PaperlayerView exact revision on arXiv

Source-linked AI summary

Evolving in Thought Space: Training a Small Model at Test Time Unlocks Better Discoveries

Chonghe Jiang, Ao Qu, Siyuan Liu, Ruoyun Ma, Zijian Zhou, Dingyi Zhuang, Bo Liu, Han Zheng, Hanfei Yu, Baichuan Mo, Jinhua Zhao, Paul Pu Liang

arXiv:2610.06269v1cs.AIcs.LG

TL;DR

Discovery-oriented test-time training adapts solution-generating models, which can be expensive for large models and makes verifier feedback span strategy and implementation. Guidance-TTT adapts compact strategic guidance while a stronger executor remains frozen, and across four domains it outperforms strongest prior-work references while remaining competitive with public leaderboard results.

  • Problem

    Adapting a solution-generating LLM can be costly when reliable generation requires a large model, while outcome-level verifier feedback combines strategy and implementation quality.

    Source: §1 · p. 2

  • Method

    Guidance-TTT trains a compact model to propose high-level, parent-conditioned changes and uses a frozen, stronger model to implement them as complete solutions.

    Source: §1 · p. 2

  • Results

    Across four distinct discovery domains, Guidance-TTT solutions outperform the strongest prior-work references while remaining competitive with public leaderboard results.

    Source: §5 · p. 10

  • Takeaways & Limitations

    Controlled comparisons show that learning strategic guidance can be substantially more effective than freezing guidance or directly adapting a solution-generating model, while avoiding the cost of training the large executor.

    Source: §5 · p. 10

    Future work includes richer guidance representations, improved credit assignment between guidance and execution, and more diversity-aware search.

    Source: §5 · p. 10

Abstract

from arXiv · show

Open-ended scientific discovery often requires repeatedly proposing and evaluating candidate solutions. LLM-based systems can support this process by generating and refining executable solutions from verifier feedback. Methods such as TTT-Discover use test-time training (TTT) to update the solution-generating LLM from verifier feedback, adapting its generation policy to improve subsequent proposals on the target problem. However, this becomes expensive when reliable execution requires a large model, since training must maintain gradients, optimizer states, and policy statistics while repeatedly generating long, structured outputs. It also complicates credit assignment: outcome-level verifier feedback must jointly evaluate the high-level strategy and its low-level implementation. In this work, we introduce Guidance-TTT, which separates these roles. A compact guidance model is trained at test time to propose high-level strategic changes, while a frozen execution model implements them as complete executable solutions. At each step, the system selects a promising previously discovered solution, proposes a change, executes and verifies it, and updates only the guidance model using an adaptive group-relative RL objective. This concentrates test-time learning on short strategic decisions while retaining the implementation capability of a substantially stronger model without adapting it. Without web access, Guidance-TTT produces strong solutions across four distinct domains: combinatorial optimization (Polyomino Packing), heuristic programming (AHC058), machine learning (Lasso), and GPU kernel optimization (TriMul). Across these tasks, it outperforms the best solutions reported in prior work while remaining competitive with state-of-the-art results on public online leaderboards. Code is available at https://github.com/Human-Agent-Society/reef/tree/guidance-ttt-support.

1 INTRODUCTION

Existing discovery-oriented test-time training adapts the model that generates complete solutions, making large-model adaptation costly and leaving verifier feedback to assess strategy and implementation together. Guidance-TTT instead adapts compact strategic guidance while a stronger executor stays frozen, and the paper reports strong results across four domains.

1 INTRODUCTION

  • Adapting a solution-generating LLM is costly when reliable generation requires a large model, and outcome-level feedback must assign credit across strategy and implementation.A promising strategy may receive poor feedback simply because its implementation fails.Source: §1 · p. 2
  • Guidance-TTT trains only a compact guidance model to propose parent-conditioned strategic changes, while a stronger execution model remains frozen and implements complete candidate solutions.The paper calls this learning in thought space, where learned actions are explicit natural-language proposals rather than complete programs or latent representations.Source: §1 · pp. 2–3
  • Polyomino scores 91.89, exceeding CORAL’s 89.40 by 2.49 points; Lasso is approximately 1.40× faster than SimpleTES, and TriMul reaches 1129 µs versus a 1128 µs public rank-1 reference.On AHC058, Guidance-TTT obtains 850.08M AtCoder points versus SimpleTES’s 849.33M.Source: §1 · pp. 2–3
  • The paper analyzes guidance learning through controlled ablations, model-scale comparisons, search trajectories, and failure-mode studies.These analyses are presented as revealing benefits of learning in thought space.Source: §1 · p. 3

2 RELATED WORK

Prior discovery systems use repeated search and verifier feedback either without changing model parameters or by adapting a solution-generating model. Guidance-TTT instead learns strategic guidance from verifier feedback while separating it from execution.

