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Input-Conditioned Search for Efficient Test-Time Scaling of Recursive Reasoners

Haochen Shi, Tianwen Fu, Qianggang Ding, Yinhan Lu, Yizhan Li, Huan Zhang, Yajie Zhao, Xingdi Yuan, Hengyang Lu, Bang Liu

arXiv:2610.18547v1cs.AI

TL;DR

Recursive reasoners can fail to find correct solutions even when inference is scaled through more updates or perturbed trajectories. ICS searches input conditions to induce alternative update maps without retraining, and achieves the highest mean accuracy across six tasks while using substantially fewer model FLOPs than evaluated baselines.

  • Problem

    Depth and perturbation-based width scaling can both converge to incorrect attractors, raising the question of how a frozen recursive reasoner can overcome such failures efficiently at test time.

    Source: §1 · pp. 1–2

  • Method

    ICS searches input conditions to induce alternative update maps from a frozen reasoner and restarts reasoning from the same initial latent state.

    Source: §1 · p. 2, §3 · p. 4

  • Results

    Across six tasks, ICS achieves the highest mean accuracy under both answer-selection regimes, tying on Light-Up; on Sudoku-Extreme it reaches 99.7%, versus EqR’s 98.8%.

    Source: §1 · p. 2, §4 · p. 8

  • Takeaways & Limitations

    ICS provides an additional test-time scaling axis without retraining, and the findings motivate further research on shaping recursive reasoning dynamics.

    Source: §7 · p. 12

Abstract

from arXiv · show

People often tackle difficult problems by thinking more deeply or exploring more possibilities. Similarly, recursive reasoners, which reason by repeatedly applying a learned update rule to latent states, can also scale inference through more updates (depth) or additional trajectories generated by perturbing latent states (width). When people hit a dead end in their thinking, they may step back and approach the problem from a fresh angle. However, neither depth scaling nor perturbation-based width scaling of recursive reasoners can reliably achieve this: both may converge to incorrect attractors. To address this, we propose to explore diverse reasoning traces by applying different update maps to the same latent state, without any additional training. We realize this idea through \emph{Input-Conditioned Search (ICS)}, which searches over input conditions to induce alternative update maps from the pretrained reasoner. Across six reasoning tasks, ICS achieves the highest mean accuracy, while requiring up to two orders of magnitude fewer model FLOPs than the evaluated scaling baselines. Controlled experiments confirm that ICS can redirect trajectories from the same state. Our theoretical analysis and rigorous numerical verification also confirm that ICS can locally eliminate incorrect attracting fixed points. Together, these findings establish ICS as an effective axis for efficient test-time scaling and motivate further research on shaping recursive reasoning during training and at test time.

1 Introduction

Recursive reasoners can spend more computation on longer trajectories or perturbed trajectories, yet both approaches may converge to incorrect attractors. Input-Conditioned Search (ICS) changes the update map by varying input conditions, and the paper reports accuracy and efficiency gains alongside theoretical and experimental evidence about trajectory redirection.

1 Introduction

  • Depth and perturbation-based width scaling retain the original input-conditioned update map, and trajectories can converge to incorrect attractors despite more updates or perturbations.This motivates asking how a frozen recursive reasoner can overcome such failures efficiently at test time.Source: §1 · pp. 1–2
  • ICS searches input conditions to induce alternative update maps from the frozen reasoner, restarting from the same initial latent state without retraining.Its operations include hypothesis injection and validity-preserving restatements; model predictions guide hypothesis expansion and answer selection, while correctness remains defined by the original task.Source: §1 · p. 2
  • Analysis and computer-assisted proofs in separately trained 4 × 4 Sudoku reasoners show that input conditioning can locally eliminate incorrect attracting fixed points.Controlled interventions also show that conditions can redirect trajectories from the same initialization; recovery depends on starting state and conditioning duration.Source: §1 · p. 2
  • Across six tasks, ICS achieves the highest mean accuracy with and without a task correctness checker, tying at 100% on Light-Up, while using 10–195× fewer model FLOPs.Without a checker, it achieves higher mean accuracy than the evaluated scaling baselines on the other five tasks.Source: §1 · pp. 2–3

2 Preliminaries

A recursive reasoner repeatedly updates a latent state under an input-conditioned map, decodes candidate answers, and produces a halting score. Test-time scaling varies update depth or trajectory count, while evaluation distinguishes discovering a valid candidate from selecting the final answer.

