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Conflicting Supervision Moves Commitment, Not Capability: A 12.29{\sigma} arrangement effect that is exactly zero under a convention-agnostic score

Wenhui Chen

arXiv:2610.00234v1cs.AI

TL;DR

When the same problems appear under incompatible but correct conventions, it is unclear what their ordering writes into model parameters. The paper separates arrangement period from schedule reach in a bound and measures both schedule settings; a natural-corpus B-only rerun at the matched learning rate raises minority install from 0.2091 to 0.3738.

  • Problem

    It is unclear what conflicting but correct supervision changes in a trained model.

    Source: §1 · p. 1

  • Method

    The paper bounds the ordering-dependent endpoint term as a product of arrangement period and schedule reach, then measures paths under decaying and constant schedules.

    Source: §8 · p. 46

  • Results

    In the natural-corpus B-only rerun at cf2’s learning rate, minority install rose from 0.2091 (sd 0.0089, eight seeds) to 0.3738 (sd 0.0181, three seeds).

    Source: §7 · p. 44

  • Takeaways & Limitations

    Below corpus scale under a decaying schedule, the path is compressed to its time-average, yielding a stochastic convention policy with capability conserved to within an eighth.

    Source: §8 · p. 46

    Only mutually exclusive conflicts are tested; style conflicts and cases where both answers can be wrong remain open.

    Source: §6 · p. 40

Abstract

from arXiv · show

"Train a model on the same problems written under two incompatible conventions, both correct, and ask what the ordering of that data writes into the parameters. The learning-rate schedule is not a background condition for that question. It is the averaging operator, and it decides the answer. We prove a bound in which the arrangement and the schedule enter the ordering effect as separate multiplied factors: the arrangement only as a block period, the schedule only as how much weight the endpoint can place on any one moment of the run. A decaying schedule cannot put a large step size and an uncontracted remainder at the same moment; a constant one does exactly that at the last step. That decay moderates ordering effects has been reported in pretraining; the mechanism, the separation, and a controlled measurement of both halves are ours. Ten orderings of one corpus, one budget, everything but the path held fixed, run twice under families differing in lr_scheduler_type and nothing else: at a constant rate the interior spans 0.2221 in allocation, 11.63 contrast floors, monotone in how blocked the arrangement is. Under the single cosine every published arm uses, the same ten arms occupy two distinguishable states where their own resolution would allow about ten. "Order matters" and "order does not matter" are the two ends of one knob. What the path writes is which convention the model commits to, and no exact-match benchmark can see it. Across twelve arms acc_A+acc_B is constant to within 9.7% while the allocation share runs 0.04 to 0.87, so the 12.29-sigma arrangement switch this paper measures is exactly zero under a convention-agnostic metric. That conservation is quoted from the decayed family throughout, the constant-rate one being a noisier place to read it. Marking the convention in the prompt collapses the switch and reaches 87.5% of the union ceiling."

1 Newton’s Apples That Disagree

Conflicting but correct conventions need not reduce what a model can solve: data order shifts which form it writes, and the learning-rate schedule controls whether that shift persists.

1 Newton’s Apples That Disagree

  • 12.29σ arrangement effect moves allocation share from 0.04 to 0.87, while acc_A+acc_B stays within 9.7% across twelve arms.The effect is exactly zero on the convention-agnostic capability coordinate.Source: §1 · pp. 2, 4
  • The ordering-dependent endpoint term is bounded by the arrangement’s block period multiplied by schedule reach, which measures how much weight the endpoint can place on one moment.A decaying schedule cannot pair a large step size with an uncontracted remainder, while a constant schedule can at the last step.Source: §1 · pp. 2–3
  • Across ten arrangements, the measured span is 11.63 contrast floors under a constant rate versus 2.20 under cosine, increasing monotonically with blocking under the constant rate.The schedule families differed only in lr_scheduler_type, with corpus, budget, and seeds held fixed.Source: §1 · p. 3
  • BLOCKED is the lowest-frequency alternation stopped at maximum swing; marking the convention in the prompt collapses the arrangement effect from 5.16 floors to 0.03 and reaches 87.5% of the union ceiling.The paper presents the key as a second channel that removes path influence, unlike schedule decay.Source: §1 · p. 3, Figure 1 · p. 5
  • Evidence is bounded to one budget on Qwen2.5-7B with two conflict constructions; Qwen3-8B-Base replicates the switch and key, but not the ladder criterion or mirror.The paper cautions that these results are not a general law of LLM training.Source: §1 · p. 4
  • Definition 2 separates capability from convention allocation using existing per-problem scores; C is corpus-comparable, while S equals convention policy only under a covariance condition that fails by 5.17 floors.The decomposition distinguishes what the model can do from which correct form it writes.Source: §1 · pp. 4–5

