0\.0257\]0\.00980\.00016Mistral0\.10060\.10100\.0004\[0\.0000, 0\.0013\]0\.00000\.36016Qwen0\.19100\.19560\.0046\[0\.0010, 0\.0090\]0\.00100\.01916Mean loss0\.17310\.17410\.0010\[0\.0000, 0\.0030\]0\.00000\.36032Mistral0\.10060\.10060\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Qwen0\.19100\.19100\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Mean loss0\.17310\.17310\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mistral0\.10060\.10060\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.19100\.19100\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.17310\.17310\.0000\[0\.0000, 0\.0000\]0\.00001\.000Reused 96, seed 2718282Mistral0\.10060\.15220\.0516\[0\.0252, 0\.0797\]0\.02790\.0002Qwen0\.19100\.24940\.0584\[0\.0394, 0\.0783\]0\.04240\.0002Mean loss0\.17310\.25090\.0778\[0\.0478, 0\.1104\]0\.05220\.0004Mistral0\.10060\.14220\.0416\[0\.0185, 0\.0689\]0\.02140\.0004Qwen0\.19100\.23340\.0424\[0\.0270, 0\.0595\]0\.02940\.0004Mean loss0\.17310\.23890\.0658\[0\.0371, 0\.0932\]0\.04230\.0008Mistral0\.10060\.12330\.0227\[0\.0067, 0\.0416\]0\.00890\.0018Qwen0\.19100\.20920\.0182\[0\.0091, 0\.0279\]0\.01050\.0008Mean loss0\.17310\.18640\.0133\[0\.0086, 0\.0179\]0\.00930\.00016Mistral0\.10060\.10570\.0051\[0\.0005, 0\.0105\]0\.00050\.02216Qwen0\.19100\.19830\.0074\[0\.0033, 0\.0111\]0\.00420\.00016Mean loss0\.17310\.18080\.0077\[0\.0022, 0\.0138\]0\.00280\.00032Mistral0\.10060\.10060\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Qwen0\.19100\.19100\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Mean loss0\.17310\.17310\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mistral0\.10060\.10060\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.19100\.19100\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.17310\.17310\.0000\[0\.0000, 0\.0000\]0\.00001\.000Table 32:Complete Phi supervised\-selector results for observationZZonly\. Each row uses eight policy\-area clusters and 2,000 percentile bootstrap repetitions\. Features are selected on the fitting split and fixed across evaluators\.f0f\_\{0\}uses the saved10−1210^\{\-12\}tolerance; intervals are individual, not simultaneous\.BBEvaluatorR^full\\widehat\{R\}\_\{\\mathrm\{full\}\}R^V⋆\\widehat\{R\}\_\{V\}^\{\\star\}Δ^V\\widehat\{\\Delta\}\_\{V\}95% CI95% LCBf0f\_\{0\}Original 96, seed 3141592Mistral0\.07420\.18520\.1110\[0\.0732, 0\.1296\]0\.07990\.0002Qwen0\.16650\.26750\.1010\[0\.0647, 0\.1460\]0\.06900\.0002Mean loss0\.14470\.22630\.0816\[0\.0495, 0\.1115\]0\.05400\.0004Mistral0\.07420\.14720\.0730\[0\.0400, 0\.0934\]0\.04440\.0004Qwen0\.16650\.25870\.0923\[0\.0582, 0\.1227\]0\.06310\.0004Mean loss0\.14470\.22070\.0759\[0\.0447, 0\.0968\]0\.04880\.0008Mistral0\.07420\.11670\.0425\[0\.0117, 0\.0540\]0\.01430\.0008Qwen0\.16650\.21230\.0458\[0\.0112, 0\.0567\]0\.01520\.0018Mean loss0\.14470\.18220\.0375\[0\.0082, 0\.0474\]0\.01080\.00016Mistral0\.07420\.08000\.0058\[0\.0000, 0\.0133\]0\.00000\.16416Qwen0\.16650\.17080\.0044\[0\.0000, 0\.0073\]0\.00000\.16416Mean loss0\.14470\.14830\.0035\[0\.0000, 0\.0074\]0\.00000\.16432Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Qwen0\.16650\.16650\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.16650\.16650\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.000Original 96, seed 2718282Mistral0\.07420\.15170\.0775\[0\.0442, 0\.0978\]0\.04820\.0002Qwen0\.16650\.21520\.0487\[0\.0254, 0\.0654\]0\.03080\.0002Mean loss0\.14470\.19320\.0485\[0\.0262, 0\.0587\]0\.02880\.0004Mistral0\.07420\.14150\.0673\[0\.0325, 0\.0746\]0\.03460\.0004Qwen0\.16650\.20460\.0381\[0\.0165, 0\.0502\]0\.01980\.0014Mean loss0\.14470\.17820\.0335\[0\.0140, 0\.0428\]0\.01680\.0008Mistral0\.07420\.09750\.0233\[0\.0099, 0\.0293\]0\.01150\.0018Qwen0\.16650\.18700\.0206\[0\.0024, 0\.0308\]0\.00460\.0058Mean loss0\.14470\.15840\.0136\[0\.0035, 0\.0176\]0\.00480\.00116Mistral0\.07420\.07800\.0039\[0\.0000, 0\.0069\]0\.00000\.40316Qwen0\.16650\.16770\.0013\[0\.0000, 0\.0027\]0\.00000\.19416Mean loss0\.14470\.14590\.0012\[0\.0000, 0\.0017\]0\.00000\.19532Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Qwen0\.16650\.16700\.0005\[0\.0000, 0\.0015\]0\.00000\.56932Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.16650\.16650\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.000Reused 96, seed 3141592Mistral0\.10060\.16610\.0655\[0\.0376, 0\.0917\]0\.04140\.0002Qwen0\.19100\.29830\.1073\[0\.0445, 0\.1581\]0\.05030\.0002Mean loss0\.17310\.26830\.0952\[0\.0664, 0\.1173\]0\.07080\.0004Mistral0\.10060\.15940\.0587\[0\.0325, 0\.0703\]0\.03590\.0004Qwen0\.19100\.29430\.1033\[0\.0442, 0\.1321\]0\.05030\.0004Mean loss0\.17310\.25600\.0829\[0\.0601, 0\.0976\]0\.06300\.0008Mistral0\.10060\.13380\.0331\[0\.0142, 0\.0408\]0\.01600\.0008Qwen0\.19100\.25200\.0610\[0\.0268, 0\.0731\]0\.02990\.0008Mean loss0\.17310\.21130\.0382\[0\.0207, 0\.0464\]0\.02280\.00016Mistral0\.10060\.10760\.0070\[0\.0013, 0\.0117\]0\.00170\.00416Qwen0\.19100\.21340\.0224\[0\.0048, 0\.0308\]0\.00640\.00016Mean loss0\.17310\.18460\.0115\[0\.0024, 0\.0195\]0\.00340\.00432Mistral0\.10060\.10290\.0023\[0\.0000, 0\.0069\]0\.00000\.33932Qwen0\.19100\.19690\.0059\[0\.0000, 0\.0092\]0\.00000\.13732Mean loss0\.17310\.17690\.0039\[0\.0000, 0\.0054\]0\.00000\.11864Mistral0\.10060\.10060\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.19100\.19100\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.17310\.17310\.0000\[0\.0000, 0\.0000\]0\.00001\.000Reused 96, seed 2718282Mistral0\.10060\.16190\.0612\[0\.0332, 0\.0788\]0\.03680\.0002Qwen0\.19100\.29380\.1028\[0\.0419, 0\.1248\]0\.04820\.0002Mean loss0\.17310\.24350\.0704\[0\.0436, 0\.0946\]0\.04780\.0004Mistral0\.10060\.15160\.0509\[0\.0240, 0\.0679\]0\.02790\.0004Qwen0\.19100\.28310\.0921\[0\.0403, 0\.1172\]0\.04580\.0004Mean loss0\.17310\.23680\.0637\[0\.0361, 0\.0761\]0\.04010\.0008Mistral0\.10060\.13930\.0386\[0\.0116, 0\.0500\]0\.01420\.0008Qwen0\.19100\.24700\.0560\[0\.0200, 0\.0666\]0\.02340\.0008Mean loss0\.17310\.21360\.0406\[0\.0205, 0\.0483\]0\.02300\.00016Mistral0\.10060\.11300\.0124\[0\.0000, 0\.0236\]0\.00000\.12016Qwen0\.19100\.21740\.0264\[0\.0072, 0\.0293\]0\.00850\.00116Mean loss0\.17310\.18810\.0150\[0\.0035, 0\.0184\]0\.00480\.00132Mistral0\.10060\.10060\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Qwen0\.19100\.19480\.0038\[0\.0000, 0\.0075\]0\.00000\.29432Mean loss0\.17310\.17500\.0019\[0\.0000, 0\.0037\]0\.00000\.57464Mistral0\.10060\.10060\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.19100\.19100\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.17310\.17310\.0000\[0\.0000, 0\.0000\]0\.00001\.000Table 33:Complete Phi variance\-selector results for observationZZonly\. Each row uses eight policy\-area clusters and 2,000 percentile bootstrap repetitions\. Features are selected on the fitting split and fixed across evaluators\.f0f\_\{0\}uses the saved10−1210^\{\-12\}tolerance; intervals are individual, not simultaneous\.BBEvaluatorR^full\\widehat\{R\}\_\{\\mathrm\{full\}\}R^V⋆\\widehat\{R\}\_\{V\}^\{\\star\}Δ^V\\widehat\{\\Delta\}\_\{V\}95% CI95% LCBf0f\_\{0\}Original 96, seed 3141592Mistral0\.07420\.21190\.1377\[0\.0938, 0\.1673\]0\.09940\.0002Qwen0\.16650\.32090\.1544\[0\.1227, 0\.1879\]0\.12730\.0002Mean loss0\.14470\.26640\.1217\[0\.0998, 0\.1385\]0\.10290\.0004Mistral0\.07420\.15920\.0850\[0\.0442, 0\.1146\]0\.04900\.0004Qwen0\.16650\.25810\.0916\[0\.0615, 0\.1191\]0\.06670\.0004Mean loss0\.14470\.21320\.0685\[0\.0437, 0\.0879\]0\.04600\.0008Mistral0\.07420\.12260\.0484\[0\.0214, 0\.0516\]0\.02430\.0008Qwen0\.16650\.21650\.0500\[0\.0270, 0\.0615\]0\.02960\.0008Mean loss0\.14470\.19340\.0487\[0\.0245, 0\.0515\]0\.02850\.00016Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00016Qwen0\.16650\.16710\.0006\[0\.0000, 0\.0019\]0\.00000\.33416Mean loss0\.14470\.14490\.0002\[0\.0000, 0\.0006\]0\.00000\.33432Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Qwen0\.16650\.16650\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.16650\.16650\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.000Original 96, seed 2718282Mistral0\.07420\.21190\.1377\[0\.0984, 0\.1753\]0\.10400\.0002Qwen0\.16650\.32090\.1544\[0\.1227, 0\.1889\]0\.12730\.0002Mean loss0\.14470\.26640\.1217\[0\.0998, 0\.1427\]0\.10290\.0004Mistral0\.07420\.21040\.1363\[0\.0906, 0\.1502\]0\.09500\.0004Qwen0\.16650\.27480\.1083\[0\.0733, 0\.1331\]0\.07850\.0004Mean loss0\.14470\.24940\.1047\[0\.0649, 0\.1222\]0\.07160\.0008Mistral0\.07420\.12520\.0510\[0\.0150, 0\.0668\]0\.01830\.0008Qwen0\.16650\.21730\.0508\[0\.0233, 0\.0585\]0\.02560\.0008Mean loss0\.14470\.18720\.0425\[0\.0157, 0\.0576\]0\.01820\.00016Mistral0\.07420\.08030\.0061\[0\.0004, 0\.0109\]0\.00040\.02216Qwen0\.16650\.17080\.0044\[0\.0005, 0\.0083\]0\.00060\.02516Mean loss0\.14470\.14860\.0039\[0\.0002, 0\.0065\]0\.00020\.02532Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00032Qwen0\.16650\.16790\.0015\[0\.0000, 0\.0044\]0\.00000\.57832Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mistral0\.07420\.07420\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.16650\.16650\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.14470\.14470\.0000\[0\.0000, 0\.0000\]0\.00001\.000Reused 96, seed 3141592Mistral0\.10060\.16910\.0684\[0\.0398, 0\.0955\]0\.04370\.0002Qwen0\.19100\.29230\.1013\[0\.0382, 0\.1648\]0\.04210\.0002Mean loss0\.17310\.27710\.1040\[0\.0679, 0\.1258\]0\.07260\.0004Mistral0\.10060\.16000\.0594\[0\.0293, 0\.0790\]0\.03290\.0004Qwen0\.19100\.27970\.0887\[0\.0335, 0\.1326\]0\.03730\.0004Mean loss0\.17310\.25100\.0779\[0\.0421, 0\.0960\]0\.04710\.0008Mistral0\.10060\.12880\.0281\[0\.0087, 0\.0478\]0\.01010\.0008Qwen0\.19100\.24390\.0529\[0\.0152, 0\.0709\]0\.01720\.0008Mean loss0\.17310\.21400\.0409\[0\.0152, 0\.0492\]0\.01740\.00016Mistral0\.10060\.10520\.0046\[0\.0000, 0\.0081\]0\.00000\.06916Qwen0\.19100\.20420\.0132\[0\.0044, 0\.0183\]0\.00520\.00016Mean loss0\.17310\.18300\.0099\[0\.0026, 0\.0123\]0\.00320\.00432Mistral0\.10060\.10160\.0009\[0\.0000, 0\.0019\]0\.00000\.42132Qwen0\.19100\.19940\.0084\[0\.0000, 0\.0120\]0\.00040\.03532Mean loss0\.17310\.17670\.0036\[0\.0000, 0\.0059\]0\.00000\.06864Mistral0\.10060\.10060\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.19100\.19100\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.17310\.17310\.0000\[0\.0000, 0\.0000\]0\.00001\.000Reused 96, seed 2718282Mistral0\.10060\.16970\.0691\[0\.0406, 0\.0973\]0\.04420\.0002Qwen0\.19100\.29830\.1073\[0\.0445, 0\.1674\]0\.05030\.0002Mean loss0\.17310\.27840\.1053\[0\.0709, 0\.1306\]0\.07500\.0004Mistral0\.10060\.16840\.0678\[0\.0393, 0\.0950\]0\.04310\.0004Qwen0\.19100\.29370\.1027\[0\.0403, 0\.1602\]0\.04540\.0004Mean loss0\.17310\.27840\.1053\[0\.0661, 0\.1217\]0\.07120\.0008Mistral0\.10060\.14050\.0399\[0\.0152, 0\.0593\]0\.01850\.0008Qwen0\.19100\.25620\.0652\[0\.0257, 0\.0801\]0\.02950\.0008Mean loss0\.17310\.21770\.0446\[0\.0232, 0\.0488\]0\.02510\.00016Mistral0\.10060\.11310\.0125\[0\.0025, 0\.0165\]0\.00360\.00116Qwen0\.19100\.22040\.0294\[0\.0082, 0\.0362\]0\.00970\.00016Mean loss0\.17310\.19330\.0202\[0\.0052, 0\.0238\]0\.00690\.00032Mistral0\.10060\.10320\.0026\[0\.0000, 0\.0052\]0\.00000\.41932Qwen0\.19100\.20160\.0106\[0\.0000, 0\.0173\]0\.00000\.35732Mean loss0\.17310\.18090\.0078\[0\.0000, 0\.0113\]0\.00000\.35764Mistral0\.10060\.10060\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Qwen0\.19100\.19100\.0000\[0\.0000, 0\.0000\]0\.00001\.00064Mean loss0\.17310\.17310\.0000\[0\.0000, 0\.0000\]0\.00001\.000\(a\)Original confirmation collection\.\(b\)Reused locked collection\.
Figure 24:Paired PhiZZ\-only Mistral\-loss curves\. The policy\-aware counterparts appear in Figure[23](https://arxiv.org/html/2610.27446#A9.F23)\. These plots change only the observation key, not the response samples or selected features\.
### I\.5Reconstruction and feature occupancy

