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Expanding LLM Reasoning
Rian Atri, Evan Luo
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Abstract
Abstract:Extra inference compute is usually spent on sampling more reasoning chains. We study where inside an existing chain an additional continuation should begin. We define expansion utility, the change in correctness from restarting a chain at a stored step, and measure it at every eligible step for nine models on six benchmarks (41 model and benchmark cells). Restart position matters: steps selected on one set of continuations beat uniform placement when scored on disjoint ones, in held-out audits on 5, 16, and 38 cells (+4.25 points [+2.51, +6.63] in a fresh five-cell audit). A fixed rule that restarts from the last eligible steps, always-last, is a strong baseline: our learned router beats uniform placement but shows no detected gain over it, and on DeepSeek-R1-Distill-Qwen-14B/MATH-500 always-last exceeds the exact self-consistency frontier at matched aggregate generated output by +0.052 [+0.008, +0.098], using 0.774x the aggregate generated output of four-sample self-consistency. Cross-fitted oracle selection still finds held-out headroom beyond declared positional classes, a target for future selectors. Finally, breaking step-label ties by earliest index flips the sign of a pointwise selector's gain over uniform placement in every seed of a five-seed diagnostic with four rollouts per step; randomized ties remove the bias.
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- Source: https://arxiv.org/abs/2610.05584v1
- Canonical: https://arxiv.org/abs/2610.05584v1
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Expanding LLM Reasoning Rian Atri Affiliation: Keiji AI Email: rianatri@keiji.ai Evan Luo Affiliation: University of California, Berkeley Email: evanluo25@berkeley.edu Abstract Extra inference compute is usually spent on sampling more reasoning chains. We study where inside an existing chain an additional continuation should begin. We define expansion utility, the change in correctness from restarting a chain at a stored step, and measure it at every eligible step for nine models on six benchmarks (41 model and benchmark cells). Restart position matters: steps selected on one set of continuations beat uniform placement when scored on disjoint ones, in held-out audits on 5, 16, and 38 cells (+4.25+4.25 points [+2.51,+6.63][+2.51,+6.63] in a fresh five-cell audit). A fixed rule that restarts from the last eligible steps, always-last, is a strong baseline: our learned router beats uniform placement but shows no detected gain over it, and on DeepSeek-R1-Distill-Qwen-14B/MATH-500 always-last exceeds the exact self-consistency frontier at matched aggregate generated output by +0.052+0.052 [+0.008,+0.098][+0.008,+0.098], using 0.774×0.774× the aggregate generated output of four-sample self-consistency. Cross-fitted oracle selection still finds held-out headroom beyond declared positional classes, a target for future selectors. Finally, breaking step-label ties by earliest index flips the sign of a pointwise selector’s gain over uniform placement in every seed of a five-seed diagnostic with four rollouts per step; randomized ties remove the bias. 1 Introduction Test-time compute is now a standard way to improve reasoning performance: models generate longer chains of thought, sample and vote over many chains, or rerank candidates with verifiers (Wei et al., 2022; Wang et al., 2023; Lightman et al., 2024; Snell et al., 2024; Brown et al., 2024). Most of these methods operate on complete trajectories, deciding how many to generate and which to keep. They leave open a finer-grained question: given an existing chain, where would one additional continuation help most? We define the correctness gain from restarting a chain at a stored step as its expansion utility (Figure 1). Estimating it requires intervening at every eligible step and observing outcomes, rather than inferring step importance from a verifier score. The resulting restart maps support four distinct measurements (Figure 2): (i) opportunity when the same draws select and score a step; (i) whether the selected steps keep their advantage under held-out scoring; (i) oracle headroom beyond declared positional classes; and (iv) what a feasible selector captures without observing outcomes. We measure expansion utility at every eligible step for nine models (8B to 123B parameters) on six benchmarks. Each model and benchmark pair is a cell, and 41 cells form the primary set. Restarting from each chain’s top-ranked steps at a 10% budget (one or two steps per chain) exceeds the mean over eligible steps by +14.80p+14.80\,p when the same draws select and score (Figure 3). This same-draw statistic is optimistic by construction, so we test the selection on held-out draws: in audits on 5, 16, and 38 cells, positions selected on one set of continuations still beat uniform placement when scored on disjoint ones (Figure 4). Where a chain is restarted changes whether it succeedsk=2k=2restart success ratexix_is1s_1s2s_2s3s_3s4s_4s5s_5s6s_6s7s_7s8s_8baseYi,base=0Y_i,base=0restart k=2k=2keeps s<2s_<2restart k=6k=6keeps s<6s_<6p^i,2=1/4 p_i,2=1/4p^i,6=3/4 p_i,6=3/4 U^i,k=p^i,k−Yi,base U_i,k= p_i,k-Y_i,base expansion utility of restarting at k Base answer is wrong, soutility equals the hit rate.early k=2k=2: U^i,2=1/4 U_i,2=1/4late k=6k=6: U^i,6=3/4 U_i,6=3/4resample rightresample wrong Figure 1: Expansion utility. Restarting a stored chain at step k keeps the prefix s<ks_<k and samples n continuations. Their success rate p^i,k p_i,k, taken over all eligible steps, forms the chain’s restart map, and U^i,k=p^i,k−Yi,base U_i,k= p_i,k-Y_i,base is the expansion utility of restarting at k. In this schematic an early restart rarely succeeds and a late one usually does. Across 2525 of 2525 split-and-refit runs on a frozen dataset, our learned router exceeds uniform placement in GR@10 but shows no detected gain over always-last, a fixed rule that restarts from the last eligible stored steps and has no learned component (Figure 5a). On DeepSeek-R1-Distill-Qwen-14B/MATH-500, always-last also clears the exact self-consistency frontier at 0.774×0.774× the aggregate generated output of four-sample self-consistency, SC(4) (Figure 6); in the one other available cell, the margin is unresolved. selectscoreUniformAlways-lastLearned router+Measurerestart + samplesEstimateseparate drawsDecidespend / skip Figure 2: Study design. We restart a stored chain at every eligible step and sample continuations (measure); estimate opportunity from the resulting restart map, both on the same draws and on held-out draws that separate selection from scoring (estimate); and compare placement rules (uniform, always-last, and a learned router), with a confidence gate that decides whether to spend a restart at all (decide). Cross-fitted oracle selection still finds held-out headroom beyond the declared positional classes (Figure 5b), a target for selectors that go beyond them. Finally, we identify a pitfall for step-level supervision: when several steps tie for the positive label, breaking ties by position can flip the sign of a pointwise selector’s gain. Randomized tie breaking removes the systematic early-position preference, and the evaluated ListNet loss is invariant to tie resolution (Figure 7). 