The 6696 Framework: A Structural Pattern in Linear Combinations of 19 and 9

A central observation is that 19 ≡ 1 (mod 9), which causes the residue class of all a-values to equal the digital root of N directly. This makes 9 the unique modulus for which all single-digit decimal values (1–9) are reachable, and the pair (19, 9) the smallest pair with m ≡ 1 (mod n) covering the complete single-digit numeral system. The report further introduces the concept of a symmetry point — a solution where dr(a) = dr(b) = dr(N) — and identifies 1679 as the minimal number yielding a complete cycle of all nine digital roots with two symmetry points. All theorems are fully proved. The paper was subjected to peer review; responses and revisions are incorporated.