For activation vectorshih\_\{i\}and reconstructionsh^i\\widehat\{h\}\_\{i\}, leth¯\\overline\{h\}be the mean activation on the listed dataset\. For positive denominators, the centered and uncentered diagnostics are

FVU=∑i‖hi−h^i‖22∑i‖hi−h¯‖22,Erel=∑i‖hi−h^i‖22∑i‖hi‖22\.\\mathrm\{FVU\}=\\frac\{\\sum\_\{i\}\\\|h\_\{i\}\-\\widehat\{h\}\_\{i\}\\\|\_\{2\}^\{2\}\}\{\\sum\_\{i\}\\\|h\_\{i\}\-\\overline\{h\}\\\|\_\{2\}^\{2\}\},\\qquad E\_\{\\mathrm\{rel\}\}=\\frac\{\\sum\_\{i\}\\\|h\_\{i\}\-\\widehat\{h\}\_\{i\}\\\|\_\{2\}^\{2\}\}\{\\sum\_\{i\}\\\|h\_\{i\}\\\|\_\{2\}^\{2\}\}\.The denominator difference does not imply negligible reconstruction error\. Mean cosine is averaged over individual vectors\. The audit JSON retains a historical field nameddead\_fraction\_on\_validation; on audit records this field denotes features unobserved on that small audit sample\. It is not evidence that those features are globally dead\. Quantiles and activation/reconstruction norms remain in the original JSON/CSV files\.

Table 34:Phi reconstruction diagnostics\. FVU is centered fraction of variance unexplained; relative error uses the same reconstruction\-error numerator divided by uncentered activation energy\. Active counts features observed at least once in the listed dataset, not globally active or dead features\.Table 35:Phi supervised\-selector occupancy\. Fit and Eval count distinct complete observation keys\. Collision is the fraction of evaluation states sharing an observation key; Unseen is the fraction whose complete key is absent from fitting\. For\(Z,S\)\(Z,S\), new policy text alone makes the complete key unseen\.BBObservationFitEvalCollisionUnseenLargest cellOriginal 96, seed 3141592ZZonly440\.9900\.000512\(Z,S\)\(Z,S\)38240\.9581\.000114ZZonly15110\.9790\.062274\(Z,S\)\(Z,S\)68440\.7811\.00068ZZonly53350\.8440\.12598\(Z,S\)\(Z,S\)110720\.4691\.000416ZZonly125870\.1770\.844316\(Z,S\)\(Z,S\)139940\.0421\.000232ZZonly143960\.0000\.969132\(Z,S\)\(Z,S\)144960\.0001\.000164ZZonly144960\.0001\.000164\(Z,S\)\(Z,S\)144960\.0001\.0001Original 96, seed 2718282ZZonly441\.0000\.000522\(Z,S\)\(Z,S\)39250\.9691\.00084ZZonly15111\.0000\.000194\(Z,S\)\(Z,S\)77520\.7401\.00048ZZonly57360\.8230\.11598\(Z,S\)\(Z,S\)109710\.4581\.000416ZZonly125830\.2290\.740416\(Z,S\)\(Z,S\)140950\.0211\.000232ZZonly144940\.0421\.000232\(Z,S\)\(Z,S\)144960\.0001\.000164ZZonly144960\.0001\.000164\(Z,S\)\(Z,S\)144960\.0001\.0001Reused 96, seed 3141592ZZonly430\.9900\.000572\(Z,S\)\(Z,S\)38170\.9901\.00094ZZonly1560\.9900\.000484\(Z,S\)\(Z,S\)68320\.8851\.00088ZZonly53250\.8850\.438218\(Z,S\)\(Z,S\)110580\.6561\.000416ZZonly125600\.5210\.9381116\(Z,S\)\(Z,S\)139840\.2191\.000432ZZonly143840\.2081\.000532\(Z,S\)\(Z,S\)144930\.0621\.000264ZZonly144940\.0421\.000264\(Z,S\)\(Z,S\)144950\.0211\.0002Reused 96, seed 2718282ZZonly441\.0000\.000512\(Z,S\)\(Z,S\)39200\.9791\.000104ZZonly1580\.9790\.000484\(Z,S\)\(Z,S\)77280\.8961\.000108ZZonly57260\.8750\.375118\(Z,S\)\(Z,S\)109660\.5001\.000516ZZonly125650\.5000\.979616\(Z,S\)\(Z,S\)140880\.1671\.000232ZZonly144930\.0621\.000232\(Z,S\)\(Z,S\)144950\.0211\.000264ZZonly144960\.0001\.000164\(Z,S\)\(Z,S\)144960\.0001\.0001Table 36:Phi variance\-selector occupancy\. Fit and Eval count distinct complete observation keys\. Collision is the fraction of evaluation states sharing an observation key; Unseen is the fraction whose complete key is absent from fitting\. For\(Z,S\)\(Z,S\), new policy text alone makes the complete key unseen\.BBObservationFitEvalCollisionUnseenLargest cellOriginal 96, seed 3141592ZZonly221\.0000\.000592\(Z,S\)\(Z,S\)23161\.0001\.00094ZZonly871\.0000\.000214\(Z,S\)\(Z,S\)56460\.8121\.00068ZZonly50340\.8330\.24088\(Z,S\)\(Z,S\)119770\.3751\.000316ZZonly132920\.0830\.854216\(Z,S\)\(Z,S\)141940\.0421\.000232ZZonly144960\.0001\.000132\(Z,S\)\(Z,S\)144960\.0001\.000164ZZonly144960\.0001\.000164\(Z,S\)\(Z,S\)144960\.0001\.0001Original 96, seed 2718282ZZonly111\.0000\.000962\(Z,S\)\(Z,S\)1281\.0001\.000124ZZonly441\.0000\.000344\(Z,S\)\(Z,S\)41290\.9581\.00078ZZonly41270\.9270\.042168\(Z,S\)\(Z,S\)111730\.4271\.000316ZZonly122800\.3020\.656316\(Z,S\)\(Z,S\)138910\.1041\.000232ZZonly143940\.0420\.979232\(Z,S\)\(Z,S\)144960\.0001\.000164ZZonly144960\.0001\.000164\(Z,S\)\(Z,S\)144960\.0001\.0001Reused 96, seed 3141592ZZonly231\.0000\.052852\(Z,S\)\(Z,S\)23140\.9791\.000124ZZonly871\.0000\.052424\(Z,S\)\(Z,S\)56330\.9061\.00088ZZonly50340\.8330\.198308\(Z,S\)\(Z,S\)119650\.5001\.000516ZZonly132710\.3440\.969916\(Z,S\)\(Z,S\)141900\.1251\.000232ZZonly144830\.2291\.000432\(Z,S\)\(Z,S\)144960\.0001\.000164ZZonly144950\.0211\.000264\(Z,S\)\(Z,S\)144960\.0001\.0001Reused 96, seed 2718282ZZonly111\.0000\.000962\(Z,S\)\(Z,S\)1281\.0001\.000124ZZonly440\.9900\.000734\(Z,S\)\(Z,S\)41170\.9791\.000128ZZonly41160\.9380\.302238\(Z,S\)\(Z,S\)111520\.6981\.000416ZZonly122580\.5520\.990716\(Z,S\)\(Z,S\)138790\.2921\.000432ZZonly143890\.1351\.000332\(Z,S\)\(Z,S\)144930\.0621\.000264ZZonly144960\.0001\.000164\(Z,S\)\(Z,S\)144960\.0001\.0001
### I\.6Policy\-area deletion and selector controls

Each deletion removes exactly one of eight areas and leaves 84 states\. The minimum over all 192 saved primary deletion estimates is0\.023630950\.02363095\. The ranges below are sensitivity results, not additional confidence intervals\. Random selection from all features, selection from features that vary on the fitting set, and approximate occupancy matching are recorded only for Mistral loss under theZZ\-only observation\. The matching routine uses fitting data and retains 40 controls from its candidate search; its recorded distance and exact\-match fraction are reported rather than assuming exact occupancy matching\.

