Landmark XVI-L Complete-source two-edge renormalization, local collision counterterms, and scale dependence Draft — Release Status: XVI-L constructs the complete-source contribution for the specified two-edge graph, conditional on the imported root hypotheses, using a coherent extension on the fixed admissible numerical span. It characterizes coherent collision freedom abstractly and constructs a smooth subtraction-scale family. All-sector contraction is computationally verified as reported on the stated residual-metric background. L-D is not claimed. L-A/L-C Constructed; L-B abstract finite-dimensional characterization

(XVI-L, this draft).{ Overall release: XVI-L constructs the complete-source contribution for the specified two-edge graph, conditional on the imported root hypotheses, using a coherent extension on the fixed admissible numerical span. It characterizes coherent collision freedom abstractly and constructs a smooth subtraction-scale family. All-sector contraction is computationally verified as reported on the stated residual-metric background. L-D is not claimed. L-A and L-C of Section sec:ladder: Constructed, conditional on (R1)–(R3). L-B: abstract finite-dimensional characterization (all numerical-kernel relations; selected constraints exhibited; raw contact counts are not dimensions). L-D is not claimed. L.0: Recorded. L.1–L.2: constructed-as-stated with complete-graph contraction computationally verified (Sections sec:l1–sec:l2). L.3–L.5: constructed-as-stated with inventory/coherence repairs as recorded (Sections sec:l3–sec:l5). L.6 (renormalization scale): constructed-as-stated — scale family ${\mathfrak{R}}_{\mathrm{L},\mu }$, first scale theorem, infinitesimal generator $A_{\mu }$, prescription versus scale-variation-coefficient distinction Proved locally; explicit nonzero scaling residue (anomaly form) Unresolved (Section sec:l6). L.7 (root attachment and exact regressions): constructed-as-stated — ${U^{{\mathfrak{R}}_{\mathrm{L}}}}_{\mathrm{L}}$, ${{\mathcal{C}}^{{\mathfrak{R}}_{\mathrm{L}}}}_{\mathrm{L}}$, source-sector K regression and separate restriction ${\mathfrak{R}}_{\mathrm{L}}{|}_{{\mathcal{V}}_{\mathrm{K}}}={\mathfrak{R}}_{\mathrm{K}}$, propagated $\Delta {\mathcal{C}}_{\mathrm{L}}$ and scale variation through $B_{f,g}$ Proved locally; truncation regressions Computationally verified (Section sec:l7). L.8 (bridge to gravitational EFT): Recorded as matching questions only — short matching section and BV boundary; L-D remains not claimed (Section sec:l8). }

(XVI-L).{ Ambient $M={\mathbb{R}}^4$, $\eta =\mathrm{diag}(1,-1,-1,-1)$, $\mathcal{P}r=r-\frac{1}{2}\eta {\mathrm{tr}}_{\eta }r$, ${\mathcal{P}}^2=\mathrm{id}$; fixed $0<\epsilon <T$; fixed preparation $f$, $g$, ${\sigma }_{\epsilon }$ (not varied); an independent compact interaction switch ${\chi }_{\mathrm{I}}\in {C_c}^{\infty }(M)$; off-shell smooth residual metric $h$; smooth scalar test $\varphi$. The pairing is $⟨A,C{⟩}_{\eta }=A_{\mu \nu }{C}^{\mu \nu }$ and the metric-dual basis is $B^i=(G^{-1}{)}^{ii}{B}_i$, $G=\mathrm{diag}(1,-2,-2,-2,1,2,2,1,2,1)$, exactly as in XVI-K. The root is the imported $B_{f,g}$ of XII-D.tex, eq:B-kernel, used only through (R1)–(R3) of the XVI-K hypothesis box (XVI-K.tex, Section sec:hyp): (R1) the composition exists; (R2) compact support; (R3) $\mathrm{WF}(B_{f,g})\subset Z_2=\{(x_1,p_1;x_2,p_2):p_1+p_2=0\}\setminus 0$.

Change relative to XVI-K. The selected source $2{C_2}^{\mu \nu \alpha \beta }{\partial }_{\alpha }{\partial }_{\beta }{\sigma }_{\epsilon }$ is replaced by the complete imported quadratic cubic source of XII-C.tex, eq:actualsource,

and all three summands are to be contracted with the full cubic Hessian ${L^{pq}}_{ij}$ of XVI-K, eq:Lpq.

Name of the object. The target object is called the complete-source contribution for the specified two-edge graph. It is not the complete one-loop gravitational correction: completing the insertion source does not complete the diagrammatic content of the theory.

