Calibration-Preserving Pruning: Compression as a Reliability Contract

# Calibration-Preserving Pruning: Compression as a Reliability Contract
Source: [https://arxiv.org/html/2608.23744](https://arxiv.org/html/2608.23744)
Ibne Farabi ShihabAdria Binte HabibAffiliation:Department of Computer Science & Engineering, Independent University of Bangladesh, BangladeshAnuj SharmaAffiliation:Department of Civil, Construction & Environmental Engineering, Iowa State University, USA

###### Abstract

Split conformal prediction, not the pruning rule, supplies finite\-sample marginal coverage once a pruned model is fixed independently of the conformal calibration split\. We study the separate efficiency problem: can pruning preserve score geometry well enough to obtain smaller valid prediction sets? Calibration\-Preserving Pruning \(CPP\) augments a base pruning score with nonconformity\-gradient saliency and uses disjoint pruning, validation\-selection, conformal\-calibration, and test splits\. Bounded score perturbations imply bounded conformal\-quantile shifts and controlled set inflation, but do not make the generic coverage theorem CPP\-specific\. Final five\-seed Qwen2\.5\-1\.5B results at 50% sparsity show the largest gains on large\-label tasks\. On DBpedia\-14, CPP\-SparseGPT reduces mean set size from10\.110\.1to8\.68\.6while changing accuracy from0\.3470\.347to0\.3660\.366; CPP\-Wanda reduces11\.211\.2to9\.09\.0with an accuracy trade\-off from0\.3100\.310to0\.2950\.295\. Across 15 dataset–sparsity cells, CPP\-SparseGPT produces smaller sets in 13 and higher accuracy in 11\. Matched controls show that generic supervised gradients explain much of the gain: true\-label CPP is not statistically resolved from matched Wanda\+SNIP, whereas threshold\-aware candidate\-label CPP reaches7\.87\.8mean set size at explicit accuracy and offline\-compute costs\. RoBERTa\-base and Llama\-3\-8B diagnostics support transfer, but our claims remain limited to reliability\-sensitive classification\.

## 1Introduction

A compressed language model can retain its top\-label accuracy while becoming less useful to a reliability\-sensitive downstream system\. The issue is not only whether the most likely label changes\. Triage, retrieval verification, and assisted decision making often consume calibrated probabilities or conformal prediction sets\. A predictor that returns ten labels at 90% coverage is materially less informative than one returning eight labels at the same coverage, even when their top\-label accuracies are similar\.

This paper separates two claims that pruning studies can easily conflate\. First, independent split\-conformal recalibration supplies finite\-sample marginal coverage for*any*fixed pruned classifier under exchangeability\([Vovk et al\., 2005](https://arxiv.org/html/2608.23744#bib.bib7);[Angelopoulos and Bates, 2023](https://arxiv.org/html/2608.23744#bib.bib8)\)\. CPP neither creates nor improves that generic validity result\. Second, pruning determines how informative the valid predictor remains\. If compression collapses separation among plausible labels, recalibration can recover coverage only by enlarging the prediction set\. Our method\-specific target is this second quantity: conformal efficiency after valid recalibration\.

Existing one\-shot pruners optimize other objectives\. Magnitude pruning removes small weights\([Han et al\., 2015](https://arxiv.org/html/2608.23744#bib.bib24)\); Wanda combines weight magnitude with activation norms\([Sun et al\., 2024](https://arxiv.org/html/2608.23744#bib.bib21)\); SparseGPT minimizes layerwise reconstruction error\([Frantar and Alistarh, 2023](https://arxiv.org/html/2608.23744#bib.bib22)\); and gradient\-enhanced variants use supervised or regional gradients\. These are strong baselines, but none directly represents the nonconformity\-score geometry around a conformal threshold\. Temperature scaling\([Guo et al\., 2017](https://arxiv.org/html/2608.23744#bib.bib12);[Platt, 1999](https://arxiv.org/html/2608.23744#bib.bib13)\)can adjust a global confidence offset, yet cannot reconstruct class\-specific separation that pruning has removed\.

Calibration\-Preserving Pruning \(CPP\) adds a score\-sensitivity term to a base pruning importance\. For weightθj\\theta\_\{j\}, true\-label CPP uses

Ical​\(θj\)=θj2​𝔼​\[\(∂s⁡\(X,Y,θ\)∂θj\)2\],I\_\{\\mathrm\{cal\}\}\(\\theta\_\{j\}\)=\\theta\_\{j\}^\{2\}\\mathbb\{E\}\\\!\\left\[\\left\(\\frac\{\\partial s\(X,Y;\\theta\)\}\{\\partial\\theta\_\{j\}\}\\right\)^\{2\}\\right\],wheressis the nonconformity score\. The factor measures first\-order displacement caused by setting that coordinate to zero\. Because true\-label sensitivity is only a proxy for a prediction set containing many candidate labels, we also evaluate top\-33, threshold\-aware, and all\-label constructions\. Every gradient is evaluated on the pruning split\. A validation\-calibration half\-split supplies only the preliminary threshold needed by the threshold\-aware construction and provisional quantiles for model selection\. The final conformal split remains untouched until the selected sparse model is frozen\.

The response\-stage controls materially narrow the claim\. Gradient\-only, Wanda\+\+, SNIP\-style, and Wanda⊕\\oplusSNIP baselines show that generic supervised\-gradient information explains a substantial part of CPP’s improvement\. True\-label CPP\-Wanda reduces DBpedia\-14 set size from9\.29\.2to9\.09\.0relative to matched Wanda⊕\\oplusSNIP, but with lower accuracy and overlapping intervals; we do not claim that difference is statistically resolved\. Threshold\-aware CPP gives a larger efficiency improvement, reaching8\.48\.4for CPP\-Wanda and7\.87\.8for CPP\-SparseGPT, with explicit accuracy and gradient\-compute costs\. The evidence supports objective\-specific efficiency effects, not uniform superiority over gradient pruning\.

Our contribution has four parts\. We first formulate reliability\-preserving compression as a constrained comparison: coverage is supplied by independent conformal calibration, average set size is the primary efficiency outcome, accuracy measures utility, and ECE is diagnostic\. We then define true\-label and candidate\-label CPP with an exact validation rule and disjoint data roles\. Third, we retain only the theory that matches those roles: the coverage theorem is the standard pruning\-rule\-agnostic split\-conformal result, while separate perturbation results connect score movement to quantile and set\-size inflation\. Finally, we report one authoritative result path with final five\-seed Qwen comparisons, matched gradient controls, candidate\-label ablations, offline cost, independent split redraws, RoBERTa\-base transfer, and a scoped Llama\-3\-8B diagnostic\. We study fixed\-label classification using complete\-sequence verbalizer scores, not free\-form conformal generation\.

## 2Related Work

#### Post\-training pruning\.

One\-shot language\-model pruning includes magnitude methods\([Han et al\., 2015](https://arxiv.org/html/2608.23744#bib.bib24)\), SparseGPT\([Frantar and Alistarh, 2023](https://arxiv.org/html/2608.23744#bib.bib22)\), Wanda\([Sun et al\., 2024](https://arxiv.org/html/2608.23744#bib.bib21)\), structured LLM\-Pruner\([Ma et al\., 2023](https://arxiv.org/html/2608.23744#bib.bib23)\), and broader surveys and scaling analyses\([Shi et al\., 2024](https://arxiv.org/html/2608.23744#bib.bib4);[Wang et al\., 2024](https://arxiv.org/html/2608.23744#bib.bib5);[Frantar et al\., 2025](https://arxiv.org/html/2608.23744#bib.bib25)\)\. Structured pruning followed by continued pretraining addresses a different compute regime from the no\-retraining setting studied here\. Parameter\-importance and retention methods in continual learning provide related gradient signals\([Mallya and Lazebnik, 2018](https://arxiv.org/html/2608.23744#bib.bib19);[Li and Hoiem, 2017](https://arxiv.org/html/2608.23744#bib.bib18);[Kirkpatrick et al\., 2017](https://arxiv.org/html/2608.23744#bib.bib1);[Zenke et al\., 2017](https://arxiv.org/html/2608.23744#bib.bib3);[Wang et al\., 2022](https://arxiv.org/html/2608.23744#bib.bib20);[Wang et al\., 2023](https://arxiv.org/html/2608.23744#bib.bib2)\)\. Our matched SNIP\-style and Wanda⊕\\oplusSNIP controls are therefore essential: they test whether CPP adds more than generic supervised\-gradient information\.