2 Preliminaries

  • A frozen recursive reasoner iterates a learned update map on a latent state, then decodes an answer and produces a halting score.For a given input, frozen parameters induce a fixed update map; the Tiny Recursive Model uses a shared network to update reasoning state and answer representation.Source: §2 · p. 3
  • Depth scaling increases updates per trajectory, whereas width scaling increases trajectory count, often by perturbing a latent state and continuing recursion.Additional steps provide further opportunities to discover a correct answer but need not improve coverage.Source: §2 · p. 3
  • Solution coverage measures whether the candidate pool contains any valid answer, separating discovery from the requirement to select an answer to return.A task certificate can identify a valid candidate in a fixed pool; model-based selection instead relies on model-derived information without access to the certificate.Source: §2 · p. 4
  • With a task certificate, evaluation measures verified-solution discovery against inference cost; without one at selection, it measures final-answer accuracy and cost.Each answer-selection regime uses its corresponding search and stopping rules.Source: §2 · p. 4

3 Input-Conditioned Search

Input-Conditioned Search (ICS) uses alternative task inputs to induce update maps in the same frozen reasoner, restarting reasoning to generate candidate answers. Its search combines tentative hypotheses and equivalent task representations, then applies staged expansion and answer-selection rules.

3 Input-Conditioned Search

  • ICS explores alternative input conditions, each inducing an update map under the same frozen model, and restarts reasoning from z0 to generate candidate answers.It uses hypothesis injection and validity-preserving restatements; predictions are mapped back to the original representation when needed, and correctness is judged against the original input.Source: §3 · p. 4

3.1 Input-conditioned Reasoning

  • Hypothesis injection adds tentative assignments while preserving original givens; validity-preserving restatements express the same task in equivalent representations.Predictions from restatements are mapped back to the original representation; the frozen reasoner need not be equivariant to the transformation, so equivalent inputs can yield different candidates.Source: §3.1 · pp. 4–5

Search and answer selection

  • ICS tries the original input first, then hypothesis-conditioned search, and explores restatements if no answer is accepted earlier.Search proposes alternatives at low-margin positions, retains children consistent with their parent hypotheses and givens, and prefers higher halting scores among ties. Acceptance uses a task certificate when available, otherwise model-derived scores or prediction consistency.Source: Search and answer selection · p. 5

4 Performance and Inference Efficiency

Across six tasks, ICS achieves the highest mean accuracy under both answer-selection regimes and uses less inference computation than the evaluated scaling baselines. Ablations show that hypothesis search contributes more than validity-preserving restatements at the evaluated settings.

Test-time scaling

  • Under both certificate-based and model-based answer selection, ICS achieves the highest mean accuracy across all six tasks and ties on Light-Up.With a task certificate, ICS reaches 95.9% on Tapa, exceeding the second-best method by 13.7 percentage points. Under model-based selection, it reaches 99.1% on Maze-Hard, compared with 91.8% for the second-best method.Source: §4 · p. 8
  • Across six tasks, ICS uses approximately 10–525× less test-time computation than evaluated scaling baselines and has the lowest measured execution time per puzzle.Figure 3 compares model computation and execution time across tasks under model-based answer selection.Source: Figure 3 · p. 6, §4 · p. 8
  • On the PPBench golden set without a task certificate at selection, ICS scores 91.7% mean accuracy versus 78.3% for the strongest evaluated LLM, a reported 13.3-point margin.The mean is across three training seeds.Source: Figure 3 · p. 6, §4 · p. 8
  • At 1.14 TFLOPs per puzzle, ICS reaches 99.70% Sudoku-Extreme accuracy and reduces errors by approximately 92% versus TRM + greedy depth at 1.20 TFLOPs.At higher budgets, ICS reaches 99.96% with width scaling at 1.94 TFLOPs and 99.91% with depth scaling at 3.10 TFLOPs.Source: §4 · p. 8

Contributions of the input-conditioning operations

  • Removing hypothesis search lowers model-selected Sudoku accuracy from 99.81% to 98.82%, while removing restatements yields 99.75%.With certificate-based selection, the corresponding accuracies are 99.94%, 98.83%, and 99.92%, respectively; hypothesis search contributes more at the tested settings.Source: §4 · pp. 8–9

5 Discussions

The discussion compares interventions on latent states with changing the input-conditioned update map. It finds that both where exploration starts and which map governs recursion affect recovery from failed trajectories.

