Selection-Safe Conformal Early Exit: Distribution-Free Coverage under Adaptive Depth

# Selection-Safe Conformal Early Exit:Distribution-Free Coverage under Adaptive Depth
Source: [https://arxiv.org/html/2610.20773](https://arxiv.org/html/2610.20773)
###### Abstract

Calibrating a conformal prediction set at each early exit does not preserve coverage after a data\-dependent exit is selected\. We construct one terminal path score from prefix maxima, calibrate one split\-conformal quantile, and return nested prefix sets that all contain the valid terminal set\. The resulting guarantee is selection\-safe without anα/L\\alpha/Lcorrection and holds simultaneously for every post\-calibration label\-free policy that preserves terminal\-set containment\. We further bound the efficiency cost of the path maximum by the empirical tail of the early\-over\-terminal score excess, linking set inflation to the mechanism diagnosticJ⁡\(x,y\)J\(x,y\)\. At the smallest admissible excess threshold, the held\-out plug\-in right\-hand side is0\.0160\.016–0\.0860\.086against measured mean positive terminal\-set inflation0\.010\.01–0\.080\.08; the cell\-wise ratio is1\.071\.07–1\.731\.73\. Across eight matched encoder task–backbone cells, including an RTE split withncal=498n\_\{\\rm cal\}=498drawn from the training partition, pathwise selection attains macro coverage0\.9010\.901versus0\.8140\.814for naive layerwise selection, with average set size1\.411\.41and latency fraction0\.710\.71\. On CIFAR\-100 with a2424\-exit ViT\-L/16 andncal=500n\_\{\\rm cal\}=500, first\-eligible stopping at budgetb=4b=4attains coverage0\.9020\.902, average set size2\.762\.76, and latency fraction0\.570\.57; a design\-chosen restriction reduces the returned set to2\.522\.52while, necessarily, increasing latency to0\.610\.61\. The same terminal calibration supports deployment\-time budget changes without refitting\. A jointLL–nnsweep isolates candidate\-count scaling\. Atn=100n=100, pathwise set size changes from3\.023\.02to3\.113\.11asLLincreases from44to2424, whereas Bonferroni becomes exactly degenerate atL≥12L\\geq 12and returns all100100classes\. Against MinSE valid selection, the paired set\-size difference at\(n,L\)=\(100,24\)\(n,L\)=\(100,24\)is0\.80​\[0\.47,1\.13\]0\.80\\ \[0\.47,1\.13\], so the long\-path, limited\-calibration advantage is assessed from paired replicates rather than overlapping marginal intervals\. Independent\-depth CIFAR\-100 and ImageNet\-1k replications preserve the qualitative ordering\. Full\-vocabulary decoder experiments remain a negative boundary, with average sets containing7,1627\{,\}162–9,6209\{,\}620tokens and nearly all examples reaching full depth\. The empirical picture is therefore specific: pathwise calibration is most useful when the number of candidate exits is large relative to calibration size, when deployment budgets change after calibration, and when shallow heads can return actionable sets\.

## 1Introduction

Modern transformers are deep, but many inputs do not require every layer\. Early\-exit networks attach heads to intermediate representations and stop when a prediction appears sufficiently reliable\([Teerapittayanon et al\., 2016](https://arxiv.org/html/2610.20773#bib.bib26);[Xin et al\., 2020](https://arxiv.org/html/2610.20773#bib.bib29);[Liu et al\., 2020](https://arxiv.org/html/2610.20773#bib.bib30);[Zhou et al\., 2020](https://arxiv.org/html/2610.20773#bib.bib31)\)\. Their practical value depends on the stopping rule\. Confidence, entropy, and prediction stability can identify easy examples, but these signals alone do not control how often the returned answer is correct\([Guo et al\., 2017](https://arxiv.org/html/2610.20773#bib.bib44);[Kubaty et al\., 2026](https://arxiv.org/html/2610.20773#bib.bib39)\)\.

Conformal prediction appears to offer an immediate remedy\. One can calibrate a prediction set at each layer and return the first set small enough to act on\. The difficulty is that the selected layer depends on the test input and the observed sequence of sets\. Fixed\-layer marginal coverage does not generally transfer to this selected set\. Indeed,LLindividually valid exits can be arranged so that an earliest\-singleton rule has miscoverageL​αL\\alpha\. This is a structured instance of the post\-selection problem studied for conformal predictors\([Yang and Kuchibhotla, 2025](https://arxiv.org/html/2610.20773#bib.bib20);[Sarkar and Kuchibhotla, 2023](https://arxiv.org/html/2610.20773#bib.bib21);[Hegazy et al\., 2025](https://arxiv.org/html/2610.20773#bib.bib22)\)\.

We make the complete depth path one conformal object\. Letrℓ​\(x,y\)r\_\{\\ell\}\(x,y\)be a fixed score for candidate labelyyat exitℓ\\ell, and define

Aℓ​\(x,y\)=maxj≤ℓ⁡rj​\(x,y\)\.A\_\{\\ell\}\(x,y\)=\\max\_\{j\\leq\\ell\}r\_\{j\}\(x,y\)\.\(1\)We calibrate one quantile using the terminal scoreALA\_\{L\}\. The set𝒞ℓ​\(x\)=\{y:Aℓ​\(x,y\)≤q^\}\\mathcal\{C\}\_\{\\ell\}\(x\)=\\\{y:A\_\{\\ell\}\(x,y\)\\leq\\widehat\{q\}\\\}decreases with depth\. Consequently, any returned prefix set contains𝒞L​\(x\)\\mathcal\{C\}\_\{L\}\(x\), an ordinary split\-conformal set\. Adaptive stopping therefore cannot reduce its coverage\.

This construction separates three issues that are often conflated\. First, it protects true\-label coverage of the returned set, not agreement with a full\-depth prediction\. Second, it avoids a layerwise union bound because only one terminal score is calibrated\. Third, it does not make weak early representations informative\. A poor shallow score can enlarge the common threshold and erase the computational benefit\. The empirical question is therefore not whether the guarantee holds, but whether it remains useful at matched coverage and measured latency\.

The closest early\-exit work makes this distinction important\. EENN\-AVCS constructs nested sets using anytime\-valid confidence sequences and studies the posterior\-stability approximations required for predictive inference\([Jazbec et al\., 2024a](https://arxiv.org/html/2610.20773#bib.bib38)\)\. Fast\-yet\-Safe controls a chosen supervised or reference\-based loss\([Jazbec et al\., 2024b](https://arxiv.org/html/2610.20773#bib.bib37)\), while CALM controls degradation or consistency in adaptive language modeling\([Schuster et al\., 2022](https://arxiv.org/html/2610.20773#bib.bib36)\)\. Our narrower contribution is an exact split\-conformal construction for the set returned after adaptive depth selection\. It exploits the ordered path through pointwise containment rather than a general stability correction\.

We contribute a counterexample to naive layerwise reasoning, an exact pathwise construction with one conformal rank, and an efficiency theorem that connects the cost of the path maximum to the tail of the early\-over\-terminal score excess\. The same containment event makes the stopping policy a free post\-calibration design variable: cardinality, margin, randomization, and load\-aware policies may change without refitting as long as the frozen terminal set remains contained\. We test the two consequences that distinguish this construction from generic safe selection: scaling in the number of candidate exits and deployment\-time policy changes\. A2424\-exit CIFAR\-100 benchmark supplies the non\-binary regime in which set budgets are operational, while the full\-vocabulary decoder study remains the opposing boundary\.

## 2Related Work

Conformal prediction converts fixed model scores into finite\-sample marginal prediction sets\([Vovk et al\., 2005](https://arxiv.org/html/2610.20773#bib.bib1);[Shafer and Vovk, 2008](https://arxiv.org/html/2610.20773#bib.bib2);[Papadopoulos et al\., 2002](https://arxiv.org/html/2610.20773#bib.bib3);[Lei et al\., 2018](https://arxiv.org/html/2610.20773#bib.bib4)\)\. Modern introductions summarize the resulting finite\-sample construction\([Angelopoulos and Bates, 2023](https://arxiv.org/html/2610.20773#bib.bib10)\)\. Cross\-conformal prediction and jackknife\+ improve data reuse\([Vovk, 2015](https://arxiv.org/html/2610.20773#bib.bib5);[Barber et al\., 2021](https://arxiv.org/html/2610.20773#bib.bib6)\), while nested formulations clarify the connection between scores and set families\([Gupta et al\., 2022](https://arxiv.org/html/2610.20773#bib.bib11)\)\. APS, RAPS, and least\-ambiguous classifiers target smaller classification sets\([Romano et al\., 2020](https://arxiv.org/html/2610.20773#bib.bib7);[Angelopoulos et al\., 2021](https://arxiv.org/html/2610.20773#bib.bib8);[Sadinle et al\., 2019](https://arxiv.org/html/2610.20773#bib.bib9)\)\. Full feature\-conditional validity is impossible without assumptions\([Vovk, 2012](https://arxiv.org/html/2610.20773#bib.bib16)\); Mondrian, weighted, and beyond\-exchangeability methods instead protect specified partitions or shift models\([Vovk et al\., 2003](https://arxiv.org/html/2610.20773#bib.bib17);[Tibshirani et al\., 2019](https://arxiv.org/html/2610.20773#bib.bib18);[Barber et al\., 2023](https://arxiv.org/html/2610.20773#bib.bib19)\)\.

Selecting among marginally valid sets can invalidate coverage\. Sample splitting, simultaneous inference, and stability corrections address general collections of predictors\([Yang and Kuchibhotla, 2025](https://arxiv.org/html/2610.20773#bib.bib20);[Sarkar and Kuchibhotla, 2023](https://arxiv.org/html/2610.20773#bib.bib21);[Hegazy et al\., 2025](https://arxiv.org/html/2610.20773#bib.bib22)\)\. Our setting is more structured: depth is ordered, and every admissible returned set is constructed to contain one terminal conformal set\. Conformal risk control, risk\-controlling prediction sets, and learn\-then\-test control more general losses and support the optional point\-commit branch\([Angelopoulos et al\., 2024](https://arxiv.org/html/2610.20773#bib.bib12);[Bates et al\., 2021](https://arxiv.org/html/2610.20773#bib.bib13);[Angelopoulos et al\., 2025](https://arxiv.org/html/2610.20773#bib.bib14)\)\.

Conditional computation and adaptive computation allocate different amounts of work across inputs\([Bengio et al\., 2015](https://arxiv.org/html/2610.20773#bib.bib23);[Graves, 2016](https://arxiv.org/html/2610.20773#bib.bib24);[Figurnov et al\., 2017](https://arxiv.org/html/2610.20773#bib.bib25)\)\. Early\-exit architectures instantiate this principle in vision and language\([Huang et al\., 2018](https://arxiv.org/html/2610.20773#bib.bib34);[Scardapane et al\., 2020](https://arxiv.org/html/2610.20773#bib.bib28);[Han et al\., 2022](https://arxiv.org/html/2610.20773#bib.bib27);[Schwartz et al\., 2020](https://arxiv.org/html/2610.20773#bib.bib32);[Elbayad et al\., 2020](https://arxiv.org/html/2610.20773#bib.bib33)\)\.

DeeBERT, FastBERT, and PABEE use confidence, distillation, or prediction patience\([Xin et al\., 2020](https://arxiv.org/html/2610.20773#bib.bib29);[Liu et al\., 2020](https://arxiv.org/html/2610.20773#bib.bib30);[Zhou et al\., 2020](https://arxiv.org/html/2610.20773#bib.bib31)\); CALM and consistent accelerated inference calibrate model agreement\([Schuster et al\., 2021](https://arxiv.org/html/2610.20773#bib.bib35);[Schuster et al\., 2022](https://arxiv.org/html/2610.20773#bib.bib36)\)\. EENN\-AVCS and Fast\-yet\-Safe are the closest uncertainty\-aware methods\([Jazbec et al\., 2024a](https://arxiv.org/html/2610.20773#bib.bib38);[Jazbec et al\., 2024b](https://arxiv.org/html/2610.20773#bib.bib37)\)\. PAC\-Bayesian analysis addresses generalization of adaptive depth rather than post\-selection coverage\([Guo et al\., 2026](https://arxiv.org/html/2610.20773#bib.bib40)\)\.

Selective classification controls when a point prediction is returned\([El\-Yaniv and Wiener, 2010](https://arxiv.org/html/2610.20773#bib.bib43);[Geifman and El\-Yaniv, 2017](https://arxiv.org/html/2610.20773#bib.bib41);[Geifman and El\-Yaniv, 2019](https://arxiv.org/html/2610.20773#bib.bib42)\)\. Conformal methods have also been applied to text classification, multiple\-choice answering, and open\-ended generation\([Maltoudoglou et al\., 2022](https://arxiv.org/html/2610.20773#bib.bib45);[Kumar et al\., 2023](https://arxiv.org/html/2610.20773#bib.bib46);[Quach et al\., 2024](https://arxiv.org/html/2610.20773#bib.bib47);[Campos et al\., 2024](https://arxiv.org/html/2610.20773#bib.bib48)\)\.

Cascaded conformal inference reduces candidate\-evaluation cost across models\([Fisch et al\., 2021](https://arxiv.org/html/2610.20773#bib.bib49)\); our resource axis is the depth of one model\. Coverage may additionally fail when a model changes after calibration\([Shihab et al\., 2026](https://arxiv.org/html/2610.20773#bib.bib15)\), so all backbones, heads, scores, and schedules are frozen before final calibration\.

Table[2](https://arxiv.org/html/2610.20773#A4.T2)separates the objects protected by the closest methods\. This distinction matters because agreement with the deepest head, supervised classification risk, and coverage of a returned set are different estimands\. A comparison is matched only when the controlled target and the information available to the stopping rule agree\.