#### Calibration and conformal prediction\.

Temperature and Platt scaling correct confidence after training\([Guo et al\., 2017](https://arxiv.org/html/2608.23744#bib.bib12);[Platt, 1999](https://arxiv.org/html/2608.23744#bib.bib13)\), while sparse\-subnetwork studies report calibration as a pruning byproduct\([Kuhn et al\., 2026](https://arxiv.org/html/2608.23744#bib.bib27);[Zhou et al\., 2022](https://arxiv.org/html/2608.23744#bib.bib28)\)\. Neural\-network pruning under inductive conformal prediction is the closest direct predecessor\([Zhao and Bellotti, 2022](https://arxiv.org/html/2608.23744#bib.bib26)\)\. CPP differs by targeting post\-training language\-model pruning and by making validation selection independent of final conformal calibration\. Split conformal prediction provides distribution\-free marginal coverage under exchangeability\([Vovk et al\., 2005](https://arxiv.org/html/2608.23744#bib.bib7);[Angelopoulos and Bates, 2023](https://arxiv.org/html/2608.23744#bib.bib8)\); language\-model applications include generation, factuality, and information\-relative certificates\([Quach et al\., 2024](https://arxiv.org/html/2608.23744#bib.bib9);[Mohri and Hashimoto, 2024](https://arxiv.org/html/2608.23744#bib.bib10);[Sanjeda Akter and Sharma, 2026](https://arxiv.org/html/2608.23744#bib.bib11)\)\. Other work optimizes conformal score functions or answer\-choice sets rather than model weights\([Vishwakarma et al\., 2025](https://arxiv.org/html/2608.23744#bib.bib29)\)\. Shift\-aware and risk\-controlling extensions\([Park et al\., 2020](https://arxiv.org/html/2608.23744#bib.bib14);[Gibbs and Candes, 2021](https://arxiv.org/html/2608.23744#bib.bib15);[Bates et al\., 2021](https://arxiv.org/html/2608.23744#bib.bib17)\)address different assumptions\. Learn\-Then\-Test\([Angelopoulos et al\., 2022](https://arxiv.org/html/2608.23744#bib.bib16)\)is a valid optional selection tool, but it was not used for any result reported here; Section[5\.2](https://arxiv.org/html/2608.23744#S5.SS2)gives the exact implemented rule\.

## 3The Reliability Contract Under Compression

Letfθ:𝒳→Δ⁡\(𝒴\)f\_\{\\theta\}:\\mathcal\{X\}\\rightarrow\\Delta\(\\mathcal\{Y\}\)be a classifier with a finite label space𝒴\\mathcal\{Y\},\|𝒴\|=K\|\\mathcal\{Y\}\|=K\. For decoder models, each labelyyhas a fixed verbalizerv⁡\(y\)=\(v1,…,vTy\)v\(y\)=\(v\_\{1\},\\ldots,v\_\{T\_\{y\}\}\)\. We score the entire teacher\-forced sequence rather than applying conformal prediction token by token:

log⁡pθ​\(y∣x\)=∑t=1Tylog⁡pθ​\(vt∣x,v<t\)\.\\log p\_\{\\theta\}\(y\\mid x\)=\\sum\_\{t=1\}^\{T\_\{y\}\}\\log p\_\{\\theta\}\(v\_\{t\}\\mid x,v\_\{<t\}\)\.The manifest fixes whether this sum is length\-normalized\. A common nonconformity score is

s⁡\(x,y,θ\)=−log⁡pθ​\(y∣x\),s\(x,y;\\theta\)=\-\\log p\_\{\\theta\}\(y\\mid x\),\(1\)where smaller values indicate greater compatibility\.

The final protocol uses four disjoint roles\.𝒟prune\\mathcal\{D\}\_\{\\mathrm\{prune\}\}constructs gradients and pruning statistics\. The 1,024\-example validation split is divided deterministically into 512\-example𝒟val​\-​cal\\mathcal\{D\}\_\{\\mathrm\{val\\text\{\-\}cal\}\}and𝒟val​\-​eval\\mathcal\{D\}\_\{\\mathrm\{val\\text\{\-\}eval\}\}subsets\. The former supplies provisional quantiles and, only for threshold\-aware CPP, the dense\-model threshold band\. The latter selectsλ\\lambda\. The independent conformal split𝒟conf=\{\(Xi,Yi\)\}i=1n\\mathcal\{D\}\_\{\\mathrm\{conf\}\}=\\\{\(X\_\{i\},Y\_\{i\}\)\\\}\_\{i=1\}^\{n\}is accessed only after the sparse model is frozen, and𝒟test\\mathcal\{D\}\_\{\\mathrm\{test\}\}is used only for evaluation\.

For a fixedθ\\thetaand target miscoverageα\\alpha, define

k=⌈\(n\+1\)​\(1−α\)⌉\.k=\\left\\lceil\(n\+1\)\(1\-\\alpha\)\\right\\rceil\.\(2\)Whenk≤nk\\leq n,q^α​\(θ\)\\hat\{q\}\_\{\\alpha\}\(\\theta\)is thekk\-th smallest calibration score; whenk\>nk\>n, we set it to\+∞\+\\infty\. The prediction set is

Cα​\(x,θ\)=\{y∈𝒴:s⁡\(x,y,θ\)≤q^α​\(θ\)\}\.C\_\{\\alpha\}\(x;\\theta\)=\\\{y\\in\\mathcal\{Y\}:s\(x,y;\\theta\)\\leq\\hat\{q\}\_\{\\alpha\}\(\\theta\)\\\}\.\(3\)
The reliability contract has a strict hierarchy\. Marginal coverage,ℙ\{Y∈Cα\(X;θ\)\}\\mathbb\{P\}\\\{Y\\in C\_\{\\alpha\}\(X;\\theta\)\\\}, is the validity constraint\. Mean set size,C¯P​\(θ\)=𝔼X​\|Cα​\(X,θ\)\|\\bar\{C\}\_\{P\}\(\\theta\)=\\mathbb\{E\}\_\{X\}\|C\_\{\\alpha\}\(X;\\theta\)\|, is the primary conformal\-efficiency outcome\. Accuracy measures task utility, and ECE is a complementary calibration diagnostic\. For a pruned modelθ′=𝒫⁡\(θ\)\\theta^\{\\prime\}=\\mathcal\{P\}\(\\theta\), contract preservation with efficiency slackεsize\\varepsilon\_\{\\mathrm\{size\}\}means

ℙ\{Y∈Cα\(X;θ′\)\}\\displaystyle\\mathbb\{P\}\\\{Y\\in C\_\{\\alpha\}\(X;\\theta^\{\\prime\}\)\\\}≥1−α,\\displaystyle\\geq 1\-\\alpha,\(4\)C¯P​\(θ′\)\\displaystyle\\bar\{C\}\_\{P\}\(\\theta^\{\\prime\}\)≤C¯P​\(θ\)\+εsize\.\\displaystyle\\leq\\bar\{C\}\_\{P\}\(\\theta\)\+\\varepsilon\_\{\\mathrm\{size\}\}\.Independent split conformal supplies the first line for any fixedθ′\\theta^\{\\prime\}\. CPP targets the second while the validation guardrail limits accuracy loss\.

## 4Why Recalibration Alone Is Not Enough

Prune\-then\-recalibrate is valid when the final conformal split is independent, but validity does not imply informativeness\. Top\-1 accuracy depends on the largest class score\. Set size depends on all candidate\-label scores around a calibration threshold\. A scalar temperature can move confidence globally\([Guo et al\., 2017](https://arxiv.org/html/2608.23744#bib.bib12);[Li and Hoiem, 2017](https://arxiv.org/html/2608.23744#bib.bib18)\), but cannot restore label\-specific separation that the sparse model no longer represents\.

Reliability degradation can also vary nonmonotonically with sparsity\([Junhao Zheng and Ma, 2025](https://arxiv.org/html/2608.23744#bib.bib6)\)\. We therefore measure every reported sparsity directly and do not interpolate unobserved cells\.

The provenance audit exposed a useful feasibility check\. For any prediction setC⊆𝒴C\\subseteq\\mathcal\{Y\},

𝔼​\|C\|≤ℙ⁡\(Y∈C\)\+\(K−1\)\.\\mathbb\{E\}\|C\|\\leq\\mathbb\{P\}\(Y\\in C\)\+\(K\-1\)\.A method returning allKKlabels on every example must therefore have coverage one\. Several submitted pilot pairs combined\|C\|=K\|C\|=Kwith coverage below one and could not come from a common evaluation path\. We retire those entries and use only final\-manifest results\. Figure[1](https://arxiv.org/html/2608.23744#S4.F1)shows the corrected Qwen2\.5\-1\.5B/DBpedia\-14 comparison at 50% sparsity\. All four pruned predictors have empirical coverage between0\.9010\.901and0\.9020\.902; the difference is set\-size efficiency and, separately, accuracy\.

