PRM-Pruned Fragment Grafting（PPFG）在何种配置下失效：来自三个推理语言模型的证据

# Characterizing a Configuration Where Inference-Time PRM-Pruned Fragment Grafting Is Inert: Evidence from Three Reasoning LMs
Source: [https://arxiv.org/html/2610.00047](https://arxiv.org/html/2610.00047)
Khawaja Murad ul Hassan††thanks:Correspondence:khawajamuradulhassan@gmail\.comMehran EbrahimiAffiliation:Faculty of Science, Ontario Tech UniversityAffiliation:Oshawa, ON, Canada

###### Abstract

Diversity collapse in parallel chain\-of\-thought has motivated a wave of inference\-time interventions, including a natural design space: when a process reward model \(PRM\) prunes one reasoning chain, the high\-PRM prefix that survived to the point of pruning is extracted and grafted verbatim as an in\-context demonstration into a still\- decoding sibling chain\. We isolate this mechanism, PRM\-Pruned Fragment Grafting \(PPFG\), as the most cost\-minimal inference\-time operationalization of cross\-trajectory step\-level transfer, and ask whether it moves reasoning trajectories at the operating point where prior fragment\-grafting work has reported gains under additional compensating ingredients\. On Qwen2\.5\-7B\-Instruct with the Math\-Shepherd PRM on full MATH500 \(n=500n=500, three seeds\), PPFG in both stagnation\- and random\-targeting variants is statistically indistinguishable from the independent parallel\-CoT baseline on every measured axis: Pass@kkatk∈\{1,2,4,8\}k\\in\\\{1,2,4,8\\\}and the answer\-mode rate all match within one seed standard error\. We characterize*why*through two complementary views of one targeting limitation\. A four\-bucket classification of all 322 stagnation\-rule injection events shows that only14%14\\%targeted a struggling chain with room to act \(first\-100 hand\-validated slice:6/556/55,11%11\\%\); the rest each landed on a chain that had already succeeded, was within two steps of answering, or sat on a high, flat PRM plateau, states where a rescue graft cannot change the outcome\. A counterfactual followed by an end\-to\-end sweep shows that no compound\-gate refinement jointly achieves well\-targeted firing and adequate injection density\. A random\-targeting control firing at2\.4×2\.4\\timesthe rate with the same parity shows the inertness is not specific to this heuristic\. The parity finding replicates across three base LMs spanning two architecture families \(Qwen2\.5\-7B\-Instruct and DeepSeek\-R1\-Distill\-Qwen\-7B are Qwen\-family; LLaMA\-3\.1\-8B\-Instruct is the cross\-family case\) and six reasoning benchmarks \(MATH500, AIME 2024\+2025, NuminaMath\-CoT, GSM8K, GPQA\-Diamond, MMLU\-Pro Law/Biology/Philosophy\), against a second multi\-domain PRM \(VersaPRM\), and across a four\-point sweep of the compatibility\-gate threshold; cross\-architecture replication is run on four of the six \(MATH500, AIME, NuminaMath\-CoT, GPQA\-Diamond\), with GSM8K and MMLU\-Pro reported on Qwen2\.5\-7B\-Instruct only as additional task\-family coverage\. Every PPFG\-versus\-independent contrast falls inside the per\-benchmark minimum detectable effect on Pass@1 and mode rate\. A pre\-registered two\-one\-sided\-tests analysis promotes the parity from non\-significance to positive equivalence on all twelve Qwen and LLaMA cells \(13 of 18 cross\-arch cells; the four inconclusive DeepSeek\-R1\-Distill cells are within\-σ\\sigmabut power\-limited by higher seed variance, Appendix[J](https://arxiv.org/html/2610.00047#A10)\)\. On Qwen, a per\-event spot\-check finds the injected chain prunes at2\.75×2\.75\\timesthe matched\-step rate, and a matched\-PRM\-trajectory control attributes only part of this to selection \(2\.59×2\.59\\timespersists\), so injection locally accelerates pruning; a surviving\-sibling counterfactual \(N=65N\{=\}65\) finds no population\-level compensation detectable at its power\. The evidence is consistent with a selection\-dominated account that the population absorbs without a Pass@kkchange \(the causal local\-hazard sub\-claim is established on Qwen only\)\. Beyond the negative result, the equivalence\-testing battery we assemble documents a template for establishing inference\-time mechanism nulls, and a hindsight oracle that per\-problem picks the better of PPFG and independent gains at most\+0\.13\+0\.13pp, giving future verbatim\-graft work a pre\-registrable number to clear\. We state the operating point explicitly, because the claim is scoped to it: 7–8B\-parameter instruction\-tuned LMs, PRM\-guided pruning with Math\-Shepherd \(VersaPRM off the math axis\), populations ofN=8N=8\(parity replicated toN=256N=256\), and English single\-answer reasoning\. Within that regime the mechanism is*inert*, not refuted in general; any future method claiming to transfer reasoning information through this mechanism class should specify which compensating ingredient its design supplies, and on what task family the claim is to be tested\.

## 1Introduction

Diversity collapse in parallel chain\-of\-thought \(CoT\) sampling is a now\-documented failure mode of strong reasoning models\([Tian et al\.,, 2026](https://arxiv.org/html/2610.00047#bib.bib23);[Yue et al\.,, 2025](https://arxiv.org/html/2610.00047#bib.bib29);[Luyten and van der Schaar,, 2026](https://arxiv.org/html/2610.00047#bib.bib16)\), and has motivated a range of inference\-time and training\-time interventions including PRM\-weighted sequential Monte Carlo resampling, mode\-conditioning frameworks\([Wu et al\.,, 2025](https://arxiv.org/html/2610.00047#bib.bib28)\), KV\-cache geometric regulation\([Du and Tanaka\-Ishii,, 2026](https://arxiv.org/html/2610.00047#bib.bib6)\), and reverse\-KL replacements for the RL training objective\.

A natural inference\-time response to diversity collapse is*step\-level fragment grafting*: when one parallel\-CoT chain is pruned by a process reward model \(PRM\), the high\-PRM prefix that the chain produced before pruning is extracted and inserted as an in\-context demonstration into a still\-decoding sibling\. Several recent papers explore versions of this design, Stitching Noisy Diffusion Thoughts \(SNDT;[Miles et al\.,, 2026](https://arxiv.org/html/2610.00047#bib.bib17)\), Recycling Search Experience \(RSE;[Wang et al\., 2026b,](https://arxiv.org/html/2610.00047#bib.bib27)\), and T\-STAR\([Li et al\.,, 2026](https://arxiv.org/html/2610.00047#bib.bib14)\), and each reports positive effects in a specific operating regime that adds at least one compensating ingredient on top of the bare graft: a separate solver pass that re\-attends to the spliced text, a training\-time gradient through the graft, or a learned distillation of failed\-trajectory patterns rather than verbatim step text\. What remains open is whether the most cost\-minimal corner of this design space, verbatim step text spliced*in\-place*into a still\- decoding sibling chain, with target selection by a population\-state heuristic that adds no extra forward pass, no gradient, no learned abstraction, and no style adaptation, moves reasoning trajectories at all on strong base LMs at the operating points where the diversity\-collapse motivation bites\. We answer that question\.

The answer carries an inferential payoff beyond the single negative cell\. The recent fragment\-grafting systems above each bundle the cross\-trajectory transfer with one or more compensating ingredients and report a net gain; none isolates the transfer step itself\. The implicit reading a practitioner could take from that literature is that moving high\-PRM partial reasoning between chains is where the benefit originates, with the extra machinery as supporting detail\. Our result bears on that reading: isolated from every compensating ingredient, the transfer is inert \(and locally mildly adverse\)\. We do not ablate any prior system—those systems differ from this corner on at least one ingredient—so this single null cannot attribute their gains; but it is consistent with the hypothesis that the gains reside in the compensating ingredients rather than in the bare transfer\. That makes the finding actionable: a designer weighing where to spend complexity should not expect the in\-place verbatim transfer to carry the benefit on its own, and should budget for whichever compensating ingredient their setting can afford\.

We definePRM\-Pruned Fragment Grafting \(PPFG\)as the operationalization of this corner\. At the moment of pruning, PPFG extracts the longest contiguous PRM\-above\-threshold prefix of the pruned chain, selects a still\-decoding sibling via a configurable targeting rule \(compatibility\-maximizing, stagnation\-targeting, or a random control\), and splices the prefix verbatim as a bracketed in\-context demonstration before the sibling’s next step\. The mechanism composes three operational properties not jointly realized by prior fragment\-grafting work: \(i\) the PRM pruning event is the trigger, extraction is conditioned on the verifier killing a chain, rather than firing on all chains or on all low\-scoring chains; \(ii\) injection is in\-place into a chain still being decoded by the same model, rather than serving as conditioning for a separate solver pass or as training data; and \(iii\) the grafted content is verbatim step text, rather than a learned skill, embedding distillation, or natural\-language summary\. Section[3](https://arxiv.org/html/2610.00047#S3)gives the procedure in full\.

On Qwen2\.5\-7B\-Instruct paired with the Math\-Shepherd PRM and run on full MATH500 at three seeds, PPFG in both stagnation\- and random\- targeting variants is statistically indistinguishable from the independent parallel\-CoT baseline on every measured axis: Pass@kkatk∈\{1,2,4,8\}k\\in\\\{1,2,4,8\\\}match within one seed standard error, mode rate matches within one seed standard error, and tokens\-per\-problem agree to within0\.3%0\.3\\%\(§[5\.1](https://arxiv.org/html/2610.00047#S5.SS1)\)\. The mechanism fires reliably \(≈16%\\approx 16\\%of problems on the fulln=500n=500set;≈31%\\approx 31\\%on the harder first\-100 slice, the rate tracking difficulty, cf\. the GSM8K drop to2\.8%2\.8\\%in §[5\.4](https://arxiv.org/html/2610.00047#S5.SS4)\), with mean fragment length 3\.2–3\.8 steps at PRM quality 0\.71–0\.78; the injections are well\-formed but do not change which answer the target chains land on\. We characterize*why*through two complementary views of one targeting limitation\. First, a four\-bucket classification of all 322 stagnation\-rule injection events shows that only14%14\\%targeted a struggling chain with room to act \(first\-100 hand\-validated slice:6/556/55,11%11\\%\); the rest each landed where a rescue graft cannot help, the chain had already succeeded, was within two steps of answering, or sat on a high, flat PRM plateau \(§[B](https://arxiv.org/html/2610.00047#A2)\)\. Second, a counterfactual followed by an end\-to\-end sweep of a five\-gate compound refinement shows that no threshold setting jointly achieves well\-targeted firing and adequate injection density \(§[B](https://arxiv.org/html/2610.00047#A2)\)\. The random\-rule control’s mode\-rate parity with both stagnation and independent indicates the targeting policy is not the binding constraint: we find no evidence that the mechanism moves chain trajectories under the heuristic targeting regime tested \(§[5\.4](https://arxiv.org/html/2610.00047#S5.SS4)\)\.

#### Contributions\.

\(1\) PPFG \(§[3](https://arxiv.org/html/2610.00047#S3)\): a minimal isolation of the most cost\-minimal corner of the cross\-chain step\-level grafting design space—three properties \(PRM\-pruned trigger, in\-place sibling injection, verbatim content\) not jointly realized by prior work—constructed expressly to test that corner in isolation\. \(2\) A six\-method matched\-compute comparison on full MATH500 at three seeds \(§[5\.1](https://arxiv.org/html/2610.00047#S5.SS1), Table[1](https://arxiv.org/html/2610.00047#S5.T1)\): PPFG\-stag and PPFG\-random both sit on the independent\-baseline frontier; Best\-of\-NNmatched\-compute lifts Pass@1 by≈\+6\\approx\{\+6\}pts uniformly with a modest diversity cost\. \(3\) A regime characterization \(§[5\.4](https://arxiv.org/html/2610.00047#S5.SS4)\) resting on three lines of evidence, which we keep distinct rather than count as independent replicates\.*\(a\) Parity*: matched\-compute Pass@kkand mode rate parity, established on Qwen MATH500 and replicated broadly \(the cross\-arch and cross\-benchmark cells are the same measurement on more cells, not separate evidence about mechanism\)\.*\(b\) Targeting\-heuristic limitation*: only14%14\\%of the 322 stagnation fires \(6/556/55,11%11\\%, on the hand\-validated first\-100 slice\) landed on a struggling chain with room to act, and the structural infeasibility of any compound\-gate refinement at adequate injection density is the same limitation viewed a second way, that population\-state\-only signals cannot separate competent\-flat from struggling\-flat chains; the random\-rule control, firing at2\.4×2\.4\\timesthe rate with the same parity, shows the inertness is not an artifact of this particular heuristic\.*\(c\) Mechanism\-level evidence \(suggestive\)*: a per\-event spot\-check \(§[5\.5](https://arxiv.org/html/2610.00047#S5.SS5)\) decomposing the post\-injection PRM depression into a format component \(Δempty=−0\.074\\Delta\_\{\\mathrm\{empty\}\}=\-0\.074, 241 events,p<10−19p<10^\{\-19\}\) and a content component \(Δstag=−0\.134\\Delta\_\{\\mathrm\{stag\}\}=\-0\.134, 322 events,p<10−19p<10^\{\-19\}; the impliedΔcontent=−0\.060\\Delta\_\{\\mathrm\{content\}\}=\-0\.060assumes additivity we do not test directly, so we read it as bounded rather than as a point estimate\), paired with a2\.75×2\.75\\timespruning\-hazard analysis and anN=65N\{=\}65surviving\-sibling counterfactual \(Appendix[L](https://arxiv.org/html/2610.00047#A12)\) that finds no detectable compensation within its power\. The evidence is consistent with a selection\-dominated account in which the stagnation rule pre\-selects already\-doomed chains and injection additionally accelerates their pruning locally \(a PRM\-matched control shows the hazard is not pure selection; Appendix[L](https://arxiv.org/html/2610.00047#A12)\), without redistributing completion across the population\. \(4\) A documented template for establishing inference\-time mechanism*nulls*\(which the field underproduces\): pre\-registered TOST bounds, an empty\-bracket confound control, a matched\-PRM\-trajectory hazard baseline, an oracle upper bound, and a surviving\-sibling counterfactual—a battery that separates “no effect” from “underpowered,” adapted here to PPFG and, we argue, to other inference\-time\-mechanism studies\.

The parity finding replicates across Qwen2\.5\-7B\-Instruct, LLaMA\-3\.1\-8B\-Instruct, and DeepSeek\-R1\-Distill\-Qwen\-7B on MATH500 \(TOST\-confirmed on Qwen and LLaMA, within\-σ\\sigmabut inconclusive on four of six DeepSeek cells; §[J](https://arxiv.org/html/2610.00047#A10)\), five further benchmarks \(AIME 2024\+2025, NuminaMath\-CoT, GSM8K, GPQA\-Diamond, MMLU\-Pro Law/Biology/Philosophy\), and a second multi\-domain PRM \(VersaPRM,[Zeng et al\.,, 2025](https://arxiv.org/html/2610.00047#bib.bib30); §[5\.6](https://arxiv.org/html/2610.00047#S5.SS6)\), with cross\-architecture coverage on four \(MATH500, AIME, NuminaMath\-CoT, GPQA\-Diamond\)\. We position the characterization as a target for future falsification \(§[6](https://arxiv.org/html/2610.00047#S6.SS0.SSS0.Px2)\); §[7](https://arxiv.org/html/2610.00047#S7)enumerates generalization axes still untested\.

