Landmark IX-B Derivative-stable triangle calibration and counterterm constraints for sharp Cauchy insertions

We calibrate a selected sharp-insertion triangle relative to the actual-functional carrier ${\mathcal{P}}_{\Sigma }$ of IX-A. We classify the eight connected contraction skeletons through three edges and identify the unique fully contracted graph for two sharp insertions and one quadratic metric observable. Using the quantified VII-D finite-part theorem, we prove stability under fixed observation differential operators and smooth spectator multipliers, with test-order loss $64\to 64+d$. We fix the sign and multiplicity of the two-factor output already admitted by IX-A, retain the observation-diagonal obstruction, and prove that the observation-derivative orbit of the residue is linearly independent. The infinite distribution orbit is compatible with the single parent conversion parameter of VII-D. Both regularized parenthesizations have the same calibrated numerical limit, but the normal-topology obstruction of VII-C remains unchanged. The manuscript does not factor the numerical finite part ${u_{\mu }}^{\rho }$ through sharp binary multiplications.

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

Figure fig:scope-IX-B records that separation. The drawing adds no hypothesis.

inherited boundaries{

1. IX-A supplies the carrier ${\mathcal{P}}_{\Sigma }$ and topology ${\tau }_{\Sigma }$. It admits $B_f$ and the selected two-factor contraction with $W_g$; it does not construct or assign the triple after either sharp association.

2. VII-C supplies the logarithmic divergence, the separated-support witness, and the normal-topology obstruction under its extension requirements (E1)–(E4).

3. VII-D supplies the strong numerical finite part, the parent conversion class $\mathbb{R}r$, and the rate controlled by $p_{K,64}$. This is a numerical prescription, not a quantum algebra construction.

4. None of these sources supplies the second sharp multiplication. No such multiplication is defined here.

}

and scope. The new arguments are the skeleton classification, the transfer of the numerical estimate to observation derivatives and smooth multipliers, the residue-orbit and weighted-class results, and the explicit normalization of the admitted two-factor coefficient. An unlabelled inventory is not an existence theorem for decorated distributional graphs. Likewise, the complete conversion theorem for the fixed parent triangle does not classify ambiguities of later decorations.

Let

and write $x=(t,\mathbf{x})$, $y=(s,\mathbf{y})$, $x^{\prime }=(t^{\prime },{\mathbf{x}}^{\prime })$. Put

The inverse ${\mathsf{E}}_0$ has zero value and zero first time derivative on ${\Sigma }_0=\{t=0\}$, and satisfies $\square {\mathsf{E}}_{0}j=j$. All configurations $(h,\varphi )$ are smooth and off shell. The insertion and middle observable are

where $f,g\in {C_c}^{\infty }(M_{+})$.

We retain the VII-C Wick-symbol convention

The sum is finite for finite polynomials; self-contractions inside a Wick symbol are excluded.

The fixed even blocks are

where

and, for fixed $m\ge 0$,

The following identities are inherited, not re-derived:

They are VII-C's tensor cancellation and IX-A's fixed tensor lemma VIIC (prop:tensor) IXA (lem:tensor). Both sources retain the mixed reflected branch:

In particular, $H_{ww}=0$ does not imply $K=0$.

As in VII-D, the same fixed radial mollifier is used on both slots of each insertion, and the middle observable is unsmeared VIID (sc:placement, prop:positive). Specifically, $\rho \in {C_c}^{\infty }({\mathbb{R}}^4)$ is real, nonnegative, even, Euclidean radial, and normalized by $\int \rho =1$. Set

With the Fourier convention $\hat{v}(k)=\int e^{-ik\cdot z}v(z)\mathrm{d}z$, define

Radiality makes the profile independent of $\mathbf{n}$. Let $K_{\epsilon }$ be (eq:K) with the multiplier ${\vartheta }_{\rho }(\epsilon |p|)$, and set

VII-D identifies the exact numerical family as

It is smooth for every $\epsilon >0$ VIID (prop:positive, eq:Texact). On real product tests, its pairing is the single VII-C coefficient

The joint-test distribution is complex-linear; conjugation in (eq:T) conjugates the kernel, not the test.

Following the VII-D residue definition, let

and

The inherited positive order-zero distribution is

These are VII-D's prescribed residue and support VIID (def:residue, eq:rdef).