WandaCPP\-WSparseGPTCPP\-SG0055101011\.211\.29910\.110\.18\.68\.6Average set sizeFigure 1:Final five\-seed Qwen2\.5\-1\.5B/DBpedia\-14 set size at 50% sparsity\. Coverage is matched at0\.9010\.901–0\.9020\.902\. CPP\-SparseGPT also improves accuracy; CPP\-Wanda trades 1\.5 accuracy points for smaller sets\.
## 5Calibration\-Preserving Pruning

CPP modifies a base importance rather than replacing the base pruner\. LetIacc​\(θj\)I\_\{\\mathrm\{acc\}\}\(\\theta\_\{j\}\)denote the coordinate importance supplied by magnitude pruning\([Han et al\., 2015](https://arxiv.org/html/2608.23744#bib.bib24)\), Wanda\([Sun et al\., 2024](https://arxiv.org/html/2608.23744#bib.bib21)\), or the SparseGPT reconstruction procedure\([Frantar and Alistarh, 2023](https://arxiv.org/html/2608.23744#bib.bib22)\)\. For a labeled pruning split, true\-label CPP first computes

Gcal​\(θj\)=1\|𝒟prune\|​∑\(x,y\)∈𝒟prune\(∂s⁡\(x,y,θ\)∂θj\)2\.G\_\{\\mathrm\{cal\}\}\(\\theta\_\{j\}\)=\\frac\{1\}\{\|\\mathcal\{D\}\_\{\\mathrm\{prune\}\}\|\}\\sum\_\{\(x,y\)\\in\\mathcal\{D\}\_\{\\mathrm\{prune\}\}\}\\left\(\\frac\{\\partial s\(x,y;\\theta\)\}\{\\partial\\theta\_\{j\}\}\\right\)^\{2\}\.\(5\)Whens=−log⁡pθ​\(y∣x\)s=\-\\log p\_\{\\theta\}\(y\\mid x\), this is a diagonal empirical\-Fisher\-like factor; for generalss, we use the narrower term nonconformity\-gradient sensitivity\([Kirkpatrick et al\., 2017](https://arxiv.org/html/2608.23744#bib.bib1);[Zenke et al\., 2017](https://arxiv.org/html/2608.23744#bib.bib3)\)\. Because pruning changesθj\\theta\_\{j\}to zero, the relevant first\-order displacement includes the weight:

Ical​\(θj\)\\displaystyle I\_\{\\mathrm\{cal\}\}\(\\theta\_\{j\}\)=θj2​Gcal​\(θj\)\\displaystyle=\\theta\_\{j\}^\{2\}G\_\{\\mathrm\{cal\}\}\(\\theta\_\{j\}\)\(6\)=1\|𝒟prune\|​∑\(x,y\)∈𝒟prune\(θj​∂s⁡\(x,y,θ\)∂θj\)2\.\\displaystyle=\\frac\{1\}\{\|\\mathcal\{D\}\_\{\\mathrm\{prune\}\}\|\}\\sum\_\{\(x,y\)\\in\\mathcal\{D\}\_\{\\mathrm\{prune\}\}\}\\left\(\\theta\_\{j\}\\frac\{\\partial s\(x,y;\\theta\)\}\{\\partial\\theta\_\{j\}\}\\right\)^\{2\}\.Per\-example gradients are required\. Squaring the gradient of a batch mean would introduce cross\-example terms and would not estimate Eq\.[6](https://arxiv.org/html/2608.23744#S5.E6)\.

Within each eligible module, CPP normalizes the base and calibration scores to unitℓ2\\ell\_\{2\}norm and uses

ICPP​\(θj\)=\(1−λ\)​I~acc​\(θj\)\+λ​I~cal​\(θj\)\.I\_\{\\mathrm\{CPP\}\}\(\\theta\_\{j\}\)=\(1\-\\lambda\)\\widetilde\{I\}\_\{\\mathrm\{acc\}\}\(\\theta\_\{j\}\)\+\\lambda\\widetilde\{I\}\_\{\\mathrm\{cal\}\}\(\\theta\_\{j\}\)\.\(7\)We prune dense attention and MLP matrices under the base pruner’s allocation rule\. Norm parameters, embeddings, tied output heads, positional mechanisms, and KV\-cache state remain dense\.

### 5\.1Candidate\-label Saliency

True\-label saliency is inexpensive but only indirectly represents set\-size inflation\. We therefore test three candidate\-aware constructions\.*Top\-33*retains the three labels with smallest dense\-model nonconformity scores for each pruning example\.*All\-label*averages over the complete label space\.*Threshold\-aware*focuses gradients near a preliminary dense\-model conformal boundary\. It estimatesq~\\tilde\{q\}from true\-label scores on𝒟val​\-​cal\\mathcal\{D\}\_\{\\mathrm\{val\\text\{\-\}cal\}\}, computes all candidate\-label distances\|s⁡\(x,y,θ\)−q~\|\|s\(x,y;\\theta\)\-\\tilde\{q\}\|on that same half\-split, and fixesbbto their 20th percentile\. On each pruning example, it retains labels withinbb, keeps the nearest label if the set is empty, caps the set at the eight nearest labels, and breaks boundary ties by fixed verbalizer order\. Gradients are still evaluated only on𝒟prune\\mathcal\{D\}\_\{\\mathrm\{prune\}\}\. This construction retains 2\.7 labels per pruning example on average\. The preliminaryq~\\tilde\{q\}is used only to construct saliency and is distinct from every candidate model’s provisional validation quantile and the final conformal quantile\.

### 5\.2Exact Validation Selection

For each base pruner, dataset, sparsity, and seed, the implemented grid isΛ=\{0,0\.1,0\.3,0\.5\}\\Lambda=\\\{0,0\.1,0\.3,0\.5\\\}\. Each candidate receives a provisional nominal\-90% quantile from the 512\-example𝒟val​\-​cal\\mathcal\{D\}\_\{\\mathrm\{val\\text\{\-\}cal\}\}split and is evaluated once on the disjoint 512\-example𝒟val​\-​eval\\mathcal\{D\}\_\{\\mathrm\{val\\text\{\-\}eval\}\}split\. A candidate is eligible when its accuracy is within two percentage points of the correspondingλ=0\\lambda=0model and empirical coverage is at least0\.880\.88\. Among eligible candidates, selection minimizes mean set size\. Ties within0\.020\.02set\-size units are broken by higher accuracy and then smallerλ\\lambda\. If no nonzero candidate is eligible, the base modelλ=0\\lambda=0is returned\. The selected model is frozen before𝒟conf\\mathcal\{D\}\_\{\\mathrm\{conf\}\}is accessed\. The broader\{0,0\.1,0\.3,0\.5,0\.7,1\.0\}\\\{0,0\.1,0\.3,0\.5,0\.7,1\.0\\\}sweep in Table[4](https://arxiv.org/html/2608.23744#S7.T4)is diagnostic only, and Learn\-Then\-Test was not used\.

Algorithm 1CPP with independent final calibration1:Dense

θ\\theta; disjoint

𝒟prune,𝒟val​\-​cal,𝒟val​\-​eval,𝒟conf\\mathcal\{D\}\_\{\\mathrm\{prune\}\},\\mathcal\{D\}\_\{\\mathrm\{val\\text\{\-\}cal\}\},\\mathcal\{D\}\_\{\\mathrm\{val\\text\{\-\}eval\}\},\\mathcal\{D\}\_\{\\mathrm\{conf\}\}; sparsity; base pruner

2:Construct true\-label or candidate\-label

IcalI\_\{\\mathrm\{cal\}\}without

𝒟conf\\mathcal\{D\}\_\{\\mathrm\{conf\}\}
3:for

λ∈\{0,0\.1,0\.3,0\.5\}\\lambda\\in\\\{0,0\.1,0\.3,0\.5\\\}do

4:Combine scores by Eq\.[7](https://arxiv.org/html/2608.23744#S5.E7)and prune

5:Fit a provisional quantile on

𝒟val​\-​cal\\mathcal\{D\}\_\{\\mathrm\{val\\text\{\-\}cal\}\}
6:Measure accuracy, coverage, and set size on

𝒟val​\-​eval\\mathcal\{D\}\_\{\\mathrm\{val\\text\{\-\}eval\}\}
7:endfor

8:Apply the accuracy/coverage guardrails, set\-size objective, tie rules, and

λ=0\\lambda=0fallback

9:Freeze

θ′\\theta^\{\\prime\}; compute

q^α​\(θ′\)\\hat\{q\}\_\{\\alpha\}\(\\theta^\{\\prime\}\)by Eq\.[2](https://arxiv.org/html/2608.23744#S3.E2)only on

𝒟conf\\mathcal\{D\}\_\{\\mathrm\{conf\}\}
10:return

θ′\\theta^\{\\prime\}and

q^α​\(θ′\)\\hat\{q\}\_\{\\alpha\}\(\\theta^\{\\prime\}\)

## 6Theory

The formal claims mirror the experimental hierarchy\. The first theorem records when validity survives data\-dependent pruning and model selection\. It is the standard split\-conformal result, not a CPP\-specific coverage theorem\. The remaining statements explain the efficiency objective: small score movement limits quantile and prediction\-set movement, and CPP is a diagonal first\-order proxy for that score movement\.

###### Assumption 1\(Exchangeability and split independence\)\.

The examples in𝒟conf\\mathcal\{D\}\_\{\\mathrm\{conf\}\}and the future test example are exchangeable\. The final pruned modelθ′\\theta^\{\\prime\}, including its mask, saliency construction, sparsity,λ\\lambda, prompt, verbalizers, and temperature, is measurable with respect to data independent of𝒟conf\\mathcal\{D\}\_\{\\mathrm\{conf\}\}and the test example\.