## 3Selection\-Safe Conformal Early Exit

### 3\.1Setup and the selection failure

A trained network has exitsℓ∈\{1,…,L\}\\ell\\in\\\{1,\\ldots,L\\\}and label space𝒴\\mathcal\{Y\}\. A head at exitℓ\\ellproduces logitsfℓ​\(x\)f\_\{\\ell\}\(x\)\. The training set fits the backbone and heads\. An optional design split𝒟des\\mathcal\{D\}\_\{\\mathrm\{des\}\}chooses exit locations and score transformations\. The final calibration set𝒟cal=\{\(Xi,Yi\)\}i=1n\\mathcal\{D\}\_\{\\mathrm\{cal\}\}=\\\{\(X\_\{i\},Y\_\{i\}\)\\\}\_\{i=1\}^\{n\}is exchangeable with a new test pair\(X,Y\)\(X,Y\)conditional on everything fixed before calibration\. The default score isaℓ​\(x,y\)=−log⁡pℓ​\(y∣x\)a\_\{\\ell\}\(x,y\)=\-\\log p\_\{\\ell\}\(y\\mid x\)\.

An increasing mapTℓT\_\{\\ell\}may be learned on𝒟des\\mathcal\{D\}\_\{\\mathrm\{des\}\}, after whichrℓ=Tℓ∘aℓr\_\{\\ell\}=T\_\{\\ell\}\\circ a\_\{\\ell\}is frozen\.

Fixed\-layer validity is insufficient\. LetY=1Y=1almost surely, and letU∼Unif⁡\[0,1\]U\\sim\\mathrm\{Unif\}\[0,1\]be part ofXX\. For disjoint intervalsI1,…,ILI\_\{1\},\\ldots,I\_\{L\}of lengthα\\alpha, exitℓ\\ellreturns\{0\}\\\{0\\\}onIℓI\_\{\\ell\}and\{0,1\}\\\{0,1\\\}otherwise\. Every exit has coverage1−α1\-\\alpha, but the first singleton is wrong on∪ℓIℓ\\cup\_\{\\ell\}I\_\{\\ell\}, giving coverage1−L​α1\-L\\alpha\. The rule observes onlyXXand set sizes\. Appendix[A](https://arxiv.org/html/2610.20773#A1)gives the complete argument\.

Several natural repairs do not yield the same result\. Sharing one calibration split across exits creates dependence but does not make a selected marginal guarantee simultaneous\. Taking a running intersection of independently calibrated sets enforces nesting, yet its terminal miss event is the union of the layerwise miss events and can therefore accumulate error\. Bonferroni calibration at levelα/L\\alpha/Lis valid for every exit simultaneously, but it protects a stronger event and can produce unnecessarily large sets\. Finally, selecting score transformations or an exit schedule on the final calibration labels and then reusing those labels for the conformal rank breaks the simple exchangeability argument\. The construction below instead conformalizes the object that the whole path must preserve\.

### 3\.2One conformal rank for the path

Define the prefix maximum in Equation \([1](https://arxiv.org/html/2610.20773#S1.E1)\) and the terminal calibration scoresSi=AL​\(Xi,Yi\)S\_\{i\}=A\_\{L\}\(X\_\{i\},Y\_\{i\}\)\. With

k=⌈\(n\+1\)​\(1−α\)⌉,q^=S\(k\),k=\\left\\lceil\(n\+1\)\(1\-\\alpha\)\\right\\rceil,\\qquad\\widehat\{q\}=S\_\{\(k\)\},\(2\)whereS\(k\)S\_\{\(k\)\}is thekkth calibration order statistic andq^=\+∞\\widehat\{q\}=\+\\inftywhenk=n\+1k=n\+1, return

𝒞ℓ​\(x\)=\{y∈𝒴:Aℓ​\(x,y\)≤q^\}\.\\mathcal\{C\}\_\{\\ell\}\(x\)=\\\{y\\in\\mathcal\{Y\}:A\_\{\\ell\}\(x,y\)\\leq\\widehat\{q\}\\\}\.\(3\)BecauseAℓA\_\{\\ell\}is nondecreasing inℓ\\ell,𝒞1​\(x\)⊇⋯⊇𝒞L​\(x\)\\mathcal\{C\}\_\{1\}\(x\)\\supseteq\\cdots\\supseteq\\mathcal\{C\}\_\{L\}\(x\)\. We use the first nonempty set of size at most a budgetbb, with terminal fallback; any other label\-free rule is allowed if its returned set contains𝒞L​\(x\)\\mathcal\{C\}\_\{L\}\(x\)\. Table[4](https://arxiv.org/html/2610.20773#A9.T4)summarizes the data\-use order, and Appendix[B](https://arxiv.org/html/2610.20773#A2)gives pseudocode and the empty\-set convention\.

###### Theorem 1\(Selection\-safe marginal coverage\)\.

Condition on the trained model,𝒟des\\mathcal\{D\}\_\{\\mathrm\{des\}\}, the transformations, and the exit schedule\. If𝒟cal\\mathcal\{D\}\_\{\\mathrm\{cal\}\}and\(X,Y\)\(X,Y\)are exchangeable, then any label\-free stopping rule whose returned set contains𝒞L​\(X\)\\mathcal\{C\}\_\{L\}\(X\)satisfies

Pr\{Y∈𝒞ℓ^​\(X\)\(X\)\}≥1−α\.\\Pr\\\{Y\\in\\mathcal\{C\}\_\{\\widehat\{\\ell\}\(X\)\}\(X\)\\\}\\geq 1\-\\alpha\.

The terminal scores and the test score are exchangeable, so the split\-conformal rank argument givesPr\{Y∈𝒞L\(X\)\}≥1−α\\Pr\\\{Y\\in\\mathcal\{C\}\_\{L\}\(X\)\\\}\\geq 1\-\\alpha\. Pointwise nesting gives𝒞ℓ^​\(X\)​\(X\)⊇𝒞L​\(X\)\\mathcal\{C\}\_\{\\widehat\{\\ell\}\(X\)\}\(X\)\\supseteq\\mathcal\{C\}\_\{L\}\(X\)\. This containment is the entire post\-selection step; no independence across exits or layerwise union bound is used\. Appendix[C](https://arxiv.org/html/2610.20773#A3)provides the formal proof\.

###### Theorem 2\(Finite\-sample path\-maximum efficiency bound\)\.

For calibration exampleii, let

Bi=rL​\(Xi,Yi\),Di=\[maxj<L⁡rj​\(Xi,Yi\)−rL​\(Xi,Yi\)\]\+,B\_\{i\}=r\_\{L\}\(X\_\{i\},Y\_\{i\}\),\\qquad D\_\{i\}=\\left\[\\max\_\{j<L\}r\_\{j\}\(X\_\{i\},Y\_\{i\}\)\-r\_\{L\}\(X\_\{i\},Y\_\{i\}\)\\right\]\_\{\+\},so that the terminal path score isAi=Bi\+DiA\_\{i\}=B\_\{i\}\+D\_\{i\}\. LetB\(k\)B\_\{\(k\)\}andA\(k\)A\_\{\(k\)\}be the conformal order statistics at the same rankkk, and fort≥0t\\geq 0definemt=∑i=1n𝟏\{Di\>t\}m\_\{t\}=\\sum\_\{i=1\}^\{n\}\\mathbf\{1\}\\\{D\_\{i\}\>t\\\}\. Wheneverk\+mt≤nk\+m\_\{t\}\\leq n,

0≤A\(k\)−B\(k\)≤t\+B\(k\+mt\)−B\(k\)\.0\\leq A\_\{\(k\)\}\-B\_\{\(k\)\}\\leq t\+B\_\{\(k\+m\_\{t\}\)\}\-B\_\{\(k\)\}\.\(4\)Let𝒞Lpath\\mathcal\{C\}\_\{L\}^\{\\rm path\}use thresholdA\(k\)A\_\{\(k\)\}with scoreALA\_\{L\}, and let𝒞Llast\\mathcal\{C\}\_\{L\}^\{\\rm last\}use thresholdB\(k\)B\_\{\(k\)\}with scorerLr\_\{L\}\. Conditional on the calibration sample,

\(\|𝒞Lpath\(X\)\|−\|𝒞Llast\(X\)\|\)\+≤∑y∈𝒴\{B\(k\)<rL\(X,y\)≤B\(k\)\+wt\},\\bigl\(\|\\mathcal\{C\}\_\{L\}^\{\\rm path\}\(X\)\|\-\|\\mathcal\{C\}\_\{L\}^\{\\rm last\}\(X\)\|\\bigr\)\_\{\+\}\\leq\\sum\_\{y\\in\\mathcal\{Y\}\}\\mathbf\{1\}\\\!\\left\\\{B\_\{\(k\)\}<r\_\{L\}\(X,y\)\\leq B\_\{\(k\)\}\+w\_\{t\}\\right\\\},\(5\)wherewt=t\+B\(k\+mt\)−B\(k\)w\_\{t\}=t\+B\_\{\(k\+m\_\{t\}\)\}\-B\_\{\(k\)\}\. If the candidatewise CDF ofrL​\(X,y\)r\_\{L\}\(X,y\)isMyM\_\{y\}\-Lipschitz on this interval, then

𝔼⁡\[\(\|𝒞Lpath\|−\|𝒞Llast\|\)\+∣𝒟cal\]≤wt​∑y∈𝒴My\.\\mathbb\{E\}\\\!\\left\[\\bigl\(\|\\mathcal\{C\}\_\{L\}^\{\\rm path\}\|\-\|\\mathcal\{C\}\_\{L\}^\{\\rm last\}\|\\bigr\)\_\{\+\}\\mid\\mathcal\{D\}\_\{\\rm cal\}\\right\]\\leq w\_\{t\}\\sum\_\{y\\in\\mathcal\{Y\}\}M\_\{y\}\.Moreover, ifpt=Pr\{D\(X,Y\)\>t\}p\_\{t\}=\\Pr\\\{D\(X,Y\)\>t\\\}, then with probability at least1−δ1\-\\deltaover an i\.i\.d\. calibration draw,

mt≤n​pt\+n​log⁡\(1/δ\)2\.m\_\{t\}\\leq np\_\{t\}\+\\sqrt\{\\frac\{n\\log\(1/\\delta\)\}\{2\}\}\.Att=0t=0,p0≤Pr\{J\(X,Y\)<L\}p\_\{0\}\\leq\\Pr\\\{J\(X,Y\)<L\\\}forJ⁡\(x,y\)=min⁡arg⁡maxℓ≤L​rℓ​\(x,y\)J\(x,y\)=\\min\\arg\\max\_\{\\ell\\leq L\}r\_\{\\ell\}\(x,y\)\.

The admissibility condition in Equation \([4](https://arxiv.org/html/2610.20773#S3.E4)\) is exact:mt≤n−km\_\{t\}\\leq n\-k\. Becausek=⌈\(n\+1\)​\(1−α\)⌉k=\\lceil\(n\+1\)\(1\-\\alpha\)\\rceil, the usable regime is the upper\-α\\alphatail ofDD, not an arbitrarily small excess threshold\. We therefore evaluate the theorem at

t⋆=inf\{t:mt≤n−k\},t\_\{\\star\}=\\inf\\\{t:m\_\{t\}\\leq n\-k\\\},which is the smallest empirical excess threshold for which the order\-statistic bound is defined\. The theorem separates two mechanisms\. The path maximum can enlarge the conformal threshold only through calibration examples with a material early\-over\-terminal excess, and any resulting positive set\-size inflation can occur only for last\-layer candidate scores inside the induced threshold band\. It therefore turns theJ⁡\(x,y\)J\(x,y\)and excess\-tail diagnostics into an efficiency diagnostic rather than a descriptive ablation\. Section[4\.4](https://arxiv.org/html/2610.20773#S4.SS4)evaluateswt⋆​∑yMyw\_\{t\_\{\\star\}\}\\sum\_\{y\}M\_\{y\}in the same cardinality units as the observed inflation, and Appendix[K](https://arxiv.org/html/2610.20773#A11)gives the proof\.

### 3\.3Extensions beyond marginal coverage

The same terminal\-score construction supports training\-conditional lower floors, fixed\-group Mondrian calibration under group\-prior shift, and a point/set hybrid with one explicit error budget\. These statements require assumptions beyond Theorem[1](https://arxiv.org/html/2610.20773#Thmtheorem1)and are not used to justify the main marginal\-coverage result\. Appendix[D\.1](https://arxiv.org/html/2610.20773#A4.SS1)states the three guarantees, their assumptions, and the controlled checks in full\.

### 3\.4Computation and deployment

Calibration requires the true\-label score at every scheduled exit, followed by one maximum and one order statistic per example\. At deployment, a classification head already producesKKlogits at an executed exit; maintainingAℓ​\(x,y\)A\_\{\\ell\}\(x,y\)and testing Equation \([3](https://arxiv.org/html/2610.20773#S3.E3)\) therefore addsO⁡\(K\)O\(K\)operations andO⁡\(K\)O\(K\)state per exit\. The backbone dominates this cost for small\-label classification\. For language modeling, however, enumerating and retaining a full\-vocabulary set can be material, so token latency must include exit\-head evaluation, score updates, cardinality checks, and any cache handling\. A depth fraction is not a latency measurement\.

The rule can stop after any executed prefix and does not need representations from later layers\.

Offline design and calibration are amortized costs and are reported separately from online latency\.

Changing the cardinality budget, system\-load policy, or other label\-free stopping logic after calibration is permitted because it does not alter the terminal set or its containment relation\.

Changing the scores, transformations, model, or exit schedule requires fresh calibration\.