Levels kept distinct. (1) source and vertex differential operators; (2) scalar numerical distributions; (3) their selected extensions; (4) unrooted collision counterterms; (5) propagated root-attached ambiguities; (6) scale variation; (7) gravitational symmetry compatibility. A result at one level is not a result at another. In this draft levels (1)–(6) are treated as constructions (L.1–L.7): source jet, graph decomposition, numerical-kernel span, coherent extension, unrooted collision counterterms, scale variation, and root-attached propagated ambiguities; level (7) (gravitational symmetry compatibility / matching) is recorded as matching questions only (L.8), with L-D not claimed. Not claimed: BRST, BV, Ward identities, all-orders renormalization, $\epsilon \to 0$, a state, an ultraviolet completion, a physical beta function for a complete gravitational quantity, or any proved match of collision counterterms to gravitational Wilson coefficients. }

Conventions are those of XVI-K.tex (quoted there from XVI-J.tex and XII-C.tex). Indices are moved with $\eta$; parentheses in index expressions include the factor $\frac{1}{2}$. Multi-indices $a,b,p,q,\gamma \in {\mathbb{N}}^4$ act by ${\partial }^a={{\partial }_0}^{a_0}\cdots {{\partial }_3}^{a_3}$, $|a|=a_0+\cdots +a_3$. The symmetric basis is $B_0,\dots ,B_9$ in the order $(00),(01),(02),(03),(11),(12),(13),(22),(23),(33)$, $(B_{(ab)}{)}_{ab}=(B_{(ab)}{)}_{ba}=1$; for a symmetric tensor $X={\sum }_i{X}^i{B}_i$, $X^i=X_{ab}$ for $i=(ab)$ (XVI-K, Section K.1). Equation labels of predecessor files are cited as “file, label”.

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

Figure fig:scope-XVI-L records that separation. The drawing adds no hypothesis.

XVI-K.tex is treated as a frozen dependency: it is cited, not edited. Its release is “Constructed, conditional on the imported root hypotheses, for the $C_2$ sector with full cubic vertex” (XVI-K.tex, title and Section sec:status). The following items are imported as stated there; nothing beyond the statements is used.

1. Edge kernel. $S:=h_0+i{E}_0^{\mathrm{A}}=-{{\Delta }_F}^{\mathrm{FR}}$, $D_F=-\mathcal{P}S$ (XVI-K, eq:S), a globally defined distribution of $z=y-w$, homogeneous of degree $-2$ (XVI-K, lem:homog).

2. Full cubic Hessian. XVI-K, Proposition prop:hess, eq:Lpq:

   $$D^2{\ell }_3(h)[A,C](w)={\sum _{i,j=0}}^9\sum _{|p|+|q|\le 2}{L^{pq}}_{ij}(w;h){\partial }^p{A}^i(w){\partial }^q{C}^j(w),$$

   with ${L^{pq}}_{ij}$ linear in ${\partial }^{r}h$, $|r|=2-|p|-|q|$, and ${L^{pq}}_{ij}={L^{qp}}_{ji}$.

3. Placement sign. XVI-K, eq:sign: ${{\partial }_w}^p[u(y-w)]=(-1{)}^{|p|}({\partial }^{p}u)(y-w)$, ${{\partial }_y}^p[u(y-w)]=({\partial }^{p}u)(y-w)$.

4. Extension map. XVI-K, Definition def:RK and Proposition prop:RK: the linear map ${\mathfrak{R}}_{\mathrm{K}}:{\mathcal{V}}_{\mathrm{K}}\to {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$ built with the $\varpi$-projections $W_{\rho }$ (XVI-K, eq:Wrho) and zero declared moments (XVI-K, eq:moments).

5. Root attachment. XVI-K, Lemma lem:attachK and Theorem thm:main, conditional on (R1)–(R3).

6. Propagated ambiguity. XVI-K, Lemma lem:amb, eq:deltaCK.

(a) Coherent versus placementwise changes. XVI-K, Remark rem:coherent, distinguishes two levels, and XVI-L keeps them apart: placementwise families (a separate same-degree extension for each labelled kernel ${q^{\circ }}_{\mathsf{k}}$; XVI-K, eq:deltaCK holds at that level), and coherent changes ${\mathfrak{R}}_{\mathrm{K}}^{\prime }-{\mathfrak{R}}_{\mathrm{K}}$, linear on the actual span ${\mathcal{V}}_{\mathrm{K}}$, for which every constant-coefficient relation ${\sum }_{\mathsf{k}}{c}_{\mathsf{k}}{q^{\circ }}_{\mathsf{k}}=0$ forces ${\sum }_{\mathsf{k}}{c}_{\mathsf{k}}{C}_{\mathsf{k}\gamma }=0$ for every $\gamma$. Only coherent changes will be called renormalization-prescription changes in XVI-L (target L.4–L.5). Within coherent changes, XVI-K distinguishes those altering the XVI-J component ($C_{\{0,0\},0}\ne 0$) from those preserving it ($C_{\{0,0\},0}=0$).