## 2Related work

#### Cross\-trajectory information flow at inference time\.

SNDT\([Miles et al\.,, 2026](https://arxiv.org/html/2610.00047#bib.bib17)\)performs PRM\-scored step\-level stitching but into a*separate*solver pass with the fragment in its prompt context, rather than into a chain still being decoded\. RSE\([Wang et al\., 2026b,](https://arxiv.org/html/2610.00047#bib.bib27)\)distills failure patterns from dead ends into a within\-problem experience bank that conditions*subsequent*rollouts; the patterns are a learned summary, not verbatim sibling text\. T\-STAR\([Li et al\.,, 2026](https://arxiv.org/html/2610.00047#bib.bib14)\)performs step grafting at*training time*via a Cognitive Tree mechanism that merges trajectory nodes across successful and unsuccessful branches\. LATS\([Zhou et al\.,, 2023](https://arxiv.org/html/2610.00047#bib.bib31)\)injects*verbal self\-reflection summaries*from failed trajectories\. Metacognitive Reuse\([Didolkar et al\.,, 2025](https://arxiv.org/html/2610.00047#bib.bib5)\)extracts reusable behaviors*across problems*into a long\-horizon behavior repository\. GoT\([Besta et al\.,, 2023](https://arxiv.org/html/2610.00047#bib.bib1)\)aggregates active branches but never targets pruned chains; the PRM pruning event central to our trigger is absent\. PPFG composes three operational properties not jointly realized by any of these: \(i\) PRM pruning as trigger, \(ii\) in\-place sibling injection mid\-decoding, \(iii\) verbatim fragment\. Table[16](https://arxiv.org/html/2610.00047#A11.T16)\(Appendix[K](https://arxiv.org/html/2610.00047#A11)\) pins each prior method against these axes\.

#### Process reward models, SMC, credit assignment\.

PRMs as inference\-time verifiers were introduced by[Lightman et al\., \(2023\)](https://arxiv.org/html/2610.00047#bib.bib15)and operationalized at scale by Math\-Shepherd\([Wang et al\., 2024a,](https://arxiv.org/html/2610.00047#bib.bib24)\), the PRM we use throughout\. SMC variants\([Puri et al\.,, 2025](https://arxiv.org/html/2610.00047#bib.bib20);[Golowich et al\.,, 2026](https://arxiv.org/html/2610.00047#bib.bib8);[Giannone et al\.,, 2025](https://arxiv.org/html/2610.00047#bib.bib7)\)use PRM as the live emission model for chain reweighting; our SMC baseline \(§[5\.1](https://arxiv.org/html/2610.00047#S5.SS1)\) instantiates this at matched compute\. Population\-level credit assignment\([Kazemnejad et al\.,, 2024](https://arxiv.org/html/2610.00047#bib.bib12);[Wang et al\., 2026a,](https://arxiv.org/html/2610.00047#bib.bib26);[Parthasarathi et al\.,, 2025](https://arxiv.org/html/2610.00047#bib.bib18)\)provides the analytical lens for §[5\.4](https://arxiv.org/html/2610.00047#S5.SS4); all those methods are training\-time\.

#### Diversity\-preserving sampling and novelty defense\.

The phenomenon we attempt to act on is well documented and largely addressed at training time\([Tian et al\.,, 2026](https://arxiv.org/html/2610.00047#bib.bib23);[Yue et al\.,, 2025](https://arxiv.org/html/2610.00047#bib.bib29);[Luyten and van der Schaar,, 2026](https://arxiv.org/html/2610.00047#bib.bib16);[Jiang et al\.,, 2025](https://arxiv.org/html/2610.00047#bib.bib11)\); concurrent inference\-time work\([Wu et al\.,, 2025](https://arxiv.org/html/2610.00047#bib.bib28);[Du and Tanaka\-Ishii,, 2026](https://arxiv.org/html/2610.00047#bib.bib6);[Petullo et al\.,, 2026](https://arxiv.org/html/2610.00047#bib.bib19)\)attacks it from different angles\. Every prior work named above succeeds in a regime that differs from ours on at least one operational dimension \(separate\-solver\-pass re\-attention, learned distillation, training\-time gradient, verbal summarization/style adaptation, cross\-problem abstraction\); §[6](https://arxiv.org/html/2610.00047#S6.SS0.SSS0.Px2)enumerates these regime asymmetries explicitly\.

## 3Method

### 3\.1Setup

A parallel chain\-of\-thought generation pass produces a*population*Pt=\{C1,…,CN\}P\_\{t\}=\\\{C\_\{1\},\\dots,C\_\{N\}\\\}ofNNreasoning chains decoded step\-by\-step from one problem promptqq\. Each chainCiC\_\{i\}is a sequence of reasoning steps\(si,1,…,si,ti\)\(s\_\{i,1\},\\dots,s\_\{i,t\_\{i\}\}\)delimited by a double\-newline step separator \(the natural reasoning\-step boundary for chat\-formatted Qwen2\.5 outputs\)\. At every step the base language modelπθ\\pi\_\{\\theta\}samples a continuationsi,ti\+1∼πθ\(⋅∣q,si,1,…,si,ti\)s\_\{i,t\_\{i\}\+1\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid q,s\_\{i,1\},\\dots,s\_\{i,t\_\{i\}\}\)for each active chain in batch, capped atLstepL\_\{\\mathrm\{step\}\}tokens per step\. A process reward model \(PRM\) then assigns a scalar quality scoreρi,t∈\[0,1\]\\rho\_\{i,t\}\\in\[0,1\]to each step using the canonical decoupled\-PRMki\-tag protocol\([Wang et al\., 2024a,](https://arxiv.org/html/2610.00047#bib.bib24);[Lightman et al\.,, 2023](https://arxiv.org/html/2610.00047#bib.bib15)\)\. The PRM is a math\-tuned Math\-Shepherd checkpoint throughout this paper; §[4](https://arxiv.org/html/2610.00047#S4)reports configuration\.

A chain transitions to a terminal status at the end of each step under three conditions: \(i\)promotedif its latest step contains a final answer matched by a balanced\-brace\\boxed\{⋅\\cdot\}parser, \(ii\)prunedif its latest PRM score satisfiesρi,t<τkill\\rho\_\{i,t\}<\\tau\_\{\\mathrm\{kill\}\}, or \(iii\)hit\-maxif its step count reaches the population\-wide ceilingTmaxT\_\{\\mathrm\{max\}\}\.

### 3\.2PPFG procedure

PPFG operates as a hook inside the per\-step population loop, invoked once at the end of each step on the set of chains pruned that step\. Each injection mutates one active sibling chain’s in\-context prompt to include a high\-PRM prefix extracted from a pruned chain \(architectural diagram in Figure[2](https://arxiv.org/html/2610.00047#A6.F2), Appendix[F](https://arxiv.org/html/2610.00047#A6)\)\. The four sub\-operations are:Extract, for each newly prunedCsrcC\_\{\\mathrm\{src\}\}, find the longest contiguous prefix with all PRM scores≥τextract\\geq\\tau\_\{\\mathrm\{extract\}\}, kept if its lengthk∗∈\[kmin,kmax\]k^\{\*\}\\in\[k\_\{\\mathrm\{min\}\},k\_\{\\mathrm\{max\}\}\]\.Target\-select, pick a targetCtgtC\_\{\\mathrm\{tgt\}\}from active candidates by the ruleR∈\{compat,stagnation,stagnation​\_​compound,random\}R\\in\\\{\\mathrm\{compat\},\\mathrm\{stagnation\},\\mathrm\{stagnation\\\_compound\},\\mathrm\{random\}\\\}\.Gate, compute compatibilityξ⁡\(F,Ctgt\)=1−cos⁡\(ϕ⁡\(ssrc,k∗\),ϕ⁡\(stgt,ttgt\)\)\\xi\(F,C\_\{\\mathrm\{tgt\}\}\)=1\-\\cos\\bigl\(\\phi\(s\_\{\\mathrm\{src\},k^\{\*\}\}\),\\phi\(s\_\{\\mathrm\{tgt\},t\_\{\\mathrm\{tgt\}\}\}\)\\bigr\)withϕ\\phia sentence\-embedding model \(all\-MiniLM\-L6\-v2\); admit ifξ≥ξmin\\xi\\geq\\xi\_\{\\mathrm\{min\}\}\.Inject, splice the fragment into the target’s prompt as a bracketed in\-place demonstration before its next step\. Full pseudocode in Algorithm[1](https://arxiv.org/html/2610.00047#alg1)\(Appendix[F](https://arxiv.org/html/2610.00047#A6)\)\.

### 3\.3Operational details

#### Fragment formatting\.

Each fragment is rendered as a bracket\-bullet block tagging its quality and length:\[Insight from a related reasoning path \(quality=ρ¯\\bar\{\\rho\}, length=k∗k^\{\*\}steps\): \- <step\_1\> \- <step\_2\> …\]\. The bracket frames the block as demonstration and the bullets prevent confusion with the target’s own step numbering\.

#### Targeting rules\.

compatmaximizesξ\\xi\.stagnationfilters to candidates whose three\-step PRM range is below the flatness thresholdτflat\\tau\_\{\\mathrm\{flat\}\}, then picks max\-ξ\\xi\.stagnation\_compoundadditionally requires \(i\) target mean recent PRM<τlow<\\tau\_\{\\mathrm\{low\}\}, \(ii\)≥h\\geq hremaining steps beforeTmaxT\_\{\\mathrm\{max\}\}, \(iii\) target’s latest PRM below the median of active siblings, \(iv\)≥ℓ\\geq\\ellactive siblings\.randompicks uniformly at random, ignoringξ\\xi\(ablation control\)\. The injection action is a population\-level move whose post\-action value is non\-local; we adopt the step\-level credit\-assignment framing\([Kazemnejad et al\.,, 2024](https://arxiv.org/html/2610.00047#bib.bib12)\)as the analytical lens for §[5\.4](https://arxiv.org/html/2610.00047#S5.SS4)but do not run a per\-injection MC value estimator in this submission\.

## 4Experimental setup

#### Base model and verifier\.

We use Qwen2\.5\-7B\-Instruct\([Qwen Team,, 2024](https://arxiv.org/html/2610.00047#bib.bib21)\)as the base reasoning model throughout\. The base model is held fixed across all baselines and the PPFG method so any measured difference is attributable to the parallel\-CoT mechanism, not to model strength\. The base model is loaded in bf16 via vLLM 0\.7\.3 on a single 40GB A100 GPU at maximum sequence length 4096\. The PRM is Math\-Shepherd\([Wang et al\., 2024a,](https://arxiv.org/html/2610.00047#bib.bib24)\)\.

#### Task\.

MATH500\([Lightman et al\.,, 2023](https://arxiv.org/html/2610.00047#bib.bib15)\)is a 500\-problem subset of the MATH dataset\([Hendrycks et al\.,, 2021](https://arxiv.org/html/2610.00047#bib.bib10)\)curated for the*Let’s Verify Step by Step*paper that introduced PRMs\. We evaluate on the full 500 problems of the MATH500 test split\. Mechanism analyses in §[5\.4](https://arxiv.org/html/2610.00047#S5.SS4)use the first\-100 slice of then=500n=500PPFG\-stag trajectories for tractable hand\-classification of injection events \(§[B](https://arxiv.org/html/2610.00047#A2)\) and the counterfactual compound\-gate dry\-run \(§[B](https://arxiv.org/html/2610.00047#A2)\); aggregate\-metric tables reportn=500n=500results\.

#### Baselines\.

We compare PPFG against three inference\-time parallel\-CoT baselines, all sharing the same base model, PRM, and the double\-newline step separator\. The step separator is held fixed at double\-newline across architectures for comparability; the PRM\-distribution audit \(Appendix[G](https://arxiv.org/html/2610.00047#A7)\) confirms the prune\-trigger rate atτkill\\tau\_\{\\mathrm\{kill\}\}remains in a usable range \(2\.42\.4–6\.4%6\.4\\%\) on all three architectures, ruling out a “PRM never fires” artifact from a non\-native step boundary\. The three baselines are: \(1\)*Independent parallel CoT*\(indep,N=8N=8\): eight chains decoded fully independently\. \(2\)*SMC / particle filter*\(smc,N=8N=8\): PRM\-weighted resampling at temperature0\.50\.5\. \(3\)*Best\-of\-NN*atN=16N=16\(2×2\\timescompute\) andN=8N=8\(matched\-compute\); theN=8N=8matched\-compute row is derived from the independentN=8N=8trajectories by argmax\(final\-step PRM\) per problem per seed\.

#### PPFG configuration\.

PPFG runs atN=8N=8with targeting ruleR∈\{stagnation,stagnation​\_​compound,random\}R\\in\\\{\\mathrm\{stagnation\},\\mathrm\{stagnation\\\_compound\},\\mathrm\{random\}\\\},τextract=0\.6\\tau\_\{\\mathrm\{extract\}\}=0\.6,kmin=2k\_\{\\mathrm\{min\}\}=2,kmax=8k\_\{\\mathrm\{max\}\}=8,M=2M=2,τkill=0\.4\\tau\_\{\\mathrm\{kill\}\}=0\.4,ξmin=0\.0\\xi\_\{\\mathrm\{min\}\}=0\.0\.

#### Metrics\.

Pass@kkfork∈\{1,2,4,8\}k\\in\\\{1,2,4,8\\\}via the unbiased Chen\-et\-al\-2021 estimator\([Chen et al\.,, 2021](https://arxiv.org/html/2610.00047#bib.bib2)\); mode rate \(fraction of problems where the modal non\-null answer share exceeds 0\.5; lower is better\); and distinct\-4 \(ratio of unique 4\-grams to total 4\-grams across all chains’ full reasoning; higher is better\)\. We additionally compute the continuous raw modal share as a supplement to the binary\-thresholded mode rate; raw modal share tracks the binary metric’s ranking on every comparison in our tables \(Spearmanρ≈0\.94\\rho\\approx 0\.94over 16 method×\\timesarchitecture×\\timesnproblemsn\_\{\\mathrm\{problems\}\}cells\), and we report the binary\-thresholded version in tables for compactness\. Tokens\-per\- problem \(the sum of generated tokens across all chains\) is the compute footing\.

#### Seeds \+ extraction \+ release\.

Seeds 42–44 \(with 45–46 for the 5\-seed Qwen cells\); per\-chain sampling seeds are deterministic in the run seed and chain index\. All Pass@kkand mode rate numbers are computed with a balanced\-brace\\boxed\{⋅\\cdot\}parser that handles nested LaTeX\. An earlier non\-balanced regex undercounted Pass@kkuniformly by≈10\\approx 10pp; the balanced parser is used throughout and PPFG\-vs\-baseline parity is preserved under either because the shift is method\-invariant\. Code \+ saved trajectories will be released under an open\-source license at de\-anonymization\.

## 5Results

### 5\.1The six\-method tradeoff space

Table 1:Pass@kk, mode rate, and distinct\-4 on full MATH500 \(n=500n=500\), Qwen2\.5\-7B\-Instruct, mean±\\pmstd\. All numbers computed with the canonical scoring path \(correctness==task\.is\_correcton baremath\_verify, no normalized\-string\-match prefilter; answers always re\-extracted from steps, never read from a cached field; see §[5\.2](https://arxiv.org/html/2610.00047#S5.SS2)\); PPFG\-vs\-baseline parity is preserved because all methods shift together under the canonical comparator\.†BoN\-selector atN=8N\{=\}8is not a distinct inference\-time procedure; it applies argmax\-PRM selection \(tie\-averaged: a tie’s expected accuracy is the mean correctness over the tied chains, replacing an earlier no\-tie\-handling convention biased to chain index 0\) to the independentN=8N\{=\}8trajectories\. It shares per\-chain compute, mode rate, distinct\-4, and tpp with the independent row by construction\. We report it for compute\-frontier visualization \(Fig\.[4](https://arxiv.org/html/2610.00047#A8.F4)\) only\.‡5 seeds \(\{42,…,46\}\\\{42,\\ldots,46\\\}\) for these rows; the corresponding 3\-seed means agree to three decimals\. Other rows \(SMC, BoNN=16N\{=\}16, BoN\-selector\) at 3 seeds \(\{42,43,44\}\\\{42,43,44\\\}\); the BoN\-selector row inherits the 3\-seed independent values by construction\.Table[1](https://arxiv.org/html/2610.00047#S5.T1)reports the matched\-compute six\-method comparison\. Three observations frame the rest of the section\.

#### Observation 1: SMC exploits at lowkk, pays at highkk\.

PRM\-weighted resampling lifts SMC’s Pass@1 by\+4\.47\+4\.47pts over indep \(0\.6343 vs 0\.5896, outside1​σ1\\sigmaat each seed\) but costs diversity: mode rate\+4\.60\+4\.60pts worse, distinct\-4−27\.52\-27\.52pts worse \(0\.2600 vs 0\.5352, annn\-gram overlap the answer\-level mode\-rate underestimates\), and Pass@8−3\.73\-3\.73pts below indep\. The SMC\>\>indep crossover falls betweenk=4k\{=\}4andk=8k\{=\}8, consistent with premature exploitation in PRM\-guided SMC\([Giannone et al\.,, 2025](https://arxiv.org/html/2610.00047#bib.bib7)\)\.