2 Related Work

The related-work discussion distinguishes order effects on capability from effects on convention allocation, and argues that the learning-rate schedule helps explain when ordering survives to the endpoint. It also situates the paper between corpus-scale mixing, within-batch interventions, and stronger order-channel results in other regimes.

Evaluation and conflicting conventions

  • The paper separates convention allocation from capability and argues that an undecomposed accuracy cannot express the coordinate moved by conflicting but correct conventions.It connects this measurement issue to work on surface-form competition and evaluation metrics.Source: §2 · p. 7

Inside the wall: batch composition, shuffling, and per-step gradient surgery

  • Batch composition and per-step gradient interventions operate inside the averaging wall, while the paper reports no allocation movement from batch-sized changes below corpus scale.Its block sweep finds momentum-window changes do not move the conflict signal; a one-shot commutator score is inverted on the tested pairs.Source: §2 · p. 8

The wall from the other side: data mixing at corpus scale

  • Corpus-scale reweighting changes source proportions in the mean field, whereas path arrangement changes order; moving the ratio from 1:1 to 2:1 shifts allocation from 0.453 to 0.312.The passage compares these measured shares with mean-field predictions of 0.500 and 0.333.Source: §2 · p. 8

Sequential training and forgetting

  • The account distinguishes conflict-driven allocation shifts at nearly fixed capability from continual-learning forgetting, which is measured as capability loss.Related work also links sequential-training recency in representations to the paper’s policy-level readout.Source: §2 · pp. 8–9

Curriculum, ordering, and their nulls

  • Ten paths on one conflicted corpus span 11.63 contrast floors at a constant rate but 2.20 under cosine, matching the schedule-dependent ordering account.The comparison holds the corpus, budget, seeds, and paths fixed while changing lr_scheduler_type.Source: §2 · pp. 6, 9
  • Prior ordering results are divided, and the paper frames schedule decay as an averaging operator that can reconcile effects observed under different schedules.It also cites a modular-arithmetic counter-regime where ordering acts as a covert information channel, unlike fine-tuning a pretrained 7B model.Source: §2 · pp. 9–10

Order effects at LLM fine-tuning scale

  • For LLM fine-tuning, model merging across orderings is described as consistent with averaging away their differences, though the cited work uses heterogeneous data.The authors explicitly characterize this comparison as consistent rather than a replication.Source: §2 · p. 10

Theory of the training path

  • The paper presents its contribution as transporting classical averaging and incremental-gradient results to convention allocation, rather than introducing a new averaging theorem.The cited theory is used for signs and orderings, not magnitude prediction.Source: §2 · pp. 8, 10

3 Method: the Path, the Channel, and the Two Coordinates

The method defines training order as a path, separates endpoint capability from convention allocation, and uses averaging results to analyze how path and schedule shape the endpoint. Its main qualifications are that the theory does not predict effect magnitudes and that capability and allocation can be coupled across problems.

