PaperlayerView exact revision on arXiv

Source-linked AI summary

Commutator Memory: Sparse, Path-Local Reading and Steering in Language Models

John Sweeney

arXiv:2609.34348v1cs.LGcs.AIcs.CL

TL;DR

Aggregate benchmark differences reveal that training order matters but not where its interaction lands in output space. The paper projects the Lie bracket of two updates into a pair-specific token readout and tests localization, interventions, and endpoint assignment. The readout is localized and actionable in the reported settings, and the bracket identifies training order across four LLMs with 91.7% accuracy.

  • Problem

    Order effects are usually measured through aggregate benchmark deltas, leaving the output-space location of interactions between training stages unobserved.

    Source: §1 · p. 1

  • Method

    The method projects the Lie bracket of two updates into signed per-token scores for a source pair and evaluation slice, then tests localization, interventions, and paired-endpoint assignment.

    Source: §1 · pp. 2–3

  • Results

    91.7% accuracy across four LLMs: the paired-endpoint statistic identifies which order produced each k=1 SGD endpoint.

    Source: §8 · p. 10

  • Takeaways & Limitations

    The framework could support diagnosis of sequential post-training interactions by estimating which evaluation slices and output tokens carry the antisymmetric residue.

    Source: §6 · p. 10

    The evidence is scoped to specific protocols: SFT uses SGD with two sources, GRPO-style results use a frozen matched-rollout surrogate on a 0.5B model, and AdamW evidence is limited.

    Source: §7 · p. 10

Abstract

from arXiv · show

Gradient updates on different data generally do not commute: training a language model on two data sources in opposite orders gives different weights, even with the same data and total exposure. Loss or benchmark deltas show that the models differ, not where. We ask whether this path dependence leaves a parametric training-history memory: a weight component that flips sign when the two sources are swapped, is localized in output space, changes the held-out loss gap between the two orders under targeted interventions, and reveals which trained model came from which order. For one small SGD step of size $\eta$ on each of sources $A$ and $B$, the weight difference $\theta_{AB}-\theta_{BA}$ is, to leading order, $\eta^2 b_{AB}$, where $b_{AB}=H_Bg_A-H_Ag_B$ is the Lie bracket of the two gradient fields at the base model. We define commutator memory by projecting the bracket through the logits into one score per vocabulary token; the scores sum to the bracket's prediction of the gap. The scores are localized: on three models, the same readout of the measured $\theta_{AB}-\theta_{BA}$, or of a bracket from disjoint batches, shares 82-99% of the original top-20 tokens, versus 35-49% for norm-matched random directions. They are causally actionable: in Qwen-3-4B SFT, downweighting the ten tokens with the largest predicted share of the gap closes a median 32% of the measured gap, while frequency-matched tokens with near-zero scores have almost no effect. The weights themselves carry the component: projecting the difference between the two trained models onto $b_{AB}$ identifies which came from which order in 92% of cases across four LLMs (chance 50%). Controlled tests also cover matched-batch DPO, a frozen-rollout GRPO-style objective, and an AdamW endpoint check. The memory is defined per source pair, not per example, and its projection on $b_{AB}$ decays with further training.

1 Introduction

The paper asks whether training-order effects leave a localized, actionable trace in model weights rather than only aggregate benchmark differences. It defines this pair-specific trace through a Lie-bracket token readout and evaluates its localization, causal effects, and endpoint readability.