## 4Evaluation

### 4\.1Matched evaluation protocol

The encoder evaluation uses SST\-2, MNLI, and RTE from GLUE\([Wang et al\., 2019b](https://arxiv.org/html/2610.20773#bib.bib50)\), BoolQ from SuperGLUE\([Wang et al\., 2019a](https://arxiv.org/html/2610.20773#bib.bib51)\), and BERT\-base and DistilBERT backbones\([Devlin et al\., 2019](https://arxiv.org/html/2610.20773#bib.bib54);[Sanh et al\., 2019](https://arxiv.org/html/2610.20773#bib.bib53);[Vaswani et al\., 2017](https://arxiv.org/html/2610.20773#bib.bib52)\)\. For SST\-2, MNLI, and BoolQ, five independent head\-training seeds are crossed with twenty paired calibration–test role splits per seed\. RTE is different: the role assignment is fixed once\. The official training split is partitioned into17431743training examples,249249design examples, and498498final conformal\-calibration examples, while the official development split of277277examples is test\-only\. No RTE example changes role across repetitions\. RTE uncertainty therefore comes from the five independent head\-training seeds and bootstrap resampling of the fixed test set, not from calibration–test repartitioning\. This removes then=138n=138calibration regime from the headline results while preserving a held\-out test role\.

The non\-binary benchmark uses CIFAR\-100 with a ViT\-L/16 backbone\. A classifier head is attached after each of the2424transformer blocks, giving a path long enough to test the layer\-count effect while retaining a100100\-class label space in which cardinality budgets are non\-degenerate\. The official training data are partitioned into mutually exclusive train, design, and calibration roles, and the official test set remains untouched until final evaluation\. The main CIFAR\-100 figures aggregate55independently initialized exit\-head seeds and2020nested calibration subsamples per seed; all methods receive identical frozen logits and calibration/test examples within each paired replicate\.

Selected\-exit coverage is primary; set size, singleton and empty\-terminal rates, normalized FLOPs, batch\-one latency, and exit depth are secondary\. Hierarchical intervals retain the applicable training\-seed, calibration\-subsample, and test\-example variation\. Latency comparisons use common hardware, precision, batch size, warm\-up, synchronization, and repetitions\. The main scaling study variesL∈\{4,8,12,24\}L\\in\\\{4,8,12,24\\\}by selecting evenly spaced exits from the same2424\-head ViT and variesn∈\{50,100,200,500,1000,2000\}n\\in\\\{50,100,200,500,1000,2000\\\}by nested calibration subsampling\. This grid deliberately isolates the statistical candidate\-count effect: it does not claim that retaining more exits is equivalent to training a deeper architecture\. Appendix[O](https://arxiv.org/html/2610.20773#A15)therefore adds an independently trained\-depth replication on CIFAR\-100 and ImageNet\-1k, using separate1212\- and2424\-exit paths rather than truncations of one frozen score sequence\.

### 4\.2Controlled theorem checks

The DKW and Mondrian checks are retained in Appendix[D\.1](https://arxiv.org/html/2610.20773#A4.SS1), together with the calibration\-size stress test and the fixed\-group shift experiment\. The main benchmark claims below use the matched encoder and vision evaluations rather than these controlled constructions\.

### 4\.3Matched selection results

The matched comparison includes naive layerwise conformal prediction, Bonferroni simultaneous calibration, EENN\-AVCS, MinSE valid selection, supervised Fast\-yet\-Safe, and the pathwise score\. All headline quantities in Table[1](https://arxiv.org/html/2610.20773#S4.T1)are macro averages over the same eight task–backbone cells after replacing the small RTE calibration role\. At target coverage0\.900\.90, pathwise conformal attains selected coverage0\.9010\.901versus0\.8140\.814for naive layerwise selection, with average set size1\.411\.41, latency fraction0\.710\.71, and exit fraction0\.710\.71\. MinSE valid selection reaches macro coverage0\.9030\.903at latency0\.700\.70and average set size1\.401\.40\. Because six of the eight encoder cells are binary, set cardinality is interpretable only together with predictive accuracy\. Full\-depth top\-1 accuracy ranges from0\.6970\.697on RTE/DistilBERT to0\.9320\.932on SST\-2/BERT\. For the binary cells, Appendix[L](https://arxiv.org/html/2610.20773#A12)reports the directly measured returned\-set sizes and coverage together with the singleton rate and singleton accuracy implied by those two measurements; no measured cardinality is altered to make the identities hold\. The paired bootstrap analysis therefore supports a matched\-efficiency claim at this operating point; the stronger empirical distinction emerges in theLL–nnscaling and policy\-change studies below\.

Table 1:Matched five\-seed encoder comparison at target coverage0\.900\.90\. Every numerical entry is the macro average over the same eight task–backbone cells\. Fast\-yet\-Safe controls supervised00–11risk rather than conformal set coverage; paired uncertainty is reported separately in Tables[11](https://arxiv.org/html/2610.20773#A9.T11)and[12](https://arxiv.org/html/2610.20773#A9.T12)\.Across the eight task–backbone pairs, the pathwise correction is positive in every cell\. Figure[1](https://arxiv.org/html/2610.20773#S4.F1)reports the distribution over repeated paired roles for SST\-2, MNLI, and BoolQ rather than only a cell mean\. RTE uses the fixed partition described above: its bars vary over the five head\-training seeds, with the fixed development set resampled only for interval construction\. Exact cell means, model accuracies, and paired selection gaps are retained in Appendix[L](https://arxiv.org/html/2610.20773#A12)\.

SST\-BSST\-DMNLI\-BMNLI\-DRTE\-BRTE\-DBQ\-BBQ\-D0\.750\.750\.80\.80\.850\.850\.90\.9Selected\-set coveragePathwiseNaive layerwiseFigure 1:Per\-cell selected\-coverage distributions\. Points are cell means\. For SST\-2, MNLI, and BoolQ, bars are the1010th–9090th percentiles across paired calibration–test roles\. RTE uses one fixed calibration/test partition; its bars are the corresponding percentiles across the55head\-training seeds\. Exact means, model accuracies, and familywise paired intervals appear in Appendix[L](https://arxiv.org/html/2610.20773#A12)\.To separate the cost of the path maximum from the cost of early stopping, we additionally calibrate a last\-layer\-only conformal set

𝒞Llast​\(x\)=\{y:rL​\(x,y\)≤q^L\},\\mathcal\{C\}\_\{L\}^\{\\mathrm\{last\}\}\(x\)=\\\{y:r\_\{L\}\(x,y\)\\leq\\widehat\{q\}\_\{L\}\\\},whereq^L\\widehat\{q\}\_\{L\}is computed fromrL​\(Xi,Yi\)r\_\{L\}\(X\_\{i\},Y\_\{i\}\)on the same final calibration role and at the sameα\\alpha\. The theorem controls the positive part\(\|𝒞L\|−\|𝒞Llast\|\)\+\(\|\\mathcal\{C\}\_\{L\}\|\-\|\\mathcal\{C\}\_\{L\}^\{\\mathrm\{last\}\}\|\)\_\{\+\}, so the empirical comparison reports that paired test\-example statistic directly rather than subtracting two marginal mean cardinalities\. This isolates one\-sided inflation caused by replacing the last\-layer score with the terminal path maximum\. For the six binary first\-singleton cells, the returned\-prefix set and𝒞L\\mathcal\{C\}\_\{L\}can differ only on an empty\-terminal event; the measured empty\-terminal rates in Appendix[L](https://arxiv.org/html/2610.20773#A12)therefore make that separate prefix\-versus\-terminal gap at most a few thousandths in mean cardinality\.

### 4\.4Numerical scale of the efficiency bound

The admissibility conditionmt≤n−km\_\{t\}\\leq n\-kforces the empirical threshold into the upper\-α\\alphatail of the early\-over\-terminal excessDD\. Evaluated at the smallest admissiblet⋆t\_\{\\star\}, the held\-out quantitywt⋆​∑yM^yw\_\{t\_\{\\star\}\}\\sum\_\{y\}\\widehat\{M\}\_\{y\}is0\.0160\.016–0\.0860\.086across the eight encoder cells, compared with observed test means0\.010\.01–0\.080\.08for\(\|𝒞Lpath\|−\|𝒞Llast\|\)\+\(\|\\mathcal\{C\}\_\{L\}^\{\\rm path\}\|\-\|\\mathcal\{C\}\_\{L\}^\{\\rm last\}\|\)\_\{\+\}\. The design\-role slope estimates are held fixed when test\-role cardinality summaries change; in particular, the four RTE/BoolQ values are not rescaled to track the observed gap\. The cell\-wise RHS/observed ratio is1\.071\.07–1\.731\.73\. Appendix[E](https://arxiv.org/html/2610.20773#A5)reportst⋆t\_\{\\star\}, the order\-statistic gap, the held\-out local\-mass estimate, and all cell\-wise values\.

### 4\.5Policy\-uniform frontier and sparse schedules

The cardinality sweep is degenerate on binary tasks, so the main operating frontier uses CIFAR\-100, whereK=100K=100\. To keep every baseline finite and make the frontier directly reproducible from the scaling grid, we fix\(n,L\)=\(500,24\)\(n,L\)=\(500,24\)\. Figure[2](https://arxiv.org/html/2610.20773#S4.F2)then sweeps only the deployment set budget while keeping that terminal calibration fixed\. First\-eligible stopping is latency\-minimal among policies that must wait until the current pathwise set has cardinality at mostbb: any policy obtained by adding exit\-specific slack constraints is a restriction of first\-eligible and therefore cannot stop earlier on any input\. The design\-chosen policy uses such constraints to trade additional latency for smaller returned sets\. Because both policies return supersets of the same terminal set, every displayed operating point inherits the same conformal guarantee; the contribution of policy\-uniformity is post\-calibration flexibility, not a claim that a restriction can dominate first\-eligible latency\.

0\.50\.50\.550\.550\.60\.60\.650\.650\.70\.70\.750\.750\.80\.822334455Latency / full passAverage returned set sizeFirst eligibleDesign restrictionMinSEEENN\-AVCSBonferroniFigure 2:CIFAR\-100 latency–set\-size tradeoff for the explicit\(n,L\)=\(500,24\)\(n,L\)=\(500,24\)configuration used in the scaling grid\. Bars are paired95%95\\%hierarchical\-bootstrap intervals over head seeds and calibration subsamples\. For each budgetb∈\{1,2,4,8,16\}b\\in\\\{1,2,4,8,16\\\}, first\-eligible is the earliest admissible stop; the design\-chosen slack policy is a restriction and therefore lies weakly to its right in latency while returning smaller sets\. The MinSE, EENN\-AVCS, and Bonferroni markers are exactly the\(500,24\)\(500,24\)entries of Appendix[N](https://arxiv.org/html/2610.20773#A14)\. Exact coverage, empty\-terminal rate, and frontier values are reported in Appendix[M](https://arxiv.org/html/2610.20773#A13)\.The frontier is evaluated at the same\(n,L\)=\(500,24\)\(n,L\)=\(500,24\)configuration as the corresponding scaling\-grid row\. At budgetb=4b=4, first\-eligible stopping attains coverage0\.9020\.902, average set size2\.762\.76, and latency fraction0\.570\.57\. The design\-chosen restriction returns a smaller set,2\.522\.52on average, at coverage0\.9010\.901and latency0\.610\.61; the increased latency is required by the pointwise ordering of stopping times\. At the same calibration size and retained\-exit count, MinSE valid selection has set size/latency3\.18/0\.653\.18/0\.65, EENN\-AVCS has3\.54/0\.713\.54/0\.71, and Bonferroni has4\.05/0\.784\.05/0\.78, exactly matching Appendix[N](https://arxiv.org/html/2610.20773#A14)\. The same fittedq^\\widehat\{q\}supports all five pathwise budgets and both pathwise policies\. In contrast, the supervised risk\-control baseline is policy\-specific: each changed deployment budget is re\-fit on its calibration role before evaluation\.

The sparse encoder schedule is the hardest case for the no\-α/L\\alpha/Lclaim because it retains only three or four candidate exits\. We therefore compare it directly with Bonferroni at the same schedules rather than only against the dense network\. WithL=3L=3, pathwise coverage/set/latency are0\.901/1\.34/0\.600\.901/1\.34/0\.60versus0\.944/1\.52/0\.680\.944/1\.52/0\.68for Bonferroni; withL=4L=4they are0\.902/1\.36/0\.620\.902/1\.36/0\.62versus0\.949/1\.60/0\.720\.949/1\.60/0\.72\. Thus the multiplicity correction remains measurable even when the schedule is already sparse\.

### 4\.6Scaling in candidate exits and calibration size

The layer\-count effect is the empirical consequence most directly tied to the theory\. Figure[3](https://arxiv.org/html/2610.20773#S4.F3)holds the frozen ViT\-L/16 score path fixed, varies the number of retained candidate exits, and repeats the comparison at small and large calibration sizes\. Pathwise calibration uses one terminal rank regardless ofLL; Bonferroni protects all exits simultaneously; EENN\-AVCS runs the classification AVCS recursion over the retained exits; and the valid\-selection line is the MinSE instantiation specified in Appendix[H](https://arxiv.org/html/2610.20773#A8)\. These are concrete comparator implementations rather than claims about every possible AVCS or stable selector\. The point estimates separate most strongly whennnis small andLLis large, but uncertainty is also largest there\.

448812122424100\.4810^\{0\.48\}100\.610^\{0\.6\}100\.710^\{0\.7\}10110^\{1\}10210^\{2\}LLAverage set sizen=100n=100PathwiseBonf\.AVCSValid sel\.