For the light-ray transform

the inherited product-test identity reads

Thus this pairing is exactly the VII-C light-ray coefficient, including every numerical factor VIIC (eq:residue) VIID (eq:rjoint, prop:ledger).

. The symbol $\mu >0$ denotes the subtraction scale and ${\vartheta }_{\rho }$ the mollifier profile, as in VII-D; VII-C used $\mu$ for the profile. Here $r$ always denotes the residue distribution, and $\lambda$ denotes a radial variable when needed. The reflected relation $\mathrm{Rel}$ is distinct from the observation equality diagonals ${\Delta }_{ij}$.

For compact $K\subset X$, let

Strong convergence in ${\mathcal{D}}^{\prime }(X)$ means uniform convergence on bounded subsets of $\mathcal{D}(X)$.

We use the following statements of VII-D as analytical input VIID (thm:finitepart, thm:conversion). There is a distribution $U_m$, independent of $\rho$, such that

For every compact $K\subset X$ and every $\Phi \in {\mathcal{D}}_K(X)$,

The estimate includes compacts meeting $\mathrm{Rel}\cap {\Delta }_{13}$. For two admissible regulators,

This is the complete parent conversion: there is no extra normal derivative, nonconstant weight, or separate term on $\mathrm{Rel}\cap {\Delta }_{13}$. The inherited identities also give

The assumption restates a theorem of the supplied source rather than introducing a binary-product hypothesis. No radial expansion, profile-integral evaluation, or wavefront estimate is re-proved here.

ledger

The source labels below refer to the delivered manuscript versions. The last column gives the local point of use. Inherited existence, topology, and ultraviolet statements are not promoted to new product theorems.

{}1.15

{c

enter}

New versus inherited. IX-B proves the combinatorics, the continuous-operation transfer, the independent derivative orbit, and the weighted-class warning. It fixes the two-factor sign and multiplicity. It does not re-prove the carrier construction, the parent finite part, or the normal-topology obstruction.

For self-contained use of the carrier name, we recall its admission criterion, not its proof. Let ${\mathcal{P}}_{\mathrm{dist}}$ denote actual finite polynomial functionals on smooth configurations, with compact symmetric distributional coefficient kernels. Let ${\mathcal{M}}_1$ be the space of compact linear metric kernels with no nonzero nonspacelike wavefront covectors, and put

Here ${G_{\mathrm{A}}}^0$ is the massless advanced inverse. IX-A defines

The empty direction list is allowed IXA (def:Padm).

The topology ${\tau }_{\Sigma }$ is the initial locally convex topology for the strong compact-distribution coefficient inclusion and all boundary-jet maps

IX-A proves that this trace is intrinsic, that ${\tau }_{\Sigma }$ is Hausdorff, and that fixed-direction differentiation is continuous IXA (lem:charge, def:topology, prop:top-basic). Its multiplication theorem concerns separately continuous classical pointwise multiplication only.

IX-A admits $B_f$, $W_g$, and the selected two-factor coefficients. This does not supply a quantum multiplication on all of ${\mathcal{P}}_{\Sigma }$, nor an assignment of either sharp triple association. A future compact polynomial output must satisfy (eq:admission); a numerical finite part alone is not a substitute for that test.

A contraction graph has one vertex per insertion and one edge per cross-contraction. Multiple edges are allowed, but self-loops are excluded by the Wick-symbol convention. Decorations specify field slots, covariance labels, and derivatives; no slot may be used twice.

Every nontrivial connected loopless contraction multigraph with at most three edges has one of the following eight skeletons:

1. one edge;

2. two parallel edges;

3. a two-edge path;

4. three parallel edges;

5. a two-edge path with one edge doubled;

6. a triangle;

7. a three-edge path;

8. a three-edge star.

Every graph with at most three edges is a disjoint union of these, with total edge count at most three, together with isolated vertices.

Proof. With one edge there is one connected graph. With two edges there are either two vertices, giving parallel edges, or three vertices, giving a path.

A connected three-edge graph has at most four vertices. On two vertices it has three parallel edges. On three vertices it either joins all three pairs, giving a triangle, or joins two pairs with multiplicities two and one. On four vertices it is a tree, hence either a path or a star.