#### Observation 2: Best\-of\-NNlifts Pass@kkconsistently but pays on diversity\.

BoN\-N=16N\{=\}16reaches Pass@16=0\.817=0\.817\(\+3\.47\+3\.47pts above indep’s Pass@8\) and leads at matchedk≤8k\\leq 8\(Pass@1\+4\.23\+4\.23, Pass@8\+1\.05\+1\.05pts\)\. Argmax\-PRM reranking over the indep population lifts Pass@1 by\+6\.45\+6\.45pts—a property of the PRM as ranker, not a distinct mechanism, and one that survives the comparator correction almost exactly \(the absolute anchors both shifted by≈\\approx5–6 pts; the delta did not\)\. distinct\-4 degrades−9\.48\-9\.48pts atN=16N\{=\}16but is within 0\.2 pts of indep atN=8N\{=\}8\. BoN is a compute\-efficient Pass@kk\-frontier method with a diversity cost growing inNN; as §[5\.2](https://arxiv.org/html/2610.00047#S5.SS2)details, part of both the SMC and BoN lift over indep is a censoring artifact of the pruned arm rather than a quality difference\.

#### Observation 3, PPFG\-stag sits on the independent frontier on every axis\.

Pass@kkΔ\\Deltarelative to indep∈\[−0\.003,\+0\.001\]\\in\[\-0\.003,\+0\.001\]acrossk=1\.\.8k=1\.\.8, each delta is within the pooled seed\-levelσ\\sigma\. mode rate is statistically identical \(0\.7900\.790vs0\.7890\.789\)\. distinct\-4 is identical to three decimal places\. Tokens\-per\-problem is identical \(within 0\.3%\)\. PPFG\-stag is the only inference\-time mechanism in Table[1](https://arxiv.org/html/2610.00047#S5.T1)that does not trade either accuracy or diversity in either direction\. PPFG\-random sits within seed\-σ\\sigmaof both indep and PPFG\-stag on every axis\. The targeting rule does not differentiate the mechanism’s aggregate effect at this regime\.

Together, at this operating point the mechanism space splits cleanly—SMC trades high\-kkcoverage for low\-kkgain, BoN improves Pass@kkuniformly at modest diversity cost, and PPFG\-stag is statistically indistinguishable from independent on every axis, the finding §[5\.4](https://arxiv.org/html/2610.00047#S5.SS4)then characterizes mechanistically\.

### 5\.2A note on the answer comparator \(this revision\)

This revision corrects three issues found in a self\-audit after the numbers below were first computed; we disclose them here rather than silently republish corrected tables, following the same practice as an earlier public correction to this project’s results\.

#### Comparator unification\.

Two independent scoring paths had been used in different scripts: one callingtask\.is\_correctdirectly \(baremath\_verify\), the other a normalized\-string\- match*union*math\_verify, whose normalization step strips\\text\{\.\.\.\}wholesale—so two distinct\\text\{\.\.\.\}\-only answers both normalize to the empty string and compare equal, and the union rescues some genuine near\-misses baremath\_verifyrejects\. The two paths disagreed by roughly 9 points of Pass@1 on identical chains\. Every number in this paper now goes through one canonical path: correctness istask\.is\_correctverbatim, answers are always re\-extracted from a chain’s steps \(never read from a cached field, which we found could hold stale values from an earlier, less robust extractor\), and Pass@kkuses one implementation of the unbiased Chen et al\. \(2021\) estimator\. PPFG\-vs\-baseline*parity*is unaffected: every method shifts together under the corrected comparator, so the deltas that drive §[5\.4](https://arxiv.org/html/2610.00047#S5.SS4)’s claim are essentially unchanged even though the absolute Pass@1 values below moved down by roughly five points across the board relative to values reported in an earlier draft\.

#### Selector tie\-handling\.

The “BoN\-selector” row in Table[1](https://arxiv.org/html/2610.00047#S5.T1)\(argmax\-PRM reranking of the independent population\) previously broke ties by taking the first chain index, which is a biased convention rather than a principled one\. It now reports the tie\-averaged expected accuracy \(the mean correctness over all chains tied for the top PRM score\)\. The corrected value is lower in absolute terms than the uncorrected one, but the*delta*over indep is close to unchanged \(§[5\.1](https://arxiv.org/html/2610.00047#S5.SS1), Observation 2\)\.

#### Censoring\.

Pass@1 is mean per\-chain accuracy, and a pruned or step\-budget\-exhausted chain that never produces a\\boxed\{\}answer is scored as incorrect by construction, not skipped\. On Qwen this affects a modest share of the population; on LLaMA\-3\.1\-8B\-Instruct and DeepSeek\-R1\-Distill\-Qwen\-7B, roughly half the population is censored this way \(LLaMA: 47% no answer; DeepSeek: 52%\), and essentially none of the pruned or step\-exhausted chains on either architecture produce an extractable answer at all\. Read against accuracy conditional on producing an answer—LLaMA0\.5940\.594, DeepSeek0\.6490\.649, both broadly in line with each model’s typical reported MATH500 performance—the low absolute Pass@1 figures in Table[13](https://arxiv.org/html/2610.00047#A7.T13)are substantially a scoring\-denominator effect, not evidence the models reason qualitatively worse than expected\. We flag this so a reader does not independently conclude the harness is broken; a fuller accounting \(including a corresponding decomposition for SMC and BoN’s lift over indep in Table[1](https://arxiv.org/html/2610.00047#S5.T1), which is partly the same effect\) is in preparation for a longer version of this work\.

#### Scope of this pass\.

The corrections above are reflected in Table[1](https://arxiv.org/html/2610.00047#S5.T1), Table[13](https://arxiv.org/html/2610.00047#A7.T13), and the TOST calibration discussion below \(the TOST verdicts themselves are computed from PPFG\-vs\-indep*deltas*, which we confirmed above are essentially unchanged by the comparator correction, so Table[15](https://arxiv.org/html/2610.00047#A10.T15)did not need re\-derivation\)\. Three secondary appendix tables reporting*absolute*Pass@kkunder the earlier comparator—the injection\-format ablation \(Table[20](https://arxiv.org/html/2610.00047#A14.T20)\), theξmin\\xi\_\{\\min\}sweep \(Table[14](https://arxiv.org/html/2610.00047#A9.T14)\), and the compound\-gate end\-to\-end measurement \(Table[9](https://arxiv.org/html/2610.00047#A2.T9)\)—have not yet been re\-derived and are flagged inline; we have no reason to expect their qualitative, within\-table relative conclusions to change, but we have not verified this, and a fully re\-derived version accompanies the extended submission of this work\.

#### Equivalence testing \(TOST\)\.

Two one\-sided Welchtt\-tests atα=0\.05\\alpha=0\.05against pre\-registered bounds equal to the 3\-seed MDE \(±0\.018\\pm 0\.018Pass@1,±0\.031\\pm 0\.031Pass@8,±0\.022\\pm 0\.022mode rate\)\. Atn=5n\{=\}5seeds per cell, both nulls reject on every Qwen and LLaMA cell \(12/1212/12“eq\.”\); DSR1 reaches “eq\.” on stag\. Pass@1, “upper” on random Pass@1, and is inconclusive on its four Pass@8/mode rate cells \(seedσ\\sigmais22–3×3\\timeslarger\)\. Per\-cell verdicts and methodology in Appendix[J](https://arxiv.org/html/2610.00047#A10)\.

#### Mechanism fires reliably\.

PPFG\-stag generates 322 injection events across the3×500×N=83\{\\times\}500\{\\times\}N\{=\}8sweep \(21\.5%21\.5\\%per\-problem;16\.0%16\.0\\%of problems receive≥1\\geq 1; mean fragment3\.80±1\.643\.80\\pm 1\.64steps, quality0\.782±0\.0770\.782\\pm 0\.077\)\. The random control fires more often \(52\.2%52\.2\\%;28\.8%28\.8\\%of problems\) lacking a stagnation filter, otherwise comparable\. The targeting rule controls*where*injections land, not*which*fragments are extracted\.

### 5\.3Cross\-architecture replication

To verify the §[5\.1](https://arxiv.org/html/2610.00047#S5.SS1)parity is not specific to Qwen, we replicate the indep/PPFG\-stag/PPFG\-random comparison on two further base models—LLaMA\-3\.1\-8B\-Instruct\([Grattafiori et al\.,, 2024](https://arxiv.org/html/2610.00047#bib.bib9)\)\(different instruct\-tuning lineage\) and DeepSeek\-R1\-Distill\-Qwen\-7B\([DeepSeek\-AI,, 2025](https://arxiv.org/html/2610.00047#bib.bib4)\)\(Qwen\-family, distilled from a reasoning model\)—at fulln=500n\{=\}500MATH500, 3 seeds,N=8N\{=\}8, same hyperparameters\. Mistral\-7B\-Instruct\-v0\.3 was excluded \(Pass@10\.0940\.094, below the comparison floor\)\.

PPFG\-stag\-vs\-independent Pass@1 deltas \(full table \+ cross\-arch parity panel in Appendix[G](https://arxiv.org/html/2610.00047#A7)\): QwenΔ=−0\.0015\\Delta\{=\}\-0\.0015, LLaMAΔ=−0\.0026\\Delta\{=\}\-0\.0026, DeepSeekΔ=−0\.0014\\Delta\{=\}\-0\.0014, all within seed\-σ\\sigmaat everykk\. mode rate deltas: Qwen\+0\.0007\+0\.0007, LLaMA\+0\.0132\+0\.0132, DeepSeek−0\.0018\-0\.0018\. The parity is preserved across the2×2\\timesPass@1 gap between Qwen \(0\.64\) and LLaMA/DeepSeek \(0\.31–0\.32\)\.

#### PRM\-distribution audit \+ LLaMA pattern\.

PPFG fires*more*often on LLaMA \(4\.3%4\.3\\%τkill\\tau\_\{\\mathrm\{kill\}\}\-trigger\) and DeepSeek \(6\.4%6\.4\\%\) than Qwen \(2\.4%2\.4\\%\), so the parity is not a “PRM never prunes” artifact \(Appendix[G](https://arxiv.org/html/2610.00047#A7)\)\. Both variants show LLaMA mode rate slightly above independent \(\+0\.0132\+0\.0132stag,\+0\.0060\+0\.0060random\), a small adverse pattern absent on Qwen/DeepSeek; magnitudes are inside then=5n\{=\}5MDE band \(±0\.022\\pm 0\.022\) and both LLaMA mode rate TOST cells reach “equivalent,” ruling out a hidden effect above±2\.2\\pm 2\.2pts\. We read it as an architecture\-specific small adverse signal bounded by the equivalence test, not a differentiating effect\.

### 5\.4Targeting\-policy analysis

The random\-rule control \(Table[1](https://arxiv.org/html/2610.00047#S5.T1)\) fires at2\.4×2\.4\\timesthe stagnation rate yet matches its aggregate metrics within seed\-σ\\sigmaon every column—the strongest evidence that targeting is not the binding constraint\. As to*why*the heuristic misfires: bucketing the 55 stag events on the first\-100 Qwen slice shows only6/556/55\(11%11\\%\) landed on a struggling chain with room to act; the wasted 49 are dominated by high\-flat\-PRM \(76%76\\%\), since the flatness check cannot separate consistent competence from consistent struggling\. The same classifier on the fulln=500n\{=\}500corpus \(322 events\) reproduces this \(14%14\\%well\-targeted,67%67\\%high\-flat\-PRM\)\. The well\-targeted share is a complement\-of\-unions residual and its precise value would shift under different plateau/near\-completion thresholds; the robust claim is the qualitative dominance of cosmetic\-plateau targets, not the exact14%14\\%\. A compound\-gate refinement confirms end\-to\-end a structural tradeoff: strict \(τlow=0\.60\\tau\_\{\\mathrm\{low\}\}\{=\}0\.60\) gives clean buckets at∼5\\sim 5events/cell, looser \(0\.700\.70\) restores density but high\-flat\-PRM creeps back to44%44\\%—both within seed\-σ\\sigmaof independent\. No targeting heuristic on population\-state\-only signals shows benefit in this regime \(Appendix[B](https://arxiv.org/html/2610.00047#A2)\)\. Stratifying the 322 events by problem difficulty separates two accounts of the low well\-targeted rate: on the hardest tercile, where the base\-rate explanation least applies, the well\-targeted share rises only to15\.6%15\.6\\%\(from14\.0%14\.0\\%overall\) while high\-flat\-PRM stays dominant at62\.3%62\.3\\%, so PRM flatness rather than base\-rate success is the binding constraint \(Appendix[B\.2](https://arxiv.org/html/2610.00047#A2.SS2)\)\.

#### Synthesis\.

Every targeting threshold we tested either admits cosmetic\-plateau false positives \(Appendix[B](https://arxiv.org/html/2610.00047#A2)\) or empties the injection budget below the level at which any aggregate metric would move; the three\-axis novelty \(PRM\-pruned trigger, in\-place sibling injection, verbatim fragment\) is operationally realized but produces no detectable benefit\. The random\-rule control’s matching parity at2\.4×2\.4\\timesthe stagnation rate \(§[5\.4](https://arxiv.org/html/2610.00047#S5.SS4), Table[1](https://arxiv.org/html/2610.00047#S5.T1)\) places the inertness on the mechanism at this corner, not the particular heuristic\. §[6](https://arxiv.org/html/2610.00047#S6)separates this from claims about the broader fragment\-grafting design space\.

#### ξmin\\xi\_\{\\min\}sensitivity\.

Sweepingξmin∈\{0\.10,0\.30,0\.50,0\.70\}\\xi\_\{\\min\}\\in\\\{0\.10,0\.30,0\.50,0\.70\\\}\(Qwen/Math\-Shepherd/MATH500\), injection rate falls monotonically from16\.0%16\.0\\%to2\.4%2\.4\\%\(gate functional\) yet no setting moves Pass@kkor mode rate outside then=3n\{=\}3MDE band: the gate rejects injections but is not the binding constraint \(Appendix[I](https://arxiv.org/html/2610.00047#A9)\)\.

#### Population size\.

Because grafting is a population\-level mechanism, we sweepNNover a32×32\\timesrange \(N=8N\{=\}8toN=256N\{=\}256,n=100n\{=\}100slice, 3 seeds/cell, matched populations\)\. Parity holds at everyNNand never turns positive; the binding cell isN=16N\{=\}16Pass@8 at−0\.27\-0\.27pp, and atN=256N\{=\}256parity is essentially exact \(indep Pass@10\.70730\.7073vs\. PPFG\-stag0\.70620\.7062, seed\-σ≤0\.0014\\sigma\\leq 0\.0014\)\. Larger populations do not make the transfer useful \(Appendix[P](https://arxiv.org/html/2610.00047#A16)\)\. Thisn=100n\{=\}100sweep is a distinct data slice from Table[1](https://arxiv.org/html/2610.00047#S5.T1)and has not yet been re\-derived under the §[5\.2](https://arxiv.org/html/2610.00047#S5.SS2)canonical comparator; we expect the parity conclusion \(a relative, within\-sweep comparison\) to be robust to the correction, as it was for then=500n\{=\}500cells, but have not verified this for this slice specifically\.

#### Compute matching\.

At matched tokens\-per\-problem \(∼4000\\sim 4000forN=8N\{=\}8methods\), BoN\-N=8N\{=\}8and indep lead Pass@4 \(0\.8090\.809\), SMC0\.7890\.789, PPFG\-stag0\.8070\.807; at2×2\\timescompute BoN\-N=16N\{=\}16reaches Pass@16=0\.891=0\.891\(pre\-§[5\.2](https://arxiv.org/html/2610.00047#S5.SS2)\-correction values; cf\. Table[1](https://arxiv.org/html/2610.00047#S5.T1)for the canonical Pass@4/Pass@16 figures on this samen=500n\{=\}500population\)\. PPFG\-stag never strictly dominates indep, SMC, or BoN at any Pass@kkunder either comparator \(Appendix[H](https://arxiv.org/html/2610.00047#A8), Figure[4](https://arxiv.org/html/2610.00047#A8.F4)\)\.