(b) Declared subtraction orders. XVI-K, Remark rem:N: $N_{\mathsf{k}}:=|p|+|q|$ is a declared order, not defined through the scaling degree; $\mathrm{sd}({q^{\circ }}_{\mathsf{k}})\le 4+N_{\mathsf{k}}$, with equality for nonzero ${q^{\circ }}_{\mathsf{k}}$, and a vanishing kernel costs nothing. XVI-L adopts the same convention for the new kernels: the declared order of a numerical kernel is its total numerical derivative count, fixed before any question of vanishing is decided (target L.3).

(c) Computational artifact. The XVI-K artifact is the folder XVI-K-artifacts/ (hessian_full.py, xvi_common.py, recompute_K_dual.py, hessian_full_out.txt); the shared routines (negP, polar_s, Q_of, C2_component, ell3_of) are named functions of xvi_common.py (XVI-K, Appendix app:script). XVI-L does not modify that folder. The XVI-L artifact is the folder XVI-L-artifacts/, which contains a copy of xvi_common.py (byte-identical; scripts check this when the original is present), source_full.py with source_full_out.txt/source_full_table.txt (L.1), two_edge_full.py with two_edge_full_out.txt (L.2), kernel_inventory.py with kernel_inventory_out.txt (L.3), counterterm_map.py with counterterm_map_out.txt (L.4–L.5), and regression_L.py with regression_L_out.txt (L.6–L.7) (Appendix app:script).

(d) Verification boundary. Three kinds of evidence are distinguished throughout, as in XVI-K, Appendix app:script: symbolic constructions (exact polynomial identities in independent jet coordinates, valid for all smooth fields); coefficientwise structural checks (properties verified monomial by monomial on such a symbolic polynomial); finite exact tests (exact rational evaluation on specified test data; agreement there is not a coefficientwise equivalence proof). A statement is called “Proved locally” only if it has a written argument in this note; a statement supported only by the script is called “Computationally verified” and its kind of evidence is named.

(e) Archive. Run commands (from any working directory; package versions are printed in the first line of each output): /workspace/tmp/venv/bin/python XVI-L-artifacts/source_full.py > XVI-L-artifacts/source_full_out.txt; /workspace/tmp/venv/bin/python XVI-L-artifacts/two_edge_full.py > XVI-L-artifacts/two_edge_full_out.txt; /workspace/tmp/venv/bin/python XVI-L-artifacts/kernel_inventory.py > XVI-L-artifacts/kernel_inventory_out.txt; /workspace/tmp/venv/bin/python XVI-L-artifacts/counterterm_map.py > XVI-L-artifacts/counterterm_map_out.txt; /workspace/tmp/venv/bin/python XVI-L-artifacts/regression_L.py > XVI-L-artifacts/regression_L_out.txt (Appendix app:out).

No BV, BRST, Ward-identity, or all-orders renormalization result is transferred to XVI-L. The locally covariant BV framework and the EFT reading of gravity are background orientation only; neither is a premise of any statement below, and neither promotes one graph coefficient to a theory. No statement of XVI-K about the $C_2$ sector is upgraded by XVI-L: XVI-K's release stays exactly as stated there.

No source derivative count is assumed in this section. The printed formulas are quoted, implemented atom by atom, and the derivative budget is then derived.