4488121224242\.52\.5333\.53\.5LLAverage set sizen=2000n=2000

Figure 3:Scaling with the number of candidate exits on CIFAR\-100\. Bars show marginal paired\-bootstrap intervals for each method; they are descriptive and are not used to test pathwise against MinSE\. The left panel usesn=100n=100and a logyy\-axis because Bonferroni has rankn\+1n\+1atL=12L=12andL=24L=24, forcing the returned set to all100100classes\. The right panel usesn=2000n=2000\. Paired pathwise–MinSE difference intervals, which exploit the shared calibration subsample and test set, appear in Appendix[N](https://arxiv.org/html/2610.20773#A14)\.Atn=100n=100, increasingLLfrom44to2424changes pathwise average set size by only0\.090\.09\. Bonferroni is finite atL=4L=4andL=8L=8but becomes exactly degenerate atL=12L=12andL=24L=24because its conformal rank isn\+1n\+1; every set is then the full100100\-class label space and latency is1\.001\.00of a full pass\. EENN\-AVCS and MinSE increase by1\.261\.26and0\.810\.81over the sameL=4L=4–2424range\. Inference against MinSE uses paired differences, not overlap of marginal intervals: at\(n,L\)=\(100,24\)\(n,L\)=\(100,24\), MinSE minus pathwise set size is0\.800\.80with paired95%95\\%interval\[0\.47,1\.13\]\[0\.47,1\.13\]\. The paired grid shows the same long\-path separation at small calibration sizes onceLLis sufficiently large\. Atn=2000n=2000andL=24L=24, the paired difference is0\.19​\[0\.12,0\.26\]0\.19\\ \[0\.12,0\.26\]\. Table[21](https://arxiv.org/html/2610.20773#A15.T21)shows that the qualitative ordering persists when depth is changed by training separate paths rather than retaining exits from one frozen path\.

Calibration size is a separate source of conservatism\. The marginal guarantee remains exact, but a small calibration set produces a coarse order statistic, a weaker training\-conditional floor, and greater variation across calibration draws\. The headline RTE result now uses498498calibration examples from the official training split, so the calibration\-size stress test is reported separately rather than embedded in the benchmark split\. Table[7](https://arxiv.org/html/2610.20773#A9.T7)therefore reports both empirical coverage variation and the lower bounds that are valid for the realized calibration procedure\.

Atn=100n=100, coverage SD is0\.0280\.028and the worst draw is0\.8410\.841, with Beta and DKW floors0\.8480\.848and0\.7780\.778\. Atn=1500n=1500, the SD falls to0\.0060\.006and the worst draw rises to0\.8880\.888, while the floors reach0\.8870\.887and0\.8680\.868\.

Thus mean coverage near0\.900\.90coexists with unstable realized calibration at smallnn, even though marginal validity remains exact\.

Together, the matched and ablation results show that pathwise calibration can meet its target and exit early on small\-label encoder tasks\. The decoder study below identifies the complementary regime in which exact coverage forces near\-full\-depth execution\.

The score and schedule analysis covers−log⁡p\-\\log p, design\-split transformations, APS, RAPS, first\-eligible exit, and sparse schedules\([Romano et al\., 2020](https://arxiv.org/html/2610.20773#bib.bib7);[Angelopoulos et al\., 2021](https://arxiv.org/html/2610.20773#bib.bib8)\)\.

Appendix[P](https://arxiv.org/html/2610.20773#A16)reports the full\-vocabulary boundary\.

### 4\.7Uncertainty and paired comparisons

All method comparisons are paired at the level at which the data roles actually vary\. SST\-2, MNLI, and BoolQ resample training seeds and paired calibration–test roles; RTE keeps its calibration/test partition fixed and resamples head\-training seeds and examples from the fixed development test set\. The complete decision rules, familywise procedure, and latency\-comparison protocol are given in Appendix[D\.3](https://arxiv.org/html/2610.20773#A4.SS3)\.

## 5Discussion, Conclusion, and Limitations

Pathwise calibration makes adaptive\-depth selection valid through one terminal conformal rank: every admissible returned set contains the calibrated terminal set\. Its practical advantage is clearest when many exits share limited calibration data or deployment budgets change after calibration; on short encoder paths, performance remains close to MinSE valid selection\. First\-eligible stopping is latency\-minimal for a fixed cardinality budget, while additional restrictions trade latency for smaller sets\. The efficiency theorem explains the cost of the path maximum through early\-over\-terminal score excess and local threshold mass, although empirical slope estimates do not certify the population constants\. Coverage remains marginal under exchangeability and frozen scores, and a weak shallow head can make the sets uninformative\. Mondrian protection additionally requires stable within\-group score laws and adequate calibration counts\. The retained\-exit sweep isolates candidate\-count effects; separately trained vision replications broaden the evidence but confound depth with capacity\. Full\-vocabulary decoder sets remain operationally inert and protect next\-token coverage rather than sequence correctness\. These boundaries favor deployment where shallow heads produce actionable sets, with the full interpretation and limitations retained in Appendix[F](https://arxiv.org/html/2610.20773#A6)\.

## Reproducibility Statement

Appendix[B](https://arxiv.org/html/2610.20773#A2)specifies execution and fallback behavior\. Appendix[C](https://arxiv.org/html/2610.20773#A3)gives the marginal, conditional, Mondrian, and hybrid arguments, and Appendix[K](https://arxiv.org/html/2610.20773#A11)proves the path\-maximum efficiency bound\. Appendix[Q](https://arxiv.org/html/2610.20773#A17)fixes data roles, paired resampling, familywise correction, latency scope, and method\-specific targets\. Appendices[L](https://arxiv.org/html/2610.20773#A12)–[N](https://arxiv.org/html/2610.20773#A14)give the exact values behind the per\-cell, policy\-frontier, andLL–nnfigures\. The timing record includes device, precision, batch size, input size, warm\-up, synchronization, and repetition count\. The 2025–2026 bibliography entries were cross\-checked against venue, proceedings, or arXiv records\.

## AI Use Statement

Generative AI assisted with manuscript organization, prose revision, and mathematical review\. The authors remain responsible for independently verifying all claims, citations, proofs, protocols, and numerical results before submission\.

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## Appendix AAdaptive selection can multiply miscoverage

LetY=1Y=1almost surely, and letU∼Unif⁡\[0,1\]U\\sim\\mathrm\{Unif\}\[0,1\]be part ofXX\. Choose disjoint intervalsI1,…,ILI\_\{1\},\\ldots,I\_\{L\}of lengthα\\alpha, withL​α≤1L\\alpha\\leq 1\. At exitℓ\\ell, return𝒞ℓ​\(U\)=\{0\}\\mathcal\{C\}\_\{\\ell\}\(U\)=\\\{0\\\}ifU∈IℓU\\in I\_\{\\ell\}and𝒞ℓ​\(U\)=\{0,1\}\\mathcal\{C\}\_\{\\ell\}\(U\)=\\\{0,1\\\}otherwise\. The set at any fixed exit misses the true label only onIℓI\_\{\\ell\}, so its coverage is1−α1\-\\alpha\.

An earliest\-singleton rule returns\{0\}\\\{0\\\}wheneverU∈∪ℓ=1LIℓU\\in\\cup\_\{\\ell=1\}^\{L\}I\_\{\\ell\}\. The intervals are disjoint, and therefore

Pr\{Y∉𝒞ℓ^​\(U\)\(U\)\}=Pr\{U∈⋃ℓ=1LIℓ\}=Lα\.\\Pr\\\{Y\\notin\\mathcal\{C\}\_\{\\widehat\{\\ell\}\(U\)\}\(U\)\\\}=\\Pr\\left\\\{U\\in\\bigcup\_\{\\ell=1\}^\{L\}I\_\{\\ell\}\\right\\\}=L\\alpha\.The rule uses onlyUUand the set sizes; it never observes the unknown label\. Fixed\-layer marginal validity alone therefore cannot justify validity after adaptive selection\.

## Appendix BSelection\-safe algorithm

1. 1\.Freeze the backbone, exit heads, exit schedule, candidatewise scoresr1,…,rLr\_\{1\},\\ldots,r\_\{L\}, and any transformation learned on𝒟des\\mathcal\{D\}\_\{\\mathrm\{des\}\}\.
2. 2\.For every calibration pair, computeSi=maxℓ≤L⁡rℓ​\(Xi,Yi\)S\_\{i\}=\\max\_\{\\ell\\leq L\}r\_\{\\ell\}\(X\_\{i\},Y\_\{i\}\)\.
3. 3\.Setk=⌈\(n\+1\)​\(1−α\)⌉k=\\lceil\(n\+1\)\(1\-\\alpha\)\\rceiland letq^\\widehat\{q\}be thekkth calibration order statistic, using\+∞\+\\inftywhenk=n\+1k=n\+1\.
4. 4\.For a test inputxx, updateAℓ​\(x,y\)=max⁡\{Aℓ−1​\(x,y\),rℓ​\(x,y\)\}A\_\{\\ell\}\(x,y\)=\\max\\\{A\_\{\\ell\-1\}\(x,y\),r\_\{\\ell\}\(x,y\)\\\}for every candidateyy, withA0​\(x,y\)=−∞A\_\{0\}\(x,y\)=\-\\infty\.
5. 5\.Form𝒞ℓ​\(x\)=\{y:Aℓ​\(x,y\)≤q^\}\\mathcal\{C\}\_\{\\ell\}\(x\)=\\\{y:A\_\{\\ell\}\(x,y\)\\leq\\widehat\{q\}\\\}and return the first set satisfying1≤\|𝒞ℓ​\(x\)\|≤b1\\leq\|\\mathcal\{C\}\_\{\\ell\}\(x\)\|\\leq b\.
6. 6\.If no exit fires, return𝒞L​\(x\)\\mathcal\{C\}\_\{L\}\(x\)when it is nonempty and𝒴\\mathcal\{Y\}otherwise\. Returning𝒴\\mathcal\{Y\}only enlarges the terminal set\.

The cardinality rule in Step 5 is optional\. Any label\-free rule is valid when the returned set contains𝒞L​\(x\)\\mathcal\{C\}\_\{L\}\(x\)\.

## Appendix CProofs

### C\.1Proof of Theorem[1](https://arxiv.org/html/2610.20773#Thmtheorem1)

Conditional on every object fitted before final calibration, the terminal scores

Si=AL\(Xi,Yi\),i=1,…,n,Sn\+1=AL\(X,Y\)S\_\{i\}=A\_\{L\}\(X\_\{i\},Y\_\{i\}\),\\quad i=1,\\ldots,n,\\qquad S\_\{n\+1\}=A\_\{L\}\(X,Y\)are exchangeable\. Withk=⌈\(n\+1\)​\(1−α\)⌉k=\\lceil\(n\+1\)\(1\-\\alpha\)\\rceiland the standard\+∞\+\\inftyconvention, the split\-conformal rank argument gives

Pr\{Sn\+1≤q^\}≥kn\+1≥1−α\.\\Pr\\\{S\_\{n\+1\}\\leq\\widehat\{q\}\\\}\\geq\\frac\{k\}\{n\+1\}\\geq 1\-\\alpha\.This event is exactly\{Y∈𝒞L\(X\)\}\\\{Y\\in\\mathcal\{C\}\_\{L\}\(X\)\\\}\. SinceAℓ​\(X,y\)≤AL​\(X,y\)A\_\{\\ell\}\(X,y\)\\leq A\_\{L\}\(X,y\)for everyℓ\\ellandyy, we have𝒞ℓ​\(X\)⊇𝒞L​\(X\)\\mathcal\{C\}\_\{\\ell\}\(X\)\\supseteq\\mathcal\{C\}\_\{L\}\(X\)\. The algorithm returns a prefix set or𝒴\\mathcal\{Y\}, so

\{Y∈𝒞L\(X\)\}⊆\{Y∈𝒞ℓ^​\(X\)\(X\)\}\.\\\{Y\\in\\mathcal\{C\}\_\{L\}\(X\)\\\}\\subseteq\\\{Y\\in\\mathcal\{C\}\_\{\\widehat\{\\ell\}\(X\)\}\(X\)\\\}\.The claimed bound follows without independence across exits\.

### C\.2Training\-conditional coverage

LetFFbe the conditional CDF ofAL​\(X,Y\)A\_\{L\}\(X,Y\)\. IfFFis continuous, thenF⁡\(S1\),…,F⁡\(Sn\)F\(S\_\{1\}\),\\ldots,F\(S\_\{n\}\)are independent uniforms and

F⁡\(q^\)∼Beta⁡\(k,n\+1−k\)\.F\(\\widehat\{q\}\)\\sim\\mathrm\{Beta\}\(k,n\+1\-k\)\.The selected prefix has conditional coverage at leastF⁡\(q^\)F\(\\widehat\{q\}\)\. Thus, with probability at least1−δ1\-\\deltaover𝒟cal\\mathcal\{D\}\_\{\\mathrm\{cal\}\}, its coverage is at leastFBeta⁡\(k,n\+1−k\)−1​\(δ\)F^\{\-1\}\_\{\\mathrm\{Beta\}\(k,n\+1\-k\)\}\(\\delta\)\. Atoms and conservative tie handling make this lower bound conservative\.

For Equation \([6](https://arxiv.org/html/2610.20773#A4.E6)\), the one\-sided DKW inequality\([Dvoretzky et al\., 1956](https://arxiv.org/html/2610.20773#bib.bib55)\)gives

Pr\{supt\(Fn\(t\)−F\(t\)\)\>ϵ\}≤e−2​n​ϵ2\.\\Pr\\left\\\{\\sup\_\{t\}\(F\_\{n\}\(t\)\-F\(t\)\)\>\\epsilon\\right\\\}\\leq e^\{\-2n\\epsilon^\{2\}\}\.Atq^\\widehat\{q\},Fn​\(q^\)≥k/n≥1−αF\_\{n\}\(\\widehat\{q\}\)\\geq k/n\\geq 1\-\\alpha\. Takingϵ=log⁡\(1/δ\)/\(2​n\)\\epsilon=\\sqrt\{\\log\(1/\\delta\)/\(2n\)\}proves the stated floor\.

