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Beyond Token Alignment: Event Completion for Cross-Tokenizer On-Policy Distillation

Jiacheng Liu, Jingwei Song, Qituan Zhang, Siheng Chen, Linfeng Zhang

arXiv:2609.34738v1cs.CLcs.AI

TL;DR

Cross-tokenizer OPD can leave a student at a partial teacher-token event, where valid next-token completions are known but their relative probabilities are not. ESCD aggregates related teacher events and supervises the total probability of byte-compatible completions. It reports consistent gains across tasks and model scales, including large-scale MoE distillation.

  • Problem

    After a student partially realizes a teacher token, multiple next tokens may satisfy the remaining byte constraint, but the teacher does not specify how probability should be divided among them.

    Source: §1 · p. 2

  • Method

    ESCD aggregates prefix-related teacher events and supervises the total probability of byte-compatible student completions without prescribing probabilities for individual completions.

    Source: §1 · p. 3

  • Results

    ESCD reports consistent gains across mathematics, code, and scientific reasoning, including gains of 3.0–5.1 points in large-scale MoE distillation.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The paper presents event entry and event completion as complementary supervision targets for cross-tokenizer knowledge transfer.

    Source: §1 · p. 3

    OPD performance and stability also depend on student capacity, initialization, training data, and rollout quality, so event-completion supervision is not a general remedy for OPD collapse.

    Source: §4.2 · p. 8

Abstract

from arXiv · show

On-policy distillation (OPD) transfers knowledge between language models through teacher supervision on student-generated trajectories. With different tokenizers, a single teacher token may require multiple student tokens to generate, creating intermediate states where the event is entered but not yet completed. Existing cross-tokenizer methods align tokens or text spans to construct comparable prediction targets. We study a complementary problem after partial generation: once the student produces a prefix of a teacher token, multiple next tokens may complete the same remaining bytes, but the teacher only specifies the required completion rather than how probability should be divided among these valid continuations. We introduce Event-Set Completion Distillation (ESCD), which complements cross-tokenizer probability alignment with completion-set supervision. ESCD aggregates prefix-related teacher events and supervises the total probability of byte-compatible one-step student completions, avoiding tokenizer-dependent probability splits among individual tokens. The method reuses student trajectories and predictions, requiring neither additional rollouts nor changes to the student vocabulary. Experiments demonstrate consistent gains in mathematics, code, and scientific reasoning across model families and tokenizers, extending to large-scale MoE distillation from a 1T teacher to a 35B student. Local analyses show that retaining completion sets better matches the reference supervision, while one-step completion covers over 99% of observed compatible teacher mass after partial event entry in the studied tokenizer pairs. These findings support event entry and event completion as complementary supervision targets for cross-tokenizer knowledge transfer. Code will be released on GitHub.

1 INTRODUCTION

Cross-tokenizer OPD must supervise a student state that may have entered a teacher token’s byte event without completing it. ESCD adds completion-set supervision to address that gap while leaving probability allocation among valid continuations unspecified.

1 INTRODUCTION

  • When tokenizers differ, a teacher token may span multiple student actions, unlike the direct next-token supervision available with a shared tokenizer.Figure 2 illustrates partial entry with the student token Br and residual bytes ead.Source: Figure 2 · p. 2
  • On-policy distillation obtains teacher feedback on student-generated trajectories, reducing mismatch between training contexts and generation states.This is the paper’s motivation for using student-induced states in autoregressive generation.Source: §1 · p. 2
  • A student may enter a teacher byte event with a partial token, leaving a residual constraint that does not specify relative probabilities among valid completions.The paper frames this as a target-design question: supervise the remaining byte constraint without prescribing allocation among individual student tokens.Source: §1 · pp. 2–3
  • ESCD aggregates prefix-related teacher events and uses their probability mass to supervise the total probability of byte-compatible student completions.It complements root-level alignment with visited-child completion supervision, without requiring counterfactual rollouts or changes to the student vocabulary.Source: §1 · pp. 2–3
  • ESCD improves performance across model families, task domains, and scales, including scientific-reasoning gains of 3.0–5.1 points in large-scale MoE distillation.In controlled experiments, it matches or exceeds the strongest compared baseline on every reported metric; Qwen-teacher pass@2 rises from 18.1% to 42.9% on LiveCodeBench and from 18.7% to 47.3% on TACO.Source: §1 · p. 3

2 RELATED WORK

The related work moves from on-policy distillation objectives and state-distribution concerns to methods for aligning supervision across tokenizers. It positions ESCD as targeting completion after partial event entry rather than only constructing comparable prediction targets.