The final statement follows by decomposition into connected components.

The fully contracted order-${\hslash }^3$ graph for

has one scalar edge between the first and third vertices and one metric edge from each sharp vertex to the middle vertex. The two assignments to the middle slots give the factor $2$ in (eq:T). No $ww$-edge occurs.

Proof. Each sharp vertex has one scalar and one metric slot, and the middle vertex has two metric slots. Since mixed scalar–metric covariances vanish, the scalar slots must contract with each other. The remaining sharp metric slots must contract with the two middle metric slots, in two possible assignments.

An edge joining the two sharp metric slots would leave the middle slots without partners, and its covariance is also zero by (eq:tensor). The inherited tensor identity selects vector index $0$ at both sharp vertices. Thus the scalar edge gives $D_{\epsilon }$, and the mixed edges give the two $K_{\epsilon }$ factors. The minus sign in the first $B_f$ and the minus sign in the third $B_{f^{\prime }}$ multiply to $+1$, as on the VII-D convention sheet VIID (sc:spectral, prop:positive).

There are three nonzero one-edge placements in this word: the scalar first–third edge and the two sharp–middle metric edges. Choosing two of them gives three two-edge placements. These are placement statements, not sharp output-composition theorems.

Each observation equality diagonal ${\Delta }_{ij}\subset X$ has dimension eight and codimension four; the total diagonal has dimension four. The inherited embedding $\Psi$ parametrizes an eight-dimensional relation $\mathrm{Rel}$. On it,

Consequently $\mathrm{Rel}\cap {\Delta }_{13}=\Psi \{t=t^{\prime }\}$ has dimension seven and codimension five in $X$. Moreover,

These are the VII-D observation relations VIID (eq:chartrelations; Section “Observation coverage'). The initial Cauchy surface is an auxiliary surface, not an observation face inside $X$.

local density. For the local transverse-degree calculation below, recall the VII-D chart

Its inherited density and residue representation are

There is no $\delta (t-t^{\prime })$ VIID (eq:density, eq:rlocal). We do not re-prove this coordinate formula. Positivity and the separated witness use the parameter measure in (eq:r), not this chart.

There are nonnegative $f,g,f^{\prime }\in {C_c}^{\infty }(M_{+})$ with pairwise disjoint supports such that

No family of counterterms supported on ${\Delta }_{12}\cup {\Delta }_{23}\cup {\Delta }_{13}$ removes the logarithmic divergence on all joint tests. The same limitation holds for observation boundary faces avoided by these compact supports.

Proof. Use the VII-C separated-support witness VIIC (lem:separated). Its centers are

with positive pairwise distinct $S,T,T^{\prime }$. The source supplies nonnegative bumps with pairwise disjoint supports and $\mathcal{C}(f,g,f^{\prime })>0$. Equation (eq:C) identifies this number with the residue pairing.

For any fixed joint test, Assumption ass:VIID gives

For the selected product test, the leading coefficient is therefore exactly $\mathcal{C}(f,g,f^{\prime })$. For fixed $\mu$, the contribution $-(\log \mu )\mathcal{C}$ is bounded, so this agrees with the VII-C form $\mathcal{C}\log ({\epsilon }^{-1})+O(1)$.

The joint support avoids all three diagonals. Any distribution supported on their union pairs to zero with this test, even if its coefficients depend on $\epsilon$. Such a subtraction cannot remove the nonzero logarithm. This is the observation-diagonal limitation already recorded in VII-C's counterterm discussion.

The theorem does not exclude counterterms supported on $\mathrm{Rel}$, nor every possible sharp quantum algebra.

Let $D={\sum }_{|\alpha |\le d}{b}_{\alpha }{\partial }^{\alpha }$ be a fixed differential operator on $X$ with smooth coefficients, and let $a\in C^{\infty }(X)$. Its distributional action is defined by the formal transpose relative to the coordinate density:

No complex conjugation occurs in this transpose. The ordered operation is

Assume ass:VIID. For every fixed $a,D$, with $d=\mathrm{ord}D$,

More precisely, for $\Phi \in {\mathcal{D}}_K(X)$,

The estimate is uniform when $a$ and the coefficients $b_{\alpha }$ range over sets bounded in the compact smooth seminorms through order $64+d$. It includes compacts meeting $\mathrm{Rel}\cap {\Delta }_{13}$.