2 Expansion Utility Let xix_i be a problem and let si,0:Kis_i,0:K_i be a model-generated reasoning chain segmented at stored step boundaries. Restarting at position k keeps the prefix si,<ks_i,<k fixed and samples a new suffix. Each chain is stored as at most 1212 segments, and a restart regenerates all text after its prefix. With correctness outcome Yi,k,j∈0,1Y_i,k,j∈\0,1\ for continuation draw j, the observed restart rate and expansion utility are p^i,k=1n∑j=1nYi,k,j,U^i,k=p^i,k−Yi,base, p_i,k= 1n _j=1^nY_i,k,j, U_i,k= p_i,k-Y_i,base, (1) where Yi,base∈0,1Y_i,base∈\0,1\ is the correctness of the original chain. Thus p^i,k p_i,k is the empirical success rate of continuations from step k, and U^i,k U_i,k is its gain over the base outcome. The primary sweep uses n=4n=4 (two or eight in three cells; Appendix G). The eligible set ℰiE_i contains restart positions 1,…,Ki−11,…,K_i-1; index 0, which would regenerate the whole chain, is excluded. At a nominal budget b=0.10b=0.10, mi=⌈b|ℰi|⌉m_i= b|E_i| , which is one step for chains with at most ten eligible positions and two for the eleven-position chains produced by the 12-segment cap. For the N included chains of a cell, the statistic compares each chain’s top-ranked steps with the mean over its eligible steps: Δ^=1N∑i=1N[1mi∑k∈Topmi(p^i)p^i,k−1|ℰi|∑k∈ℰip^i,k]. = 1N _i=1^N [ 1m_i _k _m_i( p_i) p_i,k- 1|E_i| _k _i p_i,k ]. (2) Because the same continuations select and score the chosen positions, Eq. (2) is a same-draw statistic: it measures opportunity, not what a feasible selector gains. To test whether the selection replicates, we also select positions on one set of continuation outcomes and score them on disjoint outcomes. These held-out protocols differ in how many draws select and score, so we report each separately rather than using one to correct another. We summarize concentration with two Gini coefficients (higher means more concentrated) that answer different questions: a pooled Gini, which clips negative utility and pools positive utility over all eligible positions in a cell, and a within-chain Gini, computed per chain and then averaged. 3 Experimental Setup Models and tasks We evaluate nine models from 8B to 123B parameters on MATH-500 (Hendrycks et al., 2021; Lightman et al., 2024), GPQA (Rein et al., 2023), HumanEval (Chen et al., 2021), AIME 2025, HMMT, and MBPP, with Putnam as an exploratory benchmark (Mathematical Association of America, 2025; Harvard-MIT Math Tournament Organisers, 2025; Austin et al., 2021; Mathematical Association of America, 2024). The primary set contains 41 nondegenerate model and benchmark cells with 30 to 500 problems each; not every model is run on every benchmark, and the two exploratory Putnam cells are reported separately and excluded from aggregates (Appendix A). Checkpoint identifiers, graders, and generation settings are given in Appendix G, and per-cell counts in Table 2. Uncertainty We report coverage as cell counts. Held-out scoring intervals use a crossed percentile bootstrap over shared problems and cells. Oracle-residual intervals resample examples pooled across cells for the depth class and problem clusters for the enriched class. Intervals for the router comparisons (Section 5) resample paired split seeds and describe stability across splits of one frozen dataset, not problem-population uncertainty. The self-consistency comparison uses a paired problem bootstrap that keeps every seed of a resampled problem together. Model fitting, calibration, and held-out scoring follow frozen protocols. 4 Restart Opportunity Is Concentrated and Survives Held-Out Scoring Same-draw opportunity averages +14.80p+14.80\,p across the 4141 primary cells and varies substantially across models and benchmarks (Figure 3). Because a chain’s top-ranked mean cannot fall below its eligible mean, the sign of this statistic is not informative; its magnitude, its concentration, and its survival under held-out scoring are. For utilities clipped at zero, positive restart value is highly concentrated within cells (pooled Gini 0.8960.896 on average, at least 0.800.80 in 3737 of 41 cells) and uneven within chains (mean within-chain Gini 0.5850.585 among chains with nonzero clipped utility). Cells with more concentrated positive utility have smaller advantages (r=−0.563r=-0.563 across the 41 primary cells of Appendix Table 2). MBPPAIME 2025HMMTHumanEvalMATH-500GPQAPutnam (exploratory)010203035advantage over the chain average (p) Figure 3: Same-draw restart opportunity varies widely across cells. Each dot is one model and benchmark cell: the Eq. (2) advantage of the top-ranked steps at a 10%10\% budget over the chain’s mean eligible step, in percentage points; ticks mark benchmark means. The statistic is non-negative by construction; Figure 4 tests whether the selection replicates on held-out draws. The two exploratory Putnam cells (hollow, n=23n=23) are excluded from aggregates. Per-cell values are in Appendix Table 2. 005510101515Eq. (2), all 41 primary cellsFresh 8 draws, 5 cellsStored 2/2 split, 38 cellsSeparate A/B samples, 16 cellsFresh 4 select, 4 score, 5 cellsStored winner, 8 unseen draws, 5 cells+14.80+14.80+14.25[+12.00,+16.35]+14.25\ [+12.00,\ +16.35]+11.03[+7.74,+14.54]+11.03\ [+7.74,\ +14.54]+5.48[+3.67,+7.59]+5.48\ [+3.67,\ +7.59]+4.25[+2.51,+6.63]+4.25\ [+2.51,\ +6.63]+2.01[+1.04,+3.66]+2.01\ [+1.04,\ +3.66]Same drawsselect and scoreHeld-outscoringadvantage over uniform (p)Held-out scoring results Figure 4: Held-out scoring. Advantage of selected restart positions over uniform placement when the same draws select and score (gray) and when selection and scoring use disjoint draws (blue), with 95%95\% intervals where available. The protocols estimate different selectors and cover different cells (Appendix B). Figure 4 scores selected positions on draws that did not select them. The advantage shrinks, as selection on noise predicts, and persists. In a five-cell audit of 718718 examples, positions selected on four fresh continuations have a restart-success advantage over uniform of +4.25p+4.25\,p [+2.51,+6.63][+2.51,+6.63] when scored on the complementary four, and positions selected from the original four stored continuations keep +2.01p+2.01\,p [+1.04,+3.66][+1.04,+3.66] on eight new continuations. Across 1616 cells with two separately generated continuation samples per prefix, the advantage is +5.48p+5.48\,p, positive in every cell. Across the 3838 primary cells that retain individually labeled continuation outcomes, selecting on two of each step’s four stored continuations and scoring on the other two gives +11.03p+11.03\,p [+7.74,+14.54][+7.74,+14.54], positive in 3636 of 3838 cells. These audits estimate different selectors (Appendix B, Table 3), but all place the selected positions above uniform placement in aggregate. 