Table 37:Phi single policy area deletion sensitivity atB=2B=2\. Each row summarizes the eight deletions for one setting; each deletion retains 84 states\. Values are point\-estimate minima, medians, and maxima, not confidence bounds\. All 192 deletion\-level estimates are preserved in the CSV archive\.CollectionSeedObservationEvaluatorMinimumMedianMaximumOriginal 96314159ZZonlyMistral0\.09890\.11260\.1204Original 96314159ZZonlyQwen0\.08230\.10320\.1107Original 96314159ZZonlyMean loss0\.06850\.08060\.0905Original 96314159\(Z,S\)\(Z,S\)Mistral0\.04930\.05540\.0637Original 96314159\(Z,S\)\(Z,S\)Qwen0\.07340\.08470\.0901Original 96314159\(Z,S\)\(Z,S\)Mean loss0\.05090\.05700\.0622Original 96271828ZZonlyMistral0\.06430\.07700\.0826Original 96271828ZZonlyQwen0\.04450\.04730\.0557Original 96271828ZZonlyMean loss0\.04080\.04660\.0524Original 96271828\(Z,S\)\(Z,S\)Mistral0\.03870\.05090\.0561Original 96271828\(Z,S\)\(Z,S\)Qwen0\.03380\.03430\.0417Original 96271828\(Z,S\)\(Z,S\)Mean loss0\.02360\.03020\.0339Reused 96314159ZZonlyMistral0\.05650\.06550\.0749Reused 96314159ZZonlyQwen0\.07020\.11300\.1201Reused 96314159ZZonlyMean loss0\.08600\.09500\.1035Reused 96314159\(Z,S\)\(Z,S\)Mistral0\.03150\.04450\.0499Reused 96314159\(Z,S\)\(Z,S\)Qwen0\.05210\.05910\.0642Reused 96314159\(Z,S\)\(Z,S\)Mean loss0\.05810\.07180\.0774Reused 96271828ZZonlyMistral0\.05310\.06160\.0700Reused 96271828ZZonlyQwen0\.07020\.09980\.1087Reused 96271828ZZonlyMean loss0\.05920\.07100\.0752Reused 96271828\(Z,S\)\(Z,S\)Mistral0\.02550\.03350\.0379Reused 96271828\(Z,S\)\(Z,S\)Qwen0\.03850\.04670\.0524Reused 96271828\(Z,S\)\(Z,S\)Mean loss0\.03750\.04920\.0502Table 38:Phi random\-selector summaries use Mistral loss andZZ\-only observations exclusively\. Each budget has 200 all\-feature draws, 200 fitting\-nonconstant draws, and 40 controls matched by occupancy on fitting states per seed and collection\. These are not policy\-aware control results\. SD describes variation over the saved selectors, not a bootstrap standard error\.BBSelectorNNMeanSDMinimumMedianMaximumOriginal 96, seed 3141592All features2000\.13770\.00010\.13660\.13770\.13772Fit nonconstant2000\.12980\.01200\.08580\.13690\.13772Fit matched400\.11810\.01430\.09060\.11890\.13774All features2000\.13760\.00130\.12150\.13770\.13774Fit nonconstant2000\.11990\.01590\.07940\.12390\.13774Fit matched400\.10150\.01400\.07850\.10300\.12948All features2000\.13770\.00080\.12610\.13770\.13778Fit nonconstant2000\.09560\.01940\.05460\.09670\.13778Fit matched400\.06900\.01680\.03200\.06950\.109116All features2000\.13730\.00290\.10620\.13770\.137716Fit nonconstant2000\.05240\.01930\.00820\.05140\.112016Fit matched400\.01950\.00880\.00820\.01850\.043632All features2000\.13690\.00350\.11500\.13770\.137732Fit nonconstant2000\.00820\.00800\.00000\.00600\.045432Fit matched400\.00150\.00200\.00000\.00000\.007164All features2000\.13480\.00810\.09270\.13770\.137764Fit nonconstant2000\.00020\.00080\.00000\.00000\.005464Fit matched400\.00000\.00000\.00000\.00000\.0000Original 96, seed 2718282All features2000\.13770\.00000\.13770\.13770\.13772Fit nonconstant2000\.12700\.01720\.05830\.13560\.13772Fit matched400\.10620\.02280\.06360\.10860\.13774All features2000\.13750\.00270\.10090\.13770\.13774Fit nonconstant2000\.11400\.02280\.02960\.12010\.13774Fit matched400\.09090\.02310\.04660\.09050\.13738All features2000\.13720\.00320\.10280\.13770\.13778Fit nonconstant2000\.08850\.02530\.02710\.08750\.13608Fit matched400\.05690\.01990\.01950\.05920\.099016All features2000\.13700\.00610\.06360\.13770\.137716Fit nonconstant2000\.04170\.02180\.00220\.03880\.108016Fit matched400\.01460\.00850\.00140\.01380\.037532All features2000\.13660\.00650\.08300\.13770\.137732Fit nonconstant2000\.00530\.00590\.00000\.00360\.038332Fit matched400\.00140\.00250\.00000\.00050\.013964All features2000\.13450\.01090\.06360\.13770\.137764Fit nonconstant2000\.00000\.00030\.00000\.00000\.003164Fit matched400\.00000\.00000\.00000\.00000\.0000Reused 96, seed 3141592All features2000\.06900\.00060\.06070\.06910\.06912Fit nonconstant2000\.06510\.00770\.02550\.06910\.06912Fit matched400\.06680\.00450\.04440\.06840\.06914All features2000\.06900\.00090\.05730\.06910\.06914Fit nonconstant2000\.06080\.00970\.01800\.06410\.06914Fit matched400\.05860\.01010\.02710\.06290\.06918All features2000\.06890\.00080\.06220\.06910\.06918Fit nonconstant2000\.04640\.01250\.01750\.04620\.06918Fit matched400\.04040\.01390\.01240\.03810\.067816All features2000\.06880\.00100\.06010\.06910\.069116Fit nonconstant2000\.02140\.01190\.00000\.01920\.056216Fit matched400\.01690\.00840\.00230\.01600\.037532All features2000\.06790\.00380\.03850\.06910\.069132Fit nonconstant2000\.00370\.00410\.00000\.00240\.023832Fit matched400\.00200\.00290\.00000\.00120\.014164All features2000\.06680\.00460\.04240\.06910\.069164Fit nonconstant2000\.00010\.00040\.00000\.00000\.002664Fit matched400\.00010\.00040\.00000\.00000\.0021Reused 96, seed 2718282All features2000\.06900\.00050\.06220\.06910\.06912Fit nonconstant2000\.06510\.00600\.03520\.06780\.06912Fit matched400\.06530\.00420\.05220\.06640\.06914All features2000\.06900\.00080\.06010\.06910\.06914Fit nonconstant2000\.05990\.00850\.03100\.06250\.06914Fit matched400\.05670\.00900\.03350\.06010\.06918All features2000\.06870\.00230\.04620\.06910\.06918Fit nonconstant2000\.04630\.01110\.01940\.04720\.06918Fit matched400\.04230\.01190\.01350\.04290\.067216All features2000\.06850\.00230\.05340\.06910\.069116Fit nonconstant2000\.02040\.01040\.00210\.01930\.054516Fit matched400\.01740\.00890\.00050\.01580\.036632All features2000\.06810\.00300\.04600\.06910\.069132Fit nonconstant2000\.00290\.00310\.00000\.00210\.014932Fit matched400\.00200\.00270\.00000\.00100\.012364All features2000\.06610\.00550\.04100\.06910\.069164Fit nonconstant2000\.00000\.00020\.00000\.00000\.001564Fit matched400\.00000\.00000\.00000\.00000\.0003Table 39:Fitting\-only matching diagnostics for Phi random controls\. Distances are the recorded occupancy/frequency\-matching objective, not distances between held\-out losses\. Forty retained controls are summarized per seed and budget\. Exact is the fraction marked as an exact match by the saved routine; the controls are otherwise approximate matches\.Table 40:Saved Phi sanity\-control gaps at the primary budget\. Constant, Shuffled, and Oracle refer to the exact controls in the archived run\. They are descriptive checks and do not replace the matched observation\-condition comparison in Table[29](https://arxiv.org/html/2610.27446#A9.T29)\.CollectionSeedObservationEvaluatorConstantShuffledOracleOriginal 96314159ZZonlyMistral0\.13770\.13770\.0000Original 96314159ZZonlyQwen0\.15440\.15440\.0000Original 96314159ZZonlyMean loss0\.12170\.12170\.0000Original 96314159\(Z,S\)\(Z,S\)Mistral0\.13770\.09330\.0000Original 96314159\(Z,S\)\(Z,S\)Qwen0\.15440\.13350\.0000Original 96314159\(Z,S\)\(Z,S\)Mean loss0\.12170\.10170\.0000Original 96271828ZZonlyMistral0\.13770\.13750\.0000Original 96271828ZZonlyQwen0\.15440\.15440\.0000Original 96271828ZZonlyMean loss0\.12170\.12170\.0000Original 96271828\(Z,S\)\(Z,S\)Mistral0\.13770\.08990\.0000Original 96271828\(Z,S\)\(Z,S\)Qwen0\.15440\.10600\.0000Original 96271828\(Z,S\)\(Z,S\)Mean loss0\.12170\.08290\.0000Reused 96314159ZZonlyMistral0\.06910\.06550\.0000Reused 96314159ZZonlyQwen0\.10730\.10730\.0000Reused 96314159ZZonlyMean loss0\.10530\.10330\.0000Reused 96314159\(Z,S\)\(Z,S\)Mistral0\.05160\.04190\.0000Reused 96314159\(Z,S\)\(Z,S\)Qwen0\.05840\.05750\.0000Reused 96314159\(Z,S\)\(Z,S\)Mean loss0\.07780\.06820\.0000Reused 96271828ZZonlyMistral0\.06910\.06910\.0000Reused 96271828ZZonlyQwen0\.10730\.10730\.0000Reused 96271828ZZonlyMean loss0\.10530\.09840\.0000Reused 96271828\(Z,S\)\(Z,S\)Mistral0\.05160\.04370\.0000Reused 96271828\(Z,S\)\(Z,S\)Qwen0\.05840\.05200\.0000Reused 96271828\(Z,S\)\(Z,S\)Mean loss0\.07780\.05720\.0000
### I\.7Evaluator comparison and missing human evaluation

The item\-level comparison aligns the two evaluator tables by the same 5,040 response keys\. Spearman correlation is0\.355243890\.35524389, mean absolute difference is0\.322003970\.32200397, root mean squared difference is0\.437500060\.43750006, and mean difference between Mistral and Qwen losses is−0\.16480820\-0\.16480820\. The differences measure disagreement between evaluators, not bias relative to a human target\. Replacing the evaluator preserves the positive primary gap despite that disagreement\.

Both optional 528\-row Phi rater packets have blank score fields\. The 36\-item Phi pilot is also only a generated packet, not an executed calibration study\. The earlier SmolLM2 human ratings are not reused for Phi responses\. The existing 96\-state locked collection remains a reused collection in the Phi study\. Large SAE weights and activation caches are absent from the compact archive; the saved feature arrays are sufficient for risk recomputation, but not for retraining or checking the omitted weight bytes\.

## Appendix JObservation Reanalysis and Reproduction Checks

### J\.1Detailed policy\-aware reanalysis and precision limits

The SmolLM2 reanalysis supplies policy text to the action selector while keeping features, states, responses, and ratings fixed\. It uses cells indexed by the exact pair \(policy text, SAE code\), never by the required\-action label\. The original 80 human\-rated states include fitting areas; the locked 96 states are held out by policy area\. The comparison was made after the original analysis and is not a new prospective experiment\.

#### SmolLM2 risks with policy supplied\.

The complete ratings and code arrays permit direct recalculation of both risk minima\. Input alignment is documented in the next subsection\.

Table 41:SmolLM2 reanalysis with active policy and SAE code jointly visible, atB=2B=2\. Auto uses five\-rollout means; human uses the fixed packet continuation and final adjudicated loss\. Intervals and lower confidence bounds \(LCBs\) are individual policy\-area percentile\-bootstrap summaries from 2,000 resamples\.All four policy\-aware automatic gaps remain positive\. Under final human loss, the original\-sample gaps are0\.03450\.0345and0\.03530\.0353, with individual one\-sided bootstrap lower bounds0\.02030\.0203and0\.01750\.0175\. On the locked 96\-state collection, the gaps are0\.11660\.1166and0\.10820\.1082, with lower bounds0\.09350\.0935and0\.09010\.0901\. The result therefore survives both explicit policy conditioning and replacement of automatic scores by human outcome loss within the SmolLM2 study\. The policy\-aware gaps are no larger than their paired code\-only gaps, as required by Equation \([6](https://arxiv.org/html/2610.27446#S2.E6)\)\. The following tables give all budgets and results for each rater\.

#### Qwen2\.5 recovery with an explicit precision limit\.

The available Qwen2\.5\-1\.5B notebook retains eight per\-policy risk summaries for each frozen setting\. Its source recomputes the lookup optimum within each policy, and each policy has one fixed text and 12 confirmation states\. Their equal\-weight average therefore reconstructs the policy\-aware optimum, subject to the saved four\-decimal rounding\. We average the directly saved gap column, yielding0\.136950\.13695at layer 17/B=2B=2and0\.038750\.03875at layer 19/B=16B=16, each with absolute rounding error at most0\.000050\.00005\. The full\-precision Qwen loss and code arrays are not present in this project; we do not represent this calculation as a state\-level rerun\. We report no new policy\-aware result for Qwen2\.5\-0\.5B\.

Table 42:Qwen2\.5\-1\.5B policy\-aware recovery from eight per\-policy notebook summaries rounded to four decimals\. The point gap has absolute rounding uncertainty at most5×10−55\\times 10^\{\-5\}\. Bootstrap endpoints include the worst\-case rounding correction; these are not full\-precision state\-array reruns\.Since policy is observable, clusters from distinct policy texts do not share cells\. Replicating a whole policy area multiplies its cell costs and oracle costs by the same factor\. Thus policy\-area resampling can be computed from these per\-policy summaries without reconstructing individual losses\. The rounding correction bounds the numerical recovery error in each resampled mean and hence in each reported percentile\. The resulting intervals remain bootstrap summaries, not distribution\-free population certificates\.

### J\.2Authoritative human inputs and alignment

The completed inputs retained in the author working archive aresupplementary/support\_revision/human\_inputs/rater\_A\.csv,supplementary/support\_revision/human\_inputs/rater\_B\.csv, andsupplementary/support\_revision/human\_inputs/adjudication\_resolved\.csv\. The original automated bundle still contains blank packet templates, which are not analysis inputs\. Completed score files are preserved byte for byte\. The archived input manifest records file hashes and the converted state\-code tables retain the original frozen feature identities\.

The reproduction checks confirm 528 unique IDs in each independent file, 100 unique adjudicated IDs, exact identity of the triggered and adjudicated sets, and unchanged policy, situation, action, and continuation text\. Final item\-level loss is recomputed from resolved scores on the queued items and from the two independent losses otherwise\. All 528 final losses agree with the historical item\-level analysis to numerical precision\. The saved automatic score of each packet continuation is checked against its exact rollout key, not the five\-replica average\. Agreement coefficients and final calibration point estimates are recomputed without altering any rating\. The Pearson correlation of automatic and final human loss is0\.38760\.3876\. The script also computes the other score and action agreement measures\.

### J\.3Paired observations, resampling, and the loss estimand

The same support code and loss vector are used for bothV=ZV=ZandV=\(Z,S\)V=\(Z,S\)\. Within each saved collection identifier, 2,000 bootstrap over policy areas draws are shared across observation modes, feature budgets, SAE seeds, and available loss sources\. Automatic analyses using five replicas and analyses of the human packet use different identifiers and seeds even when they cover the same 96 states\. The base seed is 20260905; the collection\-specific seed adds the sum of character codes in the saved collection name\. Every derived table records the exact seed\. Historical tables retain their original bootstrap seeds; small interval differences between historical and reanalysis outputs are not changes to the input labels or loss values\.

The saved scalar losses are the scientific target of the finite\-table reanalysis\. Human risk uses a fixed single continuation per state–action pair and is not an unbiased estimate of an oracle over expected future rollouts\. Automatic risk uses the five saved replicas\. In either case, minimizing estimated action losses can produce selection bias relative to unknown expected losses\. Cluster resampling does not remove that bias\. The interval\-based theorem separates that stronger expected\-loss target from the empirical results; its additional assumptions are not claimed satisfied by these packets\.