LLM self-evolution

  • Self-evolution methods such as FunSearch, AlphaEvolve, and CORAL use archive reuse, verifier feedback, and repeated refinement while keeping their underlying models fixed.CORAL distributes discovery across multiple fixed frontier LLMs using shared memory and asynchronous evolution.Source: §2 · p. 3

Test-time training

  • Recent discovery-oriented test-time training updates the solution-generating model, with TTT-Discover combining an entropic objective and PUCT-based reuse for max-seeking discovery.ThetaEvolve adds optional test-time RL to a program database.Source: §2 · p. 3

Role specialization in LLM systems

  • The paradigms differ in where verifier feedback drives adaptation: candidate selection in self-evolution, solution generation in TTT-Discover, and strategic guidance in Guidance-TTT.Guidance-TTT trains only compact guidance while retaining the implementation capability of a stronger frozen executor.Source: Figure 2 · p. 3
  • Role-specialized LLM systems coordinate through natural language, and some work separates idea generation from execution; Guidance-TTT additionally learns parent-conditioned guidance from verifier feedback at test time.Si et al. (2026) ground ideation in execution feedback through large-scale experiments.Source: §2 · p. 4

3 GUIDANCE-TTT

Guidance-TTT treats discovery as iterative refinement of archived executable solutions, separating strategic changes from their implementation. It selects parents, samples guidance proposals, executes and verifies them, then updates the guidance policy from grouped rewards.

3.1 PROBLEM SETUP

  • The discovery problem uses a task description and executable verifier to score complete candidate solutions, with the goal of finding the highest-reward archived solution.In Polyomino Packing, a candidate is a complete C++ solver that the verifier compiles and evaluates on benchmark instances.Source: §3.1 · p. 4

3.2 WHAT SHOULD BE LEARNED AT TEST TIME?

  • Because solution-generating TTT must choose and implement changes, its outcome feedback conflates strategy quality with implementation quality; Guidance-TTT updates compact strategic guidance while a separate executor applies changes to the exact parent.This separation focuses test-time learning on strategic decisions and avoids adapting a much larger solution generator.Source: §3.2 · pp. 4–5

3.3 THE GUIDANCE-TTT LOOP

  • For each selected archive parent, guidance samples K high-level modifications, a frozen executor applies them to produce child candidates, and verifier rewards form a sibling group for updating the guidance policy.The loop stores eligible solutions with compact summaries and verifier scores.Source: §3.3 · p. 5

3.4 GUIDANCE MODEL UPDATE VIA RL

  • The guidance policy maximizes an entropic reward objective whose exponential weighting favors high-reward candidates over simply improving mean reward.For each parent, K realized solutions provide a group-relative learning signal; β is adapted within each group and held fixed during the policy update.Source: §3.4 · p. 5

4 EXPERIMENTS

Across four discovery tasks, Guidance-TTT outperforms prior-work references without web access or adapting the execution model. The analyses examine discovered solution structures, design choices, guidance scaling, and search trajectories.

4.1 EXPERIMENTAL SETUP

  • Experiments cover Polyomino Packing, TriMul, Lasso path optimization, and AHC058, typically using Qwen3-8B guidance with frozen GLM-5.2 execution.Seeds come from one-shot execution-model calls; discovery runs offline without web access or inherited solution code.Source: §4 · p. 5, §4.1 · p. 6, TriMul GPU kernel · p. 6

4.2.1 PERFORMANCE ANALYSIS

  • Fixed Lasso solvers passed 15/15 held-out cases versus SimpleTES’s 13/15, while the TriMul kernel transferred unchanged across Triton 3.4.0 and 3.6.0.On both versions, it remained the fastest evaluated automated method.Source: §4.2.1 · p. 7

4.2.2 SOLUTION ANALYSIS

  • Across tasks, discovered solutions preserve strong algorithmic backbones while introducing targeted structural changes (Figure 4).Changes include Polyomino simulated-annealing refinement, Lasso segment-tree screening, AHC058 phase-dependent scheduling, and TriMul hybrid Triton–cuBLAS dispatch.Source: §4.2.2 · p. 7, Figure 4 · p. 7

