The Gap-3 Phenomenon: Exact Minimum-Distance Bounds for Valuation-Metric Circulant Codes over Z/p²Z and the Two-Power Boundary at Z/16Z

We establish exact values of the maximum minimum distance M_p(K,R') for circulant systematic codes over finite chain rings Z/p²Z under the valuation (Δp) metric, which assigns weight 1 to nonzero units and weight 2 to nonzero zero-divisors. The central result is a uniform gap-3 theorem: for every odd prime p, the maximum Δp-distance at parameters (n,k) = (9,6) is exactly 4, proved by direct valuation-pattern classification and certified by exhaustive computation at p=5 (244,140,625 seeds) and p=7 (13,841,287,201 seeds, zero distance-5 seeds). Companion exhaustive computations over Z/9Z and Z/4Z produce a 49-cell phase diagram, establish K-axis non-monotonicity unconditionally, and falsify a proposed cyclotomic-fingerprint mechanism via explicit counterexamples. A complete Z/16Z computation (3,072 distance-5 seeds) establishes that the gap-3 ceiling fails for two-power rings. All results are produced by a file-backed, replay-oriented verifier with per-theorem JSON artifacts archived for independent reproduction.