Proof. Write $R_{\epsilon }={T_{\epsilon }}^{\rho }-\log ((\mu \epsilon {)}^{-1})r-{u_{\mu }}^{\rho }$. Then

Differential operators and multiplication preserve the test support. Leibniz's rule gives

All coefficient bounds used here involve at most $64+d$ derivatives. Apply (eq:input).

Every bounded subset of $\mathcal{D}(X)$ has a common compact support and bounded smooth seminorms. Thus the estimate proves strong convergence and its asserted uniformity. The parent input already covers the reflected-diagonal intersection; the same Leibniz transfer covers it at order $64+d$, without a separate intersection argument.

For the same operation,

The complete regulator difference satisfies

strongly, with the corresponding order-$64+d$ remainder bound.

Proof. Apply $aD$ to the inherited identities (eq:parentconversion)–(eq:parentscale). For the last assertion, subtract the two estimates (eq:descendantrate); the logarithmic terms agree. The parent cocycle is inherited from VII-D, not re-derived from profile integrals here.

For product tests,

This is the observation-test descendant calibration for $B_{{{\partial }_t}^{n}f}$.

Directional derivatives are evaluations of functional derivatives on directions, but the three extension claims above do not follow from one another. In particular, IX-A's directional closure does not place the observation residues $r_n$ in a finite-dimensional distribution space. Conversely, the descendant finite part does not establish directional differentiation of an unconstructed sharp triple output.

Suppose a selected graph coefficient has the form $a[h,\varphi ]D{T}_{\epsilon }^{\rho }$ after fixing smooth configurations, where $a[h,\varphi ]$ is smooth on $X$. If these multipliers depend continuously on the configurations in the compact smooth seminorms required by Theorem thm:descendants, its numerical finite part is

Convergence is uniform on configuration sets for which those multiplier seminorms are bounded.

Proof. Use Theorem thm:descendants with $a=a[h,\varphi ]$ and its uniform coefficient bounds.

This applies to fixed finite products of smooth field jets and to smooth Cauchy responses when the required bounds are supplied. It neither proves carrier membership of every resulting polynomial nor permits multiplication by an additional singular kernel. Disconnected factors sharing singular variables are not justified by calling them spectators.

The following distinctions prevent a confusion between the complete VII-D parent conversion and the new derivative-orbit theorem.

The distributions

are linearly independent over $\mathbb{C}$. No finite-dimensional subspace of ${\mathcal{D}}^{\prime }(X)$ containing $r$ is invariant under $-{\partial }_{x^0}$.

Proof. Let

Then $F=0$ on $\mathrm{Rel}$, so $Fr=0$, whereas

is nowhere zero on $\mathrm{Rel}$.

For an integer $N\ge 1$ and $j<N$, every term in ${{\partial }_{x^0}}^j(F^N\Phi )$ retains a factor $F$. Consequently

At derivative order $N$, restriction to $\mathrm{Rel}$ gives

Because the transpose of $(-{\partial }_{x^0}{)}^N$ is ${{\partial }_{x^0}}^N$, this proves

The right-hand side is nonzero. Indeed, $r$ is the positive parameter measure (eq:r), and the multiplier has constant nonzero sign; a nonnegative test positive near any point of $\mathrm{Rel}$ detects it.

If ${{\sum }_{j=0}}^N{c}_j{r}_j=0$ with $c_N\ne 0$ and $N\ge 1$, multiplication by $F^N$ leaves precisely $c_{N}N!({\partial }_{x^0}F{)}^{N}r=0$, a contradiction. For $N=0$, nonvanishing of $r$ gives the conclusion directly. Thus every finite relation is trivial. An invariant finite-dimensional space containing $r$ would contain all $r_n$, which is impossible.

parent parameter, not one parameter per derivative{ Theorem thm:independence concerns the dimension of a distribution orbit, not the number of independent normalization parameters. VII-D fixes the parent conversion to $c_{ab}r$ (thm:conversion). If differentiation is imposed as a compatibility identity, the parent change $u\mapsto u+cr$ induces

with the same parent coefficient $c$. The compatible freedom can therefore be one-dimensional even though the derivative orbit is infinite-dimensional. No algebra obstruction or new independent constants are inferred.}

The real vector space

is infinite-dimensional, even after imposing first–third exchange symmetry.

