Landmark XIII-B One binary causal datum for the displayed coefficient, on the unchanged carrier Tree construction, normalization audit, and the unresolved binary quantum extension

XIII-B uses evaluation-only claims in the inherited carrier ${\mathcal{E}}_H$. It does not use a reduced algebra or a completed observable. Those readings remain Unresolved in XIII-A. The construction below uses the displayed off-shell lifts and then takes their admitted full-shell classes. Classical evaluation on ${\mathcal{C}}_W$ follows admission; it is not the domain of the product.

At fixed regulator width, we study the selected pair consisting of the switched metric–scalar interaction functional and the already admitted second-order composite. We prove admission of the vertex and construct the classical binary retarded coefficient. A balanced kernel calculation proves the joint one-line composition, properness, compact support, and microcausal admission. Its scalar branch is the inherited switched scalar response; its metric branch has the inherited inverse coefficient and agrees with the first-completion root normalization wherever that branch is used.

Two corrections are indispensable. Causal factorization is a normalization of the binary time-ordered datum, not the assertion that the retarded coefficient equals the ordinary quantum product. The latter assertion is incompatible with the stated retarded definition and field-degree normalization. Moreover, retarded support concerns the contracted attachment variables, not every uncontracted variable of a nonlocal composite.

The coefficient linear in $\hslash$ requires a joint two-line extension. We freeze a conservative total-diagonal counterterm class and identify the additional internal collision problem, but do not claim an extension that has not been constructed. The corrected quantum release therefore remains Unresolved at that gate. We give precise conditional admission and factorization statements, explicit classical zero-evaluation subclasses, and the free-differential display. No restriction quotient or gauge-consistency defect is formed.

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

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

and proof boundary{ Constructed: the vertex functional, the selected tree-level binary causal composition, its properness and compact microcausal kernel, its inherited-root overlap, and the free-differential display.

Proved obstructed: the literal normalization $R_1({\mathcal{V}}_g;\mathcal{O})={\mathcal{V}}_g{\star }_H\mathcal{O}$ on ordered supports, when combined with the specified retarded definition and contraction grading.

Unresolved through the requested quantum order: the joint two-line extension in the frozen counterterm class. Neither a failed wavefront criterion nor an unconstructed extension is called a no-go.

The carrier remains ${\mathcal{E}}_H$. No quotient by ${\mathcal{N}}_W$, completed observable, general time-ordered family, $R_2$, gauge-consistency defect, interacting differential, state, loop value, master equation, or $\epsilon \to 0$ limit is released.}

The construction has three different levels:

The second arrow does not turn the final evaluation set into a quantum quotient. It also does not automatically make a retarded operation independent of all choices of representative.

Fix a nonempty open $W$ with

The classical subclass is

Under the uniform support hypothesis

the prescribed lift satisfies ${\Delta }_{\xi }\mathcal{O}{|}_{{\mathcal{C}}_W}=0$ for every $\xi \in {\mathfrak{g}}_W$.

The coefficientwise vanishing space ${\mathcal{N}}_W$ is not a two-sided ideal of the inherited product. Compact regular smearings give

An admissible rank-one ordering change also fails to preserve coefficientwise vanishing on ${\mathcal{C}}_W$. No restricted quantum domain is handed off XIIIA.

This is the release box inherited from XIII-A: classical cancellation is Constructed; the naive coefficientwise quantum quotient is Proved obstructed; a separately constructed reduction or completion is Unresolved. We neither recompute its commutator nor modify its constraint ideal.

The endings here are attached to the selected binary problem. Constructed means that the required joint extension, normalization, and admission have actually been proved. Proved obstructed means that every candidate in a previously declared class fails a specified condition. Unresolved means that a necessary extension or compatibility theorem is missing. The last ending is not concealed by writing a formal contraction formula.

XI-B is used only for the inverse normalization and free symbol rule. X-G, VII–IX, and Landmark I are not inputs. Standard wave and distribution calculus are identified separately from manuscript handoffs.