XII-C.tex, Section sec:density (lines above eq:finite-curvature): with $\tau ={\mathrm{tr}}_{\eta }\gamma$,

XII-C.tex, eq:finite-curvature: for $n=1,2,3$,

XII-C.tex, eq:ell3:

XII-C.tex, the definitions printed before eq:J3, and eq:J3:

XII-C.tex, eq:Ccoefficients (with $C^{\mu \nu \alpha \beta }(q)=\sqrt{-q}(q^{\mu \nu }{q}^{\alpha \beta }-q^{\mu (\alpha }{q}^{\beta )\nu })=C_0+\kappa C_1+{\kappa }^2{C}_2+O({\kappa }^3)$):

XII-C.tex, eq:M2:

XII-C.tex, Theorem thm:actualsource, eq:actualsource: the selected cubic vertex $V_{3,\mathrm{D}}[\gamma ]=\int \sigma {\ell }_3[\gamma ]{\mathrm{d}}^{4}x$ has Euler source

XII-C.tex, eq:full-trace-sector, and XII-B (quoted in XII-C.tex as eq:input-trace): for $\gamma =\psi \eta$, $C_1=C_2=0$, ${\Gamma }_2=-\psi {\Gamma }_1$, and

D_^{}|_{=}=-[^{}(_)^+(^)^+(^)^

gr], _{} D_^{}|_{=}=-6(_)^ .

In XVI-L, $\sigma ={\sigma }_{\epsilon }$ is the fixed ramp of XII-C (Section sec:freeze); it is not varied.

Equation (eq:L-J3) specifies $J_3$ only implicitly: through $V_1$, $V_2$, which are built from the curvature atoms (eq:L-fc). XII-C prints no table of elementary monomials of $J_3$, and none is typed into XVI-L. The script source_full.py implements $J_3$ through exactly these printed definitions (atoms $\to$$R_n$, $r_n$$\to$$V_1,V_2$$\to$ (eq:L-J3)); the coefficient table $P_{J_3}$ below is an output of that implementation, never an input. The same holds for $C_2$ (via (eq:L-C)) and $M_2$ (via (eq:L-C), (eq:L-fc), (eq:L-M2)).

For a smooth symmetric $\gamma$ put

so that ${\mathcal{J}}_{\mathrm{D}}={\mathfrak{s}}_{J_3}+{\mathfrak{s}}_{C_2}+{\mathfrak{s}}_{M_2}$ by (eq:L-JD). For $A\in \{C_2,M_2,J_3\}$ and smooth symmetric $X,Y$, the polarization is XII-D's rule, as used in XVI-K, Definition def:a:

$X$ and $Y$ are the two contracted field slots (the increments carried by the two edges in L.2). No residual metric $h$ enters ${\mathfrak{s}}_A$: the source is quadratic in the slot fields alone, and $h$ enters the graph only through the vertex ${L^{pq}}_{ij}(w;h)$.

For each $A\in \{C_2,M_2,J_3\}$ there are unique coefficients ${P^{ab,\mu \nu }}_{A,ij}(y)$, indexed by $0\le i,j\le 9$ and multi-indices $a,b$, such that

for all smooth symmetric $X,Y$, read as an identity of polynomials in the independent jet coordinates of $X$ and $Y$. Moreover:

1. Quadratic homogeneity. Every monomial of the unpolarized ${\mathfrak{s}}_A(\gamma )$ has degree exactly $2$ in the jet of $\gamma$; hence ${\mathfrak{s}}_A(\lambda \gamma )={\lambda }^2{\mathfrak{s}}_A(\gamma )$ for $\lambda \in \mathbb{R}$.

2. Bilinearity and diagonal. ${\mathfrak{s}}_A[X,Y]$ is bilinear, and ${\mathfrak{s}}_A[X,X]={\mathfrak{s}}_A(X)$.

3. Slot symmetry. ${P^{ab,\mu \nu }}_{A,ij}={P^{ba,\mu \nu }}_{A,ji}$; also ${P^{ab,\mu \nu }}_{A,ij}={P^{ab,\nu \mu }}_{A,ij}$.

4. Distribution of derivatives (exact, not only a maximum).

   • $A=C_2$: $P_{C_2}$ is a linear form in $\partial \partial {\sigma }_{\epsilon }(y)$, and ${P^{ab}}_{C_2,ij}\ne 0$ only for $|a|=|b|=0$.

   • $A=M_2$: $P_{M_2}$ is a linear form in $\partial {\sigma }_{\epsilon }(y)$, and ${P^{ab}}_{M_2,ij}\ne 0$ only for $|a|+|b|=1$.

   • $A=J_3$: $P_{J_3}$ is a constant multiple of ${\sigma }_{\epsilon }(y)$, and ${P^{ab}}_{J_3,ij}\ne 0$ only for $|a|+|b|=2$.

   In every monomial of ${\mathcal{J}}_{\mathrm{D}}$ the number of derivatives on ${\sigma }_{\epsilon }$ plus the number on the two slots equals $2$.

