CUBIC COLLISIONS MODULO COMPOSITE INTEGERS: EXPLICIT COUNTS IN BOUNDED WINDOWS

We count cubic congruence pairs by separating the prime-power divisibility of D = x − y and H = x² + xy + y². Local four-state counts give exact masking statistics over square-free composite moduli, including the exceptional primes 2 and 3. For square-free moduli with prime factors greater than 3, we derive periodic formulas for the number of masked pairs in bounded squares, with an explicit correction for diagonal pairs. The semiprime specialization yields a quadratic quasipolynomial, explicit periodic coefficients, and a sharp first-occurrence threshold when both primes are 2 modulo 3. For primes greater than 3, a rational joint valuation series and an independent finite sum give strict-masking counts at arbitrary prime-power exponents. Exact integer enumeration checks the formulas for selected moduli and window sizes.