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Self-Retrospection Distillation: Turning Post-hoc Experiences into Prior Foresight

Haoxiang Zhang, Qinglin Chen, Hiroaki Hayashi, Zhuofeng Li, Siming Zhang, Jiaxin Zhang, Jixuan Chen, Fang Wu, Pan Lu, Silvio Savarese, Julian McAuley, Chien-Sheng Wu

arXiv:2610.08077v1cs.AIcs.CLcs.IRcs.LG

TL;DR

Group-relative RLVR can lose its learning signal when rollouts receive uniform rewards, even though their trajectories contain information about task demands and failures. The paper introduces prospective learning and SRD, which use hindsight to supervise pre-interaction foresight. Across evaluated objectives and settings, SRD improves category-level performance, while the paper cautions that benchmark gains do not establish safety or reliability in deployment.

  • Problem

    Group-relative RLVR provides no advantage signal when all rollouts in a group receive the same reward, despite the task and environment information those trajectories may reveal.

    Source: §1 · pp. 1–2
  • Method

    SRD uses completed trajectories as privileged hindsight to supervise the same policy’s predictions of task-relevant knowledge and possible failures from pre-interaction information.

    Source: §1 · p. 2
  • Results

    SRD improves category-level performance across GRPO, OPSD, and RLSD at both model scales, with one regression: WebShop under 9B OPSD (−3.61 pp).

    Source: §4.2 · p. 6
  • Takeaways & Limitations

    The results support prospective learning as a post-training design axis that uses post-hoc experience to shape pre-interaction anticipation.

    Source: §7 · p. 10

    Benchmark gains do not establish safety, fairness, or reliability in deployment, and applications beyond evaluated settings call for task-specific safety evaluations and safeguards.

    Source: ETHICS STATEMENT · p. 11

Abstract

from arXiv · show

Reinforcement learning with verifiable rewards (RLVR) turns agent experience into learning signals primarily through scalar outcome rewards after interaction. For group-relative objectives, however, this signal vanishes when all rollouts receive the same reward, even though their trajectories may reveal useful information about what the task requires and how the agent fails. We ask a complementary question: can hindsight teach an agent what it could have anticipated before acting? We introduce prospective learning, which uses post-hoc experience to supervise foresight predictions from the pre-interaction view, and instantiate it with Self-Retrospection Distillation (SRD). Intuitively, a completed trajectory reveals knowledge that would have been useful and pitfalls that should be avoided; SRD distills this privileged hindsight into trajectory-blind foresight of the same policy. Foresight serves only as a training target and need not be explicitly generated at inference time. Across 10 tool-integrated reasoning and long-horizon agentic tasks, SRD complements RLVR and self-distillation baselines with gains of up to $24.2$ pp. Its advantage is especially pronounced when reward contrast is scarce: when $37$--$98\%$ of rollout groups are reward-uniform across model scales, yet SRD can still exploit learning signal from sampled trajectories. In the 2B setting, where $98\%$ of groups are all-failure, the RLVR training ends up at $0.0\%$ success, while adding SRD reaches $60.6\%$ under the same rollout budget. Our results suggest that post-hoc agent experience is useful not only for evaluating or improving behavior, but also for shaping predictive representations before available interaction.

1 INTRODUCTION

RLVR learns from post-interaction outcome rewards, but group-relative learning receives no advantage signal when all rollouts share a reward, despite the information their trajectories contain. The paper introduces prospective learning and SRD to distill hindsight into pre-interaction foresight, complementing RLVR and self-distillation across reasoning and agentic tasks.

Motivation

  • When all rollouts in a group receive the same reward, group-relative advantages vanish even if their trajectories reveal useful task and environment information.This leaves uniform-reward experience without a learning signal under the described RLVR objective.Source: §1 · pp. 1–2

Prospective learning

  • Prospective learning uses completed trajectories as privileged supervision for predictions of task requirements and possible failures formed from pre-interaction information alone.It shifts the target from what the agent should do next to what it could have anticipated before acting.Source: §1 · p. 2