1 Introduction

  • The order-asymmetric weight residue has a concentrated vocabulary-level signature that can be estimated from the base model, sources, and evaluation slice before either training order is run.This claim applies at calibrated step sizes where the second-order expansion holds.Source: §1 · pp. 1–2
  • The proposed memory is pair-conditioned parametric training history: swapping two updates flips its sign, and it is intended to be localized, causally actionable, and readable from paired endpoints.The paper does not claim item-addressable recall, personalization, or long-horizon agent memory.Source: §1 · p. 2
  • The token readout decomposes the bracket’s order criterion into signed per-token scores and identifies tokens to downweight for a given source pair and evaluation slice.The implementation uses central finite differences for the logit readout.Source: §1 · pp. 2–3
  • The evaluation tests localization, causal effects, and paired-endpoint assignment, with additional matched-path experiments for DPO, a GRPO-style objective, and AdamW.The later regimes test whether the same object remains structured on realized update paths.Source: §1 · p. 3

2 Related Work

The paper situates commutator memory among work on training order, optimization geometry, continual learning, mechanistic localization, and training-history attribution. Its distinction is a signed, pair-specific weight-space residue decomposed over output tokens, rather than an activation trace or sample-level attribution.

Training order, curricula, and temporal traces

  • Prior studies link data or task order to performance and find activation-space recency traces; this work instead studies the weight-space noncommutativity of two updates and its attenuation under continued training.The paper frames its memory as parametric and pair-conditioned, not agent memory with storage and retrieval.Source: §2 · p. 3

Lie brackets and splitting geometry

  • The bracket identity and scalar order criterion build on prior splitting-geometry work, while this paper adds token localization, interventions, paired-endpoint readability, and an optimizer-state endpoint check.The related-work section distinguishes the AdamW check from the main SGD claims.Source: §2 · pp. 3–4

Continual learning and editing

  • Unlike continual-learning and model-editing methods, commutator memory resolves the order-specific residue of composed updates; unlike task arithmetic, it is an antisymmetric within-sequence component.Small masked bracket edits on Qwen-3-4B have near-additive loss responses, with 4.3% mean relative error.Source: §2 · p. 4

Mechanistic localization and data attribution

  • The attribution is to output tokens, not training samples or activation-level units, and endpoint assignment distinguishes training order rather than membership or initialization seed.The paper contrasts its target with mechanistic data attribution, membership inference, and seed fingerprinting.Source: §2 · p. 4

3 Background: Lie Bracket Analysis of Alternating Training

The background formalizes the difference between alternating training orders and approximates it with a Lie bracket under smoothness assumptions. It then projects that bracket into a target-loss criterion and a token-level decomposition.

3.1 Problem Setup

  • The setup compares held-out evaluation losses for endpoints produced by source orders AB and BA, with θAB and θBA denoting their final parameters.The ordering gap is the difference between those held-out losses.Source: §3.1 · p. 4

3.2 Lie Bracket Decomposition

  • Under the stated C3 smoothness and bounded-third-derivative assumption, the two-step SGD endpoint difference is approximated by η^2 times the Lie bracket bAB.The bracket is computed through Hessian-vector products; the paper cites Sweeney [2026a] for the endpoint identity.Source: §3.2 · pp. 4–5

3.3 Target Score and Trotter Reference

  • At the shared first-order Trotter reference, the bracket’s projection onto the target gradient gives a scalar order score; under stated conditions, a bracket-only step can outperform both pure orders to leading order.The condition is disadvantageous symmetric drift on the evaluation target and a correctly estimated sign of the score.Source: §3.3 · p. 5

3.4 Token-Level Decomposition

  • The token decomposition maps the parameter-space bracket through the logit Jacobian, with each τk attributing a share of the leading-order gap and positive τk disadvantaging token k under AB relative to BA.The aggregate identity is invariant to position-wise logit shifts.Source: §3.4 · p. 5

4 Experimental Setup

The experiments use several language models and two-source training pairs, with learning rates selected by an empirical locality check. Intervention and correction results are reported with safeguards for small baseline gaps and heavy-tailed closure ratios.