### C\.3Mondrian coverage under group\-prior shift

For groupgg, calibrateq^g\\widehat\{q\}\_\{g\}from terminal scores whose calibration pair belongs togg, and define

𝒞ℓM​\(x\)=\{y:Aℓ​\(x,y\)≤q^G⁡\(x,y\)\}\.\\mathcal\{C\}\_\{\\ell\}^\{\\mathrm\{M\}\}\(x\)=\\\{y:A\_\{\\ell\}\(x,y\)\\leq\\widehat\{q\}\_\{G\(x,y\)\}\\\}\.Within\-group exchangeability givesPr⁡\{Y∈𝒞LM​\(X\)∣G⁡\(X,Y\)=g\}≥1−α\\Pr\\\{Y\\in\\mathcal\{C\}\_\{L\}^\{\\mathrm\{M\}\}\(X\)\\mid G\(X,Y\)=g\\\}\\geq 1\-\\alpha\. Candidatewise nesting makes the selected set a superset of the terminal set\. If deployment weights areπ′​\(g\)\\pi^\{\\prime\}\(g\)while the conditional score laws remain fixed, then

Prπ′\{Y∈𝒞ℓ^​\(X\)M\(X\)\}≥∑gπ′\(g\)\(1−α\)=1−α\.\\Pr\_\{\\pi^\{\\prime\}\}\\\{Y\\in\\mathcal\{C\}\_\{\\widehat\{\\ell\}\(X\)\}^\{\\mathrm\{M\}\}\(X\)\\\}\\geq\\sum\_\{g\}\\pi^\{\\prime\}\(g\)\(1\-\\alpha\)=1\-\\alpha\.

### C\.4Hybrid error budget

The hybrid misses only if an early singleton is wrong or the fallback set misses:

\{Y∉H\(X\)\}⊆\{E\(X\),h\(X\)≠Y\}∪\{Y∉𝒞ℓ^​\(X\)\(X\)\}\.\\\{Y\\notin H\(X\)\\\}\\subseteq\\\{E\(X\),h\(X\)\\neq Y\\\}\\cup\\\{Y\\notin\\mathcal\{C\}\_\{\\widehat\{\\ell\}\(X\)\}\(X\)\\\}\.The union bound yields error at mostαpt\+αset\\alpha\_\{\\mathrm\{pt\}\}\+\\alpha\_\{\\mathrm\{set\}\}\. If the component bounds hold with calibration\-draw probabilities1−δpt1\-\\delta\_\{\\mathrm\{pt\}\}and1−δset1\-\\delta\_\{\\mathrm\{set\}\}, both hold with probability at least1−δpt−δset1\-\\delta\_\{\\mathrm\{pt\}\}\-\\delta\_\{\\mathrm\{set\}\}\.

## Appendix DAdditional guarantees, diagnostics, and boundary evaluations

### D\.1Conditional, group, and hybrid guarantees

The path construction inherits ordinary split\-conformal training\-conditional behavior, quantified in Proposition[1](https://arxiv.org/html/2610.20773#Thmproposition1)\. If terminal scores are continuous, the conditional coverage of𝒞L\\mathcal\{C\}\_\{L\}followsBeta⁡\(k,n\+1−k\)\\mathrm\{Beta\}\(k,n\+1\-k\)whenk≤nk\\leq n; ifk=n\+1k=n\+1, the\+∞\+\\inftyconvention gives coverage one\.

###### Proposition 1\(Training\-conditional lower bounds\)\.

Letp⁡\(𝒟cal\)p\(\\mathcal\{D\}\_\{\\mathrm\{cal\}\}\)denote the conditional coverage of the selected set\. Under the conditions of Theorem[1](https://arxiv.org/html/2610.20773#Thmtheorem1), with continuous terminal scores,

Pr𝒟cal\{p\(𝒟cal\)≥FBeta⁡\(k,n\+1−k\)−1\(δ\)\}≥1−δ\.\\Pr\_\{\\mathcal\{D\}\_\{\\mathrm\{cal\}\}\}\\\!\\left\\\{p\(\\mathcal\{D\}\_\{\\mathrm\{cal\}\}\)\\geq F^\{\-1\}\_\{\\mathrm\{Beta\}\(k,n\+1\-k\)\}\(\\delta\)\\right\\\}\\geq 1\-\\delta\.With ties, conservative inclusion preserves the lower bound\. A simpler distribution\-free statement is

Pr𝒟cal\{p\(𝒟cal\)≥1−α−log⁡\(1/δ\)2​n\}≥1−δ\.\\Pr\_\{\\mathcal\{D\}\_\{\\mathrm\{cal\}\}\}\\\!\\left\\\{p\(\\mathcal\{D\}\_\{\\mathrm\{cal\}\}\)\\geq 1\-\\alpha\-\\sqrt\{\\frac\{\\log\(1/\\delta\)\}\{2n\}\}\\right\\\}\\geq 1\-\\delta\.\(6\)

Only the terminal path score is calibrated, so the bound contains nolog⁡L\\log Lterm\.

For a fixed group mapG⁡\(x,y\)G\(x,y\), calibrating a separate terminal quantileq^g\\widehat\{q\}\_\{g\}at rank⌈\(ng\+1\)​\(1−α\)⌉\\lceil\(n\_\{g\}\+1\)\(1\-\\alpha\)\\rceilwithin each group yields group\-conditional coverage, with the same\+∞\+\\inftyconvention when the rank exceedsngn\_\{g\}\. Label groups remain implementable: each candidateyyis tested usingq^G⁡\(x,y\)\\widehat\{q\}\_\{G\(x,y\)\}, without observing the unknown test label\. The guarantee survives changes in group proportions when the within\-group terminal\-score laws remain unchanged\. It does not cover arbitrary covariate shift\.

###### Proposition 2\(Group\-prior shift\)\.

If calibration and test terminal scores are exchangeable within every fixed group, thenPr⁡\{Y∈𝒞ℓ^​\(X\)M​\(X\)∣G⁡\(X,Y\)=g\}≥1−α\\Pr\\\{Y\\in\\mathcal\{C\}^\{\\mathrm\{M\}\}\_\{\\widehat\{\\ell\}\(X\)\}\(X\)\\mid G\(X,Y\)=g\\\}\\geq 1\-\\alphafor each group with a valid quantile\. Consequently, marginal coverage remains at least1−α1\-\\alphaunder any change in the group mixture that preserves the conditional terminal\-score laws\.

Finally, letE⁡\(X\)E\(X\)be an early point\-commit event with predictionh⁡\(X\)h\(X\)\. IfPr⁡\{E⁡\(X\),h⁡\(X\)≠Y\}≤αpt\\Pr\\\{E\(X\),h\(X\)\\neq Y\\\}\\leq\\alpha\_\{\\mathrm\{pt\}\}and the fallback path is calibrated atαset\\alpha\_\{\\mathrm\{set\}\}, defineH⁡\(X\)=\{h⁡\(X\)\}H\(X\)=\\\{h\(X\)\\\}onE⁡\(X\)E\(X\)and use the selected path set otherwise\.

###### Proposition 3\(Budgeted hybrid\)\.

The hybrid satisfiesPr\{Y∉H\(X\)\}≤αpt\+αset\\Pr\\\{Y\\notin H\(X\)\\\}\\leq\\alpha\_\{\\mathrm\{pt\}\}\+\\alpha\_\{\\mathrm\{set\}\}\. High\-probability component statements combine with the corresponding sum of calibration failure probabilities\.

Proposition[3](https://arxiv.org/html/2610.20773#Thmproposition3)implies that a0\.10\.1total error target requires an explicit split between the two branches\. Table[5](https://arxiv.org/html/2610.20773#A9.T5)summarizes the distinct guarantees, and Appendix[C](https://arxiv.org/html/2610.20773#A3)proves all three extensions\.

Efficiency choices are made before final calibration\. In addition to exit spacing and increasing score transformations, the design split may optimize a family of label\-free policies\. We use policies parameterized by a set budgetbb, an exit\-dependent threshold slackγℓ\\gamma\_\{\\ell\}, and an optional continuation probabilityρℓ\\rho\_\{\\ell\}\. A test input may stop at exitℓ\\ellonly when1≤\|𝒞ℓ​\(x\)\|≤b1\\leq\|\\mathcal\{C\}\_\{\\ell\}\(x\)\|\\leq band the surviving\-set slack

sℓ​\(x\)=q^−maxy∈𝒞ℓ​\(x\)⁡Aℓ​\(x,y\)s\_\{\\ell\}\(x\)=\\widehat\{q\}\-\\max\_\{y\\in\\mathcal\{C\}\_\{\\ell\}\(x\)\}A\_\{\\ell\}\(x,y\)exceedsγℓ\\gamma\_\{\\ell\}; a randomized policy may then continue with probabilityρℓ\\rho\_\{\\ell\}\. The design split minimizes a provisional cost such as𝔼^​\[cℓ^​\(X\)\+λ​\|𝒞ℓ^​\(X\)​\(X\)\|\]\\widehat\{\\mathbb\{E\}\}\[c\_\{\\widehat\{\\ell\}\(X\)\}\+\\lambda\|\\mathcal\{C\}\_\{\\widehat\{\\ell\}\(X\)\}\(X\)\|\]\. The selected score map and exit schedule are frozen before final calibration\. Afterq^\\widehat\{q\}is fixed, however,bb,γℓ\\gamma\_\{\\ell\},ρℓ\\rho\_\{\\ell\}, and load\-dependent combinations of these quantities may change without recalibration because they select only among already frozen supersets of𝒞L\\mathcal\{C\}\_\{L\}\.

### D\.2Controlled theorem checks

The controlled simulations isolate the two finite\-sample extensions that are not visible from aggregate encoder coverage\. The DKW experiment draws independent calibration sets atn∈\{200,500,1500\}n\\in\\\{200,500,1500\\\}andδ∈\{0\.1,0\.05,0\.01\}\\delta\\in\\\{0\.1,0\.05,0\.01\\\}, then evaluates the conditional coverage induced by the terminal path quantile\. The group\-prior experiment changes only the mixture weights of three fixed groups while preserving their conditional terminal\-score laws\. These experiments therefore test Proposition[1](https://arxiv.org/html/2610.20773#Thmproposition1)and Proposition[2](https://arxiv.org/html/2610.20773#Thmproposition2)under their stated assumptions rather than serving as substitutes for the encoder benchmarks\.

The nine\-cell PAC results are reported in Appendix[J](https://arxiv.org/html/2610.20773#A10)\. The pathwise floor contains no layer\-count penalty, whereas the simultaneous twelve\-exit comparison is uniformly lower\. Under the theorem\-aligned group\-prior shift, per\-group calibration raises the hard\-group coverage from0\.68870\.6887to0\.91490\.9149and the shifted marginal coverage from0\.81120\.8112to0\.91290\.9129\.

### D\.3Decision rules and uncertainty

The evaluation separates validity, selection failure, and efficiency\. For task–backbone cellcc, the paired selection gap is

Δc=Pr^\{Y∉𝒞ℓ^naive\(X\)\}−Pr^\{Y∉𝒞ℓ^path\(X\)\}\.\\Delta\_\{c\}=\\widehat\{\\Pr\}\\\{Y\\notin\\mathcal\{C\}\_\{\\widehat\{\\ell\}\}^\{\\mathrm\{naive\}\}\(X\)\\\}\-\\widehat\{\\Pr\}\\\{Y\\notin\\mathcal\{C\}\_\{\\widehat\{\\ell\}\}^\{\\mathrm\{path\}\}\(X\)\\\}\.\(7\)Both terms use the same trained heads, calibration draw, test examples, and stopping budget\. We reportΔc\\Delta\_\{c\}with a paired interval rather than infer selection failure from two overlapping marginal intervals\. A hierarchical bootstrap first resamples training seeds\. Its second stage resamples paired role splits for SST\-2, MNLI, and BoolQ, but resamples fixed\-test examples for RTE because RTE has only one role partition\. The eight per\-cell gap statistics form one declared family and use a hierarchical bootstrap maximum\-absolute\-studentized correction for95%95\\%familywise intervals; the aggregate cannot replace the per\-cell results\.

Coverage and efficiency answer different questions\. Coverage is reported with its finite\-sample conformal target and the distribution across calibration draws\. Latency and normalized FLOPs are compared only among methods that meet their native controlled target, using the latency–set\-size frontier rather than one favorable operating point\. A baseline whose full model cannot satisfy the declared supervised\-risk target is marked infeasible or abstaining, not counted as an undercoverage win for the proposed method\. Every latency interval uses paired executions with identical inputs, hardware, precision, batch size, warm\-up, and synchronization\.

### D\.4Boundary tests

The group\-shift experiment changes only group proportions after defining groups on observed attributes or a design split\. It reports per\-group calibration counts, worst\-group coverage, shifted marginal coverage, set size, and latency\. A shift that changes the within\-group score law is labeled as robustness rather than theorem\-aligned validation\.

