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Characterizing a Configuration Where Inference-Time PRM-Pruned Fragment Grafting Is Inert: Evidence from Three Reasoning LMs

Khawaja Murad ul Hassan, Mehran Ebrahimi

arXiv:2610.00047v1cs.AIcs.CL

TL;DR

Diversity collapse in parallel chain-of-thought has motivated fragment grafting, but prior systems combine cross-trajectory transfer with other ingredients rather than isolating the transfer itself. This paper tests isolated PRM-Pruned Fragment Grafting and finds its variants indistinguishable from independent parallel-CoT on every tested cell.

  • Problem

    Prior fragment-grafting systems bundle cross-trajectory transfer with compensating ingredients, leaving the transfer step itself unisolated amid documented diversity collapse in parallel chain-of-thought.

    Source: §1 · p. 2

  • Method

    The paper isolates PRM-Pruned Fragment Grafting, using PRM-pruned prefixes injected verbatim into sibling chains, and compares targeting rules in matched-compute experiments.

    Source: §1 · p. 3, §3.3 · p. 5

  • Results

    PPFG variants were indistinguishable from independent on every tested cell, while only 14% of 322 stagnation injections targeted a struggling chain with room to act.

    Source: §1 · p. 3, §6 · p. 10

  • Takeaways & Limitations

    A hindsight oracle bounds any per-problem gain from choosing PPFG over independent at +0.13 pp, providing a pre-registrable benchmark for this graft-content class.

    Source: §6 · p. 10

    With three seeds per cell at n = 500, the minimum detectable effects were approximately 2.2 mode-rate points, 3.1 Pass@8 points, and 1.8 Pass@1 points.

    Source: §7 · p. 10

Abstract

from arXiv · show

Diversity collapse in parallel chain-of-thought has motivated inference-time interventions built on a natural design: when a process reward model (PRM) prunes a chain, its high-PRM prefix is extracted and grafted verbatim as an in-context demonstration into a still-decoding sibling. We isolate this mechanism, PRM-Pruned Fragment Grafting (PPFG), as the most cost-minimal operationalization of cross-trajectory step-level transfer, and test it at the operating point where prior fragment-grafting work reports gains only under additional compensating ingredients. On Qwen2.5-7B-Instruct with Math-Shepherd on full MATH500 (n=500, three seeds), PPFG in both stagnation- and random-targeting variants is statistically indistinguishable from an independent parallel-CoT baseline on every measured axis. We characterize why: a four-bucket classification of 322 stagnation-rule injection events shows only 14% targeted a genuinely struggling chain; the rest landed on chains that had already succeeded, were near completion, or sat on a flat PRM plateau, states a rescue graft cannot change. No compound-gate refinement jointly achieves well-targeted firing and adequate density, and a random control matches the same parity at 2.4x the firing rate, so the inertness is not heuristic-specific. The finding replicates across three base LMs, six benchmarks, a second PRM, and a compatibility-gate sweep; two-one-sided-tests analysis promotes the parity to positive equivalence on all twelve Qwen/LLaMA cells. A per-event spot-check finds injected chains prune at 2.75x the matched-step rate, but a surviving-sibling counterfactual finds no population-level compensation. A hindsight oracle bounds any per-problem gain from choosing PPFG over independent at +0.13 pp. We contribute an equivalence-testing template for establishing inference-time mechanism nulls, with every claim scoped to its tested operating point.

1 Introduction

The paper isolates bare, verbatim in-place transfer from the extra ingredients in prior fragment-grafting approaches, then tests whether it changes parallel reasoning. On the tested setting, PPFG is inert at the population level, with targeting analyses explaining why and bounding the scope of that conclusion.