4.1 Models and Domains

  • Experiments span four main SFT models, six domains, all 15 domain pairs, and three seeds for sparsity and token-prediction runs; intervention trials use fixed pairs and 30 seeds.Held-out evaluation documents are disjoint from training, and DPO and GRPO-style protocols use separate matched data or rollouts.Source: §4.1 · p. 5

4.2 Closure metric and measurement floor

  • The closure metric compares edited and baseline ordering gaps, while pre-fixed baseline floors, median and winsorized summaries, and pair-level tests address noisy ratios near the measurement floor.For DPO, 11/12 source pairs meet the pre-specified bootstrap criterion and all 12/12 pair medians are positive; the passage cautions that this sign agreement is descriptive.Source: §4.2 · pp. 5–6

5 Results

The results test whether commutator memory is localized, interpretable, causally actionable, and readable from paired endpoints. Across the reported settings, the bracket-derived support tracks order effects, targeted interventions reduce held-out ordering gaps, and paired endpoints retain an above-chance order signal.

5.1 Localized: Sparse, Pair-Specific Token Readout

  • Bracket-derived token supports are pair-specific, with cross-pair Jaccard 0.004 on Qwen-3-4B; endpoint and resampled supports are compared with norm-matched random controls.Table 2 reports top-20 overlap for measured endpoint displacements and brackets from disjoint batches, alongside random-direction overlap.Source: §5.1 · p. 6, Table 2 · p. 6

5.2 Localized: The Sparse Support Is Interpretable

  • The high-mass token categories vary by domain pair, including domain markers, digits, and BPE fragments, while token contributions are aggregated over an evaluation slice rather than scored per sentence.Prediction-only tokens account for 30% of |τ| mass, and reference or evaluation-slice choices can change surface tokens while high-mass categories remain domain-linked.Source: §5.2 · p. 7

5.3 Agreement with the Scalar Bracket Predictor

  • Bracket-sign prediction accuracy was 72% on Qwen-3-4B and 100% on Qwen-2.5-1.5B and Llama-3.1-8B, versus 11–56% for first-order baselines and 39–50% for random.The evaluation used 18 pair-seed units per model; the Qwen-3-4B Magnus-expansion validation had R2 = 0.994.Source: §5.3 · pp. 7–8

5.4 Causal: Token Interventions Close the SFT Gap

  • 32% was the median held-out ordering-gap reduction on Qwen-3-4B after intervening on the ten predicted harmful tokens; frequency-matched near-zero-score controls showed no effect.On Qwen-2.5-1.5B fp32, the predefined above-floor subset showed a +47.9% median gap reduction with random controls near zero; these cross-model magnitudes are not directly compared.Source: §5.4 · p. 8, Table 3 · p. 8

5.5 Readable: The Endpoint Pair Identifies Which Order Was Trained

  • The paired-endpoint assignment interval was 83–96%, above 50% chance, and the antisymmetric statistic stayed above chance at all four tested model scales.The antisymmetric difference removes a shared first-order reference offset and the second-order symmetric term.Source: §5.5 · pp. 8–9, Table 4 · p. 9

5.6 Controlled Tests Beyond Supervised Fine-Tuning

  • Across k = 1, 2, 4, and 8 SGD steps per source, at least 98% of k=8 empirical top-1% label tokens were in the single-step top-1% τ set.On Qwen-3-4B, τ-to-output-row displacement Spearman ρ declined from 0.83 at k=1 to 0.71 at k=8, even as the scalar BCH forecast degraded at the largest calibrated step size.Source: §5.6 · p. 9
  • Iterative projection toward the observed reverse-order endpoint closed about 90% of the gap on 5/6 above-floor SFT pairs, and a conditional bracket-only update beat the better pure order in 12 of 13 eligible units.The conditional result applies when the estimated symmetric drift is target-disadvantageous and the sign of σ is estimated correctly.Source: §5.6 · p. 9
  • Matched-batch DPO correction closed a median 77% of the held-out gap, while continued third-domain training reduced the median retained aligned trace from 27% to 7% over sixteen updates.The DPO strict-CI criterion held for 11/12 source pairs; in the continued-training study, some runs reversed sign.Source: §5.6 · p. 9

6 Discussion

The bracket is a local correction tied to the realized update path, so matched-path tests are central to interpreting its closures. Under a target-disadvantageous symmetric drift and a correctly estimated bracket sign, a bracket-only step can outperform either pure order to leading order.

