Landmark XVI-M From coherent graph renormalization to local counterterm functionals and symmetry-compatible effective quantum gravity Draft: integration of work packages A–C (final verification); Gate S-bv checked; M-C curv proved obstructed; broad M-C not achieved ( unspecified); M-C data satisfied

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.

(XVI-M, this draft).{ M.0–M.3: Constructed as stated (M.1 unconditional; M.2/M.3 use Input (S-bv), now certified; Lemma lem:bv(c) by same-sign BV product). Gate S-bv: certified (Theorem thm:wpA-Sbv: $c=1/(4{\pi }^2)$, boundary side $Q-i0$, including across the nonzero null cone); a written check of XVI-J, eq:rational; the Euclidean motivation (Remark rem:Sbv-motivation) is not the certification; the predecessor $S$ is globally defined (XVI-J), and its position-space identification at the origin is not established here; not discharged by matching. M.4: Constructed (Gate V) in the formal, $\chi$-plateau, preparation-plateau, double-plateau and time-only readings (the formal, preparation-plateau and time-only readings are universal formal-jet readings). Stage 2: fixed-data criteria, universal region comparisons and the time-only classification constructed; actual-data transfer on $(0,\epsilon )\times {\mathbb{R}}^3$ only as a proposition with explicit hypotheses; non-constant-${\chi }_{\mathrm{I}}$ regions and transitions Unresolved. M.5: Functional ledger with named theorem PlateauCurvatureSquareMatching; ledgers (1)–(2) filled; (3)–(4) Unresolved (EOM/redefs undefined). M.6: dilation identity and $A_{\mu }={\mathcal{A}}_{\mu }$ for all $N\le 4$ Constructed; $S^2$ residue $r=-i/(8{\pi }^2)$ Constructed; top-order residues exact and $\varpi$-independent; lower-order values Unresolved ($\varpi$ not fixed by XVI-L). 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 (Section sec:wpB). M.7: Constructed after audit (Gate V for the SA items; (S-bv) certified) for the exactly verified items: Lorentz action on $B_N$ and on ${\partial }_{\gamma }\delta$ with explicit $\eta$ intertwiner and boost check; equivariant dims $1,1,3,3,6$; exact $\mathrm{Equiv}\cap \mathrm{SA}$ ($1/1/12$) with projections and intersections; linear divergence identities. Obs${}_L(R_L)$ Unresolved. See the audit log, Remark rem:M7-audit. M.8–M.9: light stubs (Target II / BV Unresolved; M.8 should wait for XVI-N). Release ladder: M-A Constructed (presentation unconditional on the punctured space); M-B Constructed in the formal reading (Gate V convention); {M-C${}_{\mathrm{curv}}$} (curvature-ledger scaling criterion, Definition def:MCcurv) Proved obstructed, graph-specific (Corollary cor:wpB-MC); broad M-C not achieved (Unresolved: $\varpi$ unspecified by XVI-L, thm:L6); {M-C${}_{\mathrm{data}}$} Satisfied; M-D/M-E not claimed. XVI-L is a frozen dependency. No $\epsilon \to 0$, no loop-as-result, no UV claim, no interacting BRST/Ward identity as proved. }

arrow (domains/images/kernels/intersections/compatibility).{

The arrows need not be surjective or injective. This draft computes the first arrow (M.3) and the intersection of its collapsed image with the variational tensors (M.4). The symmetry arrow is reserved for M.7–M.9. }

Conventions: ambient $M={\mathbb{R}}^4$, $\eta =\mathrm{diag}(1,-1,-1,-1)$, $\mathcal{P}r=r-\frac{1}{2}\eta {\mathrm{tr}}_{\eta }r$, Gram $G=\mathrm{diag}(1,-2,-2,-2,$$1,2,2,1,2,1)$, dual basis $B^i=(G^{-1}{)}^{ii}{B}_i$; Hessian on field directions $B_i$; propagator/source contraction on dual slots $-\mathcal{P}{B}^i$. The XI inverse ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{E_0}^{\mathrm{R}}$ is not equated with the X-sheet ${{\Delta }_R}^{\mathrm{gr}}=-4\mathcal{P}{E_0}^{\mathrm{R}}$. Series voice: Constructed / Proved obstructed / Unresolved / Non-claims. Predecessor quotations are exact.

Figure fig:scope-XVI-M records that separation. Figure fig:split-XVI-M records one view of the same separation. Figure fig:chain-XVI-M records one view of the same separation. Figure fig:pair-XVI-M records one view of the same separation. Figure fig:gate-XVI-M records one view of the same separation. The drawings add no hypothesis.

XVI-L.tex and XVI-L-artifacts/ are a frozen dependency: they are cited and read, not edited. XVI-K-artifacts/hessian_full_out.txt is likewise frozen. The following items are imported as stated in XVI-L.