5.1 Recovery depends on where exploration begins.

  • Cold-start random perturbations recover substantially more failures than warm-start perturbations, which recover none across the evaluated scales; oracle recovery also depends on direction.Oracle directions and GRAM posterior perturbations use ground-truth labels and are diagnostic controls.Source: §5.1 · p. 9
  • ICS achieves significantly higher candidate coverage than tested label-free perturbations on both TRM and GRAM, while perturbation recovery drops sharply after two recursion steps.The recovery decline holds across three perturbation scales for both incorrect fixed-point and moving-failure cohorts.Source: §5.1 · p. 10

5.2 Input conditioning changes recursive dynamics.

  • At a wrong near-fixed point, hypothesis injection raises the relative one-step residual from approximately 2.4 × 10^-7 to 0.225, and warm- and cold-start paths reach the same correct near-fixed point.Under the original map, the illustrated trajectories approach the same incorrect near-fixed point.Source: §5.2 · p. 10
  • Input conditioning can locally eliminate an incorrect attracting fixed point through a fold bifurcation, supported by numerical continuation and computer-assisted certificates.The certificates use separately trained 4×4 Sudoku reasoners; continuation provides complementary evidence in Maze-Hard and Sudoku-Extreme models.Source: §5.2 · pp. 10–11
  • Restarting from the fresh initialization recovers solutions under input conditions where continuation from a saved failed state remains incorrect.Some retained incorrect near-fixed states have estimated local spectral radii below one, and recovery depends on whether a hypothesis corrects an erroneous prediction.Source: §5.2 · p. 11
  • Validity-preserving restatements alter recursive updates and can redirect trajectories from the same initialization.Together with hypothesis injection, they are used by ICS to search alternative update maps alongside fresh initialization.Source: §5.2 · p. 11

6 Related Work

ICS is situated among recursive reasoning, test-time candidate diversification, equilibrium-based approaches, and search over assignments or equivalent task views. Its distinguishing move is to search input-conditioned update maps while keeping model parameters frozen.

Recursive and looped reasoning

  • Recursive reasoning extends from Universal Transformers and adaptive computation time (Graves, 2017) to recursive reasoners HRM and TRM (Wang et al., 2025).ICS differs by changing the input so each condition induces a different update map under the same frozen model.Source: §6 · p. 11

Test-time scaling and answer selection

  • PTRM (Sghaier et al., 2026) and GRAM (Baek et al., 2026) diversify recursive candidates through latent perturbations or stochastic transitions, whereas ICS changes the update map.On the same failed instances, ICS recovers more solutions than label-free latent perturbations and uses less compute overall.Source: §6 · p. 11

Equilibrium and attractor views of reasoning

Search over partial assignments and equivalent views

  • ICS combines model-guided branching on tentative assignments with equivalent task views, keeping parameters frozen and using the reasoner itself to complete hypotheses.The ablation identifies hypotheses as the larger source of improvement.Source: §6 · p. 12

7 Conclusion

The paper presents Input-Conditioned Search as a test-time scaling axis that explores alternative update maps without retraining. Across six reasoning tasks, it reports strong accuracy and efficiency, alongside evidence that input conditioning redirects trajectories and can eliminate incorrect attractors locally.

7 Conclusion

  • ICS allocates test-time computation to exploring alternative update maps as an additional scaling axis without retraining the model.It searches over input conditions and combines this with fresh initialization.Source: §7 · p. 12
  • Across six reasoning tasks, ICS achieves the highest mean accuracy while using substantially less model computation and execution time than baselines.The conclusion reports this as efficient test-time scaling without additional training.Source: §7 · p. 12
  • Controlled experiments show that ICS redirects recursive updates and trajectories, while theory and rigorous numerical verification establish local elimination of incorrect attracting fixed points.The findings motivate further research on shaping recursive reasoning dynamics during training and at test time.Source: §7 · p. 12

A.1 Datasets and Splits

The appendix documents datasets and splits, shared backbones, inference protocols, ICS configurations, and cost measurement. It also explains the Maze-Hard benchmark choice and reports a matched-subset Sudoku-Extreme comparison.