Contribution

  • PPFG isolates a PRM-pruned trigger, in-place injection into a still-decoding sibling, and verbatim fragment content—three properties not jointly realized by prior fragment-grafting work.The graft is the longest contiguous prefix whose PRM scores exceed the extraction threshold, inserted as a bracketed in-context demonstration.Source: §1 · p. 2

Main result

  • On full MATH500 with Qwen2.5-7B-Instruct and Math-Shepherd, both PPFG targeting variants match independent parallel CoT across measured axes; only 14% of stagnation fires target a struggling chain with room to act.The four-bucket analysis covers 322 events; the first-100 hand-validated slice yields 6/55, or 11%, well-targeted fires.Source: §1 · pp. 2–3

Regime characterization

  • The four-bucket analysis finds other fires land on chains already successful, near answering, or on a high, flat PRM plateau; no compound-gate setting jointly provides well-targeted firing and adequate density.The random-rule control also matches stagnation and independent mode rate, so the tested targeting policy is not the binding constraint.Source: §1 · p. 3

Mechanism evidence

  • A per-event analysis reports a 2.75× pruning hazard and an N=65 surviving-sibling counterfactual with no detectable compensation; the authors describe this as consistent with a selection-dominated account.The implied content component of the PRM change assumes untested additivity and is treated as bounded rather than a point estimate.Source: §1 · p. 3

Scope

  • The paper reports parity across three base models, five further benchmarks, and VersaPRM, while noting that cross-architecture coverage spans four benchmarks and DeepSeek results are inconclusive on four of six cells.The authors present the characterization as a target for future falsification, not a general refutation of fragment grafting.Source: §1 · pp. 2–3

2 Related work

Prior inference-time and training-time methods move information across reasoning trajectories in different forms and at different stages. PPFG is distinguished by combining pruning-triggered extraction, mid-decoding sibling injection, and verbatim text.

Cross-trajectory information flow

  • SNDT uses a separate solver pass, RSE distills failure patterns into an experience bank, and T-STAR grafts trajectory nodes during training; none jointly matches PPFG’s three operational properties.PPFG uses PRM pruning as its trigger, injects into a sibling still decoding, and transfers verbatim fragment text.Source: §2 · pp. 3–4
  • Other related methods differ in target or abstraction: LATS injects reflection summaries, Metacognitive Reuse stores behaviors across problems, and GoT aggregates active branches without targeting pruned chains.The related-work comparison identifies the PRM pruning trigger as absent from GoT.Source: §2 · p. 4

PRMs and credit assignment

  • PRMs serve as inference-time verifiers and as live emission models for SMC resampling, while cited population-level credit-assignment methods are training-time approaches.The paper’s SMC baseline instantiates PRM-weighted resampling at matched compute.Source: §2 · p. 4

Diversity-preserving sampling

  • Diversity collapse has been addressed by training-time and inference-time approaches, but the cited prior methods operate with at least one dimension differing from PPFG’s bare grafting regime.The listed differences include a separate solver pass, learned distillation, training-time gradients, style adaptation, and cross-problem abstraction.Source: §2 · p. 4

3 Method

PPFG runs within a parallel chain-of-thought population loop, where a PRM scores steps and chains can be promoted, pruned, or stopped at a step ceiling. The method extracts a high-scoring prefix from a pruned chain and injects it into an active sibling under configurable targeting rules.

3.1 Setup

  • Each population contains N chains decoded step by step from one prompt; the PRM assigns each step a quality score, and chains terminate when promoted, pruned below threshold, or at the step ceiling.The setup uses Math-Shepherd as its math-tuned PRM.Source: §3.1 · p. 4

3.2 PPFG procedure

  • At each step, PPFG extracts from each newly pruned chain its longest contiguous prefix above the extraction threshold, then selects an active sibling and gates injection by compatibility.The accepted fragment is spliced into the target’s prompt as a bracketed in-place demonstration before its next step.Source: §3.2 · p. 4

3.3 Operational details

  • The stagnation rule filters targets for a low three-step PRM range and chooses the most compatible; the compound rule adds low-mean-score, remaining-step, sibling-relative-score, and active-sibling-count requirements.The random rule selects uniformly and ignores compatibility; fragments are formatted as bracketed bullet blocks.Source: §3.3 · pp. 4–5

4 Experimental setup

Experiments hold the base model and PRM fixed for the main comparison and evaluate the full MATH500 test split. The study compares PPFG variants with parallel-CoT baselines using accuracy, answer-mode, diversity, and compute metrics.

