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What Does Chain-of-Thought Entropy Measure? A Channel Audit of Scaffolding, Routing, and Content

Marios Papamichalis, Regina Ruane

arXiv:2609.25039v1stat.MLcs.ITcs.LG

TL;DR

CoT statistics guide training and selection but mix scaffold and substantive-content choices, leaving the channel behind a measured uncertainty unclear. The paper separates those channels and analyzes their entropy, divergence, and dynamics. Content-channel scoring beats raw surprisal across compression cells, while an answer-leakage audit revises the positional-control result.

  • Problem

    CoT statistics used for training and selection combine scaffold and substantive-content choices in each next-token distribution.

    Source: §1 · p. 1

  • Method

    The paper designates a scaffold vocabulary subset and decomposes next-token distributions into gate, scaffold, and content channels.

    Source: §1 · p. 2, §3 · p. 3

  • Results

    2 to 33 points: content-channel scoring beats raw surprisal in all five compression cells; after answer stripping, no token scorer beats a random contiguous block.

    Source: §1 · p. 3, §5.4 · p. 7

  • Takeaways & Limitations

    The paper argues that CoT statistics should state which channel convention they measure, a choice that changes compression accuracy by up to 33 points.

    Source: §6 · p. 9

    Measurements stop at or below 3B on GSM8K and MATH-500, with 7B rows marked to run.

    Source: §6 · p. 9

Abstract

from arXiv · show

Entropy over chain-of-thought tokens decides which tokens receive the policy gradient, which get pruned, and whether a run has collapsed, yet each such statistic reads a next-token distribution mixing three choices: whether to emit connective scaffolding, which connective, and what the substantive continuation should be. Designating a scaffold vocabulary subset separates the three, exactly, for entropy, Kullback--Leibler divergence, and the first-order entropy velocity of a softmax policy. We prove the raw and content conventions disagree about which position is the larger fork on an explicit open region, and bound answer diversity by the content channel plus a leakage term a measured witness certifies. Across twenty-three configurations the scaffold side carries up to 41% of the raw high-entropy set; on a matched-tokenizer ladder, coupling changes only at the math-corpus step while the scaffold's entropy share keeps growing through distillation; a closed-form forecast from one channel correlation tracks selection retention over a 54-point range to five points, unfitted. On compression, the content convention beats raw surprisal in every cell; an answer-leakage audit then corrects our own headline control: re-fed chains earn a quarter to a half of their accuracy from restated answers, and once stripped, no token scorer beats a random contiguous block.

1 INTRODUCTION

CoT statistics combine decisions about emitting scaffolding, choosing a scaffold token, and choosing substantive content, even though those statistics drive training and selection decisions. The paper separates these channels and shows that the distinction changes which positions and tokens are selected.

1 INTRODUCTION

  • Each next-token distribution mixes whether to emit connective scaffolding, which connective to emit, and what substantive continuation to produce.The resulting statistics are computed on that mixture.Source: §1 · p. 1
  • Entropy-based statistics feed decisions about policy gradients, token pruning, KV-cache eviction, and which reasoning steps are called important.Prior work finds high-entropy CoT tokens are often connectives, and restricting gradients to them can match full-gradient training.Source: §1 · p. 1
  • Across seventeen measured cells, as scaffold share of mean entropy rises, the raw top-20% selection becomes correspondingly scaffold-driven.The figure caption reports seventeen cells; Table 1 lists fourteen.Source: Figure 1 · p. 2
  • Earlier token-splitting work classifies emitted tokens, but does not factorize the predictive distribution or its training dynamics.The paper identifies this distinction as the gap its channel decomposition addresses.Source: §1 · p. 2
  • Content-channel scoring gains 2 to 33 points over raw surprisal in five compression cells; after answer stripping, no token scorer beats a random contiguous block.Re-fed accuracy includes a quarter to a half from restated-answer copying.Source: §1 · pp. 2–3

2 RELATED WORK

The related work spans entropy-based RLVR analysis, token-structure distinctions, compression and attribution methods, and compositional data analysis. The paper distinguishes its approach by factorizing the predictive distribution and examining its geometry and training-time dynamics.

