The Gap-3 Phenomenon Revisited: Exact Valuation-Weight Bounds for Circulant Codes over Prime-Square Chain Rings

We establish exact valuation-weight bounds for circulant systematic codes over prime-square chain rings Z/p²Z under the 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 verified by exhaustive computation at p=5 and p=7. Companion exhaustive computations over Z/9Z and Z/4Z produce a phase diagram, establish K-axis non-monotonicity unconditionally, and falsify a proposed cyclotomic-fingerprint mechanism via explicit counterexamples. A complete Z/16Z computation establishes that the gap-3 ceiling fails for two-power rings. The rank-2 period-3 enumeration is corrected to the verified degree law (q−1)3m+2, degree 10m+2. All results are produced by a file-backed, replay-oriented verifier with per-theorem artifacts archived for independent reproduction.