4.2.1 PERFORMANCE ANALYSIS

  • Polyomino 91.89 beats CORAL 89.40; Lasso Path Score 0.1739 tops SimpleTES 0.1243; AHC058 AtCoder score 850,082,731 tops 849,325,750; TriMul latency 1129 µs beats K-Search 1131 µs.Table 2 compares results across the four tasks and public references.Source: §4.2.1 · p. 7, Table 2 · p. 8, §4.3 · p. 8

4.3 ABLATIONS

  • Freezing guidance lowers Polyomino from 91.89 to 84.85 and raises TriMul latency from 1225.58 to 1256.62 µs; direct 8B solution training reaches 41.63 and 9664.91 µs.G-Top2 has opposite task effects: Polyomino falls to 82.12, while TriMul improves to 1166.82 µs. With a frozen GPT-OSS-120B executor, Guidance-TTT scores 83.19 at about $120 versus TTT-Discover’s 83.72 at about $283, a roughly 58% cost reduction.Source: §4.3 · pp. 8–9, Table 3 · p. 8

4.4 ANALYSIS: SCALING GUIDANCE IMPROVES PERFORMANCE ON SATURATED TASKS

  • On AHC058 and TriMul, 14B guidance improves over 8B and reaches state-of-the-art performance; on the TriMul probe, 8B adds incorrect branch restrictions in two of three explanations, while 14B is correct in all three.The probe counts explanations without incorrect extra conditions across three samples per model.Source: §4.4 · p. 9, Figure 5 · p. 9

4.5 ANALYSIS: GUIDANCE-TTT DYNAMICS

  • Search trajectories show structural gains followed by local refinement, with plateaus: Lasso gains after updates 7–25, AHC058 resumes gains late, and TriMul reaches its best latency at update 25.Update 0 denotes the seed; search scores and Table 2 results use different evaluation environments.Source: §4.5 · p. 9, Figure 6 · p. 10

5 CONCLUSION

Guidance-TTT separates strategic adaptation from solution implementation, training a compact guidance policy while keeping a stronger executor frozen. Across four discovery domains, it outperforms prior-work references while remaining competitive with public leaderboard results.

5 CONCLUSION

  • Guidance-TTT separates strategic adaptation from solution implementation, learning a compact guidance policy while keeping a stronger execution model frozen.Across four distinct discovery domains, it produces solutions outperforming the strongest prior-work references while remaining competitive with public leaderboard results.Source: §5 · p. 10
  • Controlled comparisons find strategic guidance learning more effective than either freezing the guidance policy or directly adapting a solution-generating model, while avoiding large-executor training costs.The paper concludes that test-time adaptation can learn to guide a frozen executor.Source: §5 · p. 10
  • The authors identify richer guidance representations, improved guidance–execution credit assignment, and more diversity-aware search as future-work directions.Source: §5 · p. 10

STATEMENT ON AI USAGE

The authors disclose uses of generative AI and describe the appendices’ additional results, methods, evaluation settings, and selected code.

STATEMENT ON AI USAGE

  • Generative AI assisted with implementation, debugging, analysis, literature checks, editing, and figure preparation; authors reviewed and verified AI-assisted code and analyses.Reported experimental results came from the stated evaluation pipelines, and the authors take responsibility for the final claims, code, and results.Source: STATEMENT ON AI USAGE · p. 10

APPENDIX OVERVIEW

  • The appendices provide additional results, guidance and failure-mode analyses, the search and training procedure, experimental settings, and selected code.Source: APPENDIX OVERVIEW · p. 13

A FURTHER RESULTS

  • Transfer evaluations keep selected programs fixed and do not use out-of-distribution feedback for further search.Source: A FURTHER RESULTS · p. 13

A.1 LASSO OOD EVALUATION

The Lasso OOD evaluation finds robust correctness across held-out cases, while the compiler-transfer, executor, and guidance-size analyses qualify where performance gains do and do not generalize. Diagnostic probes also compare how 8B and 14B guidance handle selected algorithmic mechanisms.