###### Theorem 2\(Split\-conformal coverage after pruning\)\.

Under Assumption[1](https://arxiv.org/html/2608.23744#Thmtheorem1), letθ′\\theta^\{\\prime\}be any model fixed before𝒟conf\\mathcal\{D\}\_\{\\mathrm\{conf\}\}is accessed\. Then the set in Eq\.[3](https://arxiv.org/html/2608.23744#S3.E3)satisfies

ℙ\{Yn\+1∈Cα\(Xn\+1;θ′\)\}≥1−α\.\\mathbb\{P\}\\\!\\left\\\{Y\_\{n\+1\}\\in C\_\{\\alpha\}\(X\_\{n\+1\};\\theta^\{\\prime\}\)\\right\\\}\\geq 1\-\\alpha\.\(8\)If scores are almost surely distinct andk≤nk\\leq n, coverage is less than1−α\+1/\(n\+1\)1\-\\alpha\+1/\(n\+1\)\.

The proof conditions on every pruning and validation decision and then applies the usual exchangeable\-rank argument\. It would hold for Wanda, SparseGPT, random pruning, or any other fixed model\. Its CPP\-relevant content is the independence requirement\. Finite test\-set coverage can fluctuate around population coverage; Appendix[A](https://arxiv.org/html/2608.23744#A1)records the corresponding concentration statement\.

For efficiency, define the maximum score change on a domain𝒵⊆𝒳×𝒴\\mathcal\{Z\}\\subseteq\\mathcal\{X\}\\times\\mathcal\{Y\}:

ϵ𝒵​\(θ,θ′\)=sup\(x,y\)∈𝒵\|s⁡\(x,y,θ′\)−s⁡\(x,y,θ\)\|\.\\epsilon\_\{\\mathcal\{Z\}\}\(\\theta,\\theta^\{\\prime\}\)=\\sup\_\{\(x,y\)\\in\\mathcal\{Z\}\}\|s\(x,y;\\theta^\{\\prime\}\)\-s\(x,y;\\theta\)\|\.\(9\)
###### Lemma 3\(Order\-statistic stability\)\.

Ifmaxi⁡\|ai−bi\|≤ϵ\\max\_\{i\}\|a\_\{i\}\-b\_\{i\}\|\\leq\\epsilon, then theirkk\-th order statistics obey\|a\(k\)−b\(k\)\|≤ϵ\|a\_\{\(k\)\}\-b\_\{\(k\)\}\|\\leq\\epsilon\.

###### Theorem 4\(Set\-size stability under bounded score perturbation\)\.

Fix one conformal calibration sample\. Suppose𝒴\\mathcal\{Y\}is finite and

\|s⁡\(x,y,θ′\)−s⁡\(x,y,θ\)\|≤ϵ\|s\(x,y;\\theta^\{\\prime\}\)\-s\(x,y;\\theta\)\|\\leq\\epsilon\(10\)for every calibration pair and every test pair\(x,y\)\(x,y\)\. Then

Cα​\(x,θ′\)⊆\{y:s⁡\(x,y,θ\)≤q^α​\(θ\)\+2​ϵ\}\.C\_\{\\alpha\}\(x;\\theta^\{\\prime\}\)\\subseteq\\\{y:s\(x,y;\\theta\)\\leq\\hat\{q\}\_\{\\alpha\}\(\\theta\)\+2\\epsilon\\\}\.\(11\)If eachs⁡\(X,y,θ\)s\(X,y;\\theta\)has density at mostMMbetweenq^α​\(θ\)\\hat\{q\}\_\{\\alpha\}\(\\theta\)andq^α​\(θ\)\+2​ϵ\\hat\{q\}\_\{\\alpha\}\(\\theta\)\+2\\epsilon, then

𝔼X\|Cα​\(X,θ′\)\|≤𝔼X​\|Cα​\(X,θ\)\|\+2​M​\|𝒴\|​ϵ\.\\mathbb\{E\}\_\{X\}\|C\_\{\\alpha\}\(X;\\theta^\{\\prime\}\)\|\\leq\\mathbb\{E\}\_\{X\}\|C\_\{\\alpha\}\(X;\\theta\)\|\+2M\|\\mathcal\{Y\}\|\\epsilon\.\(12\)

This is a sufficient stability condition, not a certificate that CPP will produce small sets\. It also clarifies why candidate labels near the threshold matter: only labels entering the2​ϵ2\\epsilonboundary band can inflate the pruned set under the stated uniform bound\.

###### Assumption 5\(Local smoothness along the pruning path\)\.

For every evaluated examplezz,s⁡\(z,θ\)s\(z;\\theta\)is twice differentiable on the segment between dense and pruned parameters, with Hessian operator norm at mostHzH\_\{z\}; letH=supzHz<∞H=\\sup\_\{z\}H\_\{z\}<\\infty\.

###### Proposition 6\(First\-order score displacement\)\.

LetP⁡\(𝗆\)P\(\\mathsf\{m\}\)be the coordinates removed by mask𝗆\\mathsf\{m\}, and define

Ical,𝒟0​\(θj\)=θj2\|𝒟0\|​∑z∈𝒟0\(∂s⁡\(z,θ\)∂θj\)2\.I\_\{\\mathrm\{cal\},\\mathcal\{D\}\_\{0\}\}\(\\theta\_\{j\}\)=\\frac\{\\theta\_\{j\}^\{2\}\}\{\|\\mathcal\{D\}\_\{0\}\|\}\\sum\_\{z\\in\\mathcal\{D\}\_\{0\}\}\\left\(\\frac\{\\partial s\(z;\\theta\)\}\{\\partial\\theta\_\{j\}\}\\right\)^\{2\}\.Under Assumption[5](https://arxiv.org/html/2608.23744#Thmtheorem5), define the masked linear term

a𝗆​\(z\)=∑j∈P⁡\(𝗆\)θj​∂s⁡\(z,θ\)∂θj\.a\_\{\\mathsf\{m\}\}\(z\)=\\sum\_\{j\\in P\(\\mathsf\{m\}\)\}\\theta\_\{j\}\\frac\{\\partial s\(z;\\theta\)\}\{\\partial\\theta\_\{j\}\}\.Also write the mean score displacement as

Δ¯𝒟0​\(𝗆\)=1\|𝒟0\|​∑z∈𝒟0\|s⁡\(z,θ𝗆\)−s⁡\(z,θ\)\|\.\\bar\{\\Delta\}\_\{\\mathcal\{D\}\_\{0\}\}\(\\mathsf\{m\}\)=\\frac\{1\}\{\|\\mathcal\{D\}\_\{0\}\|\}\\sum\_\{z\\in\\mathcal\{D\}\_\{0\}\}\|s\(z;\\theta\_\{\\mathsf\{m\}\}\)\-s\(z;\\theta\)\|\.Then

Δ¯𝒟0​\(𝗆\)\\displaystyle\\bar\{\\Delta\}\_\{\\mathcal\{D\}\_\{0\}\}\(\\mathsf\{m\}\)≤\[1\|𝒟0\|​∑z∈𝒟0a𝗆​\(z\)2\]1/2\\displaystyle\\leq\\left\[\\frac\{1\}\{\|\\mathcal\{D\}\_\{0\}\|\}\\sum\_\{z\\in\\mathcal\{D\}\_\{0\}\}a\_\{\\mathsf\{m\}\}\(z\)^\{2\}\\right\]^\{1/2\}\+H2​‖θ𝗆−θ‖22\.\\displaystyle\\quad\+\\frac\{H\}\{2\}\\\|\\theta\_\{\\mathsf\{m\}\}\-\\theta\\\|\_\{2\}^\{2\}\.\(13\)Ifm=\|P⁡\(𝗆\)\|m=\|P\(\\mathsf\{m\}\)\|, the first term is at most

\[m​∑j∈P⁡\(𝗆\)Ical,𝒟0​\(θj\)\]1/2\.\\left\[m\\sum\_\{j\\in P\(\\mathsf\{m\}\)\}I\_\{\\mathrm\{cal\},\\mathcal\{D\}\_\{0\}\}\(\\theta\_\{j\}\)\\right\]^\{1/2\}\.For𝒟0=𝒟prune\\mathcal\{D\}\_\{0\}=\\mathcal\{D\}\_\{\\mathrm\{prune\}\}, this is the saliency in Eq\.[6](https://arxiv.org/html/2608.23744#S5.E6)\.

The diagonal bound ignores cross\-coordinate cancellation and the Hessian remainder can dominate at high sparsity\. It justifies CPP as a local proxy, not as an optimal pruning theorem\. Proofs and an optional, unused Learn\-Then\-Test alternative appear in Appendix[A](https://arxiv.org/html/2608.23744#A1)\.

