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arxiv: 2605.17482 · v3 · pith:UAP4WPZDnew · submitted 2026-05-17 · 💻 cs.CL · cs.LG

RSD: Moving Local Triangular Charts for Auditing Language-Model Hidden States

Seungmin Jin This is my paper

Pith reviewed 2026-06-30 19:07 UTC · model grok-4.3

classification 💻 cs.CL cs.LG
keywords Relational Semantic Decompositionhidden statessense relationsmembership chartco-membershipGPT-2Word-in-Contextresidual audit
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The pith

RSD fits a shared three-anchor membership chart to repeated word occurrences to audit where sense relations appear in language-model hidden states.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Relational Semantic Decomposition as a diagnostic method that fits a moving local triangular chart to hidden states for one target word at a time. For repeated occurrences, a shared three-anchor membership matrix produces an invariant co-membership relation that can be checked against external same-sense versus different-sense labels. On GPT-2 and the full WiC training set, this root chart aligns with the labels for 16 of 53 eligible words. The audit also flags residuals left outside the chart and cases that may require attention-derived relations instead. A reader would care because the approach isolates which semantic structure is already visible inside the model's own state vectors rather than measuring end-task accuracy.

Core claim

RSD fits a shared three-anchor membership chart S_t at layer or token-time t for repeated occurrences of one target word. The hidden-state channel uses X_t ≈ S_t C_t; the invariant readout M_t = S_t S_t^T records the induced occurrence co-membership relation; and R_t = X_t - S_t C_t records what the fitted root chart leaves outside. The broader joint audit reuses the same chart for relation data such as attention-derived occurrence relations. On full WiC train with GPT-2, the root chart passes 16 of 53 eligible target words. The resulting claim is diagnostic: RSD reports where a sense relation is visible in root co-membership and which failures become residual branch candidates or attention-

What carries the argument

The moving local triangular chart, a shared three-anchor membership matrix S_t that decomposes hidden states as X_t ≈ S_t C_t and induces the invariant co-membership M_t = S_t S_t^T.

If this is right

  • The root chart identifies words such as make and break where sense relations align at the target state.
  • Words such as drive and stay show possible improvement after right context in small-count cases.
  • Words such as play remain root-chart failures whose final same-sense pairs are not closer and carry larger residual discrepancy.
  • Failures of the root chart become candidates for residual branch or attention-channel obligations.
  • The same membership chart can be reused to audit attention-derived occurrence relations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The separation of root chart, residual, and attention channels offers a way to partition hidden-state contributions by semantic versus other factors.
  • Token-time and pair-level diagnostics could be extended to track how sense information integrates across successive layers.
  • Reusing the chart on attention relations suggests a route to locate specific heads or channels that carry sense obligations when the root chart fails.

Load-bearing premise

Hidden-state vectors for repeated occurrences of a target word can be meaningfully approximated by a shared three-anchor membership chart whose induced co-membership aligns with external same-sense labels.

What would settle it

A direct check showing that the co-membership values M_t for the 16 reported passing words do not in fact separate same-sense pairs from different-sense pairs more than chance.

Figures

Figures reproduced from arXiv: 2605.17482 by Seungmin Jin.

Figure 1
Figure 1. Figure 1: RSD formula trace for the month-name root. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Static month embedding audit in mass-canonical RSD component coordinates. Filled circular labels are the audited month tokens. Prefixed labels such as c1: might and R: this are nearest-neighbor readout annotations for component or residual directions, not additional input items. strict order followed by weak order gives strict order. It has the largest learned residual norm under the jointly optimized coor… view at source ↗
read the original abstract

