Extremal Modulus Kernels for Scale-Based Distinguishability An Extremal Coverage Theorem for Natural Collision Density

This paper reinterprets modular arithmetic as a minimal distinguishability mechanism.Each modulus greater than or equal to 2 is viewed as a scale under which certain integer differences become indistinguishable. Given a finite set of moduli, we define a collision set consisting of all positive integer differences that are divisible by at least one modulus in the set. The central object of study is the natural density of this collision set, measuring the long-run proportion of differences detected by the chosen scales. The main extremal question addressed in this work is: For a fixed number of moduli (called the width), which choice of moduli maximises collision density? Main Contributions Canonical Kernel ReductionAny finite set of moduli admits a unique minimal antichain kernel under divisibility. Removing redundant moduli does not change collision density. Periodicity and RationalityFor finite kernels, the collision indicator is periodic and the density always exists and is rational. Product-Space ReformulationUsing a Chinese remainder decomposition, the density problem is reformulated on a product probability space with independent prime-power coordinates. Sharp Product Upper BoundA correlation inequality for monotone events yields a sharp upper bound for collision density in terms of the chosen moduli. Prime-Factor CompressionA structural compression lemma shows that any density maximiser can be reduced to a kernel consisting entirely of primes. Extremal Kernel TheoremFor fixed width w, the unique density maximiser is the set of the first w prime numbers.This kernel maximises distinguishability coverage among all antichain kernels of the same size. Structural Interpretation The result shows that minimal-scale distinguishability is optimally achieved by the smallest primes.The extremal density follows a classical multiplicative sieve structure and approaches full coverage slowly as the number of primes increases. Mathematical Tools Divisibility partial order and antichain theory Inclusion–exclusion principle Chinese remainder theorem Product probability spaces Correlation inequalities for monotone events