1. Printed cubic density ${\ell }_3$ (XVI-L, eq:L-ell3; XII-C, eq:ell3).

2. Complete weighted source ${\mathcal{J}}_{\mathrm{D}}={\sigma }_{\epsilon }{J}_3+2C_2\partial \partial {\sigma }_{\epsilon }-2M_2\partial {\sigma }_{\epsilon }$ (XVI-L, eq:L-JD).

3. Full cubic Hessian ${L^{pq}}_{ij}$ (XVI-K, prop:hess, eq:Lpq).

4. Metric-dual contraction on $-\mathcal{P}{B}^i$.

5. Ambient numerical span ${\mathcal{V}}_{\mathrm{adm}}$, $|{\Lambda }_{\mathrm{adm}}|=255$ (XVI-L, def:VL).

6. Selected extension ${\mathfrak{R}}_{\mathrm{L}}$ with ${\mathfrak{R}}_{\mathrm{L}}{|}_{{\mathcal{V}}_{\mathrm{K}}}={\mathfrak{R}}_{\mathrm{K}}$ (XVI-L, def:RL, prop:RL).

7. Coherent contact freedom (XVI-L, thm:L5); raw counts $14321/321/14000$ “are not dimensions”.

8. Subtraction-scale family ${\mathfrak{R}}_{\mathrm{L},\mu }$ and generator $A_{\mu }$ (XVI-L, thm:L6); explicit nonzero scaling residue Unresolved there.

9. Conditional root attachment under (R1)–(R3) (XVI-L, thm:L7).

versus new results. XVI-M does not alter any release claim of XVI-L: L-A/L-C remain as constructed there, L-B remains as stated there, and L-D remains not claimed. The new statements below are results about the objects of XVI-L (its admissible span, its contact freedom, its graph coefficients). They strengthen knowledge of the span without retroactively rewriting XVI-L. In particular, the graded presentation of M.2 is a new result resting on an explicitly checked kernel identification (Gate S-bv, now certified by Theorem thm:wpA-Sbv); it is not a correction of XVI-L, Theorem thm:L5.

Target I (unrooted counterterm matching). Determine which coherent contact assignments of ${\mathfrak{R}}_{\mathrm{L}}$ produce collision tensors that arise as variations of local metric or insertion functionals, without using the root hypotheses (R1)–(R3). Target II (gravitational perturbative-sector interpretation). Interpret those matched functionals inside a specified gravitational action, insertion, gauge choice, ghost content, and diagram inventory.

The M.4 results serve Target I. Target II begins only in skeleton Section sec:m8.

M-C (broad) is the release-ladder matched-scale criterion: it requires (i) at least one proved functional matching (Target I), and (ii) an explicitly specified cutoff $\varpi$ together with a functional representation of the entire collapsed scale variation in a stated reading. {M-C${}_{\mathrm{data}}$} is the weaker data criterion: at least one proved functional matching together with explicitly specified, prescription-dependent scaling formulas (residues, ${\nu }_k[\varpi ]$, top-order coefficients) for the collapsed scale variation; it does not require a fixed numerical cutoff. {M-C${}_{\mathrm{curv}}$} is the curvature-ledger scaling criterion: the scale variation of the graph is matched inside the span of the M.5 curvature-ledger targets (the Euler tensors $E_{R^2},E_{{\mathrm{Ric}}^2}$ at $D=4$, and the lower-degree curvature targets of the ledger), with the matched-scale data of (ii). M-C${}_{\mathrm{curv}}$ implies broad M-C, not conversely; M-C${}_{\mathrm{data}}$ is satisfied in this draft (Theorem thm:wpB-B0, Theorem thm:wpB-dp, Section sec:wpB), while broad M-C remains Unresolved because XVI-L leaves $\varpi$ unspecified (XVI-L, thm:L6; Remark rem:Lressretch). Enlarging the target ledger by non-gauge-invariant quadratic representatives (for example ${I^{\mathrm{top}}}_{\mathrm{scale}}$ of Theorem thm:wpB-dp) changes the matching domain of M-C${}_{\mathrm{curv}}$ but may remain within broad Target I; gauge-fixing/ghost or further graph contributions are a different, independent enlargement and are not the only one.

XVI-L determines the combinatorial graph factor $(1/2!)\cdot 2=1$ and leaves coupling and loop weights undetermined (XVI-L, prop:Lnorm). No number is invented below.