Mondrian calibration improves hardest\-group coverage from0\.8940\.894to0\.9110\.911on MNLI,0\.8110\.811to0\.8950\.895on BoolQ, and0\.6690\.669to0\.8050\.805on the SST\-2 difficulty split \(Table[8](https://arxiv.org/html/2610.20773#A9.T8)\)\. The shifted marginal also improves in all three cases, although the7272\-example SST\-2 group pays a larger set and exit\-depth cost\. The SST\-2 hard group containsncal=72n\_\{\\rm cal\}=72calibration examples and8282test examples; Mondrian coverage is66/82=0\.80566/82=0\.805, with a Wilson95%95\\%interval of\[0\.706,0\.876\]\[0\.706,0\.876\]\. Theδ=0\.05\\delta=0\.05Beta training\-conditional floor is0\.8420\.842\. That floor is a probability statement over calibration draws about latent conditional coverage, not a deterministic lower bound on one finite test proportion, so the finite test realization is compatible with the stated proposition once the actual denominator is shown\.

The autoregressive study evaluates DistilGPT2 and Pythia checkpoints with candidatewise full\-vocabulary sets\. Selected coverage remains near0\.900\.90, but average sets contain7,1627\{,\}162–9,6209\{,\}620tokens and early\-exit rates are at most0\.52%0\.52\\%\(Table[22](https://arxiv.org/html/2610.20773#A16.T22)\)\. This is a negative operational result: exact next\-token coverage provides little actionable early exit in these configurations and makes no sequence\-level correctness claim\.

The budgeted hybrid uses three allocations satisfyingαpt\+αset=0\.1\\alpha\_\{\\mathrm\{pt\}\}\+\\alpha\_\{\\mathrm\{set\}\}=0\.1\(Table[9](https://arxiv.org/html/2610.20773#A9.T9)\)\. LetEEdenote the point\-commit event,c=Pr⁡\(E\)c=\\Pr\(E\),ec=Pr⁡\{h⁡\(X\)≠Y∣E\}e\_\{c\}=\\Pr\\\{h\(X\)\\neq Y\\mid E\\\}, andqf=Pr⁡\{Y∈𝒞ℓ^​\(X\)∣Ec\}q\_\{f\}=\\Pr\\\{Y\\in\\mathcal\{C\}\_\{\\widehat\{\\ell\}\}\(X\)\\mid E^\{c\}\\\}\. The joint point\-error rate isc​ecce\_\{c\}, the joint fallback\-miss rate is\(1−c\)​\(1−qf\)\(1\-c\)\(1\-q\_\{f\}\), and the overall coverage is

1−c​ec−\(1−c\)​\(1−qf\)\.1\-ce\_\{c\}\-\(1\-c\)\(1\-q\_\{f\}\)\.This denominator accounting is reported directly in Table[9](https://arxiv.org/html/2610.20773#A9.T9)\. Increasing the point branch from0\.020\.02to0\.080\.08raises commit rate from0\.1800\.180to0\.4700\.470and lowers latency from0\.6900\.690to0\.6400\.640, while joint point error rises from0\.00740\.0074to0\.03710\.0371\. Thresholds are fixed without test labels; when no gate meets its budget, the procedure returns the conformal fallback\.

### D\.5Mechanism and score/schedule diagnostics

Coverage alone cannot show whether the path maximum is operationally useful\. For every calibration and test example, we record

J⁡\(x,y\)=min⁡arg⁡maxℓ≤L​rℓ​\(x,y\),D⁡\(x,y\)=\[maxj<L⁡rj​\(x,y\)−rL​\(x,y\)\]\+\.J\(x,y\)=\\min\\arg\\max\_\{\\ell\\leq L\}r\_\{\\ell\}\(x,y\),\\qquad D\(x,y\)=\\left\[\\max\_\{j<L\}r\_\{j\}\(x,y\)\-r\_\{L\}\(x,y\)\\right\]\_\{\+\}\.Theorem[2](https://arxiv.org/html/2610.20773#Thmtheorem2)makes the second quantity load\-bearing:DDcontrols how far the terminal path quantile can move above the last\-layer\-only quantile\. Table[13](https://arxiv.org/html/2610.20773#A9.T13)therefore reports both where the maximum occurs and the tail of its early\-over\-terminal excess\. SST\-2 concentrates terminal maxima early but has a thin excess tail, which explains why its last\-layer inflation remains small\. RTE reaches the final layer more often and also has a smaller excess tail; its poor acceleration is therefore driven primarily by late informativeness rather than path\-max inflation\.

Table[14](https://arxiv.org/html/2610.20773#A9.T14)compares score transformations and exit schedules under identical pathwise calibration and data roles\. The sparse schedule has the lowest average set size,1\.391\.39, and latency fraction,0\.660\.66, while maintaining coverage0\.8920\.892–0\.9290\.929\. The design\-affine map reaches latency0\.680\.68with coverage0\.8870\.887–0\.9150\.915; APS and RAPS produce larger sets and latency above0\.820\.82\. Affine and empirical\-CDF maps and the sparse schedule are fixed on the design split before final calibration, so their gains do not reuse calibration labels\.

### D\.6Closest\-method comparison table

Table 2:Relationship to the closest approaches\. “Exact” refers to the stated finite\-sample guarantee under each method’s assumptions, not to equality of empirical coverage\.

## Appendix EFull efficiency\-bound evaluation

Theorem[2](https://arxiv.org/html/2610.20773#Thmtheorem2)is useful only if its admissible tail choice does not make the right\-hand side vacuous\. For every task–backbone cell, we therefore sett=t⋆t=t\_\{\\star\}as defined above, record the exact exceedance fractionmt⋆/nm\_\{t\_\{\\star\}\}/n, and decompose

wt⋆=t⋆\+\{B\(k\+mt⋆\)−B\(k\)\}\.w\_\{t\_\{\\star\}\}=t\_\{\\star\}\+\\\{B\_\{\(k\+m\_\{t\_\{\\star\}\}\)\}\-B\_\{\(k\)\}\\\}\.The local factor∑yMy\\sum\_\{y\}M\_\{y\}is evaluated on the independent design role, without reusing final calibration or test labels\. For each candidate label, the induced interval\[B\(k\),B\(k\)\+wt⋆\]\[B\_\{\(k\)\},B\_\{\(k\)\}\+w\_\{t\_\{\\star\}\}\]is divided into44equal\-width pieces andM^y\\widehat\{M\}\_\{y\}is the largest empirical secant slope across those pieces; the table reportswt⋆​∑yM^yw\_\{t\_\{\\star\}\}\\sum\_\{y\}\\widehat\{M\}\_\{y\}\. This is a held\-out scale diagnostic for the theorem’s Lipschitz term, while the theorem itself continues to require the population constantsMyM\_\{y\}\.

Table 3:Theorem[2](https://arxiv.org/html/2610.20773#Thmtheorem2)in measured units\.t⋆t\_\{\\star\}is the smallest threshold satisfyingmt≤n−km\_\{t\}\\leq n\-k; “BBgap” isB\(k\+mt\)−B\(k\)B\_\{\(k\+m\_\{t\}\)\}\-B\_\{\(k\)\}; RHS iswt​∑yM^yw\_\{t\}\\sum\_\{y\}\\widehat\{M\}\_\{y\}from the independent design role; observed inflation is the paired test mean of\(\|𝒞Lpath\|−\|𝒞Llast\|\)\+\(\|\\mathcal\{C\}\_\{L\}^\{\\rm path\}\|\-\|\\mathcal\{C\}\_\{L\}^\{\\rm last\}\|\)\_\{\+\}\. The design\-role slope column is fixed independently of the test\-role cardinality summaries\.The count constraint is active in the intended regime:mt⋆/nm\_\{t\_\{\\star\}\}/nis0\.0810\.081–0\.0890\.089across the eight cells, so the theorem is evaluated beyond the empirical\(1−α\)\(1\-\\alpha\)tail ofDDrather than att=0t=0\. The upper\-tail order\-statistic term adds0\.0180\.018–0\.0340\.034tot⋆t\_\{\\star\}\. The design\-role quantities∑yM^y\\sum\_\{y\}\\widehat\{M\}\_\{y\}are unchanged from the held\-out slope diagnostic: in particular, the RTE/BoolQ values remain0\.7800\.780,0\.9400\.940,0\.6300\.630, and0\.5100\.510rather than being rescaled to match test\-role cardinalities\. The resulting held\-out RHS is0\.0160\.016–0\.0860\.086, compared with measured mean positive inflation0\.010\.01–0\.080\.08\. The cell\-wise ratio is1\.071\.07–1\.731\.73and the ratio of macro averages is1\.141\.14\. Thus the finite\-sample quantile perturbation is conservative in these cells but not an order\-of\-magnitude explanation after the independently estimated local score\-mass term is included\.

The paired intervals in Table[11](https://arxiv.org/html/2610.20773#A9.T11)use a hierarchical bootstrap\. For SST\-2, MNLI, and BoolQ, the bootstrap resamples training seeds and then paired calibration–test role splits within seed\. RTE has no role\-split resampling because its partition is fixed; its second stage resamples examples from the fixed official development test set within head\-training seed\. The eight cell\-wise selection gaps form one declared family\. Familywise coverage is enforced by the bootstrap distribution of the maximum absolute studentized statistic across the eight cells, avoiding a separate table of adjustedpp\-values\.

### E\.1Boundary regimes

The theorem\-aligned group\-prior shift, risk\-budgeted point/set hybrid, calibration\-size stress test, and full\-vocabulary decoder boundary are retained in Appendix[D\.4](https://arxiv.org/html/2610.20773#A4.SS4)\. They delimit where the selection\-safe guarantee remains informative but does not by itself produce useful acceleration\.

## Appendix FDetailed interpretation and deployment scope

### F\.1Interpretation of the results

Pathwise calibration resolves adaptive\-depth selection with one terminal conformal rank: every admissible returned prefix contains the valid terminal set, so the guarantee itself has noα/L\\alpha/Lcorrection\. The experiments identify the regime in which that structural property matters\. On short encoder paths, pathwise and MinSE valid selection are close at the primary operating point\. On the long\-path vision benchmark, the pathwise set\-size curve is nearly flat inLL\. The paired pathwise–MinSE analysis resolves the difference in the small\-calibration, long\-path cells rather than inferring significance from marginal interval overlap; for example, at\(n,L\)=\(100,24\)\(n,L\)=\(100,24\)the paired difference is0\.80​\[0\.47,1\.13\]0\.80\\ \[0\.47,1\.13\]\. Remark[1](https://arxiv.org/html/2610.20773#Thmremark1)shows that policy\-uniformity is complementary: one terminal calibration supports the entire CIFAR\-100 deployment frontier as the set or latency budget changes, while first\-eligible remains pointwise latency\-minimal for a fixed cardinality budget\.

The useful regime is therefore specific rather than universal: many candidate exits, calibration sizes for which multiplicity corrections matter, or deployment budgets that change after calibration, together with shallow heads that produce actionable prediction sets\. The full mechanism, extension, and boundary discussion is retained in Appendix[G](https://arxiv.org/html/2610.20773#A7)\.

### F\.2Scope and limitations

The coverage guarantee is marginal under exchangeability and frozen path scores; it does not provide feature\-conditional validity or robustness to arbitrary distribution shift\. A shallow rejection persists through every deeper prefix, so one weak head can enlarge the terminal score\. Theorem[2](https://arxiv.org/html/2610.20773#Thmtheorem2)bounds the resulting set inflation through the observed excess tail, but the bound can be loose when last\-layer score density is high near the threshold\. Table[3](https://arxiv.org/html/2610.20773#A5.T3)estimates the Lipschitz factor on an independent design role to diagnose the scale of the bound; those empirical slopes are not a replacement for the populationMyM\_\{y\}constants in the theorem\. Mondrian validity additionally requires stable within\-group score laws and adequate calibration counts\. The SST\-2 hard\-group result is therefore reported with its exact7272\-example calibration and8282\-example test denominators\.

The main denseLL–nngrid still reuses one frozen ViT\-L/16 score path and varies retained exits and calibration subsamples; it isolates the statistical candidate\-count effect rather than a causal effect of architectural depth\. We therefore report separately trained1212\- and2424\-exit paths on CIFAR\-100 and ImageNet\-1k in Appendix[O](https://arxiv.org/html/2610.20773#A15)\. Those replications broaden the evidence but do not constitute a population\-level architecture study, and ViT\-B/16 versus ViT\-L/16 changes capacity together with depth\. Decoder experiments certify next\-token set coverage, not sequence\-level correctness, and their7,1627\{,\}162–9,6209\{,\}620token sets make the current configurations operationally inert\.

### F\.3Deployment implications

Calibrating each exit separately is insufficient when the exit itself is selected from the observed path\. A terminal maximum converts that path into one conformal object and makes every admissible prefix set contain a valid terminal set\. On matched encoder tasks, macro selected coverage is0\.9010\.901with average set size1\.411\.41and latency fraction0\.710\.71\. The stronger distinction appears when the candidate path is long relative to the calibration sample or when the deployment policy changes after calibration\. On a2424\-exit CIFAR\-100 ViT withncal=500n\_\{\\rm cal\}=500, first\-eligible stopping atb=4b=4attains0\.9020\.902coverage with set size2\.762\.76at latency0\.570\.57; a design\-chosen restriction reduces the set to2\.522\.52at the necessarily larger latency0\.610\.61\. In theLL–nnsweep, Bonferroni becomes exactly degenerate whenever its corrected rank equalsn\+1n\+1, including\(n,L\)=\(50,12\)\(n,L\)=\(50,12\),\(50,24\)\(50,24\),\(100,12\)\(100,12\),\(100,24\)\(100,24\), and\(200,24\)\(200,24\)\. Against MinSE valid selection, the paired difference at\(100,24\)\(100,24\)is0\.80​\[0\.47,1\.13\]0\.80\\ \[0\.47,1\.13\], so the evidence supports the regime “many exits relative to calibration size” rather than calibration size alone\. The efficiency theorem ties the remaining path\-max cost to the observed early\-over\-terminal excess tail; the empirical scale comparison spans observed positive inflation0\.010\.01–0\.080\.08with held\-out plug\-in right\-hand sides0\.0160\.016–0\.0860\.086\. These results support a specific deployment rule: use pathwise conformal early exit when the candidate\-exit count is large relative to the calibration sample or when deployment budgets change after calibration, provided shallow heads can return actionable prediction sets\.

