Theorem: Main constant-weight substitution and regrouping

Fix the action, contractions, observations, and marked insertion specified in Sections sec:model–sec:insertion. Let $U⋐M^4$ be the separated time-ordering patch of Section sec:geometry. Let $\rho \in {C_c}^{\infty }({\mathbb{R}}^8;\mathbb{R})$ satisfy $\int \rho =1$, and let $\chi \in {C_c}^{\infty }({\mathbb{R}}^8)$ equal one near $\mathfrak{f}(\overline{U})$.

Every input in

is determined by the displayed formulas of this paper.

The seven kernels are exactly the coefficient kernels of the defined marked insertion. Their arbitrary compact joint-test actions have the cone-seed representation of Theorem thm:seed. The selected family has the exact 42-entry audit expansion of Theorem thm:ledger.

For sufficiently small $\delta >0$, the same complete family has the combined scalar-profile representation (eq:combined-seeds); this representation agrees with the audit expansion on every compact joint test.

For every $L⋐U$, there is $C_L>0$, independent of $\delta ,\lambda ,\hslash$, such that

The remaining fixed data may enter $C_L$. No raw-limit conclusion or leading scale exponent is asserted.