5. Rational coefficients. Each ${P^{ab,\mu \nu }}_{A,ij}(y)$ is a linear combination, with rational constant coefficients, of the components of the ${\sigma }_{\epsilon }$-jet of the order stated in (4). There is no other $y$-dependence; in particular neither $h$ nor ${\chi }_{\mathrm{I}}$ occurs.

6. Agreement with XVI-K. ${\mathfrak{s}}_{C_2}[X,Y]$ is the polarized source ${\mathfrak{s}}^{\mu \nu }(X,Y;S)$, $S_{\alpha \beta }={\partial }_{\alpha }{\partial }_{\beta }{\sigma }_{\epsilon }$, of XVI-K, Definition def:a (XVI-J, eq:Q), identically.

Proof. Count, for each monomial of each printed atom, its degree in $\gamma$ and its number of derivatives (on $\gamma$ and on ${\sigma }_{\epsilon }$), as in the proof of XVI-K, Proposition prop:hess. $u_n$, $b_n$ have degree $n$ and no derivative. $C_0$ is constant; $C_1$ and $C_2$ of (eq:L-C) are sums of products of $b$'s, $u$'s and $\eta$ of total degree $1$ and $2$, with no derivative. ${\Gamma }_n$ has degree $n$ and exactly one derivative. In $R_n$, $\partial {\Gamma }_n$ has degree $n$ and two derivatives (Leibniz' rule distributes them over the factors), and ${\Gamma }_a{\Gamma }_b$ has degree $a+b=n$ and two derivatives; so $R_n$, $r_n$ have degree $n$ and two derivatives. Hence $V_1$ has degree $1$, $V_2$ degree $2$ (the term $\gamma r_1$ included), both with two derivatives, and by (eq:L-J3) every monomial of $J_3$ has degree $2$ and exactly two derivatives on $\gamma$. By (eq:L-M2), every monomial of $M_2$ has degree $2$ ($C_0{\Gamma }_2$: $0+2$; $C_1{\Gamma }_1$: $1+1$) and exactly one derivative on $\gamma$. $C_2$ has degree $2$ and none. Multiplying by ${\sigma }_{\epsilon }$, ${\partial }_{\rho }{\sigma }_{\epsilon }$, ${\partial }_{\alpha }{\partial }_{\beta }{\sigma }_{\epsilon }$ respectively gives (1) and, for the unpolarized source, the statements of (4) and (5): each summand is linear in the ${\sigma }_{\epsilon }$-jet of order $0$, $1$, $2$, with no derivative of ${\sigma }_{\epsilon }$ of another order, and with $2$, $1$, $0$ derivatives on $\gamma$. All printed numerical coefficients are rational ($\pm 1,\pm 2,\pm \frac{1}{2},\frac{1}{4},\frac{1}{8}$) and $\eta$ has entries $0,\pm 1$, which gives rationality. No atom contains $h$ or ${\chi }_{\mathrm{I}}$.

A polynomial $F$ homogeneous of degree $2$ in independent jet coordinates ${\gamma }_I$ can be written $F={\sum }_{I\le J}{c}_{IJ}{\gamma }_I{\gamma }_J$; its polarization $\frac{1}{2}(F(X+Y)-F(X)-F(Y))$ equals ${\sum }_{I\le J}\frac{1}{2}{c}_{IJ}(X_I{Y}_J+X_J{Y}_I)$. This is bilinear, symmetric under $X\leftrightarrow Y$, and returns $F(X)$ on $Y=X$; this proves (2) and (3) (the $(\mu ,\nu )$ symmetry is the symmetry of $J_3$, $C_2$, $M_2$ in $\mu \nu$, immediate from (eq:L-J3)–(eq:L-M2) because $R_1$, $R_2$ are symmetric; the latter is checked by the script). A jet coordinate ${\gamma }_I$ with $I=((ij),a)$ stands for ${\partial }^a{\gamma }_{ij}$; polarization replaces the pair ${\gamma }_I{\gamma }_J$ by $X_I{Y}_J+X_J{Y}_I$, which keeps the derivative orders $|a|$, $|b|$ of the two factors. So the slot orders $(|a|,|b|)$ of (eq:jet) are the derivative orders of the two $\gamma$-factors of the unpolarized monomials, and (4) follows. The expansion (eq:jet) exists by collecting terms, and is unique because distinct jet coordinates are independent variables. (6): XVI-K, Definition def:a, polarizes $Q^{\mu \nu }=2{C_2}^{\mu \nu \alpha \beta }{\partial }_{\alpha }{\partial }_{\beta }{\sigma }_{\epsilon }$ by the same rule, with $C_2$ of the same printed formula (eq:L-C); the two definitions coincide.