Self-Retrospection Distillation

  • SRD aligns pre-interaction and hindsight-conditioned token distributions on the student’s own prospective rollout, allowing failed and successful trajectories to teach pitfalls and useful knowledge.Reward determines which lesson a trajectory teaches, rather than whether it contributes supervision.Source: §1 · p. 2

Contributions

  • SRD adds a lightweight auxiliary objective to RLVR and self-distillation and improves performance across tool-integrated reasoning and long-horizon agentic settings.The paper also reports that SRD remains informative for reward-uniform baselines where RL yields zero advantage.Source: §1 · p. 2

2 PRELIMINARIES

The section describes how RLVR converts completed trajectories into scalar-reward policy updates and why uniform-reward groups yield no group-relative advantage. It then contrasts retrospective methods, which use post-hoc information to improve behavior, with the prospective target of pre-interaction anticipation.

RLVR preliminaries

  • An agent interacts with an environment by producing a trajectory of actions and resulting observations conditioned on its interaction history.The task is drawn from a distribution, and each step’s action and observation depend on the history available at that point.Source: §2 · p. 2
  • RLVR applies a verifier to a completed trajectory, converts its scalar reward into an advantage estimate, and assigns that estimate to the actions that produced it.For group-relative methods, identical rewards give every rollout zero advantage, so uniform-reward groups are typically discarded and resampled.Source: §2 · p. 2

Retrospective learning from agent experience

  • Retrospective methods use structured post-hoc hindsight—such as feedback, selected solutions, or reusable skills—to improve the agent’s reasoning or action policy.On-policy self-distillation uses a hindsight-conditioned teacher to supervise the student’s token distributions; skill- and memory-based methods optimize downstream task performance.Source: §2 · p. 3

From retrospective to prospective learning

  • The section shifts the supervision target from the action the agent takes to what it could have anticipated before interaction.This distinguishes prospective learning from retrospective methods that use post-hoc information to improve behavior.Source: §2 · p. 3

3 LEARNING PROSPECTIVELY VIA SELF-RETROSPECTION DISTILLATION

Prospective learning trains predictions made from task and environment context before interaction to anticipate what hindsight reveals. SRD instantiates this with knowledge and pitfall perspectives, using a hindsight-conditioned self-teacher to supervise the student’s foresight rollout.

3.1 PROSPECTIVE LEARNING

  • Prospective learning defines foresight from task and environment context without access to the realized interaction, then uses hindsight to supervise predictions over the same output space.The objective internalizes patterns of failure and knowledge revealed by the agent’s own traces rather than directly optimizing its action distribution.Source: §3.1 · pp. 3–4

3.2 SELF-RETROSPECTION DISTILLATION

  • SRD uses KNOWLEDGE to describe potentially required knowledge and PITFALL to describe possible failures, with successful and failed trajectories supplying their respective supervision.Both perspectives use the same distillation procedure and differ only in the prospection instruction.Source: §3.2 · p. 4
  • For each rollout, SRD conditions hindsight on the completed trajectory and privileged context, while foresight is formed without that interaction information under the same prospective view.The post-hoc context includes the trajectory, a gold solution or ground-truth answer, and an error annotation when applicable.Source: §3.2 · pp. 4–5
  • The student samples a foresight sequence, and the hindsight teacher evaluates the same student-generated prefix to provide token-level distillation supervision.The teacher is a stopped-gradient copy of the current policy, and the paper identifies generalized Jensen–Shannon divergence as a natural, stable divergence choice.Source: §3.2 · p. 5

4 EMPIRICAL STUDY

Across the reported task categories, SRD improves post-training objectives at both model scales, with gains also reported under distribution, format, and interaction-horizon shifts. The study evaluates ten benchmarks, compares RLVR and self-distillation baselines, and reports that SRD can mitigate instability in pure self-distillation.