A.1 Datasets and Splits

  • The six tasks are Sudoku-Extreme, Maze-Hard, Light-Up, Nurikabe, Tapa, and Heyawake; PPBench tests combine golden instances with fixed-seed samples from the non-golden pool.Tapa uses 50 additional sampled instances because fewer are available; other PPBench tasks use 100.Source: §A.1 · p. 16

A.2 Models and Training

  • TRM-based methods share the same three trained checkpoints per task, while GRAM, EqR, and Attractor Models are trained separately; model parameters remain frozen during inference.Accuracy is reported as the mean and standard deviation across three training seeds.Source: §A.2 · p. 16

A.3 Maze-Unique and the choice of maze benchmark

  • Maze-Unique leaves little accuracy headroom: standard TRM reaches 100.0%, 100.0%, and 99.8%, while ICS and PTRM each reach 100.0% on all three backbones.The main maze experiments therefore use Maze-Hard, where remaining errors permit a more informative comparison.Source: §A.3 · p. 17

A.4 Inference Protocols

  • Depth scaling runs a 6,400-step budget, greedy depth reallocates a shared 6,400N budget, and PTRM and GRAM use trajectory pools with their configured selection rules.PTRM uses K D = 6,400 steps per instance; GRAM uses 100 trajectories of 64 steps.Source: §A.4 · pp. 17–18

A.5 Matched-subset comparison on Sudoku-Extreme

  • 99.95% with certificate-based selection and 99.65% with model-based selection, versus strongest baseline means of 97.48% and 97.49%, respectively, on the shared Sudoku-Extreme subset.The comparison uses the same first 3,072 test instances, but each method retains its corresponding inference budget.Source: Table 3 · p. 18, §A.5 · p. 18

A.6 ICS Configuration

  • ICS conditions rollouts on task-specific hypotheses and validity-preserving restatements, with fresh initializations and 16 recursion steps per rollout.The proposal limit caps assignments per parent, and the node budget caps hypothesis children across search levels.Source: §A.6 · pp. 18–19

A.7 Inference Cost Measurement

  • Inference cost is averaged per evaluated instance and includes unsuccessful hypotheses; model FLOPs are profiled from executed calls, while elapsed time is measured end-to-end on NVIDIA L40S GPUs.The profiler counts supported model operations, primarily matrix multiplications.Source: §A.7 · p. 20

A.8 LLM Evaluation Protocol

The LLM evaluation uses the PPBench golden set and a single-attempt protocol. The trajectory diagnostics show that latent stationarity and answer correctness can vary independently, including correct answers during continued motion.

A.8 LLM Evaluation Protocol

  • LLMs are evaluated on 15 golden instances from each of four PPBench tasks, with one attempt per configuration and no task certificate available to ICS during answer selection.The certificate is used to score returned answers; malformed, missing, and incorrect answers count as incorrect.Source: §A.8 · pp. 20–21
  • ICS solves 57, 54, and 54 of the 60 golden-set instances across its three seeds, for 91.7 ± 2.9% overall accuracy with model-based selection.The comparison reports each LLM's strongest evaluated effort setting and does not pool answers across settings.Source: §A.8 · p. 21, Table 6 · p. 21

B Latent Trajectory Diagnostics

  • The diagnostics classify trajectories using return tests for near-fixed or periodic behavior and motion tests for regular or irregular behavior, based on finite observations.The operational motion classes use the 0–1 statistic and, when available, a finite-time perturbation-growth estimate.Source: B Latent Trajectory Diagnostics · pp. 22–24
  • Correct and incorrect endpoints occur across near-stationary and periodic trajectories, and correct answers also occur during continued motion, so latent stationarity is distinct from answer correctness.The examples illustrate possible combinations rather than their prevalence across tasks.Source: B Latent Trajectory Diagnostics · pp. 22, 24, Figure 7 · p. 23

B.1 Stability and robustness of incorrect states

Incorrect recursive predictions can persist under continued iteration and substantial perturbations, including near stationary states and moving trajectories. The extended diagnostics separate answer correctness, decode preservation, and return to a recorded latent neighborhood across sampled failure and successful-reference cohorts.

