A Controlled Transfer-Function Map for Symmetry-Decomposition Detectors

Abstract

The parent paper argued from n=2 domains (LLM hidden states, earthquake catalogs) that a B4 isotypic componenthas no universal referent -- what each block reads is set by the data-generating process (DGP). Here we replace thetwo-domain anecdote with a controlled map: eleven stochastic processes of known statistical structure -- monotonetrend, random walk, sinusoid, two iid distributions, AR(1) with positive and negative coefficient, a Hawkes-likeself-exciting interval series, a burst process, a clustered renewal process, and an Omori-Utsu aftershock-intervalseries -- are each run through the identical B4 pipeline (width-4 Takens window, min-max normalize, radial project,nearest 24-cell vertex, per-block energy). Each process is a probe with a known answer, and the resulting energytable is the transfer function of the decomposition. We add the statistical rigor the parent paper's reviewers askedfor: an order-null (block-shuffle of the series before windowing) gives a two-sided permutation p per (process,component) cell, and Benjamini-Hochberg control across the whole 55-cell grid separates 'somewhere is significant'from 'this pre-named cell is order-dependent.' Two self-corrections emerged and are reported as findings. First, anaive null that shuffles windows is vacuous: 24-cell occupancy is a set-statistic over windows, invariant to their order,so the order-null must act on the series before windowing -- a small but exact statement of why the method's nullsare pre-windowing. Second, under this correct null, only drift processes (monotone, random walk) carryorder-dependent component energy that survives FDR among the synthetic battery, and the synthetic Omoriaftershock ax_a signal does not survive (z=0.9). We then resolve this against real catalogs: applying the identicalper-sequence order-null to 300 USGS and 178 JMA single aftershock sequences, real ax_a is genuinelyorder-dependent (Stouffer z=+19.3 and +6.4; sign-tests p=1.7e-28 and 3.9e-6), so the parent paper's phrasingstands for real earthquakes -- the synthetic Omori decay was simply too impoverished to reproduce it. Finally weidentify the mechanism by surrogate dissection: a Gaussian AR(1) matched only to each sequence's lag-1magnitude autocorrelation reproduces the real ax_a z, a trend-only surrogate destroys it, and time-reversalpreserves it -- so the order structure is short-lag magnitude autocorrelation (similar magnitudes clustering in time),direction-symmetric, not b-value drift, echoing the LLM ax_a reversal-invariance from the parent paper. The mapotherwise confirms and sharpens the parent result: sign responds to sign-oscillation whether or not order ispreserved (generic across the battery, drift the rare exception), resid4 owns the iid limit, and the Omori process isthe sole synthetic member spreading energy across all five non-trivial blocks. Two parameter sweeps show thetransfer functions are smooth functions of the known DGP parameter. Text CC-BY-4.0; code AGPL-3.0-or-later.