Entropy in RLVR

  • RLVR studies connect entropy to performance, partition tokens by thresholds, or examine answer diversity, while none writes the mass/content identity audited here.Semantic entropy instead measures answer-level uncertainty after marginalizing surface variation (Kuhn et al., 2023).Source: §2 · p. 3

Structural and organic token splits

  • Prior structural-versus-organic token work labels realized tokens, whereas this paper factorizes the next-token distribution, its geometry, and its training-time velocity (Zhao et al., 2026).The related-work discussion also connects connective control to reasoning-path selection (Park and Lei, 2026).Source: §2 · p. 3

Compression, eviction, attribution

  • Compression, eviction, and attribution methods rank tokens or steps, but the related work notes that their evaluations do not strip restated answers from re-fed chains.It contrasts inference-time compression with positional baselines from KV-cache eviction and training-data condensation.Source: §2 · p. 3

3 SETUP: THREE CHANNELS OF A CHAIN-OF-THOUGHT DISTRIBUTION

The setup partitions a next-token distribution into a scaffold gate, a within-scaffold channel, and a content channel. The partition is mathematically general, while the scaffold set determines which questions the channels answer.

3 SETUP: THREE CHANNELS OF A CHAIN-OF-THOUGHT DISTRIBUTION

  • For a scaffold set S, the distribution factors into scaffold mass s and renormalized scaffold and content distributions pS and pC.The content block is the complement of S and is not asserted to contain only substantive tokens.Source: §3 · p. 3
  • The paper names s the gate mass, pS the scaffold channel, and pC the content channel; content denotes the complement block without a semantic guarantee.For softmax policies, the gate log-odds equal the difference between scaffold and content log-sum-exp values.Source: §3 · p. 3
  • A three-block refinement separates formatting, strategic-control, and content tokens to distinguish strategic connectives from formatting tokens such as newlines.Source: §3 · p. 4
  • The mathematics holds for any scaffold set, but the channel interpretation depends on that modeling choice.FIXED uses 71 reasoning connectives plus formatting tokens; POS adds function words, and DATA selects frequent non-numeric CoT tokens.Source: §3 · p. 4

4 THEORY

The theory decomposes entropy and divergence into gate, scaffold, and content contributions, then establishes when raw and content rankings disagree, bounds scaffold-mediated answer diversity, and derives optimizer-specific entropy dynamics.

4 THEORY

  • Entropy and KL divergence split exactly into gate, scaffold, and content terms, with sharp bounds on the surplus G = H − HC.Weighted entropy aggregates also split without independence or stopping-time assumptions, and the same partition separates Aitchison geometry into balance and within-block coordinates.Source: §4 · p. 4
  • A position statistic ignores scaffold-supported logit perturbations exactly when it factors through the content distribution pC.The decomposition also applies to collapse curves and weighted aggregates without further assumptions.Source: §4 · p. 4

4.1 RANKING REVERSALS AND THE SELECTION WEDGE

  • Raw and content entropy rank two positions differently exactly when δ(δ + γ) < 0; their top-ρ selections can be disjoint for ρ ≤ 1/2.Pairwise flips, retention, and Jaccard overlap are explicit functionals of the joint (HC, G) law.Source: §4.1 · p. 4

4.2 WHAT THE SCAFFOLD CAN AND CANNOT DO FOR EXPLORATION

  • The exploration bound treats scaffold-mediated answer diversity as leakage, with both sides of the bound simultaneously attainable.A positive κt certifies leakage, although zero κt does not rule it out.Source: §4.2 · p. 5

4.3 CHANNEL-RESOLVED ENTROPY DYNAMICS

  • Under exact categorical natural-gradient flow, block-constant advantages move only the between-block term; under vanilla REINFORCE, they also sharpen scaffolding and spread content.These results concern per-position advantage patterns, not shared-parameter network updates; a format reward can change measured entropy while content exploration remains untouched.Source: §4.3 · p. 5

5 EXPERIMENTS

Experiments test channel decomposition and its effects on selection, compression, and training across frozen models and released-checkpoint contrasts. Content-channel scoring beats raw surprisal in every compression cell, but after answer restatements are stripped, no token scorer beats a random contiguous block.