Input (S-bv) (Input inp:Sbv) was introduced as an explicit input of XVI-M. Certification had to establish the representation for the actual predecessor kernel $S=h_0+i{E}_0^{\mathrm{A}}$ on the punctured space $\Omega$, including across the nonzero null cone—not only for a model kernel with the same wave equation, and not only off the cone. (The alternative route, a direct certification of the relations of M.2 and of their completeness, is no longer needed.) Earlier drafts proceeded conditionally on (S-bv); no downstream matching result was used to remove that condition. The unconditional general-$h$ assembly (M.1) and the wave-operator bound (Proposition prop:wave) are stated separately and never depended on this gate. Gate S-bv certified (Theorem thm:wpA-Sbv): $S=\frac{1}{4{\pi }^2}(Q-i0{)}^{-1}$ in ${\mathcal{D}}^{\prime }(\Omega )$, i.e.\ $c=1/(4{\pi }^2)$ with boundary side $Q-i0$ (Section sec:wpA-Sbv, Remark rem:Sbv-status).

Chosen Phase II convention; it is not certified by the normalization ledger.

1. Field variable: the symmetric lowered residual field $h_{\mu \nu }$, with no rescaling and no trace reversal.

2. Variation: $h\mapsto h+t$ with $t$ symmetric and compactly supported.

3. Pairing: $\delta I[h](t)=\int d^{4}y{t}_{\mu \nu }{E_I}^{\mu \nu }[h]$ with the full symmetric-index contraction over all $16$ pairs $(\mu ,\nu )$. In the ten-component coordinates $h_{\mu \nu }$, $\mu \le \nu$:

   $$t_{\mu \nu }{E}^{\mu \nu }=\sum _{\mu \le \nu }{w}_{\mu \nu }{t}_{\mu \nu }{E}^{\mu \nu },w_{\mu \mu }=1,w_{\mu \nu }=2(\mu <\nu ),$$

   so the Euler tensor of a density $\mathcal{L}$ in the independent components is

   $${E_I}^{\mu \nu }=\frac{1}{w_{\mu \nu }}\sum _{\beta }(-D{)}^{\beta }\frac{\partial \mathcal{L}}{\partial ({\partial }^{\beta }{h}_{\mu \nu })}.$$

4. External data: ${\sigma }_{\epsilon }$ and ${\chi }_{\mathrm{I}}$ are fixed under field variation, but their derivatives are kept in every integration by parts.

5. Order: $h$-linear collapsed tensors; quadratic functional ansatz modulo field-independent terms.

6. Helmholtz test: $\int r\cdot (D{E}_C[h]\cdot t)=\int t\cdot (D{E}_C[h]\cdot r)$; for homogeneous linear $E_C$ this is $\int ⟨r,E_C[t]⟩=\int ⟨t,E_C[r]⟩$.

The following are reported computational results, with exact certificates in the archive (Appendix app:script):

• the $1065$-entry specialization of M.1 and the support counts;

• the numerator ranks $d_N$, reconstruction certificates and relation vectors of M.2;

• the general-$h$ ranks $\dim U_N$ and the background/plateau kernels of M.3;

• the collapse ranks, self-adjoint subspaces, Helmholtz certificates, and matching preimages of M.4;

• the symbolic/numerical cross-checks of work package A (Lemma lem:wpA-comp); the top-order residues, coefficient files and scale-tensor collapse certificates of work package B; the time-only kernels, route 1/route 2 agreement and matching probes of work package C (Lemma lem:to-comp);

• the audit-3 checks: the separating functional on the archived target vectors, its pure-gauge sympy verification and the recollapse of $C_{\mathrm{top}}$ (Proposition prop:wpB-sep); the numerical variable-regulator pairing (Lemma lem:wpA-comp); the Beta-integral convolution of Lemma lem:bv(c).

The analytic statements (Lemma lem:bv, Lemma lem:wpA-varreg, Theorem thm:wpA-Sbv, Proposition prop:wpB-sep, Propositions prop:relations, prop:Gsplit, prop:collinj, Lemma lem:helm, Lemma lem:wpB-dw, Theorem thm:wpB-B0, Lemmas lem:to-intertwine and lem:to-indep, Corollaries cor:wpB-timeonly and cor:wpB-MC) are written proofs. The outputs are reproduced verbatim in Appendix app:out.

XVI-L assembles all-sector coefficients on the background $h_{\mu \nu }=(w^1{)}^2{\eta }_{\mu \nu }$ (XVI-L, lem:L2comp: “NOT a general-$h$ symbolic assembly”). This section retains the independent residual-metric jets of ${L^{pq}}_{ij}$.

With ${T_i}^r:=(-\mathcal{P}{B}^i{)}^r$,

and ${\widehat{a^{A\mu \nu }}}_{\lambda }$ is the accumulation over $\{a+p,b+q\}=\lambda$, exactly as in XVI-L, def:La / eq:L-apq, {with ${\chi }_{\mathrm{I}}$ suppressed}. The physical (switched) coefficient—XVI-L style—is the single product

The hat marks the switch-free factor: $a_{\lambda }$ and ${\hat{a}}_{\lambda }$ are not the same object. Here ${L^{pq}}_{ij}$ is kept as the symbolic linear forms of XVI-K in independent $h$-jet symbols $h_{ab}$, ${\partial }_c{h}_{ab}$, ${\partial }_c{\partial }_d{h}_{ab}$. The recorded table graph_general_table.txt stores the switch-free $\hat{a}$ (${\chi }_{\mathrm{I}}$ suppressed), as in XVI-L.