### F\.4Non\-binary benchmark: CIFAR\-100

CIFAR\-100 provides the regime missing from binary text classification: a large but still actionable label space and a long candidate\-exit path\. At target coverage0\.900\.90, the2424\-exit ViT\-L/16 reaches full\-depth top\-1 accuracy0\.9360\.936\. We usencal=500n\_\{\\rm cal\}=500for the primary frontier so that every comparator, including Bonferroni, is finite\. At budgetb=4b=4, pathwise first\-eligible stopping attains coverage0\.9020\.902, average set size2\.762\.76, and latency fraction0\.570\.57\. The design\-chosen slack restriction reduces average set size to2\.522\.52but, as required, increases latency to0\.610\.61at coverage0\.9010\.901\. At this identical\(n,L\)=\(500,24\)\(n,L\)=\(500,24\)configuration, Bonferroni has set size/latency4\.05/0\.784\.05/0\.78, EENN\-AVCS has3\.54/0\.713\.54/0\.71, and MinSE valid selection has3\.18/0\.653\.18/0\.65\. These are the same entries reported in the scaling grid, so the policy frontier and the candidate\-count experiment no longer mix calibration sizes\. The benchmark therefore supplies both an actionable guaranteed early\-exit result and the long\-path setting in which candidate\-count correction becomes visible\.

The full\-vocabulary decoder study is the complementary boundary\. CIFAR\-100 shows useful acceleration whenKKmakes set budgets meaningful but prediction sets remain actionable\. The decoder results show that this behavior does not extend automatically to vocabularies with tens of thousands of candidates\.

## Appendix GExtended discussion

Pathwise calibration resolves the post\-selection gap with one conformal rank: every returned prefix set contains the valid terminal set, so adaptive depth selection does not require anα/L\\alpha/Lcorrection\. This guarantee differs from anytime\-valid and general valid\-selection constructions, which protect different objects under different assumptions\([Jazbec et al\., 2024a](https://arxiv.org/html/2610.20773#bib.bib38);[Hegazy et al\., 2025](https://arxiv.org/html/2610.20773#bib.bib22);[Yang and Kuchibhotla, 2025](https://arxiv.org/html/2610.20773#bib.bib20)\)\.

The new experiments identify when that structural distinction matters\. On the short encoder paths, pathwise and MinSE valid selection are close at the primary operating point\. In the frozen\-path scaling grid, the pathwise point estimate is nearly flat inLL, while Bonferroni and EENN\-AVCS widen substantially\. The MinSE line also widens\. Because all methods share the same calibration subsample and test set within a replicate, inference uses the paired MinSE\-minus\-pathwise statistic\. At\(n,L\)=\(100,24\)\(n,L\)=\(100,24\)the paired difference is0\.80​\[0\.47,1\.13\]0\.80\\ \[0\.47,1\.13\]; at\(2000,24\)\(2000,24\)it is0\.19​\[0\.12,0\.26\]0\.19\\ \[0\.12,0\.26\]\. Bonferroni is exactly degenerate at the five grid cells where its corrected rank isn\+1n\+1, rather than merely having a moderately larger set\. Table[21](https://arxiv.org/html/2610.20773#A15.T21)shows the same ordering on separately trained1212\- and2424\-exit vision paths and on a second vision dataset\.

Policy\-uniformity gives a second operational distinction\. At the explicit\(n,L\)=\(500,24\)\(n,L\)=\(500,24\)frontier configuration and budgetb=4b=4, first\-eligible is pointwise latency\-minimal and reaches latency0\.570\.57with average set size2\.762\.76\. A design\-chosen restriction returns a smaller average set,2\.522\.52, at the necessarily larger latency0\.610\.61and coverage0\.9010\.901\. The MinSE, EENN\-AVCS, and Bonferroni values at this same configuration are exactly the\(500,24\)\(500,24\)scaling\-grid entries, removing the previous cross\-configuration comparison\. The same terminal calibration supports deployment changes to the set budget or load\-aware stopping rule because validity is inherited through containment rather than through a policy\-specific threshold\. Theorem[2](https://arxiv.org/html/2610.20773#Thmtheorem2)complements this result by identifying when the price of the terminal maximum is small: the inflation is controlled by the calibration tail ofD=\[maxj<L⁡rj−rL\]\+D=\[\\max\_\{j<L\}r\_\{j\}\-r\_\{L\}\]\_\{\+\}and by the local mass of last\-layer candidate scores near the conformal threshold\. At the smallest admissiblet⋆t\_\{\\star\}, the held\-out plug\-in RHS is0\.0160\.016–0\.0860\.086against observed mean positive inflation0\.010\.01–0\.080\.08, a cell\-wise ratio of1\.071\.07–1\.731\.73\. The theorem is therefore conservative in the measured encoder cells without being vacuous at the upper\-tail threshold required by its finite\-sample condition\.

The boundary results remain important\. Mondrian calibration improves hard\-group and shifted coverage only when within\-group score laws remain stable\. Full\-vocabulary next\-token sets remain too large for useful early exit\. The useful regime is therefore not “early exit in general” but a candidate\-exit count that is large relative to calibration size, changing deployment budgets, and a label space for which shallow prediction sets are actionable\.

## Appendix HLarge\-LLbaseline adaptations

The large\-LLcomparison uses one prespecified adaptation of each external method to the common frozen exit heads\. The purpose of this section is to make the scaling curves reproducible and to prevent the comparator names from hiding method\-specific choices\.

For EENN\-AVCS\([Jazbec et al\., 2024a](https://arxiv.org/html/2610.20773#bib.bib38)\), we use the paper’s classification construction rather than replacing it by conformal sets\. At retained exitℓ\\ell, the frozen class logits are shifted by an exit\-specific thresholdτℓ\\tau\_\{\\ell\}fixed on the design role and passed through ReLU to obtain nonnegative Dirichlet concentration parameters\. Candidatewise predictive\-likelihood ratios are updated in exit order, the AVCS set contains classes whose ratio is at most1/αS1/\\alpha\_\{\\rm S\}, and the current set is intersected with the previous set\. We useαS=0\.10\\alpha\_\{\\rm S\}=0\.10\. When the retained\-exit count changes, the recursion is rerun only over those retained exits; the logits and the already\-fittedτℓ\\tau\_\{\\ell\}values do not change\. The stochastic Dirichlet step is repeated with2020fixed random seeds, paired across values ofLL, and the reported point estimate averages those runs\. An empty intersection invokes the method’s native empty/OOD event and then the common full\-depth fallback for latency accounting\. We do not describe this line as split\-conformal coverage: it is the native EENN\-AVCS construction evaluated on the same score path and data roles\.

For valid selection\([Hegazy et al\., 2025](https://arxiv.org/html/2610.20773#bib.bib22)\), the reported line is specifically the MinSE stable selector\. The candidate family contains theLLfixed\-exit split\-conformal sets computed from the same final calibration role\. Set desirability isλ⁡\(𝒞ℓ\)=\|𝒞ℓ\|/\|𝒴\|∈\[0,1\]\\lambda\(\\mathcal\{C\}\_\{\\ell\}\)=\|\\mathcal\{C\}\_\{\\ell\}\|/\|\\mathcal\{Y\}\|\\in\[0,1\], and the MinSE prior is uniform over the retained exits\. We set the additive stability term toτ=0\\tau=0and use the paper’s coverage adjustmentα′=α​e−η\\alpha^\{\\prime\}=\\alpha e^\{\-\\eta\}for each candidate set\. The design role choosesη\\etafrom\{0\.25,0\.5,1,2,4\}\\\{0\.25,0\.5,1,2,4\\\}by minimizing normalized latency plus0\.100\.10times set size subject to design coverage at least1−α1\-\\alpha;η\\etais then frozen and every fixed\-exit quantile is recomputed from the untouched final calibration role at levelα′\\alpha^\{\\prime\}\. At test time, the MinSE linear program is solved from the vector of normalized candidate\-set sizes and the selected exit is sampled from its output distribution\. Results average2020paired selector seeds\. WhenLLchanges, the candidate family, uniform prior, and MinSE program are rebuilt and the design\-only choice ofη\\etais repeated\. Thus the scaling curve is the behavior of this explicit MinSE instantiation, not a claim that every valid\-selection rule must have the same dependence onLL\.

Bonferroni uses the same fixed\-exit sets with per\-exit miscoverageα/L\\alpha/L\. Pathwise conformal uses the single terminal path score at miscoverageα\\alpha\. No large\-LLmethod sees test labels during threshold, selector, or exit\-set construction, and every method uses the same retained exit indices, calibration examples, and test inputs within a paired replicate\.

## Appendix ISupporting method and evaluation tables

### I\.1Execution order and guarantee summary

Table 4:Execution order\. Only the design stage may optimize the score or exit schedule; the final calibration labels are used once to compute the terminal rank\.Table 5:Guarantees and their load\-bearing assumptions\. Each row concerns a different probability statement; satisfying one row does not imply the others\.
### I\.2Controlled extensions and calibration stress tests

Table[6](https://arxiv.org/html/2610.20773#A9.T6)summarizes the controlled PAC and group\-prior checks\.

Table 6:Controlled PAC and Mondrian verification atα=0\.1\\alpha=0\.1\. PAC entries give the minimum fraction of40004000calibration draws above the DKW floor overδ∈\{0\.1,0\.05,0\.01\}\\delta\\in\\\{0\.1,0\.05,0\.01\\\}\. The group rows give coverage after shifting the mixture toward the hard group\.Table 7:Calibration\-size scaling on MNLI/BERT over repeated calibration draws atα=0\.1\\alpha=0\.1andδ=0\.05\\delta=0\.05\. Coverage is mean±\\pmSD; Beta and DKW columns are training\-conditional lower floors\.Table 8:Group\-prior shift with the hardest group’s test weight tripled while within\-group examples and score laws remain unchanged\.Table 9:Risk\-budgeted point\-prediction hybrid\. Conditional commit error uses only committed examples; fallback coverage uses only theEcE^\{c\}denominator\. The two joint error columns are measured on all inputs and sum directly to one minus overall coverage\.
### I\.3Paired encoder, path\-cost, and mechanism diagnostics

The path\-maximum diagnostic does not reconstruct terminal means from a coverage floor\. Table[10](https://arxiv.org/html/2610.20773#A9.T10)instead reports the directly measured returned\-set cardinality, the observed empty\-terminal rate before the∅→𝒴\\varnothing\\to\\mathcal\{Y\}fallback, and the paired positive\-part statistic required by Theorem[2](https://arxiv.org/html/2610.20773#Thmtheorem2)\. For a binary first\-singleton rule, ifeeis the empty\-terminal rate, then

0≤𝔼​\|𝒞ℓ^\|−𝔼​\|𝒞L\|≤2​e\.0\\leq\\mathbb\{E\}\|\\mathcal\{C\}\_\{\\widehat\{\\ell\}\}\|\-\\mathbb\{E\}\|\\mathcal\{C\}\_\{L\}\|\\leq 2e\.The six binary rows havee=0\.11%e=0\.11\\%–0\.1​–​0\.2%0\.1\\text\{\-\-\}0\.2\\%, so their returned\-prefix and path\-terminal means differ by at most0\.00220\.0022–0\.00400\.0040\. This is why the theorem evaluation uses the pointwise positive part against a separately calibrated last\-layer set rather than a difference of reconstructed means\.

Table 10:Cost diagnostic for the terminal path maximum\. Returned\-set size is a direct test\-role measurement; the binary empty\-terminal column reports the run\-level range supplied for these cells\. Positive inflation is the paired statistic bounded by Theorem[2](https://arxiv.org/html/2610.20773#Thmtheorem2),𝔼⁡\[\(\|𝒞Lpath\|−\|𝒞Llast\|\)\+\]\\mathbb\{E\}\[\(\|\\mathcal\{C\}\_\{L\}^\{\\rm path\}\|\-\|\\mathcal\{C\}\_\{L\}^\{\\rm last\}\|\)\_\{\+\}\]\. No terminal or last\-layer mean cardinality is reconstructed from a coverage lower bound\.Table 11:Paired selection\-gap inference\. Intervals are familywise95%95\\%hierarchical paired\-bootstrap intervals obtained from the maximum absolute studentized statistic across the eight declared cells\.Table 12:Paired differences relative to pathwise conformal at the primary operating point\. All point estimates subtract macro averages over the same eight cells in Table[1](https://arxiv.org/html/2610.20773#S4.T1); positive latency and set differences mean pathwise is larger\. Brackets give95%95\\%hierarchical paired\-bootstrap intervals\.Table 13:Mechanism diagnostics pooled across backbones\.D=\[maxj<L⁡rj−rL\]\+D=\[\\max\_\{j<L\}r\_\{j\}\-r\_\{L\}\]\_\{\+\}is the excess entering Theorem[2](https://arxiv.org/html/2610.20773#Thmtheorem2);Q\.95​\(D\)Q\_\{\.95\}\(D\)is its9595th percentile on true labels\.Table 14:Score and schedule ablations under common pathwise calibration and data roles\. Coverage ranges span the eight encoder task–backbone cells\.

## Appendix JAdditional Statistical Results

### J\.1Complete DKW grid

Table[15](https://arxiv.org/html/2610.20773#A10.T15)reports all nine\(n,δ\)\(n,\\delta\)cells over40004000calibration draws each\. Mean conditional coverage is0\.90010\.9001–0\.90040\.9004, and the DKW floor is met with frequency0\.99850\.9985–1\.01\.0\.

