Same Beliefs, Different Policies: Learning Reusable Representations for Early Stopping in LLM Reasoning

1 INTRODUCTION

• For a frozen reasoner and decoding rule, token price changes how outcomes are valued but not the prediction targets, allowing one predictive representation to support different prices.BELT fits a linear stopping policy on frozen features, either by refitting for a price or shifting a threshold on one fitted direction.Source: §1 · pp. 1–2
• BELT jointly predicts four outcomes—current correctness, natural-completion correctness, whether a later answer can be correct, and agreement with the natural-completion answer—to build beliefs for stopping control.Later includes a correct answer at a later boundary even if the final answer is wrong; Same measures answer agreement regardless of correctness.Source: §1 · p. 2
• Across five OOD math benchmarks, BELT saves 41% of generated tokens on Qwen3 and 51% on OLMo3 without loss of accuracy, and its math-trained probes transfer to coding and agentic tasks.The reported transfers require no code- or agent-training data; reusing one probe across five prices reduces OLMo3 controller training by approximately 55× in GPU-hours.Source: §1 · p. 2
• Belief-trained features outperform raw-state controls and end-to-end stopping-controller features in OOD utility at all five prices; Later and Same each improve stopping beyond Now and End.Policy diagnostics indicate that adapting to a new price reweights the same beliefs.Source: §1 · p. 2

2 BACKGROUND AND PROBLEM FORMULATION

• At each candidate boundary, a controller observes the hidden states up to that point and chooses whether to stop and answer or continue toward natural completion.Stopping yields an answer, its correctness, and total token cost; natural completion supplies the final answer, correctness, and cost.Source: §2 · pp. 2–3
• For token price λ, utility is correctness minus λ times token cost, and the objective maximizes expected utility over trajectories and stopping decisions.The price values correctness against computation without imposing a hard token budget.Source: §2 · p. 3
• Direct utility training jointly learns a representation and stopping head for a token price and current policy, so adaptation to a new price can require another controller-training run.Rarely reached or saturated boundaries can provide weak learning signals despite available outcome labels.Source: §2 · p. 3
• BELT instead probes a frozen LLM’s internal states for outcome probabilities conditioned on the observed prefix, separating price-independent prediction targets from price-dependent stopping decisions.Its outcome predictions remain unchanged across prices for a fixed reasoner and decoding rule.Source: §2 · p. 3

3 BELT: BELIEF TRAINING AND POLICY FITTING

• BELT predicts outcome labels independently of token price, then uses the frozen probe features to fit stopping decisions for the price-dependent accuracy-cost trade-off.The training targets are outcome labels, while policy fitting optimizes the recorded-trajectory utility.Source: §3 · pp. 3–4
• Now and End predict correctness of the current and natural-completion answers; Later predicts whether any later boundary can offer a correct answer, and Same predicts agreement with the natural-completion answer.Later can be positive even when the final answer is wrong; Same is independent of correctness and agreement at intervening boundaries.Source: §3 · p. 3
• The probe predicts the four outcome probabilities from observed prefixes, then its features are frozen for linear policy fitting on trajectories with recorded correctness and token costs.The B interface uses the four beliefs and observed prefix length; H uses their shared representation, while the OLMo3 B interface omits prefix length.Source: §3 · pp. 3–4
• Refit variants solve the stopping objective at each price, whereas threshold variants fit once at a reference price and vary the threshold on a hard stopping rule.Both variants reuse the belief encoder; B/H identify the feature interface, and R/T identify refit/threshold control.Source: §3 · p. 4

