Abstract

We study a local regularization of the complete retained Einstein–scalar contraction while keeping its internal kernel exact. The aligned-null locus admits explicit normal coordinates in which the three cone functions have differential rank two along that locus. Their restriction to the supported response incidence has rank one, with a definite quadratic transverse defect. We compute the coordinate inverse, ambient density, and a weighted dilation adapted to these equations.

The density-before-variation source has a finite symbolic specification. An explicit tensor-to-component conversion and a jet-polynomial Hessian determine all source coefficients without a printed component list. On the retained chronological patch, the source-ramp terms disappear by locality. A displayed unreduced coordinate-response expression also reduces exactly by causal support and the existing retarded convolution.

Transposition of the complete mixed vertex produces a compact, test-dependent distribution in a relative translation variable. The regularized contraction, dyadic increments, and logarithmic scale derivatives are exactly its evaluations on specified compact profiles. This construction retains every internal row and vertex term and respects all reconstruction identities.

For a separate scalar cone regularization, we compute an order-zero limit with an order-one convergence rate. Its response-normal derivative has an explicit inverse-scale coefficient, an explicit finite distribution, and an order-one remainder. A single fixed test detects divergence whenever that coefficient is nonzero.

The scalar regularization differs from the physical eight-relative-variable smoothing. The complete physical origin coefficient and its controlled remainder remain unevaluated. Accordingly, the results are geometric, symbolic, and analytic reductions, not a convergence or failure theorem for the complete physical family.