Use

The fixed smooth ramp ${\sigma }_{\epsilon }$ is zero for $t\le 0$ and one for $t\ge \epsilon$, with $0<\epsilon <T$. The switch $g\in {C_c}^{\infty }({M_T}^{\circ };[0,1])$ is external. All configurations are ambient smooth multiplets; spatial decay is not required.

The operators are

Parentheses in indexed expressions include $\frac{1}{2}$. The complete free shell is

The auxiliary field remains independent off shell.

The named composite is

with

It is an even, ghost-number-zero homogeneous cubic polynomial: its sectors have degrees $(2,1)$ and $(0,3)$ in $(\gamma ,\varphi )$. Its compact polarized kernel has the covector-sum-zero bound $Z_3$ proved in XII-D and XII-E XIID,XIIE. Its metric derivatives include the admitted insertion's metric slots. Nonlocality is not removed by suppressing those derivatives.

The inherited carrier is

The relevant action inverses are

in the even sector. Although ${E_m}^{\mathrm{R}}$ inverts the scalar equation, the scalar action Hessian is $-P_m$. This sign enters the binary response.

The causal part satisfies

on the even blocks used here. Admissible differences $d$ are smooth graded-symmetric shell bisolutions, not necessarily compactly supported. XIII-A's rank-one example consequently remains within the admissible class XIIA.

For later comparison, the inherited roots are

They are not newly selected product data.

For XII-F's detecting data, the supplied value is

The parameter is compact, (CP) holds, and the initial configuration lies on the complete free shell but outside the detected cancellation subclass XIIF. The factor $8$ and the cylinder integral are copied. No binary normalization is chosen after this witness, and the witness is not inserted into an odd Ward identity.

The retained action boundary is

Gravitational interaction vertices are omitted from the selected calculation, not from the action XIIC.

The only interaction entry is

where

Its parameter $\lambda$ counts this interaction insertion only.

The vertex belongs to ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$, is even and has ghost number zero, and has functional support contained in $\mathrm{supp}g$. Its derivatives are

with

Proof. The cubic polarized kernel is a finite sum of derivatives of three-field diagonal distributions, multiplied by $g$ and constant fiber tensors. Its support is compact and its wavefront covectors sum to zero. A nonzero tuple wholly in either pointed closed causal cone cannot have zero sum. Hence the kernel is microcausal.

The fields entering the vertex are even and have ghost number zero. Variation of the displayed density gives the metric derivative. Compact integration by parts in the scalar variation gives the second formula, retaining derivatives of $g$.

The proposed expansion is printed once:

We match $\lambda$ to one additional $\kappa$ power: since $\mathcal{O}$ is the displayed ${\kappa }^2$ coefficient, its selected response would contribute at ${\kappa }^3$ when $\lambda =\kappa$. This does not assert completeness of the full ${\kappa }^3$ expansion.

Both entries have field degree three. A single cross-contraction has degree four and a double cross-contraction has degree two. The latter is the requested coefficient linear in $\hslash$ after the retarded division by $\hslash$. Triple contractions lie beyond the order in (eq:order). No $R_2$ is introduced.

The Euler operator $V_{1,g}$ appears only in derivative calculations. It is not the first argument of a time-ordered product. Similarly, contractions involving metric derivatives of $\mathcal{O}$ are contractions of the admitted composite. They do not define a separate quantum product with ${\mathcal{J}}_{\mathrm{D}}$ as an argument.

We adopt

The zeroth-order term of ${{\mathsf{T}}_2}^{\mathrm{ren}}$ is the ordinary product ${\mathcal{V}}_g\mathcal{O}$. On regular kernels the even time-ordered line in this convention is

For the vertex variables $y$ and observable variables $x$, it equals $H(y,x)$ when the vertex is later and $H^{\mathsf{T}}(y,x)$ when the observable is later. This follows directly from $H-H^{\mathsf{T}}=i({G_{\mathrm{act}}}^{\mathrm{R}}-{G_{\mathrm{act}}}^{\mathrm{A}})$ and causal support.

The usual distinction between a time-ordered normalization and the corresponding retarded difference is made in the functional formulation of causal perturbation theory FR (Sections 6–7). Here the consequences are checked with the stated signs.