4 EXPERIMENTAL SETUP

• Probe training uses three math datasets, with 3,918 Qwen3 training questions and up to eight trajectories per question, while OLMo3 uses up to four.A separate 320-question bank selects the probe checkpoint, and disjoint 160-question halves fit policies and select hyperparameters.Source: §4 · pp. 4–5
• Evaluation covers held-out math and five OOD math benchmarks, LiveCodeBench coding, and OLMo3 agent tasks in ALFWorld and ScienceWorld.The same math-trained controllers are tested on coding without code-side training and on per-turn agent thinking.Source: §4 · p. 5
• The study compares test-frontier accuracy at fixed token-saving levels and maximum savings at fixed accuracy tolerances, alongside dev-selected test utility.Math natural-completion grading adds one seeded answer request of at most 32 tokens for closed trajectories without a parseable answer; all results use this corrected protocol.Source: §4 · p. 5
• BELT is evaluated on Qwen3-1.7B and OLMo-3-7B-Think against fixed caps, training-free controls, correctness or stability controls, and OSP variants.OSP-1 is the single-encoder comparator with one checkpoint and 13 score-quantile thresholds; full OSP uses five price-specific checkpoints.Source: §4 · p. 5

Qwen3 math

• 50.64% token saving on Qwen3 ID and 45.98% on OOD at a one-point accuracy tolerance exceed OSP-1 and are within seed variation of full OSP.The corresponding OSP-1 savings are 46.59% and 36.89%; full OSP gives 49.72% and 45.19%.Source: §5 · p. 5, Table 1 · p. 6

Math-to-code transfer

• 29.91% and 43.78% token savings on OLMo3 LiveCodeBench before and from August 2024 exceed OSP-1’s 21.94% and 35.35%.The same math-trained probe transfers without code-side training.Source: §5 · p. 6

OLMo3 math

• 54.09% token saving at a one-point tolerance on OLMo3 exceeds full OSP’s 49.86% and OSP-1’s 47.54%.At natural accuracy, BELT saves 51.22% against 45.92% for full OSP and 39.83% for OSP-1.Source: §5 · p. 5, Table 3 · p. 7

Math-to-code transfer

• 38.25% token saving on Qwen3 LiveCodeBench Pre matches full OSP and exceeds OSP-1’s 15.25%, using one encoder instead of five.The probes and policies transfer from mathematics without code-side training.Source: §5 · p. 6, Table 3 · p. 7

Recommended configuration

• H-T is the recommended configuration: one fitted direction with price thresholds matches or exceeds full OSP in 40 of 48 feasible readings.It exceeds OSP-1 in 35 of 49 readings; refit with threshold expansion adds at least one percentage point in every B-interface math reading but only 3 of 35 H-interface cells.Source: §5 · p. 8

6.1 BELIEF-TRAINED REPRESENTATIONS FOR STOPPING

• BELT features yield higher OOD stopping utility than raw-state and other fixed-feature controls at all five prices, including matched OSP-1 features.A linear probe trained on the four beliefs also beats every fixed-feature control at all five prices.Source: §6.1 · p. 8

6.2 TRANSFER TO AGENTIC TASKS

• In agent tasks, every BELT family saves 85% of ScienceWorld tokens at or above natural score; in ALFWorld, refit H has the highest mean at every reported reading.The transferred OLMo3 math probes, fitted policies, and OSP-1 checkpoint are used unchanged; thinking raises ALFWorld success from 4.5% to 33.6%.Source: §6.2 · p. 8

6.3 NECESSITY OF THE Later AND Same BELIEFS

• Adding Same to Now/End raises utility at all five prices, while removing it from the fixed four-belief probe lowers utility by 0.0005–0.0068.Each comparison gains 0.006–0.007 at λ = 2 × 10−5; Same marks prefixes whose current answer matches the natural-completion answer.Source: §6.3 · p. 9
• Adding Later as a prediction target raises token saving by 1.7–2.0 pp at all three accuracy tolerances; with it as an input, the full set leads on six of seven constrained readings.Later represents a correct answer becoming available before natural completion.Source: §6.3 · p. 9

6.4 INTERPRETING THE BELT-B-R LOGISTIC POLICY

• As price rises, BELT-B-R shifts weight from prefix length to Same and End, stopping when End is low relative to Now and continuing when End is high.Later becomes negative at the highest prices, while Now remains slightly positive; the policy reweights the same four beliefs.Source: §6.4 · p. 9

