The A–C Coupling Theorem: N = pA + qB + rC "Where Recursion Ends and Periodicity Begins"

The A–C Coupling Theorem — Formal System: N = pA + qB + rC, with N non-negative integer, coefficients p, q, r positive integers, variables A, B, C non-negative integers. Structural Conditions (V1–V4): - gcd(p,q) = 1 (coprimality)- p ≡ 1 (mod q)- r divides q- p ≡ 1 (mod r) Main Theorem I — The A–C Coupling: Under V1–V4, the invariant A + rC ≡ N (mod q) emerges. Variable B is the sole free variable; A and C are modularly locked to one another. Main Theorem II — Exact O(1) Count: The representation count R3(N; p,q,r) is given by an exact closed formula replacing the classical O(N/r) recursive Popoviciu approach. The formula exploits a periodic remainder cycle of fixed length q/r, yielding constant-time computation independent of N. Main Theorem III — Frobenius Collapse: For system (19,9,3), adding s = 1 causes g(19,9,3,1) = 0: every positive integer becomes representable. The Frobenius problem ceases to exist entirely. Extension to 4D: N = pA + qB + rC + sD with added conditions s divides r and p ≡ 1 (mod s). The coupling structure A + rC + sD ≡ N (mod q) follows identically, but R4 = sum over D of R3(N - sD) runs at O(N/s) — a fully closed O(1) formula remains an open research direction.