#### Ceiling\-regime replication \(GSM8K\)\.

On GSM8K\([Cobbe et al\.,, 2021](https://arxiv.org/html/2610.00047#bib.bib3)\)at Pass@1≈0\.87\\approx 0\.87\(Qwen2\.5\-7B\-Instruct,n=500×3n\{=\}500\{\\times\}3seeds\), indep, PPFG\-stag, and PPFG\-random produce Pass@1 identical to three decimals \(max\|Δ\|=0\.03\|\\Delta\|\{=\}0\.03pp, within the 5\-seed MDE±1\.4\\pm 1\.4pp\), Pass@8 within0\.00\.0pp, mode rate within0\.270\.27pp\. Injection rates fall to2\.8%2\.8\\%/9\.5%9\.5\\%\(vs16\.0%16\.0\\%/28\.8%28\.8\\%on MATH500\) since chains rarely dip belowτkill=0\.4\\tau\_\{\\mathrm\{kill\}\}\{=\}0\.4on arithmetic; the mechanism fires in proportion to task difficulty \(Appendix[C](https://arxiv.org/html/2610.00047#A3)\)\.

### 5\.5Mechanistic spot\-check: post\-injection PRM delta

For every PPFG\-stag injection event across the three Qwen2\.5\-7Bn=500n\{=\}500seeds \(N=322N\{=\}322\), we compute the change in the target chain’s process\-reward score around the injection step:Δ=PRM¯\[k,k\+3\)−PRM¯\[k−3,k\)\\Delta=\\overline\{\\mathrm\{PRM\}\}\_\{\[k,\\,k\+3\)\}\-\\overline\{\\mathrm\{PRM\}\}\_\{\[k\-3,\\,k\)\},kkthe grafting step index \(windows bounded by chain end; no event skipped\)\.

The meanΔ\\Deltais−0\.134\-0\.134\(median−0\.093\-0\.093, IQR\[−0\.215,−0\.031\]\[\-0\.215,\\,\-0\.031\]\); 272/322 events \(84\.5%\) haveΔ<0\\Delta<0\. A Wilcoxon signed\-rank test againstH0:median⁡\(Δ\)=0H\_\{0\}\\\!:\\,\\mathrm\{median\}\(\\Delta\)\{=\}0rejects decisively \(W=3497\.5W\{=\}3497\.5,p≈×10−40p\\approx 1\\\!\\times\\\!10^\{\-40\}\); Cliff’sδ=−0\.699\\delta=\-0\.699\(large effect\)\. This sharpens rather than contradicts the inertness verdict: the per\-event PRM signal is not noisy around zero but reliably depressed \(∼\\sim0\.09–0\.13 lower three steps after injection, despite mean fragment quality0\.78±0\.080\.78\\pm 0\.08\), consistent with the regime\-asymmetry account \(§[6](https://arxiv.org/html/2610.00047#S6.SS0.SSS0.Px2)\)—a donor fragment that scored highly natively lands as a foreign\-context stylistic discontinuity the same PRM penalizes\. This is inconsistent with the charitable “helps locally but averages out” reading: grafts do not appear to help locally either\.

#### Chain dynamics after injection\.

Bucketing the 322 events by the injected chain’s status three steps later,25\.8%25\.8\\%reach a pruned terminal status vs\. a9\.4%9\.4\\%matched\-step baseline on non\-injected siblings \(CI\[9\.0,9\.8\]\[9\.0,9\.8\]\), a2\.75×2\.75\\timesrate \(2\.59×2\.59\\timeson the 289\-event PRM\-trajectory\-matched subset, so the hazard is not pure selection; Appendix[L](https://arxiv.org/html/2610.00047#A12)\)\. A surviving\-sibling counterfactual on6565matched problem\-seed pairs finds no detectable compensation \(no promotion liftΔ=−0\.011\\Delta\{=\}\{\-\}0\.011, pop\-correctnessΔ=\+0\.015\\Delta\{=\}\{\+\}0\.015, per\-chain tokens slightly*lower*Δ=−20\.2\\Delta\{=\}\{\-\}20\.2, opposite a freed\-budget account\)\. The evidence is consistent with a selection\-dominated account: stagnation chains were pre\-selected for low/flat PRM and would have been pruned anyway; PPFG pulls pruning forward by≈3\\approx 3steps without redistributing completion\.

#### Empty\-bracket confound control\.

To test whether theΔ<0\\Delta<0effect is the out\-of\-distribution bracket wrapping rather than the foreign\-style step content, we ran an empty\-bracket PPFG\-stag variant \(identical targeting/extraction/gate; injected text replaced by a content\-free wrapper\) at full MATH500 \(n=500n\{=\}500, 3 seeds\)\. Over 241 events the meanΔ\\Deltais−0\.0736\-0\.0736\(W=4594W\{=\}4594,p≈5×10−20p\\approx 5\{\\times\}10^\{\-20\}; Cliff’sδ=−0\.515\\delta=\-0\.515\), so the depression has both a format and a content component; under \(untested\) additivity the content\-only term isΔcontent=−0\.060\\Delta\_\{\\mathrm\{content\}\}=\-0\.060, a≈55%/45%\\approx 55\\%/45\\%split we read as bounded between format\- and content\-dominant rather than a point estimate \(§[7](https://arxiv.org/html/2610.00047#S7)item 10\)\. A seamless variant \(no wrapper\) and a re\-attention variant \(post\-graft self\-check\) both leave Pass@kkwithin seed\-σ\\sigmaof independent, so neither remedy alone unlocks a benefit here \(full numbers: Appendices[D](https://arxiv.org/html/2610.00047#A4),[N](https://arxiv.org/html/2610.00047#A14)\)\. The residual content channel—the PRM mis\-pruning grafted chains that would otherwise have answered correctly, which would let a real benefit hide behind flat Pass@kk—is closed separately: granting injected chains a 3\-step exemption from threshold pruning measurably reduces pruning, yet graft content adds−0\.04\-0\.04pp at Pass@1 over a matched exemption\-only control with no spliced content \(Appendix[O](https://arxiv.org/html/2610.00047#A15)\)\. The elevated\-hazard pattern itself replicates descriptively on both further architectures \(2\.18×2\.18\\times,1\.83×1\.83\\times; Appendix[L\.1](https://arxiv.org/html/2610.00047#A12.SS1)\)\.

### 5\.6Replication across benchmarks \(Qwen\)

To establish inertness is not a MATH500 artifact, we replicate the three\-method sweep \(Qwen2\.5\-7B\-Instruct\) on four further benchmarks: AIME 2024\+2025 \(n=120n\{=\}120, near\-floor\), NuminaMath\-CoT\([Li et al\.,, 2024](https://arxiv.org/html/2610.00047#bib.bib13)\)\(n=99n\{=\}99\), GPQA\-Diamond\([Rein et al\.,, 2023](https://arxiv.org/html/2610.00047#bib.bib22)\)\(n=198n\{=\}198, both Math\-Shepherd and VersaPRM\([Zeng et al\.,, 2025](https://arxiv.org/html/2610.00047#bib.bib30)\)\), and MMLU\-Pro Law/Biology/Philosophy\([Wang et al\., 2024b,](https://arxiv.org/html/2610.00047#bib.bib25)\)\(n=500n\{=\}500, both PRMs\)\. VersaPRM is a multi\-domain LoRA adapter on Llama\-PRM800K; a held\-out smoke check confirmed calibration on math \(\+0\.076\+0\.076gap\) and law \(\+0\.423\+0\.423\) before any end\-to\-end run\.

The headline metrics across all four benchmarks are summarized in Appendix[A](https://arxiv.org/html/2610.00047#A1)\(Table[2](https://arxiv.org/html/2610.00047#A1.T2)\), with full 9\-cell per\-benchmark tables\. The absolute capability range spans0\.04170\.0417\(AIME\) to0\.41160\.4116\(Numina\); under VersaPRM, GPQA drops to0\.11090\.1109as the multi\-domain PRM is a stricter scorer at this depth\. Across all four benchmarks×\\timesboth PRMs, every PPFG\-vs\-independent contrast on Pass@1 and mode rate falls inside the per\-benchmark MDE\. The AIME \(0\.04170\.0417\) and VersaPRM\-GPQA \(0\.11090\.1109\) cells sit near the capability floor, so their parity is*corroborating*\(PPFG does not rescue a near\-floor regime\) rather than*discriminating*; the discriminating cells have non\-floor base rates \(NuminaMath, Math\-Shepherd\-GPQA, MMLU\-Pro, and the MATH500 headline\)\.

#### Cross\-architecture replication on the extended benchmarks\.

LLaMA\-3\.1\-8B\-Instruct and DeepSeek\-R1\-Distill\-Qwen\-7B extend the matrix on AIME, NuminaMath\-CoT, and GPQA\-Diamond \(Math\-Shepherd PRM\); cross\-architecture replication on GSM8K and MMLU\-Pro Law/Biology/Philosophy is deferred \(§[7](https://arxiv.org/html/2610.00047#S7)item 3\)\. Every PPFG\-vs\-indep contrast is inside the per\-benchmark MDE \(Appendix[E](https://arxiv.org/html/2610.00047#A5)\)\.

## 6Discussion

#### Practical implications\.

For parallel\-CoT inference with a math\-tuned PRM and a strong base LM, Best\-of\-NNdominates Pass@kkat matched compute and PPFG variants are indistinguishable from independent on every cell tested\.*Inert*here is population\-level \(no Pass@kkor mode rate effect\); per\-event the mechanism is mildly*adverse*\(§[5\.5](https://arxiv.org/html/2610.00047#S5.SS5):−0\.134\-0\.134PRM,2\.75×2\.75\\timeshazard\), a footprint the selection account explains and the population absorbs\.

#### Target for future falsification\.

The configuration we characterize is the corner where none of the five compensating ingredients prior work relies on \(separate\-solver\-pass re\-attention, learned distillation, training\-time gradient, verbal summarization/style adaptation, cross\-problem abstraction; §[2](https://arxiv.org/html/2610.00047#S2)\) is present\. PPFG’s corner \(PRM\-pruned trigger×\\timesin\-place sibling×\\timesverbatim fragment×\\timesheuristic targeting\) pays for none; these ingredients were derived inductively from those systems and are not exhaustive\. Two added in isolation \(the seamless and re\-attention variants, Appendix[N](https://arxiv.org/html/2610.00047#A14)\) also stay within seed\-σ\\sigmaof independent, consistent with the gains requiring more than a single ingredient\. Future work here should pre\-register which ingredient it supplies and test the prediction that a separate\-solver\-pass variant moves Pass@1\.

#### A null benchmark future work can pre\-register against\.

This paper is diagnostic: it characterizes a mechanism null and does not propose a positive method, and we decline to manufacture one we do not have\. What a diagnosis of this kind can supply instead is a number to beat, and the oracle analysis \(Appendix[M](https://arxiv.org/html/2610.00047#A13)\) is that number\. A hindsight oracle that gets to choose, per problem, between running PPFG\-stag and running independent—an upper bound on any possible per\-problem deployment gate over this mechanism—exceeds the independent baseline by at most\+0\.13\+0\.13pp acrossk∈\{1,2,4,8\}k\\in\\\{1,2,4,8\\\}, inside seed\-σ\\sigma\. Optimal deployment of bare verbatim grafting buys essentially nothing, so whatever leverage exists in this design space is not in deciding*when*to graft\.

Concretely, we propose two things of future inference\-time fragment\-grafting work\. \(a\) Name explicitly which compensating ingredient the design supplies, from the taxonomy above or beyond it\. \(b\) Report against the oracle ceiling: a Pass@kkgain within\+0\.13\+0\.13pp of independent has not beaten optimal per\-problem deployment of this graft\-content class, and a gain above it should be credited to the ingredient named in \(a\) rather than to the transfer itself\. The bound is scoped—it covers per\-problem deployment, not event\-level targeting \(injecting only a hand\-picked subset of events\) and not ingredient\-side modifications—but within that scope it converts a negative result into a pre\-registrable comparison\. Of the ingredients we were able to test cheaply, the prompt\-level ones do nothing: both seamless rendering and a post\-graft self\-check directive stay within seed\-σ\\sigma\(Appendix[N](https://arxiv.org/html/2610.00047#A14)\), which points at the heavier ingredients—training\-time integration, cross\-problem abstraction, or a genuine separate solver pass—rather than at a prompt\-level fix\. A separate\-solver\-pass re\-attention variant in the manner of[Miles et al\., \(2026\)](https://arxiv.org/html/2610.00047#bib.bib17)is not ruled out by anything we report here and is, on our reading, the most informative next experiment\.