B.1 Stability and robustness of incorrect states

  • Among 239 incorrect near-fixed points, 128 passed the spectral-radius convergence check, with estimates from 0.7384 to 0.9877; the other 111 remained unconverged.The estimates provide numerical evidence of local attraction for a subset, not certified stability at exact fixed points.Source: §B.1 · p. 25
  • Across 4,073 irregular moving failures, approximately 99.8–100% of endpoints met the reference-set return criterion, but only 0.13% yielded valid answers at σ=1.These finite-horizon measurements do not identify a unique invariant attracting set.Source: §B.1 · p. 25
  • 100% of perturbed endpoints retained the incorrect answer, while latent return within 64 updates decreased as perturbation scale grew.Failure to meet the finite-horizon return test alone does not establish escape.Source: §B.1 · p. 25, Table 7 · p. 26

C.2 Comparisons Across Failure and Successful States

  • At σ=1, full-state perturbations retained both strict correctness and the original decode in all successful near-fixed-point trials, but only 3.2% returned to the saved latent neighborhood.For a regular-moving Nurikabe reference, all full-state trials remained strictly correct, while 3.1% reproduced the complete raw reference decode.Source: §C.2 · p. 47
  • Across failed references, recovery varied with numerical class and intervention direction; smaller cohorts support descriptive examples rather than task-wide estimates.The extended figures retain periodic, regular-moving, irregular-moving, and unresolved cohorts alongside larger fixed-point groups.Source: §C.2 · p. 47

C.3 Hypothesis Content, Initialization, and Persistence of Near-Fixed Points

Paired interventions test how hypothesis content and initialization relate to recovery under conditioned maps. Cold starts can recover solutions while warm starts preserve incorrect near-fixed states, and conditioned maps can also retain existing solutions.

C.3 Hypothesis Content, Initialization, and Persistence of Near-Fixed Points

  • Cold starts recovered 445 of 602 compatible near-fixed-point pairs (73.9%), whereas warm starts recovered no solutions.Across all 3,824 hypotheses in this cohort, 98.3% of warm endpoints remained near-fixed and decoded the same incorrect answer.Source: Table 9 · p. 48, §C.3 · pp. 48–49
  • Correcting an erroneous cell outperformed repeating an already-correct value by 25.8 percentage points for near-fixed failures and 47.2 points for moving failures.These paired recovery gains use 1,000 bootstrap resamples of instances.Source: Table 10 · p. 49, §C.3 · p. 49
  • For moving failures, compatible hypotheses recovered 50.4% of warm pairs and 61.3% of cold pairs.The benefit of restarting depends on the starting state and hypothesis, rather than applying uniformly to every intervention.Source: §C.3 · p. 49
  • All 585 retained incorrect endpoints under compatible hypotheses had estimated spectral radius below one, with median 0.9085, but these finite-iteration estimates are not certified stability bounds.For 422 compatible pairs, warm and cold starts yielded retained incorrect and correct near-fixed endpoints, respectively, under the same conditioned map.Source: Table 11 · p. 50, §C.3 · p. 50
  • Among 6,896 incompatible hypotheses on solved references, warm continuation kept all answers valid and 99.1% near-fixed with the same decoded solution, while cold starts recovered 15.1% of pairs.These tests establish endpoint correspondence at the measured tolerance, not continuous tracking between maps.Source: §C.3 · p. 50

C.4 Sudoku component and proposal ablations

The experiments examine how input conditioning changes recursive dynamics, from Sudoku search components to fixed-point branches in trained reasoners. They report useful proposal information, state-dependent trajectory redirection, and local folds that can remove incorrect attracting branches.