6 Discussion

  • The bracket is a local correction for a realized path; matched-batch SFT/DPO and matched-rollout GRPO closures use the updates that produced the endpoints.Calibration on unmatched GRPO rollouts is the negative control.Source: §6 · p. 10
  • The practical aim is controlled measurement of sequential post-training interactions, not a deployment tool: the bracket remains tied to realized data, optimizer state, and a calibrated step-size window.The proposed pipeline estimates affected evaluation slices and output tokens before testing reordering, token-local reweighting, or matched-path correction.Source: §6 · p. 10
  • When symmetric drift on the evaluation target is disadvantageous and the bracket sign is estimated correctly, the bracket-only step beats the better pure order by η^2µ to leading order.The closure identity θAB − η^2bAB = θBA + O(η^3) motivates the matched-batch DPO edit.Source: §6 · p. 10

7 Limitations

The framework measures a path-local, pair-conditioned antisymmetric component, with evidence bounded to specific training and evaluation settings.

7 Limitations

  • The evidence is scoped to two-source SGD paths with k = 1–8, matched-batch fp32 DPO, and a frozen matched-rollout KL surrogate on a 0.5B model with analytic rewards; the trace attenuates with continued training.The paper claims neither item-addressable recall nor broad capability gains; AdamW evidence is limited and uses an optimizer-state commutator.Source: §7 · p. 10

8 Conclusion

Commutator memory uses one bracket to forecast order effects, report sparse token-level scores, define a path-local parameter direction, and assign paired endpoints to their training order.

8 Conclusion

  • Across four LLMs, the paired-endpoint statistic identifies which order produced each k=1 SGD endpoint correctly 91.7% of the time.The statistic projects the endpoint difference onto the bracket direction.Source: §8 · p. 10

Broader impact and release statement

The framework locates training-update noncommutativity in vocabulary space and may support more interpretable diagnosis of capability shifts in sequential post-training pipelines. The release is limited to order-gap measurement.

Broader impact and release statement

  • The authors have not evaluated bypassing safety training, private-data recovery, or selective capability editing.Released material is limited to order-gap measurement.Source: Broader impact and release statement · p. 14
  • The supplementary archive includes source code, fixed protocol definitions, result files, and analysis scripts; experimental settings are documented in Appendix AE.Token-level interventions, DPO correction, and the GRPO-style test follow the fixed protocols.Source: Broader impact and release statement · p. 14

A Bracket derivation and optimizer connection

The appendix derives the SGD order effect from a bracket, develops conditional bracket-based loss and endpoint results, and details the DPO readout and correction conventions.

A Bracket derivation and optimizer connection

  • For two single-step SGD sources, the endpoint difference is η^2b_AB + O(η^3), with b_AB = H_Bg_A − H_Ag_B; this bracket is the optimizer connection’s curvature.The η^2 bracket is the antisymmetric order defect, whereas the η^2 symmetric component is shared by both orders.Source: A Bracket derivation and optimizer… · p. 14
  • When μ>0 and η is sufficiently small, a correctly signed bracket-only step at the drift-matched reference has lower target loss than the better pure order.The leading improvement is η^2μ; the empirical subset with μ>0 had 12/13 successful trials.Source: A Bracket derivation and optimizer… · pp. 15–16
  • Paired endpoint subtraction cancels the leading symmetric drift; across four models, the statistic achieved 66/72 correct assignments (91.7%; Wilson 95% CI [83%, 96%]).The finite-step conclusion requires the leading signal to dominate remainders and estimator error.Source: A Bracket derivation and optimizer… · pp. 16–17