## 7Experiments

#### Final protocol\.

The authoritative study uses Qwen2\.5\-1\.5B on AG News, TREC, DBpedia\-14, Banking77, and CLINC150 at 30%, 50%, and 70% unstructured sparsity\. Each dataset–sparsity–method cell uses five seeds\. Dedicated matched\-gradient, candidate\-label, RoBERTa\-base, and Llama\-3\-8B controls use three seeds where stated\. Fixed label verbalizers are scored as complete teacher\-forced sequences\. The scope is classification; free\-form conformal generation is not evaluated\. Appendix[B](https://arxiv.org/html/2608.23744#A2)gives the full protocol, and Appendix[D](https://arxiv.org/html/2608.23744#A4)reports the scoped Llama diagnostic\.

Unless a dataset is too small, each partition has 1,024 pruning examples, 1,024 validation examples divided 512/512, 1,024 final conformal examples, and at least 5,000 test examples\. The manifest records exact exceptions\. Saliency,λ\\lambda, sparsity, prompts, and temperature never use𝒟conf\\mathcal\{D\}\_\{\\mathrm\{conf\}\}\. All final tables report mean empirical coverage after split\-conformal calibration, mean set size, and accuracy with 95% intervals across the stated seeds\. Coverage is the constraint, set size the primary outcome at matched coverage, accuracy the utility outcome, and ECE a diagnostic\.

A provenance audit found that several submitted pilot entries mixed result paths\. In particular, full sets with coverage below one violated the feasibility inequality in Section[4](https://arxiv.org/html/2608.23744#S4)\. Those entries, the projected Banking77/CLINC150 cells, and the old 10\-cell exclusion are retired\. Tables and figures below use one final\-manifest family\. The corrected small\-label values and all 15 cell\-level signs appear in Appendix[C](https://arxiv.org/html/2608.23744#A3)\.

#### Large\-label results\.

Table[1](https://arxiv.org/html/2608.23744#S7.T1)reports the primary 50% sparsity comparison\. Empirical coverage lies between0\.9000\.900and0\.9020\.902in all displayed cells\. Set\-size reductions relative to the corresponding base pruner range from 10\.4% to 19\.6%\. CPP\-SparseGPT improves accuracy in all three datasets\. CPP\-Wanda improves accuracy on Banking77 and CLINC150, but loses 1\.5 points on DBpedia\-14\. We report that trade\-off directly rather than collapsing the metrics into one score\.

Table 1:Final five\-seed Qwen2\.5\-1\.5B results at 50% sparsity \(mean±\\pm95% CI\)\. Coverage is the validity constraint; compare set size only at matched coverage\.Across the complete five\-dataset, three\-sparsity Qwen grid, CPP\-Wanda has higher mean accuracy in 12/15 cells and smaller mean sets in 14/15\. CPP\-SparseGPT has higher accuracy in 11/15 and smaller sets in 13/15\. These are directions of five\-seed means, not significance claims\. The exact sign table is Table[13](https://arxiv.org/html/2608.23744#A3.T13); Table[11](https://arxiv.org/html/2608.23744#A3.T11)reports the totals\.

#### What generic gradients explain\.

The matched control in Table[2](https://arxiv.org/html/2608.23744#S7.T2)uses Qwen2\.5\-1\.5B/DBpedia\-14 at 50% sparsity, three seeds, identical per\-example gradient access, the sameλ\\lambdagrid, and the same validation rule\. Gradient\-only ranks by𝔼i​\[gi​j2\]\\mathbb\{E\}\_\{i\}\[g\_\{ij\}^\{2\}\], SNIP\-style by𝔼i​\[\|θj​gi​j\|\]\\mathbb\{E\}\_\{i\}\[\|\\theta\_\{j\}g\_\{ij\}\|\], and Wanda⊕\\oplusSNIP combines Wanda with the latter\. Generic gradients recover much of the efficiency gap\. True\-label CPP\-Wanda reaches9\.09\.0rather than9\.29\.2, but has lower accuracy and overlapping intervals; the data do not resolve that difference\. Threshold\-aware CPP\-Wanda produces the smallest sets in this control,8\.48\.4, while retaining less accuracy than Wanda⊕\\oplusSNIP\. CPP is therefore objective\-specific, not uniformly better than gradient pruning\.

Table 2:Matched gradient\-information controls \(three seeds, 95% CI\)\.
#### Candidate\-label objective\.

Table[3](https://arxiv.org/html/2608.23744#S7.T3)isolates the saliency construction for CPP\-SparseGPT\. Top\-33and threshold\-aware gradients improve both accuracy and set size relative to true\-label CPP\. All\-label gradients reduce set size only0\.10\.1beyond threshold\-aware saliency while increasing saliency\-construction time from2\.7×2\.7\\timesto8\.9×8\.9\\times\. The relative cost excludes the unchanged base\-pruner and final\-calibration stages\.

Table 3:Candidate\-label CPP\-SparseGPT on DBpedia\-14 at 50% sparsity\.Table 4:Selection grid versus the broader diagnostic sweep\. Learn\-Then\-Test is not used\.
#### Transfer, robustness, and cost\.

A three\-seed RoBERTa\-base/DBpedia\-14 experiment at 50% sparsity \(Table[7](https://arxiv.org/html/2608.23744#S7.T7)\) reduces set size from2\.31±0\.142\.31\\pm 0\.14for magnitude pruning to1\.88±0\.111\.88\\pm 0\.11for CPP\-Magnitude and raises accuracy from0\.934±0\.0060\.934\\pm 0\.006to0\.946±0\.0050\.946\\pm 0\.005, at matched0\.9000\.900–0\.9010\.901coverage\. This supports encoder transfer but is one model, one task, and one base pruner\. The Llama\-3\-8B diagnostic likewise remains scoped: CPP\-SparseGPT changes WikiText\-2 perplexity from7\.257\.25to7\.287\.28, DBpedia accuracy from0\.610\.61to0\.650\.65, and set size from8\.98\.9to7\.27\.2, at matched coverage\.

Five independently redrawn four\-way Qwen DBpedia\-14 partitions give10\.27±0\.4110\.27\\pm 0\.41versus8\.79±0\.348\.79\\pm 0\.34set size for SparseGPT and CPP\-SparseGPT, with0\.9004±0\.00680\.9004\\pm 0\.0068versus0\.9026±0\.00570\.9026\\pm 0\.0057coverage \(Table[6](https://arxiv.org/html/2608.23744#S7.T6)\)\. Nested calibration\- and pruning\-size diagnostics show the same direction; they are point\-estimate sensitivity sweeps on one fixed partition, not additional variance estimates\. Appendix[C](https://arxiv.org/html/2608.23744#A3)reports all rows\.

CPP adds no inference\-time parameters or computation, but its offline cost is substantial\. CPP\-Wanda is4\.0×4\.0\\timesWanda, CPP\-SparseGPT is1\.9×1\.9\\timesSparseGPT, and the Llama\-3\-8B CPP\-SparseGPT diagnostic is2\.3×2\.3\\timesits base\. Threshold\-aware CPP\-Wanda is8\.8×8\.8\\timesWanda and2\.2×2\.2\\timestrue\-label CPP\-Wanda\. Wall time and peak memory appear in Table[5](https://arxiv.org/html/2608.23744#S7.T5)\. These costs are part of the method’s trade\-off, not implementation footnotes\.

Table 5:Offline pruning cost on one NVIDIA A100 80GB\. Ratios compare each method with its own base unless stated otherwise\.Table 6:Independent Qwen/DBpedia split redraws at 50% sparsity\.Table 7:Three\-seed RoBERTa\-base/DBpedia\-14 transfer at 50% sparsity\.

## 8Conclusion

CPP should be read as an efficiency\-oriented pruning objective under an independently restored validity constraint\. Split conformal prediction supplies marginal coverage for any fixed pruned model\. CPP changes which weights are retained so that, after the same valid recalibration, candidate\-label scores remain more separated and prediction sets can be smaller\. The final Qwen results support that goal most clearly on large\-label classification tasks\.

The matched controls also define the boundary of the contribution\. Generic supervised gradients explain much of the gain, and true\-label CPP is not resolved from a matched Wanda⊕\\oplusSNIP control\. Candidate labels near the conformal threshold better align saliency with set\-size inflation, but cost more offline computation\. The practical result is therefore conditional: CPP can improve conformal efficiency, especially with threshold\-aware saliency, while coverage still comes from split independence and accuracy must be reported alongside set size\.

The corresponding deployment workflow is deliberately conservative\. Construct saliency and chooseλ\\lambdausing only pruning and validation data; freeze the sparse model and every preprocessing choice; then compute a fresh quantile on the untouched conformal split\. Compare pruners by set size only after checking that they attain comparable empirical coverage, and keep accuracy as a separate utility guardrail\. If the checkpoint, verbalizer, prompt, temperature, mask, sparsity, or saliency construction changes, recalibration must be repeated\. This workflow, rather than the pruning score alone, is the reliability contract\.

Evidence strength follows the protocol\. The five\-seed, 15\-cell Qwen grid supports the primary claim; three\-seed controls isolate generic\-gradient effects and candidate\-label cost\. RoBERTa\-base and Llama\-3\-8B remain transfer diagnostics\. Together, the results support a narrow conclusion: after generic validity is restored, pruning can be optimized for conformal informativeness\.