2.1 DISTILLATION FOR LANGUAGE MODELS

2.2 CROSS-TOKENIZER DISTILLATION

  • Cross-tokenizer methods align tokens, spans, or distributions, while ESCD explicitly supervises total probability over valid one-step completions after partial event entry.It combines representative prefix constraints and their aggregated teacher mass with completion supervision at visited student child states.Source: §2.2 · p. 4

3 METHOD

ESCD extends byte-aligned root probability projection with residual-event completion supervision, aggregating teacher events and supervising the total probability of valid next-token completions.

3.1 FROM EVENT ENTRY TO EVENT COMPLETION

  • A sampled student token can leave residual bytes at a visited child state, where the remaining teacher event is not yet complete.For example, sampling Br leaves ead to complete the teacher event Bread.Source: §3.1 · p. 5, Figure 3 · p. 5
  • After a student token partially realizes a teacher event, ESCD supervises the aggregate probability of tokens that complete the residual bytes, not their individual probabilities.For residual ead, both ead and eads qualify as one-step completions; their probabilities are not assigned a prescribed split.Source: §3.1 · p. 5, §3.2 · p. 6

3.2 CONSTRUCTING COMPLETION SETS

  • Prefix-related teacher candidates are grouped under a shared representative prefix, and their combined teacher mass weights completion of that constraint.Aggregation preserves the group’s total teacher mass while coarsening longer members’ constraints to the shortest member’s prefix.Source: §3.2 · pp. 5–6

3.3 TRAINING OBJECTIVE

  • ESCD combines root alignment with child completion loss, reusing student predictions at visited states without additional rollouts.The root and child losses are summed under a common training reduction.Source: §3 · p. 4, §3.3 · p. 7

4 EXPERIMENTS

Experiments compare ESCD with baselines using Qwen3.5-2B under two teachers and extend evaluation to heterogeneous MoE pairs. ESCD improves across benchmarks, including large-scale transfer with a 1T teacher.

4.1 EXPERIMENT SETTINGS

  • The controlled experiments use Qwen3.5-2B as student with Qwen3-32B and GLM-Z1-9B teachers, evaluating mathematics and code generation.The teacher pairs differ in tokenizer, and the GLM pair also differs in model family.Source: §4.1 · p. 7

4.2 DISTILLATION PERFORMANCE

  • ESCD consistently outperforms representative baselines across both teacher settings, with particularly strong gains on code-generation benchmarks.With Qwen3-32B, gains reach +15.6/+24.8 points on LiveCodeBench and +18.5/+28.6 points on TACO; with GLM-Z1-9B, gains include +6.3/+9.4 on LiveCodeBench and +6.2/+6.0 on TACO.Source: §4.2 · p. 7
  • ESCD improves over BPM in heterogeneous MoE distillation with 397B and 1T teachers; the Kimi comparison uses the same SFT-initialized checkpoint.In the Kimi setting, ESCD raises FrontierScience Olympiad accuracy from 68.0% to 72.0% and PHYRD-40 mean score from 66.2 to 74.3.Source: §4.2 · pp. 7–8, Table 2 · p. 8
  • ESCD addresses tokenizer-boundary supervision, but OPD stability also depends on student capacity, initialization, training data, and rollout quality.The authors frame ESCD as complementary to initialization and optimization strategies, not a general remedy for OPD collapse.Source: §4.2 · p. 8

5 ABLATION AND MECHANISM ANALYSIS

The analysis compares local completion objectives against a byte-event reference and measures how often one-step child supervision is available on visited trajectories. Event better matches the reference than Root or Single in the examined contexts, while one-step completion covers nearly all visited residual constraints.