5 Where to Restart Versus Whether to Spend We evaluate a learned router that, without observing restart outcomes, scores each eligible step from text and structure features, process reward model (PRM) summaries, and token log-probability summaries. It is trained with a listwise loss over the steps of a chain and a learned depth prior, and it restricts candidates to a benchmark-conditional late support: the last third of the chain (the middle third for Putnam). A separate confidence gate decides whether to spend a restart on a chain at all. We compare placement rules on held-out problems by gap recovered at a 10%10\% budget, GR@10=rule−uniformoracle−uniform,GR@10= rule-uniformoracle-uniform, where each term is the mean observed restart rate at the positions a policy selects. Uniform selects eligible positions uniformly at random (GR@10=0=0), and oracle is the same-draw top-mim_i selector of Eq. (2) (GR@10=1=1), so GR@10 is the fraction of the oracle’s advantage over random placement that a rule recovers. The simplest fixed rule, always-last, selects the last mim_i eligible positions. GR@10 is computed on held-out problems whose restart rates vary across positions; when the gate skips a chain, that chain contributes its base outcome to every term. 0.000.000.100.10uniformalways-last ungatedalways-last gatedrouteralways-lastRouterminusGate effectonGR@10 difference000.0250.0250.050.05A to B to A to B to A14/1614/1615/1615/1615/1615/1616/1616/16Depth classEnrichedclassrestart-success residual(a) Placement and spending(b) Oracle headroom Figure 5: Placement, spending, and oracle headroom. (a) Paired GR@10 differences over 2525 split seeds: the router against uniform placement (both ungated) and against always-last (both ungated, and both gated), and the gate’s effect on each policy (orange involves the router, blue is always-last). Intervals are split-seed stability intervals on one frozen store, not problem-population intervals. (b) Held-out restart-success residual of cross-fitted oracle selection beyond the declared 21-policy depth class and the enriched positional class, in both directions (A to B: sample A selects, sample B scores); numbers at right count cells with a positive residual out of 1616. Intervals resample examples pooled across cells (depth class) or problem clusters (enriched class). This is an oracle target, not a feasible-policy gain. Exact values are in Appendix Tables 7, 5, and 6. Figure 5a separates where to restart from whether to spend over 25 split seeds. Ungated, the router beats uniform placement by +0.127+0.127 GR@10 [+0.108,+0.146][+0.108,+0.146], in 2525 of 2525 seeds. Against always-last, however, it shows no detected gain: the paired difference is −0.006-0.006 [−0.016,+0.005][-0.016,+0.005] ungated and +0.008+0.008 [−0.023,+0.037][-0.023,+0.037] gated. Gate-on minus gate-off GR@10 is +0.056+0.056 [+0.020,+0.092][+0.020,+0.092] for the router and +0.042+0.042 [+0.005,+0.083][+0.005,+0.083] for always-last, and the paired difference between the two is +0.013+0.013 [−0.020,+0.044][-0.020,+0.044]. A separate fixed-composition audit, comparing learned gating with random chain selection under always-last, finds an unresolved accuracy contrast (Appendix Table 7). The router’s gain over uniform is therefore not separable from late position. Component controls agree: dropping the late support entirely (while keeping the learned depth prior) changes little, removing the depth prior improves GR@10 by +0.0215+0.0215, and adding PRM, log-probability, or hidden-state features shows no detected improvement in the tested selector within the evaluated scopes (Appendix Table 8). Always-last is therefore the reference that learned placement must beat: the router beats uniform placement in every seed yet shows no detected gain over always-last. Gates must also be applied to every compared policy, because the gate raised GR@10 for both policies (+0.056+0.056 and +0.042+0.042); comparing a gated learned policy with an ungated baseline would credit spending to placement. 6 Oracle Headroom Beyond Declared Positional Classes Cross-fitted oracle selection measures the held-out restart-success residual over the rule selected from each declared positional class; the residual is an oracle target, not a feasible selector’s gain. We use the two continuation samples collected at each prefix, A and B, from the point- and block-restart sweeps (Appendices B and C). Sample A fits a declared positional class and selects the oracle’s restart positions; sample B scores both, and the roles are then swapped. On the restart positions 11 to Ki−3K_i-3 that the two sweeps share, across 1616 cells, the oracle exceeds the rule selected from a 21-policy depth class by +0.0275+0.0275 [+0.0183,+0.0368][+0.0183,+0.0368] and +0.0248+0.0248 [+0.0157,+0.0340][+0.0157,+0.0340] in held-out restart-success rate, positive in 1414 and 1515 of 1616 cells (Figure 5b; Appendix Table 5). The residual remains positive against a richer positional learner, which scores relative position separately by chain length (length-by-rank tables) with model selection nested inside the training folds; that learner does not outperform the original class out of sample (Table 6). Because continuation outcomes select the oracle’s positions, the residual is an oracle target for feasible selectors, lying beyond the declared positional classes (the richer family does not span every positional function). Whether features can predict it is open: the PRM, log-probability, and hidden-state summaries we added showed no detected improvement in the tested selector (Table 8). 