Whenever the full functional supports of the entries are strictly time ordered, a binary time-ordered datum satisfying causal factorization must obey

The supports here are functional supports, not just $\mathrm{supp}f$.

Proof. Substitute the two time-ordered factorization values in (eq:R-definition). No extension on a collision diagonal can alter these values on the corresponding separated ordered region.

For nonzero displayed entries, the additional requirement

is incompatible with (eq:R-definition), ordinary zeroth-order time ordering, and the contraction grading in (eq:order). No counterterm in the class below repairs this incompatibility.

Proof. The classical symbol of the right side is the homogeneous degree-six polynomial ${\mathcal{V}}_g\mathcal{O}$. The classical retarded coefficient on the left is a one-contraction polynomial of degree four. Equality forces the degree-six polynomial to vanish identically.

For two nonzero bosonic polynomial functionals their ordinary product is nonzero: restrict to a finite-dimensional space on which both restrictions are nonzero, and use the integral-domain property of the polynomial algebra there. The displayed entries contain no odd variables. Thus the required equality is impossible. All counterterms allowed below enter the degree-two quantum coefficient and cannot alter this classical comparison.

Zero observations give the trivial zero pair and are not covered by the nonzero assertion. For the detecting data, the supplied witness ensures that $\mathcal{O}$ is not the zero polynomial. The correction in (eq:ordered-values) is necessary, not a choice made to remove a witness.

A nonlocal composite has no unique spacetime “entry point.” Therefore a statement about the total diagonal must specify the graph presentation to which it refers.

Use the finite rooted graph presentations of $\mathcal{O}$ supplied by XII-D and XII-E, together with the local vertex (eq:vertex). The total graph diagonal is the locus where all spacetime vertices in a graph coincide.

The class ${\mathfrak{C}}_{\mathrm{tot}}$ permits changes only at order ${\hslash }^2$ in ${{\mathsf{T}}_2}^{\mathrm{ren}}$, hence at order $\lambda \hslash$ in (eq:order). After pushforward these changes are finite sums

where:

1. $A,B\in \{\gamma ,\varphi \}$, with constant fiber contractions;

2. the total field-derivative order is at most eight;

3. each $P$ is a polynomial of total degree at most eight in the listed external jets, with fixed complex coefficients depending on the frozen mass;

4. no higher derivative of $g$ or ${\sigma }_{\epsilon }$ is admitted as an independent coefficient;

5. the unintegrated change is supported on the total graph diagonal and preserves the degree-two Wick expansion;

6. the tree inverse and off-diagonal first-completion root normalization are unchanged.

The numerical bounds in this conservative class are definitions, not claims that power counting has classified all possible renormalizations. The class is frozen before any evaluation is used. It does not include arbitrary auxiliary or ghost corrections.

A smooth coefficient supported in the ramp is legal when it can be built from the listed ramp jets. The classical no-ramp companion lemma does not prohibit such a quantum counterterm.

Each counterterm in (eq:counterterms) is itself compact and microcausal: its quadratic local kernel has balanced covectors. This proves admission of the allowed ambiguities, not existence of a distribution to which they can be added.

In particular, an internal partial collision diagonal is not the total graph diagonal. If extension at a partial collision requires a larger ambiguity class, that class must be declared separately; it is not silently included in C_{tot}.

Define the off-shell tree coefficient by

Only the metric and scalar diagonal blocks contribute.

Let $F,G$ have compact homogeneous cubic kernels with wavefront sets contained in $Z_3$. A contraction of one derivative slot from each kernel through a balanced retarded Green line exists. The remaining degree-four kernel is compact and has wavefront set contained in $Z_4$.

Proof. Denote the contracted variables by $y,x$ and the four uncontracted variables by $z_1,\dots ,z_4$. The derivative kernels have covectors

whose sums vanish; the line has covectors $(k,-k)$ at $(x,y)$.

A forbidden cancellation in the full distribution product would have zero covectors at every uncontracted variable. The two sum-zero relations then force $q_y=q_x=0$. The line covector must also vanish, contradicting a nonzero cancellation. The product therefore exists by the distribution product criterion.