5 ABLATION AND MECHANISM ANALYSIS

  • Event sums probability across all valid one-step completions, whereas Single selects one valid completion and Root adds no child loss.The controlled comparison holds student predictions, teacher masses, and root supervision fixed.Source: §5 · p. 8
  • COUF rises from 0.7383 to 0.8495 for Qwen and from 0.8848 to 0.9341 for GLM with Event versus Root; Single scores below Root in both settings.The comparison evaluates local logit-gradient agreement, not downstream performance after independent training.Source: §5 · p. 9, Table 3 · p. 9
  • Boundary-crossing tokens account for 2.10% and 0.63% of completion probability; removing them leaves gradient cosine similarities of 0.9832 and 0.9887 to full-set gradients.The diagnostic covers naturally visited strict-prefix events with available child logits.Source: §5 · p. 9

5.1 ON-POLICY AVAILABILITY OF CHILD SUPERVISION

  • 81.83% and 83.29% of branchable teacher mass has compatible children visited; among visited mass, 99.01% and 99.43% admits one-step completion.The availability analysis uses full teacher vocabularies at aligned non-whitespace positions in 16 frozen student trajectories per pair.Source: §5.1 · p. 9, Table 4 · p. 9

6 CONCLUSION

The conclusion presents event entry and completion as complementary supervision targets, with reported gains across tasks and scales. The accompanying statements document the evaluation boundary, reproducibility materials, and implementation practices for cross-model training.

6 CONCLUSION

  • ESCD complements root-level alignment with completion-set supervision and shows gains across mathematics, code, and scientific reasoning, including MoE distillation from a 1T teacher to a 35B student.The conclusion also reports that one-step-completable residual constraints cover over 99% of observed compatible teacher mass.Source: §6 · p. 9

AI USE STATEMENT

  • The authors report using generative AI for language polishing, translation assistance, and limited code assistance, but not to generate results, fabricate data, or make final scientific claims.They state that AI-assisted material was manually reviewed, revised, and verified.Source: AI USE STATEMENT · p. 10

ETHICS STATEMENT

  • The evaluation focuses on reasoning accuracy and does not assess safety or bias; the authors recommend separate safety evaluation before deployment.The statement also notes that model-use licenses should be respected when distilling.Source: ETHICS STATEMENT · p. 10

REPRODUCIBILITY STATEMENT

  • The reproducibility materials include method and pipeline descriptions, training and evaluation details, and a planned release of code, prompts, problem sets, rubrics, scripts, and grading prompts.The authors point to the main paper and appendices for specific formulations, settings, and protocols.Source: REPRODUCIBILITY STATEMENT · p. 10

A TRAINING FRAMEWORK AND SETTINGS

  • The appendix organizes implementation details across rollout and optimization, model-native chat templates and special tokens, training settings and data, evaluation, and shared generation protocols.The training-framework overview identifies on-policy rollout, teacher serving, student optimization, and a unified baseline implementation.Source: A TRAINING FRAMEWORK AND SETTINGS · p. 14

A.1 TRAINING FRAMEWORK

  • Slime coordinates student rollout generation, retokenized-response teacher scoring, and distributed student optimization, updating only the student.Compared methods share data, rollout settings, and optimization budgets while implementing their own alignment objectives.Source: §A.1 · p. 14

A.2 CROSS-MODEL CHAT TEMPLATES AND SPECIAL-TOKEN HANDLING

  • Each model renders the structured conversation using its own native chat template, and model-specific special tokens are handled semantically rather than translated between tokenizers.The appendix warns that copying token IDs or serialized prompts can misalign role boundaries, duplicate termination symbols, or misrepresent truncation.Source: §A.2 · p. 14

A.2.1 SPECIAL-TOKEN HANDLING

  • Termination uses each model’s declared terminal semantics, with a model-specific bridge stop after retokenization; native thinking prefixes also vary by checkpoint.GLM-Z1-9B and Qwen3-32B do not prefill an opening <think>, unlike the other illustrated thinking configurations; generated reasoning markers are retokenized as response text.Source: §A.2.1 · pp. 14–15, Figure 4 · p. 15, Table 5 · p. 15

A.3 TRAINING SETTINGS

The appendix describes the model configurations used for small- and large-scale OPD experiments and notes that compared methods share training and optimization settings within each model pair.