6.5 REFIT AND THRESHOLD COVER DIFFERENT OPERATING POINTS

• On the H interface, refit and threshold policies are within 1 pp at most readings; on B, refitting improves math saving or accuracy by at least 1 pp on every seed.Adding thresholds to H-R raises OOD one-point saving from 45.25% to 46.75%.Source: §6.5 · pp. 9–10

6.6 WHY DIRECT UTILITY TRAINING COLLAPSES

• Direct stopping-utility training collapses to never-stop or always-stop checkpoints with OOD utility 0.6269 or 0.1814, while BELT reaches 0.6689.The utility gradient vanishes as the policy collapses; BELT's label residual remains large when a belief prediction is wrong.Source: §6.6 · p. 10

6.7 TRAINING COST AND REUSE ACROSS TOKEN PRICES

• On OLMo3, BELT’s probe has 21.2× fewer parameters and needs approximately 55× fewer training GPU-hours than five OSP checkpoints.On GH200, training requires approximately 11× fewer GPU-hours than one OSP checkpoint; the probe has 21.2× fewer parameters than one checkpoint.Source: §6.7 · p. 10

• Prior early-stopping methods use answer confidence, convergence, or hidden-state overthinking detectors, while OS-Pruner jointly trains a stopping representation at a chosen token price.Thought Calibration learns correctness or final-answer agreement scores and separately calibrates thresholds with statistical risk control.Source: §7 · pp. 10–11
• BELT builds on ReProbe to control stopping within one trajectory and reuses its belief-trained representation across token prices.The paper compares representations and distinguishes target retraining from fixed-probe input ablations.Source: §7 · p. 11
• Other predictive-control approaches estimate intermediate correctness, future tokens, reward and remaining cost, or recovery from cumulative response sets.Examples include ReProbe, cross-regressive decoding, ZIP-RC, and AERA.Source: §7 · p. 11

• BELT separates learning what further reasoning could achieve from deciding whether its cost is worthwhile.One belief probe supports different token prices through a small CPU policy fit or threshold shift; its features exceed raw-state controls and matched OSP-1 features in OOD mean utility.Source: §8 · p. 11
• At a one-point tolerance, BELT saves 45.98% of Qwen3 OOD tokens, matching OSP’s 45.19% with one encoder instead of five.On OLMo3 it saves 4.23 pp more than full OSP and 6.55 pp more than OSP-1; on Qwen3 LiveCodeBench Pre it matches full OSP without code-side training.Source: §8 · p. 11
• The OLMo3 math probe saves 85% of ScienceWorld tokens at or above natural score, and one fitted direction with price thresholds saves more than OSP-1 at all 21 tolerance readings.The 21 readings cover seven math and coding settings.Source: §8 · p. 11

• The grading parser strips whitespace, dollar delimiters, outer boxed wrappers, and trailing periods, then accepts only an optional minus sign followed by digits.Fractions, symbolic expressions, decimals, and comma-separated numerals fail; empty, over-2,000-character, and specified image-markup questions are excluded.Source: §A.1 · p. 16
• Admission depends on reference format rather than method performance, and missing method coverage does not change the shared roster.Methods share trajectories, strict grading, token accounting, and operating-point constraints.Source: §A.1 · p. 16
• The ID bank admits two disjoint batches of 1,500 DAPO-Math, 1,000 NuminaMath, and 1,000 DeepScaleR questions, split 60%/15%/25% into fitting, selection, and reporting sets.The reporting portions form the ID set after prior-use exclusions, normalized-text deduplication, and overlap screening.Source: §A.1 · p. 16
• After integer and figure filtering, MATH-500 keeps 295 questions and Gaokao keeps 191; OlympiadBench keeps 366 single-answer candidates and excludes 93 multi-answer entries.Some AIME-2025 and AMC-2023 records lack identities for exhaustive semantic deduplication.Source: §A.1 · p. 16
• TC, ROM, CoDE-Stop, and DEER use adapted released implementations, while OSP was implemented from its published architecture and expected-utility objective.The reproduction’s performance is checked separately.Source: §A.1 · p. 16
• Full OSP uses five Qwen3 checkpoints per run, while BELT refit variants provide 17 policies and threshold variants use 13 thresholds on one fitted policy.BELT and OSP use three independent training runs; the comparison grid also includes 13 OSP-1 threshold policies.Source: §A.1 · p. 17