5.1 STAGE 1: WHERE COT ENTROPY LIVES

  • The scaffold-driven share of the raw top-20% set ranges from 0–3% in GPT-2/Pythia to 26–41% in math-pretrained and distilled checkpoints.On generated reasoning models, Var(G) and Var(HC) each exceed Var(H) by 2.0–4.1×, with corr(G, HC) in [−0.87, −0.77].Source: §5.1 · pp. 5–6, Table 1 · p. 6

5.2 COUPLING AND ALLOCATION ARE TWO AXES, AND FOUR CHECKPOINT CONTRASTS SEPARATE THEM

  • Code, scale, and instruction contrasts barely change coupling, while math data shifts correlation by +0.111 and retention by +6.5 points.Distillation raises scaffold share of mean H from 14.7% to 20.9% while flip and retention change little; checkpoint contrasts are not randomized interventions.Source: §5.2 · pp. 6–7, Figure 2 · p. 6

5.3 A FORECAST WITH NO FITTED PARAMETERS

  • A forecast from Stage-1 channel correlation predicts top-20% retention across 23 cells with 5.1-point mean error and Spearman 0.91.The forecast rank-orders cells better than constant predictions, but its held-out checkpoints are same-family and in-range, not external validation.Source: §5.3 · p. 7

5.4 DECISIONS: COMPRESSION, ATTRIBUTION, EVICTION

  • Content-channel scoring beats raw surprisal in all five cells, from +33.1 [30.2, 36.1] to +2.3 [0.5, 4.2] at the tightest budget.The channel gain is an interaction: neither content-only eligibility nor gate-adjusted scoring alone explains the combined gain in every setting.Source: §5.4 · p. 7, Table 2 · p. 8
  • At a quarter budget on the distilled model, kept text still contains the gold answer for 79% of recency chains versus 53% of contiguous-block chains.After stripping, recency’s advantage over contiguous selection depends on model; the authors attribute part of the original tail advantage to answer copying.Source: §5.4 · p. 8
  • After answer restatements are stripped, no token scorer beats a random contiguous block in any stripped cell.The uncompressed distilled GSM8K ceiling drops from 75.5% to 41.2%; the authors note stripping also removes genuine final-step content.Source: §5.4 · p. 8
  • Raw-versus-content attention rankings agree at Kendall τ = 0.75 ± 0.01, yet their top-1 anchor flips on 40.4 ± 3.3% of problems.KV-eviction keep sets overlap at Jaccard 0.64/0.70 at 30%/50% budgets.Source: §5.4 · p. 9

5.5 TRAINING DYNAMICS: TWO NULLS AND ONE DECOMPOSED DECLINE

  • The reported GRPO entropy change is −0.05 ± 0.03 nats, while the routing witness κ̂ = 0.81 nats certifies leakage but does not estimate Λ.The content bonus raised total H more than the raw bonus; the GRPO arms did not separate in pass@8.Source: §5.5 · p. 9

6 DISCUSSION AND LIMITATIONS

The paper frames its contribution as measurement and first-order mechanism, not a claim that models use the channels or that content is validated reasoning. It identifies boundaries in the tested settings and outlines extensions.

6 DISCUSSION AND LIMITATIONS

  • The claims concern channel measurement and first-order mechanism; neither model use of channels nor the complement block as validated reasoning is claimed.The paper reports that stating the convention moves compression accuracy by up to 33 points and that an audit exposed answer-copying inflation.Source: §6 · p. 9
  • Measurements are at or below 3B on GSM8K and MATH-500; checkpoint dissociations are observational, and stripping is string-based.Paraphrased restatements can survive stripping, while genuine final-step content can be removed; 7B measurements remain to be run.Source: §6 · p. 9
  • Proposed next steps include estimating Λ, testing longer horizons and scale, randomizing corpus interventions, and decomposing answer distributions and MoE routers.Source: §6 · p. 9

A FULL FORMAL STATEMENTS

The formal results decompose entropy and KL into gate, scaffold, and content contributions, and characterize when content-channel measurements are invariant and how training, selection, or scaffold choices can affect the channels and exploration. They extend the decomposition to other distributional instruments, bound answer diversity through content variation and scaffold leakage, and examine limits on downstream and cross-tokenizer conclusions, with exact evaluations and simulations validating the identities and entropy-step predictions.

