Arithmetic Wake Memory in Self-Interacting Lattice Walkers

This paper consolidates a reproducible computational program on deterministic self-interacting lattice walkers with accumulating integer-valued memory. The empirical results are supported by versioned Zenodo archives. During preparation of this consolidation, selected critical claims were additionally regenerated or recomputed from the archived materials rather than being accepted solely from the conclusions of the earlier papers. A reproducibility registry in the paper identifies the public bundle and verification status associated with each result family. The walkers deposit an oriented integer-valued wake through a finite body stencil and subsequently read that accumulated history through a cost-normalized local decision rule. Passive finite audits expose arithmetic structure in the growing wake, including occupied gcd, reduced integer arrays, modular residues, occupied grain, and audit capacity. Three result families are unified. Single walkers form exact recurrent flip clocks whose later recurrence interval is tuned by deposition lag, with phase-locked audits revealing gcd coarsening at invariant reduced readout. Forced transport words show that endpoint pose and total wake mass can discard behaviorally active information: endpoint-equivalent preparations can produce different positive-margin autonomous branches. Recurrent multi-walker systems write repeated integer packets into a shared wake; in a three-walker period-14 sector, formation residue can mask exact packet coarsening in inclusive audits, while pairwise reductions reveal base-gcd splitting. In every tested base-2 case, the doubled factor is already present in the deposited pair packet through cell-by-cell mod-2 registration of the two constituent histories, while closely matched recurrent controls retain a structured nonzero parity residue. The paper also develops a common finite-audit formalism for these results, including the finite-audit form of the Occupied Grain Theorem. Register coordinates are treated only as discrete implementation variables and are not identified with physical space or metric time.