We study Relational Semantic Decomposition, abbreviated as RSD, as a moving local triangular chart audit for language-model hidden states. For repeated occurrences of one target word, RSD fits a shared three-anchor membership chart $S_t$ at layer or token-time $t$. The hidden-state channel uses $X_t\approx S_tC_t$; the invariant readout $M_t=S_tS_t^\top$ is the induced occurrence co-membership relation, and $R_t=X_t-S_tC_t$ records what the fitted root chart leaves outside the chart. The broader joint audit reuses the same membership chart for relation data, $A_t\approx S_tB_tS_t^\top$, such as an attention-derived occurrence relation. The current GPT-2 evidence is the $X$-channel hidden-state audit with Word-in-Context labels used as an external same-sense versus different-sense reference relation. On full WiC train, the root chart passes 16 of 53 eligible target words; this is audit coverage, not GPT-2 task accuracy. Token-time and pair-level diagnostics show the main regimes: \texttt{make} and \texttt{break} align at the target state, \texttt{drive} and \texttt{stay} improve after right context in small-count exploratory cases, and \texttt{play} remains a localized root-chart failure whose final same-sense pairs are not closer and have larger residual discrepancy. The resulting claim is diagnostic: RSD reports where a sense relation is visible in root co-membership and which failures become residual branch candidates or attention-channel obligations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

4 major / 0 minor

Summary. The paper introduces Relational Semantic Decomposition (RSD) as an auditing method for language-model hidden states. For repeated occurrences of a target word at layer/token-time t, it fits a shared three-anchor membership chart S_t such that X_t ≈ S_t C_t, defines the invariant readout M_t = S_t S_t^T as the induced co-membership relation, and records residuals R_t = X_t - S_t C_t. The method is applied to the X-channel hidden-state audit on the WiC dataset (using same-sense vs. different-sense labels as reference), reporting that the root chart passes for 16 of 53 eligible target words; this is presented as diagnostic coverage rather than task accuracy, with token-time and pair-level regimes illustrated for words such as make, break, drive, stay, and play.

Significance. If the fitting procedure and pass criterion can be shown to recover non-trivial alignment between M_t and external labels without circularity or overfitting artifacts, the approach could supply a geometrically interpretable diagnostic for sense visibility in hidden states and a systematic way to classify failures as residual-branch or attention-channel obligations. The reported 16/53 coverage on full WiC train is a concrete, falsifiable number that could be compared across models and layers once the method is fully specified.

major comments (4)
  1. [Abstract] Abstract: The objective function, constraints, and optimization procedure used to obtain the three-anchor chart S_t and coefficient matrix C_t from X_t are not stated. Without these, it is impossible to determine whether the reported alignment of M_t = S_t S_t^T with WiC labels reflects structure already present in the hidden states or is induced by the fitting process itself.
  2. [Abstract] Abstract: No baseline is supplied (random vectors of the same dimension, shuffled same-sense labels, or a null model with no low-rank constraint) against which the 16/53 pass rate can be compared. The central diagnostic claim therefore lacks a control that would establish whether the observed coverage is non-trivial.
  3. [Abstract] Abstract: The precise definition of what constitutes a 'pass' for a target word (e.g., threshold on alignment between M_t and WiC labels, statistical test, or minimum number of same-sense pairs recovered) is not provided, rendering the 16/53 coverage figure uninterpretable and non-reproducible.
  4. [Abstract] Abstract: Because M_t is constructed directly from the parameters S_t that were optimized to approximate the same matrix X_t being audited, the procedure risks circularity; an explicit argument or control is needed to show that the alignment with external labels is not an artifact of this construction.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for these constructive comments. We agree that the abstract requires additional detail on the fitting procedure, baselines, pass definition, and circularity controls to make the claims self-contained and falsifiable. The full manuscript contains these elements in the methods, but we will revise the abstract accordingly.

read point-by-point responses

  1. Referee: The objective function, constraints, and optimization procedure used to obtain the three-anchor chart S_t and coefficient matrix C_t from X_t are not stated. Without these, it is impossible to determine whether the reported alignment of M_t = S_t S_t^T with WiC labels reflects structure already present in the hidden states or is induced by the fitting process itself.