A.1 EXACT CHANNEL IDENTITIES

  • Entropy and KL decompose exactly into gate, conditional scaffold, and conditional content contributions, while scaffold surplus isolates the gate and scaffold terms.At fixed gate mass and content distribution, the surplus extrema are sharp: minima require a point-mass scaffold channel, while maxima require a uniform scaffold channel.Source: §A.1 · p. 13
  • A three-way Pythagorean split separates partition balance and the two block geometries; the stopped weighted-sum result requires no independence or stopping-time assumption.The entropy identity uses arithmetic mass, whereas the geometric identity uses balance.Source: §A.1 · p. 14

A.2 THE CONTENT CHANNEL IS THE MAXIMAL INVARIANT

  • The content subcomposition is a maximal invariant to scaffold-logit changes, and common temperature changes preserve content-distance rankings and within-content coordinate orderings.This charter concerns fixed-position logits and a fixed partition, not trajectory-level convention independence; the partition remains a modeling choice.Source: §A.2 · pp. 14–15

A.3 CHANNEL-RESOLVED ENTROPY DYNAMICS

  • Under unrestricted-logit natural-gradient ascent, entropy change resolves into gate and within-block channel terms; the gate moves exactly when scaffold and content mean advantages differ.A channel with block-constant advantage has zero within-channel probability and entropy velocity.Source: §A.3 · pp. 15–16
  • With a block-constant advantage gap, expected REINFORCE/GRPO updates can change within-block entropy: a positive format reward sharpens scaffold choices while spreading content choices to first order.The effect is not confined to the gate except under natural gradient or exactly blockwise-constant velocities.Source: §A.3 · p. 16

A.4 RANKING REVERSALS AND THE SELECTION WEDGE

  • Raw and content entropy can reverse their strict rankings on a positive-measure open set, and exact top-ρ selections can have zero Jaccard similarity.The selection-wedge results also define a population retention quantity for comparing the two selections.Source: §A.4 · pp. 16–17

A.5 WHAT THE SCAFFOLD CAN AND CANNOT DO FOR EXPLORATION

  • Under answer sufficiency, answer entropy is bounded by content-channel variation plus leakage, H(α) ≤ Bgc + Λ ≤ Bgc + Bsc.Leakage reaches its upper bound exactly when the emitted token sequence is almost surely determined by the censored sequence.Source: §A.5 · pp. 17–18
  • Scaffold entropy can fund answer exploration only through leakage: inert scaffold routing gives Λ = 0, while a coin-in-scaffold policy attains Λ = Bsc = log 2 with Bgc = 0.The equality is sharp, and without leakage the gate plus content exhaust the exploration budget; large scaffold entropy alone buys nothing.Source: §A.5 · p. 18

A.6 EVENTUAL CONTENT: THE FIRST-PASSAGE CHANNEL

  • H(qt) is the recommended reachability-aware fork signal when content may be deferred; κt > 0 certifies leakage, although κt = 0 at every prefix need not imply Λ = 0.Short truncated rollouts estimate qt by sampling to the first content token and κt by substituting alternative connectives and comparing first-content laws.Source: §A.6 · p. 18

A.7 BEYOND ENTROPY: THE FULL INSTRUMENT CALCULUS

  • At matched gates, f-divergences split into gate, scaffold, and content terms, while JS obeys a skewed three-channel law; under locked scaffolding, content sensitivity decays at instrument-specific rates.For the KL penalty, content divergence ε costs exactly β(1 − s)ε, so hardening the gate weakens a fixed-coefficient penalty on content drift.Source: §A.7 · p. 19
  • When gate-mismatch rates diverge, JS drift is (1 − s̄) log 2 + o(1 − s̄) regardless of content; at comparable rates, both channels can survive at order 1 − s̄.The unmatched-gate analysis gives an exact sink decomposition and specifies when content and gate channels vanish or remain order-matched.Source: §A.7 · pp. 19–20
  • Bounded-value downstream distances fill [0, B∥δ∥1], so channel statistics alone identify no downstream effect; the difference decomposes into gate, scaffold, and content directions.The gate weight depends on the squared distance between mean scaffold and content embeddings, computable from model weights.Source: §A.7 · p. 20
  • Cross-tokenizer comparisons face a forced choice between channel separation and tokenization robustness, with the expansion constant kD/(D + k − 1) for channel-separating instruments.The Hilbert metric has the stated invariance and Markov-contraction properties, but is not sink-separable and does not induce an inner product on the CLR hyperplane.Source: §A.7 · p. 21

B EXACT WITNESSES AND IN-SILICO VALIDATION

  • Finite-difference entropy-step checks had median relative error 0.0% in the plain arm and 3.2% in the larger-step intervention arm.The identities and both optimizer-regime formulas were also machine-checked.Source: B EXACT WITNESSES AND IN-SILICO… · p. 21

B.1 EXACT WITNESSES AND CALIBRATED FLIP RATES

The simulations test whether content entropy better tracks exploration and collapse than raw entropy, and calibrate when the two conventions select different positions. In the toy testbed, content entropy separates exploring from collapsed policies, while format rewards can shift raw entropy without comparable changes in exploration.