B τ implementation: native-logit convention and finite differences

  • The token readout decomposes the bracket’s logit effect by vocabulary item; its aggregate is invariant to position-wise logit shifts, but individual scores depend on native logit coordinates.The logit-space JVP is computed by central finite differences after exact HVPs; the aggregate approaches η^2σ as ε→0.Source: B τ implementation: native-logit… · p. 17

C Precision policy for HVP computation

  • The Qwen-2.5-1.5B intervention used full fp32 because its gap was about 10^-4, and reports only the predefined above-floor subset (28/90 trials).For Qwen-3-4B, the reported intervention used bf16 storage with fp32 HVPs and perturbed parameters.Source: C Precision policy for HVP computation · p. 18

D Matched-batch DPO correction: per-pair table and figure (Qwen-3-4B fp32)

  • Table 5 reports matched-batch bracket-correction closure for six original DPO source pairs on Qwen-3-4B fp32, with λ=1 edit summaries and wrong-sign and random controls.Its three-seed cluster intervals are descriptive diagnostics, not formal 95% inference.Source: Table 5 · p. 18

E Matched-batch DPO correction: λ-sweep and control-direction signs

  • λ=1 peaks at 0.78 median closure; λ=0.5 closes about half the gap, while λ=2 averages −0.01.Random directions have closure approximately zero; wrong-sign edits have winsorized mean closure −1.21, and first-order norm-matched controls −0.93.Source: E Matched-batch DPO correction:… · pp. 18–19

F DPO closure-ratio statistical convention

  • All 12/12 DPO source-pair medians are positive for bracket-minus-random closure, but only 11/12 pass the flat-bootstrap strict-CI criterion.Each pair has three seed-index clusters, so cluster intervals are descriptive; qualitative control rankings do not depend on the statistic choice.Source: F DPO closure-ratio statistical… · p. 19

G DPO extended methodology and per-pair detail

The DPO analyses relate token-level readouts to parameter corrections, check the token-sum identity, examine endpoint closure, and test whether bracket scores recover empirically affected tokens.

G DPO extended methodology and per-pair detail

  • DPO token reweighting and direct parameter correction use the same bracket but intervene differently, so their magnitudes are not directly comparable.Within each protocol, bracket-versus-random separation is comparable; together the tests support τ as a readout of a parameter-level direction controlling the aggregate order gap.Source: G DPO extended methodology and… · p. 21

H DPO τ token-sum identity check (original pairs, per pair)

I Per-pair endpoint-correction closure (Qwen-3-4B, k = 1)

J Token-level ordering prediction: support recovery and rank-correlation discussion

  • On Qwen-3-4B, predicted top-1% tokens recover 53% of the empirical top-1% set on average, versus a 1% equal-size-set baseline; full-rank Spearman is uninformative under sharp concentration.On Qwen-2.5-1.5B fp32, full-vocabulary Spearman is ρ=0.52; parameter-row tests give per-pair ρ=0.78–0.90.Source: J Token-level ordering prediction:… · p. 22, Table 9 · p. 22

K Paired-endpoint training-order readability

The paired antisymmetric statistic reads training order from model endpoints, while DPO, triplet-edit, SVD, and null-control tests probe the scope and specificity of the same bracket-based account.

K Paired-endpoint training-order readability

  • 91.7% of paired endpoints were assigned correctly across four unique models using the sign(∆s) estimator.The table counts 66/72 assignments; the dual-precision count is a replication check.Source: Table 10 · p. 23

L DPO commutator memory: full protocol and per-pair table

M DPO token reweighting on fresh pairs

  • The fresh-pair token-reweighting arm had only three evaluable pairs and is treated as descriptive, with six-pair matched-batch correction serving as the confirmatory fresh-pair evidence.The three available pairs used the full token-reweighting protocol as a partial replication.Source: M DPO token reweighting on fresh pairs · p. 24