7 Always-Last Versus the Self-Consistency Frontier We compare always-last with the standard way of spending extra samples, self-consistency (SC), which samples N complete chains and returns the most common parsed answer. We compare against the exact SC frontier. At each policy’s aggregate generated output, we solve for the best randomized mixture of SC(1) to SC(4) with the same output, the best comparator in the tested SC(1) to SC(4) family at that budget. The panel is a frozen DeepSeek-R1-Distill-Qwen-14B/MATH-500 set of 100100 problems and 33 generation seeds (Appendix D). AccuracyGenerated output0.760.780.800.820.8400.51.0generated output (× SC(4))SC(4)always-lastrouterSC at same output+0.052+0.0541.000×0.774×0.885× Figure 6: Always-last is more accurate than self-consistency at matched aggregate generated output in the tested cell (DeepSeek-R1-Distill-Qwen-14B/MATH-500, 100100 problems, 33 seeds). Left: hollow circles are the best mix of SC(1) to SC(4) at each policy’s own generated output; filled circles are the policy. Margins are +0.052+0.052 [+0.008,+0.098][+0.008,+0.098] for always-last and +0.054+0.054 [+0.010,+0.103][+0.010,+0.103] for the router (95%95\% paired bootstrap; Appendix Table 10); their difference is not detected. Right: aggregate generated output relative to SC(4), not total tokens or wall-clock time. Always-last reaches 0.8300.830 accuracy at 0.774×0.774× the aggregate generated output of SC(4) and exceeds the exact SC(1) to SC(4) frontier at matched aggregate output by +0.052+0.052 [+0.008,+0.098][+0.008,+0.098]; it lies above the frontier in 33 of 33 generation seeds (Figure 6; Appendix Table 10). The router also clears its frontier, in 33 of 33 seeds, and the paired router-minus-always-last difference is unresolved, so the experiment supports late continuation rather than learned restart placement, in one cell and under one cost ledger. In a separate two-seed, 100-problem H200 assay, always-last used 0.912×0.912× the aggregate input-plus-output tokens and 1.108×1.108× the wall-clock time of parallel SC, with unresolved accuracy; these results do not establish a latency or total-token benefit. Always-last requires no training, verifier, or features, and a late restart regenerates only the tail of an existing chain, which matters when generated output is the binding cost. It runs after the base chain, so latency and total tokens must be measured separately, and in the one other available cell, Qwen3-32B/MATH-500, its margin is unresolved. 8 Label Ties Can Manufacture Position Effects Our pointwise step selector is trained on binary labels: with four rollouts per step, the top mim_i steps of each chain are labeled positive. Several steps often tie at the selection cutoff, and breaking those ties by earliest index favors earlier steps, injecting an artificial early-position signal into the labels (Figure 7a). In a separate five-seed diagnostic without positional support, mean pointwise binary cross-entropy (BCE) GR@10 moves from −0.175-0.175 under earliest-index ties to +0.095+0.095 under randomized ties, and the sign flips in all 55 seeds (ranges −0.218-0.218 to −0.137-0.137 and +0.050+0.050 to +0.134+0.134). The evaluated ListNet formulation (Cao et al., 2007), which trains on within-chain relative scores, is exactly invariant (+0.123+0.123 under both rules; Figure 7b). (a) Tie rulecorrect / 4earliest indexrandomized14024134earlylate(b) GR@10−0.2-0.2000.20.2uniformBCE, earliest indexBCE, randomizedListNet, either rule−0.175-0.175+0.095+0.095+0.123+0.123GR@10 Figure 7: Label ties can flip a pointwise selector. (a) Schematic: one chain’s correct continuations out of four at each step (darker cells are higher; outlined cells tie at the maximum). Earliest-index tie breaking always labels the first tied step, which injects an early-position signal; randomized tie breaking labels one tied step at random. (b) GR@10 in a five-seed diagnostic (bars are means; whiskers span the minimum to maximum across seeds), where zero is uniform placement: pointwise BCE moves from −0.175-0.175 with earliest-index ties to +0.095+0.095 with randomized ties, while ListNet is +0.123+0.123 under either rule. Fair-label loss contrasts are in Appendix Table 12. In the separate primary 24-seed comparison with randomized ties, ListNet minus BCE is +0.0128+0.0128 [−0.0033,+0.0293][-0.0033,+0.0293] without positional support and −0.0016-0.0016 [−0.0161,+0.0133][-0.0161,+0.0133] within the benchmark-conditional late support, using a problem-cluster bootstrap (Appendix Table 12). Both intervals cross zero, so we conclude only that ListNet is invariant to tie resolution, not that one loss ranks above the other or that they are equivalent within the support. After the primary results were available, we locked the code, data, and configuration and ran a 64-seed robustness extension. It meets its frozen criteria for a positive contrast without positional support and for equivalence within ±0.02± 0.02 GR@10 inside the late support; it is a post-lock robustness extension, not independent confirmation. This pitfall and the attenuation in Section 4 both bear on process supervision, where steps are often labeled with Monte Carlo rollouts from prefixes (Wang et al., 2024; Luo et al., 2024). On the same five audited cells, selecting and scoring on the same eight continuations gives +14.25p+14.25\,p, whereas selecting on four and scoring on the complementary four gives +4.25p+4.25\,p. The selectors differ, but the attenuation is in the direction predicted by selection on noise, so rollout-derived step importance should be validated on held-out rollouts before it is used for training or reported. Because ties are common with four rollouts per step, randomized tie breaking removes systematic earliest-index labeling and the evaluated ListNet loss is invariant to tie resolution; neither requires additional rollouts. 9 Related Work Self-consistency allocates compute across independent chains (Wang et al., 2023), and inference-scaling studies compare search and verification strategies under budgets (Snell et al., 2024; Brown et al., 2024). Process supervision and PRMs score intermediate reasoning (Lightman et al., 2024; Uesato et al., 2022), while tree-based methods branch over partial trajectories (Yao et al., 2023; Luo et al., 2024). Resampling from intermediate prefixes has also been used to locate forking tokens and influential reasoning sentences (Bigelow et al., 2025; Bogdan et al., 2025), and adaptive computation methods learn when to halt or route computation (Graves, 2016; Schuster et al., 2022). We instead measure interventional restart effects at every stored prefix, then test how much of that opportunity fixed placement, declared positional classes, and a feasible selector capture. 10 Discussion Where to continue is its own allocation decision Test-time scaling is usually framed as how many independent chains to sample or how to rank complete candidates. Our measurements quantify a complementary axis: where inside an existing chain to spend a continuation. Restart value along this axis is uneven within chains, and its advantage survives held-out scoring across models and benchmarks at reduced magnitude (Section 4). Where to continue is therefore a separate allocation decision: always-last is the reference baseline, and same-draw optimism and tie-induced position bias are its characteristic failure modes. Always-last as the reference The learned router we evaluated shows no detected gain over a rule with no learned component (Section 5), and that rule clears matched self-consistency in the one tested cell under aggregate generated output (Section 7). Claims of learned within-chain placement should be measured against it. We recommend that such methods state whether a reported gap is same-draw, held-out, or achieved by a feasible selector; report paired contrasts against both uniform placement and always-last, with any gate applied to every policy; and report generated output, total tokens, and wall-clock time separately. Restart value as a supervision target Restart value is an interventional quantity rather than a judgment of whether a step is correct, and adding Math-Shepherd PRM summaries showed no detected improvement in the tested selector. Restart value may therefore be a useful supervision target, and the held-out residual beyond declared positional classes (Section 6) is the reference against which a selector trained on it should be measured. 