A.2 CORRECTION OF NATURAL-COMPLETION GRADING

• For closed math chains without a parseable final answer, a seeded boxed-answer request can recover an answer; results in this version use the corrected grading.Recovered answers set terminal correctness and agreement labels and affect End, Later, and Same supervision; LiveCodeBench grading is unchanged.Source: §A.2 · p. 17
• The corrected natural-completion outcomes change some training labels, and several submitted headline claims are withdrawn because they do not hold under the correction.In Qwen3 fit data, End changes for 1,853 of 39,035 chains (4.75%); in OLMo3 fit data, it changes for 62 of 15,672 (0.40%).Source: §A.2 · pp. 18–19

A.4 TRAINING TRAJECTORIES AND BELIEF LABELS

• BELT probes are trained on outcome labels, then frozen while linear stopping policies are fitted; exported probe features are reused across token prices.The OLMo3 replication trains a new layer-18 probe rather than transferring Qwen3 weights; BELT policy variants use refitting or thresholds on a fitted direction.Source: §A.4 · p. 20, §A.5 · p. 20, §A.6 · p. 22

A.6 EFFICIENT PROBE PROCESSING FOR LONG TRAJECTORIES

• The appendix specifies pinned model revisions and hardware, and the long-trajectory procedure reuses frozen reasoner features and exported BELT features.Trajectory batches are right-padded; the appendix lists NVIDIA GH200, RTX 4090, and RTX PRO 6000 GPUs.Source: §A.3 · pp. 19–20, §A.6 · pp. 21–22, §A.9 · p. 22

A.7 METRICS, AGGREGATION AND UNCERTAINTY

• Accuracy and generated-token cost define test frontiers, while constrained readings report the cheapest point within an accuracy tolerance or accuracy at a token-saving target.The appendix also specifies aggregation and uncertainty conventions, and notes that validation supports selection while test data determine frontier readings after families are frozen.Source: §A.8 · p. 22, §A.7 · p. 22

A.11 MATH TO CODE DOMAIN TRANSFER

• On LiveCodeBench, math-trained controllers are evaluated on code without code-side training, using unit tests for forced and natural programs.The Pre set may overlap Qwen3’s training period; both Pre and Post sets precede OLMo3’s data cutoff.Source: §A.11 · p. 23

A.12 PER-TURN THINKING CONTROL IN ALFWORLD

• In ALFWorld, math-trained controllers are calibrated on seen natural trajectories without reading outcomes, then evaluated on 134 unseen-split games.Calibration targets projected savings of 20%, 35%, 50%, and 65%; realized savings are within 6.1 percentage points of those targets.Source: §A.12 · pp. 23–24

A.13 PER-TURN THINKING CONTROL IN SCIENCEWORLD

• ScienceWorld evaluation uses 90 test games, with calibration on development trajectories and accuracy measured by simulator progress.The 95% saving target exceeds the calibration-split ceiling, and realized savings fall 3.2–20.0 percentage points short of targets; the reported operating range is 50–80% saving.Source: §A.13 · p. 24

B DETAILED RESULTS BY DATASET AND DISTRIBUTION

• Across 49 readings in seven settings, fixed-direction H exceeds OSP-1 in 35, ties in 10, and trails in 4; refit H exceeds it in 38 and trails in none.The settings include Qwen3 ID/OOD, LiveCodeBench Pre/Post, OLMo3 mathematics, and both OLMo3 LiveCodeBench sets.Source: B DETAILED RESULTS BY DATASET AND… · p. 25
• Against full OSP, fixed-direction H matches or exceeds it in 40 of 48 feasible readings, while refit B wins 11, ties 30, and trails in 7.The OSP reading at 70% saving on Qwen3 Pre is infeasible and excluded; a tie means the difference of seed means is within the larger seed SD.Source: B DETAILED RESULTS BY DATASET AND… · p. 25
• BELT uses one approximately 16M-parameter probe rather than five approximately 101M-parameter OSP checkpoints; the archived reference schedules estimate 5.1× fewer training GPU-hours.The GPU-hour estimate is 10.0 versus 50.9; per-request OSP executes only one selected checkpoint.Source: B DETAILED RESULTS BY DATASET AND… · p. 25

