Finite Audit Structure and Masked GCD Coarsening in a Three-Walker Shared-Wake Motif
Year: 2026
Type: preprint
Abstract: An autonomous deterministic three-walker run is shown to enter a recurrent sector with observed period P=14 after a nonperiodic formation interval. In the recorded continuation, the relative walker state repeats in successive 14-step blocks. Each completed cycle deposits the same integer wake increment G_14. Consequently, after n completed cycles, corresponding to 14n comparison steps beyond the sector entry kappa_0, H(kappa_0 + 14n) = R + n·G_14, where R is the fixed formation residue and G_14 is the wake deposited during one complete recurrent cycle. After subtracting the formation residue, the tested finite audits satisfy the periodic-sector scaling law m_W(n·G_14) = n·m_W(G_14) = n, because m_W(G_14) = 1 in the tested windows. Audits of the inclusive field remain primitive, with m_W(R + n·G_14) = 1. The masking mechanism is the entrywise congruence O_W(R + n·G_14) ≡ O_W(R) (mod n). Thus the run contains both an explicitly recurring three-walker route and a coherently accumulating wake sector, while the nonperiodic formation history masks that arithmetic structure in inclusive total-history audits. The paper formalizes this distinction using finite relational audit operators and proves a general periodic-sector gcd-scaling theorem. The accompanying reproducibility archive contains the recorded row logs, audit tables, deterministic validation scripts, visualization notebooks, interactive trace-tube views, and figure-generation code. Notation-harmonized revision (June 23, 2026). This version aligns the paper's symbols with the Integer Convolution Locking series: the array-valued audit output previously written Q_W(X) is now O_W(X), and its gcd-reduced array previously written C_W(X) is now Õ_W(X). The symbols Q, m, and C are reserved for their established scalar meanings (total mass, gcd, and occupied-grain capacity). The change is purely notational — the audited arrays, gcd calculations, figure data, and results are unchanged from the original version.
Author: James, John R.
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7164997283
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