4.1 EXPERIMENT SETUP

  • The evaluation covers four categories and ten benchmarks, including held-out distribution, format, and interaction-horizon shifts across domains.Baselines include GRPO, on-policy self-distillation, and RLSD; each is also evaluated with an auxiliary SRD loss.Source: §4.1 · p. 5

4.2 MAIN RESULTS

  • SRD improves category-level performance across GRPO, OPSD, and RLSD at both scales, with one regression: WebShop under 9B OPSD (−3.61 pp).For 4B GRPO, SRD raises Math, Code, and Search averages by 10.03, 8.33, and 8.05 pp, respectively; it also improves ALFWorld by 3.25 pp and WebShop by 5.92 pp.Source: Table 1 · p. 6, §4.2 · p. 6
  • At 9B, OPSD+SRD raises the Math average from 39.70% to 56.92% (+17.22 pp), while SRD improves five of six BrowseComp-Plus settings by up to 9.52 pp.BrowseComp-Plus uses an interaction horizon of 10× or more tool calls compared with training rollouts.Source: §4.2 · p. 6
  • At 9B, OPSD alone finishes below the untrained policy on HotpotQA, 2Wiki, and LCB-v6, while OPSD+SRD restores increasing Math performance from 4B to 9B.The respective baseline deficits are 5.50, 3.76, and 2.18 pp; with SRD, the Math averages are 48.11% at 4B and 56.92% at 9B.Source: §4.2 · p. 6
  • SRD remains effective at 9B, although its gains do not increase monotonically with model scale.The best +SRD result rises from 4B to 9B in every task category, with particularly strong improvements in Math and Agentic tasks.Source: §4.2 · p. 7

5 RESEARCH QUESTIONS

The research questions examine how SRD’s target choice, token shifts, and use of reward-uniform trajectories relate to learning outcomes. They also assess policy changes and whether making foresight explicit at inference improves results.

RQ1: WHICH FORESIGHT IS MORE INFORMATIVE?

  • Pitfall-only SRD improves AIME26 under both raw-trace and hindsight prefixes, while adding KNOWLEDGE costs more and can underperform the baseline.The pitfall channel’s divergence falls during training, whereas the knowledge channel stays flat in the two runs described.Source: RQ1: WHICH FORESIGHT IS MORE… · p. 7

RQ2: WHICH TOKENS DOES SRD REALLOCATE?

  • On code validation, SRD shifts expected tokens toward connective prose (+8.6) and planning language (+3.7), away from inline notation (−6.7), while tool-call syntax remains unchanged (+0.03).Across lexical classes, the net change is +0.6 tokens per rollout.Source: RQ2: WHICH TOKENS DOES SRD REALLOCATE? · p. 8, Figure 3 · p. 8

RQ3: CAN SRD LEARN FROM REWARD-UNINFORMATIVE TRAJECTORIES?

  • At 2B, baseline final success is 0.0%, while SRD reaches 60.6% in training under the same rollout budget.Reward-uniform groups consume a substantial fraction of sampled experience across capability levels; the usable share never exceeds 63%.Source: RQ3: CAN SRD LEARN FROM… · p. 8

RQ5: DOES TEST-TIME FORESIGHT HELP?

  • Explicit test-time foresight barely changes three of four checkpoints, while the fourth gains 5.00 pp; effects vary by base objective and channel.For the three near-unchanged checkpoints, differences are 2, 3, and 11 rollouts out of 800.Source: Table 3 · p. 9, RQ5: DOES TEST-TIME FORESIGHT HELP? · p. 9

RQ4: HOW DOES SRD SHAPE THE PROSPECTION DISTRIBUTION?

  • On held-out code, +SRD travels 30% farther along GRPO’s direction and adds an orthogonal displacement of comparable size.The reported alignment is positive at 77% of positions for GRPO and 94% for OPSD; the host objectives themselves disagree in direction.Source: RQ4: HOW DOES SRD SHAPE THE… · p. 9

RQ5: DOES TEST-TIME FORESIGHT HELP?

  • No arm improves all six ALFWorld task types, and average results conceal task-specific swings spanning up to 60.07 pp.The reported best-to-worst task-type spread ranges from 10.16 pp to 60.07 pp.Source: RQ5: DOES TEST-TIME FORESIGHT HELP? · p. 10

6 RELATED WORK

Prior work increases the information retained from interaction through outcome, process, and self-distillation supervision, while predictive methods use learned expectations to support interaction. SRD redirects hindsight to trajectory-blind foresight rather than to behavior.

