Abstract

We formulate and prove coefficient-level decision theorems for a fixed retained contraction whose crossed array is regularized in its actual eight relative variables. A complete finite power–logarithm expansion with a compact-uniform vanishing remainder determines raw convergence, fixed-test raw failure, and the mathematical finite part. The nonsummable coefficients and finite part are unique. The principal theorem does not require a homogeneous translation germ.

Three sufficient verification routes are distinguished: scale-integrability, direct complete-family expansion, and a logarithmically homogeneous translation-origin certificate. Localization and Taylor estimates connect explicit calculations to these routes. Physical identification, witness evaluation, reference agreement, subtraction authorization, normalization, and microlocal output are separate records.

An evaluated complete degenerate specialization of the retained seven-row architecture is calculated on arbitrary compact joint tests. Five rows vanish by its declared observations; both remaining internal rows and all three vertex types are retained. The apparent order ${\delta }^{-2}$ vanishes, a nonzero order ${\delta }^{-1}$ survives, and the subtracted family converges strongly with an order-six $O(\delta )$ remainder. A further normal-derivative cancellation is exposed in the complete inverse-scale coefficient. The singular coefficient and finite part have normal orders at most two and four, respectively, and wavefront sets contained in the conormal bundle of their smooth attachment. The finite part is independent of the specified even factorized profile class.

The specialization is not identified with the physical Einstein–scalar contraction. A physical identification ledger and a six-task verification protocol state what is still required for that application. No physical characteristic decision, authorized physical counterterm, or quantum admission is inferred from the mathematical example.