N Matched-batch DPO correction on fresh pairs

O Seed-clustered bootstrap diagnostics for the DPO correction

  • With three seed-index clusters per pair, seed-clustered bootstrap intervals are descriptive diagnostics rather than formal 95% inference.Under this bootstrap, 3/6 original-pair intervals exclude zero; all 12/12 pair medians remain positive across the original and fresh grids.Source: O Seed-clustered bootstrap diagnostics… · pp. 24–25

P Triplet additivity of bracket edits

Q Descriptive SVD of the cross-pair τ-matrix

  • The top two singular directions carried 96.1% of the six-vector τ-matrix Frobenius mass, but the SVD is descriptive for one model, seed, and parameter subspace.The rows were not norm-matched, and the principal directions are not interpreted semantically.Source: Q Descriptive SVD of the cross-pair… · p. 26

R Null baselines for support specificity

  • Empirical endpoint and resampled-bracket directions preserved 82–99% of bracket top-20 support across three models, versus 35–49% for random directions.Across models, the stable claim is support specificity rather than concentration alone; top-20 support was also stable across tested precision settings.Source: R Null baselines for support specificity · p. 27

S Observable/null decomposition of the training commutator

The observable/null analysis estimates how much of the training commutator is represented by an aggregated logit readout, then details calibration and four intervention approaches.

S Observable/null decomposition of the training commutator

  • 77–89% of the bracket’s squared norm was assigned to the top-K observable component in the selected subspace, with 5–8% further observable outside top-K and 6–15% unexplained.These are approximate empirical fractions from three Qwen-3-4B pairs; the unexplained component mixes readout-null directions and numerical approximation error.Source: S Observable/null decomposition of the… · p. 28, Table 16 · p. 28
  • The decomposition applies only to the last-block MLP plus LM-head subspace and is approximate, using a four-batch operator subset, finite CG, nonzero damping, and one seed across three pairs.The reported projection coefficients are batch-subset-specific estimates.Source: S Observable/null decomposition of the… · pp. 28–29

T.1 Calibration regimes

  • Calibration targets a BCH-local window where bracket signs are stable, bracket displacement remains subordinate to the first-order update, and wrong-sign controls follow local antisymmetry.Too-small steps yield undetectable signals; saturated steps indicate that second-order truncation is no longer the right local model.Source: §T.1 · p. 29

T Learning Rate Calibration Details

  • The procedure probes short runs across candidate learning rates, classifies scaling regimes, and searches for the cube-root boundary; reported target-model SFT rates use fixed calibrated values.All target models are reported in the cube-root regime; GPT-2 is reference only.Source: §T.2 · p. 29, §T.3 · p. 29, Table 17 · p. 29

U.1 Loss Reweighting (Main Method)

  • Loss reweighting downweights positions labeled with the ten highest-|τk| harmful tokens while mean-preserving normalization keeps the mean position weight at one.The main runs use α = 0.1, a 90% downweight; harmful tokens are selected by the sign of the measured baseline gap.Source: §U.1 · p. 30

U.2 Learning Rate Scaling

  • Learning-rate scaling changes only gradients for the harmful tokens’ lm_head rows, avoiding the hidden-state pathway affected by loss reweighting.The passage presents this as a potentially less noisy targeting choice for small-gap models.Source: §U.2 · p. 30

U.3 Projection (PCGrad-Inspired)

  • Projection of harmful-token gradients orthogonal to helpful-token gradients had weaker effects than loss reweighting on Qwen-3-4B.The dose-response sweep reports a positive response at every tested α < 1 for harmful-token downweighting, while random controls remain near zero.Source: §U.3 · p. 31, Figure 4 · p. 31

U.4 Regularization

  • Regularization pulls harmful-token lm-head rows toward initialization with λ = 10^-3, but was less effective than direct gradient manipulation.The comparison is with loss reweighting and learning-rate scaling.Source: §U.4 · p. 31

U.5 Comparative Results

Targeted token edits reduce ordering gaps, while the remaining gap and a separate bracket-validation test show that the leading commutator captures substantial order-dependent structure. The supplementary trial records document the intervention runs.