Proposition prop:jet(4) gives upper bounds and the exact grading. It does not by itself say that each graded piece is nonzero; that is the content of the computational verification below.

The script XVI-L-artifacts/source_full.py implements (eq:L-fc)–(eq:L-M2) atom by atom on independent jet coordinates (order $\le 2$) of $\gamma$, of the two slots $X,Y$, of $\sigma$, and of a scalar $\psi$, in exact rational arithmetic (sympy sparse polynomial ring over $\mathbb{Q}$); it exits with status $0$, and every check raises an error on failure. Its output XVI-L-artifacts/source_full_out.txt records:

Symbolic identities (valid for all smooth fields).

1. (S1)

   For all ten components, ${\mathcal{E}}_{\gamma }(\sigma {\ell }_3{)}^{\mu \nu }=\sigma {J_3}^{\mu \nu }+2{C_2}^{\mu \nu \alpha \beta }{\partial }_{\alpha }{\partial }_{\beta }\sigma -2{M_2}^{\mu \nu \rho }{\partial }_{\rho }\sigma$, as a polynomial identity in the independent $2$-jets of $\gamma$ and $\sigma$. Here ${\mathcal{E}}_{\gamma }$ is the Euler operator ${\sum }_a(-D{)}^a\partial /\partial ({\partial }^a{\gamma }_{\mu \nu })$ in the symmetric convention $\delta \int L=\int {\mathcal{E}}^{\mu \nu }\delta {\gamma }_{\mu \nu }$ (a factor $\frac{1}{2}$ for $\mu \ne \nu$), and ${\ell }_3$ is (eq:L-ell3). In particular ${\mathcal{E}}_{\gamma }({\ell }_3)=J_3$ of (eq:L-J3), with sign $+$ in this convention. The Euler computation never required a third-order jet.

2. (S2)

   At $\gamma =\psi \eta$: $C_1=C_2=0$, ${\Gamma }_2=-\psi {\Gamma }_1$, and both identities (eq:L-trace) hold.

3. (S3)

   ${\mathfrak{s}}_{C_2}[X,Y]$ equals XVI-K's routine xvi_common.polar_s$(X,Y,\partial \partial \sigma )$ identically in symbolic $X$, $Y$, $\partial \partial \sigma$; and on XVI-K's dual-slot tensors $X=-\mathcal{P}{B}^i$, $Y=-\mathcal{P}{B}^j$ (all $55$ pairs $i\le j$) it equals XVI-K's $\mathfrak{s}(-\mathcal{P}{B}^i,-\mathcal{P}{B}^j;S)$.

Coefficientwise structural checks.

1. (C1)

   Unpolarized: every monomial of ${\mathfrak{s}}_{C_2}$ (546 monomials), ${\mathfrak{s}}_{M_2}$ (1152), ${\mathfrak{s}}_{J_3}$ (1068) has $\gamma$-degree $2$, $\sigma$-jet order exactly $2$, $1$, $0$, and exactly $0$, $1$, $2$ derivatives on $\gamma$; no foreign generator occurs. Every monomial of ${\mathcal{J}}_{\mathrm{D}}$ has total derivative weight $2$. $J_3$ contains both monomial types $\gamma \partial \partial \gamma$ (534) and $\partial \gamma \partial \gamma$ (534), and no other.

2. (C2)

   The two polarization routes (the defining formula, and monomial polarization) agree for every summand and component, and ${\mathfrak{s}}_A[X,X]={\mathfrak{s}}_A(X)$.

3. (C3)

   Every monomial of ${\mathfrak{s}}_A[X,Y]$ is exactly linear in the $X$-jet and in the $Y$-jet; ${P^{ab}}_{A,ij}={P^{ba}}_{A,ji}$ entry by entry; $J_3$, $2C_2\partial \partial \sigma$, $-2M_2\partial \sigma$ and $R_1$, $R_2$ are symmetric in $(\mu ,\nu )$.