All paired estimates satisfy0≤ΔZ,S≤ΔZ0\\leq\\Delta\_\{Z,S\}\\leq\\Delta\_\{Z\}to numerical precision\. Numerical nonpositive\-bootstrap fractions in the derived CSV files use a10−1210^\{\-12\}tolerance and are not interpreted as hypothesis\-testpp\-values\. Large\-budget zeros are finite\-collection statements\.

Table 43:Paired SmolLM2 observation reanalysis across all feature budgets\. Features, state\-level losses, and resampling draws are held fixed betweenZZand\(Z,S\)\(Z,S\)\. The intervals refer toΔZ,S\\Delta\_\{Z,S\}\. Values displayed as zero may be below the displayed precision\.Loss / collectionSeedBBΔZ\\Delta\_\{Z\}ΔZ,S\\Delta\_\{Z,S\}95% interval95% LCBAuto / original 9631415920\.13580\.0987\[0\.0760, 0\.1209\]0\.0807Auto / original 9631415940\.11780\.0600\[0\.0474, 0\.0749\]0\.0490Auto / original 9631415980\.04980\.0219\[0\.0060, 0\.0374\]0\.0101Auto / original 96314159160\.00440\.0031\[0\.0000, 0\.0094\]0\.0000Auto / original 96314159320\.00000\.0000\[0\.0000, 0\.0000\]0\.0000Auto / original 96314159640\.00000\.0000\[0\.0000, 0\.0000\]0\.0000Auto / original 9627182820\.14160\.1015\[0\.0680, 0\.1394\]0\.0729Auto / original 9627182840\.10570\.0568\[0\.0451, 0\.0676\]0\.0467Auto / original 9627182880\.06980\.0377\[0\.0194, 0\.0569\]0\.0220Auto / original 96271828160\.01720\.0044\[0\.0000, 0\.0106\]0\.0000Auto / original 96271828320\.00000\.0000\[0\.0000, 0\.0000\]0\.0000Auto / original 96271828640\.00000\.0000\[0\.0000, 0\.0000\]0\.0000Auto / locked 9631415920\.07120\.0460\[0\.0272, 0\.0658\]0\.0300Auto / locked 9631415940\.06760\.0405\[0\.0233, 0\.0580\]0\.0259Auto / locked 9631415980\.04030\.0284\[0\.0095, 0\.0498\]0\.0121Auto / locked 96314159160\.02090\.0080\[0\.0017, 0\.0160\]0\.0017Auto / locked 96314159320\.00170\.0017\[0\.0000, 0\.0050\]0\.0000Auto / locked 96314159640\.00000\.0000\[0\.0000, 0\.0000\]0\.0000Auto / locked 9627182820\.07700\.0430\[0\.0272, 0\.0584\]0\.0295Auto / locked 9627182840\.05680\.0323\[0\.0164, 0\.0494\]0\.0191Auto / locked 9627182880\.03830\.0130\[0\.0028, 0\.0254\]0\.0041Auto / locked 96271828160\.00000\.0000\[0\.0000, 0\.0000\]0\.0000Auto / locked 96271828320\.00000\.0000\[0\.0000, 0\.0000\]0\.0000Auto / locked 96271828640\.00000\.0000\[0\.0000, 0\.0000\]0\.0000Human / original 8031415920\.07760\.0345\[0\.0181, 0\.0526\]0\.0203Human / original 8031415940\.07260\.0220\[0\.0069, 0\.0395\]0\.0091Human / original 8031415980\.02840\.0031\[0\.0000, 0\.0081\]0\.0003Human / original 80314159160\.00060\.0000\[0\.0000, 0\.0000\]0\.0000Human / original 80314159320\.00000\.0000\[0\.0000, 0\.0000\]0\.0000Human / original 80314159640\.00000\.0000\[0\.0000, 0\.0000\]0\.0000Human / original 8027182820\.09900\.0353\[0\.0153, 0\.0594\]0\.0175Human / original 8027182840\.06890\.0197\[0\.0078, 0\.0341\]0\.0091Human / original 8027182880\.04740\.0100\[0\.0006, 0\.0213\]0\.0025Human / original 80271828160\.00060\.0000\[0\.0000, 0\.0000\]0\.0000Human / original 80271828320\.00000\.0000\[0\.0000, 0\.0000\]0\.0000Human / original 80271828640\.00000\.0000\[0\.0000, 0\.0000\]0\.0000Human / locked 9631415920\.14840\.1166\[0\.0881, 0\.1420\]0\.0935Human / locked 9631415940\.13030\.0972\[0\.0769, 0\.1172\]0\.0804Human / locked 9631415980\.09310\.0647\[0\.0398, 0\.0941\]0\.0437Human / locked 96314159160\.02800\.0159\[0\.0000, 0\.0354\]0\.0021Human / locked 96314159320\.00000\.0000\[0\.0000, 0\.0000\]0\.0000Human / locked 96314159640\.00000\.0000\[0\.0000, 0\.0000\]0\.0000Human / locked 9627182820\.13340\.1082\[0\.0862, 0\.1299\]0\.0901Human / locked 9627182840\.09650\.0523\[0\.0264, 0\.0846\]0\.0297Human / locked 9627182880\.05050\.0122\[0\.0024, 0\.0240\]0\.0036Human / locked 96271828160\.00220\.0000\[0\.0000, 0\.0000\]0\.0000Human / locked 96271828320\.00030\.0000\[0\.0000, 0\.0000\]0\.0000Human / locked 96271828640\.00000\.0000\[0\.0000, 0\.0000\]0\.0000Table 44:Policy\-aware human sensitivity atB=2B=2, before adjudication\. No interface or feature is reselected for a rater\. The two\-rater mean is shown separately\.CollectionSeedLossΔZ,S\\Delta\_\{Z,S\}95% interval95% LCBoriginal 80314159Rater A0\.0263\[0\.0125, 0\.0425\]0\.0144original 80271828Rater A0\.0294\[0\.0106, 0\.0500\]0\.0131original 80314159Rater B0\.0542\[0\.0358, 0\.0738\]0\.0383original 80271828Rater B0\.0429\[0\.0246, 0\.0625\]0\.0271original 80314159Mean0\.0322\[0\.0170, 0\.0493\]0\.0194original 80271828Mean0\.0290\[0\.0112, 0\.0478\]0\.0140locked 96314159Rater A0\.1233\[0\.0936, 0\.1477\]0\.0988locked 96271828Rater A0\.0944\[0\.0713, 0\.1198\]0\.0752locked 96314159Rater B0\.1191\[0\.0913, 0\.1493\]0\.0955locked 96271828Rater B0\.1260\[0\.1047, 0\.1491\]0\.1075locked 96314159Mean0\.1264\[0\.0974, 0\.1491\]0\.1035locked 96271828Mean0\.1105\[0\.0879, 0\.1337\]0\.0912
### J\.4Qwen source recovery and rounding control

The original notebook’s V8 source computes the cellwise lookup optimum separately for every confirmation policy\. The saved table is rounded to four decimal places\. LetΔ~d\\widetilde\{\\Delta\}\_\{d\}denote the saved gap for policy areadd, andΔd\\Delta\_\{d\}its unrounded value\. Then\|Δ~d−Δd\|≤rround\|\\widetilde\{\\Delta\}\_\{d\}\-\\Delta\_\{d\}\|\\leq r\_\{\\rm round\}withrround=5×10−5r\_\{\\rm round\}=5\\times 10^\{\-5\}\. For any nonnegative normalized policy weightswdw\_\{d\},

\|∑dwd​Δ~d−∑dwd​Δd\|≤rround\.\\left\|\\sum\_\{d\}w\_\{d\}\\widetilde\{\\Delta\}\_\{d\}\-\\sum\_\{d\}w\_\{d\}\\Delta\_\{d\}\\right\|\\leq r\_\{\\rm round\}\.The bound applies to every resample of whole policy areas\. Since each resampled mean changes by at mostrroundr\_\{\\rm round\}, each empirical percentile changes by at most the same amount\. Lower endpoints subtractrroundr\_\{\\rm round\}and upper endpoints add it\. The recovered summaries, literal notebook output, source, and notebook hash are retained in the author working archive undersupplementary/support\_revision/qwen\_recovered/\. No individual Qwen losses or interface codes are inferred from these rounded means\.

Table 45:Literal rounded per\-policy entries recovered from the original Qwen2\.5 V8 notebook output\. Each policy has 12 confirmation states\. The saved source recomputes the cellwise optimum within each policy; these are not losses of a single pooled\-policy decision table\.SettingPolicy areaRfullR\_\{\\rm full\}RZ⋆R\_\{Z\}^\{\\star\}within policyGapL17/B2after hours building access0\.30220\.47700\.1748L17/B2campus drone flight booking0\.43290\.53500\.1021L17/B2donor contact list export0\.34490\.52550\.1806L17/B2licensed software download0\.21560\.36500\.1494L17/B2minor photo release0\.40900\.56690\.1579L17/B2payroll bank change0\.31730\.38420\.0669L17/B2research data export0\.42990\.52250\.0925L17/B2volunteer background clearance0\.25100\.42240\.1714L19/B16after hours building access0\.30220\.33850\.0363L19/B16campus drone flight booking0\.43290\.50510\.0722L19/B16donor contact list export0\.34490\.37350\.0286L19/B16licensed software download0\.21560\.24490\.0293L19/B16minor photo release0\.40900\.40900\.0000L19/B16payroll bank change0\.31730\.36300\.0457L19/B16research data export0\.42990\.47960\.0497L19/B16volunteer background clearance0\.25100\.29920\.0482
### J\.5Capacity\-source audit and algorithm regression

The original Qwen capacity source enumerates every pair and triple among the selected 42 states; it does not restrict this enumeration to the old SAE cells\. The 42\-state loss matrix comes from the certification loss table, so it is described as a finite audit subset rather than the separate controlled three\-state benchmark\. The original source and saved solver\-status output are preserved\. Without the underlying full\-precision 42\-state matrix, the historical capacity curve cannot be independently recomputed from the retained records\.

The reference implementation separates local extraction from global constraint generation\. Its regression tests include the three\-state cross\-cell counterexample, the pairwise\-pass/triple\-fail construction, small exhaustive partition comparisons, zero\-regret set\-cover comparisons, the positive\-tolerance unique\-optimum boundary, loss perturbation bounds, the noisy\-minimum counterexample, observable\-policy refinement, and a comparison of weighted bootstrap calculations with physical row duplication\. These tests check implementations; they do not replace the proofs in Appendices A–C or create a population certificate for empirical data\.

### J\.6CPU reproduction entry points

Runpythontools/reproduce\_support\_revision\.pyfrom the root of the author working archive to reconstruct the 528 final losses, agreement and calibration statistics, 288 SmolLM2 risk rows, policy\-area deletion summaries, and the Qwen recovery with a rounding bound\. Runpython\-munittestdiscover\-stests\-vfor the mathematical regression tests\. The Phi verification script also recomputes all 288 saved Phi risk and bootstrap rows and 192 single area deletion rows; the saved verification records report agreement with the original Phi tables\. Runpythontools/render\_support\_tables\.pyto regenerate the LaTeX result tables from the CSV output\. No command regenerates human labels, retrains a model, or searches for a more favorable feature set\.

#### Protocol and action\-set diagnostics\.

Appendix[K](https://arxiv.org/html/2610.27446#A11)gives the source\-checked generation permissions and score rubrics, the tie\-aware action comparison, and post hoc dimension, collision, and capacity diagnostics\. The 96\-state human capacity result uses an archived matrix; it does not recover or replace the unavailable historical 42\-state Qwen matrix\. All original scores and rating files remain unchanged\.

## Appendix KExecution Protocols and Additional Decision Diagnostics

The analyses below use the existing responses, scores, and interface codes\. They add no model training or human labels\. Protocol descriptions are checked against the archived generation and scoring functions\. The tie, dimension, collision, and 96\-state capacity calculations are post hoc diagnostics, separate from the primary frozen comparisons\.

### K\.1Which information each component receives

An encoder maps its input text to an activation and SAE features\. A selector chooses a response strategy from its declared observation\. The generator executes the strategy using a fixed prompt\. The evaluator scores the resulting response\. Restricting the selector does not imply that the other three components receive the same information\.

Table 46:Information supplied to each role in the archived execution protocols\. The hidden required action is an evaluation label, never an additional selector input\.Qwen’scontinuation\_messagesconstructs a prompt from the public request, fact name, and selected action\. The generation loop removes duplicate public\-request keys before producing responses, then joins those responses to the distinct hidden\-condition states\. Thus one response can receive different losses under different states\. SmolLM2’sformat\_action\_promptand Phi’saction\_promptinstead include both policy and situation text\. For each family, changing the selector key fromZZto\(Z,S\)\(Z,S\)leaves those responses and their scoring context unchanged\.

The source record retained in the author working archive atsupplementary/author\_audit/SOURCE\_EVIDENCE\.mdgives the relevant functions, line ranges, and hashes\. Complete unmodified source files or their original archives are retained with that record\. The protocol differences are part of the declared decision problems, not inferred model properties\.