## 7Limitations

1. 1\.Statistical power\.With three seeds per cell atn=500n=500, the minimum detectable effect at 80% power \(two\-sided,α=0\.05\\alpha=0\.05\) is approximately 2\.2 absolute points for mode rate, 3\.1 points for Pass@8, and 1\.8 points for Pass@1\. The §[5\.1](https://arxiv.org/html/2610.00047#S5.SS1)and §[5\.4](https://arxiv.org/html/2610.00047#S5.SS4)“indistinguishable from independent” claims should be read as “no improvement larger than 2\.2 points on mode rate, 3\.1 points on Pass@8, or 1\.8 points on Pass@1 detectable\.” The four\-bucket targeting analysis in §[B](https://arxiv.org/html/2610.00047#A2)was originally hand\-classified on the first\-100 slice \(55 events\); we have since reproduced it with the identical deterministic classifier on the fulln=500n=500corpus \(322 events\), with a consistent14%14\\%well\-targeted rate \(§[B](https://arxiv.org/html/2610.00047#A2)\), so the diagnostic is no longer slice\-bound\. A higher\-seed re\-run would further tighten the aggregate MDE bounds but is out of scope\.
2. 2\.PRM domain transfer\.Math\-Shepherd is math\-tuned with Mistral\-family training data; the §[5\.3](https://arxiv.org/html/2610.00047#S5.SS3)cells apply it unchanged to LLaMA\-3\.1\-8B\-Instruct and DeepSeek\-R1\-Distill\-Qwen\-7B outputs\. The Table[12](https://arxiv.org/html/2610.00047#A7.T12)audit shows that absolute score levels differ across architectures, but the prune\-trigger rate atτkill=0\.4\\tau\_\{\\mathrm\{kill\}\}=0\.4varies modestly \(2\.4% / 4\.3% / 6\.4%\), and the LLaMA and DeepSeek cells actually fire PPFG*more*often than Qwen\. The cross\-architecture parity is therefore not an artifact of the PRM failing to trigger\. §[5\.6](https://arxiv.org/html/2610.00047#S5.SS6)additionally substitutes VersaPRM \([Zeng et al\.,, 2025](https://arxiv.org/html/2610.00047#bib.bib30); a multi\-domain PEFT\-LoRA adapter on Llama\-PRM800K\) for Math\-Shepherd on GPQA\-Diamond and MMLU\-Pro Law/Biology/Philosophy and observes the same inertness pattern, empirically ruling out “the math\-tuned PRM is the wrong scorer” as an explanation of the negative finding at this regime\. Absolute accuracy on LLaMA\-3\.1\-8B\-Instruct and DeepSeek\-R1\-Distill\-Qwen\-7B in Table[13](https://arxiv.org/html/2610.00047#A7.T13)is substantially depressed by scoring censorship \(roughly half of each population produces no extractable answer and is scored incorrect by construction\); see §[5\.2](https://arxiv.org/html/2610.00047#S5.SS2)for the conditional\-accuracy figures and the answer\-comparator correction underlying all numbers in this revision\.
3. 3\.Base model coverage\.Experiments span Qwen2\.5\-7B\- Instruct, LLaMA\-3\.1\-8B\-Instruct, and DeepSeek\-R1\-Distill\-Qwen\-7B \(the latter is Qwen\-family but distilled from a reasoning model\)\. We note that DeepSeek\-R1\-Distill\-Qwen\-7B shares its base LM lineage with Qwen2\.5\-7B\-Instruct; LLaMA\-3\.1\-8B\-Instruct is therefore the genuinely cross\-family replication, while DeepSeek\-R1\-Distill\-Qwen\-7B provides a same\-family reasoning\-distilled replication at substantially different absolute Pass@kk\. Mistral\-7B\-Instruct\-v0\.3 was attempted but excluded\. Generalization to non\-Qwen, non\-LLaMA architecture families \(Gemma, OLMo, Mixtral\-of\-Experts variants\) untested\. Cross\-architecture replication on GSM8K and MMLU\-Pro Law/Biology/Philosophy is untested in this submission; these two benchmarks are reported on Qwen2\.5\-7B\-Instruct only, so the off\-math\-axis claim rests on Qwen alone for the two largest non\-math cells\.
4. 4\.Compute scale and injection density\.The headline cells run PPFG atN=8N=8\(N=16N=16appears only for the Best\-of\-NNbaseline\)\. Behavior at larger populations is no longer untested: a sweep toN=256N=256on then=100n\{=\}100slice \(Appendix[P](https://arxiv.org/html/2610.00047#A16)\) finds parity at every population size and no delta exceeding0\.270\.27pp in magnitude, with the effect never turning positive, so the “N=8N\{=\}8is too small for a population mechanism” hypothesis is not supported across a32×32\\timesrange\. That sweep runs atn=100n\{=\}100rather thann=500n\{=\}500, so it is a sensitivity analysis rather than a second headline result, and populations beyondN=256N\{=\}256remain untested\. The per\-chain injection capM=2M=2was fixed at small\-scale smoke testing and not re\-swept, so per\-chain density remains a candidate binding constraint the random\-rule control \(which raises per\-*population*density2\.4×2\.4\\times\) does not fully close\. We note, however, that raising density is an implausible route to benefit given the per\-event footprint: each injection locally depresses PRM by−0\.134\-0\.134and elevates the pruning hazard2\.75×2\.75\\times\(§[5\.5](https://arxiv.org/html/2610.00047#S5.SS5)\), so applying*more*of a locally\-adverse graft is not expected to help; anMMsweep would test this directly\.
5. 5\.English math, science, and graduate\-MCQ only\.Benchmarks span MATH500, AIME 2024\+2025, NuminaMath\-CoT, GSM8K \(math\), GPQA\-Diamond \(graduate science\), and MMLU\-Pro Law/Biology/Philosophy \(mixed\-domain MCQ\)\. All are English\-language with single\-answer ground\-truth \(numeric or letter\)\. Open\-ended generation, multilingual reasoning, and tasks without a checkable gold answer are untested\.
6. 6\.Heuristic policy\.A learned target\-selection policy is future work\.
7. 7\.No human evaluation of fragment quality or graft naturalness\.The §[5\.5](https://arxiv.org/html/2610.00047#S5.SS5)empty\-bracket control and the post\-injection PRM\-delta provide a verifier\-side proxy for graft naturalness, but human ratings of whether grafted text reads as natural continuation versus stylistic break would test the “stylistic discontinuity” interpretation of §[5\.5](https://arxiv.org/html/2610.00047#S5.SS5)more directly; the present paper relies on the PRM\-side proxy and does not commit to human evaluation\.
8. 8\.Per\-injection credit\-assignment analysis\(PopMC\-GRPO or comparable MC value estimation\) is not included in this submission; the §[5\.4](https://arxiv.org/html/2610.00047#S5.SS4)characterization rests on the three lines of evidence enumerated in §[1](https://arxiv.org/html/2610.00047#S1)\(parity, broadly replicated; targeting\-heuristic limitation, of which the false\-positive rate and compound\-gate infeasibility are two views; and suggestive mechanism\-level evidence from the PRM\-delta decomposition and the surviving\-sibling counterfactual\), none of which substitutes for the per\-injection MC value estimate this item defers\.
9. 9\.Compatibility\-gateξmin\\xi\_\{\\min\}was set to0\.00\.0in §[5\.1](https://arxiv.org/html/2610.00047#S5.SS1)–§[5\.4](https://arxiv.org/html/2610.00047#S5.SS4)results\. The four\-threshold sweep in Appendix[I](https://arxiv.org/html/2610.00047#A9)\(Table[14](https://arxiv.org/html/2610.00047#A9.T14)\) confirms inertness is preserved across non\-trivial gate settings \(injection rate falls monotonically from 16\.0% atξmin=0\\xi\_\{\\min\}=0to 2\.4% atξmin=0\.70\\xi\_\{\\min\}=0\.70with no Pass@kkor mode rate delta outside then=3n=3MDE band\), but the headline tables in §[5\.1](https://arxiv.org/html/2610.00047#S5.SS1)use theξmin=0\.0\\xi\_\{\\min\}=0\.0configuration\. The interpretation is that the gate’s role in this regime is to filter pathological inputs \(zero\-norm embeddings, fragment\-equals\-target degenerate cases\), not to alter the mechanism’s aggregate behavior\.
10. 10\.Mechanistic analysisis limited\. §[5\.5](https://arxiv.org/html/2610.00047#S5.SS5)reports the post\-vs\-pre injection PRM delta distribution and the empty\-bracket format/content decomposition\. The*causal*sub\-claim—that injection locally accelerates pruning beyond selection, resting on the2\.59×2\.59\\timesmatched\-PRM\-trajectory ratio \(Appendix[L](https://arxiv.org/html/2610.00047#A12)\)—is established on Qwen only\. The*descriptive*elevated\-hazard pattern does replicate on both further architectures \(2\.18×2\.18\\timeson DeepSeek\-R1\-Distill\-Qwen\-7B,1\.83×1\.83\\timeson LLaMA\-3\.1\-8B\-Instruct, CIs non\-overlapping from baseline in both; Appendix[L\.1](https://arxiv.org/html/2610.00047#A12.SS1)\), but a descriptive replication of the ratio is not a replication of the matched\-PRM\-trajectory control, and we do not treat it as one\. Deeper mechanistic probes, attention\-pattern analysis around the spliced span, token\-level logit comparison between counterfactual injection and no\-injection generations, are out of scope for this submission\. Finer\-grained ablations of the bracket markup—seamless \(no markup\) and re\-attention \(self\-check directive appended\)—were run as robustness checks; see Appendix[N](https://arxiv.org/html/2610.00047#A14)\.
11. 11\.Both PRMs may share a formatting sensitivity\.Two independent controls bound the confound that the PRM penalizes spliced*formatting*rather than the transfer being inert: seamless rendering removes all markup and leaves Pass@kkat parity \(Appendix[N](https://arxiv.org/html/2610.00047#A14)\), and a prune\-immunity arm with a matched null\-content control closes the content channel \(Appendix[O](https://arxiv.org/html/2610.00047#A15)\)\. Neither excludes the deeper possibility that Math\-Shepherd and VersaPRM*both*react to mid\-chain in\-context grafts as out\-of\-distribution text, in a way a differently\-trained scorer would not\. Our PRMs are two of a small family trained on broadly similar step\-level supervision; a PRM\-free pruning criterion, or a scorer trained with spliced\-context augmentation, would be needed to exclude this, and we did not run one\.
12. 12\.Equivalence testing\(Appendix[J](https://arxiv.org/html/2610.00047#A10)\) is conducted atn=5n\{=\}5seeds per cell across all three architectures \(Qwen2\.5\-7B, LLaMA\-3\.1\-8B, DeepSeek\-R1\-Distill\-Qwen\-7B\), against bounds pre\-registered from the 3\-seed MDE\. Both one\-sided nulls reject atα=0\.05\\alpha=0\.05on every Qwen and every LLaMA cell; on DSR1 only stag\. Pass@1 reaches positive equivalence \(random Pass@1 rejects only the upper\-side null and the four DSR1 Pass@8 / mode rate cells remain inconclusive because seed\-levelσ\\sigmaon this architecture is22–3×3\\timesthat of the other two\)\. The positive\-equivalence claim therefore covers13/1813/18cross\-arch cells outright; the four inconclusive DSR1 cells are within\-σ\\sigmabut not formally rejected, and we report them as such rather than widen the bound\.

## 8Ethics, broader impact

PPFG is an inference\-time mechanism evaluated on a fixed math\- reasoning benchmark; it introduces no new training data, no new model weights, and no new annotated corpus\. Its operational cost \(a sentence\-embedding compatibility score plus an in\-context demonstration splice on a chain decoded by a model the system already runs\) is small relative to the parallel\-CoT generation it conditions\. Because the mechanism characterization in §[5\.4](https://arxiv.org/html/2610.00047#S5.SS4)finds no detectable improvement at the configurations tested, there is no behavioral capability newly introduced by the method that requires safety review beyond what already attaches to the underlying base LMs and to Math\-Shepherd as a process reward model\. The paper’s primary broader\-impact contribution is methodological: a regime characterization of a frequently\-proposed mechanism class \(cross\-trajectory step\-level grafting at inference time\), which we hope will reduce the prevalence of speculative claims about inference\-time mechanism efficacy\. We release the full code, configs, analysis scripts \(targeting classifier, TOST, oracle, hazard\), CPU smoke tests, and saved trajectories at[https://github\.com/Khawaja\-Murad/ppfg\-anon](https://github.com/Khawaja-Murad/ppfg-anon)\.

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## Appendix APer\-benchmark replication details

This appendix contains the full per\-benchmark tables and prose summarized in §[5\.6](https://arxiv.org/html/2610.00047#S5.SS6)\.

Table 2:Replication summary across four further benchmarks \(Qwen2\.5\-7B\-Instruct, 3 seeds, indep vs PPFG\-stag; full per\-benchmark tables below\)\. AllΔ\\Deltavalues within MDE\.### A\.1AIME 2024\+2025 \(very\-hard math, near\-floor regime\)

The mechanism characterization rests on inertness holding across the capability range\. AIME 2024\+2025 \(120 problems combined\) places the base model near the difficulty floor: Qwen2\.5\-7B\-Instruct’s Pass@1 on this set is0\.040\.04\(N=120N\{=\}120, 3 seeds;∼\\sim15 chains of 360 ever produce the gold integer\)\. At this regime the question “could PPFG lift the floor?” is sharpest\. Table[3](https://arxiv.org/html/2610.00047#A1.T3)reports the 9\-cell sweep at the same operating point used elsewhere\. PPFG\-stag matches independent on Pass@1 to within0\.000\.00pp \(matched to four decimals:0\.04170\.0417vs\.0\.04170\.0417\), with mode rate\+1\.60\+1\.60pp; the random control sits on the same frontier\. We note that Pass@1 parity at this floor is weak evidence on its own: the estimator is deterministic on perfectly\-failed problems, so AIME corroborates that PPFG does not*rescue*a near\-floor regime but does not by itself discriminate inertness from task unsolvability\. We treat it as corroborating rather than as one of the discriminating cells\.

Table 3:AIME 2024\+2025 \(n=120n\{=\}120, 3 seeds, Qwen2\.5\-7B\-Instruct\)\. MDE±2\.4\\pm 2\.4pp \(Pass@1\) /±4\.7\\pm 4\.7pp \(mode rate\)\. The identical0\.04170\.0417across methods reflects55problems×3\\times\\,3seeds=15=15problem\-seed instances solved consistently at this floor regime; the unbiased Chen\-et\-al\-2021 Pass@1 estimator is deterministic on perfectly\-solved or perfectly\-failed problems, so differences between methods can only register on partial\-success problems, of which there are essentially none at Pass@1≈0\.04\\approx 0\.04\.
### A\.2NuminaMath\-CoT \(olympiad\-tier mid\-difficulty math\)

NuminaMath\-CoT\([Li et al\.,, 2024](https://arxiv.org/html/2610.00047#bib.bib13)\)fills the difficulty point between MATH500 \(Qwen Pass@1=0\.64=0\.64\) and AIME \(Qwen Pass@1=0\.04=0\.04\): mixed olympiad / AOPS / competition sources, 99 test problems with gold answers parseable from\\\\backslashboxed in the reference solutions\. PPFG\-stag matches independent on Pass@1 to four decimals \(0\.41160\.4116\); mode rateΔ=\+1\.02\\Delta=\+1\.02pp, within MDE \(±5\.2\\pm 5\.2pp\)\.

Table 4:NuminaMath\-CoT test \(n=99n\{=\}99, 3 seeds, Qwen2\.5\-7B\-Instruct\)\. MDE±2\.6\\pm 2\.6pp /±5\.2\\pm 5\.2pp\.
### A\.3GPQA\-Diamond \(cross\-PRM, off the math axis\)

GPQA\-Diamond\([Rein et al\.,, 2023](https://arxiv.org/html/2610.00047#bib.bib22)\)extends the characterization off the math axis: 198 expert\-graduate\-level MCQ across physics, biology, and chemistry\. We run the cross\-PRM cell with both Math\-Shepherd and VersaPRM\([Zeng et al\.,, 2025](https://arxiv.org/html/2610.00047#bib.bib30)\)\. Under Math\-Shepherd, Qwen2\.5\-7B’s Pass@1 is0\.23630\.2363; under VersaPRM it drops to0\.11090\.1109\(the multi\-domain PRM is the stricter scorer at this depth, but the ordering of methods is preserved\)\. All PPFG contrasts within MDE under both PRMs \(Table[5](https://arxiv.org/html/2610.00047#A1.T5)\)\. The PRM\-domain confound is empirically ruled out at this regime\.

Table 5:GPQA\-Diamond cross\-PRM \(n=198n\{=\}198, 3 seeds, Qwen2\.5\-7B\-Instruct\)\. MDE±1\.9\\pm 1\.9pp /±3\.7\\pm 3\.7pp\.
### A\.4MMLU\-Pro Law / Biology / Philosophy \(stratifiedn=500n\{=\}500\)

MMLU\-Pro\([Wang et al\., 2024b,](https://arxiv.org/html/2610.00047#bib.bib25)\)carries the regime characterization fully off the math axis at then=500n\{=\}500scale:≈167\\approx 167problems each from law, biology, and philosophy \(seed=42=42\), three\- method sweep under both PRMs\. Under Math\-Shepherd, Qwen2\.5\-7B Pass@1=0\.394=0\.394; under VersaPRM,0\.3250\.325\. All PPFG contrasts within MDE on Pass@1 and mode rate under both PRMs \(Table[6](https://arxiv.org/html/2610.00047#A1.T6)\)\. Per\-subject \(biology/law/philosophy\) Pass@1 deltas under Math\-Shepherd are all within±0\.20\\pm 0\.20pp of zero; inertness holds per\-subject\.

Table 6:MMLU\-Pro Law/Biology/Philosophy stratified subset \(n=500n\{=\}500, 3 seeds\)\.

## Appendix BTargeting\-policy diagnostics in detail

This appendix contains the full four\-bucket classification and the compound\-gate dry\-run \+ end\-to\-end measurement summarized in §[5\.4](https://arxiv.org/html/2610.00047#S5.SS4)\.

### B\.1Four\-bucket false\-positive classification

All 55 PPFG\-stag injection events from the first\-100 slice of then=500n\{=\}500Qwen corpus were classified along four mutually\-non\-exclusive buckets defined as deterministic functions of saved trajectory data:target\-succeeded\(target chain was scored correct\),high\-flat\-PRM\(target’s three\-step pre\-injection PRM had mean≥0\.6\\geq 0\.6and range<0\.05<0\.05\),near\-completion\(target produced its answer within two steps after injection\), andplausibly\-helped\(the complement of the union of the other three\)\. The classifier is deterministic and the bucket definitions are reproduced by an independent re\-implementation; we do not claim inter\-annotator agreement\.

Figure 1:Four\-bucket classification of the 55 PPFG\-stag injection events\. Buckets are not mutually exclusive\. Only6/556/55\(11%11\\%\) events fall in*plausibly\-helped*\(the complement of the union of the other three\); equivalently,49/5549/55\(89%89\\%\) fall into at least one state, \{high\-flat\-PRM, near\-completion, target\-succeeded\}, in which a rescue graft cannot change the outcome\. We report the6/556/55well\-targeted share as the primary quantity and the complement as the share of wasted fires\.Table 7:Four\-bucket counts on the first\-100 slice \(55 events\); buckets are not mutually exclusive\.#### Full\-corpus replication \(n=500n\{=\}500, 322 events\)\.

