Mode Locking Without Dynamics: Integer Convolution on Finite Graphs

We study Integer Convolution Locking (ICL) on finite grid graphs, a purely combinatorial mechanism in which a binary mask convolved with a nonnegative integer kernel generates an overlap multiset Oₜ. Normalizing pₜ = Oₜ / Q (Q = Σₜ Oₜ) and letting m = gcdOₜ induces an occupied rational probability lattice with effective grain Δpₑff = m / Q and capacity C = Q / m (Occupied Grain Theorem). On this lattice we define two coherence diagnostics: (i) overlap–residue coherence A (q), constructed from residues Oₜ mod q weighted by pₜ, and (ii) probability–phase coherence C (m), constructed from lattice coordinates nₜ = Oₜ / m. Locking is summarized by a preferred denominator q*. Two geometry-only control sweeps are analyzed in detail. (1) A double-cone mask translated on a 96×96 grid with a 3×2 kernel under wrap versus fill boundary conditions: along fixed cuts, q* forms long plateaus while fill-mode mass decays smoothly, followed by abrupt jumps when the active support reorganizes. (2) A cone–wall inner-support morph inside an aperture (no boundary truncation): q* again shows plateau-plus-jump behavior driven purely by internal incidence changes. A hierarchy of null tests—spatial shuffles, value–occupancy randomizations at fixed overlap alphabet kᵢ, total mass Q, and number of occupied sites n, and pathwise random integer multisets at fixed (Q, n) —preserves the underlying lattice arithmetic but does not reproduce the observed locking transitions. All constructions are base-independent: results depend only on integer structure (gcds and rational lattices) ; decimal plotting choices are purely visual. Additional cuts and variants are reported in the appendices. Code companion: DOI 10. 5281/zenodo. 18840950