Both derivative kernels are supported in fixed compact sets. Their product with the line has compact support, so projection over $x,y$ is proper. The wavefront pushforward condition sets the contracted output covectors to zero. Summing all covectors leaves ${{\sum }_{j=1}}^4{p}_j=0$. Thus the output bound is $Z_4$. Fiber sums and normalized symmetrization preserve these conclusions.

This proof includes the nonlocal derivative kernel of $\mathcal{O}$. It is not a theorem only for two local entries. The product and proper-pushforward criteria used in the argument are the standard wavefront calculus Hormander.

Equation (eq:tree-definition) defines an even, ghost-number-zero compact microcausal polynomial of degree four. Its class

is well defined for the displayed lifts. It is independent of the smooth ordering difference $d$ at classical-symbol order.

Moreover, the binary time-ordered coefficient

exists through the displayed order and yields (eq:tree-definition) under (eq:R-definition).

Proof. Proposition prop:vertex and the inherited bound for $\mathcal{O}$ supply the hypotheses of Lemma lem:one-line. The action inverse has balanced translation-invariant lines. Consequently the contraction exists, is compact, and satisfies $Z_4$, which implies microcausality.

The star contraction $C_1(H)$ is already defined by the inherited domain theorem. The additional contraction $C_1(i{G}_{\mathrm{act}}^{\mathrm{A}})$ is the transpose of the constructed retarded contraction. Thus

This proves the sign and the stated binary coefficient. Neither ${G_{\mathrm{act}}}^{\mathrm{R}}$ nor this classical contraction depends on $d$. The inherited quotient map supplies the class.

No extension was required for this one-line calculation. It is a causal composition, not a newly renormalized product.

Substituting (eq:vertex-derivatives) and (eq:action-inverse) gives

Both the metric inverse coefficient $-1$ and the scalar Hessian sign have been retained.

Every occurrence of the inherited first-completion branch with metric argument ${G_{\mathrm{gr}}}^{\mathrm{R}}w$ obeys

In particular,

The tree datum uses these same response and root normalizations. It is not identified with ${\mathcal{C}}_{3,\mathrm{D}}$.

Proof. Equation (eq:root-overlap) is XII-D's transposition identity with the inherited sign, not a new root construction. Since $r_g={G_{\mathrm{gr}}}^{\mathrm{R}}(g{T}_0/2)$, substitution gives (eq:stress-overlap). In (eq:tree-branches) the same metric response is contracted with the complete metric derivative of $\mathcal{O}$, and the scalar branch is its inherited switched response. Differentiating any displayed first-completion branch uses the same identity.

This is the meaningful overlap: a branch common to the constructions has unchanged normalization. The entire binary response has degree four, whereas ${\mathcal{C}}_{3,\mathrm{D}}$ has degree three. Requiring equality of those objects would confuse operations and contradict their typing.

The constructed value has a class in ${\mathcal{E}}_H$. This does not establish a binary map on arbitrary representatives of the quotient. A simple equation-functional check explains why.

For a compact scalar test $a$, let $E_a(\Phi )=\int a{P}_m\varphi$. It vanishes on ${\mathcal{C}}_{\mathrm{sol}}$. The tree response to this alternative observable lift is

Indeed $P_m{E}_m^{\mathrm{R}}=\mathrm{id}$ after transposition on the compact test. The right side need not vanish on the free shell.

Thus retarded contraction is not automatically invariant under adding free equation functionals to an entry. Here the prescribed lifts are frozen, so (eq:representative-test) does not invalidate Theorem thm:tree. It prevents the stronger, unproved claim that we have constructed a representative-independent operation on all of ${\mathcal{E}}_H$. No new ideal is introduced to hide this distinction.

For regular kernels, the coefficient needed next is

with the standard slot pairings and symmetrizations. Using (eq:HT), the difference contains

The expression is a specification of contractions, not a proof that their joint product exists.

There is now one uncontracted field slot in each entry and two contracted slots. In a proposed forbidden product cancellation, the uncontracted covectors may vanish while the two contracted covectors in an entry are opposite and nonzero. Their sum is zero, so the $Z_3$ bounds do not force them individually to vanish.

This is the exact place where Lemma lem:one-line no longer applies. It is not evidence that the distribution cannot exist; it identifies a hypothesis needing additional analysis or extension.