3 Method: the Path, the Channel, and the Two Coordinates

  • The data path orders a fixed two-source multiset, and the division ladder varies same-source block length from fine alternation to coarser arrangements.Each source has 864 examples; the training budget is 324 steps, with BLOCKED using two contiguous 162-step stages.Source: §3 · pp. 10–11
  • The endpoint decomposition reports capability as acc_A+acc_B and allocation as the share assigned to convention B, while the measured across-problem covariance limits interpreting allocation as a pure convention policy.The solved/half-solved split differs by 0.0988 in B share, so capability conservation remains reported but the coordinates are not fully orthogonal.Source: §3 · pp. 11–12
  • As step size and alternation period shrink with fixed training horizon, the trajectory converges to a mean-field flow depending on source proportions, whose contested-input minimizer is a mixture.The parameter-space identification additionally assumes the model family can represent that mixture.Source: §3 · p. 12
  • Interventions preserving the period-average field leave the endpoint unchanged in the infinite-division limit, including fixed-block batch composition and time-symmetric within-block reorderings.The paper identifies both as pre-registered mechanisms ruled out by the averaging result.Source: §3 · p. 12
  • Reordering cannot move the optimum because it does not change what a held-out query conditions on, but it can move a finite run’s endpoint; a query-time key changes the optimal policy.Source: §3 · pp. 14–15
  • The ordering-dependent endpoint term separates arrangement period from schedule reach, with the bound proportional to block period and the endpoint weight assignable to any moment.The schedule’s weight reflects how much a deviation can persist to the endpoint; the two factors enter separately.Source: §3 · p. 13
  • The ladder effect is expected to grow with block length, while BLOCKED is a full-period corner whose displacement depends on the whole schedule-weight profile.BLOCKED differs by 1.0 floor between schedule families, versus 17.0 floors at L=54.Source: §3 · pp. 13–14
  • A decaying schedule has less reach than a constant schedule at the same peak rate because large step size and an uncontracted remainder cannot coincide under decay.At fixed block period, the measured interior span is 2.20 floors under cosine versus 11.63 under constant rate.Source: §3 · p. 14

4 The Instrument, and What the Path Writes

This section examines how training arrangements and learning-rate schedules shape convention allocation while aggregate capability remains approximately conserved, and establishes the instrument’s scope and limits. It compares cosine and constant-rate behavior, tests mixture weights, and reports that a designed palindromic schedule does not reduce the conflict penalty.

4.1 The Instrument

  • The instrument uses synthetic shifted-answer and natural numeral-versus-spelled-answer conflicts, with matched halves and same-convention controls, trained as full-parameter SFT on Qwen2.5-7B.The halves are disjoint in problems, so convention disagreement belongs to the corpus rather than an individual example; evaluation measures conformance to each convention, not mathematical ground truth on cf2. The 200 held-out problems are a skill-sorted prefix, omitting calculus and physics and truncating combinatorics.Source: §4.1 · pp. 15–17
  • On the first build, the intended conflict arm moved only 0.10σ while the supposedly inert control moved 2.41σ; the shifted convention had lost to the pretrained prior.Pooling rows across skills and increasing epochs made the conflicting convention learnable, and the authors caution that the 12.29σ result depends on conflict being learnable at the chosen budget.Source: §4.1 · p. 18

4.2 Allocation Moves, Capability Does Not

  • 12.29× the noise floor, the exact-match switch is −0.2346, while crediting either convention makes it exactly zero under the capability sum rule.Across twelve synthetic arms, acc_A+acc_B stays within 9.7% of its mean as allocation share ranges from 0.04 to 0.87; natural and ratio corpora reproduce the pattern.Source: §4.2 · p. 18, Table 3 · p. 19
  • Among the 386 instances solved in both SHUF and BLOCKED, 48.2% change favored convention, with 179 A→B flips versus 7 B→A flips.The solved set barely changes (7.3% newly solved, 8.5% lost); committed arms put 26–30% of problems at 4/4 samples, while mixed arms put 64–68% in intermediate bins, with 25–30% unsolved in both.Source: §4.2 · p. 19, Table 4 · p. 20, Figure 6 · p. 20
  • Capability’s trajectory dip precedes the commitment shift: it spends 97.4% of its excursion in the first four steps, while allocation spends only 4.5%.Capability falls from 0.4900 to 0.3463 and later returns to 0.4191; the later allocation sweep runs from 0.03 to 0.72, and a drop in the sum reflects answers matching neither convention, though accuracy alone cannot identify why.Source: §4.2 · pp. 20–21, Figure 7 · p. 21