A.3 TRAINING SETTINGS

  • Small-scale experiments use Qwen3.5-2B as student with Qwen3-32B or GLM-Z1-9B as teacher; large-scale experiments include two additional teacher–student pairs.For Kimi-K2.7-Code to Qwen3.6-35B-A3B, BPM and ESCD start from the same SFT-initialized student checkpoint.Source: §A.3 · p. 15, Table 6 · p. 16
  • Within each teacher–student pair, compared methods share training data, rollout budget, optimizer settings, and distributed-training configuration, differing in their cross-tokenizer objectives.GPU totals include teacher serving, student optimization, and student rollout, excluding separately scheduled evaluation jobs.Source: §A.3 · p. 16, Table 6 · p. 16

A.4 TRAINING DATA

Training data vary by scale: small-scale distillation uses mathematics and code prompts, while large-scale distillation uses SciDeriv prompts for scientific derivation reasoning. The examples illustrate endpoint assessment, reconstruction, and distinguishing entailment from consistency.

A.4 TRAINING DATA

  • Small-scale distillation uses 20,000 balanced prompts: 10,000 DAPO mathematics problems and 10,000 TACO programming problems.Large-scale SciDeriv contains 17,577 prompts for conclusion assessment and conclusion reconstruction, without stored reference completions or scalar reward labels.Source: §A.4 · pp. 16–17, Table 7 · p. 17, Table 8 · p. 17

Case 1: Endpoint exposed and verifiable

Case 2: Endpoint withheld and reconstructed

  • The endpoint-withheld example asks the model to reconstruct functional equations, trace their supporting intermediate relations, and identify whether additional conditions are needed for uniqueness.The corpus frames this setting as reconstructing the strongest conclusion justified by supplied relations.Source: Case 2: Endpoint withheld and… · pp. 17–18

Case 3: Consistent but not entailed

  • The differential-equation example asks whether a proposed endpoint is entailed, merely consistent, or inconsistent, and whether the derivation omits a special solution.The prompt also requests the strongest supported solution family and any additional condition needed to recover the endpoint.Source: Case 3: Consistent but not entailed · p. 18

A.5 BENCHMARKS AND EVALUATION PROTOCOL

The supplied sections specify benchmark tasks and generation protocols, then define cross-tokenizer events and quantify direct vocabulary overlap. They establish the visited-child setting that motivates later completion supervision.

A.5 BENCHMARKS AND EVALUATION PROTOCOL

  • Mathematics, code, and scientific reasoning are evaluated on fixed benchmark sets with protocols held constant across compared models.The benchmarks include AIME, HMMT, LiveCodeBench, TACO, FrontierScience Olympiad-100, and PHYRD-40.Source: §A.5 · pp. 18–19

A.6 COMMON GENERATION PROTOCOL

  • All models use matched decoding settings within each setting, while scientific reasoning receives higher temperature and larger context and output limits.Mathematics and code use temperature 0.6; scientific reasoning uses 1.0. Limits are 32,768/27,648 tokens for math and code, and 262,144/256,000 for scientific reasoning.Source: §A.6 · p. 19, Table 9 · p. 19

Mathematical reasoning prompt (used for training and evaluation)

PHYRD-40 prompt (used for evaluation)

B CROSS-TOKENIZER DISTILLATION UNDER A UNIFIED VIEW

  • The section organizes cross-tokenizer distillation around vocabulary correspondence, supervision objects and states, ESCD, a worked example, and a training algorithm.Its basic definitions introduce byte realizations, shared and non-shared tokens, and root and visited child states.Source: B CROSS-TOKENIZER DISTILLATION UNDER A… · p. 21

B.1 BASIC DEFINITIONS

  • Shared tokens support direct coordinate remapping, whereas non-shared tokens are outside that one-to-one mapping and are not necessarily without a single-token byte counterpart.ESCD uses a separate byte-based criterion to select child-event candidates.Source: §B.1 · p. 22
  • ESCD can supervise the unfinished remainder of a Teacher event after the Student naturally emits a compatible prefix, at the visited child state.The sampled Student action determines the child state and the remaining bytes; this does not require a counterfactual branch.Source: §B.1 · p. 22

B.2 HOW MUCH OF THE VOCABULARY IS DIRECTLY SHARED?

