Minimum Distance of Valuation-Weight Circulant Codes over Two-Power Chain Rings: Beyond the Gap-3 Ceiling

We study the minimum distance of circulant systematic codes over two-power chain rings Z/2eZ under the valuation (Δ2) metric. Whereas the odd-prime-square family obeys a uniform gap-3 ceiling, exhaustive computation over Z/16Z at (n,k)=(9,6) already exhibits seeds of distance 5, so the ceiling does not extend to p=2. A second complete exhaustion establishes by exhaustive computation that the cap rises to 6 over Z/32Z. We show the two-power obstruction is governed by nilpotent depth: a square-class classification over Z/2eZ, a targeted-family Z/32Z product-congruence theorem, and a conjectural support-family taxonomy describing lift behaviour at higher depths. We give the intrinsic upper bound dΔ ≤ 2R'+1, exhaustive certificates at small depth, and bounded computational evidence with explicit conjectures at the frontier. All computational claims are produced by a file-backed, replay-oriented verifier.