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.
John Robert James (Tue,) studied this question.