The 19-9 System: A0 = dr(N).

The 19-9 System: N = 19A + 9B is a careful, non-sensational record of what emerges when one linear Diophantine equation is taken seriously and explored systematically. The paper proves that for every integer N \geq 171 , the smallest non-negative solution A_0 in N = 19A + 9B equals the digital root of N — the A0 Theorem. From this single algebraic fact (using 19 \equiv 1 \mod 9 ), the entire structure unfolds: an exact closed-form representation count R(N) , thresholds spaced by 171, family boundaries at 171k + 144 , and the unique anchor family 1675–1683 where each number has exactly as many representations as its digital root. Three structural numbers — 366, 592, and 958 — satisfy multiple independent conditions simultaneously and are linked by 366 + 592 = 958 and near-exact golden ratio scaling ( 366 \times \phi \approx 592 , 592 \times \phi \approx 958 ). The paper also examines perfect numbers (digital root always 1 for P \geq 28 ), Fibonacci numbers (Pisano period 18 gives F(18) = 2584 with B = 0 ), and Newton-Raphson convergence to \sqrt{5} and \phi . The results are observed without claiming deeper causality. Python implementations are included. No numerology — only systematic observation.