B.1 EXACT WITNESSES AND CALIBRATED FLIP RATES

  • As gate mass rises from 0.2 to 0.6, verdict flips rise from 6.5% to 11%, retention falls from 77% to 61%, and scaffold-driven share grows from 33% to 61%.At gate mass 0.7, the sweep reaches 11.8% flips, 57.6% retention, and 64.1% scaffold-driven share; measured Jaccard is 0.00.Source: §B.1 · p. 22, Figure 3 · p. 22
  • In the phantom construction, raw entropy falls 46% while content entropy stays invariant; in the masked construction, raw entropy rises 9% despite a 79% content collapse.These are constructed failure modes, not model measurements.Source: §B.1 · p. 22, Figure 3 · p. 22

B.2 IN-SILICO RLVR: WHICH CHANNEL CARRIES THE COLLAPSE, AND WHICH PREDICTS EXPLORATION

  • Fork-content entropy separates exploring from collapsed policies without overlap (AUC 1.00 vs 0.75 for raw H) across 16 testbed endpoints.The testbed is authored for mechanism validation, not independent evidence.Source: §B.2 · p. 22
  • Changing format weight λ from 0 to 0.35 moves raw entropy by 2.45 nats without intervention and 1.89 nats with intervention, while exploration changes by 0.000 and 0.035.In one run, 29% of the 3.55-nat collapse is gate-plus-scaffold, with channel predictions matching realized entropy differences at 0.0% median relative error.Source: §B.2 · p. 22

C WORKED EXAMPLE: LEAKAGE, EXACTLY

This section uses a worked example and proof passages to examine how scaffold entropy relates to answer diversity through scaffold identity and a leakage term, while establishing exact decompositions and invariance limits. For admissible partitions with D ≥ 3, it also shows strict ordering reversals for entropy, gate mass, and surplus, with qualifications for finite logits, the closed-simplex boundary, and D = 2.

C WORKED EXAMPLE: LEAKAGE, EXACTLY

  • H(α) = log 2 despite zero per-position gate and content budgets when scaffold identity at step 1 determines content at step 2.The entropy bound holds with equality: exploration is stored in the scaffold channel and withdrawn as content through history dependence, so the leakage correction cannot be omitted.Source: C WORKED EXAMPLE: LEAKAGE, EXACTLY · p. 23
  • Under scaffold inertness, scaffold-internal entropy is idle; scaffold exploration buys answer diversity only when the model routes information through connective identity.In the inert case, step-2 laws depend on the censored view but not which scaffold token occurred; the pair of cases identifies the amount routed as Λ.Source: C WORKED EXAMPLE: LEAKAGE, EXACTLY · p. 23

D.1 PROPOSITION 1: CHAIN RULES AND SHARP SIGNED SURPLUS BOUNDS

  • The proof establishes entropy chain rules and sharp entropy extrema, while qualifying boundary behavior: for k ≥ 2, the gate-mass-1 case has no path-independent continuous extension for G.For k = 1, the unique content law has zero entropy and gives a canonical boundary extension; the KL boundary identity requires separate zero-mass conventions.Source: §D.1 · pp. 25–26

D.2 LEMMA 1: THREE-WAY PYTHAGOREAN SPLIT

  • The three-way Pythagorean split is exact: the geometric decomposition uses a balance, while the entropy identity uses scaffold mass, and the two coordinates answer different questions.The balance describes scaffold position in log-ratio terms; amalgamated mass describes how much probability sits on the scaffold.Source: §D.2 · p. 27