4. (C4)

   Nonzero ordered coefficients ${P^{ab,\mu \nu }}_{A,ij}$ ($\mu \le \nu$), by $(|a|,|b|)$: $C_2$: $(0,0)$: 688; $M_2$: $(1,0)$: 1152, $(0,1)$: 1152; $J_3$: $(2,0)$: 534, $(1,1)$: 996, $(0,2)$: 534. The ordered slot placements $(a,b)$ that occur are: $1$ for $C_2$; all $8$ with $|a|+|b|=1$ for $M_2$; all $36$ with $|a|+|b|=2$ for $J_3$. Each $P_A$ is a rational linear form in the $\sigma$-jet of the order of Proposition prop:jet(4) and nothing else. The full table (5056 entries) is written to XVI-L-artifacts/source_full_table.txt.

Finite exact test (not a coefficientwise equivalence).

1. (F1)

   The density ${\ell }_3$ of this script equals XVI-K's xvi_common.ell3_of on two random rational fields $\gamma (x^1)$ quadratic in $x^1$ (exact polynomial identity in $x^1$).

Proof. These are the printed lines of the output file; Appendix app:out reproduces them.

Sample entries, copied from the output (${\sigma }_{\alpha }={\partial }_{\alpha }{\sigma }_{\epsilon }$, ${\sigma }_{\alpha \beta }={\partial }_{\alpha }{\partial }_{\beta }{\sigma }_{\epsilon }$; slots $i,j$ are labelled by $(ab)$; multi-indices are written as $e_{\alpha }$):

The complete source ${\mathcal{J}}_{\mathrm{D}}$ of XII-C.tex, eq:actualsource, has the following budget, where “slot order” is $|a|+|b|$ in (eq:jet):

In particular, the maximum source-slot derivative order of the complete source is $2$, attained only in the ${\sigma }_{\epsilon }{J}_3$ summand; and the anticipated table of the XVI-L plan ($C_2$: $0$; $M_2$: $\le 1$; $J_3$: $\le 2$) holds, in the sharper form that each summand is homogeneous in the slot order.

Proof. Grading and preparation factors: Proposition prop:jet(4). Attainment of every listed $(|a|,|b|)$: Lemma lem:L1comp(C4).

1. Combined with the vertex bound $|p|+|q|\le 2$ of XVI-K, Proposition prop:hess, Theorem thm:budget suggests numerical derivative counts at most $2$, $3$, $4$ in the $C_2$, $M_2$, $J_3$ sectors, hence declared orders at most $2$, $3$, $4$. This is proved in L.2–L.3: graph decomposition and placement of source-slot derivatives on the edge kernels (Proposition prop:QL), and product/scaling statements (Proposition prop:Lsd, Theorem thm:L3).

2. The source-slot derivatives are derivatives in the source variable $y$. By XVI-K, eq:sign, ${{\partial }^a}_y[S(y-w)]=({\partial }^{a}S)(y-w)$ carries no sign, while vertex derivatives carry $(-1{)}^{|p|}$; this is the L.2 sign rule, re-derived for the complete source in Lemma lem:Lsign.

3. The source is a pointwise polynomial in jets. By (S1) it is the Euler derivative of the printed ${\sigma }_{\epsilon }{\ell }_3$. No replacement of the printed density by an integration-by-parts-equivalent representative is performed. XII-C records that replacing ${\ell }_3$ by ${\ell }_3+{\partial }_{\mu }{B}^{\mu }$ changes the weighted source by $-{\mathcal{E}}_{\gamma }(({\partial }_{\mu }{\sigma }_{\epsilon })B^{\mu })$; no such replacement is made.

4. Nothing in L.1 concerns the vertex, the edge kernels, the extension, counterterms, scale, or gravitational symmetry.

The graph has the four pieces of XVI-K, Section K.2: source point $y$; interaction point $w$; root $B_{f,g}(y,x)$, attached only in the target L.7; scalar test $\varphi (x)$. Both edges run from $y$ to $w$. Each edge carries $D_F(y,w)=-\mathcal{P}S(y-w)$, with $S$ of XVI-K, eq:S; the factor $-\mathcal{P}$ acts on tensor indices and $S$ is the scalar numerical part. The change relative to XVI-K is that the source end is the complete polarized source ${\mathfrak{s}}_A$ of Definition def:sA, $A\in \{C_2,M_2,J_3\}$, rather than the $C_2$ summand alone.

Let $z=y-w$ and let $u\in {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$. On $y\ne w$,

In particular, for the edge kernel $S$,

where

The source-end derivatives carry no sign; the vertex-end sign is $(-1{)}^{|p|+|q|}$, exactly as in XVI-K, eq:sign, for the $w$-derivatives alone.