  • 53.05–94.24% of the Teacher vocabulary is covered by shared coordinates across evaluated directions, leaving a non-shared subset for explicit treatment.Table 10 reports static overlap under the deterministic one-to-one canonical-surface mapping.Source: §B.2 · p. 23, Table 10 · p. 23

B.3 EXISTING METHODS UNDER THE UNIFIED VIEW

Existing methods construct cross-tokenizer supervision through aligned coordinates, spans, ranks, or token targets. ESCD adds an explicit loss on sets of valid residual completions at naturally visited Student states.

B.3 EXISTING METHODS UNDER THE UNIFIED VIEW

  • ULD transfers distributional shape by matching ranked probability magnitudes, without preserving textual correspondence between coordinates.Its rank-matching term does not itself resolve mismatched text boundaries.Source: §B.3 · pp. 23–24
  • GOLD combines sequence alignment and merging with shared-coordinate matching and rank-based comparison of unmatched coordinates.Its hybrid matching objective differs from an auxiliary loss over valid completions of a specified residual byte event.Source: §B.3 · p. 24
  • X-Token projects vocabulary distributions over aligned chunks, while BPM maps Teacher mass to Student-token targets through byte prefixes, including at positions inside a Teacher token.BPM’s targets do not sum probability over all Student tokenization paths; ESCD adds aggregate residual-completion supervision.Source: §B.3 · pp. 24–25
  • SimCT normalizes length-adjusted scores over shared tokens and minimal aligned continuation units, rather than preserving native next-token probability mass.Its candidate continuations can use intermediate states not visited by the actual rollout.Source: §B.3 · pp. 24–25
  • ESCD’s distinction is an explicit auxiliary set-probability loss for valid residual completions at a naturally visited child state, not simply supervision at later Student positions.Existing distribution-matching objectives may constrain aggregate probabilities indirectly.Source: §B.3 · p. 25

B.4 ESCD: EVENT-SET COMPLETION DISTILLATION

  • ESCD groups prefix-related Teacher events under their shortest representative constraint, then sums native Student probabilities for tokens beginning with the residual bytes.The completion set may include tokens extending past the representative boundary, and probabilities are not renormalized over the set.Source: §B.4 · p. 27
  • The child loss weights each residual-completion event by its retained root Teacher mass and requires no additional Teacher query.Groups without a valid one-step completion contribute no child loss.Source: §B.4 · p. 28
  • ESCD supervises representative residual constraints, only when a valid one-token completion exists; sparse candidate selection can change the supervision itself.It does not preserve every original Teacher token’s full byte requirement.Source: §B.4 · p. 29

B.5 A WORKED EXAMPLE OF CROSS-TOKENIZER SUPERVISION

A worked example contrasts how SimCT, BPM, and ESCD treat cross-tokenizer continuations after the Student emits a shared prefix. The accompanying algorithm reuses existing rollouts and predictions, while the additional analyses examine event coverage and behavior across models.

B.5 A WORKED EXAMPLE OF CROSS-TOKENIZER SUPERVISION

  • ESCD groups Bread and Breads under the representative Bread constraint, while BPM retains separate residual targets and their Teacher preference.The completion set avoids prescribing an allocation among valid Student tokens, though those tokens need not represent equivalent full continuations.Source: §B.5 · pp. 29, 31
  • Halving Student probabilities leaves the local SimCT loss unchanged but raises ESCD’s child loss from 1.1067 to 1.7305.SimCT’s normalized candidate distribution is unchanged even as native completion probability falls.Source: §B.5 · pp. 30–32
  • After Br, BPM conditions the compatible Teacher mass of 0.9 and assigns distinct targets to ead, eads, and eak.The resulting targets are 4/9, 2/9, and 3/9, derived from the root prediction rather than a new child-state Teacher query.Source: §B.5 · p. 30
  • The example’s root and interior losses sum to approximately 1.1492 before the common response-token reduction, rather than being averaged separately by position type.This calculation describes the illustrated BPM example.Source: §B.5 · p. 31
  • 0.5 to 0.6 completion probability lowers ESCD’s child loss from 1.1067 to 0.9973, while BPM’s loss increases as the allocation moves further from Teacher token preferences.The change reallocates probability between ead and eads while keeping eak fixed.Source: §B.5 · p. 32