## Limitations

The paper studies reliability\-sensitive classification, not free\-form generation\. Decoder labels are fixed verbalizer sequences and multi\-token labels are scored jointly\. Extending CPP to open\-ended outputs would require a different output space, nonconformity construction, and coverage event\. The RoBERTa\-base result covers one encoder, dataset, sparsity, and magnitude baseline\. The Llama\-3\-8B result is likewise a diagnostic on one checkpoint and one classification task, not a scaling law\.

Split\-conformal validity requires exchangeability and strict independence of the final calibration split from every pruning and selection decision\. Model, prompt, verbalizer, temperature, mask, sparsity, saliency variant, andλ\\lambdamust be frozen first\. Distribution shift, adaptive reuse of calibration data, or deployment on a different label population can invalidate the stated guarantee\. Empirical test coverage can also fall below 0\.90 in a finite sample even when population marginal coverage is valid\.

CPP requires labeled pruning data and per\-example gradients\. True\-label CPP is a proxy for a set\-size objective over all candidate labels\. Threshold\-aware and all\-label variants align more directly with that objective but increase offline cost: threshold\-aware CPP\-Wanda is8\.8×8\.8\\timesWanda and all\-label saliency is8\.9×8\.9\\timesthe true\-label gradient stage in the reported control\. CPP adds no inference\-time parameters or dense computation, but sparse\-kernel latency and energy gains are not benchmarked\.

The perturbation theory gives sufficient, not tight, conditions\. Uniform score bounds across all test\-label pairs are strong, the density bound is local and distribution\-dependent, and the diagonal first\-order saliency ignores coordinate interactions\. The Hessian remainder can dominate at high sparsity\. None of these results proves that CPP is optimal or that smaller validation sets will persist under arbitrary shift\.

Finally, the provenance audit removed mixed pilot and projected rows\. The final numerical claims are limited to values recoverable from the final manifests and the stated three\- or five\-seed protocols\. Exact reproducibility still depends on releasing split manifests, checkpoint and tokenizer versions, verbalizers, prompts, selectedλ\\lambdavalues, per\-run metrics, hardware and software versions, and the scripts that generate every table and figure\.

## Ethical Considerations

Smaller conformal sets can make compressed classifiers more useful, but marginal coverage is not subgroup, conditional, or harm\-weighted coverage\. A system may satisfy the global 90% target while failing on a rare or high\-cost subgroup\. Deployment should therefore audit accuracy, coverage, and set size by relevant subgroup and under realistic shifts\. Compression should not be presented as preserving reliability when the final calibration population does not match deployment\.

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## Appendix AProof Details and Remarks

The proofs in Section[6](https://arxiv.org/html/2608.23744#S6)are largely self\-contained; this appendix collects the remaining derivations and two auxiliary statements\. The first separates finite\-test variation from population coverage\. The second records a valid Learn\-Then\-Test alternative that was*not*used for the reported model selection\.

###### Theorem 7\(Empirical coverage concentration\)\.

For a fixed conformal predictor with population coverageppand an independent test set of sizeNN, letCov^N\\widehat\{\\mathrm\{Cov\}\}\_\{N\}be empirical coverage\. Then, for everyδ∈\(0,1\)\\delta\\in\(0,1\),

ℙ\{\|Cov^N−p\|\>log⁡\(2/δ\)2​N\}≤δ\.\\mathbb\{P\}\\\!\\left\\\{\|\\widehat\{\\mathrm\{Cov\}\}\_\{N\}\-p\|\>\\sqrt\{\\frac\{\\log\(2/\\delta\)\}\{2N\}\}\\right\\\}\\leq\\delta\.\(14\)

###### Theorem 8\(Optional finite\-grid Learn\-Then\-Test rule\)\.

For bounded validation lossℓg∈\[0,1\]\\ell\_\{g\}\\in\[0,1\], population riskR⁡\(g\)R\(g\), empirical riskR^​\(g\)\\hat\{R\}\(g\), targetrr, and finite grid𝒢\\mathcal\{G\}, define

pg=\{exp⁡\{−2​\|𝒟val\|​\(r−R^​\(g\)\)2\},R^​\(g\)<r,1,R^​\(g\)≥r\.p\_\{g\}=\\begin\{cases\}\\exp\\\{\-2\|\\mathcal\{D\}\_\{\\mathrm\{val\}\}\|\(r\-\\hat\{R\}\(g\)\)^\{2\}\\\},&\\hat\{R\}\(g\)<r,\\\\ 1,&\\hat\{R\}\(g\)\\geq r\.\\end\{cases\}\(15\)Accepting only candidates withpg≤δ/\|𝒢\|p\_\{g\}\\leq\\delta/\|\\mathcal\{G\}\|ensures, with probability at least1−δ1\-\\delta, that every accepted candidate hasR⁡\(g\)≤rR\(g\)\\leq r\.

Theorem[8](https://arxiv.org/html/2608.23744#Thmtheorem8)is included only to preserve the valid optional analysis from the submission\. Algorithm[1](https://arxiv.org/html/2608.23744#alg1)uses the deterministic guardrail and tie\-breaking rule in Section[5\.2](https://arxiv.org/html/2608.23744#S5.SS2), not this theorem\.

###### Proof of Theorem[2](https://arxiv.org/html/2608.23744#Thmtheorem2)\.

Condition on the data used to construct and selectθ′\\theta^\{\\prime\}\. Under Assumption[1](https://arxiv.org/html/2608.23744#Thmtheorem1), then\+1n\+1scoresSi​\(θ′\)=s⁡\(Xi,Yi,θ′\)S\_\{i\}\(\\theta^\{\\prime\}\)=s\(X\_\{i\},Y\_\{i\};\\theta^\{\\prime\}\)fori=1,…,n\+1i=1,\\ldots,n\+1are exchangeable conditional onθ′\\theta^\{\\prime\}\. The conformal set fails to contain the test label exactly whenSn\+1​\(θ′\)\>q^α​\(θ′\)S\_\{n\+1\}\(\\theta^\{\\prime\}\)\>\\hat\{q\}\_\{\\alpha\}\(\\theta^\{\\prime\}\), and withq^α\\hat\{q\}\_\{\\alpha\}equal to thekk\-th order statistic of the firstnnscores, the rank ofSn\+1S\_\{n\+1\}among then\+1n\+1exchangeable scores is uniformly distributed up to ties\. The failure probability is therefore at mostα\\alpha, giving Eq\.[8](https://arxiv.org/html/2608.23744#S6.E8)\. When ties occur, the conservative quantile convention preserves the lower bound; when scores are almost surely distinct, the standard rank argument gives the stated upper slack\. The final step removes the conditioning by taking expectation over the pruning and validation data\. ∎

###### Proof of Theorem[7](https://arxiv.org/html/2608.23744#Thmtheorem7)\.

Conditional on the fitted conformal predictor, the indicators

𝟏\{Yi∈Cα\(Xi;θ′\)\}\\mathbf\{1\}\\\{Y\_\{i\}\\in C\_\{\\alpha\}\(X\_\{i\};\\theta^\{\\prime\}\)\\\}are independent Bernoulli random variables with meanpp\. Hoeffding’s inequality gives

ℙ\[\|Cov^N−p\|\>t\]≤2exp\(−2Nt2\)\\mathbb\{P\}\[\|\\widehat\{\\mathrm\{Cov\}\}\_\{N\}\-p\|\>t\]\\leq 2\\exp\(\-2Nt^\{2\}\)for allt\>0t\>0; settingt=log⁡\(2/δ\)/\(2​N\)t=\\sqrt\{\\log\(2/\\delta\)/\(2N\)\}proves Eq\.[14](https://arxiv.org/html/2608.23744#A1.E14)\. The one\-sided consequenceCov^N≥1−α−log⁡\(2/δ\)/\(2​N\)\\widehat\{\\mathrm\{Cov\}\}\_\{N\}\\geq 1\-\\alpha\-\\sqrt\{\\log\(2/\\delta\)/\(2N\)\}with probability at least1−δ1\-\\deltafollows by combining this concentration event withp≥1−αp\\geq 1\-\\alpha\. ∎

###### Proof of Lemma[3](https://arxiv.org/html/2608.23744#Thmtheorem3)\.

For everyii, we have the bounded condition

ai−ϵ≤bi≤ai\+ϵ\.a\_\{i\}\-\\epsilon\\leq b\_\{i\}\\leq a\_\{i\}\+\\epsilon\.At leastkkof theaia\_\{i\}’s are no larger thana\(k\)a\_\{\(k\)\}, so at leastkkof thebib\_\{i\}’s are no larger thana\(k\)\+ϵa\_\{\(k\)\}\+\\epsilon, givingb\(k\)≤a\(k\)\+ϵb\_\{\(k\)\}\\leq a\_\{\(k\)\}\+\\epsilon\. Reversing the roles ofaaandbbyieldsa\(k\)≤b\(k\)\+ϵa\_\{\(k\)\}\\leq b\_\{\(k\)\}\+\\epsilon\. ∎