B.1 PER-DATASET RESULTS

• The per-dataset tables report token saving at accuracy tolerances and accuracy at token-saving budgets, with ID, OOD, and pooled panels.The OLMo3 tables retain both micro and macro aggregation schemes for token-saving and accuracy comparisons.Source: Table 14 · p. 25, Table 15 · p. 26, Table 16 · p. 26

B.2 OLMO3 MICRO- AND MACRO-AVERAGE RESULTS

• The OLMo3 result tables separately report micro- and macro-average token savings and accuracy.The provided passages identify the table panels but do not state their cell values.Source: Table 28 · p. 32, Table 29 · p. 32, Table 30 · p. 33

B.3 LIVECODEBENCH POST RESULTS

• BELT-B-R exceeds OSP-1 at all seven LiveCodeBench Post readings; against full OSP, the closest BELT variant is within 0.45 pp at four budgets, while OSP saves 1.07–1.46 pp more at three accuracy tolerances.The tables report token saving at accuracy tolerances and accuracy at token budgets.Source: §B.3 · p. 34, Table 32 · p. 34, Table 33 · p. 34
• 32.68 ± 0.15% is BELT-B-R’s maximum accuracy across native-family seed frontiers, versus 30.05 ± 0.51% for full OSP, 28.81 ± 0.24% for OSP-1, and 28.67% for natural completion.Every B-R seed reaches 31% and 32% accuracy, with mean token savings of 25.53% and 12.75%, respectively; no native full-OSP seed reaches either accuracy.Source: §B.3 · p. 34, Figure 4 · p. 35

B.4 OLMO3 MATH-TO-CODE TRANSFER

• On LiveCodeBench, OLMo3’s math-trained fixed-direction H saves more tokens than OSP-1 at all three accuracy tolerances on both sets, without code-side training.Before August 2024, refit H exceeds full OSP at five readings; from August 2024, full OSP has the higher mean at the one-point tolerance than every BELT variant.Source: §B.4 · p. 35, Table 34 · p. 35, Table 35 · p. 36

B.5 ALFWORLD RESULTS

• 12.4 ± 13.9% token savings at natural success and 25.4 ± 13.0% within two points: ALFWorld BELT-B-T exceeds OSP-1’s 0.0% and 11.5%, respectively.Across 20–50% savings, BELT-H-T’s mean success is within 0.9 pp of OSP-1’s; three BELT members match or exceed natural success with fewer tokens. At 65% projected savings, BELT variants and OSP-1 keep the natural action at 40–42% of stops and change it at 21.2–22.2 stops per game.Source: §B.5 · p. 36, Figure 6 · p. 37, Figure 5 · p. 37

B.6 SCIENCEWORLD RESULTS

• Every ScienceWorld controller member scores at or above natural completion in 60 of 63 cases, and every BELT family remains within 1.2 points of OSP-1 at each budget.All paired intervals against OSP-1 include zero; at 50%, 60%, and 70% savings BELT-H-R has the highest score, while BELT-H-T does at 80%.Source: §B.6 · p. 38, Table 40 · p. 39, Table 41 · p. 39

B.7 UTILITY OF DEV-SELECTED POLICIES

• BELT-B-R has higher development-selected utility than OSP at the three lowest math OOD prices and two lowest ID prices, but lower utility at the other prices.On OLMo3, BELT-B-R exceeds full OSP at all five prices; both BELT interfaces exceed OSP-1 at every reported Qwen3 OOD and OLMo3 math price.Source: §B.7 · p. 39, Table 42 · p. 40

B.8 INFERENCE-TIME THRESHOLD SWEEPS

• BELT exceeds OSP+T in 17 of 28 constrained readings after threshold expansion, with all seven accuracy deficits below 1 pp and the largest at 0.54 pp.On Qwen3 OOD, BELT-H-T saves 45.98% versus BELT-H-R’s 45.25% at the one-point tolerance; OSP-1+T leaves 14 of 15 tolerance-saving means unchanged.Source: §B.8 · p. 41, Table 43 · p. 42, Table 44 · p. 42