C.4 Sudoku component and proposal ablations

  • The Sudoku reference uses depth-20 TRM; hypothesis search and symmetry restatements add successive search stages, with candidate selection using either certificates or model scores.Search outputs are evaluated for validity against the original Sudoku rules.Source: §C.4 · p. 51
  • Omitting hypotheses lowers Sudoku accuracy by 1.11 points with certificate selection and 0.99 points under model selection; omitting restatements lowers it by 0.02 and 0.06 points, respectively.Randomizing proposal values lowers accuracy by 1.00 point, while certificate selection remains above 99.85%.Source: §C.4 · p. 51

D.1 Construction of the update-residual visualization

  • The visualization overlays recorded trajectories on interpolated update-residual surfaces, but displayed widths, distances, and slopes are not latent-space metric measurements.The surfaces represent update residuals, not an energy function whose gradient generates the dynamics.Source: §D.1 · pp. 52, 54
  • At the same saved failed state, the original map has relative residual about 2.41 × 10^-7, while the hypothesis-conditioned map has residual about 0.22465.Both conditioned trajectories return valid answers and end at states differing by only 2.40 × 10^-7 in relative norm.Source: §D.1 · p. 54

D.2 Input conditioning induces folds in incorrect fixed-point branches

  • Input conditioning produces local folds in incorrect fixed-point branches in trained Maze-Hard and Sudoku-Extreme models.At a fold, two branches meet and terminate locally on one side of a critical input condition.Source: §D.2 · p. 54
  • Maze-Hard instance 461 has a refined fold at λc ≈ 0.796498891, with numerical measurements supporting a local saddle-node involving the incorrect attracting branch.The relative fixed-point residual is 2.03 × 10^-15.Source: §D.2 · p. 54
  • Sudoku instance 60398 has two folds and an intervening unstable branch, while other tested controls reach full conditioning without detected folds.The tracked incorrect branch can therefore terminate locally or persist under the final condition, depending on the input condition.Source: §D.2 · p. 54, Figure 27 · p. 55

D.3 Certified elimination of incorrect attracting fixed points

Complete certificates in separately trained small Sudoku models establish that input conditioning can locally eliminate incorrect attracting fixed points. Post-fold warm-start outcomes vary by hypothesis consistency, and the certificates do not establish global elimination or recovery.

Certified local elimination

  • 51 complete certificates show that input conditioning can locally eliminate incorrect attracting fixed points across selected 4 × 4 Sudoku model–instance–hypothesis cases.The certificates cover 42 solution-consistent and nine inconsistent hypotheses; all nine detected folds under inconsistent hypotheses have complete certificates.Source: §D.3 · p. 55, Table 13 · p. 56

Recovery after a certified fold

  • 31 of 42 solution-consistent cases reach the correct solution in the 512-update post-fold warm-start diagnostic; the other 11 end at incorrect decodes.Ten end at another incorrect decode and one at the same incorrect decode, although that latent state has moved and remains nonstationary.Source: §D.3 · p. 56, Figure 28 · p. 56
  • Among nine inconsistent cases, one reaches the solution, seven another incorrect decode, and one the same incorrect decode; at λ = 1, counts are one, six, and two.These are endpoint outcomes, not counts of cases that ever visit a correct prediction.Source: Figure 28 · p. 56, §D.3 · p. 57

Certificate construction and replay

  • Complete certification verifies fold conditions, transverse stability, and a strictly wrong prediction throughout a neighborhood; the real-valued map is evaluated with numerical enclosures.All 54 saved candidate certificates pass augmented-root, nondegeneracy, and wrong-decoding checks; 51 also pass transverse-stability verification, while three remain unverified at that stage rather than being classified unstable.Source: §D.3 · p. 57

Scope and interpretation

  • The result is local: certificates cover neighborhoods of critical points along specified interpolations, not every incorrect attractor or the final input condition.The large-model continuation results are numerical evidence; the rigorous certificates concern separately trained small models, and the connection from the original failed state to the critical branch is tracked numerically rather than certified throughout.Source: §D.3 · p. 58

D.4 Restatement-induced dynamics in aligned coordinates

This section examines how restatements, hypotheses, solution multiplicity, and supervision choices relate to recursive inference dynamics and behavior. It uses training interventions and diagnostics of latent movement, recovery, solution preservation, decoded behavior, and task accuracy to characterize these effects.