Models and task

  • The main setup uses Qwen2.5-7B-Instruct with Math-Shepherd and evaluates all 500 MATH500 test problems; mechanism analyses use a first-100 slice while aggregate tables use n=500.The base model is held fixed across baselines and PPFG.Source: §4 · p. 5

Baselines

  • The baselines are independent parallel CoT and PRM-weighted SMC at N=8, plus Best-of-N at N=16 and a matched-compute N=8 variant.The matched-compute Best-of-N row selects the highest final-step PRM trajectory from independent N=8 runs per problem and seed.Source: §4 · p. 5

Configuration and seeds

  • PPFG runs at N=8 with stagnation, compound-stagnation, or random targeting, using fixed extraction, fragment-length, compatibility, and pruning thresholds.Seeds 42–44 are used, with 45–46 for five-seed Qwen cells; the balanced-brace answer parser is used throughout.Source: §4 · p. 5

Metrics

  • The evaluation reports Pass@k for k∈{1,2,4,8}, binary mode rate, distinct-4, and total generated tokens per problem as the compute measure.Lower mode rate and higher distinct-4 are preferred; raw modal share is reported as a supplement.Source: §4 · p. 5

5 Results

On full MATH500, PPFG sits on the independent baseline’s frontier, while other matched-compute methods trade diversity for Pass@k gains. The paper tests this parity across architectures, targeting policies, mechanisms, and benchmarks, and reports no detectable PPFG benefit in the tested settings.

5.1 The six-method tradeoff space

  • PPFG-stag matches independent within seed-level uncertainty on every Table 1 axis, while PPFG-random is likewise within seed-σ.SMC gains at Pass@1 but falls below independent at Pass@8; BoN improves Pass@k while paying a diversity cost.Source: §5.1 · pp. 5–6, Table 1 · p. 6

5.2 A note on the answer comparator (this revision)

  • The canonical scoring path uses task.is_correct directly and re-extracts answers from chain steps; the earlier scoring paths differed by roughly nine Pass@1 points on identical chains.The authors also flag that three secondary appendix tables under the earlier comparator have not been re-derived, so their relative conclusions remain unverified.Source: §5.2 · pp. 6–7
  • At five seeds, two one-sided tests reject both nulls on all 12 Qwen and LLaMA cells; DSR1 is equivalent on stagnation Pass@1 but inconclusive on four Pass@8 or mode-rate cells.The analysis uses bounds set to the three-seed minimum detectable effects.Source: §5.2 · p. 7

5.3 Cross-architecture replication

  • PPFG-stag Pass@1 deltas versus independent are −0.0015 for Qwen, −0.0026 for LLaMA, and −0.0014 for DeepSeek, all within seed-σ across k.The parity persists despite a roughly twofold Pass@1 gap between Qwen and the other two models.Source: §5.3 · p. 7

5.4 Targeting-policy analysis

  • Only 14% of 322 stagnation-rule events targeted a struggling chain with room to act; compound-gate settings either yield about five events per cell or retain 44% high-flat-PRM targets.A random control fires at 2.4× the stagnation rate yet matches aggregate metrics within seed-σ, so the parity is not specific to that targeting heuristic.Source: §5.4 · p. 8
  • Across the ξmin sweep, population sizes N=8–256, and matched-compute comparisons, no tested PPFG setting gains a Pass@k or mode-rate advantage over independent.On GSM8K, Pass@1 is identical to three decimals and Pass@8 is within 0.0 pp; injection rates fall to 2.8%/9.5%.Source: §5.4 · p. 8

5.5 Mechanistic spot-check: post-injection PRM delta

  • Injected chains reach pruned terminal status at 25.8% versus a 9.4% matched-step baseline, while a surviving-sibling counterfactual finds no detectable compensation.The authors interpret this pattern as consistent with PPFG pulling pruning forward by about three steps without redistributing completion.Source: §5.5 · p. 9

5.6 Replication across benchmarks (Qwen)

  • Across four further Qwen benchmarks and two PRMs, every PPFG-versus-independent Pass@1 and mode-rate contrast falls inside the per-benchmark MDE.AIME and VersaPRM-GPQA are near the capability floor, while NuminaMath, Math-Shepherd-GPQA, MMLU-Pro, and MATH500 provide non-floor comparisons.Source: §5.6 · p. 9

6 Discussion

The discussion frames PPFG’s null result as specific to bare verbatim grafting at the tested operating point, not as a refutation of fragment grafting broadly. It proposes a scoped benchmark for future work and asks designs to identify their compensating ingredients.

