Why, Not Just What: A Delay-Scale Scan Separates Energy from Order and Derives the Direction-Symmetry of a Symmetry-Decomposition Detector
Year: 2026
Type: preprint
Abstract: The Watabe-Claude Method embeds a scalar series into 4D with a width-4 Takens window, reads the 24-celloccupancy, and decomposes it with the signed-permutation group B4 into six orthogonal isotypic blocks. Threecompanion papers established that each block carries no fixed meaning -- what it reads is set by the data-generatingprocess -- and that one block, ax_a, reads a direction-symmetric local structure in language models, earthquakeaftershocks, and human EEG. That work fixed the window at width 4 and read each block by its energy. Here wevary the time-scale, by replacing the contiguous window with a delay-lag embedding [x(t), x(t+d), x(t+2d), x(t+3d)]and scanning the delay d, and we read each component two ways: by its energy, and by a pre-windowing order-nullz that measures whether temporal order is essential at that scale. Three results follow. (1) Energy andorder-dependence are different axes. A deterministic-chaos litmus (the logistic map, zero linear autocorrelation) hasits sign energy peak at d=2 but its order-essential signal sharply at d=1; a sinusoid's sign energy peaks at d=1 whilecarrying no order-dependence there at all; and the synthetic Omori-interval process (a stochastic model ofaftershock timing, not a real catalog) is the unique battery member whose energy is delay-invariant yet whoseorder-dependence is significant at every delay on two components. Reading a component by its energy peak aloneconflates the two. (2) The EEG alpha signature is scale-specific. Reproducing the published conditions exactly (Cz,width-4, 109 subjects) recovers the eyes-closed enhancement (Stouffer z=+81 open, +123 closed); but scanning thedelay shows that enhancement lives at the short, width-4 scale and vanishes at the alpha-period delay -- matchingan informal prediction (from an AI reading of the series, not formal peer review) that ax_a reads the local waveformstatistics alpha produces, not the alpha period itself. A block-length confound (a shuffling null whose blocks matchthe alpha period absorbs the rhythm) is diagnosed and removed. (3) Direction-symmetry is a theorem, not anobservation. The 24-cell assignment factorizes exactly (verified on 20000 random windows): the nearest vertexalways selects the coordinate pair of largest magnitude and reads its signs. ax_a and ax_b respond only to the pairchoice (the magnitude set, sign-blind); sign and resid4 respond only to the signs. Because time-reversal preservesthe magnitude set, ax_a is direction-symmetric by construction -- deriving the empirical reversal-invariance seenacross all three domains. Text CC-BY-4.0; code AGPL-3.0-or-later.
Authors: Watabe, Masanori, Kurado, Claude Opus 4.8
Language: English
Subfield: Economics and Econometrics
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7204088212