11 Limitations and Broader Impact Eq. (2) is a same-draw statistic, and the held-out audits that check it cover fewer cells and estimate different selectors. Results are conditional on a designed rather than sampled grid of models and benchmarks, on automatic graders, and on stored step boundaries. The resolved self-consistency advantage is on one panel and is measured in aggregate generated output; total tokens and wall-clock time are reported separately, and the residual of Section 6 is an oracle target. On the tested panel, continuing from stored prefixes was more accurate than the self-consistency frontier at lower generated output; a miscalibrated selector could instead damage correct answers or waste compute. The work uses public benchmarks and publicly released models and collects no human-subject data. 12 Conclusion Restart maps expose a second axis of test-time allocation: where to continue an existing chain. Across 41 model and benchmark cells, restart value is uneven within chains, and held-out audits on 5, 16, and 38 cells each retain a positive advantage over uniform placement. Always-last is a strong fixed baseline: our learned router shows no detected gain over it, and on DeepSeek-R1-Distill-Qwen-14B/MATH-500 it exceeds the exact self-consistency frontier by +0.052+0.052 at matched aggregate generated output. Cross-fitted oracle analysis shows that the declared positional classes do not exhaust restart value, and randomized ties remove an early-position labeling bias without additional rollouts. Future restart selectors should report held-out utility, paired comparisons with always-last under matched gating, and costs in explicit units. Acknowledgments, Funding, and Competing Interests The authors thank the anonymous reviewers and area chairs for the controls that strengthened this analysis. Funding: This work was self-funded by the authors and received no external funding. Competing interests: The authors declare no competing interests. References Austin et al. [2021] Jacob Austin, Augustus Odena, Maxwell Nye, Maarten Bosma, Henryk Michalewski, David Dohan, Ellen Jiang, Carrie Cai, Michael Terry, Quoc Le, and Charles Sutton. Program synthesis with large language models. arXiv preprint arXiv:2108.07732, 2021. Bigelow et al. [2025] Eric J. Bigelow, Ari Holtzman, Hidenori Tanaka, and Tomer Ullman. Forking paths in neural text generation. In International Conference on Learning Representations (ICLR), 2025. Bogdan et al. [2025] Paul C. Bogdan, Uzay Macar, Neel Nanda, and Arthur Conmy. Thought anchors: Which LLM reasoning steps matter? arXiv preprint arXiv:2506.19143, 2025. Brown et al. 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[2022] Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Brian Ichter, Fei Xia, Ed Chi, Quoc V Le, and Denny Zhou. Chain-of-thought prompting elicits reasoning in large language models. Advances in Neural Information Processing Systems, 35:24824–24837, 2022. Yao et al. [2023] Shunyu Yao, Dian Yu, Jeffrey Zhao, Izhak Shafran, Thomas L. Griffiths, Yuan Cao, and Karthik Narasimhan. Tree of thoughts: Deliberate problem solving with large language models. In Advances in Neural Information Processing Systems (NeurIPS), 2023. Zhipu AI / THUDM [2025] Zhipu AI / THUDM. GLM-4.5-Air. https://huggingface.co/zai-org/GLM-4.5-Air, 2025. Appendix A Measurement Definitions and Cell Inclusion Eq. (2) is the budgeted eligible-position advantage (+14.80p+14.80\,p). The maximum-observed-utility statistic, which averages the largest observed utility per example, is a distinct quantity (+21.34p+21.34\,p). All main-text tables and figures are generated from checksummed JSON by a single build script (Appendix G). A chain enters a cell when it has at least two eligible restart positions. A cell is degenerate when its chains are effectively single-step (Llama-Nemotron-70B on HMMT) or the model emits no usable multistep traces (Nemotron-Nano-8B on GPQA); of the 54 model and benchmark pairs, these two are degenerate and 11 were not run (GPT-OSS-20B on the five benchmarks other than HMMT, and MBPP for the six models other than GPT-OSS-20B and the two 8B models). The primary roster therefore contains 38 nondegenerate main-matrix cells, one GPT-OSS HMMT cell, and two MBPP cells. The two Putnam rows have n=23n=23 and are displayed as exploratory, producing 43 displayed rows but 41 primary rows. Aggregates weight cells equally; weighting each primary cell by its number of problems gives +18.27p+18.27\,p instead of +14.80p+14.80\,p. Both Gini coefficients use the standard sorted-vector formula on utilities clipped at zero over eligible positions. The pooled Gini includes every chain of a cell; the within-chain mean omits chains whose clipped eligible utility is zero everywhere, for which the Gini is undefined. Thirteen primary cells were collected again with per-request sampling seeds, and two 8B MATH-500 rows use their full n=500n=500 membership. Table 1: Measurement summary. The primary set contains 4141 cells; the complete display contains two additional exploratory Putnam cells with n=23n=23. Pooled Gini is computed after pooling positive utility over eligible positions within a cell. The final row is a different estimand from Eq. (2). Quantity Estimate Coverage Formal Eq. (2) advantage +14.80+14.80 p 41/4141/41 positive Pooled eligible-position Gini 0.8960.896 37/4137/41 cells ≥0.80≥ 0.80 Mean within-example eligible Gini 0.5850.585 4141 cells Maximum observed utility +21.34+21.34 p different estimand Table 2: Per-cell measurement record. Putnam rows are exploratory and are excluded from the 41-cell primary aggregate. The max-utility column averages each example’s largest observed utility, a different estimand from Eq. (2) that can fall below it because utility subtracts the base outcome. Model Benchmark n Eq. (2) p pooled Gini max-utility p DS-R1-Llama-8B MATH-500 500 +14.69 0.778 +28.90 DS-R1-Llama-8B GPQA 198 +17.20 0.836 +25.63 DS-R1-Llama-8B HumanEval 164 +9.87 0.854 +19.82 DS-R1-Llama-8B AIME 2025 30 +2.77 0.984 +2.50 DS-R1-Llama-8B HMMT 30 +9.83 0.928 +11.67 Nemotron-Nano-8B MATH-500 500 +16.17 0.760 +29.60 Nemotron-Nano-8B HumanEval 164 +13.91 0.799 +33.08 Nemotron-Nano-8B AIME 2025 30 +1.82 0.941 +3.33 Nemotron-Nano-8B HMMT 30 +8.39 0.852 +13.33 DS-R1-Qwen-14B MATH-500 500 +19.37 0.775 +39.55 DS-R1-Qwen-14B GPQA 198 +26.81 0.845 +39.14 DS-R1-Qwen-14B HumanEval 164 +19.42 0.803 +38.72 DS-R1-Qwen-14B AIME 2025 30 +8.71 0.939 +16.67 DS-R1-Qwen-14B HMMT 30 +21.35 0.911 +31.67 GLM-4.5-Air MATH-500 500 +33.01 0.819 +48.00 GLM-4.5-Air GPQA 198 +30.40 0.859 +46.46 GLM-4.5-Air HumanEval 164 +20.01 0.826 +35.82 GLM-4.5-Air AIME 2025 30 +5.00 0.971 +11.67 GLM-4.5-Air HMMT 30 +19.20 0.948 +25.00 Gemma-3-27B MATH-500 500 +6.17 0.953 +8.50 Gemma-3-27B GPQA 198 +23.94 0.893 +26.01 Gemma-3-27B HumanEval 164 +16.16 0.874 +20.27 Gemma-3-27B AIME 2025 30 +7.14 0.970 +10.00 Gemma-3-27B HMMT 30 +18.66 0.929 +27.50 Qwen3-32B MATH-500 500 +29.27 0.815 +42.25 Qwen3-32B GPQA 168 +32.14 0.887 +34.38 Qwen3-32B HumanEval 60 +16.67 0.867 +26.67 Qwen3-32B AIME 2025 30 +3.65 0.950 +7.50 Qwen3-32B HMMT 30 +13.87 0.916 +23.33 Llama-Nemotron-70B MATH-500 500 +15.62 0.931 +17.50 Llama-Nemotron-70B GPQA 198 +33.11 0.856 +36.36 Llama-Nemotron-70B HumanEval 164 +7.07 0.958 +6.71 Llama-Nemotron-70B AIME 2025 30 +4.21 0.981 +6.67 Mistral-Large-2411 MATH-500 500 +13.94 0.934 +15.95 Mistral-Large-2411 GPQA 198 +18.33 0.906 +13.64 Mistral-Large-2411 HumanEval 164 +19.32 0.984 +3.05 Mistral-Large-2411 AIME 2025 30 +8.87 0.937 +11.67 Mistral-Large-2411 HMMT 30 +7.29 0.967 +13.33 GPT-OSS-20B HMMT 30 +7.49 0.876 +16.67 DS-R1-Llama-8B MBPP 105 +2.32 0.979 +2.38 Nemotron-Nano-8B MBPP 106 +3.44 0.962 +4.25 DS-R1-Llama-8B Putnam† 23 +11.53 0.843 +20.65 Nemotron-Nano-8B Putnam† 23 +14.36 0.798 +29.35 † Exploratory n=23n=23 row, displayed but excluded from primary aggregates. Appendix B Held-Out Scoring Audits Table 3: Held-out scoring audits estimate different selectors. The five-cell audit has 718718 exact-prefix examples. The matrix-wide 2/2 audit covers the 3838 primary cells that retain individually labeled continuation outcomes. Values are cell-equal percentage-point advantages over the exact uniform-candidate comparator. Protocol Gap [95%CI][95\%\ CI] Fresh-eight same-draw selection/scoring +14.25[+12.00,+16.35]+14.25\ [+12.00,+16.35] Stored winner, eight unseen scoring draws +2.01[+1.04,+3.66]+2.01\ [+1.04,+3.66] Fresh four select, disjoint four score +4.25[+2.51,+6.63]+4.25\ [+2.51,+6.63] Exhaustive tie-neutral stored 2/2 +11.03[+7.74,+14.54]+11.03\ [+7.74,+14.54] The fresh-eight same-draw row uses the same eight new outcomes to select and score and therefore remains selected on noise. The stored-winner row freezes the original four-draw choice and scores it on eight unseen continuations. The fresh 4/4 row selects on four fresh continuations and scores on the complementary four. The 2/2 matrix audit exhausts all three complementary partitions and both directions, with analytic tie-neutral inclusion at the budget cutoff. Each protocol estimates a different selector, so we report them separately. Pooled Gini itself is reproducible across paired halves, with Pearson range from 0.9890.989 to 0.9960.996. The median disjoint-half Gini-gap correlation is −0.431-0.431 [−0.677,−0.167][-0.677,-0.167]. Table 4: Separately generated samples and boundary density. The symmetric 4/4 result uses separately generated continuation samples at identical prefixes. Boundary thinning changes the candidate set and therefore the magnitude of the same-draw Eq. (2) statistic; it does not test semantic segmentation. The boundary-density sources report point estimates only, not population intervals. Protocol Cell-equal gap [95%CI][95\%\ CI] Coverage Positive cells Separate A/B samples, symmetric 4/4 +5.48[+3.67,+7.59]+5.48\ [+3.67,+7.59] p 34213421 examples 16/1616/16 Every second exact boundary +11.87+11.87 p 4141 cells 41/4141/41 Every third exact boundary +8.66+8.66 p 4141 cells 41/4141/41 Four fixed normalized anchors +12.17+12.17 p 4141 cells 41/4141/41 The separately generated 4/4 audit (Table 4) covers 34213421 examples in 1616 cells and gives +5.48p+5.48\,p [+3.67,+7.59][+3.67,+7.59], positive in every cell. Its cross-sample cell-Gini reliability is r=0.994r=0.994. The boundary-density controls change the candidate set and thereby the magnitude of same-draw opportunity; they do not test semantic segmentation. Appendix C Policy-Class Protocol The declared depth class contains 21 policies indexed by a target relative depth q∈0,0.05,…,1q∈\0,0.05,…,1\: policy q selects the eligible positions whose relative depth is closest to q (ties to the earlier position), and q is chosen by five-fold cross-validation on the selection sample. Candidate selection uses one continuation sample and final scoring uses the other; the direction is then reversed. Sample A is the point restart at position j and sample B is the radius-one block restart centered at j+1j+1, which regenerates from the same prefix; both are separately generated with identical sampling settings and graders. Candidates are positions 11 to Ki−3K_i-3, where both samples exist. The enriched sensitivity analysis nests selection over the original grid and regularized length-by-rank tables. In both cases, continuation outcomes remain in the oracle selection rule. These experiments therefore quantify residual opportunity, not predictability from features. Table 5: Held-out oracle-information residual beyond a declared 21-policy depth class. Five-fold cross-fitting, 10%10\% within-chain budget, 34213421 records and 1616 cells; intervals resample examples pooled across cells. This is a held-out oracle residual over the selected depth rule, not a feasible selector’s gain or a deployed-accuracy result. Direction Residual [95%CI][95\%\ CI] Positive cells A selects, B scores +0.0275[+0.0183,+0.0368]+0.0275\ [+0.0183,+0.0368] 14/1614/16 B selects, A scores +0.0248[+0.0157,+0.0340]+0.0248\ [+0.0157,+0.0340] 15/1615/16 Table 6: Nested-selected enriched positional sensitivity. The candidate family contains the original grid and regularized length-by-rank tables. It does not cover the full arbitrary positional class, and the selected enriched learner trails the original grid out of sample. Intervals cluster by problems or cells as labeled. Direction Mean Problem-cluster 95%95\% CI Cell-cluster 95%95\% CI Positive cells A selects, B scores +0.03228+0.03228 [+0.02409,+0.04048][+0.02409,+0.04048] [+0.01941,+0.05055][+0.01941,+0.05055] 15/1615/16 B selects, A scores +0.02956+0.02956 [+0.02111,+0.03818][+0.02111,+0.03818] [+0.01730,+0.04869][+0.01730,+0.04869] 16/1616/16 The evaluated router uses within-example engineered text and structure features together with PRM and token-log-probability summaries, a listwise objective, a learned depth prior, positional support, and a confidence gate. Attribution rests on the paired always-last controls. The evaluated router is a two-hidden-layer MLP with widths 64 and 32, GELU activations, and dropout 0.10 after the