B.9 BUDGET CONTROL WITHOUT POLICY MIXTURES

• Selecting one feasible policy without interpolation can leave budget unused, whereas a mixture can reach costs between adjacent frontier members.Exports retain realized costs for both approaches.Source: §B.9 · p. 45

B.10 IN-DISTRIBUTION ACCURACY–COST TRADE-OFFS

• Qwen3 ID frontiers enlarge the central accuracy–cost trade-off in an inset.Source: Figure 7 · p. 45

B.11 ACCURACY–COST TRADE-OFFS BY DATASET

• Dataset-level frontiers use different axes across panels, with each inset enlarging the 38–52% token-saving range.Source: Figure 8 · p. 46

B.12 OLMO3 ACCURACY–TOKEN COST TRADE-OFFS

• The OLMo3 frontiers show operating points from Table 3 evaluated on each method’s test frontier.Source: §B.12 · p. 47, Figure 9 · p. 47

C.1 FEATURES, POLICY CAPACITY, AND BELIEF ABLATIONS

• Removing Same lowers mean utility at all five prices by 0.00052–0.00684, while MLP policies show no uniform gain over linear policies.Adding Later to Now/End lowers utility at four of five prices but raises it at the highest; in constrained readings, its effect is small and constraint-dependent. Adding Same improves all seven constrained means from Now/End and Now/End/Later.Source: §C.1 · pp. 48–49, Table 52 · p. 49, Table 51 · p. 49

C.2 LINEAR BELIEF PROBE ACCURACY–COST TRADE-OFFS

• BELT-L has the highest mean utility at the three lowest prices, but its advantage does not extend across all accuracy–cost constraints.At a one-point tolerance, it saves less than H-R on Qwen3 ID and OOD and less than OSP on Post, while its Code Pre accuracy exceeds the primary methods’ at 70% and 60% saving. Its advantage is therefore concentrated in low-price OOD utility and two code budgets.Source: §C.2 · p. 51, Table 55 · p. 51

C.3 MATCHED FEATURE COMPARISON AND TRANSFER

• BELT-H-R exceeds OSP-1 features at all five OOD prices, by 0.0032–0.0094, and at all five LiveCodeBench Pre prices under both fitting budgets.BELT-B-R also exceeds OSP-1 features on OOD under both budgets; on LiveCodeBench Post, H-R leads at the four lower prices with the main fit but trails at the highest. No code-side fitting or selection was used.Source: §C.3 · p. 52, Table 57 · p. 53, Table 58 · p. 53

C.4 FIXED-DIRECTION VERSUS PRICE-SPECIFIC POLICIES

• Adding thresholds to native H-R policies raises Qwen3 OOD one-point savings from 46.82% to 48.44%, above H-T’s 47.83%.Across random-initialized probes, refit plus thresholds beats the threshold family by at least 1 pp on all three seeds in 21 of 70 readings, while the reverse occurs in none. The comparison also changes deterministic versus stochastic stopping, and threshold-family advantages vary by interface and task.Source: §C.4 · pp. 55–56, Table 63 · p. 56

C.5 BELT PROBE CAPACITY AND INPUT SUBSETS

• At λ = 10^-5, adding prefix length to Now raises mean utility from 0.6766 to 0.7050, and the complete input set raises it to 0.7072.On Qwen3 OOD, BELT’s development-selected utility exceeds Now + length at all five prices by 0.0014–0.0316. The added benefit of beliefs depends on price and input combination.Source: §C.5 · pp. 56–57, Table 65 · p. 57
• Against Now + length, BELT saves 13.29, 17.49, and 9.21 pp more at a one-point tolerance on Qwen3 OOD, Qwen3 ID, and OLMo3, respectively.On Qwen3 OOD, BELT also exceeds the dense cap at every constrained reading and price; its one-point saving advantage is 24.63 pp. On LiveCodeBench Pre it is competitive with Now + length but leads OSP at 6 of 6 jointly feasible readings; on Post it saves 4.10 pp more than Now + length at one point. On OLMo3 it leads Now + length by 9.21 pp at one point.Source: §C.5 · pp. 57–58, Table 66 · p. 58