B.6 TRAINING ALGORITHM

  • ESCD reuses the Student rollout, training forward pass, and aligned Teacher outputs, adding sparse completion-set operations at child states already visited.The computation accesses selected Student log-probabilities without dense cross-vocabulary child targets or additional model execution.Source: §B.6 · p. 32

C ADDITIONAL ANALYSIS OF ESCD

  • The additional analyses examine event structures, boundary-crossing completions, one-step coverage, reasoning patterns, OPD degradation, and SFT-associated changes.They also assess comparisons of naturally visited and unvisited child states.Source: C ADDITIONAL ANALYSIS OF ESCD · p. 33

C.1 DOES THE CHOICE OF CHILD STATE MATTER?

The analysis compares supervision from sampled and counterfactual child states, but does not establish which is preferable. It also characterizes the event structures encountered and finds high one-step completion availability in the analyzed mass.

C.1 DOES THE CHOICE OF CHILD STATE MATTER?

  • Only 32 Qwen and 37 GLM contexts could be matched, and none allowed both branches to complete the same event in one step.Thus, the available data cannot simultaneously control event identity and one-step completion depth.Source: §C.1 · p. 33, Table 13 · p. 34
  • The counterfactual branch has higher COUF than the visited branch in both model pairs, but the comparison also allows deeper completion.The relaxed comparison does not control completion depth, so the difference cannot isolate child-state choice.Source: Table 14 · p. 34, §C.1 · p. 34
  • ESCD uses visited child states because their logits are already available from the training forward pass, avoiding additional rollouts.The analysis does not establish an independent supervision advantage over other compatible states.Source: §C.1 · p. 34

C.2 WHAT EVENT STRUCTURES DOES ESCD ENCOUNTER?

  • Root-compatible first actions and child-state completions are distinct counts: a single compatible first action can leave zero, one, or multiple valid completions.The analysis measures root choices separately from one-step completion availability after the sampled action.Source: §C.2 · p. 34
  • 99.01% for Qwen and 99.43% for GLM is the overall conditional coverage with at least one valid one-step completion.Groups with multiple teacher candidates but one compatible student first token account for 88.21% of Qwen mass and 90.10% of GLM mass.Source: Table 15 · p. 35, §C.2 · p. 35

C.3 DOES THE COMPLETION SET BECOME TOO COARSE AT THE EVENT BOUNDARY?

The completion-set objective accepts tokens that complete the residual even when they extend beyond its boundary, leaving their within-set allocation unidentified. In the analyzed subsets, crossing probability is small and removing crossing completions leaves local update directions highly aligned with the full target.

C.3 DOES THE COMPLETION SET BECOME TOO COARSE AT THE EVENT BOUNDARY?

  • Boundary-crossing tokens can extend beyond the residual without being treated as semantically equivalent to exact completions.The objective supervises the byte-prefix event of completing the residual, not termination at the same token boundary.Source: §C.3 · p. 35
  • 2.10% for Qwen and 0.63% for GLM is the boundary-crossing share of completion probability, despite Qwen’s average completion-set size exceeding 35.The crossing share is measured on naturally visited strict-prefix events with available child logits.Source: §C.3 · p. 35
  • 0.9832 for Qwen and 0.9887 for GLM is the gradient cosine between Exact/Minimal and Full targets in the frozen natural-child subsets.Every evaluated event has an exact one-step completion, making Minimal and Exact identical in this diagnostic.Source: Table 17 · p. 36, §C.3 · p. 36
  • Redistributing probability among tokens in the same completion set leaves the child loss unchanged when their total probability stays fixed.The objective identifies completion-event probability, not its allocation among valid student realizations.Source: §C.3 · p. 36
  • The frozen analysis treats boundary crossing as local target coarsening, not an observed failure; testing stricter targets requires native runs under a common protocol.Events without exact one-step completion are needed to distinguish Minimal from Exact.Source: §C.3 · p. 36

C.4 HOW OFTEN IS CHILD SUPERVISION AVAILABLE, AND IS ONE STEP ENOUGH?