Table 15:Controlled PAC verification\. “Fraction above” is the fraction of40004000calibration draws whose conditional coverage exceeded the DKW floor\. The final column shows the corresponding floor after a twelve\-exit simultaneous correction\.
### J\.2Timing configuration

All methods are timed on identical inputs after the stopping policy is frozen\. Online time includes intermediate\-head evaluation, score updates, candidate\-set construction, cardinality checks, and fallback execution\. Offline design and calibration are excluded from online latency and reported separately\.

Table 16:Latency measurement configuration used for every paired encoder timing comparison\.
### J\.3Paired baseline and mechanism summaries

Table[11](https://arxiv.org/html/2610.20773#A9.T11)gives the cell\-wise paired selection\-gap intervals with the declared familywise bootstrap correction\. Table[12](https://arxiv.org/html/2610.20773#A9.T12)gives target\-matched differences against safe baselines\. Table[13](https://arxiv.org/html/2610.20773#A9.T13)reportsJ⁡\(X,Y\)J\(X,Y\)and candidate\-survival diagnostics, and Appendix[M](https://arxiv.org/html/2610.20773#A13)gives the exact operating frontier used in Figure[2](https://arxiv.org/html/2610.20773#S4.F2)\. These analyses use the same five training seeds as the main encoder evaluation\. SST\-2, MNLI, and BoolQ use twenty paired calibration–test role splits per seed; RTE uses the one fixed calibration/test partition and varies only the head\-training seed, with fixed\-test resampling used for interval construction\.

## Appendix KProof of the path\-maximum efficiency bound

WriteAi=Bi\+DiA\_\{i\}=B\_\{i\}\+D\_\{i\}withDi≥0D\_\{i\}\\geq 0\. Fixt≥0t\\geq 0and letIt=\{i:Di≤t\}I\_\{t\}=\\\{i:D\_\{i\}\\leq t\\\}\. Among thek\+mtk\+m\_\{t\}calibration indices with the smallestBiB\_\{i\}values, at mostmtm\_\{t\}lie outsideItI\_\{t\}\. Hence at leastkkof those indices satisfy bothBi≤B\(k\+mt\)B\_\{i\}\\leq B\_\{\(k\+m\_\{t\}\)\}andDi≤tD\_\{i\}\\leq t, which impliesAi≤B\(k\+mt\)\+tA\_\{i\}\\leq B\_\{\(k\+m\_\{t\}\)\}\+t\. ThereforeA\(k\)≤B\(k\+mt\)\+tA\_\{\(k\)\}\\leq B\_\{\(k\+m\_\{t\}\)\}\+t\. SinceAi≥BiA\_\{i\}\\geq B\_\{i\}coordinatewise, order\-statistic monotonicity givesA\(k\)≥B\(k\)A\_\{\(k\)\}\\geq B\_\{\(k\)\}, proving Equation \([4](https://arxiv.org/html/2610.20773#S3.E4)\)\.

If a candidate belongs to𝒞Lpath​\(x\)\\mathcal\{C\}\_\{L\}^\{\\rm path\}\(x\)but not𝒞Llast​\(x\)\\mathcal\{C\}\_\{L\}^\{\\rm last\}\(x\), thenrL​\(x,y\)\>B\(k\)r\_\{L\}\(x,y\)\>B\_\{\(k\)\}and

rL​\(x,y\)≤rL​\(x,y\)\+D⁡\(x,y\)=AL​\(x,y\)≤A\(k\)≤B\(k\)\+wt\.r\_\{L\}\(x,y\)\\leq r\_\{L\}\(x,y\)\+D\(x,y\)=A\_\{L\}\(x,y\)\\leq A\_\{\(k\)\}\\leq B\_\{\(k\)\}\+w\_\{t\}\.Thus every positive contribution to the cardinality difference lies in the band in Equation \([5](https://arxiv.org/html/2610.20773#S3.E5)\)\. Summing the candidatewise probabilities and applying theMyM\_\{y\}\-Lipschitz condition yields the expectation bound\. Finally,mtm\_\{t\}is a sum of i\.i\.d\. Bernoulli indicators with meanptp\_\{t\}; Hoeffding’s inequality gives the stated high\-probability bound\. IfD⁡\(X,Y\)\>0D\(X,Y\)\>0, some earlier exit strictly exceeds the terminal score, so the earliest maximizing exit satisfiesJ⁡\(X,Y\)<LJ\(X,Y\)<L, provingp0≤Pr\{J<L\}p\_\{0\}\\leq\\Pr\\\{J<L\\\}\.

## Appendix LExact encoder cell results

Table 17:Exact means underlying Figure[1](https://arxiv.org/html/2610.20773#S4.F1)\. Path coverage and returned\-set size are direct test\-role measurements\. For the six binary cells, the empty\-terminal column reports the supplied run\-level range before the∅→𝒴\\varnothing\\to\\mathcal\{Y\}fallback; MNLI is left unreported here because it is not needed for the binary identity\. RTE uses one fixed training\-split calibration role withncal=498n\_\{\\rm cal\}=498and the fixed official development test set; its repeated estimates vary over head\-training seeds rather than role assignments\.Table[18](https://arxiv.org/html/2610.20773#A12.T18)supplies the full\-depth and singleton context for the returned sets\.

Table 18:Encoder point\-prediction context for interpreting set sizes\. Full\-depth accuracy is the deepest head’s top\-1 accuracy\. For the six binary rows, singleton rate and singleton accuracy are algebraic audit summaries implied by the directly measured returned\-set size and coverage in the preceding table; the MNLI singleton columns are measured directly\. RTE uses the fixed partition described in Section[4](https://arxiv.org/html/2610.20773#S4)\.For every binary cell, the fallback rule leaves only singleton and two\-label returned sets\. Hence the direct measurements in Table[17](https://arxiv.org/html/2610.20773#A12.T17)imply

s=2−𝔼​\|𝒞ℓ^\|,as=Pr\{Y∈𝒞ℓ^\}−\(1−s\)s\.s=2\-\\mathbb\{E\}\|\\mathcal\{C\}\_\{\\widehat\{\\ell\}\}\|,\\qquad a\_\{s\}=\\frac\{\\Pr\\\{Y\\in\\mathcal\{C\}\_\{\\widehat\{\\ell\}\}\\\}\-\(1\-s\)\}\{s\}\.We use these identities only as an audit and to report the two derived singleton summaries; they do not overwrite the measured coverage or cardinality\. For example, BoolQ/BERT has measured returned\-set size1\.511\.51, sos=0\.490s=0\.490, and measured coverage0\.9010\.901then givesas=0\.798a\_\{s\}=0\.798\. BoolQ/DistilBERT analogously givess=0\.510s=0\.510andas=0\.800a\_\{s\}=0\.800\. This direction of calculation preserves the directly measured set\-size and latency results\.

## Appendix MPolicy\-uniform frontier values

Table[19](https://arxiv.org/html/2610.20773#A13.T19)reports the exact operating points used in Figure[2](https://arxiv.org/html/2610.20773#S4.F2)\.

Table 19:Exact CIFAR\-100 values underlying Figure[2](https://arxiv.org/html/2610.20773#S4.F2)for\(n,L\)=\(500,24\)\(n,L\)=\(500,24\)\. Parentheses are paired95%95\\%hierarchical\-bootstrap intervals\. All pathwise rows reuse one terminal calibration; empty\-terminal is measured before the∅→𝒴\\varnothing\\to\\mathcal\{Y\}fallback\. The three safe\-baseline set\-size/latency pairs are identical to the\(500,24\)\(500,24\)scaling\-grid row\.
## Appendix NLayer\-count and calibration\-size grid

For Bonferroni, each exit is calibrated at miscoverageα/L\\alpha/L, so its split\-conformal rank is

kBonf=⌈\(n\+1\)​\(1−αL\)⌉\.k\_\{\\rm Bonf\}=\\left\\lceil\(n\+1\)\\left\(1\-\\frac\{\\alpha\}\{L\}\\right\)\\right\\rceil\.WhenL\>α⁡\(n\+1\)L\>\\alpha\(n\+1\), this rank is exactlyn\+1n\+1and the\+∞\+\\inftyconvention makes every fixed\-exit set equal to the full label space\. Atα=0\.1\\alpha=0\.1this occurs in exactly the five displayed cells\(n,L\)=\(50,12\),\(50,24\),\(100,12\),\(100,24\),\(200,24\)\(n,L\)=\(50,12\),\(50,24\),\(100,12\),\(100,24\),\(200,24\)\. Their set size is therefore deterministically100100and their latency fraction is1\.001\.00under the common cardinality\-based stopping rule\. Table[20](https://arxiv.org/html/2610.20773#A14.T20)reports the complete paired grid\.

Table 20:CIFAR\-100 scaling grid at target coverage0\.900\.90\. Method entries are average set size±\\pmmarginal paired\-bootstrap half\-width / latency fraction±\\pmhalf\-width\. The final column is the directly paired MinSE\-minus\-pathwise set\-size difference with its paired95%95\\%interval, computed from the shared calibration subsample and test set; significance is assessed from this column, not from overlap of marginal intervals\. Bonferroni entries marked100/1\.00100/1\.00are exact rank\-n\+1n\+1degeneracies and therefore have zero sampling interval\.
## Appendix OIndependent\-depth and dataset replication

The mainLL–nngrid intentionally reuses one frozen2424\-head ViT path, so it isolates candidate\-count and calibration\-size effects but cannot by itself establish scaling with trained architectural depth\. We therefore repeat the comparison on independently trained multi\-exit paths\. The1212\-exit and2424\-exit models are trained separately; neither is obtained by discarding exits from the other\. We also repeat the experiment on ImageNet\-1k so the long\-path conclusion is not tied to CIFAR\-100\. Each row uses independent training, design, and calibration roles and reports paired95%95\\%bootstrap half\-widths over head\-training seeds and calibration subsamples\.

Table 21:Replication beyond one frozen score path\. “Path” and “MinSE” entries are average set size / latency fraction at target coverage0\.900\.90; Bonferroni is shown for the same trained path\.The replication preserves the qualitative ordering while changing both the trained path and the dataset\. It should not be read as a controlled causal estimate of architectural depth because ViT\-B/16 and ViT\-L/16 differ in capacity as well as block count\. Its role is narrower: the main retained\-exit effect is not unique to one frozen CIFAR\-100 score path\.

## Appendix PFull\-Vocabulary Decoder Results

### P\.1Decoder boundary

Table 22:Candidatewise next\-token sets over the full vocabulary\. Coverage and accuracy are mean±\\pmSD over five independent head\-training seeds\.These results establish the decoder boundary: selected coverage remains near0\.900\.90, but the candidate sets are too large for actionable early exit and the three models execute essentially their full depth\. We therefore do not use decoder wall\-clock latency to support the efficiency claim; the operational conclusion follows from the measured full\-depth execution rate and full\-vocabulary set cardinality\.

## Appendix QExperimental Protocol

The final encoder study uses four nonoverlapping data roles whenever tuning is required: model training, design, conformal calibration, and test\. Identity scores with a prespecified schedule may omit the design split\. No final\-test label is used for model selection, score transformation, threshold selection, or hyperparameter tuning\.

Training\-seed variability and calibration\-split variability are distinct\. The independent unit for model training is the seed\. Within each seed, methods share every available calibration/test role split\. SST\-2, MNLI, and BoolQ use2020paired role splits per seed\. RTE uses one fixed role assignment for all seeds; no calibration or test example is reassigned across RTE repetitions\.

Selected coverage is estimated per task and backbone, and aggregate comparisons use a hierarchical paired bootstrap that first resamples training seeds\. The second stage resamples paired role splits for SST\-2, MNLI, and BoolQ and resamples examples from the fixed RTE development test set for RTE\. Every remaining±\\pmentry is identified in its table caption\. Eight per\-cell selection\-gap statistics use the declared hierarchical bootstrap maximum\-statistic familywise correction\.

For the numerical evaluation of Theorem[2](https://arxiv.org/html/2610.20773#Thmtheorem2),t⋆t\_\{\\star\}is computed only from the final calibration excessesDiD\_\{i\}through the deterministic count conditionmt≤n−km\_\{t\}\\leq n\-k\. The local slope factor is estimated on the independent design role: the interval\[B\(k\),B\(k\)\+wt⋆\]\[B\_\{\(k\)\},B\_\{\(k\)\}\+w\_\{t\_\{\\star\}\}\]is partitioned into44equal\-width pieces, the maximum candidatewise empirical secant slope is taken within that interval, and these slopes are summed over labels\. The test role is used only for the observed cardinality inflation reported in Table[10](https://arxiv.org/html/2610.20773#A9.T10)\.

The main resource outcomes are normalized FLOPs and measured latency, not exit depth alone\. Final timing uses the common configuration in Table[16](https://arxiv.org/html/2610.20773#A10.T16)and includes exit\-head, candidate\-set, cardinality\-check, synchronization, and fallback costs\. Offline design and calibration costs are reported separately and are not counted as online savings\.

Appendix[H](https://arxiv.org/html/2610.20773#A8)gives the exact large\-LLEENN\-AVCS and MinSE valid\-selection adaptations, including the score\-to\-Dirichlet construction, running\-intersection rule, stability adjustment, selector objective, design\-only tuning grid, randomness, and the action taken whenLLchanges\. Fast\-yet\-Safe is used only in the short\-path encoder comparison and is evaluated under its native supervised\-risk target\. A method that cannot meet a10%10\\%supervised risk target because the full model itself exceeds that error is reported as infeasible or abstaining, not as an undercoverage result\.