4.3 The Ladder: What the Path Moves, and What This Design Can See

  • The cosine-schedule ladder’s flat interior replicated across Qwen2.5 and Qwen3, but the registered threshold was mis-specified and the post-hoc cluster count was two states in each family.The natural-corpus ladder was also flat at a third of the step size, but because it used cosine it does not test schedule effects.Source: §4.3 · pp. 23–24
  • Both schedules ran to the same step, and the matched ten-arm, one-corpus, one-budget, same-path, same-seed ladder showed a constant-rate interior span of 0.2221 (11.63 floors), resolved, versus cosine’s 0.0420 (2.20 floors), below design resolution.Source: Table 5 · p. 24
  • The constant-rate family’s capability spans 0.0680 across ten arms, versus 0.0246 for cosine, making it a noisier basis for fixed-capability claims.For that reason, the section reports the conservation result from the cosine family rather than the constant-rate family.Source: §4.3 · p. 25
  • The 2.20-floor cosine interior span is below the design’s 3.72-floor MDE80 at k=4 and remains below 2.33 floors even with infinite evaluation samples.The paper treats 0.6074 as a generous upper bound on evaluation sampling’s share; its harness largely cancels evaluation sampling across seeds, so increasing k does not resolve the interior there.Source: §4.3 · pp. 25–26
  • Paired TOST certifies equivalence for only one of seven interior arms, while six are neither shown equivalent nor different.The 90% intervals are two to four times wider than the frozen 0.0468 equivalence bound.Source: §4.3 · p. 26
  • The difficulty coupling varies by 0.0817 across interior arms, nearly twice the interior span, so an interior-sized difference cannot be attributed to convention allocation.The between-arm allocation-share ICC is −0.076 (95% bootstrap interval [−0.409, +0.166]; permutation p = 0.61), supporting only that between-arm dispersion does not exceed within-arm seed noise.Source: §4.3 · p. 26
  • 21.99 floors separate BLOCKED from the interior, and its corner displacement is schedule-free because the whole weight profile sets it.The cosine averages arrangements in the tail, whereas at the corner τL = T no part of the profile can be deleted by a schedule; the central corner result is unchanged across families.Source: §4.3 · pp. 24, 25, 27
  • The stopping-phase amplitude turns on at L=54 on Qwen2.5 but remains within the flat region on Qwen3, while the blocked end is undiminished on both.The reported values are 0.4922 versus 0.407–0.449 on Qwen2.5, 0.4547 versus 0.424–0.479 on Qwen3, and 0.8907 against 0.8692 at the blocked end.Source: §4.3 · p. 27
  • At parity and 2:1, the shuffled arm’s allocation shares are 0.453 and 0.312 versus mean-field predictions 0.500 and 0.333, both within the frozen 0.05 tolerance.The 3:1 test was excluded by its learnability guard and redesigned at 2:1; the blocked arm instead remains at 0.731/0.699 across ratios.Source: §4.3 · pp. 27–28, Table 6 · p. 27, Table 7 · p. 28
  • The palindromic schedule is roughly four times worse than blocked, and refining the step size 16× moves the penalty only in the fifth decimal.The registered S1 and S2 predictions fail; the authors locate the penalty in coarse arrangement structure, which palindrome symmetrization does not shorten.Source: §4.3 · pp. 29–30, Table 8 · p. 29

5 The Two Non-Average Terms: the Stopping Phase and the Key

The stopping phase explains BLOCKED as an endpoint on an alternation cycle, while the prompt key removes convention-selection effects but cannot recover a convention the model never learned. The stopping-phase reading is post hoc, and the key’s remaining coverage explanation is supported but not established.

5 The Two Non-Average Terms: the Stopping Phase and the Key

  • Table 9 places the stopping phase at 23.89 contrast floors, while the block-length ladder ranks fifth of six at 4.46 floors.The path is a narrow channel, and three axes not tied to path properties move allocation further than the ladder.Source: §5 · p. 30
  • At L=54, reversing the final block shifts acc_A by +0.0850 and acc_B by −0.0875 while capability moves 0.0025, consistent with an antisymmetric stopping-phase shift.At L=27, both constructions sit at the floor and the mirror produces no shift.Source: §5 · p. 31, Figure 10 · p. 31
  • The stopping-phase reading treats BLOCKED as the lowest-frequency alternation, stopped at maximum displacement, rather than as a distinct mechanism.The reading is post hoc; the ladder cannot refine the region where the stopping phase turns on without changing what is learned.Source: §5 · pp. 30–32
  • The mirror relocates allocation across the mean-field point, from 0.8692 to 0.0343, while capability moves 0.0020, inside its 0.0270 floor.The mirror’s unequal shift magnitudes remain unresolved because measured small-accuracy drift is 34% of their gap.Source: §5 · pp. 32–33