From outcome supervision to hindsight-rich distillation

  • RLVR uses trajectory-level outcomes and process supervision increases feedback’s temporal resolution, while self-distillation enriches completed-rollout information through privileged context.The cited context variants include solutions, self-revisions, skills, procedural memory, and outcome-RL terms.Source: §6 · p. 10
  • Prior analyses report that privileged teacher conditioning can alter calibration, self-correction, epistemic suppression, and the teacher behavior being distilled.These concerns motivate separating what hindsight contains from what it supervises.Source: §6 · p. 10

Prospective Learning and Foresight

  • Unlike methods that imagine futures for planning, SRD uses completed interaction to supervise trajectory-blind predictions and does not require explicit foresight at inference.The related predictive lineage includes model-based methods, world models, hindsight relabeling, and recent language-agent foresight work.Source: §6 · p. 10

7 CONCLUSION

The conclusion frames SRD as a way to distill hindsight into trajectory-blind foresight, complementing RLVR and self-distillation across reasoning and agentic tasks. It presents prospective learning as a post-training design axis for anticipating interaction demands and possible failures.

Conclusion

  • SRD distills post-hoc experience into trajectory-blind foresight and complements RLVR and self-distillation across reasoning and agentic tasks.The conclusion highlights settings where objective-specific filtering leaves much sampled experience unused.Source: §7 · p. 10
  • The authors propose prospective learning as a post-training design axis and identify anticipation of interaction demands and failures as a direction toward more capable agents.This is presented as the authors’ view of a key direction, not as a demonstrated deployment outcome.Source: §7 · p. 10
  • The conclusion extends post-hoc experience from behavioral revision to pre-interaction anticipation.Source: §7 · p. 10

ETHICS STATEMENT

The ethics statement covers post-training for language-model agents in mathematical reasoning, programming, retrieval, and simulated interactive tasks. It warns that benchmark gains do not establish safe or reliable deployment and recommends task-specific safeguards.

ETHICS STATEMENT

  • The evaluated application scope is mathematical reasoning, programming, retrieval, and simulated interactive tasks.Source: ETHICS STATEMENT · p. 11
  • Improved task performance may increase the effectiveness of agents used for harmful purposes, and self-hindsight can reinforce existing errors and biases.Source: ETHICS STATEMENT · p. 11
  • The statement recommends task-specific safety evaluations, sandboxed execution, restricted tool permissions, human oversight, and safeguards for sensitive interaction traces.Source: ETHICS STATEMENT · p. 11

A.1 OVERVIEW

The overview argues that group-relative reward learning is structurally gated by mixed outcomes, whereas prospective supervision has no such gate. It analyzes how this affects usable rollout budgets and estimation costs as capability approaches determinism.

A.1 OVERVIEW

  • The reward direction is nonzero only on mixed-outcome groups, whose probability is at most Gπ(p); this gate creates a π(p)^−1 estimation-cost penalty as π(p) approaches zero.π(p) := min{p, 1 − p} measures distance to determinism.Source: §A.1 · p. 16
  • At G = 8, homogeneous success rates from 0.3 to 0.7 predict less than 6% discarded budget, but the measured floor is 37%, revealing task-level heterogeneity.The reward-usable share also collapses toward both saturation limits, where groups become reward-uniform.Source: §A.1 · p. 16
  • On reward-uniform groups, the reward direction is exactly zero, while the prospective direction is not structurally forced to vanish, regardless of teacher quality.On all-failed groups, no scalar rescaling can recover the zero reward direction.Source: §A.1 · p. 16
  • Estimating the reward direction pays a π(p)^−1 gating cost, whereas the prospective direction’s bound has no such factor, even in the worst case Geff = 1.The prospective bound accounts for group-level pitfall-hindsight dependence through Geff.Source: §A.1 · p. 16

A.2 SETTING

  • The setting defines the prospective update H from foresight-token divergence gradients and combines it with the reward update as θ+ = θ + η(UR + λH).The teacher is the stopped-gradient current policy.Source: §A.2 · p. 16

A.3 BUDGET ACCOUNTING

This section describes how reward gating affects group-normalized updates and estimation budgets, contrasts this with prospective learning on reward-silent groups, and specifies skill-construction and grading procedures. It also outlines three LLM-as-judge graders that assess extracted answers against references, emphasizing semantic correctness, relevance to the question, and rejection of unsupported guesses.