###### Proof of Theorem[4](https://arxiv.org/html/2608.23744#Thmtheorem4)\.

By Lemma[3](https://arxiv.org/html/2608.23744#Thmtheorem3), the calibration quantiles satisfyq^α​\(θ′\)≤q^α​\(θ\)\+ϵ\\hat\{q\}\_\{\\alpha\}\(\\theta^\{\\prime\}\)\\leq\\hat\{q\}\_\{\\alpha\}\(\\theta\)\+\\epsilon\. Ify∈Cα​\(x,θ′\)y\\in C\_\{\\alpha\}\(x;\\theta^\{\\prime\}\), thens⁡\(x,y,θ′\)≤q^α​\(θ′\)s\(x,y;\\theta^\{\\prime\}\)\\leq\\hat\{q\}\_\{\\alpha\}\(\\theta^\{\\prime\}\), and Eq\.[10](https://arxiv.org/html/2608.23744#S6.E10)givess⁡\(x,y,θ\)≤s⁡\(x,y,θ′\)\+ϵs\(x,y;\\theta\)\\leq s\(x,y;\\theta^\{\\prime\}\)\+\\epsilon, hences⁡\(x,y,θ\)≤q^α​\(θ\)\+2​ϵs\(x,y;\\theta\)\\leq\\hat\{q\}\_\{\\alpha\}\(\\theta\)\+2\\epsilon\. This proves the containment in Eq\.[11](https://arxiv.org/html/2608.23744#S6.E11)\. Define the per\-example set\-size inflation as

ΔC​\(x\)=\|Cα​\(x,θ′\)\|−\|Cα​\(x,θ\)\|\.\\Delta\_\{C\}\(x\)=\|C\_\{\\alpha\}\(x;\\theta^\{\\prime\}\)\|\-\|C\_\{\\alpha\}\(x;\\theta\)\|\.Subtracting the dense set leaves only labels whose dense scores lie in the band above the dense threshold, so

ΔC​\(x\)\\displaystyle\\Delta\_\{C\}\(x\)≤∑y∈𝒴\{q^α\(θ\)<s\(x,y;θ\)\\displaystyle\\leq\\sum\_\{y\\in\\mathcal\{Y\}\}\\mathbf\{1\}\\\!\\left\\\{\\hat\{q\}\_\{\\alpha\}\(\\theta\)<s\(x,y;\\theta\)\\right\.≤q^α\(θ\)\+2ϵ\}\.\\displaystyle\\left\.\\leq\\hat\{q\}\_\{\\alpha\}\(\\theta\)\+2\\epsilon\\right\\\}\.Taking expectations and using the density bound for each label gives Eq\.[12](https://arxiv.org/html/2608.23744#S6.E12)\. ∎

###### Proof of Proposition[6](https://arxiv.org/html/2608.23744#Thmtheorem6)\.

For a binary pruning mask𝗆\\mathsf\{m\}, letθ𝗆=𝗆⊙θ\\theta\_\{\\mathsf\{m\}\}=\\mathsf\{m\}\\odot\\theta; the perturbationδ=θ𝗆−θ\\delta=\\theta\_\{\\mathsf\{m\}\}\-\\thetasatisfiesδj=−θj\\delta\_\{j\}=\-\\theta\_\{j\}forj∈P⁡\(𝗆\)j\\in P\(\\mathsf\{m\}\)and zero otherwise\. Taylor’s theorem gives

s⁡\(z,θ𝗆\)−s⁡\(z,θ\)=∇θs​\(z,θ\)⊤​\(θ𝗆−θ\)\+rz,s\(z;\\theta\_\{\\mathsf\{m\}\}\)\-s\(z;\\theta\)=\\nabla\_\{\\theta\}s\(z;\\theta\)^\{\\top\}\(\\theta\_\{\\mathsf\{m\}\}\-\\theta\)\+r\_\{z\},with the residual bounded by

\|rz\|≤12​Hz​‖θ𝗆−θ‖22\.\|r\_\{z\}\|\\leq\\tfrac\{1\}\{2\}H\_\{z\}\\\|\\theta\_\{\\mathsf\{m\}\}\-\\theta\\\|\_\{2\}^\{2\}\.Becauseθ𝗆−θ\\theta\_\{\\mathsf\{m\}\}\-\\thetais nonzero only on pruned coordinates, its linear term is

−∑j∈P⁡\(𝗆\)θj∂s⁡\(z,θ\)∂θj\.\-\\sum\_\{j\\in P\(\\mathsf\{m\}\)\}\\theta\_\{j\}\\frac\{\\partial s\(z;\\theta\)\}\{\\partial\\theta\_\{j\}\}\.Averaging absolute values and applying Cauchy–Schwarz over examples yields Eq\.[13](https://arxiv.org/html/2608.23744#S6.E13)\. For the diagonal bound, we apply the identity

\(∑j=1maj\)2≤m​∑j=1maj2\\left\(\\sum\_\{j=1\}^\{m\}a\_\{j\}\\right\)^\{2\}\\leq m\\sum\_\{j=1\}^\{m\}a\_\{j\}^\{2\}for eachzz, average over𝒟0\\mathcal\{D\}\_\{0\}, and substitute the definition ofIcal,𝒟0I\_\{\\mathrm\{cal\},\\mathcal\{D\}\_\{0\}\}\. ∎

###### Proof of Theorem[8](https://arxiv.org/html/2608.23744#Thmtheorem8)\.

For a fixedggwithR⁡\(g\)\>rR\(g\)\>r, Hoeffding’s inequality implies

ℙ\{R^\(g\)≤r−t\}≤exp\(−2\|𝒟val\|t2\)\\mathbb\{P\}\\\{\\hat\{R\}\(g\)\\leq r\-t\\\}\\leq\\exp\(\-2\|\\mathcal\{D\}\_\{\\mathrm\{val\}\}\|t^\{2\}\)for allt\>0t\>0, showing thatpgp\_\{g\}is a valid conservative p\-value for the null hypothesisR⁡\(g\)\>rR\(g\)\>r\. A union bound over the finite grid𝒢\\mathcal\{G\}shows that, with probability at least1−δ1\-\\delta, no candidate withR⁡\(g\)\>rR\(g\)\>ris accepted at the Bonferroni thresholdδ/\|𝒢\|\\delta/\|\\mathcal\{G\}\|\. Any data\-dependent choice among accepted candidates therefore satisfiesR⁡\(g\)≤rR\(g\)\\leq ron the same event\. ∎

## Appendix BAdditional Experimental Details

### B\.1Implementation Details

Distribution name:reliable\-prune\. Python import:reliable\_prune\. We use these forms consistently\. The API keeps pruning, validation, and final conformal data separate:

fromreliable\_pruneimport\(

CalibrationPreservingPruner\)

pruner=CalibrationPreservingPruner\(

model=model,

prune\_dataset=prune\_data,

validation\_dataset=val\_data,

conformal\_dataset=conf\_data,

target\_sparsity=0\.5,

target\_coverage=0\.9,

lambda\_grid=\[0\.0,0\.1,0\.3,0\.5\],

base\_pruner="wanda"\)

pruned,q=pruner\.prune\(\)

The final implementation supports the Pythia\- and Qwen\-style modules used during development\. The completed Llama\-3\-8B CPP\-SparseGPT diagnostic uses the same module\-streaming principle and is no longer described as an engineering target\. RoBERTa\-base uses magnitude pruning as the base because it is the fastest exact encoder adaptation and isolates architecture transfer\. Mistral, Mamba, tensor\-parallel execution, structured sparsity, and sparse\-kernel latency remain future work\.

CPP computes per\-example gradients with microbatching or vectorized Jacobian products and streams one module at a time\. It stores only the current module’s saliency\. Eligible Qwen/Llama modules areq\_proj,k\_proj,v\_proj,o\_proj,gate\_proj,up\_proj, anddown\_proj\. Pythia uses attention and GELU\-MLP projection matrices\. Norm parameters, embeddings, tied output heads, positional mechanisms, and KV\-cache state are excluded\. Every matched baseline uses the same eligible coordinates and sparsity allocation\.

### B\.2Data Roles and Selection

Unless data availability requires a smaller partition,𝒟prune\\mathcal\{D\}\_\{\\mathrm\{prune\}\},𝒟val\\mathcal\{D\}\_\{\\mathrm\{val\}\}, and𝒟conf\\mathcal\{D\}\_\{\\mathrm\{conf\}\}each contain 1,024 examples, while𝒟test\\mathcal\{D\}\_\{\\mathrm\{test\}\}contains at least 5,000\. The validation split is divided deterministically into 512\-example validation\-calibration and validation\-evaluation subsets\. For true\-label, top\-33, and all\-label variants, saliency and gradient evaluation use only𝒟prune\\mathcal\{D\}\_\{\\mathrm\{prune\}\}\. Threshold\-aware CPP additionally fixes the dense\-model preliminary threshold and band width on validation\-calibration; gradients still use only the pruning split\. Validation\-evaluation is reserved for candidate evaluation andλ\\lambda\-selection\. The final conformal split is never used until the model is frozen\.