5.1 The Key: Training’s Index Channel

  • The marker’s contribution is its use in contradictory supervision: it collapses the arrangement effect, while the residual shortfall reflects coverage of the minority convention at write time.The authors connect the residue to 0.3767 versus 0.0025 in the marked-only evaluations.Source: §5.1 · p. 34
  • The prompt key reduces the arrangement effect from D = −0.0985 (5.16 floors) to −0.0006 (0.03 floors) and reaches 87.5% of the additive ceiling.On Qwen3-8B-Base, D falls from −0.0875 (4.58 floors) to −0.0125 (0.66 floors).Source: §5.1 · pp. 34, 36
  • The key cannot recover an unwritten convention: marked B-ONLY scores 0.3767 on numeral evaluation, while marked A-ONLY scores 0.0025 when asked for a word.Numerals are the pretrained default, whereas the spelled form must be acquired.Source: §5.1 · p. 34, Figure 11 · p. 35
  • The coverage explanation remains a necessary-condition reading, not a proof, because Qwen3’s other pretraining differences could also explain its larger residue.The difficulty-matched control remains missing.Source: §5.1 · pp. 36–37

6 Threats to Validity

The paper qualifies its claims through cross-family replication checks, a voided schedule-confound control, and limits on the corpus and readouts. Several registered results remain unresolved or require stricter controls.

Registered claims

  • Table 12 records the registered claims as a distribution: three mechanisms died, four predictions held, one control was void, and one instrument build inverted.The ledger includes the full registered record, with rules frozen in advance and commit and interval details in Appendix D.Source: §6 · p. 37, Table 12 · p. 38

Scope

  • The ladder, mirror, ratio, and key experiments use one model, one budget, and two conflict constructions; only the switch has broader companion-paper measurements.The companion reports the switch at 3B, 7B, and 14B and on a second family, but its learnability guard is close to the floor where the earlier build failed.Source: §6 · pp. 39–40

Cross-family checks

  • The ladder’s registered cross-family result misses its two-floor bar, while the key replicates; under Proposition 1’s post-hoc criterion, both families resolve to two endpoint states.The ladder threshold was checkable before the run but was not checked against the published Qwen2.5 row.Source: Table 11 · p. 38, §6 · p. 40

Schedule confound

  • The registered single-cosine control is void: its total accuracy is 0.4442 against 0.5125 for BLOCKED, a 2.5-capability-floor gap.The arms were not comparable in what they learned, so the allocation comparison is reported as an instrument result rather than repaired by adjusting the budget.Source: §6 · pp. 39–40

Schedule control

  • With the path held identical, restarting the cosine moves allocation from 0.4071 to 0.2912 at L=27, while capability changes remain below the two-floor bar.The measured movement is Δ = −0.116, t = −10.1, or 6.07 contrast floors; both tested rungs move in the same direction, excluding the specified stopping-phase account.Source: §6 · p. 41, Table 13 · p. 42

Open tests

  • The difficulty-matched allocation readout and predictable-conflict corpus remain open tests; the latter could refute the mixture reading if a mixture persists.The paper says the difficulty-matched control is needed to distinguish an unwritten convention from one that is merely harder.Source: §6 · pp. 41–42

Predictable-conflict test

  • The predictable-conflict test is pending: its pooled readout was voided, and the paired within-problem contrast returned Δ = +0.0018, t = 0.23.A rebuild requires an orthogonality certificate, including a worst-stratum minority share of at least 0.20, checked on the pure arms before mixed training.Source: §6 · p. 42

7 Discussion: One Stream, Two Channels

The discussion frames training and inference as bounded-state systems with compression and indexed-access channels, then identifies what the evidence supports and leaves untested. Its generalizations are limited by borrowed context claims, measurement choices, and open cross-predictions.

One stream, two channels

  • The discussion maps CCH’s bounded state, compression, and indexed access onto training-time objects, with disagreement making channel choice visible in behavior.The correspondence is described as structural identification rather than analogy.Source: §7 · p. 43

Related work

  • The three related papers address query-time service, recipe-level movement, and what the training path writes through each channel.In the coherent-domain limit, the recipe-level paper’s output is path-independent within single-run noise.Source: §7 · p. 43

Corpus scope

  • The paper’s experiments cover competition mathematics with integer answers, and the convention is surface form; the both-correct natural corpus is the weaker instrument.Headline figures use cf2, where one half’s answers are shifted, while nat spells the same number differently; the published comparison also uses different learning rates.Source: §7 · pp. 43–44