Proof. The exchange-symmetric weights $a_n=(x^0+x^{\prime 0}{)}^n$, $n\ge 0$, give linearly independent distributions. If $P(x^0+x^{\prime 0})r=0$ for a polynomial $P$, the positive pushforward density implies $P(t+t^{\prime })=0$ throughout the parameter space: a nonzero value would persist on an open set and be detected by a compact test. Since $t+t^{\prime }$ ranges over $(0,\infty )$, $P$ is identically zero.

This warning does not declare these weights admissible counterterms. Support, reality, and exchange symmetry alone do not yield a finite-dimensional ambiguity class; additional restrictions would have to be specified.

The recalled chart (eq:density) represents the residue as a smooth tangential coefficient times a delta distribution in four normal variables $v=(\sigma ,\mathbf{w})$. Its transverse scaling degree is four. A nonzero normal derivative of order $k$ with smooth tangential coefficient has transverse scaling degree $4+k$, since

and each normal derivative adds one inverse power of $\lambda$.

This degree calculation is not a complete ambiguity classification for ${u_{\mu }}^{\rho }$ or later graphs. A normal-degree bound alone does not restrict arbitrary tangential weights.

For every $\epsilon >0$, the fully contracted order-${\hslash }^3$ coefficients of

agree and equal the single VII-C coefficient

Their common logarithmic subtraction has the numerical limit $⟨{u_{\mu }}^{\rho },f\otimes g\otimes f^{\prime }⟩$. On arbitrary joint tests the corresponding distributional convergence is strong.

Proof. Introduce independent field variables at the three vertices. The cross-contraction operators ${\mathcal{P}}_{ij}$ have field-independent coefficients and commute on the positive-cutoff old-domain expressions. Both parenthesizations therefore expand as

followed by identification of the field variables. The middle Wick symbol is an old bulk polynomial; the two insertion kernels are regular at positive cutoff, so the relevant old products are defined.

Proposition prop:unique identifies the fully contracted graph and its multiplicity. Equations (eq:T)–(eq:G) give the common coefficient, and Assumption ass:VIID gives the limit. Association does not split this one graph divergence into two different logarithms.

VII-C's counterterm discussion distinguishes a numerical subtraction from an extension preserving the old Wick product. Subtracting from the positive-cutoff coefficients changes products already fixed in ${\mathcal{P}}_{\mathrm{mc}}$. Assigning a subtracted value only at the sharp kernel does not restore sequential continuity of the unsubtracted coefficient map VIIC (thm:coefficient; Section “Counterterms and the scope of the conclusion'). VII-D explicitly retains this obstruction VIID (Section “Scope audit'). Accordingly, the finite part here is not a counterterm inserted inside the unchanged old polynomial product.

The selected one-contraction coefficient for the ordered pair $(B_f,W_g)$ is

It is an actual continuous bilinear functional with a compact distributional kernel and belongs to ${\mathcal{P}}_{\Sigma }$. Its admission is supplied by IX-A; (eq:Qpartial) fixes the previously unspecified combinatorial coefficient and sign.

Proof. IX-A's selected contraction-output proposition, item 3, establishes existence and admission of this two-factor coefficient IXA (prop:knownoutputs, item 3). It remains to identify the coefficient in that admitted expression.

The metric derivative of $W_g$ supplies the factor $2$ from its two metric slots. The inherited identity (eq:tensor) selects the $0$-component of $Lh$ when contracted with $h_{00}$, giving $K(x,y){\partial }_t\varphi (x)$. There is exactly one insertion $B_f$ in this partial coefficient, so its single defining minus sign gives the sign $-2$. This proves (eq:Qpartial) without re-proving admission.

The formula is the admitted two-factor output, not an element produced by a newly defined quantum product on ${\mathcal{P}}_{\Sigma }$. In the full triangle there are instead two insertion minus signs, one from $B_f$ and one from $B_{f^{\prime }}$, and their product is $+1$. Thus the partial sign $-2$ and the positive assignment factor in VII-D are consistent.