D.4 Restatement-induced dynamics in aligned coordinates

  • Median aligned-update distances were 0.259 for Sudoku and 0.482 for Maze, with corresponding update cosine similarities of 0.966 and 0.884.Thus, restated-input updates differed even after undoing the spatial coordinate transformation.Source: §D.4 · p. 59
  • Median relative residual rose from 6.02 × 10^-7 to 0.0494 for 239 Sudoku and 9.22 × 10^-7 to 0.672 for 42 Maze near-fixed states, not establishing recovery.The recorded states ceased to be nearly stationary under restated maps, but this alone does not show removal of an entire incorrect attractor.Source: §D.4 · p. 59
  • Cold restatements recovered 35.41% of Sudoku near-fixed-point pairs versus none warm, and 72.45% of Maze pairs versus 43.88% warm.Cold-start coverage across any tested restatement was 53.97% for Sudoku and 92.86% for Maze; these were diagnostic cohorts from one checkpoint per task, not full-test evaluations.Source: §D.4 · pp. 59–60, Figure 29 · p. 60

D.5 Initialization and hypothesis retention

  • Retention effects were model-dependent: Sudoku MLP had zero warm-start recoveries, while the full Maze pool yielded 27 correct endpoints at m=16 or 64 versus 20 at m=512.For Sudoku MLP, all 37 cold-successful hypotheses remained correct at the cold-start 512-step endpoint, yet none recovered from saved incorrect states; Sudoku attention benefited from longer tested retention.Source: §D.5 · p. 63

D.6 Matched-norm perturbations and solution round trips

  • Across three models, none of 128 random-direction continuations succeeded, while the one-step input intervention succeeded on 6 of 16 Maze instances and none of the Sudoku instances.Matching displacement magnitude alone therefore did not reproduce conditioned-update recovery on the sampled Maze instances.Source: §D.6 · p. 64
  • All selected round trips recovered a strictly correct endpoint with the same decoded answer; latent return held for all selected Sudoku MLP and Maze cases, but only five of seven Sudoku attention cases.Correct-answer preservation and return to a recorded latent neighborhood were distinct outcomes.Source: §D.6 · p. 64

D.7 One-step response to hypothesis injection

  • Median relative updates rose from order 10^-7 under original inputs to 2.80% for Sudoku MLP, 6.67% for Maze attention, and 16.50% for Sudoku attention under hypothesis conditioning.This displacement need not produce a correct answer: Sudoku MLP retained its original incorrect endpoint decode throughout tested retention schedules.Source: §D.7 · p. 65

E Training Supervision and Inference Behavior

  • Incorrect Sudoku predictions frequently continued changing under recursion, Maze predictions were predominantly persistent, and Nurikabe, Tapa, and Heyawake showed mixtures of these behaviors.The original Sudoku-Extreme training and test puzzles have unique solutions.Source: E Training Supervision and Inference… · p. 65
  • Figure 31 reports that training supervision changes decoded behavior in reference-mismatch cohorts after 16 recursion steps.It compares subsequent decoded changes and return to the original decoded prediction after perturbation; Maze mismatches may include valid alternatives.Source: Figure 31 · p. 66
  • Maze instances can have multiple shortest paths, although each training instance uses one reference path selected by the dataset’s path-generation procedure.Source: E Training Supervision and Inference… · p. 66
  • Training interventions test how selecting or combining valid-solution targets affects decoded behavior and task accuracy on Maze-Hard and Sudoku-Extreme.Maze-Hard keeps training inputs fixed while changing shortest-path targets; Sudoku-Extreme introduces multiple solutions by removing givens and varies target selection.Source: E Training Supervision and Inference… · p. 66

E.1 Maze-Hard: changing supervision on fixed inputs

Maze supervision changes decoded behavior even when puzzles and their solution sets are fixed, while Sudoku ambiguity or deterministic targets alone do not reproduce Maze-like persistent failures. Conditional analysis establishes conditions for locally eliminating incorrect attracting fixed points, but departure from a near-fixed point does not guarantee reaching a correct answer.