6 Discussion

  • For parallel-CoT inference with a math-tuned PRM and strong base LM, PPFG variants are indistinguishable from independent on every tested cell, while Best-of-N dominates Pass@k at matched compute.The population-level parity coexists with a mildly adverse per-event PRM signal and a 2.75× pruning hazard.Source: §6 · p. 10
  • A hindsight oracle choosing per problem between PPFG-stag and independent exceeds independent by at most +0.13 pp across Pass@k, within seed-σ.This is an upper bound on per-problem deployment gating for this mechanism, not event-level targeting or ingredient-side changes.Source: §6 · p. 10
  • Future designs should name the compensating ingredient they supply and report against the oracle ceiling; the tested seamless and self-check prompt variants remain within seed-σ.The authors note that their ingredient taxonomy is inductive and not exhaustive, and suggest a separate-solver-pass variant as an informative next test.Source: §6 · p. 10

7 Limitations

The reported parity is bounded by statistical power and by the tested model, scorer, task, and inference-time settings. Mechanistic attribution and generalization beyond those settings remain limited.

Statistical power and equivalence

  • At n=3, 80%-power MDEs are 1.8 Pass@1, 3.1 Pass@8, and 2.2 mode-rate points; n=5 TOST confirms equivalence in 13/18 cross-architecture cells.Four DeepSeek cells remain inconclusive because seed-level variability is higher; they are reported as within-σ, not formally equivalent.Source: §7 · pp. 10, 12

Coverage

  • Generalization beyond Qwen and LLaMA families is untested; GSM8K and MMLU-Pro are Qwen-only, and open-ended or multilingual tasks are outside the evaluation.Cross-architecture replication covers four benchmarks, while the two additional task families are evaluated on Qwen alone.Source: §7 · p. 11

Compute and injection density

  • PPFG headline cells use N=8; the N=256 sweep is an n=100 sensitivity analysis, populations above 256 are untested, and the fixed per-chain cap M=2 was not re-swept.The authors identify per-chain injection density as a possible constraint not fully closed by the random-rule control.Source: §7 · p. 11

Task and policy scope

  • A learned target-selection policy remains future work, and the evaluated tasks use English with checkable single-answer ground truth.Tasks without checkable gold answers were not tested.Source: §7 · p. 11

Evaluation

  • Human ratings of graft naturalness were not collected; the paper uses PRM-side proxies rather than directly testing whether grafts read as stylistic breaks.The proposed human evaluation would more directly test the stylistic-discontinuity interpretation.Source: §7 · p. 11

Mechanistic evidence

  • Per-injection credit assignment is absent, and the matched-PRM-trajectory evidence for locally accelerated pruning is established only on Qwen.The elevated-hazard pattern is descriptively replicated on LLaMA and DeepSeek, but the matched-trajectory control is not.Source: §7 · pp. 11–12

Compatibility gate

  • The headline configuration sets ξmin=0.0; a four-threshold sweep preserves parity as injection rates fall from 16.0% to 2.4%.The paper interprets the gate as filtering pathological inputs rather than changing aggregate behavior in this regime.Source: §7 · p. 12

Verifier dependence

  • The PRM evidence remains limited: Math-Shepherd is math-tuned, and neither tested PRM rules out shared sensitivity to mid-chain grafts as out-of-distribution text.A PRM-free pruning criterion or a scorer trained with spliced-context augmentation was not tested.Source: §7 · pp. 10–12

8 Ethics, broader impact

The paper frames PPFG’s broader impact as a regime-specific methodological characterization, while its replication appendices test whether inertness persists across benchmarks, PRMs, and targeting refinements.