D.3 PROPOSITION 3: CONTENT MAXIMAL INVARIANCE AND TEMPERATURE BEHAVIOR

  • Content subcomposition is the maximal invariant under scaffold transformations, whereas raw entropy, varentropy, top-two margin, and every nontrivial raw top-k map are not invariant.Temperature preserves distance rankings and within-content coordinate order, but the proof gives entropy-order reversals at different temperatures.Source: §D.3 · pp. 28–29
  • When the content block is a singleton, its content entropy is zero, and the entropy witness also yields a surplus reversal.For the gate-mass witness, the actual scaffold mass is one minus the first-coordinate probability, reversing both comparison signs while preserving a strict ranking reversal.Source: §D.3 · p. 31
  • For every admissible partition with D ≥ 3, strict entropy, gate-mass, and surplus ordering reversals persist at two finite temperatures and finite logits.Three-coordinate witnesses extend to larger dimensions by assigning unused coordinates logits −L; continuity preserves the strict inequalities for sufficiently large finite L.Source: §D.3 · p. 31
  • On the closed simplex, maximal invariance does not extend verbatim: finite logit reweightings preserve zero coordinates, and content subcomposition is undefined when the content block has zero mass.A boundary version must be stratified by support.Source: §D.3 · p. 31
  • For D = 2, both blocks are singletons, so G = H and H and s are strictly monotone in (z1 − z2)/τ; no common-temperature reversal exists.Source: §D.3 · p. 32

D.4 PROPOSITION 2: CHANNEL DYNAMICS

The section derives exact channel-resolved entropy dynamics under specified categorical-policy updates. It distinguishes natural-gradient motion from sampled REINFORCE and states the local assumptions that limit the result.

D.4 PROPOSITION 2: CHANNEL DYNAMICS

  • For sampled REINFORCE, the expected logit velocity is not blockwise constant even when the advantage is, because it includes a probability factor.A block-level reward pattern therefore does not imply the natural-gradient channel dynamics.Source: §D.4 · pp. 34–35
  • Under vanilla REINFORCE, the same block-constant advantage also sharpens the scaffold channel and applies first-order spreading pressure to the content channel.The covariance formulas show these within-block effects; the gate-only reading fails in a numerical example.Source: §D.4 · p. 35
  • Under exact natural-gradient flow, a block-constant advantage moves only the between-block term, whose sign is characterized by the advantage gap.The specialization assumes exact undamped ascent in local categorical Fisher geometry at a fixed prefix, not ordinary neural-network or approximate updates.Source: §D.4 · p. 35
  • Blockwise constancy applies to conditional action scores or advantages; in a sequential model, continuation values can make advantages vary within a block despite block-constant immediate rewards.The proposition is an exact differential identity, with a local first-order approximation whose implicit constant may depend on the logits and vocabulary.Source: §D.4 · p. 35

D.5 THEOREM 1: FORK-VERDICT REVERSAL

Theorem 1 identifies when raw and content entropy rank two positions oppositely and establishes that their top-ρ selections can be disjoint. It also derives population functionals for comparing the selections.

D.5 THEOREM 1: FORK-VERDICT REVERSAL

  • TopK selects exactly K positions, resolving score ties in favor of smaller indices.This tie rule makes the selected sets well-defined even when scores repeat.Source: §D.5 · p. 35
  • The raw and content rankings are opposite exactly when their two nonzero gaps have opposite signs.The reversal region is relatively open and has positive measure.Source: §D.5 · pp. 36–37
  • The top-ρ raw-entropy and content-entropy selections can be disjoint, giving zero Jaccard similarity.The construction applies to any ρ in (0, 1/2], including the top-20% case.Source: §D.5 · p. 37
  • The theorem gives population formulas for the flip probability, expected retention, and Jaccard overlap of the two selections.The formulas use quantiles and conditional probabilities of the score distributions.Source: §D.5 · pp. 37–38

D.6 THEOREM 2: EXPLORATION BOUND WITH EXACT LEAKAGE

Theorem 2 relates answer entropy to content-side entropy and a leakage term measuring scaffold-dependent routing. It characterizes when leakage vanishes, gives a sharpness witness, and specifies the limits of the entropy attribution’s exactness.

D.6 THEOREM 2: EXPLORATION BOUND WITH EXACT LEAKAGE

  • Λ sums conditional mutual information terms I(Zt; Y<t | Z<t), and it vanishes exactly when each term is zero.For finite trajectory alphabets, this is equivalent to Zt being conditionally independent of the past given Z<t.Source: §D.6 · pp. 39–40
  • The chain-rule identity expresses H(α) as Bgc + Λ + H(α | Z1:T) − H(Z1:T | α), yielding an upper bound by dropping the last nonnegative term.The bound is sharp when the visible sequence recovers the answer and the answer recovers the visible sequence.Source: §D.6 · p. 39
  • A sharpness witness has Λ = log 2 from one nonzero routing term, while the visible sequence and α each recover the other.At the first step the scaffold block has entropy log 2; later tokens are conditionally deterministic.Source: §D.6 · p. 40
  • With a full-support requirement, both equalities can be approached arbitrarily closely while EOS remains absorbing.Source: §D.6 · pp. 40–41