first hidden layer. It is fitted with AdamW at learning rate 3×10−43× 10^-4, weight decay 10−410^-4, mini-batch size 256, and at most 80 epochs. Early stopping uses patience 12. Each split keeps all positions from one example on the same side, reserves 20% of examples for outer evaluation, and reserves 15% of the remaining training examples for inner validation. The evaluated configuration uses ListNet temperature 1, a three-bin depth-prior floor of 0.1, and four retained outcomes per action. The multi-seed controls repeat the complete split-and-fit procedure at their reported seeds; no test outcome is used to choose a threshold or architecture. The confidence gate is a small MLP over chain-level structure, text, and PRM summary features, trained to predict base-chain correctness. Its threshold is selected on inner validation, maximizing mean GR across budgets subject to routing at least 10% of chains, and is then frozen for testing. On held-out problems the gate routes a mean of 23%23\% of chains, ranging from 6%6\% to 96%96\% across split seeds, so its operating point varies substantially between splits. Table 7: Separating placement from spending. GR@10 intervals are nominal paired split-seed bootstrap stability intervals on one frozen store, not problem-population intervals. The router-fitted gate threshold is applied unchanged to both placement rules. Crossing-zero intervals are reported as no detected paired difference, not as equivalence. Contrast Estimate [95%CI][95\%\ CI] Interpretation Router −- uniform, ungated +0.127[+0.108,+0.146]+0.127\ [+0.108,+0.146] stable in-distribution diagnostic Router −- always-last, ungated −0.006[−0.016,+0.005]-0.006\ [-0.016,+0.005] no detected increment Gate effect on router +0.056[+0.020,+0.092]+0.056\ [+0.020,+0.092] gate-on minus gate-off Gate effect on always-last +0.042[+0.005,+0.083]+0.042\ [+0.005,+0.083] gate-on minus gate-off Difference in gate effects +0.013[−0.020,+0.044]+0.013\ [-0.020,+0.044] no detected interaction Router −- always-last, gated +0.008[−0.023,+0.037]+0.008\ [-0.023,+0.037] no detected increment Fixed-composition accuracy audit +0.003[−0.010,+0.015]+0.003\ [-0.010,+0.015] problem-cluster bootstrap Table 8: Component and feature controls. Component contrasts use matched split seeds and GR@10. The log-probability and hidden-state rows are matched-coverage 10-seed mean contrasts; their source artifact did not promote a population interval. Negative or crossing-zero results are reported as scoped nulls, not as evidence that a feature family is generally uninformative. Control Estimate [95%CI][95\%\ CI] Scope Benchmark-conditional −- no support −0.0002[−0.0029,+0.0019]-0.0002\ [-0.0029,+0.0019] 1010 matched seeds Remove learned depth prior +0.0215[+0.0032,+0.0398]+0.0215\ [+0.0032,+0.0398] 1010 matched seeds PRM-feature −- text-feature variant, ungated −0.0038[−0.0127,+0.0053]-0.0038\ [-0.0127,+0.0053] 2525 matched seeds PRM-feature −- text-feature variant, gated −0.0066[−0.0252,+0.0110]-0.0066\ [-0.0252,+0.0110] 2525 matched seeds Log-probability −- text feature set −0.0030-0.0030 1010-seed matched coverage Hidden-state −- text feature set −0.0038-0.0038 1010-seed matched coverage The leave-domain-out control (Table 9) removes all target-domain benchmarks from fitting and validation, disables benchmark identity, and applies a shared late-only support. Its positive contrasts are against unconstrained uniform placement; paired differences from always-last remain unresolved. The control therefore establishes within-suite transfer of the late-support policy, not general content-based restart localization. Table 9: Within-suite domain-held transfer. The target domain is removed from training and validation, benchmark identity is disabled, and one shared late-only support is used. Estimates are router GR@10 over uniform with problem-cluster intervals. They do not establish improvement over always-last or transfer outside the benchmark suite. Held-out domain GR@10 over uniform [95%CI][95\%\ CI] Positive fits Code +0.216[+0.138,+0.292]+0.216\ [+0.138,+0.292] 15/1515/15 Mathematics +0.120[+0.094,+0.146]+0.120\ [+0.094,+0.146] 15/1515/15 Science +0.046[+0.008,+0.084]+0.046\ [+0.008,+0.084] 15/1515/15 Appendix D Exact Self-Consistency Frontier Protocol The panel is the first 100 MATH-500 problems with identifiers sorted as strings, run with three generation seeds and an 8192-token cap per request. SC returns the plurality parsed answer, with ties broken toward the earliest sample; unparsable samples do not vote but their tokens count, and a pool with no parsable answer is scored incorrect. Generated output counts every generated token, including the fresh base chain of always-last and the router. For each problem and seed, all subsets for SC(1) to SC(4) are enumerated. For a target aggregate output budget, a linear program chooses the globally randomized mixture over adjacent frontier points. Policy accuracy is paired with the same problem’s frontier construction, and the percentile bootstrap resamples problem clusters with their seeds intact. The four integer SC points are (0.250,0.771)(0.250,0.771), (0.500,0.766)(0.500,0.766), (0.750,0.778)(0.750,0.778), and (1.000,0.780)(1.000,0.780). These results are conditional on the frozen DeepSeek-R1-Distill-Qwen-14B/MATH-500 panel. Table 10: Exact aggregate generated-output SC frontier on DeepSeek-R1-Distill-Qwen-14B/MATH-500. The panel contains 100100 problems and 33 generation seeds. Ratios use aggregate generated output only; they do not measure total tokens or latency. CIs use paired problem bootstrap. Policy Acc. Output ratio Frontier Delta [95%CI][95\%\ CI] Always-last 0.8300.830 0.7740.774 0.7780.778 +0.052[+0.008,+0.098]+0.052\ [+0.008,+0.098] Router 0.8330.833 0.8850.885 0.7790.779 +0.054[+0.010,+0.103]+0.054\ [+0.010,+0.103] The two-model MATH result is a fixed-panel sensitivity using the two available cells from a planned four-cell model-by-benchmark panel. The Qwen3-32B cell alone is unresolved, and the missing GPQA cells prevent the planned panel from being identified. The PRM best-of-N, tree, and H200 rows of Table 11 are reported with their own cohorts and ledgers rather than pooled with the exact DeepSeek frontier result. Table 11: Scoped inference-scaling controls. The first two rows are a partial fixed two-model MATH sensitivity, not the incomplete planned 2-by-2 model-by-benchmark panel. PRM best-of-N, the shallow tree, and the H200 assay use their own stated cohorts and ledgers and are not pooled with the exact DeepSeek frontier result. Control Estimate [95%CI][95\%\ CI] Scope Always-last over exact SC frontier +0.0261[+0.0020,+0.0506]+0.0261\ [+0.0020,+0.0506] cell-equal, two available MATH cells Qwen3-32B/MATH-500 alone +0.0008[−0.0168,+0.0170]+0.0008\ [-0.0168,+0.0170] individually unresolved PRM best-of-N −- SC, DeepSeek −0.013[−0.037,+0.013]-0.013\ [-0.037,+0.013] 100100 problems, 33 seeds PRM best-of-N −- SC, Qwen3-32B −0.003[−0.037,+0.023]-0.003\ [-0.037,+0.023] 100100 problems, 33 seeds Shallow PRM tree −- SC −0.020[−0.060,+0.017]-0.020\ [-0.060,+0.017] 0.720×0.720× generated output H200 always-last −- SC accuracy +0.030[−0.005,+0.068]+0.030\ [-0.005,+0.068] 0.912×0.912× total tokens; 1.108×1.108× time Appendix E Loss and Feature Diagnostics The primary fair-label comparison freezes 24 split seeds and bootstraps problem clusters. The unmasked problem-population interval crosses zero even though a separate seed-stability interval is positive; the main text uses the problem-cluster result for inference. Inside the benchmark-conditional late support, both problem- and seed-level contrasts are unresolved. We do not use an equivalence claim for this primary comparison because its protocol did not freeze and power an equivalence margin. Table 12: Fair-label loss contrast and locked seed extension. The 24-seed comparison is primary; the 64-seed estimate is a post-lock robustness extension, not an independent confirmation. Intervals cluster by problem. Protocol Unmasked ListNet −- BCE In benchmark-conditional support Primary, 24 seeds +0.0128[−0.0033,+0.0293]+0.0128\ [-0.0033,+0.0293] −0.0016[−0.0161,+0.0133]-0.0016\ [-0.0161,+0.0133] Locked extension, 6464 seeds +0.0184[+0.0040,+0.0330]+0.0184\ [+0.0040,+0.0330] +0.0022[−0.0093,+0.0138]+0.0022\ [-0.0093,+0.0138] The extension froze the additional seed labels, code, data, and configuration before running seeds 24 to 63, but after the original 24-seed artifact was available. Under its frozen decision rule, the mask-free contrast is positive under both its problem-cluster interval (Table 12) and its paired-seed interval [+0.0105,+0.0263][+0.0105,+0.0263], and the contrast inside the benchmark-conditional late support is equivalent within the frozen ±0.02± 0.02 margin (paired-seed interval [−0.0038,+0.0081][-0.0038,+0.0081]). This is post-lock stability evidence rather than independent confirmation; the primary inference remains the 24-seed problem-cluster interval. Direct feature additions under the evaluated objective show no detected improvement: Math-Shepherd PRM summaries [Wang et al., 2024], token log-probabilities, and hidden-state representations do not improve the evaluated selector on matched coverage. PRM best-of-N and a shallow PRM tree also do not beat self-consistency in their scoped controls. These diagnostics identify the current bottleneck without excluding other representations, objectives, or search algorithms. Appendix F Additional Analyses and Scope A predeclared feasible-spend analysis did not pass its randomization gate, so it establishes no learned-spend effect. Provider total-token accounting and additional always-last analyses are reported as diagnostics. The domain-transfer, H200 timing, tree-search, and fixed-panel controls are reported separately because their cohorts, cost ledgers, or uncertainty units differ from the primary results. Appendix G Reproducibility and Release The per-cell measurement records, utility maps, 41-cell inclusion record, router split seeds, and the JSON certificates used by this manuscript are available from the authors on request, together with a hash registry and a script that verifies every registered input and regenerates the tables. Figures are compiled directly from TikZ/PGFPlots. For the 28 historical primary rows, settings are reconstructed from preserved commands and logs; the original runs did not embed per-cell manifests, so these settings are not cryptographically bound. They are temperature 0.70.7, top-p=0.95p=0.95, no explicit top-k restriction, and a 10241024-token generation cap (12801280 for Qwen3-32B/HumanEval and 40964096 for the legacy Mistral-Large GPQA and HumanEval rows). The 13 recollected rows use temperature 0.70.7, top-p=0.95p=0.95, no explicit top-k restriction, a 10241024-token cap, and four distinct deterministic request seeds. Fresh exact-prefix audits reuse the frozen original base chains and prefixes rather than regenerating them; only their new continuations use the 10241024-token audit protocol. All primary measurement statistics use four retained continuation outcomes per action, except Qwen3-32B/HumanEval (two) and the legacy Mistral-Large GPQA and HumanEval rows (eight). The separate exact-frontier assay uses a newly generated, 81928192-token-cap protocol. Repository checkpoint identifiers, benchmark versions, prompts, local graders, candidate segmentation, seeds, and retry rules are recorded for the recollected and later collections. MATH-500, GPQA Diamond, HumanEval, AIME 2025, HMMT, MBPP, and Putnam use their published problem sets or splits (per-cell problem counts in Table 2) and pinned local graders. Numeric tasks use normalized extracted answers, GPQA uses parsed multiple-choice indices, and code tasks execute the published tests against extracted Python code. Commands for the measurement aggregation, held-out scoring audits, policy decomposition, positional residuals, tie-resolution analysis, and exact SC frontier are also available on request. Checkpoints and execution The evaluated repository identifiers are deepseek-ai/DeepSeek-R1-Distill-Llama-8B [DeepSeek-AI, 2025], nvidia/Llama-3.1-Nemotron-Nano-8B-v1 [NVIDIA, 2025], deepseek-ai/DeepSeek-R1-Distill-Qwen-14B [DeepSeek-AI, 2025], openai/gpt-oss-20b [OpenAI, 2025], zai-org/GLM-4.5-Air [Zhipu AI / THUDM, 2025], google/gemma-3-27b-it [Gemma Team, Google DeepMind, 2025], Qwen/Qwen3-32B [Qwen Team, 2025], nvidia/Llama-3.1-Nemotron-70B-Instruct-HF [NVIDIA, 2024], and mistralai/Mistral-Large-Instruct-2411 [Mistral AI, 2024]. Repository revisions are reported where preserved; the historical collections do not record a revision for every checkpoint. The historical and recollected campaigns use model-compatible multi-GPU inference in bfloat16 except for the native quantized GPT-OSS checkpoint. Exact per-model tensor-parallel assignments are not recorded for every run. The recollected and later campaigns retain campaign-specific seeds, generation settings, token counts, and available job metadata. Older rows lack uniform per-action manifests, so row-level provenance for the historical sweep is incomplete. Router fitting is CPU-scale; continuation collection is the GPU-dominant stage. The exact-frontier experiment uses three independently generated seeds, and the order-balanced timing assay uses the same H200 hardware for both arms. Asset and assistance disclosure The paper identifies the evaluated model repositories and benchmark datasets. We do not provide a complete per-asset license inventory, as reflected in the checklist. No human-subject data are collected. Language models were used for prose editing and code review; the authors verified every generated table, figure, and numerical statement against the frozen sources and hash manifest.