E.1 Maze-Hard: changing supervision on fixed inputs

  • Sample K=1 and K=8 yield decode-change fractions of 53.3% and 41.4%, and decoded-output returns of 51.6% and 71.6%, respectively; their rule-valid accuracies are 53.1% and 23.0%.The puzzles and valid solution sets remain unchanged; only the training targets change.Source: §E.1 · pp. 66–67
  • Min-over-8 reaches 64.2% rule-valid accuracy, versus 57.5% for min-over-64 and 7.8% for marginal targets, so broader valid-path supervision does not improve accuracy monotonically.After three rounds of self-imitation, accuracy changes from 64.2% to 62.8%, despite decode-change decreasing from 45.1% to 23.6%.Source: §E.1 · pp. 67–68
  • For Sample K=1’s 469 initial failures, continuation recovers 11.3%, while noisy restart yields valid candidates for 24.5% at s_raw=10^-3 and 30.9% at s_raw=0.3.Restart measures whether any trial endpoint is valid; it does not require a model-based selector to choose that candidate.Source: §E.1 · p. 68

E.2 Sudoku-Extreme: introducing solution ambiguity

  • Ambiguous Sudoku training lowers original-test accuracy to 60.38 ± 1.72%, versus 86.0 ± 0.6% for standard models; on clue-removed inputs, accuracy is 49.80% versus 99.25%.The accuracy reduction also appears on evaluation inputs that admit multiple solutions.Source: §E.2 · p. 68
  • Sudoku ambiguity and deterministic target conventions do not reproduce Maze-like persistent failures; the results identify supervision as a contributing factor, not a simple link between solution multiplicity and persistence.Incorrect predictions remain highly changeable under ambiguity-trained and raw min-lex models.Source: §E.2 · pp. 68–69

F Conditional Analysis of Input-Conditioned Reasoning

  • The analysis establishes sufficient conditions for input conditioning to locally eliminate an incorrect attracting fixed point, verified by computer-assisted certificates in trained small reasoners.The section also characterizes responses to input changes and conditions for reaching a correct solution from a chosen initialization.Source: F Conditional Analysis of… · pp. 69–70

F.1 One-step response at an incorrect near-fixed point

  • At an exact fixed point, any nonzero input-induced update change removes fixed-point status; for an approximate fixed point, the conditioned residual must exceed the tolerance.When the original residual is small relative to the update change, the conditioned residual tracks that change and can initiate motion without directly perturbing the state.Source: §F.1 · p. 70
  • Departure from stationarity at one state does not establish escape from a region of incorrect predictions or guarantee reaching a correct solution; those require additional conditions.The conditioned trajectory may converge to another state that still decodes incorrectly.Source: §F.1 · p. 70

F.2 Local annihilation of an incorrect attracting fixed point

The section analyzes how varying an input embedding can alter fixed-point structure and distinguishes persistence of a correct solution from access to it. Its local guarantees are conditional: verified folds remove particular incorrect attractors, while correct-state convergence applies only within a specified neighborhood.

F.2 Local annihilation of an incorrect attracting fixed point

  • 51 computer-assisted certificates in separately trained 4 × 4 Sudoku reasoners verify sufficient conditions for local elimination of incorrect attracting fixed points.The certificates concern particular fixed points, not all incorrect attractors or the entire input-interpolation path.Source: §F.2 · pp. 71, 73
  • When ab>0, increasing λ through λc locally annihilates an incorrect attracting fixed point and its unstable companion.This is the stated saddle-node outcome near the critical point.Source: §F.2 · p. 71
  • The fold certificates establish disappearance near a critical point, not that discrete ICS eliminates the original failure branch at its endpoint.They also do not establish the absence of other incorrect attractors or a validated connection along the full path.Source: §F.2 · p. 73

F.3 Local persistence of a correct solution

  • With valid decoding, contraction on a ball, and sufficiently small center displacement, the conditioned map has a unique correct fixed point attracting every trajectory initialized in that ball.The fixed point may move from the original solution; the guarantee is local to the specified ball.Source: §F.3 · p. 73

F.4 Persistence does not imply accessibility

  • Correct-solution persistence does not guarantee access: neither a fresh initialization nor a previously incorrect state is assured to enter the required neighborhood.Local contraction controls convergence after entry, not the entry time.Source: §F.3 · p. 74, §F.4 · p. 74
  • The same conditioned map can yield correct cold-start reasoning while retaining an incorrect outcome reachable from a previously failed state.ICS evaluates hypotheses from the fresh initialization, so warm-start recovery tests a different accessibility question.Source: §F.4 · p. 75
  • Which hypotheses ICS discovers and the computation needed to find and select a correct answer remain empirical outcomes.The analysis identifies mechanisms but does not settle those search and cost questions.Source: §F.4 · p. 75
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