The A-C Coupling Theorem: Extension from Three to Four Variables

This paper is a continuation of the earlier work on the linear Diophantine equation N = pA + qB + rC, which established the A-C Coupling Theorem and the exact O(1) representation count R₃(N; p, q, r) under the conditions p ≡ 1 (mod q), gcd(p,q) = 1, r | q, and p ≡ 1 (mod r). The present paper extends that framework to four variables by introducing a fourth denominator s with s | r and p ≡ 1 (mod s). The outer D-sum is closed through an explicit step-function representation of J(D), three Beatty sums in O(1) per term, and a block decomposition of the remainder correction Σ₄. The result is Theorem 4.2-bis: an exact O(p + q/s) formula for R₄(N; p, q, r, s), independent of N. All eight test systems are verified against brute-force enumeration. The Frobenius chain g(19,9) = 143 → g(19,9,3) = 35 → g(19,9,3,1) = 0 is confirmed. The derivation establishes an inductive template extensible to k dimensions.