8 Ethics, broader impact

  • The paper identifies its broader-impact contribution as a regime characterization, with no new training data, model weights, annotated corpus, or detectable capability improvement at tested configurations.It says no additional safety review is required beyond that for the underlying LMs and Math-Shepherd.Source: §8 · p. 12

A Per-benchmark replication details

  • Across AIME, NuminaMath-CoT, GPQA-Diamond, MMLU-Pro, and GSM8K, PPFG contrasts remain within reported minimum detectable effects, including under two PRMs on GPQA-Diamond and MMLU-Pro.AIME’s floor-regime parity is corroborating rather than discriminating evidence; the GSM8K variants also match Pass@1 to three decimals.Source: §A.1 · p. 14, §A.2 · p. 14, §A.3 · p. 14

B.1 Four-bucket false-positive classification

  • 45/322 stagnation fires (14.0%) were plausibly helped; high-flat-PRM remained the dominant disqualifier at 67.1%.On the first-100 slice, only 6/55 (11%) were well-targeted.Source: §B.1 · p. 15, Figure 1 · p. 15

B.2 Difficulty-stratified targeting

  • 15.6% of fires in the hardest tercile were well-targeted, barely above 14.0% overall, while high-flat-PRM disqualified 62.3%.The authors interpret this as evidence that PRM flatness, rather than base-rate success, constrains targeting quality in this regime.Source: §B.2 · p. 15, Table 8 · p. 15

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

  • Compound gating at τlow = 0.60 and 0.70 admitted 9.1% and 16.4% of dry-run events, yet both end-to-end variants stayed within seed-σ of independent.The gates also required a low recent PRM mean, at least four steps before Tmax, a latest PRM below the active-sibling median, and at least three active siblings.Source: §B.3 · p. 15

C GSM8K replication (full table)

  • On GSM8K, independent, PPFG-stag, and PPFG-random Pass@1 were identical to three decimals (maximum absolute delta 0.03 pp).Injection rates fell to 2.8% and 9.5% for stagnation and random targeting, respectively, versus 16.0% and 28.8% on MATH500.Source: C GSM8K replication (full table) · p. 15

D Empty-bracket confound control (full per-event detail)

  • Across 241 empty-bracket injections, mean per-event PRM change was −0.0736; 75.5% of events had negative change, but the content-component estimate relies on untested additivity.The empty-bracket count differs from PPFG-stag because the population state diverges after the first injection, changing subsequent targeting decisions.Source: D Empty-bracket confound control (full… · p. 16

E Cross-architecture replication on the extended benchmarks (full table)

The extended-benchmark sweep reports all contrasts within per-benchmark minimum detectable effects, while the MATH500 cross-architecture panel shows PPFG parity with the independent baseline. PPFG uses PRM-pruned prefixes injected into selected still-decoding siblings.

E Cross-architecture replication on the extended benchmarks (full table)

F PPFG algorithm + architectural diagram

  • PPFG extracts a PRM-qualified prefix from a pruned chain, selects a still-decoding sibling, gates compatibility, and injects the fragment into its prompt.The algorithm applies these operations after pruning and returns a majority-vote or highest-mean-PRM answer.Source: F PPFG algorithm + architectural diagram · p. 17, Figure 2 · p. 18

G Cross-architecture replication on MATH500 (full table + PRM-distribution audit)

  • The PRM-distribution audit reports more frequent PPFG firing on LLaMA and DeepSeek than on Qwen, ruling out a “PRM never prunes” artifact.The full MATH500 table reports independent rows at five seeds and LLaMA/DeepSeek PPFG rows at three seeds, with a relaunch ambiguity unresolved for those cells.Source: Table 12 · p. 18, G Cross-architecture replication on… · p. 18, Table 13 · p. 19
  • PPFG-stag and PPFG-random sit within seed-σ of the independent baseline at every Pass@k on all three MATH500 architectures.The panel covers full MATH500 (n = 500, 3 seeds).Source: Figure 3 · p. 19

H Compute matching figure

The compute-matching figure compares Pass@4 with tokens per problem on MATH500. The compatibility-threshold sweep reduces firing as the threshold rises, without moving Pass@k or mode rate outside the minimum detectable effect band.

H Compute matching figure

  • At matched compute, BoN dominates the matched-tpp column; BoN at N=16 reaches Pass@16=0.89 at 2× compute.The figure plots Pass@4 against tokens-per-problem on Qwen2.5-7B MATH500.Source: Figure 4 · p. 19
  • As ξmin rises, injection rate falls monotonically, but no setting moves Pass@k or mode rate outside the MDE band.The sensitivity sweep is on Qwen2.5-7B and MATH500 (n = 500, 3 seeds per cell).Source: I ξmin sensitivity sweep (full table) · p. 19, Table 14 · p. 20

J TOST equivalence: methodology and per-cell discussion

The paper applies preregistered two one-sided tests to assess whether PPFG effects fall within bounds set at the three-seed minimum detectable effect. Qwen and LLaMA cells receive equivalence verdicts, while some DeepSeek cells remain inconclusive.