A.1 LASSO OOD EVALUATION

  • Both GLM-5.2 Lasso solvers passed all 15 OOD cases; on 11 real datasets, both passed all 11, and 8B was 1.28× faster geometrically across nine mutually valid comparisons.SimpleTES was faster on the four real datasets in the separate 15-case suite, while 8B+GLM had the lowest stress-suite latency among solvers passing every case.Source: Table 4 · p. 13, §A.1 · pp. 13–14, Table 5 · p. 14

A.2 TRIMUL COMPILER TRANSFER

  • The 14B-guided kernel was selected on Triton 3.3.1 and evaluated unchanged on Triton 3.4.0 and 3.6.0; CUDA/C++ extension submissions fall outside the shared Triton scope.The published TTT-Discover latency uses its original runtime; its released kernel measures 1262 µs in the Triton 3.3.1 environment used here.Source: Table 6 · p. 14, §A.2 · p. 14

A.3 EXECUTION-MODEL COMPARISONS

  • With a 30-update budget, GPT-OSS-120B scores 83.19 on Polyomino, below TTT-Discover’s 83.72; the main text characterizes this comparison as a lower-cost, slightly lower-score trade-off.Source: §A.3 · p. 15

B FURTHER ANALYSIS

  • The analyses use stored candidate histories and fixed-solution evaluations; individual search trajectories do not establish causal scaling laws.Source: B FURTHER ANALYSIS · p. 15

B.1 GUIDANCE MODEL SIZE

  • 14B guidance improves TriMul on feasibility, parent-relative improvement, and final performance, but Polyomino has higher validity alongside lower useful yield and final score.Source: §B.1 · p. 16
  • Lasso validity rises from 86.48% to 89.19% as useful yield shifts from 7.21% to 7.08%; AHC058’s best training score rises 0.46% despite raw validity decline amid HTTP 402 failures.For AHC058, excluding service failures gives validity of 86.52% for 8B and 88.07% for 14B; removing failures from the denominator does not restore missing candidates or undo subsequent-update effects.Source: §B.1 · p. 16

B.1.1 ABSOLUTE STATISTICS AND ADDITIONAL TRAJECTORIES

  • Retained results use committed 30-update archives, and useful yield requires improvement over the exact selected parent; one search per configuration provides no between-run uncertainty estimates.Source: §B.1.1 · p. 16, Table 9 · p. 16

B.1.2 DIAGNOSTIC PROBES OF ALGORITHMIC UNDERSTANDING

  • Reasoning-enabled probes show both models pass explicit AHC058 and TriMul code-condition checks, but 14B more consistently explains TriMul branch conditions; the errors do not directly account for the benchmark gap.All checks passed were 15/15 on AHC058 and 18/18 on TriMul; only one of three 8B explanations correctly described the relevant TriMul branches, versus all three 14B explanations.Source: §B.1.2 · p. 17

B.1.3 SELECTED DIAGNOSTIC COMPARISONS

Selected probes contrast 8B and 14B guidance on objective evaluation, execution-rule reasoning, and search reachability. Parent attachment also produces different outcomes for Polyomino and TriMul, consistent with—but not proving—differences in transferability.

B.1.3 SELECTED DIAGNOSTIC COMPARISONS

  • For a 2 × 2 fill, the scoring function returns 0.55, not 1: two of three 8B responses equate geometric coverage with the full score, while 14B computes correctly in two of three.Source: §B.1.3 · pp. 17–18
  • On TriMul, one 8B response mistakes an element-level zero for grounds to prune a whole tile, while all three 14B responses reject that shortcut.Tile pruning is valid only when all relevant contributions are zero.Source: §B.1.3 · p. 18
  • On AHC058, one 8B response misclassifies both strict-inequality boundary cases and two claim greedy search can reach a joint upgrade; all three 14B responses handle both correctly.Neither individual upgrade improves the initial reward, so a procedure accepting only individually improving moves cannot reach the higher-reward combination.Source: §B.1.3 · p. 18
  • Together, the probes test objective fidelity, valid execution rules, and whether a decision is reachable under the specified search dynamics.Source: §B.1.3 · p. 18

B.2 PARENT ATTACHMENT AND TRANSFERABILITY

  • G-Top2 lowers Polyomino validity to 33.83% and useful yield to 7.71%, whereas on TriMul useful yield rises from 8.70% to 9.17% and search reaches 1166.82 µs.The contrast is consistent with greater transferability between nearby TriMul solutions than between Polyomino states, but the single-trajectory comparison does not establish the mechanism causally.Source: §B.2 · p. 18

B.3 FAILURE MODES

Guidance-TTT’s histories expose two recurring failure modes: execution errors can obscure the value of a proposal, and local improvements can accumulate without advancing the global best. These patterns appear across tasks, but the observed trajectories do not isolate their causes.

