Diophantine Interferometry Without Dynamics: A Proof Note on Integer Convolution Locking

This proof note establishes the core mathematical structure behind Integer Convolution Locking (ICL) — a purely combinatorial mechanism for generating rational coherence, Devil’s staircases, and mode locking on discrete graphs without any underlying dynamics.

We prove that for any nonzero binary mask and a nonnegative, finitely supported kernel on the integer lattice, the inner-support convolution over a finite placement set yields integer overlaps whose normalized values lie on a rational lattice. The fundamental step of this lattice, given by the ratio of the greatest common divisor of the overlaps to their total sum, defines a minimal projection unit for the field.

From this structure we derive:

• An Occupied Grain Theorem, establishing this step as the smallest nonzero increment,

• A Farey Projection Principle, organizing locking plateaus without dynamics,

• A Ridge Law, governing the slope of coherence ridges under driven control.

We also demonstrate invariance properties under integer rescaling, translation, observer frame shifts, and reparameterization. These results ground ICL as a general principle for structuring rational fields over graphs — with implications for quantization, coherence, and emergent order in discrete systems.

Version note (v0.2): adds a short section on capacity pruning, clarifies the number–theoretic status of the Occupied Grain Theorem vs convolution, discusses related work and updates the discussion of numerical results.