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

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 $X_t\approx S_t{C}_t$; the invariant readout $M_t=S_t{S}_t^{\mathrm{\top }}$ is the induced occurrence co-membership relation, and $R_t=X_t-S_t{C}_t$ records what the fitted root chart leaves outside the chart. The broader joint audit reuses the same membership chart for relation data, $A_t\approx S_t{B}_t{S}_t^{\mathrm{\top }}$, 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: $\mathtt{\text{make}}$ and $\mathtt{\text{break}}$ align at the target state, $\mathtt{\text{drive}}$ and $\mathtt{\text{stay}}$ improve after right context in small-count exploratory cases, and $\mathtt{\text{play}}$ 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.