B.3 FAILURE MODES

  • Across tasks, verifier outcomes conflate guidance and implementation quality, while parent-relative improvements may leave the global best unchanged.Polyomino and TriMul logs include many invalid candidates; Lasso and AHC058 also show related proposals with different implementation outcomes and local gains that do not advance the global best.Source: §B.3 · p. 18
  • A Polyomino density-adaptive proposal failed to compile in one realization but scored 91.3173 in another; the mechanism had helped earlier and remained in the update-27 best.On stronger parents, correctly realized versions no longer helped; the entropic objective rewards sibling differences but does not separate strategic value from realization difficulty.Source: §B.3 · p. 19, Figure 10 · p. 20
  • During lineage-bound plateaus, parent improvements failed to raise the frontier in Polyomino, TriMul, Lasso, and AHC058.The trajectories show continued local refinement within a shared ancestor’s descendants, with later step-best candidates remaining below the retained frontier.Source: §B.3 · pp. 19–22
  • In Lasso, same-parent alignment proposals yielded zero reward after a pointer-conversion error or scored 0.22552 with aligned storage, so the timing gain does not isolate alignment’s contribution.The two implementations differed, and the failed one’s zero reward cannot distinguish a type error from an unhelpful proposal.Source: §B.3 · p. 21, Figure 12 · p. 21
  • In AHC058, one planning proposal failed from a declaration-order error, while a paired implementation passed all 150 instances and scored 5,467,815.79.The implementations differed, so the successful case shows the planning direction was realizable, not that the two executions were identical.Source: §B.3 · pp. 21–22, Figure 13 · p. 22
  • These examples do not isolate the causal effects of policy updates, parent selection, evaluation variation, and service interruptions; 167 HTTP 402 failures also reduced AHC058 candidates.Progress resumed after the plateau, so the histories do not establish terminal collapse.Source: §B.3 · pp. 21–22

C METHOD DETAILS

The method allocates search across parent groups, records candidate feedback separately from archive eligibility, and trains only the guidance model using sibling-relative rewards. Task-specific score mappings and fixed rollout configurations define the evaluation and update budget.

C.1 BUDGET AND PARENT GROUPS

  • Each update selects B parents and samples K actions per parent, so U updates yield N = UBK search attempts; the default 30 × 8 × 16 budget gives 3,840 attempts.Parents come from the archive visible before the update, and newly generated children become eligible at the next update; repeated parents remain possible.Source: §C.1 · p. 23

C.2 TRAINING RECORDS AND SEARCH ARCHIVE

  • Every attempt retains its rollout record, while archive admission requires a valid verifier status, an available score, and a nonduplicate program.AHC058 requires acceptance on all 150 instances for archive eligibility, although rejected candidates may still contribute partial training reward; TriMul retains up to two eligible children per parent.Source: §C.2 · p. 23

C.3 TASK SCORES AND REWARDS

  • Rewards are oriented so higher is better, with task-specific mappings for packing score, inverse Lasso solve time, reciprocal TriMul latency, and scaled AHC058 mean score.Invalid Polyomino, Lasso, and TriMul programs receive zero reward; AHC058 may retain partial reward, but only fully accepted candidates contribute archive bests.Source: §C.3 · p. 23

C.4 ADAPTIVE ENTROPIC ADVANTAGE

  • For each parent, an adaptive inverse temperature is selected from the K realized verifier rewards before computing a leave-one-out advantage for nonconstant-reward groups.The bounded numerical search retains its upper bound if the target cannot be reached; the estimator compares each candidate with its siblings, not its parent, and is not a z-score.Source: §C.4 · p. 24

C.5 ACTOR LOSS AND UPDATE SCHEDULE

  • A token-mean actor loss updates guidance from the sampled batch, using fixed advantages, masked tokens and groups, and a detached REINFORCE importance weight when correction is needed.Each complete BK-rollout batch receives one optimizer update and one pass; only guidance LoRA parameters change, while the executor stays frozen.Source: §C.5 · p. 24

D.1 PRIMARY CONFIGURATIONS

  • Primary configurations use an 8 × 16 rollout shape for every TriMul Guidance-TTT result and eight parents, 16 rollouts per parent, and 30 updates for the additional tasks.The AHC058 120B-executor run stopped after 28 updates.Source: Table 11 · p. 25, Table 12 · p. 25, §D.1 · p. 25

D.2 AHC EVALUATION PROTOCOL

AHC058 candidates are evaluated on the 150 public instances, with scores summarized as an arithmetic mean. The archive requires full acceptance, while training can still use partial reward from rejected candidates.