C.6 INFORMATION PROVIDED TO THE CONTROLLER BY THE FOUR BELIEFS

• The mean normalized length coefficient shifts from +0.931 at the lowest price to −0.116 at the highest, while positive Same and negative End gain relative weight at higher prices.Across three probe runs, the mean directional threshold falls from 2.16 to −0.33; direction and threshold change simultaneously. The coefficient study fixes κ = 10^-4 and interprets coefficients conditionally because the inputs are correlated.Source: §C.6 · p. 59, Table 68 · p. 60

C.7 PREDICTION QUALITY ACROSS QUESTIONS AND POSITIONS

• Now and Same achieved within-question AUROCs of 0.855 and 0.926, while End and Later distinguished between-question boundaries more strongly than within-question boundaries.These diagnostics characterize outcome prediction; policy-input ablations separately quantify contributions to stopping utility.Source: §C.7 · p. 60, Figure 12 · p. 60

C.8 REPROBE INITIALIZATION

• 45.25% to 48.43% (+3.19 pp) was the H-R OOD one-point saving after ReProbe rather than random initialization, positive on all three seed pairs.Across 48 cells, H-R improved on all three paired seeds in 22 cells, with no cell worsening on all three; the shared ReProbe pretraining run took 2.36 hours on one four-GPU GH200 node.Source: §C.8 · p. 60, Table 69 · p. 61

D.1 INFERENCE ARITHMETIC

• BELT’s 15,957,765-parameter probe has a leading per-token cost of about 3.07 × 10^7 + 6.14 × 10^3t FLOPs, with weight, attention, and cache terms at 0.89%, 2.68%, and 5.36% of the reasoner’s.The probe uses cached layer-16 states, width 768, and two causal Transformer layers; four belief readouts add about 6,144 FLOPs per boundary.Source: §D.1 · pp. 60–61
• OSP’s approximate per-token cost is 2.01 × 10^8 + 1.64 × 10^4t FLOPs; its dense-attention counter is 347.6M FLOPs per banked token versus BELT’s 85.5M, about 4.1× lower for BELT.The dense-attention counter counts full-sequence attention rather than causal token-by-token arithmetic.Source: §D.1 · p. 61

D.2 REFERENCE-SCHEDULE TRAINING ESTIMATES

• 55× fewer training GPU-hours than five OSP checkpoints, and 11× fewer than one, are estimated for BELT on OLMo3.The estimates include validation but exclude shared trajectory generation, feature export, and CPU policy fitting.Source: §D.2 · p. 61

D.3 PREFERENCE ADAPTATION AND PARAMETER STORAGE

• For K deployed policies, refit stores the probe plus K policy parameter sets, while threshold control stores one policy plus K scalar cutoffs; full OSP stores K checkpoints.The comparison excludes the shared reasoner and standardization statistics.Source: §D.3 · p. 62, Table 70 · p. 62

E.1 STABILITY OF TEST-FRONTIER READINGS

• BELT’s split-half test-frontier readings are stable: maximum absolute shrinkage is 0.36 pp for Qwen3 BELT-B-R and 0.20 pp for BELT-H-R, versus 5.66 and 7.53 pp for full OSP and OSP-1.The reported OSP operating points can shift by several points even where median shrinkage is small.Source: §E.1 · p. 62, Table 71 · p. 65

E.2 SINGLE-VARIANT COMPARISONS AND PAIRED BOOTSTRAP

• BELT wins robustly on selected readings, including Qwen3 ID accuracy at 0 pp tolerance (46.52% vs. 44.77% against full OSP) and LiveCodeBench Pre token saving at 2 pp tolerance (51.40% vs. 43.73% against OSP+T).These are individual comparisons, not an across-reading or across-variant summary.Source: §E.2 · p. 64