In the studied tokenizer pairs, most visited representative residuals admit one-step completion, while compatible-child visitation covers over 80% of branchable teacher mass. The section also presents PCA projections as a qualitative view of response-level reasoning variation across benchmarks and models.

C.4 HOW OFTEN IS CHILD SUPERVISION AVAILABLE, AND IS ONE STEP ENOUGH?

  • Root adds no child loss, Single supervises one selected valid completion, and Event supervises the completion set’s total probability.Single and Event differ in their supervision targets, not in completion depth.Source: Figure 6 · p. 37, §C.4 · p. 37
  • 81.83% for Qwen and 83.29% for GLM is the branchable teacher mass with a naturally visited compatible child.This measures visitation, not whether the next sampled token completes the event.Source: Table 18 · p. 37, §C.4 · p. 37
  • 99.01% for Qwen and 99.43% for GLM is the visited mass whose representative residuals admit completion by one additional student token.The remaining 0.99% and 0.57% lack one-step completion and contribute no auxiliary child loss under ESCD.Source: Table 19 · p. 38, §C.4 · p. 38
  • The one-step coverage result is scoped to the studied tokenizer pairs and trajectories; it does not establish completion of every original teacher token or longer group members’ full byte requirements.Residuals without one-step completion remain a small but nonzero share of observed mass.Source: §C.4 · p. 38

C.5 ROLLOUT-LEVEL REASONING GEOMETRY

  • PCA projections qualitatively show response-level reasoning variation across Math, Code, PHYRD-40, and FrontierScience, with distances comparable only within each panel.PCA is fitted separately in each panel.Source: Figure 7 · p. 38, §C.5 · p. 38

C.6 OPD DEGRADATION UNDER LARGE TEACHER–STUDENT SCALE DISPARITY

The appendix documents instability during direct Kimi-to-Qwen OPD, reports stronger benchmark performance after SFT, and describes qualitative differences in derivation organization. Its summary also reports that completion-set targets better match local reference supervision and that one-step completion covers over 99% of observed compatible teacher mass in the studied tokenizer pairs.

Repetitive continuation and premature termination

  • At steps 20–21, repetitive, length-truncated responses coexist with premature termination, including one response that ends after 12 tokens without substantive reasoning.Three responses reach the 80,000-token limit; their repetitions include the symbol Z, a repeated phrase, or malformed LaTeX fragments.Source: §C.6 · p. 39

C.6 OPD DEGRADATION UNDER LARGE TEACHER–STUDENT SCALE DISPARITY

  • 10,587 to 8 tokens: mean response length falls from step 16 to step 32, when all 16 retained responses contain at most 20 tokens and fail to provide substantive solutions.The shortening reflects termination before substantive reasoning, not more concise solutions.Source: §C.6 · p. 39, Repetitive continuation and premature… · p. 39

C.7 REASONING PERFORMANCE AND STRUCTURE AFTER SFT

  • SFT raises the student’s PHYRD-40 mean rubric score from 59.2 to 66.2 and FrontierScience Olympiad accuracy from 61.0% to 68.0%.The SFT checkpoint is described as a more capable starting point for subsequent OPD.Source: §C.7 · p. 40

C.7.1 MATCHED-PROMPT REASONING PATTERNS

  • Across three matched prompts, SFT responses more consistently make normalization choices, boundary conditions, limiting cases, and local checks explicit before ending.The examples are qualitative and show no uniform change in the mathematical endpoint.Source: §C.7.1 · p. 42

C.8 SUMMARY

  • Over 99% of observed compatible teacher mass is covered by one-step completion in the studied tokenizer pairs, while local gradient comparisons favor event-set over single-completion targets.The analyses also report practical availability of visited-child supervision.Source: §C.8 · p. 43

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