U.5 Comparative Results

  • 32% median gap closure on Qwen-3-4B contrasts with approximately 0% for random tokens under held-out evaluation.The intervention downweights predicted harmful tokens; Table 18 distinguishes the held-out loss-reweighting result from earlier runs without held-out evaluation.Source: §U.5 · p. 31, Table 18 · p. 31
  • +47.9% harmful closure and −0.003% random closure on Qwen-2.5-1.5B fp32 apply only to 28/90 above-floor trials.This is a conditioned fp32 result, and its magnitude is not directly comparable to Qwen-3-4B.Source: Table 18 · p. 31, §U.5 · p. 32

V What remains after the token edit

  • After the edit, the leading-order commutator predicts 71% of the pre-intervention measured gap, but this differs from the 68% complement of median closure.The step from the actual-update linearized gap to the endpoint gap reduces the remainder for all three source pairs.Source: V What remains after the token edit · p. 32

W Magnus expansion validation

  • Mean R2 = 0.994 for the k² fit; all 12 signs agree, while the measured-to-predicted ratio declines from 1.004 at k = 1 to 0.74 at k = 8.The validation uses three seeds and four step counts; higher-order BCH terms account for the pooled slope being below one.Source: W Magnus expansion validation · p. 32, Figure 5 · p. 33

X Per-Trial Results

  • Supplementary per-trial records cover 90 Qwen-3-4B trials across three domain pairs and 90 Qwen-2.5-1.5B fp32 trials.The records include baseline and intervened gaps and percentage gap reduction.Source: X Per-Trial Results · p. 33

Y Interpretability extended

The extended analyses examine the scope and limits of order-specific commutator readouts, including their token and domain localization, iterative correction, persistence, and behavior under continued training and optimizer variants. They also evaluate a frozen-rollout GRPO-style surrogate’s matched-path predictions and block-sequential correction, alongside curvature approximations, scaling, and calculation costs.

Y Interpretability extended

  • On Qwen-3-4B, top tokens are domain-specific: code-vs-legal is dominated by digit tokens, while code-vs-biomedical includes BPE suffix fragments.Across code-vs-news, 17/20 top tokens are complete words and carry 87% of |τk| mass; across code-vs-legal, 9/20 are single digits and contribute 92% of mass, with token “0” alone at τ = +23.1. The three categories—domain markers, BPE fragments, and digit asymmetry—reflect large error-weighted logit responses; 30% of mass comes from unobserved tokens, and cancellation is rare (11% of tokens).Source: Y Interpretability extended · pp. 33, 35, Figure 6 · p. 34
  • 19–3700× larger |τk| magnitudes occur in Qwen-3-4B than Qwen-2.5-1.5B, while within-family top-token overlap is 28.6× above the equal-size baseline on Qwen pairs.Overlap is rank-order agreement, not absolute-magnitude convergence: top-20 mean |τ| differs by over four orders of magnitude across four models, with an approximately 30,000× residual after η2 normalization. Cross-family comparisons use semantic categories because Qwen and Llama use different tokenizers.Source: Y Interpretability extended · pp. 33–35

Z Tables: ordering baselines, multi-step persistence, scaling in k

  • τ-vs-∆W rank correlation decays as ρ(k) ≈0.83 −0.055 · log k across six Qwen-3-4B domain pairs for k ∈{1, 4, 8}.The mean decay rate is 0.055 ± 0.015 (r2 > 0.95); with fresh batches per step, the multi-step persistence gap grows 3.1× at k=8, a different check from the repeated-batch k2 analysis.Source: Z Tables: ordering baselines,… · p. 35, Table 21 · p. 36