To verify the well\-targeted rate is not an artifact of the first\-100 slice, we applied the identical deterministic classifier to the full three\-seedn=500n\{=\}500PPFG\-stag corpus \(322 injection events\)\. The well\-targeted \(plausibly\-helped\) share is45/322=14\.0%45/322=14\.0\\%\(vs\.10\.9%10\.9\\%on the slice\); high\-flat\-PRM remains the dominant disqualifier at67\.1%67\.1\\%\(vs\.76\.4%76\.4\\%\), with near\-completion48\.8%48\.8\\%and target\-succeeded18\.9%18\.9\\%\. The picture is unchanged at the larger scale: roughly seven in eight stagnation fires land where a rescue graft cannot help, and the cosmetic\-plateau bucket dominates\. The classifier is byte\-identical to the slice analysis \(scripts/reclassify\_buckets\_n500\.pyreuses theinter\_annotator\_classify\.pyevent\-gathering and classification functions\); output inresults/bucket\_classification\_n500\.json\.

### B\.2Difficulty\-stratified targeting

At an operating point where independent Pass@1 is≈0\.64\\approx 0\.64, the “86%86\\%of fires do not need rescue” statistic admits two readings that the aggregate does not separate\. Under a*base\-rate*account, the target\-succeeded and near\-completion buckets are inflated mechanically because the model already succeeds on most chains; under a*PRM\-landscape*account, the binding constraint is that the PRM score surface is flat and high even on chains that are struggling, so population\-state signals cannot identify a rescue target\. The two make opposite predictions on hard problems, where base\-rate success is low: under the base\-rate account the well\-targeted share should rise substantially; under the PRM\-landscape account it should not\.

We stratify all 322 events by problem difficulty\. Terciles are built from pooled independent\-baseline Pass@1 \(24 chains per problem, seeds 42–44, fixed extractor\), so the stratification is computed independently of the PPFG population being classified\. Event gathering and bucket classification are unmodified, so totals reconcile against the full\-corpus numbers above\.

Table 8:Targeting quality by problem\-difficulty tercile \(322 PPFG\-stag events, Qwen2\.5\-7B\-Instruct \+ MATH500n=500n\{=\}500, 3 seeds\)\. Terciles are defined by pooled independent\-baseline Pass@1\. The stagnation rule concentrates firing on hard problems, but the well\-targeted share barely rises there, while high\-flat\-PRM remains dominant\.#### Results\.

Two things follow from Table[8](https://arxiv.org/html/2610.00047#A2.T8)\. First, the stagnation rule is not wasting its budget on easy problems:66%66\\%of events land in the hardest tercile, and only2%2\\%in the easiest\. Second, and bearing directly on the two accounts, restricting to the hardest tercile—where base\-rate inflation least applies—the well\-targeted share rises only to15\.6%15\.6\\%, against14\.0%14\.0\\%over the full corpus\. That is the direction a base\-rate account predicts, but far short of its magnitude: if the null were primarily a base\-rate artifact, the hard tercile should show a materially higher well\-targeted rate\. Meanwhile high\-flat\-PRM remains the dominant disqualifier at62\.3%62\.3\\%on the hardest problems, outweighing well\-targeting roughly4:14\{:\}1\. This makes precise the pattern already visible in the first\-100\-versus\-full\-500 split \(11%11\\%vs\.14%14\\%\): PRM flatness, not base\-rate success, is the binding constraint on targeting quality in this regime\. Output inresults/targeting\_by\_difficulty\_stratum\.json; code inscripts/targeting\_by\_difficulty\_stratum\.py\.

### B\.3Compound\-gate refinement \(dry\-run \+ end\-to\-end\)

The compound version additionally required \(a\) target mean recent PRM belowτlow\\tau\_\{\\mathrm\{low\}\}, \(b\)≥4\\geq 4steps beforeTmaxT\_\{\\mathrm\{max\}\}, \(c\) target’s latest PRM below the median of active siblings, and \(d\)≥3\\geq 3active siblings\. A dry\-run on the 55 events atτlow∈\{0\.60,0\.70\}\\tau\_\{\\mathrm\{low\}\}\\in\\\{0\.60,0\.70\\\}admits9\.1%9\.1\\%and16\.4%16\.4\\%respectively\. The end\-to\-end measurement atn=500×3n\{=\}500\{\\times\}3seeds confirms the dry\-run: both thresholds produce deltas within seed\-σ\\sigmaof independent \(Table[9](https://arxiv.org/html/2610.00047#A2.T9)\)\.

Table 9:End\-to\-end measurement of the compound\-gate variant at two thresholds \(3 seeds×\\times500 problems each\)\. Pre\-comparator\-correction values \(§[5\.2](https://arxiv.org/html/2610.00047#S5.SS2)\); the within\-table parity finding is not expected to change\.

## Appendix CGSM8K replication \(full table\)

To rule out a MATH500\-specific artifact, we replicate at the same operating point \(Qwen2\.5\-7B\-Instruct,n=500n\{=\}500, 3 seeds,N=8N\{=\}8\) on GSM8K\([Cobbe et al\.,, 2021](https://arxiv.org/html/2610.00047#bib.bib3)\), intentionally the easier task \(Pass@1≈0\.87\\approx 0\.87, near ceiling\), where the strong\-base\-LM caveat \(§[7](https://arxiv.org/html/2610.00047#S7)item 1\) is most acute\. indep, PPFG\-stag, and PPFG\-random produce Pass@1 identical to three decimals \(max\|Δ\|=0\.03\|\\Delta\|\{=\}0\.03pp, within the 5\-seed MDE±1\.4\\pm 1\.4pp\)\. The proportional drop in injection rate \(2\.8%2\.8\\%/9\.5%9\.5\\%vs16\.0%16\.0\\%/28\.8%28\.8\\%on MATH500\) is itself characterization: the mechanism fires in proportion to how often the PRM identifies pruneable chains, which is itself a function of task difficulty\.

Table 10:GSM8K replication \(n=500n\{=\}500, 3 seeds; tpp≈1891\\approx 1891for all rows\)\.
## Appendix DEmpty\-bracket confound control \(full per\-event detail\)

Over 241 empty\-bracket injection events across three seeds, the per\-eventΔ\\Deltadistribution has mean−0\.0736\-0\.0736\(median−0\.0434\-0\.0434, IQR\[−0\.133,−0\.005\]\[\-0\.133,\-0\.005\], std0\.11250\.1125\); 182/241 events \(75\.5%75\.5\\%\) haveΔ<0\\Delta<0and 58 \(24\.1%24\.1\\%\) haveΔ\>0\\Delta\>0\. Wilcoxon signed\-rank againstH0:median⁡\(Δ\)=0H\_\{0\}\\\!:\\,\\mathrm\{median\}\(\\Delta\)\{=\}0rejects \(W=4594W\{=\}4594,p≈5×10−20p\\approx 5\{\\times\}10^\{\-20\}\); Cliff’sδ=−0\.515\\delta=\-0\.515\(large effect\)\.Δstag\\Delta\_\{\\mathrm\{stag\}\}andΔempty\\Delta\_\{\\mathrm\{empty\}\}are each individually significant \(p<10−19p<10^\{\-19\}\)\. The derivedΔcontent=Δstag−Δempty=−0\.060\\Delta\_\{\\mathrm\{content\}\}=\\Delta\_\{\\mathrm\{stag\}\}\-\\Delta\_\{\\mathrm\{empty\}\}=\-0\.060is presented under the additivity assumption stated in §[5\.5](https://arxiv.org/html/2610.00047#S5.SS5)and bounded between format\-dominant and content\-dominant readings as discussed there; we do not assignΔcontent\\Delta\_\{\\mathrm\{content\}\}its own significance test in this submission\. Under that assumption a portion of the post\-injection PRM depression is attributable to the bracketed\-demonstration wrapping that any in\-context graft would carry, and the rest to spliced step content scored as stylistically discontinuous by the PRM; consistent with the main\-text reading we treat this only as bounded between format\-dominant and content\-dominant, not as a≈\\approx50/50 point estimate\.

The empty\-bracket event count \(241\) is lower than PPFG\-stag \(322\) at matchednnand rule because population state diverges after the first injection, so subsequent targeting decisions differ; the rule definition is identical but the population\-state context at each step is not\.

## Appendix ECross\-architecture replication on the extended benchmarks \(full table\)

Full3×3\\times\-benchmark×\\times2\-architecture sweep summarized in §[5\.6](https://arxiv.org/html/2610.00047#S5.SS6); Math\-Shepherd PRM throughout, 3 seeds per cell\.

ArchMethodPass@1Pass@2Pass@4Pass@8Mode rateΔ\\DeltaP@1 vs indep*AIME 2024\+2025 \(n=120n\{=\}120\)*MDE±2\.4/4\.7\\pm 2\.4/4\.7ppLLaMA\-3\.1\-8B\-Instructindependent0\.01080\.01860\.02970\.04440\.6020—PPFG\-stag0\.01010\.01750\.02820\.04170\.6219−0\.07\-0\.07PPFG\-random0\.00940\.01610\.02560\.03890\.6534−0\.14\-0\.14DeepSeek\-R1\-Distill\-Qwen\-7Bindependent0\.00940\.01730\.03010\.05000\.6647—PPFG\-stag0\.00900\.01660\.02870\.04720\.6764−0\.04\-0\.04PPFG\-random0\.00970\.01790\.03090\.05000\.6521\+0\.03\+0\.03*NuminaMath\-CoT \(n=99n\{=\}99\)*MDE±2\.6/5\.2\\pm 2\.6/5\.2ppLLaMA\-3\.1\-8B\-Instructindependent0\.25380\.32030\.37960\.43430\.6626—PPFG\-stag0\.25250\.31810\.37650\.43100\.6676−0\.13\-0\.13PPFG\-random0\.25040\.31540\.37170\.42090\.6500−0\.34\-0\.34DeepSeek\-R1\-Distill\-Qwen\-7Bindependent0\.22560\.30480\.38120\.46130\.6073—PPFG\-stag0\.22260\.29980\.37460\.45450\.6050−0\.30\-0\.30PPFG\-random0\.22600\.30650\.38550\.46800\.6257\+0\.04\+0\.04*GPQA\-Diamond \(n=198n\{=\}198\)*MDE±1\.9/3\.7\\pm 1\.9/3\.7ppLLaMA\-3\.1\-8B\-Instructindependent0\.10230\.17200\.26870\.39730\.6001—PPFG\-stag0\.09550\.15990\.24850\.36700\.6154−0\.68\-0\.68PPFG\-random0\.09110\.15160\.23290\.33670\.6347−1\.12\-1\.12DeepSeek\-R1\-Distill\-Qwen\-7Bindependent0\.13700\.22370\.33480\.46460\.5903—PPFG\-stag0\.13450\.22160\.33460\.46630\.5997−0\.25\-0\.25PPFG\-random0\.12710\.20810\.31270\.43770\.5932−0\.99\-0\.99

Table 11:Cross\-architecture sweep on extended benchmarks\. All contrasts within per\-benchmark MDE\.
## Appendix FPPFG algorithm \+ architectural diagram

Figure 2:The PPFG mechanism\. When a chain is pruned by the PRM \(red status box\), its high\-PRM prefix is extracted; the four PPFG sub\-operations \(extract / target\-select / gate / inject\) select a still\-decoding sibling chain and splice the fragment into the sibling’s prompt context as a bracketed in\-place demonstration\.Algorithm 1PPFG in a parallel\-CoT population\.1:problem

qq, base model

πθ\\pi\_\{\\theta\}, PRM

ρ\\rho, sentence embedder

ϕ\\phi, params

\(N,τkill,τextract,kmin,kmax,M,ξmin,R,Tmax\)\(N,\\tau\_\{\\mathrm\{kill\}\},\\tau\_\{\\mathrm\{extract\}\},k\_\{\\mathrm\{min\}\},k\_\{\\mathrm\{max\}\},M,\\xi\_\{\\mathrm\{min\}\},R,T\_\{\\mathrm\{max\}\}\)
2:final answer \(or

⊥\\bot\)

3:init

P0←NP\_\{0\}\\leftarrow Nchains with prompt

qq, status

==ACTIVE

4:for

t=1,2,…,Tmaxt=1,2,\\dots,T\_\{\\mathrm\{max\}\}do

5:

At←\{C∈Pt:C\.𝑠𝑡𝑎𝑡𝑢𝑠=ACTIVE\}A\_\{t\}\\leftarrow\\\{C\\in P\_\{t\}:C\.\\mathit\{status\}=\\mathrm\{ACTIVE\}\\\}
6:if

At=∅A\_\{t\}=\\emptysetthenbreak

7:endif

8:for

C∈AtC\\in A\_\{t\}in batchdo

9:

C\.𝑎𝑝𝑝𝑒𝑛𝑑⁡\(sample​st∼πθ\)C\.\\mathit\{append\}\(\\mathrm\{sample\}\\ s\_\{t\}\\sim\\pi\_\{\\theta\}\)
10:

C\.𝑝𝑟𝑚\_𝑠𝑐𝑜𝑟𝑒𝑠\.𝑎𝑝𝑝𝑒𝑛𝑑\(ρ\(q,C\.𝑠𝑡𝑒𝑝𝑠\)\)C\.\\mathit\{prm\\\_scores\}\.\\mathit\{append\}\(\\rho\(q,C\.\\mathit\{steps\}\)\)
11:endfor

12:

𝑛𝑒𝑤𝑙𝑦​\_​𝑝𝑟𝑢𝑛𝑒𝑑←\[\]\\mathit\{newly\\\_pruned\}\\leftarrow\[\\,\]
13:for

C∈AtC\\in A\_\{t\}do

14:if

𝑑𝑒𝑡𝑒𝑐𝑡\_𝑓𝑖𝑛𝑎𝑙\_𝑎𝑛𝑠𝑤𝑒𝑟\(C\.𝑠𝑡𝑒𝑝𝑠\[−1\]\)≠⊥\\mathit\{detect\\\_final\\\_answer\}\(C\.\\mathit\{steps\}\[\-1\]\)\\neq\\botthen

C\.𝑠𝑡𝑎𝑡𝑢𝑠←PROMOTEDC\.\\mathit\{status\}\\leftarrow\\mathrm\{PROMOTED\}
15:elseif

C\.𝑝𝑟𝑚​\_​𝑠𝑐𝑜𝑟𝑒𝑠​\[−1\]<τkillC\.\\mathit\{prm\\\_scores\}\[\-1\]<\\tau\_\{\\mathrm\{kill\}\}then

C\.𝑠𝑡𝑎𝑡𝑢𝑠←PRUNEDC\.\\mathit\{status\}\\leftarrow\\mathrm\{PRUNED\};

𝑛𝑒𝑤𝑙𝑦​\_​𝑝𝑟𝑢𝑛𝑒𝑑\.𝑎𝑑𝑑⁡\(C\)\\mathit\{newly\\\_pruned\}\.\\mathit\{add\}\(C\)
16:elseif

\|C\.𝑠𝑡𝑒𝑝𝑠\|≥Tmax\|C\.\\mathit\{steps\}\|\\geq T\_\{\\mathrm\{max\}\}then

C\.𝑠𝑡𝑎𝑡𝑢𝑠←HIT​\_​MAXC\.\\mathit\{status\}\\leftarrow\\mathrm\{HIT\\\_MAX\}
17:endif

18:endfor

19:for

Csrc∈𝑛𝑒𝑤𝑙𝑦​\_​𝑝𝑟𝑢𝑛𝑒𝑑C\_\{\\mathrm\{src\}\}\\in\\mathit\{newly\\\_pruned\}do⊳\\trianglerightPPFG hook

20:

F←F\\leftarrowlongest prefix of

CsrcC\_\{\\mathrm\{src\}\}with all PRM

≥τextract\\geq\\tau\_\{\\mathrm\{extract\}\}, length

∈\[kmin,kmax\]\\in\[k\_\{\\mathrm\{min\}\},k\_\{\\mathrm\{max\}\}\]
21:if

F=⊥F=\\botthencontinue

22:endif

23:

Ctgt←𝑠𝑒𝑙𝑒𝑐𝑡​\_​𝑡𝑎𝑟𝑔𝑒𝑡​\(F,𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒𝑠⁡\(At,M\),R\)C\_\{\\mathrm\{tgt\}\}\\leftarrow\\mathit\{select\\\_target\}\(F,\\mathit\{candidates\}\(A\_\{t\},M\),R\)
24:if

Ctgt=⊥C\_\{\\mathrm\{tgt\}\}=\\botthencontinue

25:endif

26:if

ξ⁡\(F,Ctgt\)<ξmin\\xi\(F,C\_\{\\mathrm\{tgt\}\}\)<\\xi\_\{\\mathrm\{min\}\}thencontinue

27:endif

28:inject

FFinto

CtgtC\_\{\\mathrm\{tgt\}\}’s prompt

29:endfor

30:endfor

31:

𝑎𝑛𝑠←\{C\.𝑓𝑖𝑛𝑎𝑙\_𝑎𝑛𝑠𝑤𝑒𝑟:C\.𝑠𝑡𝑎𝑡𝑢𝑠=PROMOTED\}\\mathit\{ans\}\\leftarrow\\\{C\.\\mathit\{final\\\_answer\}:C\.\\mathit\{status\}=\\mathrm\{PROMOTED\}\\\}
32:return

𝑚𝑎𝑗𝑜𝑟𝑖𝑡𝑦​\_​𝑣𝑜𝑡𝑒​\_​𝑜𝑟​\_​ℎ𝑖𝑔ℎ𝑒𝑠𝑡​\_​𝑚𝑒𝑎𝑛​\_​𝑝𝑟𝑚​\(𝑎𝑛𝑠\)\\mathit\{majority\\\_vote\\\_or\\\_highest\\\_mean\\\_prm\}\(\\mathit\{ans\}\)

## Appendix GCross\-architecture replication on MATH500 \(full table \+ PRM\-distribution audit\)

§[5\.3](https://arxiv.org/html/2610.00047#S5.SS3)summarizes the PPFG\-stag\-vs\-independent Pass@1 deltas for LLaMA\-3\.1\-8B\-Instruct and DeepSeek\-R1\-Distill\-Qwen\-7B\. Table[12](https://arxiv.org/html/2610.00047#A7.T12)gives the PRM\-score\-distribution audit underlying that section’s “PRM never prunes” check; Figure[3](https://arxiv.org/html/2610.00047#A7.F3)plots the parity panel; and Table[13](https://arxiv.org/html/2610.00047#A7.T13)gives the full Pass@kk/mode rate/distinct\-4/tpp breakdown per architecture and method underlying both\.

