The Same Symmetry Lens Reads Different Structure in Each Domain

Abstract

The Watabe-Claude Method (WCM) embeds a scalar time series into 4D via a Takens window and reads theoccupancy of the 24-cell regular polytope. The signed-permutation group B4 (order 384) decomposes that24-dimensional occupancy space into six mutually orthogonal isotypic blocks of dimensions 1, 6, 3, 2, 8, 4 (uniform,sign, axispair_a, axispair_b, resid8, resid4). We ask whether these components carry a domain-independentmeaning by running the identical decomposition, unchanged, on real Large-Language-Model (Llama-3.2-3B-Instruct)hidden-state sequences and on real earthquake catalogs (USGS 1986-2026, JMA 1998-2023). They do not. Thesign component opens only when the series oscillates in sign (rate of sign-agreement anti-correlates withsign-component energy across 120 layer/projection conditions, r=-0.37, p=2.7e-5); a smooth series (LLM layerdirection, adjacent cosine 0.906; or an autoregressive generation trajectory) leaves it closed. The ax_a component ismore striking: in static LLM token sequences it fires on local change, whereas in single physical aftershocksequences it fires on flat magnitude persistence -- opposite structure from the same lens. Along the way we rejectedthree tempting unifications, each under an explicit control: ax_a as subject-predicate syntax (dissolved by awindow-position confound); ax_a as a domain-independent sequence-position detector (absent in syntheticaftershocks); and the earthquake signal as a magnitude-quantization artifact (unchanged under jitter). Our centralclaim is a scope result: a symmetry-decomposition component has no single universal referent; what each blockreads is set by the data-generating process, and this is precisely the applicability condition for the method. TextCC-BY-4.0; code AGPL-3.0-or-later. ※ The License for codes is AGPL 3.0-or-later