For every assembled switch-free coefficient ${\widehat{a^{A\mu \nu }}}_{\lambda }(y,w;h)$:

1. it is linear in the residual-metric jet of order $2-v$, where $v=|p|+|q|$;

2. it is linear in the preparation jet of order $2-s$, where $s=|a|+|b|$ ($s=0,1,2$ for $C_2,M_2,J_3$);

3. $N=s+v$, and every monomial is one preparation-jet symbol times one $h$-jet symbol, with a rational coefficient;

4. it is symmetric in $(\mu ,\nu )$;

5. it is consistent under edge interchange, $(A;a,b;p,q)\leftrightarrow (A;b,a;q,p)$;

6. no unintended generators occur, and the arithmetic is exact rational.

The supports satisfy ${{\Lambda }_{\mathrm{graph}}}^{\mathrm{background}}\subseteq {{\Lambda }_{\mathrm{graph}}}^{\mathrm{formal}}\subseteq {\Lambda }_{\mathrm{adm}}$ with cardinalities $235\le 255\le 255$ (sectorwise $C_2$$25/25/25$, $M_2$$84/84/84$, $J_3$$202/250/250$).

Proof. (1)–(3) and (6) are inherited from XVI-K, prop:hess, and XVI-L, prop:jet, through Definition def:aGeneral. (4) holds because $P_A$ is symmetric in $(\mu ,\nu )$. (5) holds because the defining sum is symmetric under simultaneous interchange of the two edges. The supports, and a coefficientwise confirmation of (1)–(6), are the reported computation of Lemma lem:M1comp.

After the specialization $h_{\mu \nu }=(w^1{)}^2{\eta }_{\mu \nu }$ (with ${\chi }_{\mathrm{I}}$ suppressed as in XVI-L), the general-$h$ table equals two_edge_coeff_table.txt entry by entry. All $1065$ nonzero grouped coefficients $(A,\lambda ,(\mu ,\nu ))$ agree as polynomials in the preparation jet and $w^1$, and the nonzero key sets coincide.

graph_general.py loads $P_A$ from the frozen XVI-L source table and ${L^{pq}}_{ij}$ from the frozen XVI-K Hessian list. It assembles the switch-free $\hat{a}$ of Definition def:aGeneral in exact $\mathbb{Q}$ arithmetic and exits with status $0$, recording the following.

1. (G1)

   Proposition prop:M1(1)–(6). The edge-interchange check is exhaustive: it runs over every enumerated ordered placement whose swap is distinct, which gives $1000$ unordered pairs.

2. (G2)

   The supports of Proposition prop:M1. Twenty labels lie in ${{\Lambda }^{\mathrm{formal}}}_{\mathrm{graph}}\setminus {{\Lambda }^{\mathrm{background}}}_{\mathrm{graph}}$.

3. (G3)

   Theorem thm:M1spec ($1065/1065$ entries).

4. (G4)

   An independent $C_2$ route via xvi_common.polar_s ($250$ comparisons).

5. (G5)

   $3282$ nonzero $(A,\lambda ,(\mu ,\nu ))$ lines. The sector-summed table has $2322$ entries with $18646$ bilinear terms.

XVI-L does not compute the full relation space of its $255$ admissible labels (XVI-L, thm:L5). This section constructs the presentation $\pi :{\mathbb{C}}^{{\Lambda }_{\mathrm{adm}}}\to {\mathcal{V}}_{\mathrm{adm}}$, $e_{\lambda }\mapsto {q^{\circ }}_{\lambda }$, degree by degree.

On $\Omega :={\mathbb{R}}^4\setminus \{0\}$,

where $S=h_0+i{E}_0^{\mathrm{A}}$ is the predecessor edge kernel (XVI-K, eq:S) and $(Q-i0{)}^{-1}$ is the boundary-value distribution $T_1$ of Lemma lem:bv. In its original formulation the constant $c$ was left unspecified; the graded results of M.2–M.4 and M.7 use only $c\ne 0$ and the side $Q-i0$. Theorem thm:wpA-Sbv certifies the input with $c=1/(4{\pi }^2)$.