Table 12:PRM\-distribution audit: PPFG fires*more*often on LLaMA and DeepSeek than on Qwen, ruling out a “PRM never prunes” artifact\.Figure 3:Cross\-architecture parity panel\. Pass@kkvskkon full MATH500 \(n=500n=500, 3 seeds, mean±\\pmseed\-σ\\sigma\) for three base architectures\. PPFG\-stag and PPFG\-random sit within seed\-σ\\sigmaof the independent baseline at everykkon every architecture\.Table 13:Cross\-architecture results on full MATH500 \(n=500n=500\), mean±\\pmstd, canonical scoring path \(§[5\.2](https://arxiv.org/html/2610.00047#S5.SS2)\)\.*independent*rows: 5 seeds \(\{42,…,46\}\\\{42,\\ldots,46\\\}\); LLaMA/DeepSeek PPFG rows: 3 seeds \(a seed\-45/46 relaunch ambiguity for those cells is unresolved\)\. LLaMA/DeepSeek’s low absolute Pass@1 is substantially a censoring effect, not weaker per\-answer accuracy—see §[5\.2](https://arxiv.org/html/2610.00047#S5.SS2)\.
## Appendix HCompute matching figure

Figure 4:Pass@4 vs tokens\-per\-problem on MATH500 \(Qwen2\.5\-7B,n=500n\{=\}500, log\-xx\)\. BoN at matched compute dominates the matched\-tpp column; BoN atN=16N\{=\}16reaches Pass@16=0\.89=0\.89at2×2\\timescompute\.
## Appendix Iξmin\\xi\_\{\\min\}sensitivity sweep \(full table\)

§[5\.4](https://arxiv.org/html/2610.00047#S5.SS4)summarizes this sweep: injection rate falls monotonically asξmin\\xi\_\{\\min\}rises, yet no setting moves Pass@kkor mode rate outside the MDE band\. Table[14](https://arxiv.org/html/2610.00047#A9.T14)below gives the full per\-threshold breakdown\.

Table 14:ξmin\\xi\_\{\\min\}sensitivity sweep on Qwen2\.5\-7B \+ MATH500 \(n=500n=500, 3 seeds per cell\)\. Values in percentage points except tpp\. Pre\-comparator\-correction values \(§[5\.2](https://arxiv.org/html/2610.00047#S5.SS2)\); the within\-column null finding is not expected to change\.
## Appendix JTOST equivalence: methodology and per\-cell discussion

The Table[1](https://arxiv.org/html/2610.00047#S5.T1)PPFG\-vs\-independent deltas all sit within one seed standard error of zero, but a within\-1​σ1\\sigmawindow is not by itself an equivalence test: a reviewer is entitled to ask whether the data also rule out a meaningful hidden effect just below the variance floor\. We address this with two one\-sided Welchtt\-tests \(TOST\) atα=0\.05\\alpha=0\.05against*pre\-registered*bounds equal to the 3\-seed minimum detectable effect sizes from §[7](https://arxiv.org/html/2610.00047#S7)item 1,±0\.018\\pm 0\.018for Pass@1,±0\.031\\pm 0\.031for Pass@8, and±0\.022\\pm 0\.022for the answer\-mode rate\. These bounds were locked before any TOSTpp\-value was computed\.

On the Qwen rows the 5\-seed procedure positively rejects*both*one\-sided nulls atα=0\.05\\alpha=0\.05on every metric for both injection rules: at the pre\-registered bounds, the data are statistically inconsistent with PPFG hiding a positive or negative effect of more than±1\.8\\pm 1\.8points on Pass@1,±3\.1\\pm 3\.1on Pass@8, or±2\.2\\pm 2\.2on mode rate\. With the cross\-architecture cells also extended to 5 seeds \(\{42,…,46\}\\\{42,\\ldots,46\\\}\), all six LLaMA\-3\.1\-8B cells \(stag\. and random, both Pass@kkand mode rate\) now likewise reject*both*one\-sided nulls and return “equivalent” verdicts\. The DSR1\-Qwen\-7B cells remain the noisiest: stag\. Pass@1 reaches equivalence at 5 seeds and random Pass@1 rejects only the upper\-side null \(meanΔ=−0\.36\\Delta=\-0\.36pp\), but the four Pass@8 and mode rate cells stay inconclusive because the seed\-to\-seed standard deviations on DSR1 are 2–3×\\timesthe corresponding Qwen / LLaMA values\. We report each cell at its actual verdict \(eq\. / upper / lower / incon\.\) rather than widen the bound to manufacture equivalence; the inconclusive cells are consistent with the same within\-σ\\sigmaframing used throughout §[5\.3](https://arxiv.org/html/2610.00047#S5.SS3)rather than evidence of a hidden effect\.

#### Are the bounds practically negligible, or merely below detection?

The bounds are pre\-registered at the experiment’s MDE, which invites the objection that they measure our power rather than a smallest effect size of practical interest—particularly since 1–2 points of Pass@1 can matter in inference\-time\-scaling settings\. Two things answer this\. First, from the point estimate directly: the 5\-seed PPFG\-stag Pass@1 effect is−0\.12\-0\.12pp with a≈90%\\approx 90\\%CI of\[−1\.2,\+0\.9\]\[\-1\.2,\+0\.9\]pp\. The upper end of that interval sits*below*the\+1\+1pp threshold at which a practitioner would begin to care, so a gain of that size is excluded by the interval, not merely undetected\. Second, as an external calibration, the same margin is smaller than three real positive effects measured on this identical model/task/pipeline \(Table[1](https://arxiv.org/html/2610.00047#S5.T1)\): SMC at\+4\.47\+4\.47pp \(2\.5×2\.5\\timesthe Pass@1 bound\), Best\-of\-NNat2×2\\timescompute at\+4\.23\+4\.23pp \(2\.4×2\.4\\times\), and a zero\-extra\-compute argmax\-PRM reranking of the same chains at\+6\.45\+6\.45pp \(3\.6×3\.6\\times; this delta is essentially unchanged by the §[5\.2](https://arxiv.org/html/2610.00047#S5.SS2)comparator correction even though its two absolute anchors both moved\)\. The bound would have comfortably detected any of these; PPFG\-stag and PPFG\-random sit at−0\.13\-0\.13and−0\.19\-0\.19pp\. The margin is therefore calibrated against effects this setup demonstrably detects, not merely against its noise floor\. \(SMC and Best\-of\-NN’s lift over indep is itself partly a censoring artifact of the pruned arm, §[5\.2](https://arxiv.org/html/2610.00047#S5.SS2); we use them here only as a detection\-power calibration, not as evidence about their own mechanisms\.\)

Table 15:TOST equivalence verdicts\. Bounds pre\-registered at the 3\-seed MDE\. “eq\.” = both one\-sided nulls rejected \(positive equivalence\); “upper” = only upper\-side null rejected; “lower” = only lower; “incon\.” = neither rejected at thisnn/variance\. All three architectures now atn=5n\{=\}5seeds \(\{42,…,46\}\\\{42,\\ldots,46\\\}\)\.

## Appendix KDifferentiator table \(full\)

Table 16:Differentiator table\. PPFG composes three operational properties \(PRM\-pruned trigger, in\-place sibling injection mid\-decoding, verbatim fragment\) not jointly realized by prior fragment\-grafting work\.
## Appendix LSurviving\-sibling counterfactual analysis \(full procedure\)

This appendix substantiates the §[5\.5](https://arxiv.org/html/2610.00047#S5.SS5)adjudication between*compensation*and*selection*readings of the2\.75×2\.75\\timespost\-injection pruning hazard\.

#### Matched problem\-seed pairs\.

Let a*treated unit*be a pair\(p,s\)\(p,s\)such that the PPFG\-stag run at seedsscontains at least one chain on problemppthat \(i\) received an injection and \(ii\) reachedprunedstatus withink\+3k\+3steps of that injection \(the26%26\\%bucket of §[5\.5](https://arxiv.org/html/2610.00047#S5.SS5)\)\. Let the matched control be the independent baseline at the same\(p,s\)\(p,s\)\. We restrict to the three Qwen2\.5\-7B\-Instruct MATH500 cells at seeds4242–4444,n=500n\{=\}500,N=8N\{=\}8\. The treated population yieldsN=65N=65matched pairs \(22\+24\+1922\+24\+19across seeds\), of which the322322injection events of §[5\.5](https://arxiv.org/html/2610.00047#S5.SS5)are a superset \(multiple events can land on the same population\)\.

#### Three outcome variables\.

For a treated unit, compute over the*non\-injected siblings*\(the chains in the population that received no injection\): \(i\)pop\_correctness∈\{0,1\}\\in\\\{0,1\\\}, did any chain produce the gold\-matching boxed answer, identical to the Pass@N=8 contribution from that problem\-seed; \(ii\)per\_chain\_promotion, mean of𝟙\[status=promoted\]\\mathbb\{1\}\[\\text\{status\}=\\texttt\{promoted\}\]over non\-injected siblings; \(iii\)per\_chain\_tokens, mean ofn\_tokens\_generatedover non\-injected siblings\. For the matched control all eight chains are non\-injected by construction\.

#### Bootstrap CIs\.

Compute deltasΔ=A−B\\Delta=A\-Bon each of the three outcomes; percentile bootstrap CI95over50005000resamples of the6565matched pairs\.

#### Results\.

Table[17](https://arxiv.org/html/2610.00047#A12.T17)summarizes\.

Table 17:Surviving\-sibling counterfactual: deltas between PPFG\-stag treated populations \(injection\-and\-prune\-within\-3\) and matched\-problem\-seed independent baseline\. Mean and percentile bootstrap CI95over50005000resamples ofN=65N=65matched pairs\.
#### Interpretation\.

Two of three outcomes have CI95including zero \(pop\_correctness, per\_chain\_promotion\); the third \(per\_chain\_tokens\) is statistically significant in the*adverse*direction, surviving siblings in the PPFG\-treated populations consume≈0\.5%\\approx 0\.5\\%*fewer*tokens than the same\-problem\-seed independent chains, the opposite direction from a freed\-decoding\-budget compensation account\. AtN=65N\{=\}65matched pairs these three deltas constitute a failure to detect compensation rather than its formal exclusion: the pop\_correctness CI \(\[−0\.031,\+0\.062\]\[\-0\.031,\+0\.062\]\) would not detect a promotion lift below≈6\\approx 6absolute points on surviving siblings\. Within that power, there is no detectable uptick on surviving siblings that would explain how the receiving chain’s2\.75×2\.75\\timespruning hazard reconciles with population\-level Pass@kkparity, and the one significant outcome \(per\_chain\_tokens\) points in the*adverse*direction\. The evidence is consistent with a selection account: the receiving chain was on a doomed trajectory pre\-selected by the stagnation rule; injection pulls its pruning forward by approximately three steps; the rest of the population behaves as it would have without injection\. The−20\.2\-20\.2\-token delta is in the opposite direction from a freed\-budget compensation account and is small in magnitude \(≈0\.5%\\approx 0\.5\\%of per\-chain tpp\); we read it as consistent with the selection account \(chains entering the treated set are sampled from problems where the injected chain stagnated early\) but do not test this attribution directly\.

#### Matched\-PRM\-trajectory baseline\.

The naive matched\-step baseline of §[5\.5](https://arxiv.org/html/2610.00047#S5.SS5)\(prune rate9\.4%9\.4\\%across all non\-injected siblings at the same step indices\) averages over both healthy and stagnating chains\. The honest counterfactual for a stagnation\-rule\-injected chain is the prune rate of non\-injected siblings whose pre\-injection PRM trajectory matches the stagnation\-trigger profile: their latest\-3\-step PRM range is below the flatness thresholdτflat=0\.10\\tau\_\{\\mathrm\{flat\}\}\{=\}0\.10\. Over414414such matched sibling slots across the three Qwen seeds, the prune\-within\-3 rate is9\.9%9\.9\\%, only0\.50\.5absolute points above the naive baseline; the post\-vs\-matched\-PRM ratio is2\.59×2\.59\\times\(vs naive2\.72×2\.72\\timeson this289289\-event subset; the2\.75×2\.75\\timesquoted in §[5\.5](https://arxiv.org/html/2610.00047#S5.SS5)is the naive ratio over all322322events\)\. The selection account does not fully account for the receiving\-chain hazard: even chains with comparable pre\-injection PRM trajectories prune at less than half the post\-injection rate\. Injection is therefore locally*causing*accelerated pruning on the receiving chain, but the population\-level surviving siblings absorb none of this local effect into a measurable outcome change\. This is the finding the §[5\.5](https://arxiv.org/html/2610.00047#S5.SS5)chain\-dynamics paragraph characterizes: a measurable local footprint that does not translate into a population\-level effect\.

#### Reproduction\.

The full computation is inscripts/surviving\_sibling\_counterfactual\.py\. Outputs:results/surviving\_sibling\_counterfactual\.json\(summary\) andresults/surviving\_sibling\_per\_problem\.csv\(per\-unit detail\)\.

### L\.1Cross\-architecture replication of the pruning hazard

The2\.75×2\.75\\timespost\-injection pruning hazard of §[5\.5](https://arxiv.org/html/2610.00047#S5.SS5)is the paper’s sharpest per\-event mechanistic observation, and in the submitted version it rested on Qwen alone\. We replicate it on the two further architectures, re\-using the already\-collected MATH500 trajectories with the unmodified hazard methodology \(no new generation\): post\-injection prune\-within\-3\-steps rate against a matched\-step baseline computed over non\-injected siblings in the same seeds\.

Table 18:Cross\-architecture replication of the post\-injection pruning hazard on MATH500\. The matched\-step baseline is the prune\-within\-3 rate over non\-injected siblings at the same step indices\. CIs are95%95\\%; in all three architectures the post\-injection and baseline intervals are non\-overlapping\.#### Results\.

The direction replicates on all three architectures \(Table[18](https://arxiv.org/html/2610.00047#A12.T18)\): injection locally elevates the pruning hazard relative to a matched non\-injected baseline, with non\-overlapping CIs in every case\. DeepSeek\-R1\-Distill reaches the same qualitative classification as Qwen \(elevated post\-injection pruning, ratio\>2\>2\); LLaMA’s1\.83×1\.83\\timesfalls just below that threshold while remaining clearly elevated\.

