Representations of Integers: N = pA + qB with p ≡ 1 (mod q)"

This paper develops a complete theory for linear Diophantine systems of the form N = pA + qB, where p and q are coprime integers satisfying p ≡ 1 (mod q). The central result — the Main Theorem — establishes that the minimal coefficient A₀ always equals the digital root of N, a direct consequence of the congruence structure of the system. The theory is built around three core results. The Main Theorem (A₀ = dr(N)) provides a closed formula for all non-negative representations and their count R(N), with threshold values lying on arithmetic sequences with step constant pq. The Additivity Theory shows that addition of numbers propagates simultaneously through three layers: the numbers themselves, their coordinates, and their digital roots. The Matryoshka Theory shows that the full reduction chain of any N is encoded within its representations — every intermediate digit sum and the digital root appear as A-coordinates.