For a nonlocal composite, the contracted observable variables need not coincide with the observation point. Consequently extension is not automatically reduced to one diagonal between two local densities. Standard local time-ordered constructions specify maps on local entries and their subsequently defined multilocal domain; they do not supply the present joint nonlocal extension merely from star-product admission FR (Sections 6–7).

The scalar-sourced coordinate part of $\mathcal{O}$ contains a branch with schematic form

Two scalar contractions can join the stress slots at $z$ to the two scalar slots of the interaction at $y$. The resulting unintegrated expression contains two differentiated time-ordered scalar lines between $y$ and $z$, together with the already inherited retarded branch from $z$ to $x$.

The collision $y=z$ may occur while $x\ne z$. It is therefore a partial graph diagonal. A counterterm supported only where $x=y=z$ does not by its support alone normalize this internal collision. Other branches contain additional internal vertices. They require a joint check, not deletion because the final output is called binary.

This observation does not prove that every extension in C_{tot} fails. A proof of such failure would need an explicit unavoidable normalization incompatibility. None is inferred from the presence of the collision.

For each supplied rooted graph of $\mathcal{O}$, attach the compactly switched vertex and either one or two cross-lines. On the closed support set determined by the existing graph weights and retarded internal lines, projection over all internal vertices is proper. A support-preserving extension does not spoil this property.

Proof. XII-D and XII-E provide compact support sets for their rooted graphs. Internal retarded vertices are bounded in time below by the ramp or compact switch, above by the compact observation, and spatially by the corresponding causal regions.

The new local interaction vertex lies in $\mathrm{supp}g$, a compact set. Adding cross-lines restricts the product support inside the product of these existing compact graph sets with $\mathrm{supp}g$. Its closure is compact. Projection from this set is proper. An extension supported in the closure of the same graph support, including allowed total-diagonal terms, preserves compactness.

This proves the relevant support control without proving the singular joint distribution. Properness cannot repair an undefined distribution product, and an extension theorem cannot be substituted for properness.

Suppose the two-line graph distributions of the selected pair admit a joint extension with all the following properties:

1. agreement with the defined off-collision contractions and with the corrected ordered-support values;

2. the frozen ambiguity class C_{tot};

3. support preservation and compatibility with the inherited root branch;

4. a balanced wavefront bound on the complete graph variables, or an independently proved microcausal bound after pushforward.

Then the resulting ${R_1}^{(1)}$ is a compact even, ghost-number-zero polynomial of degree two, and

Proof. Each double contraction leaves two field slots. Properness follows from Proposition prop:properness. A balanced complete graph bound, after setting internal pushforward covectors to zero, leaves a sum-zero bound on the two external covectors. That bound is microcausal. Alternatively, the separately proved output bound in (iv) gives the same admission directly.

Allowed counterterms are already admitted by Definition def:counterterms. The inherited quotient map then gives the stated class. All entries and lines are even in the relevant sector, and no ghost slots are generated.

The hypotheses of this theorem have not been jointly proved. It is deliberately conditional.

For the corrected binary problem, the release at this gate is Unresolved. The literal factorization error was separately Proved obstructed in Theorem thm:literal; it is not relabelled as an obstruction to every corrected extension.

The established tree attachment has support

Thus the interaction attachment can affect the contracted observable slot only to its future. If the entire interaction support lies later than the entire functional support of $\mathcal{O}$, the tree datum is zero. The analogous complete binary statement is conditional on the extension normalization in Theorem thm:conditional.

There is also the valid compact output bound

The uncontracted slots are projections of the two derivative-kernel supports.

Equation (eq:retarded-attachment) does not imply

An untouched metric slot of the nonlocal observable may lie earlier than the interaction while another slot receives the retarded attachment. A claim about every functional-support variable requires its own proof. We therefore use the attachment support condition, not the unsupported stronger inclusion.

The constructed classical coefficient can be evaluated on ${\mathcal{C}}_W$. Its degree sectors follow from (eq:tree-branches):

The tree evaluation vanishes on either of the following subsets of ${\mathcal{C}}_W$:

Also, under {(UCP${}_W$)}, the independent inherited identity ${\Delta }_{\xi }\mathcal{O}(u)=0$ remains valid for all $u\in {\mathcal{C}}_W$.