#### Scope\.

These are*descriptive*naive\-ratio replications\. The stronger*causal*sub\-claim—that the elevation is not pure selection—rests on the matched\-PRM\-trajectory control \(2\.59×2\.59\\times\) above, which we ran on Qwen only; the cross\-architecture cells extend the descriptive pattern, not the causal attribution \(§[7](https://arxiv.org/html/2610.00047#S7), item 10\)\. Outputs:results/post\_injection\_prune\_analysis\_llama\.jsonand\.\.\.\_deepseek\.json; code inscripts/post\_injection\_prune\_analysis\_crossarch\.py\.

## Appendix MOracle targeting analysis \(upper\-bound\)

This appendix substantiates a stronger version of the §[5\.4](https://arxiv.org/html/2610.00047#S5.SS4)targeting\-policy claim than the four\-bucket classification \(§[B](https://arxiv.org/html/2610.00047#A2)\) and the compound\-gate end\-to\-end \(§[B](https://arxiv.org/html/2610.00047#A2)\) together supply: not only does no*tested*heuristic move Pass@kk, but even a hindsight*oracle*that gets to choose between the PPFG\-stag population and the independent baseline on a per\-problem basis does not move Pass@kkoutside the seed\-sigma band\. The analysis runs post\-hoc on the three Qwen2\.5\-7B\-Instruct MATH500n=500n\{=\}500trajectory files \(seeds4242–4444\) and requires no further compute\.

#### Oracle construction\.

For each matched problem\-seed pair\(p,s\)\(p,s\), let𝐱p,sppfg∈\{0,1\}N\\mathbf\{x\}^\{\\mathrm\{ppfg\}\}\_\{p,s\}\\in\\\{0,1\\\}^\{N\}and𝐱p,sindep∈\{0,1\}N\\mathbf\{x\}^\{\\mathrm\{indep\}\}\_\{p,s\}\\in\\\{0,1\\\}^\{N\}be the per\-chain correctness vectors under PPFG\-stag and independent respectively \(N=8N\{=\}8\)\. The oracle population𝐱p,soracle\\mathbf\{x\}^\{\\mathrm\{oracle\}\}\_\{p,s\}is the element\-wise OR:xp,s,ioracle=xp,s,ippfg∨xp,s,iindepx^\{\\mathrm\{oracle\}\}\_\{p,s,i\}=x^\{\\mathrm\{ppfg\}\}\_\{p,s,i\}\\lor x^\{\\mathrm\{indep\}\}\_\{p,s,i\}\. This is the loosest defensible upper bound, the oracle does not get to mix\-and\-match across methods on a finer grain than the chain level, which would require running unobserved counterfactuals\. Pass@kkuses the Chen et al\. \(2021\) unbiased estimator and is averaged across the500500problems per seed\.

#### Results\.

Across the three seeds \(Table[19](https://arxiv.org/html/2610.00047#A13.T19)\), the oracle exceeds the independent baseline by at most0\.130\.13percentage points on any ofk∈\{1,2,4,8\}k\\in\\\{1,2,4,8\\\}, well within seed\-σ\\sigma\(0\.0030\.003–0\.0080\.008\)\. Even with perfect per\-problem method choice on this configuration, the union of PPFG\-stag and indep adds essentially zero architecturally new correct answers relative to indep alone\.111The absolute Pass@kklevels in this appendix \(and in Appendix[L](https://arxiv.org/html/2610.00047#A12)\) were computed with an earlier answer\-extraction pass that systematically under\-counts Qwen final\-answer parses by roughly1313points relative to the corrected extraction used in the main results \(Table[1](https://arxiv.org/html/2610.00047#S5.T1)\); this is why indep Pass@1 reads0\.510\.51here versus0\.640\.64in the body\. The oracle and per\-injection analyses are interpreted only through*within\-analysis*deltas \(Oracle−\-indep; injected vs\. non\-injected siblings\), which are invariant to the extraction offset because all three populations are scored by the identical parser; we therefore did not re\-run them under the corrected extraction\.

Table 19:Oracle targeting upper bound vs\. PPFG\-stag and independent on Qwen2\.5\-7B\-Instruct \+ Math\-Shepherd \+ MATH500 \(n=500n\{=\}500, 3 seeds\)\. The oracle is the per\-chain OR across methods; the Oracle−\-indep delta column is the maximum Pass@kka hindsight method\-chooser could extract beyond the independent baseline\. All deltas fall inside the seed\-σ\\sigmaband \(0\.0030\.003–0\.0080\.008\)\.
#### Per\-injection success\-rate baseline\.

A complementary tighter probe asks whether the chains that actually received injections succeed at a different rate than the chains that did not\. Across289289PPFG\-stag injection events atn=500×3n\{=\}500\{\\times\}3seeds, the receiving\-chain correctness rate is16\.6%16\.6\\%\(CI excluding zero\); the matched non\-injected sibling success rate in the same populations is51\.4%51\.4\\%; the delta is−34\.8\-34\.8pp \(CI95\[−39\.2,−30\.3\]\[\-39\.2,\-30\.3\]\)\. Chains entering the treated set succeed at roughly one\-third the rate of their non\-injected siblings, the direction the selection account \(§[5\.5](https://arxiv.org/html/2610.00047#S5.SS5), Appendix[L](https://arxiv.org/html/2610.00047#A12)\) predicts and the opposite of what an*oracle\-targeting\-unlocks\-gains*hypothesis would require\.

#### Reading\.

The two bounds together constrain the mechanism’s upper potential under hindsight selection: even granted perfect per\-problem method choice, the union does not exceed indep; even granted that the heuristic happened to pick the chains the oracle would have picked, the chains receiving those injections succeed at a depressed, not elevated, rate\.

#### Reproduction\.

The full computation is inscripts/oracle\_targeting\_analysis\.py\. Outputs:results/oracle\_targeting\_analysis\.json\(summary, including per\-seed Pass@kkand per\-injection deltas\) andresults/oracle\_targeting\_per\_problem\.csv\(per\-problem correctness counts under PPFG\-stag, indep, and the oracle\)\.

## Appendix NInjection\-format ablation \(seamless and re\-attention\)

§[5\.5](https://arxiv.org/html/2610.00047#S5.SS5)reports an empty\-bracket control that decomposes the post\-injection PRM depression into format \(55%\) and content \(45%\) components\. A complementary concern is whether the*bracket markup itself*influences downstream generation in a way that confounds the Pass@kkcomparison—i\.e\., that the model performs differently not because of the grafted reasoning content but because the injection renders the prompt stylistically unusual\.

To test this, we ran two format variants at full scale \(Qwen2\.5\-7B\-Instruct \+ Math\-Shepherd, MATH500,n=500n\{=\}500, 3 seeds each\):

- •Seamless \(E1\)\.The fragment is rendered as a native “StepNN:” continuation with no brackets, no quality label, and no visual separator\. The model receives the grafted steps as if they were its own prior output\.
- •Re\-attention \(E2\)\.The standard bracket\-bullet injection is retained, but a brief self\-check directive is appended immediately after the fragment, asking the model to evaluate alignment before continuing \(one short step, then continue normally\)\.

Table 20:Injection\-format ablation on Qwen2\.5\-7B\-Instruct \+ Math\-Shepherd \+ MATH500 \(n=500n\{=\}500, 3 seeds\)\. Seed\-σ\\sigma≈0\.005\\approx 0\.005for Pass@1,0\.0080\.008for Pass@8\. Pre\-comparator\-correction values \(§[5\.2](https://arxiv.org/html/2610.00047#S5.SS2)\); the parity finding is not expected to change\.#### Results\.

Both format variants are within seed\-σ\\sigmaof indep on every metric \(Table[20](https://arxiv.org/html/2610.00047#A14.T20)\)\. The seamless variant \(E1\) showsΔ\\DeltaPass@1=\+0\.002=\+0\.002vs\. indep andΔ\\DeltaPass@8=0\.000=0\.000; the re\-attention variant \(E2\) showsΔ\\DeltaPass@1=\+0\.003=\+0\.003andΔ\\DeltaPass@8=\+0\.001=\+0\.001\. Neither departs from the canonical PPFG\-stag result\.

#### Reading\.

The format\-pollution alternative is not supported: removing the bracket markup entirely leaves performance unchanged, confirming that the inertness finding is not an artifact of the bracketed\-demonstration rendering\. The re\-attention directive likewise adds nothing: instructing the model to explicitly evaluate the grafted content before continuing does not unlock any latent benefit\. Both results are consistent with the selection account developed in §[5\.4](https://arxiv.org/html/2610.00047#S5.SS4)and Appendix[L](https://arxiv.org/html/2610.00047#A12)\.

#### Reproduction\.

Configs:configs/qwen\_n500\_ppfg\_stag\_seamless\.yaml\(E1\),configs/qwen\_n500\_ppfg\_stag\_reattention\.yaml\(E2\)\. Code:src/hyp\_forest/ppfg/injector\.py\(injection\_format="seamless"\) andsrc/hyp\_forest/ppfg/policy\.py\(reattention\_pass=True\)\.

## Appendix OPrune\-immunity content control

Appendix[N](https://arxiv.org/html/2610.00047#A14)closes the*format*channel by which the PRM could be reacting to the graft’s rendering rather than to transfer being inert\. A second channel remains: the PRM could be mis\-pruning grafted chains on their*content*, killing chains that would otherwise have reached a correct answer, so that a real transfer benefit is exactly offset by induced pruning\. Flat Pass@kkis in principle consistent with both “no benefit” and “a benefit cancelled by mis\-pruning\.” This appendix reports the experiment that discriminates them\.

#### Design\.

Three arms on the first\-100 MATH500 slice \(Qwen2\.5\-7B\-Instruct \+ Math\-Shepherd,N=8N\{=\}8, 3 seeds, identical targeting/extraction/gate settings\):

- •\(a\) Independent, no immunity\.The standard baseline, scored on the same 100 problems\.
- •\(b\) Immunity\+\+null graft\.An injected chain is exempted fromτkill\\tau\_\{\\mathrm\{kill\}\}\-threshold pruning for 3 steps, but the spliced content is empty\. This holds the exemption’s extra decoding budget constant while removing graft content\.
- •\(c\) Immunity\+\+real graft\.The identical 3\-step exemption, with the real verbatim fragment spliced in\.

Arm \(c\) minus arm \(a\) measures what suppressing the mis\-pruning channel buys; arm \(c\) minus arm \(b\) isolates graft*content*with that channel held closed\.

#### Manipulation check\.

The exemption engages as intended: the population\-wide pruned\-chain fraction falls from0\.1104±\.00830\.1104\\pm\.0083under canonical PPFG\-stag \(and0\.1075±\.00650\.1075\\pm\.0065under independent\) to0\.0804±\.00750\.0804\\pm\.0075\(c\) and0\.0808±\.00760\.0808\\pm\.0076\(b\), a≈27%\\approx 27\\%relative reduction, outside seed\-σ\\sigmain both immune arms\. Chains that would have been pruned are demonstrably kept alive\.

Table 21:Prune\-immunity content control on the first\-100 MATH500 slice \(Qwen2\.5\-7B\-Instruct \+ Math\-Shepherd,N=8N\{=\}8, 3 seeds\)\. All four arms are scored by one identical extractor/comparator on the identical 100 problems\. The immunity lift \(c−\-a\) and the graft effect \(c−\-b\) both fall inside seed\-σ\\sigma; canonical PPFG\-stag is shown for reference\.
#### Results\.

Two readings follow from Table[21](https://arxiv.org/html/2610.00047#A15.T21)\. First, suppressing the mis\-pruning channel does not by itself rescue Pass@kk: arm \(c\) exceeds independent by\+0\.71\+0\.71pp at Pass@1, inside the seed\-σ\\sigmaband \(σ≈\.005\\sigma\\approx\.005–\.010\.010\), even though the manipulation check confirms substantially fewer chains were pruned\. Chains rescued from pruning do not convert into correct answers, which is what the selection account of §[5\.5](https://arxiv.org/html/2610.00047#S5.SS5)predicts: stagnation\-targeted chains were pre\-selected for low, flat PRM and were not on a path to a correct answer, so keeping them alive changes the population’s composition without changing its output\. Second, and more directly on the confound, graft content adds nothing on top of the exemption: \(c\)−\-\(b\) is−0\.04\-0\.04pp at Pass@1 and small\-negative at higherkk, if anything favouring the null\-content arm—the opposite of the sign a masked benefit would produce\. Restricted to the chains the arms actually fire on, real\-graft chains also convert no better than null\-content chains\.

#### Reading\.

Even in the exact condition under which a masked benefit should surface—the struggling chain kept alive, its pruning suppressed—the graft does not convert survival into correctness\. This closes the content channel as Appendix[N](https://arxiv.org/html/2610.00047#A14)closes the format one\. The test is small \(n=100n\{=\}100, three seeds\) and we read it as corroboration rather than proof; within its power, the evidence runs against content\-driven mis\-pruning concealing a real transfer benefit\.

#### Correction to the ARR response record\.

An earlier version of this analysis, circulated during the ARR May 2026 response period, reported an immunity lift of\+6\.96\+6\.96pp at Pass@1 for arm \(c\) over arm \(a\)\. That figure was an artifact of two mismatches, both corrected here: the \(a\) baseline was drawn from a run over MATH500 problems 400–499 while arms \(b\) and \(c\) ran on problems 0–99, and the arms were scored under two different answer\-comparator configurations\. Recomputed on matched problems under one comparator, the lift is\+0\.71\+0\.71pp and is not distinguishable from zero\. The load\-bearing \(c\)−\-\(b\) contrast is unaffected—both arms were always matched on problems and comparator— and the conclusion of the experiment is unchanged, but the “the exemption is a large lever” framing of the earlier text does not survive: the correct manipulation check is the pruned\-chain fraction reported above, not a Pass@1 lift\.

#### Reproduction\.

Configs:configs/qwen\_n100\_ppfg\_stag\_prune\_immune\.yaml\(c\),configs/qwen\_n100\_ppfg\_stag\_prune\_immune\_nullctrl\.yaml\(b\)\. The immunity counter ispopulation\.post\_injection\_prune\_immunity\_steps; the null\-content arm setsppfg\.injection\_format="null"\. The full table, including the manipulation check, is regenerated byscripts/prune\_immunity\_analysis\.py, which re\-scores all arms in one environment on matched problems\.

## Appendix PPopulation\-size sweep \(N=8N=8toN=256N=256\)

Because grafting is a population\-level mechanism, a natural objection to anN=8N\{=\}8null is thatN=8N\{=\}8may simply be too small a population for cross\-chain transfer to matter: with few siblings, few chains are pruned, few fragments are produced, and any benefit could be starved\. We test this directly rather than argue it, sweeping the population size over a32×32\\timesrange on the first\-100 MATH500 slice \(Qwen2\.5\-7B\-Instruct \+ Math\-Shepherd, 3 seeds per cell, matched independent and PPFG\-stag populations at everyNN, fixed extractor\)\.

Table 22:Population\-size sweep: PPFG\-stag minus independent, in percentage points, on the first\-100 MATH500 slice \(3 seeds per cell, matched populations\)\. Every Pass@kkdelta is≤0\.27\\leq 0\.27pp in magnitude and none is positive at anyNN\. The mode rate row changes sign acrossNN, consistent with a diversity rather than an accuracy metric\.#### Results\.

Parity replicates at every population size tested \(Table[22](https://arxiv.org/html/2610.00047#A16.T22)\), and the effect never turns positive\. The binding cell isN=16N\{=\}16Pass@8 at−0\.27\-0\.27pp, comparable to cross\-seed scatter\. AtN=256N\{=\}256, where sampling noise is nearly averaged out \(seed\-σ≤0\.0014\\sigma\\leq 0\.0014\), parity is essentially exact: independent Pass@10\.7073±\.00020\.7073\\pm\.0002versus PPFG\-stag0\.7062±\.00060\.7062\\pm\.0006, and Pass@80\.89630\.8963versus0\.89620\.8962\. The “N=8N\{=\}8is too small a population” hypothesis predicts a benefit emerging asNNgrows; across a32×32\\timesrange no such trend appears, and no adverse blow\-up appears either\.

#### Scope\.

This sweep runs on then=100n\{=\}100slice rather than full MATH500, so its per\-cell precision is lower than the headline tables; it is reported as a sensitivity analysis on population size, not as a replacement for then=500n\{=\}500results in §[5\.1](https://arxiv.org/html/2610.00047#S5.SS1)\.