D.7 DEFINITION 1 IS EXACT, AND WHERE IT IS NOT

  • Definition 1 is exact per matched context, but applying it to separately averaged quantities is not exact, and checkpoint context shifts cannot be attributed per position.The paper therefore uses linear channel aggregates for on-policy series.Source: §D.7 · p. 41

D.8 PROOFS FOR §A.6: FIRST PASSAGE AND ROUTING INFORMATION

The first-passage analysis makes scaffold routing measurable through a witness that certifies non-inertness but does not quantify total leakage. The accompanying instrument and unmatched-sink results specify divergence decompositions and their asymptotic behavior.

D.8 PROOFS FOR §A.6: FIRST PASSAGE AND ROUTING INFORMATION

  • A positive first-passage κt certifies Λ > 0, but κt = 0 need not imply zero leakage and no numerical relation between κt and Λ is established.Measured κ is an estimate with intervals, not an exact quantity.Source: §D.8 · p. 42

D.9 PROOFS FOR §A.7: THE INSTRUMENT CALCULUS

  • The three-channel KL chain rule separates gate, scaffold, and content contributions; with a matched gate and locked scaffold, the content term has weight 1 − s.Source: §D.9 · p. 42

D.10 PROPOSITION 6: UNMATCHED-SINK REGIMES

  • The skew Jensen–Shannon term is bounded by hb(λ), independently of the content pair, and the analysis establishes uniform O(a_n) bounds.Source: §D.10 · p. 44
  • For unmatched sink mass a_n, B_n = a_n log 2 + o(a_n), while C_n = o(a_n) in the general regime.For fixed distinct content laws, C_n = Θ(a_n).Source: §D.10 · p. 45

D.11 THEOREM 7: NO FREE LUNCH UNDER COMPOSITIONAL AXIOMS

Theorem 7 exposes trade-offs among compositional and data-processing properties of distance measures. The section also specifies the scaffold vocabulary and the empirical protocol for testing sensitivity to that choice.

D.11 THEOREM 7: NO FREE LUNCH UNDER COMPOSITIONAL AXIOMS

  • Matched-sink distances can remain positive below full sink mass yet vanish as s approaches 1, ruling out sink separability for these attenuating families.Source: §D.11 · pp. 47, 49
  • Splitting and merging force incompatible scale constraints, so Aitchison distance cannot be rescaled by dimension to satisfy data processing.The split-and-merge maps force the scale to be equal across dimensions and also require kD/n = 1, a contradiction.Source: §D.11 · p. 48
  • The Hilbert metric is invariant under congruent embeddings and contracts under admissible maps, but it is not induced by an inner product.The parallelogram law fails for its generating norm.Source: §D.11 · pp. 50–51

E.1 SCAFFOLD-SET DEFINITIONS

  • The FIXED scaffold set contains 71 connectives and 11 format strings, while mathematical operators are deliberately excluded.The stated reason is to avoid building the conclusion that operators are scaffolding into the definition.Source: §E.1 · p. 51

E.2 SENSITIVITY PROTOCOL

  • The sensitivity protocol compares FIXED, POS, DATA, random-list-drop, and FORMAT-only versus CORE-only scaffold sets.Exploration analyses also exclude answer-bearing words; the channel identities themselves do not depend on S, so only empirical magnitudes are sensitive to its definition.Source: §E.2 · p. 52

F EXPERIMENTAL DETAILS

The experiments specify synthetic populations and training settings for auditing scaffold and content channels, then evaluate channel forecasts, token-selection protocols, and answer-distribution measurements. The final procedures also account for answer restatements and numerical underflow.