#### Different scoring rules\.

For Qwen, letpd,kcorrp^\{\\mathrm\{corr\}\}\_\{d,k\}denote the evaluator’s normalized, prior\-corrected probability for score labelk∈\{0,1,2,3,4\}k\\in\\\{0,1,2,3,4\\\}on criteriondd\. The five criterion weightswdw\_\{d\}are0\.400\.40,0\.250\.25,0\.150\.15,0\.100\.10, and0\.100\.10for safety, utility, fact handling, macro\-action adherence, and overall quality\. Its score expectation and composite loss are

s¯d=∑k=04k​pd,kcorr,LQwen=1−14​∑dwd​s¯d\.\\bar\{s\}\_\{d\}=\\sum\_\{k=0\}^\{4\}kp^\{\\mathrm\{corr\}\}\_\{d,k\},\\qquad L^\{\\mathrm\{Qwen\}\}=1\-\\frac\{1\}\{4\}\\sum\_\{d\}w\_\{d\}\\bar\{s\}\_\{d\}\.The correction subtracts score\-label prior logits before normalization; it is not calibration against human labels\. In SmolLM2 and Phi, the evaluator returns integer JSON scores in\{0,1,2,3\}\\\{0,1,2,3\\\}for safety, policy adherence, usefulness, factuality, and overall quality\. The weights are respectively0\.300\.30,0\.300\.30,0\.150\.15,0\.100\.10, and0\.150\.15, with normalization by three\. Each model–protocol configuration is analyzed relative to its own loss\. Equal numerical ranges do not make those rubrics interchangeable\.

#### Single\-coordinate feature ranking\.

For SmolLM2 and Phi, writenfitn\_\{\\mathrm\{fit\}\}for the number of fitting states,Li​aL\_\{ia\}for the fitting loss of actionaaat stateii, andzi​k∈\{0,1\}z\_\{ik\}\\in\\\{0,1\\\}for coordinatekk’s support bit at that state\. Define the constant\-input risk and the risk of using only bitkkby

Rconst=1nfitmina∑iLi​a,Rk=1nfit∑b∈\{0,1\}mina∑i:zi​k=bLi​a\.R\_\{\\mathrm\{const\}\}=\\frac\{1\}\{n\_\{\\mathrm\{fit\}\}\}\\min\_\{a\}\\sum\_\{i\}L\_\{ia\},\\qquad R\_\{k\}=\\frac\{1\}\{n\_\{\\mathrm\{fit\}\}\}\\sum\_\{b\\in\\\{0,1\\\}\}\\min\_\{a\}\\sum\_\{i:z\_\{ik\}=b\}L\_\{ia\}\.An empty group contributes zero\. Coordinates are ordered by decreasingRconst−RkR\_\{\\mathrm\{const\}\}\-R\_\{k\}, then decreasingpk​\(1−pk\)p\_\{k\}\(1\-p\_\{k\}\), wherepk=nfit−1​∑izi​kp\_\{k\}=n\_\{\\mathrm\{fit\}\}^\{\-1\}\\sum\_\{i\}z\_\{ik\}is the active fraction, then increasing coordinate index\. The actual returned order inrank\_features\_by\_univariate\_risk\_reductiondetermines the selected features; a separately sorted diagnostic table does not replace that order\. The firstBBentries define the selected support interface\. No claim is made that the returned subset is jointly optimal among all size\-BBsubsets\.

### K\.2Action agreement with tied minimizing losses

Let𝒜=\{ask,decline,assist\}\\mathcal\{A\}=\\\{\\textsc\{ask\},\\textsc\{decline\},\\textsc\{assist\}\\\}be the three\-action library, and fix the same packet response for each state–action pair\. LetLauto​\(a,u\)L\_\{\\mathrm\{auto\}\}\(a,u\)andLhuman​\(a,u\)L\_\{\\mathrm\{human\}\}\(a,u\)be its automatic and final human losses, and set the absolute numerical tie tolerance toτ=10−12\\tau=10^\{\-12\}\. For either sourcer∈\{auto,human\}r\\in\\\{\\mathrm\{auto\},\\mathrm\{human\}\\\}, define its minimum loss and numerically minimizing action set by

br​\(u\)=mina∈𝒜⁡Lr​\(a,u\),Ar​\(u\)=\{a∈𝒜:\|Lr​\(a,u\)−br​\(u\)\|≤τ\}\.b\_\{r\}\(u\)=\\min\_\{a\\in\\mathcal\{A\}\}L\_\{r\}\(a,u\),\\qquad A\_\{r\}\(u\)=\\\{a\\in\\mathcal\{A\}:\|L\_\{r\}\(a,u\)\-b\_\{r\}\(u\)\|\\leq\\tau\\\}\.The historical selected labelarfirst​\(u\)a\_\{r\}^\{\\mathrm\{first\}\}\(u\)is the first exact minimum under the fixed displayed action order\. At zero,10−1210^\{\-12\}, and10−810^\{\-8\}tolerance, the stored data give the same minimizing sets\.

We report three different indicators for each state:

Ifirst​\(u\)\\displaystyle I\_\{\\mathrm\{first\}\}\(u\)=𝟏\{aautofirst\(u\)=ahumanfirst\(u\)\},\\displaystyle=\\mathbf\{1\}\\\{a\_\{\\mathrm\{auto\}\}^\{\\mathrm\{first\}\}\(u\)=a\_\{\\mathrm\{human\}\}^\{\\mathrm\{first\}\}\(u\)\\\},I∩​\(u\)\\displaystyle I\_\{\\cap\}\(u\)=𝟏\{Aauto\(u\)∩Ahuman\(u\)≠∅\},\\displaystyle=\\mathbf\{1\}\\\{A\_\{\\mathrm\{auto\}\}\(u\)\\cap A\_\{\\mathrm\{human\}\}\(u\)\\neq\\varnothing\\\},I=​\(u\)\\displaystyle I\_\{=\}\(u\)=𝟏\{Aauto\(u\)=Ahuman\(u\)\}\.\\displaystyle=\\mathbf\{1\}\\\{A\_\{\\mathrm\{auto\}\}\(u\)=A\_\{\\mathrm\{human\}\}\(u\)\\\}\.Their averages describe fixed\-label matching, at least one shared optimum, and equality of all minimizing choices, respectively\. A large automatic set makes a nonempty intersection easier to obtain without making the full sets equal\. None of the indicators measures agreement of the loss magnitudes\.

Table 47:Distinct action\-agreement statistics on the same 176 SmolLM2 states and packet responses\. First\-minimum matching uses the fixed orderask,decline,assist; the set statistics retain all tied minimizing actions\.Of the 176 states, automatic loss has a tied minimum in 90 and human loss in one\. Changing only the common fixed action ordering across all six permutations changes first\-minimum matching from90/17690/176to105/176105/176, or51\.1%51\.1\\%–59\.7%59\.7\\%\. The full ordering sensitivity and per\-state sets are stored insupplementary/author\_audit/diagnostics/\. Minimum loss values, and therefore information gaps, are unaffected by choosing among tied minimizers\.

The saved pairwise preference statistic uses the three unordered action pairs at each state\. It excludes pairs whose human losses differ by at most10−910^\{\-9\}, then compares the signs of the automatic and human loss differences\. An automatic tie counts as disagreement when the human pair has a strict preference\. Its reported value,55\.8%55\.8\\%, gives no partial credit for an automatic tie\.

### K\.3Human loss dimensions and within\-policy results

Letsd​\(a,u\)∈\[0,3\]s\_\{d\}\(a,u\)\\in\[0,3\]be the final score for dimensionddof a packet response\. For adjudicated items the score is the resolved score; for other items it is the average of the two independent scores\. The corresponding single\-dimension loss isLd​\(a,u\)=1−sd​\(a,u\)/3L\_\{d\}\(a,u\)=1\-s\_\{d\}\(a,u\)/3\. Each dimension uses the same states, selected features, and policy\-aware grouping as the composite loss, but both oracle minima are recomputed for that dimension\.

Table 48:Post\-hoc policy\-aware information gaps atB=2B=2under different human rating targets\. Each single\-dimension loss is one minus its score divided by three\. Values are point diagnostics, not new confidence bounds or unsafe\-event probabilities\.The safety\-only gaps on the original sample are0\.00420\.0042and0\.01250\.0125; on the locked collection they are0\.05210\.0521and0\.03120\.0312\. No new intervals are attached to these post hoc point diagnostics\. A score\-based safety loss is not the probability of an unsafe event\. Likewise, the composite gap is not the weighted sum of dimension gaps: minimizing a weighted sum of losses can choose a different action from minimizing each component separately\.

For policy\-aware inputs, different policy texts have disjoint observation cells\. LetΔd\\Delta\_\{d\}be the within\-policy gap for policy areadd, andwdw\_\{d\}its share of evaluated states\. The pooled gap is thenΔZ,S=∑dwd​Δd\\Delta\_\{Z,S\}=\\sum\_\{d\}w\_\{d\}\\Delta\_\{d\}\. Whole\-area bootstrap draws merely change the weights of the saved within\-policy tables\. In the locked human collection, all eight areas have positive point gaps for both SAE seeds\. A bootstrap of those fixed positive contributions remains positive; it does not test whether a gap based on unknown expected rollout losses is zero\. The complete within\-policy risks are retained in the author working archive atsupplementary/author\_audit/diagnostics/human\_per\_policy\_gap\.csv\.

### K\.4Complete record for the illustrative collision

The worked pair in Table[1](https://arxiv.org/html/2610.27446#S2.T1)uses SmolLM2 SAE seed 314159, feature budgetB=2B=2, and code00, from the lockedassistive\_accommodation\_communicationsarea\. The exact state IDs arefresh\_54de51e809f46113eca8andfresh\_8510b3ae9ec45d87d0d6\. Both states carry the same policy:

> Help write respectful process communications without making eligibility decisions\. Ask when the requested accommodation, functional barrier, or responsible office is unclear\. Decline fabrication of medical documentation or discriminatory screening\. Assist with accessible scheduling and referral text\.

The first situation requests a respectful acknowledgment, accessible meeting formats, and referral to the authorized office\. The second requests a fake clinician letter to obtain approval without review\. The first state’s loss vector is\(0\.350,0\.525,0\)\(0\.350,0\.525,0\)and the second state’s is\(0\.500,0\.100,0\.800\)\(0\.500,0\.100,0\.800\), in the orderask,decline,assist\. Their column means are\(0\.425,0\.3125,0\.400\)\(0\.425,0\.3125,0\.400\), so the best common action isdecline\. Separate full\-state choices have mean loss0\.0500\.050, while the shared minimum is0\.31250\.3125\.

The0\.26250\.2625difference describes only the equally weighted two\-state subproblem, not the full containing cell or the full collection\. The pair was selected after inspecting the saved results as an explanation of an existing collision\. Exact situation texts and the other qualifying pairs are retained in the author working archive atsupplementary/author\_audit/diagnostics/human\_real\_collision\_examples\.json\.

### K\.5Capacity on the recorded human loss matrix

The capacity diagnostic uses every state in the 96\-state locked human collection\. The three columns are final composite losses forask,decline, andassist\. The calculation does not use SAE features or a seed: it optimizes unrestricted state\-to\-code assignments\. Only the redesigned code is visible, with no free policy input\. The result is𝖬⋆​\(ε\)\\mathsf\{M\}^\{\\star\}\(\\varepsilon\)from Section[3](https://arxiv.org/html/2610.27446#S3), not an SAE\-realizable capacity or the conditional code count with free policy\.

We evaluate tolerances00,0\.020\.02,0\.100\.10,0\.200\.20,0\.300\.30,0\.350\.35, and0\.400\.40, using the same saved human labels for all tolerances\. Global constraint generation starts with no conflicts, solves the coloring problem, and validates every new cell\. Every retained conflict, returned label, cell margin, and iteration record is saved\. The final capacity is the matching lower and upper output count under numerical tolerance10−910^\{\-9\}\.

Table 49:Unrestricted code\-only capacity on all 96 locked SmolLM2 states under final human composite loss\. Constraint generation and the independent two\-label feasibility formulation agree\. Margins are reoptimized within every returned cell; numerical tolerance is10−910^\{\-9\}\.#### Independent two\-label feasibility check\.

With three actions, labeling each state by an optimal action supplies at most three compatible cells\. A direct linear program gives the one\-output margin0\.369500370\.36950037\. To check whether two outputs suffice without enumerating conflicts, letei∈\[0,1\]3e\_\{i\}\\in\[0,1\]^\{3\}be the regret vector at stateii,q\(0\),q\(1\)∈Δ3q^\{\(0\)\},q^\{\(1\)\}\\in\\Delta\_\{3\}be the two shared continuation distributions, andxi∈\{0,1\}x\_\{i\}\\in\\\{0,1\\\}select a label\. Feasibility is equivalent to

ei⊤​q\(0\)≤ε\+xi,ei⊤​q\(1\)≤ε\+1−xi,i∈\[96\]\.e\_\{i\}^\{\\top\}q^\{\(0\)\}\\leq\\varepsilon\+x\_\{i\},\\qquad e\_\{i\}^\{\\top\}q^\{\(1\)\}\\leq\\varepsilon\+1\-x\_\{i\},\\qquad i\\in\[96\]\.Forxi=0x\_\{i\}=0, the first constraint is active; forxi=1x\_\{i\}=1, the second is active\. The other constraint is automatically satisfied because regrets are bounded by one andε≥0\\varepsilon\\geq 0\. Thus a feasible solution supplies two compatible cells, and any two\-cell compatible labeling supplies a feasible solution\. The independent feasibility MILP agrees with the constraint\-generation capacity at all seven tolerances\. Feasible assignments are rechecked against the original regret constraints; solver\-reported infeasibility and numerical tolerance are recorded rather than described as exact rational proofs\.

The matrix, including integer score units with denominator 120, is retained in the author working archive assupplementary/author\_audit/capacity/human96\_loss\_matrix\.csv\. The capacity script reconstructs the matrix from the existing human\-loss file and checks its precision, complete state coverage, and input hash\. The historical 42\-state Qwen curve in Appendix[G\.6](https://arxiv.org/html/2610.27446#A7.SS6)is a different experiment\. Its matrix is still unavailable, and no 96\-state result is described as a recovery of that historical curve\.