XVI-K quotes $D_F=-\mathcal{P}(h_0+i{E}_0^{\mathrm{A}})$ and states “with $h_0=-W_0$ (XI-E.tex, eq:h)”. XI-E, eq:h, reads $h_{\mu }=-W_{\mu }$, $h_{\mu }-{h_{\mu }}^{\mathsf{T}}=i{\delta }_{\mu }$. Hence $S=h_0+i{E}_0^{\mathrm{A}}=-W_0+i{E}_0^{\mathrm{A}}$, and XVI-J, lem:identify, identifies this with $-{{\Delta }_F}^{\mathrm{FR}}$. A Euclidean calculation motivates the candidate form. Substituting $x^0-y^0=-i{\tau }_{\mathrm{E}}$ (${\tau }_{\mathrm{E}}>0$) into the $W_0$ integral and using ${{\int }_0}^{\infty }{e}^{-p{\tau }_{\mathrm{E}}}\sin (pr)dp=r/({{\tau }_{\mathrm{E}}}^2+r^2)$ (checked symbolically in kernel_relations.py) gives $1/(4{\pi }^2({{\tau }_{\mathrm{E}}}^2+r^2))$, which suggests the motivational value $c=1/(4{\pi }^2)$. {This Euclidean calculation motivates the candidate; it does not certify the real-time identification of $S$ on $\Omega$ as a single boundary value.} The certification is the real-time argument of Theorem thm:wpA-Sbv, which uses the predecessor Fourier family of $W_0$ and the step representation of ${E_0}^{\mathrm{A}}$, not this Euclidean slice. XVI-K states: “No explicit position-space formula for $S$ is used.” {Boundary value versus $\epsilon$.} The symbol $i0$ denotes a distributional boundary value. It is unrelated to the fixed preparation parameter $\epsilon$. The series rule “no $\epsilon \to 0$” (XVI-I, prop:signs(5): “There is no $i\epsilon$ prescription and no limit $\epsilon \to 0$”) does not forbid the $i0$ notation, and Input (S-bv) introduces no prescription and changes no definition.

What predecessors supply. XVI-K, eq:S, defines $S:=h_0+i{E}_0^{\mathrm{A}}=-W_0+i{E}_0^{\mathrm{A}}=-{{\Delta }_F}^{\mathrm{FR}}$ through two globally defined distributions, with Fourier $z^0$-family formulae (XVI-K, lem:families / eq:W-fam, eq:E-fam) quoted from XVI-J and XI-E, eq:h ($h_0=-W_0$). Homogeneity of degree $-2$ is proved (XVI-K, lem:homog); $\square S=0$ on $\Omega$ (XVI-K, lem:grading(3)); the Hörmander product of two copies exists on $y\ne w$ (XVI-J / XVI-K, lem:prod). XVI-K states explicitly: “No explicit position-space formula for $S$ is used.” XI-E, eq:W and eq:h, likewise give the Fourier definition of $W_{\mu }$ and the normalization $h_{\mu }=-W_{\mu }$, with no $(Q-i0{)}^{-1}$ formula. XVI-J, however, already prints the position-space formula: eq:DF sets $S=h_0+i{E}_0^{\mathrm{A}}=-d$ with $d=W_0-i{E}_0^{\mathrm{A}}$, and eq:rational states $d(z)=-\frac{1}{4{\pi }^2}(Q_{\mathrm{geom}}(z)-i0{)}^{-1}$ on $z\ne 0$, including nonzero null points; XVI-J, prop:q, uses it (quotations in Section sec:wpA-identify). Its printed justification is “direct Fourier evaluation”.

What was missing. A written real-time derivation of XVI-J, eq:rational, including the boundary-value argument at nonzero null points. The Euclidean Wick calculation of Remark rem:Sbv-motivation motivates the candidate $c=1/(4{\pi }^2)$; it is not a real-time identification of $S$ with $c{T}_1$ on $\Omega$, and it does not settle equality across the nonzero null cone. Lemma lem:bv establishes the model calculus for $T_m$; it does not identify $S$ with $c{T}_1$. No matching probe, collapse rank, or Helmholtz certificate discharges (S-bv).

Certification (this cycle): a written check of XVI-J, eq:rational, not a new extension. Theorem thm:wpA-Sbv establishes, from the predecessor Fourier families, all four required items on $\Omega$: (i) the distributional representation $S=c(Q-i0{)}^{-1}$; (ii) the boundary-value side, the same as for $T_1$ (namely $Q-i0$, not $Q+i0$); (iii) the normalization $c=1/(4{\pi }^2)\ne 0$; (iv) equality across the nonzero null cone, not merely on $\{Q\ne 0\}$. The alternative route (certifying the $185$ relations and their completeness directly) is therefore not needed. Consequently every graded M.2–M.4 (and M.7) conclusion that uses the presentation of Theorem thm:M2 holds unconditionally on $\Omega$; only $c\ne 0$ and the single side were ever used. Statements that also use Gate V remain relative to that chosen convention.

