CERTIFIED COMPUTATION · FRESH WITNESSES

NS-45: the existing block comparison fails

22 September 2026 · version 1 · λ=5, N=25

New exact dyadic witnesses pass interval verification at 320 and 448 bits in both parities. The first three existing blocks—26:50, 51:100 and 101:200—already suffice. The fourth block is unnecessary.

H is symmetric, nonnegative and has zero diagonal. Each entry is a verified strict lower bound for a normalized block-pair coupling, provided the criterion's required positive block metrics exist. Every pair's two energies are strictly positive as intervals. The Rayleigh quotient uses the exact vector (1,1,1).

The verifier recomputes all analytic coefficient enclosures in an empty temporary cache. Separate signed-index expansions reproduce the energies and cross pairings and pass their own strict interval gates. Float Cholesky factors and singular vectors are used only to propose the frozen integer-over-2⁶⁰ coordinates.

If the required metrics 0 < Mj ≤ Tjj exist, the diagonal blocks are positive definite and the established comparison lemma applies: these rational matrices rule out every admissible smaller metric and every positive scalar row reweighting for this partition. If the metrics do not exist, the criterion already fails its prerequisite. No full block inertia claim is needed.

These are new witnesses. They do not recover the absent v1.28 artifacts or certify its original displayed lower numbers. The failure is of this fixed norm-comparison criterion; it is not a negative Weil direction or a complete λ=5 positivity result. Other partitions, G2 and RH are not settled.

Python diagnostics: zero errors and zero warnings. No manuscript, map, version, window or tail metric was changed.

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