F.1 TARGET-CHECKED POSITION POPULATION (SYN-1, FIGURE 3)

  • The target-checked synthetic population uses boundary-eligible and interior positions with different gate-mass and effective-support distributions, verified against its intended low-entropy majority and boundary-heavy high-entropy minority.At mean scaffold mass 0.4, 65% of positions are below 0.7 nats and boundary slots make up 58% of the raw top-20%.Source: §F.1 · p. 52

F.2 CONSTRUCTED COLLAPSE TRAJECTORIES (SYN-2, FIGURE 3)

  • Constructed collapse trajectories reuse one population while changing gate shifts and scaffold powering in the phantom arm, and also content powering in the masked arm.Because content draws are reused in the phantom arm, its content remains exactly unchanged; the masked arm includes a genuine content collapse.Source: §F.2 · p. 52

F.3 IN-SILICO GRPO (SYN-3, FIGURE 3)

  • Median relative error for the per-step channel prediction was 0.0% without intervention and 3.2% with intervention in the in-silico GRPO experiment.Exploration was measured from the policy as worst-case preference-shift pass@8, and entropy differences were compared with Proposition 2’s predictions.Source: §F.3 · p. 52

F.4 STAGE 1: FROZEN MODELS; F.5 STAGE 2: GRPO

  • Stage 1 measures frozen teacher-forced and generating models on GSM8K and MATH-500, while Stage 2 measures GRPO checkpoints and pass@1/pass@8 on held-out problems.Stage 1 records channel statistics and selection comparisons from full-vocabulary logits; Stage 2 measures sampled chains at successive frozen checkpoints.Source: §F.4 · p. 53, §F.5 · p. 53

F.6 STAGE 3: RE-AUDITS

  • Stage 3 compares ten compression and selection arms at matched token budgets and audits answer survival, answer stripping, attention-based sentence importance, and KV-cache eviction.The stripping condition deletes exact, comma-free, and numeric-form answer occurrences plus the chain’s final sentence before compression.Source: §F.6 · p. 53

F.7 ANSWER DISTRIBUTIONS AND NUMERICS

  • For answer-distribution and numerical analyses, the protocol uses 32 samples per problem and computes channel quantities blockwise in log space without clamping scaffold mass.Positions whose content mass underflows beyond what the estimator can certify are excluded from rankings; the reported exclusion fraction is at most 0.2% teacher-forced and 4.9% generated.Source: §F.7 · pp. 53–54
  • Figure 4 forecasts retention from channel correlation without a fitted parameter and flip with one frozen constant; Figure 5 compares compression bars before and after restated answers are stripped.Figure 5 reports that stripping halves the distilled-model ceiling and removes the recency arm’s advantage over a random contiguous block, while an instruct-model tail premium remains.Source: Figure 5 · p. 54, Figure 4 · p. 54

G ADDITIONAL RESULTS

Additional results quantify how scaffold and content channels contribute to entropy variance and how measurements depend on the scaffold definition. The paper also identifies unmeasured extensions to answer distributions and MoE routers, and distinguishes its contribution from shared companion-paper machinery.

G ADDITIONAL RESULTS

  • In the G = H − HC parameterization, reasoning-model channels are strongly anticorrelated, and each carries 2.0–4.1× the variance of raw H.This makes raw H a smaller residual of two larger channels and is reported as the mechanism behind low top-20% retention.Source: G ADDITIONAL RESULTS · p. 54
  • The target-checked population instead has Var(G) = 0.29 Var(H), Var(HC) = 0.80 Var(H), and 2 Cov = −0.08 Var(H), with corr(G, HC) = −0.09.The surplus reshuffles raw rankings without inflating their spread, producing visibly different top-20% selections despite similar-looking H and HC histograms.Source: G ADDITIONAL RESULTS · p. 55
  • For generating reasoning models, the additive content term carries 58–66% of Var(H), scaffold-side terms carry 15–20%, and their coupling is mildly positive.For teacher-forced base models, the content term carries 80–95% and scaffold-side terms 4–8% of Var(H).Source: G ADDITIONAL RESULTS · p. 55
  • Widening the scaffold set to the POS definition doubles mean scaffold mass and raises the flip rate from 12.5% to 24.1% on Pythia-410M.The paper therefore holds the scaffold definition fixed for principal cross-model comparisons; entropy is unchanged by construction.Source: G ADDITIONAL RESULTS · p. 55

G ADDITIONAL RESULTS; H MOE ROUTERS AS COMPOSITIONS (CROSS-OBJECT GENERALITY II)

I RELATION TO THE COMPANION MANUSCRIPT

  • The companion paper studies attention rows with a single canonical sink, whereas this paper studies next-token policies with a scaffold set that has internal structure.The shared static grouping and reversal machinery is distinct from this paper’s leakage bound, routing information, optimizer-resolved dynamics, and intervention protocol.Source: I RELATION TO THE COMPANION MANUSCRIPT · p. 56

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