The admitted output ${\mathcal{Q}}_{f,g}\in {\mathcal{P}}_{\Sigma }$, the equality of regularized associations, and the parent finite-part assignment do not determine either sharp composition

Here $\mathsf{B}(f)$ denotes the actual admitted $B_f$, while ${\star }_{\Sigma }$ is only a placeholder for a future product. Such a construction still requires:

1. binary quantum maps on specified stages of the IX-A carrier, including multiplication of partial outputs by sharp insertions;

2. verification of the IX-A admission test for all resulting polynomial coefficients and directional descendants;

3. graph-specific subtraction and normalization conditions compatible with the fixed old products;

4. proof that those maps factor the calibration and satisfy the required coherence identities.

Proof. IX-A supplies the carrier and selected partial outputs, not their second quantum multiplication. VII-D assigns a numerical distribution to a particular fully contracted three-vertex graph. Its arguments are joint observation tests, not arbitrary intermediate elements of ${\mathcal{P}}_{\Sigma }$. Neither statement defines a binary map acting on ${\mathcal{Q}}_{f,g}$ and $B_{f^{\prime }}$.

A common subtracted limit of two existing regularized compositions specifies a ternary numerical coefficient. It does not factor that coefficient through the missing binary maps. Moreover, the constant fully contracted component is only one coefficient of a full product; it does not verify (eq:admission) for its nonconstant components. Therefore neither displayed sharp composition is determined.

The carrier is no longer unspecified. The unresolved step is quantum multiplication on that carrier, not the naming of a space for the selected partial coefficient.

Let $\tau (x,y,x^{\prime })=(x^{\prime },y,x)$, and define

VII-D proves the parent identities

We use these as inherited symmetry statements VIID (Section “Adjoint symmetry').

For a differential–multiplier operation $\mathcal{O}$, define its exchanged-conjugate operation by

Then, using (eq:adjoint),

This is the descendant exchange rule. A first-observation derivative is exchanged with the corresponding third-observation derivative, with conjugation of smooth coefficients; it is not generally an individually self-adjoint operation.

The first row is VII-C's coefficient obstruction. Its algebra consequence assumes all of (E1)–(E4): retention of the old bulk Wick algebra, exact sharp admission, normal-topology sequential admission continuity agreeing on regular kernels, and bounded-set hypocontinuous multiplication with continuous field-independent coefficient extraction VIIC (ass:extension, cor:noalgebra). No relaxation of those requirements is silently introduced here.

The second row is IX-A's explicit nonconvergence and compatibility boundary IXA (thm:IX1, cor:compatibility). It is consistent with the different-topology possibility that VII-C explicitly does not exclude. No claim is made that ${\tau }_{\Sigma }$ is globally finer than every normal topology on every polynomial stage, or that it supports a hypocontinuous quantum product.

The third row is Assumption ass:VIID and Theorem thm:descendants. As VII-D's scope audit states, this numerical subtraction leaves the VII-C obstruction unchanged. A limit in the strong topology of joint-test distributions is not admission continuity for a sharp binary algebra.

Retain the exact off-shell insertion, fixed covariance, prescribed regulator placement, the IX-A carrier, and Assumption ass:VIID. Then:

1. the eight unlabelled connected skeletons through three edges are exactly those in Theorem thm:skeletons;

2. the selected fully contracted word has the unique triangle coefficient (eq:T);

3. its numerical finite part and complete conversion transfer to every fixed observation differential operator and smooth spectator multiplier, with order loss $64\to 64+d$;

4. the inherited diagonal limitation is detected by the exact coefficient $⟨r,f\otimes g\otimes f^{\prime }⟩=\mathcal{C}(f,g,f^{\prime })>0$;

5. the residue derivatives $r_n$ are linearly independent, while compatible conversions can retain the single parent parameter;

6. the admitted two-factor output in ${\mathcal{P}}_{\Sigma }$ is exactly (eq:Qpartial);

7. both regularized associations have the same coefficient ${\mathcal{G}}_{\epsilon }$ and the same subtracted numerical limit.

These results neither remove the VII-C obstruction nor construct either sharp second multiplication.

Proof. The assertions follow respectively from Theorem thm:skeletons, Proposition prop:unique, Theorem thm:descendants and Corollary cor:conversion, Theorem thm:diagonal, Theorem thm:independence and the inherited parent conversion, Proposition prop:partial, and Theorem thm:regularassociation. Proposition prop:missing and Section sec:continuity establish the stated logical boundaries.