D.2 AHC EVALUATION PROTOCOL

  • The AHC evaluator runs each candidate on all 150 public instances and reports their arithmetic-mean raw score.Search trajectories convert that mean to a total over 150 instances.Source: §D.2 · p. 25
  • A candidate enters the solution archive only if all 150 instances are accepted, although rejected candidates can receive partial policy-training reward.Rejected candidates' scores are never reported as archive bests.Source: §D.2 · p. 26
  • AHC058 evaluates four candidates concurrently, using eight case workers per candidate; the controlled OpenEvolve run starts from the corresponding one-shot GLM-5.2 program and uses 500 iterations..Source: §D.2 · p. 26

D.3 FIXED-SOLUTION EVALUATION AND TRANSFER

Fixed-solution evaluations use task-specific repeat protocols, while historical controls are distinguished from final-program measurements. The discovered-code listings retain design-relevant portions and omit supporting implementation details.

D.3 FIXED-SOLUTION EVALUATION AND TRANSFER

  • Lasso's main result repeats the frozen program on a 17-case evaluator five times with newly generated inputs, using one CPU thread; invalid invocations count as zero.OOD evaluations use fixed programs without further training or selection, and the real/stress and extended-suite protocols differ and must not be pooled because DNA and leukemia overlap.Source: §D.3 · p. 26
  • TriMul fixed kernels receive three evaluations on an isolated H100, passing 18 correctness tests and reporting the geometric mean of seven case latencies.Compiler-transfer evaluations change Triton versions without modifying the selected program, so they test software-environment shifts rather than new input distributions.Source: §D.3 · p. 26
  • Historical TriMul archive endpoints and several comparison values are not matched-repeat measurements, so they do not support matched-repeat uncertainty estimates.This includes the 8B/frozen comparison's archived endpoints; Lasso's TTT-Discover and OpenEvolve values also retain original measurements.Source: §D.3 · p. 26
  • AHC058's reported results are official-platform submission scores, not means from a matched repeated-submission protocol.The cited scores are TTT-Discover's 848,414,228, SimpleTES's maximum of ten submissions at 849,325,750, and the recorded official score of 850,082,731.Source: §D.3 · p. 26

E DISCOVERED SOLUTION CODE

  • The listings preserve implementation portions for the discussed design choices but omit input handling, bookkeeping, repeated launch code, and unrelated helpers.The repository contains full implementations; excerpts are lightly condensed for typesetting.Source: E DISCOVERED SOLUTION CODE · p. 27

E.1 RELATIONSHIP TO PRIOR SOLVERS

Across the discovered solvers, the paper identifies extensions to established algorithmic backbones, including refinement, shape specialization, and state-dependent scoring. The excerpts describe the corresponding mechanisms in Polyomino, TriMul, Lasso, and AHC058.

E.1 RELATIONSHIP TO PRIOR SOLVERS

  • Guidance-TTT's Polyomino solver extends skyline construction with simulated annealing and timed restarts, while its TriMul kernel specializes a hybrid Triton–cuBLAS pipeline by channel regime and sequence shape.These are comparisons of final designs, not evidence that the solutions inherited prior source code.Source: §E.1 · p. 27

E.2 POLYOMINO PACKING

  • The Polyomino solver evaluates valid orientations and horizontal positions using height, waste, slope, position, and piece area, then anneals piece priorities and board width while retaining the best complete packing.The piece-placement key combines resulting height, wasted space, boundary slope, vertical position, and piece area.Source: §E.2 · pp. 27–28

E.3 TRIMUL GPU KERNEL

  • For TriMul at C = 128, one Triton kernel fuses normalization, projections, gating, masking, and layout conversion, while contraction dispatches between cuBLAS and persistent Triton by shape.The selected kernel records a search-time latency of 1110.85 µs and a controlled three-run mean of 1128.91 µs.Source: §E.3 · pp. 29–30

E.4 LASSO REGULARIZATION PATH

  • The Lasso solver follows a homotopy path by updating an active-set Cholesky factor, screening inactive features with cached correlations, and interpolating coefficients at requested regularization parameters.The described implementation uses a segment tree for lazy correlation screening.Source: §E.4 · pp. 31–32

E.5 AHC058: APPLE PRODUCTION PLANNING

  • The AHC058 solver scores upgrades using cost, remaining production horizon, and interactions across production levels, with a two-stage sigmoid schedule blending scoring strategies.Its score incorporates state-dependent benefits and a phase-dependent blend between single-level and broader scoring.Source: §E.5 · pp. 33–34
Loading 2610.06269v1…