E.3 CROSS-FITTED OPERATING POINTS

• BELT’s cross-fitted Qwen OOD saving at 0 pp tolerance is 39.05%, versus 31.19% for OSP and 15.96% for OSP-1.The corresponding advantages are +7.86 pp over OSP and +23.08 pp over OSP-1.Source: §E.3 · p. 64, Table 71 · p. 65

E.4 DEVELOPMENT-SPLIT SELECTION AT A FIXED TOLERANCE

• For OLMo development-split selection, BELT saves 60.49% of tokens, exceeding full OSP by 5.72 pp and OSP-1 by 7.90 pp, but its realized accuracy drop is 3.50 pp.Every method’s seed-mean realized drop exceeds the 1 pp development tolerance on the test population.Source: §E.4 · p. 66

E.5 DIRECT UTILITY TRAINING OF THE BELT PROBE

• From ReProbe initialization, one of three direct-training seeds reaches utility within 0.0010 of BELT with the same initialization, but later training reduces utility in two seeds.BELT holds 0.6682 ± 0.0010 across its three seeds under the same initialization.Source: §E.5 · p. 68
• The reported direct-training comparison uses a documented three-seed recipe that was not tuned.Near-never-stop and always-stop checkpoints occur in the random-initialization runs.Source: §E.5 · pp. 69–70
• Direct training from random initialization reaches test-family maximum utility 0.627 in all three seeds, versus 0.669 for both BELT-B-R and BELT-H-R.The Now/End warm-up does not materially improve utility at the reported precision or remove saturation.Source: §E.5 · p. 70, Table 87 · p. 70
• The direct-utility logit gradient contains a sigmoid-derivative factor that vanishes at saturation, while the binary-belief-loss residual lacks that factor and policy reach.The comparison concerns gradient structure, not matched parameter-gradient magnitudes; attenuation is one mechanism consistent with observed saturation, and generalization also affects utility.Source: §E.5 · p. 71

E.6 DIRECT-TRAINING SENSITIVITY TO INITIALIZATION

• Seed 2’s mean absolute utility advantage falls from 0.0443 to 0.0379 while mean absolute logit credit falls from 3.81×10^-6 to 1.80 × 10^-28.The logit-credit change is a probe-weighted coordinate signal, not an optimizer parameter gradient or causal intervention.Source: §E.6 · pp. 71–72
• Overfitting remains a possible contributor: one seed’s epoch-average online training loss improves while held-out utility deteriorates.Belief training is also exposed to this risk; nine corrected ablations selected epoch 1 and had higher validation loss at epoch 3.Source: §E.6 · p. 72

E.7 AGENT PROBE RUNS

• In ALFWorld, BELT-H-T’s three runs range from 4.5% to 24.6% token saving within two points of natural success, while ScienceWorld runs agree at tolerance readings.ALFWorld tolerance readings are set by members nearest natural success and move in steps; its game bootstrap resamples games, not probe runs.Source: §E.7 · p. 72

E.8 OS-PRUNER REPRODUCTION

• The authors independently implemented OS-Pruner from its published architecture and objective, and report it remains stronger than TC, ROM, and training-free rules at one-point Qwen3 tolerances.Full OSP generally improves budget accuracy over OSP-1, but not at every reported operating point.Source: §E.8 · p. 72

E.9 COMPARATORS ON OLMO3

• On OLMo3, full OSP uses approximately 405M parameters per checkpoint and improves on OSP-1 at most operating points.OLMo3 BELT-B-R omits prefix length; the reported table includes both BELT interfaces.Source: §E.9 · p. 72

E.10 EVALUATION SETS

• Two OOD questions repeat DAPO-Math training problems with different LaTeX formatting, and one MATH-500 question shares a template with a different DeepScaleR problem.The three identified overlaps remain in the fixed evaluation roster.Source: §E.10 · p. 73

F FUTURE WORK

• Future work includes full serving-stack evaluation, beliefs about remaining computation cost, and broader coverage of symbolic-answer mathematics and open-ended tasks.The current evaluations cover integer-answer math, coding, and text-game environments.Source: F FUTURE WORK · p. 73