This lemma concerns the model family $T_m$ itself; it does not yet identify $T_m$ with derivatives of the predecessor kernel $S$. For $m\ge 1$ let $T_m:=(Q-i0{)}^{-m}:={\lim }_{\tau \downarrow 0}(Q-i\tau {)}^{-m}$ in ${\mathcal{D}}^{\prime }(\Omega )$. The limit exists, and:

1. (a)

   $Q{T}_{m+1}=T_m$;

2. (b)

   ${\partial }_{\rho }{T}_m=-m({\partial }_{\rho }Q)T_{m+1}$;

3. (c)

   $T_m{T}_n=T_{m+n}$, where the product is the existing Hörmander product on $\Omega$ (wavefront criterion).

Proof. {Points with $Q(z_0)\ne 0$.} Near such a point $Q\ne 0$ and $(Q-i\tau {)}^{-m}\to Q^{-m}$ in $C^{\infty }$, so $T_m=Q^{-m}$ there and (a)–(c) are identities of smooth functions.

Nonzero null points. Let $Q(z_0)=0$ with $z_0\ne 0$. Then $dQ(z_0)=2\eta (z_0,\cdot )\ne 0$, because $\eta$ is nondegenerate and $z_0\ne 0$. Hence $Q$ is a local coordinate: there is a neighbourhood $U$ and a diffeomorphism $\Phi :U\to (-\delta ,\delta )\times V^{\prime }$ with ${\Phi }_1=Q$. On $(-\delta ,\delta )$, $(s-i\tau {)}^{-m}\to (s-i0{)}^{-m}$ in ${\mathcal{D}}^{\prime }$. This is the boundary value from $\{\mathrm{Im}\zeta <0\}$ of the holomorphic function ${\zeta }^{-m}$, and $(s-i0{)}^{-m}=\frac{(-1{)}^{m-1}}{(m-1)!}{{\partial }_s}^{m-1}(s-i0{)}^{-1}$. Pullback by the submersion $\Phi$ (Hörmander, Thm. 6.1.2) is continuous, so $T_m{|}_U={\Phi }^{\ast }((s-i0{)}^{-m}\otimes 1)$ and the limit exists on $U$.

Its Fourier transform satisfies $\widehat{(s-i0{)}^{-m}}(\sigma )=\frac{2\pi i(-i\sigma {)}^{m-1}}{(m-1)!}\theta (-\sigma )$ for $\hat{u}(\sigma )=\int e^{-is\sigma }u(s)ds$. This transform is supported in the single half-line $\sigma \le 0$, the same for every $m$. Hence $\mathrm{WF}((s-i0{)}^{-m})\subset \{(0,\sigma ):\sigma <0\}$, and by the pullback theorem (Hörmander, Thm. 8.2.4)

with one sign for all $m$.

Compatibility. All $T_m$ are defined by the same limit $\tau \downarrow 0$ from the same side, globally on $\Omega$. The local descriptions therefore agree on overlaps, and with $Q^{-m}$ off the cone.

(a) $Q(Q-i\tau {)}^{-m-1}=(Q-i\tau {)}^{-m}+i\tau (Q-i\tau {)}^{-m-1}$. In the local coordinate, $\tau (s-i\tau {)}^{-m-1}$ is a constant times ${{\partial }_s}^m(\tau (s-i\tau {)}^{-1})$, and $\tau (s-i\tau {)}^{-1}={\partial }_s(\tau \log (s-i\tau ))$ with $\tau \log (s-i\tau )\to 0$ in ${L^1}_{\mathrm{loc}}$; so the remainder tends to $0$ in ${\mathcal{D}}^{\prime }$.

(b) Derivatives commute with limits in ${\mathcal{D}}^{\prime }$, and ${\partial }_{\rho }(Q-i\tau {)}^{-m}=-m({\partial }_{\rho }Q)(Q-i\tau {)}^{-m-1}$.

(c) Same-sign boundary-value product. Since $\mathrm{WF}(T_m),\mathrm{WF}(T_n)\subset {\Gamma }_{-}$ and ${\Gamma }_{-}$ contains no opposite covectors $(z,\xi ),(z,-\xi )$, Hörmander's criterion (Thm. 8.2.10) defines the product $T_m{T}_n$ in ${\mathcal{D}}^{\prime }(\Omega )$. It remains to identify that product with $T_{m+n}$.