{}1.12

The carrier and its admission criterion are already specified by IX-A. A successor must work relative to that carrier and fix:

1. allowed vertex valences and independently differentiated decorations;

2. bounded derivative stages or a precisely declared derivative module;

3. the regulator placement for each new decorated graph;

4. graph-specific extension conditions and restrictions on tangential weights, normal derivatives, and intersection terms;

5. quantum maps on partial outputs and proof of output admission;

6. compatibility with old products, the parent calibration, involution, presentation relations, and associativity.

A binary factorization reproducing ${u_{\mu }}^{\rho }+cr$ is a further construction, not a consequence of the calibration alone.

conclusion{ IX-B supplies a source-locked derivative- and smooth-spectator-stable numerical triangle calibration. The carrier is ${\mathcal{P}}_{\Sigma }$, the admitted partial coefficient is fixed by (eq:Qpartial), and the second sharp multiplication remains undefined.

The VII-D class $\mathbb{R}r$ is the complete parent regulator-conversion class, not the distribution space of all descendants or all later decorations. The VII-C normal-topology obstruction remains in force. No ternary numerical limit is advertised as either sharp binary association.}

The analytic existence and rate of the parent finite part are the VII-D input, stated in Assumption ass:VIID. Its ultraviolet analysis, normalization ledger, profile integrals, and intersection estimates are not re-proved here. The carrier, intrinsic boundary traces, and nonconvergence theorem are IX-A inputs. The normal-topology obstruction and separated witness are VII-C inputs.

The new analytical transfer uses only the exact identity

and its finite-order Leibniz estimate. The residue-orbit proof uses the defining function $F$ and the positive parameter measure $c_0{\Psi }_{\ast }\nu$; it does not rely on a new wavefront or radial analysis.

No all-graph ultraviolet estimate, quantum hypocontinuity theorem in ${\tau }_{\Sigma }$, or complete coherence theorem is claimed. No second-order geometric insertion, retarded product, switched realization, or connection to a conical construction is introduced.

For the calibrated parent,

The field degree before full contraction is six. Observation differentiation preserves insertion count and contraction order. Smooth spectator attachment changes residual field degree but does not by itself add a contraction. A bound on $\hslash$-order does not bound derivative decorations, isolated spectator vertices, or word length.

of the result. This is a calibration theorem conditional on the explicitly restated VII-D analytical input, with new combinatorial, continuous-operation, and residue-space consequences. The IX-A and VII-C boundaries are inherited and identified at their points of use.

status. The source is designed for two-pass pdfLaTeX compilation and uses no external bibliography, figures, or code. Compilation has not been executed in this interface; no proof-assistant verification or independent review is claimed.

VIIC

Landmark VII-C: A mixed-Wick obstruction to a normal-topology boundary extension for sharp Cauchy harmonic insertions. Delivered manuscript, VII-C.tex. Relevant source labels: prop:tensor, eq:K, eq:Gdef, eq:residue, thm:log, lem:separated, lem:normalconv, thm:coefficient, ass:extension, and cor:noalgebra; also the subsection “Counterterms and the scope of the conclusion”.

VIID

Finite part of the regulated Cauchy-reflection triangle: A strong numerical distributional limit for the VII-C family. Delivered manuscript, VII-D.tex. Relevant source labels: sc:placement, sc:spectral, prop:positive, eq:Texact, def:residue, eq:rdef, eq:rjoint, prop:ledger, eq:chartrelations, eq:density, eq:rlocal, thm:finitepart, and thm:conversion; also “Scope audit”, “Adjoint symmetry”, and “Reference profile, dilation, and cocycle”.

IXA

N. Cheung. Landmark IX-A: An off-shell admission domain and boundary-sensitive topology for sharp Cauchy relational insertions. Delivered manuscript, IX-A.tex. Relevant source labels: def:Padm, lem:charge, def:topology, thm:closure, prop:top-basic, thm:IX1, cor:compatibility, lem:tensor, and prop:knownoutputs, item 3; also “Preservation is recorded, not extended by fiat” and “Output reservation and the admission criterion”.