A.3 BUDGET ACCOUNTING

  • Mixed-group rarity raises the expected attempts for n mixed groups to n/qG(p), at least n/π(p); the rollout-cost curve also reflects task-difficulty heterogeneity.The elevated middle-range floor reflects a Jensen gap across prompts, while the divergence at both ends reflects saturation in success or failure; improving average capability alone does not necessarily leave this regime.Source: §A.3 · p. 17

A.4 WHAT GROUP NORMALIZATION OPTIMIZES

  • Group normalization’s expected reward direction ascends a strictly increasing, task-reweighted success utility rather than mean success.The expected reward direction does not acquire a qG(p) factor: conditional update magnitude cancels mixed-group rarity, leaving a coefficient bounded away from zero uniformly in p.Source: §A.4 · p. 17

A.5 THE REWARD-SILENT EVENT

  • At 2B, 98% of groups were uniform; GRPO ended at 0.0% success, while LSRD reached 60.6% under the same budget.On all-failed groups, the reward update is identically zero, whereas the prospective loss is defined and unconstrained, though not guaranteed to be nonzero.Source: §A.5 · p. 18

A.6 HOW MANY GROUPS EACH DIRECTION COSTS TO ESTIMATE

  • Reward-update estimation inherits a π(p)^-1 gating penalty, while prospective-update estimation remains independent of π(p), even when Geff = 1.The gating cost comes from reward updates being zero off mixed groups, not from a particular policy-gradient estimator.Source: §A.6 · p. 18

A.7 A REMARK ON THE TWO CHANNELS

  • PITFALL divergence falls by 30–47% while KNOWLEDGE divergence remains similar, consistent with KNOWLEDGE-channel redundancy and PITFALL-only SRD as a compute-efficient default.The authors caution that the divergence bound is one-sided and its constants are unmeasured, so the observation does not establish that low divergence lacks task-relevant signal.Source: §A.7 · p. 19

B.1 KNOWLEDGE AND PITFALL CONSTRUCTION

  • SRD constructs transferable KNOWLEDGE skills from correct traces and concrete PITFALL lessons from failures, aggregating recurring failures into a concise shared list.The teacher receives trace-conditioned hindsight, while the student receives only the problem and pre-interaction information; distinct prompt labels separate correct solutions from observed mistakes.Source: §B.1 · pp. 20, 23

B.2 LLM-AS-JUDGE

  • Grading is deterministic-first, with judge use limited to math fallback and search EM misses; BrowseComp-Plus uses the judge as its primary grader.All three domains use gpt-5.6-luna as judge.Source: §B.2 · p. 24
  • The general grader accepts mathematically or chemically equivalent answers, but requires an answer supported by reasoning and a final one-word CORRECT/INCORRECT verdict.It judges answer meaning rather than wording or format and treats blank, missing, or placeholder answers as incorrect.Source: §B.2 · p. 24
  • The Search / Multi-hop QA fallback accepts equivalent phrasings and specificity, but rejects blank answers and unsupported guesses; it requests brief analysis followed by a CORRECT/INCORRECT verdict.Its prompt compares the extracted answer’s meaning with one or more acceptable references.Source: §B.2 · p. 24
  • The BrowseComp-Plus grader extracts an exact final answer, compares it with the correct answer, and reports reasoning, correctness, and confidence.It treats absent final answers as None, allows a small numerical margin, and assigns 100 confidence when none is stated.Source: §B.2 · p. 25

C.1 TRAINING DATA CONSTRUCTION

The experiments use shared, fixed training mixtures and matched training scaffolds to compare RLVR, self-distillation, and SRD. Dynamic sampling filters groups according to each objective’s available learning signal, while SRD can learn from groups those filters would discard.