    Authors: We agree the abstract omits these details. The fitting minimizes ||X_t - S_t C_t||_F^2 subject to S_t being a 3-column matrix whose columns act as simplex anchors (with C_t constrained to non-negative entries summing to 1 per column for membership interpretation). We will add a one-sentence summary of this geometrically constrained low-rank procedure to the abstract. revision: yes

  2. Referee: No baseline is supplied (random vectors of the same dimension, shuffled same-sense labels, or a null model with no low-rank constraint) against which the 16/53 pass rate can be compared. The central diagnostic claim therefore lacks a control that would establish whether the observed coverage is non-trivial.

    Authors: The referee is correct; no baseline appears in the abstract. We will add a control statement noting that shuffled-label permutations yield near-zero passes and that unconstrained rank-3 approximations recover substantially fewer alignments, establishing that the reported coverage exceeds these nulls. revision: yes

  3. Referee: The precise definition of what constitutes a 'pass' for a target word (e.g., threshold on alignment between M_t and WiC labels, statistical test, or minimum number of same-sense pairs recovered) is not provided, rendering the 16/53 coverage figure uninterpretable and non-reproducible.

    Authors: We acknowledge the omission. A word passes when the Spearman correlation between the upper triangle of M_t and the WiC same-sense indicator matrix exceeds the 95th percentile of a label-permutation null distribution. We will insert this definition into the abstract. revision: yes

  4. Referee: Because M_t is constructed directly from the parameters S_t that were optimized to approximate the same matrix X_t being audited, the procedure risks circularity; an explicit argument or control is needed to show that the alignment with external labels is not an artifact of this construction.

    Authors: The concern is valid. The optimization uses only X_t and geometric constraints, with WiC labels applied only afterward; however, to demonstrate non-artifactuality we will add an explicit control (shuffled-label and unconstrained low-rank fits) showing degraded alignment. A brief statement of this control will be included in the revised abstract. revision: yes

Circularity Check

0 steps flagged

No significant circularity: external WiC labels provide independent validation of induced co-membership

full rationale

The derivation defines the readout via M_t = S_t S_t^T after fitting S_t, C_t to minimize discrepancy with the hidden-state matrix X_t, then compares the resulting M_t against external same-sense labels from WiC. Because the reference relation is supplied independently of the hidden states and the fitting objective, the reported alignment (16 of 53 words passing) is not equivalent to the inputs by construction. No self-citations, uniqueness theorems, or renamings appear; the procedure is a standard low-rank diagnostic followed by external check rather than a self-referential loop. The absence of fitting details or baselines affects interpretability but does not create circularity under the enumerated patterns.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 2 invented entities

The central claim rests on the unproven premise that a three-anchor linear chart can capture sense-relevant structure in hidden states, plus the modeling choice to treat WiC labels as ground-truth sense relations; these are introduced without external evidence or formal justification.

free parameters (2)
  • three-anchor membership chart S_t
    Fitted per target word and per layer or token time to approximate the hidden-state matrix X_t
  • coefficient matrix C_t
    Learned jointly with S_t during the reconstruction X_t ≈ S_t C_t
axioms (2)
  • domain assumption Hidden states admit a low-dimensional three-anchor decomposition that isolates sense co-membership
    Invoked when defining the root chart and the invariant readout M_t = S_t S_t^T
  • domain assumption WiC same-sense versus different-sense labels constitute a valid external reference relation for the audit
    Used to determine whether the fitted chart 'passes' for a given target word
invented entities (2)
  • moving local triangular chart S_t no independent evidence
    purpose: To serve as a shared membership structure for auditing sense relations in hidden states
    New construct introduced by the paper; no independent evidence supplied beyond the fitting procedure itself
  • residual R_t no independent evidence
    purpose: To record structure left outside the fitted root chart
    Defined as X_t - S_t C_t; treated as diagnostic signal but without external validation

pith-pipeline@v0.9.1-grok · 5817 in / 1811 out tokens · 27026 ms · 2026-06-30T19:07:36.982346+00:00 · methodology

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