Iterative correction: per-position trajectories and downstream metrics

  • 90% of the held-out gap closes on five of six pairs with iterative bracket correction, versus 13.5% for a single-step full-bracket edit; the near-noise-floor news-vs-biomedical pair diverges.The iterative projection reaches over 85% closure within 20 iterations on five pairs; the reported 90% is averaged over those five pairs, while 13.5% averages the single-step edit over all pairs and step counts.Source: Figure 9 · p. 36
  • 85% mean per-position convergence is reached across five domain pairs, with top-gap tokens reaching 93–97% of their θBA targets.For a representative code-vs-math pair at k=1, teacher-forced NLL closes 14.2% on GSM8K and 20.4% on HumanEval; discrete accuracy changes by at most 7 percentage points and are not tested for significance. Across 34 non-divergent experiments, median selectivity ratio |movementB/movementA| is 1.04.Source: Iterative correction: per-position… · pp. 36–37

AdamW augmented-state commutator: endpoint check

  • 79.5% mean NLL-gap closure (median 83.8%, 95% CI [72.6%, 85.9%]) is achieved at λ=1 by the lifted-state AdamW edit.Across 18 experiments, the predicted endpoint difference has mean parameter cosine 0.998 and relative endpoint error 0.068; random and wrong-sign controls produce approximately null and strongly negative closure, respectively.Source: AdamW augmented-state commutator:… · p. 39

Continuation study: retention and attenuation

  • Four later SGD updates pass the pre-set continuation rule, but eight and sixteen do not; median retained trace falls from 0.27 to 0.14 to 0.07.The aligned trace persists over a short horizon and then attenuates in the median, with sign reversals in some runs; this does not establish behavioral retrieval or long-term memory.Source: Table 22 · p. 40, Continuation study: retention and… · p. 41

Cost accounting and cheaper curvature approximations

  • The exact matrix-free bracket calculation costs about six forward–backward equivalents per source pair, plus one held-out gradient and three forward-only passes for the token readout.For N sources sharing a base model and evaluation loss, reusing one HVP per source ranks all pairs with N rather than N(N −1) HVPs; reused and direct scores differ by at most 1.6 × 10−5 on the Qwen-3-4B code-vs-news test.Source: Cost accounting and cheaper curvature… · p. 41
  • A streamed diagonal approximation for the final vocabulary layer accumulates its Hessian diagonal in one pass but discards cross-coordinate interactions; reported scores use exact matrix-free HVPs.Block-diagonal or low-rank generalized Gauss–Newton approximations retain more interactions, but are not automatically HVP-free; fidelity should be tested on held-out ordering signs and largest token effects.Source: Cost accounting and cheaper curvature… · p. 41

Frozen-rollout GRPO-style test: protocol and controls

  • The GRPO-style test freezes shared sampled rollouts and uses analytic reward sources for math correctness and brevity/format compliance.The bracket is computed at θ0 on the realized rollouts, and endpoints are trained on matched rollout records.Source: Frozen-rollout GRPO-style test:… · p. 43
  • In the single-step matched-rollout BCH check across 20 trials in 5 seed clusters, identity cosine had median 0.999 and projection coefficient ζ had median 1.002; all checks passed together in 17/20 trials.Source: Frozen-rollout GRPO-style test:… · p. 43
  • T⋆ = 16 was selected: all evaluable block lengths passed the fixed identity, projection, and parameter-space closure criteria.The selection rule used the largest T in {2, 4, 8, 16} meeting the criteria, without reward-outcome tuning.Source: Frozen-rollout GRPO-style test:… · p. 44, Table 25 · p. 44
  • The GRPO-style bracket produced a near-exact matched-path closure, but a bracket from one realized path is not expected to transfer to another.Wrong-sign edits obeyed antisymmetry; random controls stayed near zero in log-probability and loss, while static cross-rollout controls failed.Source: Frozen-rollout GRPO-style test:… · p. 44

Related papers

Loading 2609.34348v1…