All $T_m$ are boundary values from the same lower half-plane. Locally near a nonzero null point, in the coordinate $s=Q$ of the first paragraph,

The one-variable distributions $(s-i0{)}^{-m}$ and $(s-i0{)}^{-n}$ therefore arise as boundary values of the holomorphic functions $\zeta \mapsto {\zeta }^{-m}$ and $\zeta \mapsto {\zeta }^{-n}$ from $\{\mathrm{Im}\zeta <0\}$. We identify their Hörmander product by an explicit Fourier computation. Write $u_m:=(s-i0{)}^{-m}$ and use the one-variable convention $\hat{u}(\sigma )=\int e^{-is\sigma }u(s)\mathrm{d}s$, with inverse $u(s)=\frac{1}{2\pi }\int e^{is\sigma }\hat{u}(\sigma )\mathrm{d}\sigma$; then $\widehat{uv}=\frac{1}{2\pi }\hat{u}\ast \hat{v}$ (Hörmander's $(2\pi {)}^{-n}$ with $n=1$). As above, ${\hat{u}}_m(\sigma )=\frac{2\pi i(-i\sigma {)}^{m-1}}{(m-1)!}\theta (-\sigma )$ (for $m=1$: $\widehat{\mathrm{PV}\frac{1}{s}}=-i\pi \mathrm{sgn}\sigma$ and $\widehat{i\pi \delta }=i\pi$; then $\widehat{{\partial }_{s}u}=i\sigma \hat{u}$).

Beta integral. For $\sigma <0$ the convolution runs over the finite interval $[\sigma ,0]$, where both factors are supported; with ${\sigma }^{\prime }=\sigma x$, $x\in [0,1]$, $\mathrm{d}{\sigma }^{\prime }$ over $[\sigma ,0]$ becomes $|\sigma |\mathrm{d}x$, and

Since $B(m,n)=\frac{(m-1)!(n-1)!}{(m+n-1)!}$ and $i|\sigma |=-i\sigma$ for $\sigma <0$, this equals $\frac{2\pi i(-i\sigma {)}^{m+n-1}}{(m+n-1)!}={\hat{u}}_{m+n}(\sigma )$; for $\sigma >0$ the integrand vanishes. So $\frac{1}{2\pi }{\hat{u}}_m\ast {\hat{u}}_n={\hat{u}}_{m+n}$, the integral being absolutely convergent.

Identification with the Hörmander product. For $\phi \in {C^{\infty }}_c(\mathbb{R})$ the Hörmander product is characterized locally by $\widehat{{\phi }^{2}uv}=\frac{1}{2\pi }\widehat{\phi u}\ast \widehat{\phi v}$, the integral converging absolutely under the wavefront condition (Hörmander, proof of Thm. 8.2.10). Here $\widehat{\phi u_m}=\frac{1}{2\pi }\hat{\phi }\ast {\hat{u}}_m$. Because $\hat{\phi }$ is Schwartz while ${\hat{u}}_m,{\hat{u}}_n$ are polynomially bounded and supported in $\sigma \le 0$, the iterated integral $\int \int \int |\hat{\phi }(a)\hat{\phi }(b){\hat{u}}_m(\sigma -a-b-c){\hat{u}}_n(c)|$ runs in $c$ over an interval of length at most $|\sigma -a-b|$ and converges absolutely; Fubini allows regrouping:

Hence ${\phi }^2(u_m{u}_n)={\phi }^2{u}_{m+n}$ for every $\phi$, i.e.

Transfer to the chart. With $p(s,y^{\prime })=s$, $u_m\otimes 1=p^{\ast }{u}_m$, and pullback is multiplicative on distributions satisfying the product condition (Hörmander, Thms. 8.2.4 and 8.2.10: the product is the pullback of the tensor product by the diagonal map, and pullbacks compose); thus $p^{\ast }{u}_m\cdot p^{\ast }{u}_n=p^{\ast }{u}_{m+n}$, and, pulling back further by the diffeomorphism $\Phi$, $T_m{T}_n=T_{m+n}$ on $U$. Off the cone the identity is the smooth identity $Q^{-m}{Q}^{-n}=Q^{-(m+n)}$. On overlaps the two local descriptions agree by the global same-side limit $\tau \downarrow 0$. Thus $T_m{T}_n=T_{m+n}$ on $\Omega$.

By Theorem thm:wpA-Sbv, Input (S-bv) holds. Then, using Lemma lem:bv(a),(b), ${\partial }^{\alpha }S=c{p}_{\alpha }{T}_{1+|\alpha |}$ on $\Omega$. Here $p_0=1$ and $p_{\alpha +e_{\rho }}=Q{\partial }_{\rho }{p}_{\alpha }-(1+|\alpha |)p_{\alpha }{\partial }_{\rho }Q$, so $p_{\alpha }$ is homogeneous of degree $|\alpha |$. For $\lambda =\{\alpha ,\beta \}$ of order $N=|\alpha |+|\beta |$, Lemma lem:bv(c) gives

Proof. By induction, ${\partial }_{\rho }(p_{\alpha }{T}_k)=({\partial }_{\rho }{p}_{\alpha })T_k-k{p}_{\alpha }({\partial }_{\rho }Q)T_{k+1}$ by (b), and $T_k=Q{T}_{k+1}$ by (a); this gives the recursion. The product is the Hörmander product used for $q^{\circ }$ on $y\ne w$ (XVI-J: “the pointwise product of two copies of ${{\Delta }_F}^{\mathrm{FR}}$ exists”), and (c) applies. kernel_relations.py computes $p_{\alpha }$ from the same recursion, realised as ${\partial }^{\alpha }(1/Q)=p_{\alpha }/Q^{1+|\alpha |}$, for all $|\alpha |\le 4$.