K Differentiator table (full)

  • The TOST bounds were preregistered at the three-seed MDE, and PPFG combines a PRM-pruned trigger, in-place sibling injection, and verbatim fragments.These three operational properties are not jointly realized by prior fragment-grafting work, according to the differentiator table.Source: J TOST equivalence: methodology and… · p. 20, Table 16 · p. 21

J TOST equivalence: methodology and per-cell discussion

  • At five seeds, both PPFG rules are equivalent to independent on every Qwen metric, and all six LLaMA cells also receive equivalent verdicts.The preregistered bounds are ±1.8 points for Pass@1, ±3.1 for Pass@8, and ±2.2 for mode rate.Source: J TOST equivalence: methodology and… · p. 20, Table 15 · p. 20
  • DeepSeek remains partly inconclusive: four Pass@8 and mode-rate cells stay unresolved because seed-to-seed standard deviations are 2–3× Qwen/LLaMA values.DeepSeek stagnation Pass@1 reaches equivalence; random Pass@1 rejects only the upper-side null.Source: J TOST equivalence: methodology and… · p. 20
  • PPFG-stag and PPFG-random show Pass@1 effects of −0.13 and −0.19 pp, respectively; the margin would have comfortably detected positive effects measured on the identical model/task/pipeline.Those effects were SMC at +4.47 pp, Best-of-N at 2× compute at +4.23 pp, and zero-extra-compute argmax-PRM reranking at +6.45 pp.Source: J TOST equivalence: methodology and… · p. 20

L Surviving-sibling counterfactual analysis (full procedure)

The matched surviving-sibling counterfactual finds no detectable population-level compensation, though its power does not formally exclude it. A separate matched-trajectory control indicates injection locally accelerates pruning, and the descriptive hazard direction replicates across architectures.

L Surviving-sibling counterfactual analysis (full procedure)

  • The counterfactual matches 65 treated problem-seed pairs to independent runs and measures population correctness, non-injected-sibling promotion, and tokens.The analysis uses bootstrap confidence intervals over 5,000 resamples of the matched pairs.Source: L Surviving-sibling counterfactual… · p. 21
  • The surviving-sibling counterfactual detects no correctness or promotion lift; its correctness interval would not detect a lift below approximately 6 absolute points.The significant token outcome is adverse: surviving siblings use about 0.5% fewer tokens, not more.Source: L Surviving-sibling counterfactual… · p. 21, Table 17 · p. 21
  • The Qwen matched-PRM-trajectory control finds a 2.59× post-injection pruning ratio, supporting local acceleration beyond comparable pre-injection trajectories.Across 414 matched sibling slots, the prune-within-3 rate is 9.9%; the text says surviving siblings show no measurable population-level outcome change.Source: L Surviving-sibling counterfactual… · p. 21

L.1 Cross-architecture replication of the pruning hazard

  • The post-injection pruning-hazard direction replicates across all three architectures, with non-overlapping confidence intervals against matched-step baselines.DeepSeek has a ratio above 2; LLaMA’s 1.83× is elevated but below that threshold.Source: §L.1 · p. 22, Table 18 · p. 22
  • Cross-architecture hazard results are descriptive; the stronger claim that elevation is not pure selection rests on the matched-PRM control run on Qwen only.The replication reuses existing MATH500 trajectories and the unmodified hazard method.Source: §L.1 · pp. 21–22

M Oracle targeting analysis (upper-bound)

A per-problem hindsight chooser yields essentially no gain over independent decoding, and injected chains succeed less often than matched non-injected siblings.