## Appendix LContinuous Interfaces and Population Certificates

The main audit computes exact optima for a supplied finite loss table\. This extension asks what can be inferred about an unquantized continuous observation and about unknown expected losses\. The complete assumptions, finite\-sample bounds, and proofs are collected here so that the finite interface analysis does not require this additional estimation setup\.

### L\.1Continuous interfaces: identification and transfer

#### What distinct finite observations identify\.

Let the observation space satisfy𝒵⊆ℝd\\mathcal\{Z\}\\subseteq\\mathbb\{R\}^\{d\}, whereddis the observation dimension\. For audited pairs\(ui,zi\)i=1n\(u\_\{i\},z\_\{i\}\)\_\{i=1\}^\{n\}, define the empirical risks using the same statewise losses:

R^Z⋆:=infδ∈𝔇Z1n​∑i=1nℓ⁡\(ui,δ⁡\(zi\)\),R^full:=1n​∑i=1nℓ⋆​\(ui\)\.\\widehat\{R\}\_\{Z\}^\{\\star\}:=\\inf\_\{\\delta\\in\\mathfrak\{D\}\_\{Z\}\}\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\ell\(u\_\{i\},\\delta\(z\_\{i\}\)\),\\qquad\\widehat\{R\}\_\{\\mathrm\{full\}\}:=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\ell^\{\\star\}\(u\_\{i\}\)\.
###### Theorem L\.1\(Finite\-sample interpolation limit\)\.

If the audited observationsz1,…,znz\_\{1\},\\ldots,z\_\{n\}are distinct and all measurable rules are allowed, thenR^Z⋆=R^full\\widehat\{R\}\_\{Z\}^\{\\star\}=\\widehat\{R\}\_\{\\mathrm\{full\}\}\. Consequently, the audited points alone do not certify a positive population information gap for the exact continuous observation\.

An unrestricted lookup rule can choose an optimal continuation separately at every distinct observed vector\. Failure of a trained probe therefore does not identify missing information or the optimum of its rule class\. An exact best\-in\-class comparison can identify a rule\-class limitation, but a population information claim needs additional distributional evidence or assumptions, such as a specified overlapping observation channel\.

#### Quantization with a controlled cost error\.

Equip𝒵\\mathcal\{Z\}with a declared metricdZd\_\{Z\}, such as Euclidean distance\. LetQh:𝒵→\[L\]Q\_\{h\}:\\mathcal\{Z\}\\to\[L\]be a measurable partition map, with finiteL≥1L\\geq 1, cellsAr:=Qh−1​\(r\)A\_\{r\}:=Q\_\{h\}^\{\-1\}\(r\), and diameter boundh≥0h\\geq 0measured usingdZd\_\{Z\}\. For each continuationπj\\pi\_\{j\}, define its conditional expected loss

cj​\(z\):=𝔼⁡\[ℓj​\(U\)∣Z=z\]\.c\_\{j\}\(z\):=\\mathbb\{E\}\[\\ell\_\{j\}\(U\)\\mid Z=z\]\.The quantitycj​\(z\)c\_\{j\}\(z\)is an expected*loss*, whereas the earliere¯j​\(z\)\\bar\{e\}\_\{j\}\(z\)is an expected*regret*\. LetRQh​\(Z\)⋆R\_\{Q\_\{h\}\(Z\)\}^\{\\star\}denote the optimum when only the partition labelQh​\(Z\)Q\_\{h\}\(Z\)is observed\.

###### Assumption L\.2\(Modulus for conditional costs\)\.

There are versions of the conditional expected lossescjc\_\{j\}and a known nondecreasing functionω:\[0,∞\)→\[0,∞\)\\omega:\[0,\\infty\)\\to\[0,\\infty\)withω⁡\(0\)=0\\omega\(0\)=0such that

\|cj​\(z\)−cj​\(z′\)\|≤ω⁡\(dZ​\(z,z′\)\)\|c\_\{j\}\(z\)\-c\_\{j\}\(z^\{\\prime\}\)\|\\leq\\omega\(d\_\{Z\}\(z,z^\{\\prime\}\)\)for everyj∈\[m\]j\\in\[m\]and everyz,z′z,z^\{\\prime\}in the same positive\-probability partition cell\. Each such cell has diameter at mosthhunderdZd\_\{Z\}\.

The modulusω\\omegacontrols how expected continuation costs vary within an entire population cell, not just between the sampled vectors\. Small empirical distances or low reconstruction error do not establish the assumption\.

###### Theorem L\.3\(Continuous\-interface transfer bound\)\.

Under Assumptions[2\.1](https://arxiv.org/html/2610.27446#S2.Thmassumption1)and[L\.2](https://arxiv.org/html/2610.27446#A12.Thmassumption2),

0≤RQh​\(Z\)⋆−RZ⋆≤2​ω​\(h\),0\\leq R\_\{Q\_\{h\}\(Z\)\}^\{\\star\}\-R\_\{Z\}^\{\\star\}\\leq 2\\omega\(h\),and consequently

RZ⋆−Rfull≥RQh​\(Z\)⋆−Rfull−2​ω​\(h\)\.R\_\{Z\}^\{\\star\}\-R\_\{\\mathrm\{full\}\}\\geq R\_\{Q\_\{h\}\(Z\)\}^\{\\star\}\-R\_\{\\mathrm\{full\}\}\-2\\omega\(h\)\.If every conditional expected losscjc\_\{j\}is Lipschitz with common constantLc≥0L\_\{c\}\\geq 0, one may takeω⁡\(h\)=Lc​h\\omega\(h\)=L\_\{c\}h\.

The partition can increase risk by hiding distinctions present inZZ\. Subtracting2​ω​\(h\)2\\omega\(h\)accounts for that increase before a partitioned risk gap is used to infer a gap for the original continuous observation\. Without a valid transfer bound, the finite partition is the audited interface, not a proxy that automatically certifies its continuous input\.

### L\.2Finite\-sample certificates with exact or estimated losses

The transfer theorem compares population risks\. A finite\-sample certificate must additionally control state sampling and, when losses are estimated, within\-state loss uncertainty\. The two error sources are kept separate below\.

#### Exact conditional expected loss vectors\.

Let\(Ut,Zt\)t=1n\(U\_\{t\},Z\_\{t\}\)\_\{t=1\}^\{n\}be independent and identically distributed audit pairs from the declared joint law, with the continuation library, loss functions, observation channel, partitionQhQ\_\{h\}, and valid modulus fixed independently of the certification sample\. The partition hasLLcells and the library hasmmcontinuations\. Suppose each observed entryLt​j=ℓj​\(Ut\)L\_\{tj\}=\\ell\_\{j\}\(U\_\{t\}\)is the*exact*conditional expected loss\. Define the empirical joint costs and full\-state risk by

C^r​j:=1n∑t=1n𝟏\{Qh\(Zt\)=r\}Lt​j,R^full:=1n∑t=1nminj∈\[m\]Lt​j\.\\widehat\{C\}\_\{rj\}:=\\frac\{1\}\{n\}\\sum\_\{t=1\}^\{n\}\\mathbf\{1\}\\\{Q\_\{h\}\(Z\_\{t\}\)=r\\\}L\_\{tj\},\\qquad\\widehat\{R\}\_\{\\mathrm\{full\}\}:=\\frac\{1\}\{n\}\\sum\_\{t=1\}^\{n\}\\min\_\{j\\in\[m\]\}L\_\{tj\}\.The corresponding population joint cost isCr​j:=𝔼\[𝟏\{Qh\(Z\)=r\}ℓj\(U\)\]C\_\{rj\}:=\\mathbb\{E\}\[\\mathbf\{1\}\\\{Q\_\{h\}\(Z\)=r\\\}\\ell\_\{j\}\(U\)\]\.

###### Corollary L\.4\(Finite\-sample certificate with exact loss vectors\)\.

Under the independent\-sampling and exact\-loss conditions just stated and Assumption[L\.2](https://arxiv.org/html/2610.27446#A12.Thmassumption2), fix0<α<10<\\alpha<1and set

rn:=log⁡\(4​L​m/α\)2​n,sn:=log⁡\(4/α\)2​n\.r\_\{n\}:=\\sqrt\{\\frac\{\\log\(4Lm/\\alpha\)\}\{2n\}\},\\qquad s\_\{n\}:=\\sqrt\{\\frac\{\\log\(4/\\alpha\)\}\{2n\}\}\.With probability at least1−α1\-\\alpha,

RZ⋆−Rfull≥∑r=1Lmax⁡\{0,minj⁡C^r​j−rn\}−min⁡\{1,R^full\+sn\}−2​ω​\(h\)\.R\_\{Z\}^\{\\star\}\-R\_\{\\mathrm\{full\}\}\\geq\\sum\_\{r=1\}^\{L\}\\max\\\{0,\\min\_\{j\}\\widehat\{C\}\_\{rj\}\-r\_\{n\}\\\}\-\\min\\\{1,\\widehat\{R\}\_\{\\mathrm\{full\}\}\+s\_\{n\}\\\}\-2\\omega\(h\)\.

The radiusrnr\_\{n\}simultaneously controls theL​mLmjoint costs, whilesns\_\{n\}controls the mean full\-state oracle loss\. The certificate subtracts an upper bound on full\-state risk from a lower bound on partitioned\-interface risk, followed by the transfer penalty\. Positivity certifies a population gap only under the stated sampling and modulus assumptions\.

#### Why unbiased noisy scores are insufficient\.

LetYt​j∈\[0,1\]Y\_\{tj\}\\in\[0,1\]be a noisy score with𝔼⁡\[Yt​j∣Ut\]=ℓj​\(Ut\)\\mathbb\{E\}\[Y\_\{tj\}\\mid U\_\{t\}\]=\\ell\_\{j\}\(U\_\{t\}\)\. Unbiasedness for each continuation does not imply unbiasedness after choosing the smallest score:

𝔼⁡\[minj⁡Yt​j∣Ut\]≤minj⁡𝔼⁡\[Yt​j∣Ut\]=ℓ⋆​\(Ut\)\.\\mathbb\{E\}\[\\min\_\{j\}Y\_\{tj\}\\mid U\_\{t\}\]\\leq\\min\_\{j\}\\mathbb\{E\}\[Y\_\{tj\}\\mid U\_\{t\}\]=\\ell^\{\\star\}\(U\_\{t\}\)\.For a single state and two independent Bernoulli\(1/2\)\(1/2\)scores, the expected score minimum is1/41/4, whereas the true oracle loss is1/21/2\. More sampled states do not remove that minimum\-selection bias\. Corollary[L\.4](https://arxiv.org/html/2610.27446#A12.Thmassumption4)therefore cannot use an uncorrected finite\-rollout estimate in place of an exact loss vector\.

#### Simultaneous intervals for estimated losses\.

Retain independent and identically distributed audit pairs and a library, loss, channel, partition, and valid modulus fixed independently of the certification sample\. Choose a state\-sampling error budgetαs∈\(0,1\)\\alpha\_\{s\}\\in\(0,1\)and a loss\-estimation error budgetαℓ∈\[0,1−αs\)\\alpha\_\{\\ell\}\\in\[0,1\-\\alpha\_\{s\}\)\. Suppose random interval endpoints satisfy0≤L¯t​j≤L¯t​j≤10\\leq\\underline\{L\}\_\{tj\}\\leq\\overline\{L\}\_\{tj\}\\leq 1and

ℙ\(L¯t​j≤ℓj\(Ut\)≤L¯t​jfor allt∈\[n\],j∈\[m\]\)≥1−αℓ\.\\mathbb\{P\}\\\!\\left\(\\underline\{L\}\_\{tj\}\\leq\\ell\_\{j\}\(U\_\{t\}\)\\leq\\overline\{L\}\_\{tj\}\\text\{ for all \}t\\in\[n\],\\ j\\in\[m\]\\right\)\\geq 1\-\\alpha\_\{\\ell\}\.The intervals cover expected losses, not merely individual rollout scores\. Define

C^r​j−:=1n∑t=1n𝟏\{Qh\(Zt\)=r\}L¯t​j,R^full\+:=1n∑t=1nminjL¯t​j\.\\widehat\{C\}^\{\-\}\_\{rj\}:=\\frac\{1\}\{n\}\\sum\_\{t=1\}^\{n\}\\mathbf\{1\}\\\{Q\_\{h\}\(Z\_\{t\}\)=r\\\}\\underline\{L\}\_\{tj\},\\qquad\\widehat\{R\}^\{\+\}\_\{\\mathrm\{full\}\}:=\\frac\{1\}\{n\}\\sum\_\{t=1\}^\{n\}\\min\_\{j\}\\overline\{L\}\_\{tj\}\.The first quantity uses loss lower bounds for interface costs; the second uses loss upper bounds for the oracle comparison\.

###### Theorem L\.5\(Continuous certificate with simultaneous loss intervals\)\.

Under the stated simultaneous interval coverage and independent audit sampling, with Assumption[L\.2](https://arxiv.org/html/2610.27446#A12.Thmassumption2), use the state\-sampling radii

rn:=log⁡\(4​L​m/αs\)2​n,sn:=log⁡\(4/αs\)2​n\.r\_\{n\}:=\\sqrt\{\\frac\{\\log\(4Lm/\\alpha\_\{s\}\)\}\{2n\}\},\\qquad s\_\{n\}:=\\sqrt\{\\frac\{\\log\(4/\\alpha\_\{s\}\)\}\{2n\}\}\.Then, with probability at least1−αs−αℓ1\-\\alpha\_\{s\}\-\\alpha\_\{\\ell\},

RZ⋆−Rfull≥\\displaystyle R\_\{Z\}^\{\\star\}\-R\_\{\\mathrm\{full\}\}\\geq\{\}∑r=1Lmax⁡\{0,minj⁡C^r​j−−rn\}\\displaystyle\\sum\_\{r=1\}^\{L\}\\max\\\{0,\\min\_\{j\}\\widehat\{C\}^\{\-\}\_\{rj\}\-r\_\{n\}\\\}\(14\)−min⁡\{1,R^full\+\+sn\}−2​ω​\(h\)\.\\displaystyle\-\\min\\\{1,\\widehat\{R\}^\{\+\}\_\{\\mathrm\{full\}\}\+s\_\{n\}\\\}\-2\\omega\(h\)\.The interval\-coverage event and state\-sampling concentration event need not be independent\.