C.1 TRAINING DATA CONSTRUCTION

  • Training mixtures and prompt order are fixed across methods: Math, Code, and Search use equal-ratio domain mixtures, while ALFWorld and WebShop use a 1:1 mixture.This controls training data and ordering across compared objectives.Source: §C.1 · p. 25
  • Math, code, search, and agentic training draw respectively on DAPO-Math-17K, stdin LiveCodeBench, HotpotQA/2Wiki questions, and standard ALFWorld/WebShop training splits.Evaluation uses separate held-out math and search data, different code instances and format, and out-of-distribution ALFWorld layouts and object placements.Source: §C.1 · p. 25

C.2 TRAINING DETAILS

  • Qwen3.5-4B and Qwen3.5-9B use 16 prompts and 8 on-policy rollouts per prompt, with matched accepted trajectories per step; self-distillation uses an EMA teacher.The EMA update rate is 0.05 per optimization step; teacher distributions use top-k log-probabilities and a clipped importance-sampling ratio of 2.0.Source: §C.2 · p. 26

C.3 COMPARED BASELINES

  • The comparison includes Vanilla, unmodified GRPO, OPSD, and hybrid RLSD; +SRD adds a prospective distillation loss to each trained base method using structured hindsight.All baselines share the same scaffold, data stream, rollout budget, and optimizer budget, differing only in learning target.Source: §C.3 · pp. 26–27

C.4 DYNAMIC SAMPLING

  • GRPO and RLSD retain groups with reward variation, while OPSD requires at least one successful rollout; without dynamic sampling, SRD continues learning from otherwise discarded groups.The filters are kept in place across compared arms, so each step commits the same usable batch-size trajectories.Source: §C.4 · pp. 27–28

C.5 EVALUATION BENCHMARKS AND PROTOCOL

Evaluation covers math, code, search, and agentic benchmarks under a shared avg@8 protocol and fixed evaluation sets. The benchmarks include held-out problems, formats, environments, or longer-horizon tasks as specified for each category.

C.5 EVALUATION BENCHMARKS AND PROTOCOL

  • The shared protocol samples eight independent trajectories per prompt at temperature 1.0 and top-p 1.0, then averages success fractions across prompts.Methods use fixed evaluation sets and task interfaces, with task-specific graders and success signals.Source: §C.5 · p. 28
  • Math evaluation uses AIME 2024, 2025, and 2026 plus AMO-Bench, with AIME 2026 postdating the backbone’s pretraining cutoff.AMO-Bench serves as the most challenging mathematical reasoning benchmark in this evaluation.Source: §C.5 · p. 28
  • Code evaluation uses LiveCodeBench-v6 functional Python and OJBench, testing transfer across problem instances and submission formats from stdin training.Credit requires the executed program to pass all retained unit tests; partial credit is not awarded.Source: §C.5 · p. 28
  • Search evaluation spans short-horizon HotpotQA and 2WikiMultiHopQA subsets and long-horizon BrowseComp-Plus, which is never used for training.The HotpotQA and 2Wiki subsets are fixed 100-question held-out development samples calibrated to avoid nearly always solved or failed examples.Source: §C.5 · p. 28
  • Agentic evaluation uses 100 out-of-distribution ALFWorld games and 100 held-out WebShop sessions, with success determined by each environment’s terminal criterion.WebShop requires an exact match to requested product attributes and options.Source: §C.5 · p. 28

D.1 ABLATION STUDY λ

The λ ablation finds a stable aggregate mean across a broad weight sweep, but domain scores trade off, motivating a shared compromise value. The policy probe compares paired checkpoints while masking tool responses and filtering low-entropy positions.

D.1 ABLATION STUDY λ

  • 63.5–64.2 is the mean range across the five benchmarks over three orders of magnitude of λ, versus 39.3 for the untrained policy.The sweep uses OPSD with Qwen3.5-4B on a mixed math/code/search stream; λ = 0.01 is the value used in the main paper.Source: Table 6 · p. 29, §D.1 · p. 29
  • At λ = 1.0, mathematics scores are 72.1 and 65.8 while code scores 48.8; at λ = 0.01, mathematics scores are 63.3 and 61.7 while code scores 57.3.The authors use λ = 0.01 as a balanced shared setting for one policy serving three tool interfaces, rather than tuning λ separately by domain.Source: §D.1 · p. 29