Borrowed context

  • The paper borrows scale, prevalence, and harness claims from other work; every arm here is 7B or 8B, and harness-specific noise may understate external replication noise.The authors call scale the largest external-validity gap and note that the cited supporting works were not peer-reviewed at writing.Source: §7 · p. 44

Future work

  • Future work includes separating constant-rate order sensitivity mechanisms, increasing seeds, testing non-mathematical conflicts, and matching budgets for the switch.The discussion also flags a trajectory capability excursion, difficulty-weighted allocation, and an unrun constant-rate mirror as unresolved tests.Source: §7 · pp. 44–45

Mean-field test

  • The mean-field point tests miss toward the pretrained prior, but cannot distinguish prior bias from a failure of mean field.The authors identify a directional 1:1, 2:1, 4:1 sweep as a better test and did not run it.Source: §7 · p. 45

Cross-predictions

  • The proposed mixture-capacity wall and replay-as-deferral links are future tests, not claims; neither has been run.The proposed tests vary the number of mutually exclusive conventions or mix first-stage data into the final stage, respectively.Source: §7 · p. 45

Where the correspondence stops

  • The two-coordinate decomposition is written for parameters trained once over a fixed corpus, and its conservation is approximate and measured.The paper does not claim that the same decomposition applies to an accumulating store, where retrieval rules could determine allocation.Source: §7 · pp. 45–46

8 Conclusion

The conclusion presents an averaging theorem as the framework for understanding ordering effects and argues that conflicting supervision can shift convention commitment without necessarily changing capability. It also cautions that exact-match evaluation can mistake commitment for capability.

Conclusion

  • The ordering-dependent endpoint component is bounded by separate arrangement-period and schedule-reach factors; under decay, the path is compressed to its time-average.The conclusion attributes the non-average path term to where training stops, and says a query-time key collapses the arrangement effect.Source: §8 · p. 46

Evaluation

  • Exact-match evaluation reports commitment as if it were capability; benchmarks with multiple correct forms can use per-problem scores to recover the two coordinates.The conclusion says this correction requires no new experiment from the paper.Source: §8 · p. 46

A Supporting Measurements

The supporting tests find budget-dependent sign reversals in-model, but the one-shot commutator proxy does not predict their locations or order effects. Additional budget, learning-rate, and positive-control measurements clarify the regimes and the first build’s failure.

The commutator proxy, measured and rejected

The stopping term in-model: real, and not where the theory says

The same effect at three budgets, and at three learning rates

  • ASYM reverses from +0.0779 at the published budget to −0.0619 at the higher budget, showing that the measured order effect does not persist across budgets.At one epoch and lr 3×10−5, fine-tuning also lowers three of five skills relative to zero-shot, limiting interpretation to that regime.Source: The same effect at three budgets, and… · pp. 47–48
  • |D| rises monotonically from 0.0236 to 0.0653 as the learning rate grows, while the highest rate lowers capability accuracy to 0.268 against 0.340.The table reports three learning rates at three epochs, with three seeds each; the higher rate moves farther from the lazy regime.Source: The same effect at three budgets, and… · p. 48, Table 16 · p. 48

The commutator proxy, measured and rejected

  • At layer fraction 0.75, the commutator ranks |ASYM| with Spearman −0.543 at one epoch and −0.600 at three, the wrong direction at both budgets.The proxy is computed once at initialization; four of six measured pairs flip sign between budgets.Source: The commutator proxy, measured and… · p. 48, Table 17 · p. 49

The positive control’s first build, in full

B The Clustering Rule, and Why It Is Not a Channel Rate

The paper counts distinguishable endpoint clusters rather than treating a logarithmic count as a channel rate. The count is stable over a stated resolution range, while the range’s supported capacity and the path family’s realized states remain distinct quantities.