For every base\-pruner, dataset, sparsity, and seed, the reported selection grid is\{0,0\.1,0\.3,0\.5\}\\\{0,0\.1,0\.3,0\.5\\\}\. A candidate is eligible when validation\-evaluation accuracy is within 0\.02 of itsλ=0\\lambda=0base and empirical coverage is at least 0\.88\. Selection minimizes set size, breaks differences within 0\.02 by higher accuracy and then smallerλ\\lambda, and falls back toλ=0\\lambda=0if no nonzero candidate is eligible\.

### B\.3Prompting and Complete\-Sequence Scoring

Each dataset uses a fixed prompt and verbalizer list\. The model is evaluated by teacher\-forced probability of the complete label verbalizer\. Multi\-token scores are summed and length\-normalized only when the fixed manifest says so; conformal prediction is over labels, never individual tokens\.

For AG News, the prompt asks for one of*World*,*Sports*,*Business*, or*Technology*\. For TREC, the verbalizers are*Abbreviation*,*Entity*,*Description*,*Human*,*Location*, and*Numeric*\. DBpedia\-14 uses*Company*,*EducationalInstitution*,*Artist*,*Athlete*,*OfficeHolder*,*MeanOfTransportation*,*Building*,*NaturalPlace*,*Village*,*Animal*,*Plant*,*Album*,*Film*, and*WrittenWork*\. Banking77 and CLINC150 use the canonical intent names in fixed lexical order\. Boundary ties in candidate selection use this same order\.

Temperature scaling, when evaluated, minimizes validation negative log likelihood over one positive scalar\. The temperature is frozen before its model\-specific conformal quantile is computed on𝒟conf\\mathcal\{D\}\_\{\\mathrm\{conf\}\}\.

### B\.4Offline Compute

All timing measurements use one NVIDIA A100 80GB GPU without model parallelism\. Table[5](https://arxiv.org/html/2608.23744#S7.T5)includes pruning and saliency construction but not inference, because CPP adds no inference\-time parameters or dense operations\.

### B\.5Release Checklist

The release contains the four\-way split manifests, random seeds, checkpoint identifiers, tokenizer versions, prompts, verbalizers, selectedλ\\lambdavalues, per\-run metrics, final sign table, table and figure scripts, timing scripts, hardware identifiers, CUDA/PyTorch versions, and response\-stage control code\. Every revised table and figure is generated from the same final\-manifest family\. Earlier pilot projections are not included as results\.

## Appendix CExtended Empirical Results

### C\.1Corrected Small\-Label Results

The original multi\-model pilot table is retired because some rows mixed evaluation paths\. To preserve the original table reference without preserving invalid values, Table[8](https://arxiv.org/html/2608.23744#A3.T8)now reports the final five\-seed Qwen small\-label results\. Every coverage–set\-size pair satisfies the feasibility inequality in Section[4](https://arxiv.org/html/2608.23744#S4)\. These tasks remain useful controls, but their small label spaces limit set\-size resolution\.

For completeness, the final correction log records two random baselines that are not used in the aggregate comparison: Pythia/AG News random has coverage0\.899±0\.0070\.899\\pm 0\.007and set size3\.72±0\.103\.72\\pm 0\.10, while Qwen/AG News random has0\.899±0\.0070\.899\\pm 0\.007and3\.74±0\.093\.74\\pm 0\.09\. Qwen/AG News dense accuracy is0\.6500\.650; at 30% sparsity, Wanda and CPP\-Wanda accuracies are0\.5120\.512and0\.5260\.526, respectively\. These values replace the incompatible submitted entries\.

Table 8:Final five\-seed Qwen2\.5\-1\.5B small\-label results\. Entries are empirical coverage and mean set size±\\pm95% CI\.
### C\.2Large\-Label Efficiency Accounting

Table[9](https://arxiv.org/html/2608.23744#A3.T9)expresses the final 50% results as relative set\-size and absolute accuracy changes\. Table[10](https://arxiv.org/html/2608.23744#A3.T10)records the corrected values behind Figure[1](https://arxiv.org/html/2608.23744#S4.F1)\. The DBpedia CPP\-Wanda row is the only displayed large\-label comparison with an accuracy loss\.

Table 9:CPP change relative to its matched base at 50% sparsity\.Table 10:Final values backing Figure[1](https://arxiv.org/html/2608.23744#S4.F1)\.
### C\.3Aggregate Win\-Rate Comparisons

A cell is one Qwen dataset–sparsity configuration\. W means the five\-seed mean improves relative to the corresponding base; it is not a significance claim\. Unlike the submitted accounting, all 15 set\-size cells are included; Table[12](https://arxiv.org/html/2608.23744#A3.T12)states the denominator explicitly\.

Table 11:Final directional win accounting\.
### C\.4Cell\-Level Accounting and Denominators

Table 12:Accounting for Table[11](https://arxiv.org/html/2608.23744#A3.T11)\.Table 13:Final Qwen cell\-level directions\. W/L compares the CPP mean with its matched base and does not assert significance\.
### C\.5Split Redraws and Sample\-Size Diagnostics

Five independently redrawn four\-way partitions provide the variance estimate in Table[6](https://arxiv.org/html/2608.23744#S7.T6)\. Tables[14](https://arxiv.org/html/2608.23744#A3.T14)and[15](https://arxiv.org/html/2608.23744#A3.T15)are nested diagnostics on one fixed final\-manifest partition\. They are point estimates, not additional independent\-seed intervals\.

Table 14:Nested conformal\-calibration\-size diagnostic: accuracy/coverage/set size\.Table 15:Nested pruning\-size diagnostic: accuracy/coverage/set size\.
### C\.6RoBERTa\-Base Transfer

Table[7](https://arxiv.org/html/2608.23744#S7.T7)is kept in the main paper because it directly answers the architecture\-transfer question; this appendix does not introduce additional RoBERTa cells\.

### C\.7Regenerated Figures

Figure[2](https://arxiv.org/html/2608.23744#A3.F2)uses only final\-manifest five\-seed means\. Figure[3](https://arxiv.org/html/2608.23744#A3.F3)provides the matching set\-size view\. Figure[4](https://arxiv.org/html/2608.23744#A3.F4)replaces the stale ECE plot with the final large\-label set\-size reductions because no corrected ECE cell values are inferred from the response record\. Figure[5](https://arxiv.org/html/2608.23744#A3.F5)displays the final 15\-cell directional rates\.

7\.57\.5888\.58\.5999\.59\.5101010\.510\.5111111\.511\.50\.280\.280\.30\.30\.320\.320\.340\.340\.360\.360\.380\.38Average set sizeAccuracyWandaCPP\-WSparseGPTCPP\-SGThresh\. CPP\-SGFigure 2:Qwen2\.5\-1\.5B/DBpedia\-14 accuracy–efficiency means at 50% sparsity\. Threshold\-aware CPP\-SparseGPT is a three\-seed ablation; the other points use five seeds\.WandaCPP\-WSparseGPTCPP\-SG0055101011\.211\.29910\.110\.18\.68\.6Average set sizeFigure 3:Final DBpedia\-14 set\-size means at matched empirical coverage\.DBpediaBanking77CLINC1500055101015152020Set\-size reduction \(%\)CPP\-WandaCPP\-SparseGPTFigure 4:Final large\-label set\-size reduction relative to the matched base at 50% sparsity\.000\.10\.10\.20\.20\.30\.30\.40\.40\.50\.50\.60\.60\.70\.70\.80\.80\.90\.911CPP\-SG setCPP\-W setCPP\-SG accCPP\-W acc0\.870\.870\.930\.930\.730\.730\.80\.8Directional win rateFigure 5:Final directional win rates from Table[11](https://arxiv.org/html/2608.23744#A3.T11)\.

## Appendix DLlama\-3\-8B Diagnostic

We use Llama\-3\-8B only as a scoped diagnostic, not as evidence for free\-form conformal generation or broad scale invariance\. At 50% sparsity, Table[16](https://arxiv.org/html/2608.23744#A4.T16)reports zero\-shot WikiText\-2 perplexity alongside DBpedia\-14 classification accuracy, empirical conformal coverage, and set size\. CPP\-SparseGPT changes perplexity only from7\.257\.25to7\.287\.28relative to SparseGPT, while increasing classification accuracy from0\.610\.61to0\.650\.65and reducing mean set size from8\.98\.9to7\.27\.2at matched coverage\. This single three\-seed experiment is consistent with objective\-dependent sparse\-model behavior\([Frantar et al\., 2025](https://arxiv.org/html/2608.23744#bib.bib25)\), but it does not establish a scaling law or a guarantee for generative tasks\.

Table 16:Llama\-3\-8B evaluation at 50% sparsity\. Set size \(\|C\|\|C\|\) and accuracy are reported on DBpedia\-14; perplexity is reported on WikiText\-2\. Means±\\pm95% CI over three seeds\.