E.1 PROBE SETTING

  • The probe compares GRPO and GRPO+SRD, and OPSD and OPSD+SRD, against a common untrained base, with each pair differing only by LSRD.Only model-emitted tokens are scored; tool-response positions and positions with predictive entropy below 0.1 nats are excluded.Source: §E.1 · pp. 29–30
  • Held-out evidence is code-only and uses LCB-v6 functional plus OJBench medium, while the mixed training split is weaker evidence because it is not prompt-paired.The held-out split differs from training in both test format and benchmark; the authors do not give the training split equal footing.Source: §E.1 · p. 30
  • The probe covers 5,152 rollouts and 2,617,027 generated positions; 1,093,803 positions pass the entropy gate, including 626,992 held-out positions.The entropy-gated positions constitute 42% of generated positions.Source: §E.1 · p. 30

E.2 WHAT THE POLICY EMITS DIFFERENTLY

This section examines how SRD changes emitted behavior and policy updates, and how hindsight can diagnose failures in reward-uniform groups and guide foresight. It illustrates this with a token-budget-exhausting implementation-planning rollout, distinguishing an efficiency problem from a conceptual error, while qualifying update comparisons and retracting a misleading alignment statistic.

E.2 WHAT THE POLICY EMITS DIFFERENTLY

  • 2888 tool calls versus 2243 and a median of four versus three assistant turns characterize +SRD’s held-out behavior; lexical shifts are not global sharpening.Mean |move| per occurrence remains 0.390–0.469 across lexical classes, while signed shifts separate the classes. Two independent estimates of the per-token shift agree at Pearson r = +0.9976 with 99.75% sign agreement (n = 801).Source: §E.2 · p. 30

E.3 WHERE THE UPDATE POINTS

  • cos(A, E) = +0.814 and cos(S, ES) = +0.980, while cos(A, S) = −0.783; the corresponding displacement norms are 0.387 → 0.857 and 1.966 → 0.568.These held-out medians show SRD’s displacement aligned with each host update, while the GRPO and self-distillation host updates point in opposite directions.Source: §E.3 · p. 31, Table 8 · p. 31

E.4 WHY SRD LENGTHENS THE RL STEP AND SHORTENS THE SELF-DISTILLATION ONE

  • SRD lengthens the GRPO displacement from 0.39 to 0.86 but shortens the OPSD displacement from 1.97 to 0.57, consistent with adding a dense signal to sparse GRPO and an anchor to dense self-distillation.The OPSD displacement stays near the same ray (ES ≈ 0.28 S), whereas the GRPO displacement grows along A and into an orthogonal direction. Endpoint norms conflate step size with effective step count, a limitation the appendix notes.Source: §E.4 · p. 31

E.5 WHY THE KEPT FRACTION VARIES SO MUCH ACROSS DOMAINS

  • The roughly constant orthogonal component (0.29–0.41) yields different kept fractions because GRPO’s update norm varies threefold; held-out code and training code also differ because the latter uses stale, non-final-policy prefixes.Training code’s kept fraction is 0.11 versus held-out code’s 1.30 despite similar ∥A∥; training-mix rows are weaker evidence, so the main text quotes the held-out split.Source: §E.5 · p. 32

E.6 A STATISTIC WE RETRACTED

  • The retracted statistic cos(F − A, A) cannot establish anti-alignment: its measured −0.36 implies 74% alignment, and the appendix instead decomposes updates from a common base.The authors verified the identity analytically and by Monte Carlo.Source: §E.6 · p. 32

F CASE STUDY: A MECHANISM-LEVEL AUDIT

  • Across 686 scanned groups, 47.2% were reward-uniform, and 29.9% of PITFALL records mentioned an interface fact absent from foresight, a lexical proxy rather than a measure of correct advice.The corpus comprised 5,216 prospection records; the appendix reports no blind human annotation of whether such advice was correct or useful.Source: F CASE STUDY: A MECHANISM-LEVEL AUDIT · p. 33, Table 9 · p. 33

