Abstract

We study a fixed-profile regularization of a frozen seven-row Einstein–scalar distributional family. The complete supported response sector has an expansion with three possible inverse powers, a degree-zero logarithm, and a finite distribution, with a compact-uniform $O(\delta (1+|\log \delta |))$ remainder. Its proof uses an exact whole-support radial reduction, including the massive causal tail, and an averaged resolvent estimate with explicit parameter seminorms.

The new coefficient calculation concerns the non-tip positive $K_5$ placement. Derivative placement reduces the possible crossed order of both $47$ and positive $K_5$ to three, and their inverse-cubic coefficients vanish by an explicit tensor contraction. For positive $K_5$, a Klein–Gordon identity further reduces the exact crossed order to two. We evaluate its full inverse-quadratic coefficient, including the mass vertex, and construct a compact joint test detecting it.

On a noncollinear positive-$K_5$ incidence, support geometry excludes all six other rows. The detecting test therefore gives a complete-seven-row nonconvergence certificate for a nonempty class of admissible fixed data, without residual-tip matching. This is not an all-data classification or a global complete-family expansion. The general output-plateau supported inverse-quadratic tensor sum, the aligned negative placement, and tip matching remain open.