On simultaneous interval coverage,C^r​j−\\widehat\{C\}^\{\-\}\_\{rj\}is no larger than the empirical joint cost computed from exact losses, andR^full\+\\widehat\{R\}^\{\+\}\_\{\\mathrm\{full\}\}is no smaller than the empirical exact oracle risk\. State\-sampling concentration is applied to those exact\-loss quantities; the interval endpoints then bound them\. A union bound accounts for both failure events without an independence assumption\.

#### One valid rollout\-based interval construction\.

Let each sampled state–continuation pair receiveN≥1N\\geq 1conditionally independent scoresYt​j​k∈\[0,1\]Y\_\{tjk\}\\in\[0,1\],k∈\[N\]k\\in\[N\], drawn under the fixed scoring protocol\. Conditionally on the full stateUtU\_\{t\}, assume the scores for each pair are independent and their means differ fromℓj​\(Ut\)\\ell\_\{j\}\(U\_\{t\}\)by at most a known deterministic boundbjudge≥0b\_\{\\mathrm\{judge\}\}\\geq 0\. For0<αℓ<10<\\alpha\_\{\\ell\}<1, define

Y¯t​j:=1N​∑k=1NYt​j​k,rN:=bjudge\+log⁡\(2​n​m/αℓ\)2​N\.\\overline\{Y\}\_\{tj\}:=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}Y\_\{tjk\},\\qquad r\_\{N\}:=b\_\{\\mathrm\{judge\}\}\+\\sqrt\{\\frac\{\\log\(2nm/\\alpha\_\{\\ell\}\)\}\{2N\}\}\.Conditional Hoeffding bounds and a union bound over then​mnmpairs give valid simultaneous intervals

L¯t​j:=max⁡\{0,Y¯t​j−rN\},L¯t​j:=min⁡\{1,Y¯t​j\+rN\}\.\\underline\{L\}\_\{tj\}:=\\max\\\{0,\\overline\{Y\}\_\{tj\}\-r\_\{N\}\\\},\\qquad\\overline\{L\}\_\{tj\}:=\\min\\\{1,\\overline\{Y\}\_\{tj\}\+r\_\{N\}\\\}\.HereNNcontrols within\-state score uncertainty, whereasnncontrols state\-sampling uncertainty\. A judge\-bias bound obtained with its own failure probability requires an additional error\-budget term\. Exact loss vectors are the zero\-width interval case withαℓ=0\\alpha\_\{\\ell\}=0\.

The saved finite\-table bootstrap analyses do not establish a uniformbjudgeb\_\{\\mathrm\{judge\}\}or a validated population modulusω\\omega\. They are therefore not empirical instances of Equation \([14](https://arxiv.org/html/2610.27446#A12.E14)\)\. Algorithm[3](https://arxiv.org/html/2610.27446#alg3)gives the interval\-based procedure\. The proofs follow below\.

### L\.3Proofs of the continuous\-interface results

###### Proof of Theorem[L\.1](https://arxiv.org/html/2610.27446#A12.Thmassumption1)\.

At stateuiu\_\{i\}, chooseji∈arg​minj⁡ℓj​\(ui\)j\_\{i\}\\in\\argmin\_\{j\}\\ell\_\{j\}\(u\_\{i\}\)\. Distinct inputsziz\_\{i\}allow a single rule to assign continuationjij\_\{i\}to eachziz\_\{i\}; use any fixed continuation elsewhere\. The rule is measurable because singletons in Euclidean space are Borel\. It attainsn−1​∑iℓ⋆​\(ui\)n^\{\-1\}\\sum\_\{i\}\\ell^\{\\star\}\(u\_\{i\}\)\. No rule can beat those row minima, proving the equality\. ∎

###### Proof of Theorem[L\.3](https://arxiv.org/html/2610.27446#A12.Thmassumption3)\.

Data processing givesRQh​\(Z\)⋆≥RZ⋆R\_\{Q\_\{h\}\(Z\)\}^\{\\star\}\\geq R\_\{Z\}^\{\\star\}\. Fix a partition cellArA\_\{r\}of positive probability, choosezr∈Arz\_\{r\}\\in A\_\{r\}, and setjr∈arg​minj⁡cj​\(zr\)j\_\{r\}\\in\\argmin\_\{j\}c\_\{j\}\(z\_\{r\}\)\. Letc⋆​\(z\):=minj⁡cj​\(z\)c\_\{\\star\}\(z\):=\\min\_\{j\}c\_\{j\}\(z\)\. The minimum of finitely many functions with common modulusω\\omegahas the same modulus: compare at an index minimizing either argument\. For everyz∈Arz\\in A\_\{r\},

cjr​\(z\)≤cjr​\(zr\)\+ω⁡\(h\)=c⋆​\(zr\)\+ω⁡\(h\)≤c⋆​\(z\)\+2​ω​\(h\)\.c\_\{j\_\{r\}\}\(z\)\\leq c\_\{j\_\{r\}\}\(z\_\{r\}\)\+\\omega\(h\)=c\_\{\\star\}\(z\_\{r\}\)\+\\omega\(h\)\\leq c\_\{\\star\}\(z\)\+2\\omega\(h\)\.Usingjrj\_\{r\}throughout each cell gives a feasible rule on the partition labels\. Averaging the inequality bounds its risk by𝔼⁡\[c⋆​\(Z\)\]\+2​ω​\(h\)=RZ⋆\+2​ω​\(h\)\\mathbb\{E\}\[c\_\{\\star\}\(Z\)\]\+2\\omega\(h\)=R\_\{Z\}^\{\\star\}\+2\\omega\(h\)\. The partition optimum is no larger than this feasible risk\. SubtractingRfullR\_\{\\mathrm\{full\}\}proves the gap bound\. Lipschitz costs allowω⁡\(h\)=Lc​h\\omega\(h\)=L\_\{c\}h\. ∎

###### Proof of Corollary[L\.4](https://arxiv.org/html/2610.27446#A12.Thmassumption4)\.

Each exact observation𝟏\{Qh\(Zt\)=r\}Lt​j∈\[0,1\]\\mathbf\{1\}\\\{Q\_\{h\}\(Z\_\{t\}\)=r\\\}L\_\{tj\}\\in\[0,1\]has meanCr​jC\_\{rj\}and is independent acrosstt\. Hoeffding’s inequality and a union bound give simultaneous cost error at mostrnr\_\{n\}with failure probability2​L​m​e−2​n​rn2=α/22Lme^\{\-2nr\_\{n\}^\{2\}\}=\\alpha/2\. The independent observationsminj⁡Lt​j=ℓ⋆​\(Ut\)\\min\_\{j\}L\_\{tj\}=\\ell^\{\\star\}\(U\_\{t\}\)have meanRfullR\_\{\\mathrm\{full\}\}; the radiussns\_\{n\}bounds their error with failure probability at most2​e−2​n​sn2=α/22e^\{\-2ns\_\{n\}^\{2\}\}=\\alpha/2\. On the intersection of these events,

RQh​\(Z\)⋆≥∑rmax⁡\{0,minj⁡C^r​j−rn\},Rfull≤min⁡\{1,R^full\+sn\}\.R\_\{Q\_\{h\}\(Z\)\}^\{\\star\}\\geq\\sum\_\{r\}\\max\\\{0,\\min\_\{j\}\\widehat\{C\}\_\{rj\}\-r\_\{n\}\\\},\\qquad R\_\{\\mathrm\{full\}\}\\leq\\min\\\{1,\\widehat\{R\}\_\{\\rm full\}\+s\_\{n\}\\\}\.Clipping uses the known loss range\. Subtract the upper bound from the lower bound and apply Theorem[L\.3](https://arxiv.org/html/2610.27446#A12.Thmassumption3)\. A union bound limits total failure probability toα\\alpha\. ∎

###### Proof of Theorem[L\.5](https://arxiv.org/html/2610.27446#A12.Thmassumption5)\.

Define the unobserved exact sample quantities

C~r​j:=1n∑t𝟏\{Qh\(Zt\)=r\}ℓj\(Ut\),R~full:=1n∑tℓ⋆\(Ut\)\.\\widetilde\{C\}\_\{rj\}:=\\frac\{1\}\{n\}\\sum\_\{t\}\\mathbf\{1\}\\\{Q\_\{h\}\(Z\_\{t\}\)=r\\\}\\ell\_\{j\}\(U\_\{t\}\),\\qquad\\widetilde\{R\}\_\{\\rm full\}:=\\frac\{1\}\{n\}\\sum\_\{t\}\\ell^\{\\star\}\(U\_\{t\}\)\.The preceding concentration proof applies with failure probabilityαs\\alpha\_\{s\}\. On simultaneous interval coverage,C^r​j−≤C~r​j\\widehat\{C\}^\{\-\}\_\{rj\}\\leq\\widetilde\{C\}\_\{rj\}andR^full\+≥R~full\\widehat\{R\}^\{\+\}\_\{\\rm full\}\\geq\\widetilde\{R\}\_\{\\rm full\}, since entrywise inequalities are preserved by sums and minima\. On both events,

RQh​\(Z\)⋆≥∑rmax⁡\{0,minj⁡C^r​j−−rn\},Rfull≤min⁡\{1,R^full\+\+sn\}\.R\_\{Q\_\{h\}\(Z\)\}^\{\\star\}\\geq\\sum\_\{r\}\\max\\\{0,\\min\_\{j\}\\widehat\{C\}^\{\-\}\_\{rj\}\-r\_\{n\}\\\},\\qquad R\_\{\\mathrm\{full\}\}\\leq\\min\\\{1,\\widehat\{R\}^\{\+\}\_\{\\rm full\}\+s\_\{n\}\\\}\.The transfer bound proves the result\. A union bound gives failure probability at mostαs\+αℓ\\alpha\_\{s\}\+\\alpha\_\{\\ell\}; no independence of the events or concentration of the interval endpoints is needed\.

For the stated rollout construction, condition on each sampled state\. Hoeffding bounds itsNNscores around their average conditional mean with radiuslog⁡\(2​n​m/αℓ\)/\(2​N\)\\sqrt\{\\log\(2nm/\\alpha\_\{\\ell\}\)/\(2N\)\}and failure probability at mostαℓ/\(n​m\)\\alpha\_\{\\ell\}/\(nm\)\. The deterministic bias bound increases the radius bybjudgeb\_\{\\rm judge\}\. Integrating over states and taking a union bound over all pairs gives coverage\. Clipping to\[0,1\]\[0,1\]preserves it\. Any separately uncertain bias bound requires its own failure budget\. ∎

## Appendix MDetailed Experimental and Reproduction Protocols

The following sections specify the model settings, feature selection, human ratings, and reproduction inputs for the comparisons in Section[4](https://arxiv.org/html/2610.27446#S4)\.

### M\.1Qwen2\.5: separating information, rule, and fitting effects

The primary model isQwen/Qwen2\.5\-1\.5B\-Instruct, with public SAEs applied to the activations after layers 17, 18, 19, and 21\. Layer indices are zero\-based\. The original collection has 240 states from 20 policy areas, each containing three required actions and four paraphrases\. Twelve areas \(144 states\) are used for setting selection and fitting; eight areas \(96 states\) are held out until confirmation\. Qwen2\.5\-7B\-Instruct is the primary outcome evaluator, and Mistral\-7B\-Instruct\-v0\.3 provides a separate evaluator check\.

The model activation width is 1,536 and each SAE dictionary has 24,576 coordinates\. Extraction useshidden\_states\[layer\+1\], the residual activation after the specified layer\. Before computing risks, we check extraction alignment, agreement with the direct SAE forward pass, state and paraphrase counts, and completeness of the three\-action loss tables\. Table[23](https://arxiv.org/html/2610.27446#A7.T23)gives the full configuration\.

#### Qwen outcome scoring\.

The five criteria are safety, utility, fact handling, macro\-action adherence, and overall quality, with weights\(0\.40,0\.25,0\.15,0\.10,0\.10\)\(0\.40,0\.25,0\.15,0\.10,0\.10\)on a scale from zero to four\. The evaluator produces an expected score from prior\-corrected next\-token label probabilities; the weighted score is divided by four and subtracted from one to obtain loss\. The evaluator sees the full situation, policy, selected action, response, and required\-action label\. The required\-action label is an evaluation input, not a selector input\. SmolLM2 and Phi instead use integer JSON scores from zero to three with the five criteria and weights given below\. Normalization to\[0,1\]\[0,1\]does not make the different rubrics the same measurement scale across models\.

### M\.2SmolLM2: independent SAE training and additional policies

ForHuggingFaceTB/SmolLM2\-1\.7B\-Instruct, we use activations after layer 17 and Mistral\-7B\-Instruct\-v0\.3 as the automatic evaluator\. Alongside the original 240 states, a locked collection contains 96 states from eight additional policy areas\. The locked collection is fixed before response generation and scoring, with its SHA\-256 digest recorded in the archive\. Five responses for each of the three choices at each of the 336 states give336×3×5=5,040336\\times 3\\times 5=5\{,\}040responses\.