Why it is not a channel rate

  • Rcap changes by exactly one bit when resolution is halved, but Rreal can change by a cluster; at 1σ the first family reads 1.58 bits rather than 2.This difference is why range-supported capacity cannot be reported as the path family’s realized rate.Source: B The Clustering Rule, and Why It Is… · pp. 50–51

The clustering rule

  • Two endpoint clusters persist in both families from 2σ resolution upward, while the first family resolves a third state at 1σ.The between-cluster gap exceeds the largest within-cluster gap by 8.7× in the first family and 20.7× in the second.Source: Table 19 · p. 51, Figure 16 · p. 52
  • Rcap counts distinguishable values supported by the read-out range, whereas Rreal counts distinguishable endpoint values occupied by the paths actually run.The write channel is the induced map from path π to behavioral read-out Φ(π).Source: B The Clustering Rule, and Why It Is… · p. 51

Why it is not a channel rate

  • Neither rate is a mutual information: estimating one would require an explicit path prior and a noise model for seed dispersion, neither of which is estimated.The paper treats the logarithm of the realized cluster count as a naming convention, not a measured information rate.Source: B The Clustering Rule, and Why It Is… · pp. 51–52
  • Ten paths span a range that could support about ten distinguishable values at 2σ resolution, yet occupy only two endpoint states.The paper defends this gap as a count-based result, not as a specific integer independent of the resolution choice.Source: B The Clustering Rule, and Why It Is… · pp. 51–52

C Corrections to Earlier Versions of This Paper

The paper revises several earlier claims after additional measurements or corrected analyses. The corrections narrow what the results establish, including the schedule comparison, resolution-dependent counts, and limits of the evaluation design.

Corrections to earlier versions

  • The constant-rate ladder spans 0.2221 in the interior, or 11.63 contrast floors, against 0.0420 and 2.20 under cosine; the interior is monotone with blockage.The paper withdraws the earlier attribution of the cosine’s flat interior to the data path and reports four occupied states under constant rate.Source: C Corrections to Earlier Versions of… · p. 53
  • The earlier “1722 bits in, 1 out” compression claim was withdrawn because it compared the entropy of all orderings with a measurement from only ten executed paths.Ten codewords carry at most log2 10 = 3.32 bits.Source: C Corrections to Earlier Versions of… · p. 53
  • The detectable-effect figure and the claim that the interior span was unresolvable were both scoped to the evaluation design actually run.The earlier power figure used the wrong dispersion and degrees of freedom; at k=32, a span of this size is resolvable.Source: C Corrections to Earlier Versions of… · p. 53
  • The paper corrects claims: Remark 1 constrains the optimum, not a finite-run endpoint; Rreal reads 1.58 rather than 2 at 1̂σ on the first family; the 0.3204 marked mean is unreproduced.Source: C Corrections to Earlier Versions of… · p. 53

D The Registration Index: Every Threshold, and When It Was Frozen

The registration index records thresholds, their result files, and the outcomes of the registered tests. It reports both the lead time and evidence quality, including exceptions and corrections to the index itself.

The Registration Index

  • Seven of seventeen registrations confirmed at their primary branch, two decided later, two were partial, three voided on a guard, and three returned no branch.The index includes outcomes because thresholds alone could invite readers to assume that they were met.Source: D The Registration Index: Every… · p. 54, Table 20 · p. 54
  • The index distinguishes confirmed, partial, void, and missing outcomes rather than treating registration status as a grade.For example, CL-2 was the largest positive result while CM-4 was void; the branch labels do not rank them.Source: D The Registration Index: Every… · p. 54
  • Two intervals use shared-volume file timestamps rather than commits, which are weaker evidence; only NI remains recomputable because CM’s timestamp was overwritten.NI’s file was left untouched after its verdict so its interval could be recomputed.Source: D The Registration Index: Every… · p. 54, Table 20 · p. 54
  • Fifteen registrations were frozen about an hour to nearly three days before their result; Figure 17 measures the gap from threshold commit to verdict-file time.The index defines registration by the first commit carrying the threshold and uses the verdict file’s modification time as the result time.Source: Table 20 · p. 54, D The Registration Index: Every… · p. 54, Figure 17 · p. 57
  • The palindromic check’s thresholds and answer entered the repository in one commit, so it asserts an ordering but cannot demonstrate prior registration.The amplitude-onset registration was withdrawn before training because its named rungs could not be built at this budget.Source: Table 20 · p. 54, Figure 17 · p. 57
  • An audit corrected a half-row mismatch between the registered artefact and result file, changing the ladder’s interval from +23h 41m to +10h 30m.Both intervals were positive, but the earlier number flattered the authors; a script now rebuilds the table from git and the result base.Source: D The Registration Index: Every… · p. 56

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