Use Theorem thm:wpA-Sbv and fix $N$. For constants $r_{\lambda }$:

1. if ${\sum }_{\lambda }{r}_{\lambda }{n}_{\lambda }=0$ as a polynomial, then ${\sum }_{\lambda }{r}_{\lambda }{q^{\circ }}_{\lambda }=0$ in ${\mathcal{D}}^{\prime }(\Omega )$;

2. if ${\sum }_{\lambda }{r}_{\lambda }{q^{\circ }}_{\lambda }=0$ in ${\mathcal{D}}^{\prime }(\Omega )$, then ${\sum }_{\lambda }{r}_{\lambda }{n}_{\lambda }=0$ as a polynomial.

Relations mixing different $N$ split into homogeneous components, because ${q^{\circ }}_{\lambda }$ is homogeneous of degree $-4-N$. Hence the nullspace of the numerator map is exactly the relation space of $\pi$.

Proof. (1) By Proposition prop:num, $\sum r_{\lambda }{q^{\circ }}_{\lambda }=c^2(\sum r_{\lambda }{n}_{\lambda })T_{2+N}=0$. (2) Restrict to the nonempty open set $\{Q<0\}\subset \Omega$ (it contains $(0,1,0,0)$). There $T_{2+N}=Q^{-2-N}$ is smooth and nowhere zero, so $c^2(\sum r_{\lambda }{n}_{\lambda })Q^{-2-N}=0$ pointwise. Hence the polynomial $\sum r_{\lambda }{n}_{\lambda }$ vanishes on a nonempty open set and is therefore zero. For mixed $N$, dilations $z\mapsto tz$ act on $\Omega$, and distinct homogeneity degrees separate.

With Input (S-bv) certified (Theorem thm:wpA-Sbv):

The numerator map is onto ${\mathrm{Poly}}_N$. A basis ${\mathcal{B}}_N$ extends a basis of ${\mathcal{V}}_{\mathrm{K}}={\mathcal{H}}_4\oplus {\mathcal{H}}_5\oplus {\mathcal{H}}_6$ ($\dim 15$). Every label reconstructs as ${q^{\circ }}_{\lambda }={\sum }_{b\in {\mathcal{B}}_N}{R}_{\lambda b}{q^{\circ }}_b$ with exact rational $R$.

Proof. Propositions prop:num and prop:relations, together with the exact $\mathbb{Q}$-linear algebra reported in Lemma lem:M2comp.

Without using Input (S-bv), ${\sum }_{\rho ,\sigma }{\eta }^{\rho \sigma }{q^{\circ }}_{\{\alpha ,\beta +e_{\rho }+e_{\sigma }\}}=0$ on $\Omega$, since $\square S=0$ there (XVI-K, lem:grading(3)). The family spans a subspace of dimensions $(0,0,1,8,36)$, so $\dim {\mathcal{V}}_{\mathrm{adm}}\le 210$ without any kernel identification. With (S-bv) (certified), $140$ further independent relations appear.

With $\eta =\mathrm{diag}(1,-1,-1,-1)$, ${\partial }_{0}Q=2z_0$ and ${\partial }_{1}Q=-2z_1$, so $p_{e_0}=-2z_0$, $p_{e_1}=2z_1$, $p_{e_0}{p}_{e_1}=-4z_0{z}_1$, and $p_{e_0+e_1}=-8z_0{z}_1$. Hence $({\partial }_{0}S)({\partial }_{1}S)=\frac{1}{2}S{\partial }_0{\partial }_{1}S$ on $\Omega$ (Theorem thm:wpA-Sbv); this is the relation q{1, 0} - (1/2) q{(), 01} = 0. The script checks exactly this example (require: “sign example”). The relation is not generated by $\square S=0$.

kernel_relations.py exits with status $0$ and records:

• the table of Theorem thm:M2;

• reconstruction certificates for all labels (kernel_basis_table.txt, which also lists every numerator row $n_{\lambda }$);

• $185$ certified relation vectors and $45$ wave-operator generators (kernel_relation_matrix.txt);

• the sign example.

regression_M.py independently spot-checks every relation (on $({\partial }^{\alpha }S)({\partial }^{\beta }S)$ at three rational points).