Abstract

We reorganize the local fixed-kernel problem around the integrals that must actually be calculated. The complete retarded inventory is retained, while the crossed array is smoothed in eight flat relative variables. Exact adjoints and the compact translation representation are collected in supporting appendices.

The main text gives a row–sector contribution ledger, exact non-tip integration formulas, and a cone-radius treatment of the coincidence transition. For a fixed smooth localization profile, the scalar seed measures supported by $R(b-a)R(a-z)$ and $R(b-z)R(z-a)$ have explicitly calculated order-two coincidence-localization coefficients. The seed supported by $R(b-a)R(b-z)$ has an order-three coefficient. Uniform remainders are proved directly. These powers concern an independent internal localization parameter, not the physical smoothing scale.

The same transition calculations are checked through the translation-distribution construction. A complete derivative certificate, if obtained for identified crossed data, determines the singular primitive under an explicit scalar absolute continuity hypothesis. The smooth-data expansion is retained only as a consistency benchmark.

The supplied excerpts do not determine an executable inherited physical input package. Accordingly, no physical singular-data sector is evaluated, no physical complete coefficient or integrable smoothing remainder is obtained, and no physical limit, authorized correction, or quantum admission is asserted. A completion ledger distinguishes these unresolved tasks from the mathematical calculations proved here.