Abstract

XVI-M begins the passage from the coherent graph renormalization of XVI-L (a frozen dependency) toward local counterterm functionals. (M.1, unconditional) The complete-source two-edge coefficients are assembled with independent residual-metric jets. The general table specializes entry by entry to the XVI-L background table (all $1065$ grouped entries), and the formal label support is all $255$ admissible labels ($235$ on the XVI-L background). (M.2) Gate S-bv is certified (Theorem thm:wpA-Sbv): on ${\mathbb{R}}^4\setminus \{0\}$ the predecessor edge kernel is $S=h_0+i{E}_0^{\mathrm{A}}=-W_0+i{E}_0^{\mathrm{A}}=\frac{1}{4{\pi }^2}(Q-i0{)}^{-1}$, including across the nonzero null cone, so $c=1/(4{\pi }^2)$ with boundary side $Q-i0$. This formula is already printed in XVI-J (eq:rational, with $S=-d$ by eq:DF); XVI-M supplies a written real-time check of it, including a local boundary-value argument at nonzero null points (Lemma lem:wpA-varreg). The predecessor $S$ is globally defined (XVI-J); what is not established here is its position-space identification at the origin. With a boundary-value calculus lemma this gives a two-direction equivalence between numerator identities and distributional relations, and hence, unconditionally on the punctured space, the graded presentation $d_N=(1,4,10,20,35)$, $\dim {\mathcal{V}}_{\mathrm{adm}}=70$, with $185$ certified relations (only $c\ne 0$ and the single boundary side are used). (M.3) The coherent contact space has dimension $3321$, against $14321$ raw slots. The graph-contact map is injective in the formal universal-jet reading, while $1750$ coherent directions are invisible on the XVI-L background. (M.4) Under a stated variational contract (Gate V), the contact-collapse map is computed exactly. It is injective in the formal reading, and its formally self-adjoint (variational) subspace is $26$-dimensional there and consists only of ultralocal potentials ${\chi }_{\mathrm{I}}{\sigma }_{\epsilon }K(h,h)$; the formal universal-jet collapse image of this selected graph contains no nonzero variational tensor of total derivative degree $D\ge 1$. The same holds in the universal time-only reading ${\sigma }_{\epsilon }=\sigma (t)$; if in addition ${\chi }_{\mathrm{I}}\equiv 1$, there are $546$ further spatial-only, non-Lorentz variational directions. In the double-plateau reading (${\sigma }_{\epsilon }\equiv 1$, ${\chi }_{\mathrm{I}}\equiv 1$) the variational image has dimension $1325$; it contains the linearized Euler tensors of $\int R^2$ and $\int R_{\mu \nu }{R}^{\mu \nu }$, with certified preimages, while the specified quadratic Einstein–Hilbert Euler tensor is not in that selected image. These curvature-square matchings do not survive the time-only reading once ${\sigma }_{\epsilon }^{\prime }$, ${\sigma }_{\epsilon }^{\prime \prime }$ are retained. Stage 2: fixed-data criteria, universal region comparisons and the time-only classification are constructed; the transfer to the actual XII-C ramp on $(0,\epsilon )\times {\mathbb{R}}^3$ is a proposition with explicit hypotheses; regions where the actual ${\chi }_{\mathrm{I}}$ is non-constant, and transitions, remain Unresolved. M.5 records the named matching theorem PlateauCurvatureSquareMatching with explicit domain, order, pairing, and equivalence; ledgers (3)–(4) remain Unresolved (EOM/redefs undefined). (M.6) The $S^2$ scaling residue is $r=-2i{\pi }^2{c}^2=-i/(8{\pi }^2)$. For the XVI-L prescription the scale derivative equals the dilation defect at every grade $N\le 4$ ($A_{\mu }={\mathcal{A}}_{\mu }$); top-order residues are $\varpi$-independent, while lower orders depend on the cutoff $\varpi$, which XVI-L does not fix. In the double plateau, the cutoff-independent $D=4$ component is variational and Lorentz covariant, but not linearized-gauge invariant. The full scale tensor has cutoff-dependent lower-degree components and need not be variational. The $D=4$ component lies outside $\mathrm{span}(E_{R^2},E_{{\mathrm{Ric}}^2})$, certified by an exact separating functional ($\ell (v_{\mathrm{scale}})=-\frac{1}{16}$, $\ell (v_{R^2})=\ell (v_{{\mathrm{Ric}}^2})=0$). Hence the curvature-ledger scaling criterion M-C${}_{\mathrm{curv}}$ is Proved obstructed for this graph, while broad M-C (Definition def:MCcurv: matching plus an explicit cutoff and a functional representation of the entire collapsed scale variation) is not achieved and remains Unresolved, because XVI-L leaves $\varpi$ unspecified (thm:L6); M-C${}_{\mathrm{data}}$ is satisfied (not a claim about gravity generally). M.7 (audited): with the contact distributions ${\partial }_{\gamma }\delta$ transported by the metric intertwiner, the Lorentz-equivariant contact assignments have dimension $14=5+9$; their exact intersection $V$ with the variational subspace has $\dim V=1$ (formal, $\chi$-plateau; $V\subset {\mathcal{C}}_{\mathrm{reg}}$) and $\dim V=12$ in the double plateau ($\dim V\cap {\mathcal{C}}_{\mathrm{K}}=3$, $\dim V\cap {\mathcal{C}}_{\mathrm{reg}}=7$); the curvature-square Euler tensors satisfy the linear divergence identity ${\partial }_{\mu }{E}^{\mu \nu }=0$ (this is not a Ward identity). No Ward or BV statement is claimed; scale-variation coefficients are not physical beta functions.