Proof. XVI-K, Lemma lem:sign (imported): ${{\partial }_w}^p[u(y-w)]=(-1{)}^{|p|}({\partial }^{p}u)(y-w)$ and ${{\partial }_y}^a[u(y-w)]=({\partial }^{a}u)(y-w)$. These commute on $y\ne w$ (sequential continuity of pullback by the linear submersion $\Phi (y,w)=y-w$, Hörmander, Theorem 8.2.4), and ${{\partial }_y}^a{{\partial }_w}^p={{\partial }_w}^p{{\partial }_y}^a$ gives (eq:L-yw). The product identity follows as in XVI-K, Lemma lem:sign: Hörmander products on $y\ne w$ are sequentially continuous, so the two signs multiply.

For $A\in \{C_2,M_2,J_3\}$ and constant symmetric tensors $X,Y$, write

for the $(a,b)$-bihomogeneous piece of ${{\mathfrak{s}}_{\mathfrak{A}}}^{\mu \nu }[X,Y](y)$ (Proposition prop:jet). For multi-indices $a,b$ in the support of $P_A$ and vertex multi-indices with $|p|+|q|\le 2$, put

(The jet orders $a,b$ are spent on the edge kernels by Lemma lem:Lsign; the dual-slot tensors are constant, as in XVI-K, Definition def:a.) Set

An unordered numerical label is $\lambda =\{\alpha ,\beta \}$ with $\alpha =a+p$, $\beta =b+q$, and ${q^{\circ }}_{\lambda }:={q^{\circ }}_{\alpha \beta }$. The corresponding coefficient ${a^{A\mu \nu }}_{\lambda }$ is obtained by accumulating every ordered contribution with $\{a+p,b+q\}=\lambda$ (no universal combinatorial factor $2$ is inserted for $\alpha \ne \beta$; when both orderings contribute equally the factor $2$ appears by summation, as in XVI-K, eq:ak). Write $\lambda =(A;a,b;p,q)$ for an ordered placement when the sector and both ends are kept labelled.

The preparation jet of ${\sigma }_{\epsilon }$ (order $2,1,0$ for $C_2,M_2,J_3$) sits in $P_A(y)$, hence in $a_{\lambda }$. The residual metric jet of $h$ sits in ${L^{pq}}_{ij}(w;h)$, with $|r|=2-|p|-|q|$ (XVI-K, Proposition prop:hess(1)). The switch ${\chi }_{\mathrm{I}}(w)$ is undifferentiated (XVI-K, Proposition prop:hess(5)). No silent integration by parts is performed: the printed representatives of ${\ell }_3$ and of ${\mathcal{J}}_{\mathrm{D}}$ are kept.

The combinatorial graph factor of the specified two-edge graph, with the complete source ${\mathcal{J}}_{\mathrm{D}}$ at $y$ and the full cubic vertex at $w$, is

It carries no power of $\kappa$ or $\hslash$. Coupling and loop weights are not determined in this note and are kept separate from (eq:L-factor).

Proof. Re-derive from the same first principles as XVI-I, without transferring XVI-K's factor as a black box.

1. Two identical graviton edges. The sector joins the two tensor slots of $F_t$ to one compactly switched copy of ${\ell }_3$ by two graviton edges, each carrying $D_F$ (XVI-I, Proposition prop:sector). Fredenhagen–Rejzner, arXiv:1208.1428v2, equation (82), place $l_{ij}!$ in the denominator for $l_{ij}$ lines between the same pair of vertices; here $l_{ij}=2$, so the coefficient is $1/2!$, declared once (XVI-I, Proposition prop:signs(6)).

2. {Second variation of $F_t$.} ${D_h}^2{F}_t$ contributes the factor $2$ (XVI-I, eq:Dh): the count of the two identical tensor slots, not a hand-inserted normalization.

3. Independence of the source completion. Replacing the selected source $2C_2\partial \partial {\sigma }_{\epsilon }$ by the complete ${\mathcal{J}}_{\mathrm{D}}={\mathfrak{s}}_{C_2}+{\mathfrak{s}}_{M_2}+{\mathfrak{s}}_{J_3}$ changes only the source-end contraction at $y$. It does not change the number of edges, the identity of the two edges, or the second tensor derivative of $F_t$. Hence neither factor in (eq:L-factor) is altered.

4. The product is $1$, as in XVI-I, Proposition prop:sector, and as used for the $C_2$ sector in XVI-K, Proposition prop:QK.

No coupling constant and no loop weight enters this count; if such weights are fixed later they multiply the whole amplitude and do not redefine (eq:L-factor).

On $y\ne w$,