Oracle construction

  • The oracle forms a per-chain OR of PPFG-stag and independent correctness vectors for each matched problem-seed pair, yielding a hindsight upper bound.Pass@k is estimated with the Chen et al. (2021) unbiased estimator and averaged over 500 problems per seed.Source: M Oracle targeting analysis… · p. 22

Results

  • At most 0.13 percentage points, the oracle’s advantage over independent on any Pass@k, stayed within seed-σ.This post-hoc analysis uses three Qwen2.5-7B-Instruct MATH500 seeds; Table 19 reports the oracle as the per-chain OR across methods.Source: M Oracle targeting analysis… · p. 22, Table 19 · p. 22

Scope

  • The appendix’s absolute Pass@k levels use an earlier answer-extraction pass that systematically under-counted Qwen final-answer parses.The authors interpret the oracle and per-injection analyses through within-analysis deltas scored by the same parser.Source: M Oracle targeting analysis… · p. 22

Per-injection success-rate baseline

  • Receiving chains were correct 16.6% of the time versus 51.4% for matched non-injected siblings, a −34.8 pp delta.The comparison covers 289 PPFG-stag injection events across n=500 and three seeds; the reported 95% CI for the delta is [−39.2, −30.3].Source: M Oracle targeting analysis… · p. 22

Reading

  • The two bounds constrain potential gains: even perfect per-problem method choice adds essentially no correct answers, while injected chains have depressed success.The passage frames these as complementary bounds on the mechanism’s upper potential under hindsight selection.Source: M Oracle targeting analysis… · p. 22

N Injection-format ablation (seamless and re-attention)

Across format, prune-immunity, and population-size checks, PPFG remains at parity with independent decoding; suppressing pruning or increasing population size does not reveal a benefit.

N Injection-format ablation (seamless and re-attention)

  • The format test removed brackets and quality labels in E1, while E2 retained bracketed bullets and added a self-check directive before decoding continued.The variants tested whether rendering or an explicit alignment check changed outcomes.Source: N Injection-format ablation (seamless… · pp. 22–23
  • An earlier answer-extraction pass undercounted Qwen parses by roughly 13 points, so the appendix interprets within-analysis deltas only.Those deltas use the same parser across compared populations; the absolute Pass@1 level differs from the corrected main results.Source: N Injection-format ablation (seamless… · p. 22
  • +0.002 (E1) and +0.003 (E2) were the ∆Pass@1 values versus indep; ∆Pass@8 was 0.000 and +0.001, respectively.Both format variants stayed within seed-σ on every metric, and neither departed from the canonical PPFG-stag result.Source: N Injection-format ablation (seamless… · p. 23, Table 20 · p. 23

O Prune-immunity content control

  • The prune-immunity test compared independent decoding, immunity with an empty graft, and immunity with a real graft to separate exemption from content effects.Both immunity arms exempted an injected chain from τkill-threshold pruning for three steps; the null graft held extra decoding budget constant.Source: O Prune-immunity content control · p. 23
  • About 27% fewer chains were pruned in both immune arms than under canonical PPFG-stag or independent, confirming the exemption engaged.The pruned-chain fraction fell from 0.1104 ± .0083 under canonical PPFG-stag and 0.1075 ± .0065 under independent to 0.0804 ± .0075 and 0.0808 ± .0076.Source: O Prune-immunity content control · p. 23
  • +0.71 pp at Pass@1 for real-graft immunity versus independent stayed within seed-σ; real graft versus null graft was −0.04 pp.The graft-content contrast was small-negative at higher k as well; the passage reads this as evidence against content-driven mis-pruning concealing a transfer benefit, within this test’s power.Source: O Prune-immunity content control · p. 23, Table 21 · p. 23

P Population-size sweep (N = 8 to N = 256)

  • The population-size sweep addressed whether N=8 starved cross-chain transfer, using matched populations on the first 100 MATH500 problems with three seeds per size.It spans a 32× range and is reported as a sensitivity analysis, not a replacement for the full-MATH500 results.Source: P Population-size sweep (N = 8 to N =… · p. 24
  • Across N=8–256, no Pass@k delta was positive; at N=256, Pass@1 was 0.7073 versus 0.7062 and Pass@8 was 0.8963 versus 0.8962.The paired values are independent versus PPFG-stag; the sweep found no benefit trend across a 32× population-size range.Source: P Population-size sweep (N = 8 to N =… · p. 24, Table 22 · p. 24
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