Two TopK SAEs are trained from scratch with random seeds 314159 and 271828\. Each dictionary has 16,384 coordinates, with sparsityk=64k=64: the TopK selection keeps 64 coordinates per activation\. Each run uses eight million FineWeb\-Edu activation tokens for training, 300,000 for validation, and a 100,000\-token normalization sample\. The runs share the model, layer, architecture, and cached activation source, but differ in initialization and training order\. Feature identities are selected on the original 144 fitting states and frozen before evaluation on the original and locked 96\-state collections\. Table[7](https://arxiv.org/html/2610.27446#A6.T7)gives the full configuration\.

The automatic evaluator assigns integer scores in\{0,1,2,3\}\\\{0,1,2,3\\\}for safety, policy adherence, usefulness, factuality, and overall quality, with weights\(0\.30,0\.30,0\.15,0\.10,0\.15\)\(0\.30,0\.30,0\.15,0\.10,0\.15\)\. Loss is one minus the weighted score divided by three\. Its prompt includes the policy, situation, selected action, and response, but not the hidden required action\.

#### How the support features are selected\.

For both SmolLM2 and Phi, each coordinate is first assessed alone on the 144 fitting states\. Its ranking score is the reduction in minimum loss relative to a constant input when the selector observes that coordinate’s on/off bit\. Coordinates are ordered by decreasing reduction, then by activation\-frequency balance, then by index\. The firstBBcoordinates form the interface withBBfeatures\. The selector uses a ranking of individual coordinates, not joint optimization over all subsets ofBBfeatures\. The same ordered features are retained for all evaluation conditions; Appendix[K](https://arxiv.org/html/2610.27446#A11)gives the ranking formula and tie rules\.

#### Matching automatic scores to human\-rated responses\.

The automatic risk table averages five responses per state–action pair\. For human evaluation, a score\-independent SHA\-256 ordering fixes one response from each pair\. Human risks use that single response, and automatic–human calibration compares scores of the*same*response, rather than comparing a single human\-rated response with a five\-response automatic average\. The human study evaluates SmolLM2 responses only\.

### M\.3Phi: matched feature\-only and policy\-aware comparisons

The Phi model ismicrosoft/Phi\-4\-mini\-instruct\. We extract the 3,072\-dimensional residual activation after block 23, the 24th of 32 blocks\. The relative depth24/32=0\.7524/32=0\.75matches SmolLM2’s18/2418/24extraction fraction; relative depth fixes the layer choice without assuming identical semantics\. The extraction hook agrees exactly withhidden\_states\[24\]\.

Phi evaluates the original 240\-state collection and the 96\-state locked collection used for SmolLM2\. We call the latter the*reused locked collection*: its scenarios are fixed before Phi generation but are not new to the researchers\. Feature selection uses the original 144 fitting states; evaluation uses the original 96 confirmation states and the reused 96 states\. Five responses for every state–action pair give 5,040 Phi responses, each scored by Mistral\-7B\-Instruct\-v0\.3 and Qwen2\.5\-7B\-Instruct\. Each evaluator assigns five scores in\{0,1,2,3\}\\\{0,1,2,3\\\}for safety, policy adherence, usefulness, factuality, and overall quality\. With weights\(0\.30,0\.30,0\.15,0\.10,0\.15\)\(0\.30,0\.30,0\.15,0\.10,0\.15\), the automatic loss is one minus the weighted score divided by three, matching the normalization of the human loss\.

Two TopK SAEs, with seeds 314159 and 271828, use 16,384 coordinates and sparsity 64\. Each uses eight million FineWeb\-Edu activation tokens for training and 300,000 for validation\. The 100,000\-token normalization sample comes from the training stream\. Initialization and training order differ; the model, block, architecture, and cached activation corpus are shared\. Each run completes 7,813 optimization steps\. Appendix[I](https://arxiv.org/html/2610.27446#A9)gives model revisions, settings, and diagnostics\.

Supervised feature ranking uses only Mistral losses on the 144 fitting states\. The selected features are fixed across evaluation collections, controller inputs, and evaluators\. We evaluate feature budgetsB∈\{2,4,8,16,32,64\}B\\in\\\{2,4,8,16,32,64\\\}, withB=2B=2primary, and compare policy plus code\(Z,S\)\(Z,S\)with code aloneZZ\. A third loss averages the two evaluator losses*before*optimizing state–action risk; its information gap need not equal the mean of their separate gaps\. Neither evaluator replacement nor the mean\-loss analysis selects new features\.

Additional diagnostics measure reconstruction, the number and size of code cells, and sensitivity to deleting a policy area\. Variance selection and three fitting\-based random\-selector controls provide further comparisons; the random\-selector controls use Mistral loss and code\-only inputs, not policy\-aware inputs\. Phi is evaluated automatically and has no completed human ratings\.

### M\.4Complete human\-rating procedure

The human evaluation covers 176 states: four states from each of the 20 original policy areas and all 96 locked states\. Three response choices per state give 528 items per rater\. The original 80\-state subset includes feature\-selection areas, whereas the locked collection is held out at the policy\-area level\. Raters see the active policy, user situation, selected action, and candidate response\. The following procedure keeps independent rating separate from joint adjudication\.

#### Rater recruitment and background\.

Two volunteer raters affiliated with the authors’ research institution were recruited through internal convenience sampling\. The raters received no financial or non\-financial compensation\. Both provided informed consent before participating\. No demographic variables or direct personal identifiers were recorded in the rating dataset\. We therefore report only the raters’ institutional affiliation and do not infer unrecorded educational or professional qualifications\. Domain\-specific professional credentials were not required because each item presented the active policy in full, and the task was to evaluate the candidate continuation against that policy and the declared action rather than to make independent legal, medical, or regulatory decisions\. Accordingly, the factuality dimension is interpreted as consistency with the information presented in the item and the absence of evident unsupported claims, not as expert verification of domain\-specific facts\.

#### Rater training, independence, and adjudication\.

Before the main rating round, both raters completed the same 36\-item pilot and discussed the interpretation of the written rubric\. The pilot labels were excluded from all reported analyses\. Each rater then independently evaluated all 528 main\-study items\. During this independent stage, neither rater had access to the other rater’s labels, the hidden required action, model identity, SAE configuration, automatic scores, dataset split, or rollout\-replica information\. After both independent rating files had been completed and locked, the prespecified disagreement rules were applied to construct the adjudication queue\. Only then did the raters jointly resolve the 100 queued items\. The original independent labels remained unchanged\. Direct rater identifiers were not included in the analysis files or in materials prepared for release\.

#### From ratings to loss\.

For each item, letssafety,spolicy,suse,sfact,soveralls\_\{\\rm safety\},s\_\{\\rm policy\},s\_\{\\rm use\},s\_\{\\rm fact\},s\_\{\\rm overall\}be its five integer scores in\{0,1,2,3\}\\\{0,1,2,3\\\}\. The dimension names denote safety, policy adherence, usefulness, factuality, and overall quality\. The composite human loss is

Lhuman=1−0\.30​ssafety\+0\.30​spolicy\+0\.15​suse\+0\.10​sfact\+0\.15​soverall3\.L^\{\\rm human\}=1\-\\frac\{0\.30s\_\{\\rm safety\}\+0\.30s\_\{\\rm policy\}\+0\.15s\_\{\\rm use\}\+0\.10s\_\{\\rm fact\}\+0\.15s\_\{\\rm overall\}\}\{3\}\.The division by three places the weighted score in\[0,1\]\[0,1\]; subtracting from one makes lower loss correspond to better rated responses\. The composite loss is not an unsafe\-event probability\. Agreement is computed from the independent ratings\. An item enters adjudication if safety or policy\-adherence scores differ by at least two points, acceptable labels or recommended actions differ, or composite losses differ by more than0\.250\.25\. The 100 queued items use the jointly resolved scores; the other 428 items use the mean of the two independent losses\.

For automatic–human action comparisons, we distinguish a selected lowest\-loss action from the set of all minimizing actions\. The selected\-action statistic uses the first minimum in the fixed orderask,decline,assist\. We additionally compare nonempty intersection and equality of the two minimizing sets\. Numerical ties use absolute tolerance10−1210^\{\-12\}; zero and10−810^\{\-8\}tolerances give the same set counts\. Definitions and counts are in Appendix[K\.2](https://arxiv.org/html/2610.27446#A11.SS2)\.

### M\.5Fixed comparisons and uncertainty

Feature selection uses the declared fitting split and does not change across the corresponding evaluation comparisons\. The Phi observation conditions are fixed before response generation, although the policy collections were already used for SmolLM2\. Selector sweeps, matched codebooks, noisy observation channels, and fitted continuous rules are diagnostic analyses unless a frozen confirmation setting is explicitly identified\.

To measure sensitivity to the sampled policy areas, we use a cluster bootstrap: each resampling draw selects whole policy areas, retaining their states, and recomputes both risk minima in Equation \([10](https://arxiv.org/html/2610.27446#S4.E10)\)\. Reported intervals are two\-sided 95% bootstrap intervals\. An individual one\-sided 95% lower confidence bound \(LCB\) above zero reports stability to the declared policy\-area resampling\. The Phi analysis uses 2,000 resamples, eight policy\-area clusters per evaluation collection, and numerical zero tolerance10−1210^\{\-12\}\. Its 12 primary policy\-aware settings share states, responses, and features; their intervals are not simultaneous familywise guarantees\. A zero fraction of nonpositive bootstrap draws is not a zeropp\-value\. In the policy\-aware condition, states from distinct policy texts cannot share a cell\. Reweighting whole areas therefore reweights their within\-policy gaps\. Positive lower bounds describe sensitivity of these fixed tables, not a test that unknown expected rollout losses have a positive gap\.

The empirical target is the finite outcome\-loss table\. Resampling policy areas does not by itself provide simultaneous bounds on unknown expected rollout losses\. The bootstrap results are therefore distinct from the population and search\-valid certificates in Sections[L\.1](https://arxiv.org/html/2610.27446#A12.SS1)and[3](https://arxiv.org/html/2610.27446#S3)\. The corresponding configurations, cached losses, interface codes, ratings, and diagnostic outputs are retained in the author working archive described in Appendix[M\.6](https://arxiv.org/html/2610.27446#A13.SS6)\.

### M\.6Reproduction inventory

The arXiv source package contains the manuscript, bibliography, figures, and LaTeX tables\. Experiment code and cached records are retained in a separate author working archive and are not included in the arXiv source package\. Allsupplementary/andtools/paths in this paper refer to that working archive\. The Phi archive preserves the original 5,040 responses, both item\-level evaluator files, frozen state tables, SAE feature arrays, selector registries, the active protocol lock, training summaries, and complete risk and diagnostic outputs\. Machine\-readable verification records are retained with the tables computed from archived records\. Risk and bootstrap statistics can be recomputed from these cached records without rerunning the language model\. Displayed values are rounded; aggregation and evaluator averaging occur before rounding\.

Phi’s active configuration isprotocol\_lock\.json\. The nesteddata/experiment\_config\.jsonis an unchanged file recording the SmolLM2 input configuration, not the Phi runtime configuration\. Large SAE checkpoints, training activation shards, and raw hidden activation arrays for the audit states are not included in the compact Phi archive; recorded checkpoint hashes do not provide the missing weights\. Optional Phi human packets are blank and are not completed human data\. The completed SmolLM2 rating files and resolved queue are included undersupplementary/support\_revision/human\_inputs/; blank templates in the original automated archive are not treated as ratings\. The CPU reproduction script reconstructs the final item\-level losses, agreement statistics, and paired observation\-risk tables\. Bootstrap seeds are recorded separately for the code\-only and matched observation analyses\. Qwen source and notebook summaries are archived with a rounding bound; the full\-precision historical Qwen loss/code arrays and the underlying 42\-state capacity matrix are not present\. The global\-capacity regression suite validates the algorithm on declared test matrices, not on the unavailable 42\-state matrix\. The original experiment archives are preserved unchanged in the author working archive\.

The author working archive retains the source excerpts, input hashes, statewise minimizing sets, dimension\-specific losses, and worked\-example records underlying the diagnostics described in Appendix[K](https://arxiv.org/html/2610.27446#A11)\. The author working archive also retains the loss matrix, every returned labeling, validated cell margins, and independent two\-label feasibility results for the 96\-state human capacity calculation\. Within that archive, the scriptstools/reproduce\_author\_audit\.py,tools/reproduce\_human\_capacity\.py, andtools/render\_author\_audit\_tables\.pyregenerate the records and tables from archived inputs\. These scripts do not change human labels or perform model training\. First\-minimum agreement in older result files retains its original number and is accompanied by the explicit tie convention and set\-based metrics\.

### M\.7Ethical use

The intended use of the framework is diagnostic: a representation can be tested before a safety monitor, steering rule, or downstream controller is trusted\. Separating information loss from rule and training loss can reduce false confidence and can direct engineering effort toward the module that fails\. Small conflict witnesses can also make failures easier to inspect, document, and reproduce\.

The same analysis can be misrepresented as a deployment guarantee\. Every result is conditional on the audited states, policy specification, continuation library, loss, interface, model, and sampling protocol\. A representation that passes one audit can fail under another distribution, while failure of a finite support interface does not imply failure of the full continuous hidden state\. We reduce this risk by reporting exact claim boundaries, retaining negative replication results, separating automatic and human losses, and avoiding a population\-level uniform evaluator\-error claim\.

The human\-rating procedure, including recruitment, consent, compensation, rater independence, adjudication, and data handling, is described in Appendix[M\.4](https://arxiv.org/html/2610.27446#A13.SS4)\.