F.1 FROM ZERO REWARD CONTRAST TO A SHARPER PROSPECTION

  • In an all-fail group, GRPO has zero pointwise gradient while SRD can use hindsight; the case audit shows failures from both an incorrect DP state and exhausting the response budget.The example’s eight rollouts all failed; hindsight identifies the state-transition error, while a second all-fail group had six rollouts stop at the token cap without tool calls.Source: §F.1 · pp. 34–36
  • At 8,192 tokens, the rollout is still describing an implementation and ends without a code call or answer.It proposes an approach for counting compatible mochi pairs but never executes it.Source: §F.1 · p. 36
  • For the sorted-input example, the extracted lesson is to avoid complex matching approaches and recognize that a direct count suffices.The annotation notes that the permutation requirement is automatically satisfied because the array is sorted and the task allows choosing K pairs.Source: §F.1 · pp. 36–37
  • The hindsight extractor classifies failures as TRUNCATED, FORMAT, or WRONG and uses the class to route the corresponding lesson.A truncated trace is treated as potentially on track, with its pitfall more likely to concern efficiency than conceptual correctness.Source: §F.1 · p. 37

F.2 AN INTERACTION CONTRACT THE TASK STATEMENT CANNOT REVEAL

This section shows how sampled interactions reveal execution contracts and environment constraints absent from task statements, and how successful trajectories can provide route-specific supervision despite reaching the answer by different paths. In all-success groups, the KNOWLEDGE channel is complementary because PITFALL has no failure to diagnose.

F.2 AN INTERACTION CONTRACT THE TASK STATEMENT CANNOT REVEAL

  • 4/8 code rollouts were correct, but sample-passing hardcoded input failed hidden tests; blind foresight identified algorithmic complexity while missing the stdin execution contract.The failing rollout spent six turns debugging hardcoded input and indexing errors, with three tool calls returning tracebacks, yet none read stdin.Source: §F.2 · pp. 37–40

F.3 INTERACTION-GROUNDED FORESIGHT IN AN AGENT ENVIRONMENT

  • 1/8 rollouts won the eight-action ALFWorld task; seven failures split between four exhaustive furniture searches and three rollouts blocked by the carry limit.The one-object carry constraint is implicit in changes to the admissible-action list, and hindsight states the resulting order: place one card before collecting the next.Source: §F.3 · pp. 40–43

F.4 ALL-SUCCESS GROUPS AND THE KNOWLEDGE CHANNEL

  • In an all-success group, reward uniformity makes group-relative advantages vanish and leaves PITFALL without a failure to diagnose; the blind KNOWLEDGE student predicts a 120° Law-of-Cosines/Fermat-point construction.All eight rollouts received reward 1, while KNOWLEDGE foresights sampled independently from the problem alone agree on that construction.Source: §F.4 · p. 43
  • The exact algebraic route yields 3√xyz = 56/13 and m + n = 69, and explicitly does not use the geometry.It derives relationships among the variables through pairwise subtraction and selects S = 13 by positivity.Source: §F.4 · p. 44
  • The three successful routes are symbolic elimination, numeric solving with rational reconstruction, and parameterizing then pruning; all reach 56/13, unlike the foresight-proposed area double-count.The area-based approach proposed in every foresight was not used by any of the three routes.Source: §F.4 · p. 44
  • Eight rollouts with identical zero advantages provide distinct demonstrations of what the problem admits, and LSRD targets each rollout’s own available route.The knowledge teacher is privileged with that rollout’s trajectory, pairing stable anticipation with different realizations.Source: §F.4 · p. 44

F.5 HOW EACH CHANNEL’S FORESIGHT CHANGES OVER TRAINING

Within mathematics, later foresights shift from isolated facts or generic advice toward ordered, problem-specific guidance.

F.5 HOW EACH CHANNEL’S FORESIGHT CHANGES OVER TRAINING

  • To isolate changes in prediction content, the comparison holds the domain to mathematics and contrasts early with late records within each channel.Source: §F.5 · p. 45
  • Early KNOWLEDGE foresights often stop at one transferable fact, such as transforming x ◦ y = x + y + xy into multiplication by shifting values by 1.Source: §F.5 · p. 45
  • Late KNOWLEDGE foresights compose Vieta, a discriminant constraint, and an optimum condition into a route where each fact licenses the next.The section contrasts this chain with an earlier example that states an identity and stops.Source: §F.5 · pp. 45–46
  • Early PITFALL foresights give general exam advice, whereas later ones identify traps tied to an expression’s structure, such as the n − 1 denominator factor.Source: §F.5 · p. 46
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