Proof. At $\gamma =0$, ${{\mathcal{O}}^{\prime }}_{\gamma }=0$ because its metric sector is quadratic, and $p_1[\gamma ;\varphi ]=0$. Thus both terms of (eq:tree-branches) vanish.

At $\varphi =0$, both $r_g[\varphi ]$ and $p_1[\gamma ;\varphi ]$ vanish. The same formula gives zero. The last assertion is the inherited restriction theorem, applied only to classical evaluation.

No vanishing of ${R_1}^{(0)}$ on the whole of ${\mathcal{C}}_W$ has been proved. The cancellation of ${\Delta }_{\xi }\mathcal{O}$ does not imply that the interaction response vanishes. The response directions $r_g[\varphi ]$ and $p_1[\gamma ;\varphi ]$ are sourced zero-data fields; they are generally not tangent to the free shell or its restricted subclass. Differentiating an evaluation identity along such a direction is not licensed.

The statement also does not supply evaluation of an unconstructed ${R_1}^{(1)}$. In particular, one cannot conclude that its degree-two symbol vanishes on the two subclasses merely from the degree-four tree statement.

The witness (eq:witness) remains the boundary of the original full-shell invariance claim. It is neither removed by evaluation elsewhere nor transformed into a binary quantum anomaly.

The free symbol rule is

It is odd and raises ghost number by one. The following calculations concern the two entries, not a gauge-consistency identity for the binary datum.

Using symmetry of $T_0$ and compact integration by parts,

On the free scalar shell,

The ghost need not be supported in $W$, so the switching term does not vanish merely because $g=1$ on $W$. There is no coordinate ramp in this vertex.

We give a compact explicit form of XIII-A's entry derivative. Set

Let $X_2,p_2$ be the inherited second completion at $(\gamma ,k;\varphi )$. The composite is

Write $\dot{k}={r_{\mathrm{D}}^{\prime }}_{\gamma }[z]$. Since the actual source is ${\mathcal{J}}_{\mathrm{D}}={\sigma }_{\epsilon }{J}_3+2C_2{\sigma }_{\epsilon }^{\prime \prime }-2M_2{\sigma }_{\epsilon }^{\prime }$ in the time-only ramp convention, its inherited derivative is

The tensors $C_2,M_2$ are exactly the finite quadratic source coefficients supplied by XII-D. This equation differentiates the existing response; it introduces no quantum argument ${\mathcal{J}}_{\mathrm{D}}$.

Define

For any tensor coefficient $a$ and scalar or coordinate component $v$, put

Thus every external derivative of the weights is visible.

On the free shell, $Dz=0$ and, with $v=c^{\mu }{\partial }_{\mu }\varphi$,

The coordinate layer and scalar switch terms survive for unrestricted homogeneous ghosts.

Finally put $Y=s_0{X}_2$ and $p_{2,z}=s_0{p}_2$. Their complete divergence-form derivatives are

Together with (eq:dotk), these formulas retain the source ${\sigma }_{\epsilon }^{\prime },{\sigma }_{\epsilon }^{\prime \prime }$ terms as well as all coordinate and scalar switching terms.

The free differential of the composite is

It is odd, has ghost number one, and belongs to the inherited polynomial domain.

Proof. Apply the graded product rule to the displayed completion. All undifferentiated coefficient factors are even. The first and second variations above are their ordinary directional derivatives in the odd direction $z$. Admission follows from XII-A's preservation of ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ by the frozen differential XIIA.

Neither (eq:sV-shell) nor (eq:sO) is set to zero. The auxiliary remainder ${\mathfrak{B}}_{\xi }$ is even and has ghost number zero; it is not a term in these ghost-number-one formulas. A compact finite shift $K\xi$ is not $s_0$, and the finite two-jet term $\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$ is not an infinitesimal ghost variation.

We do not apply the differential to the binary datum and do not form the binary gauge-consistency defect. That separate question remains closed, including in the region where the tree datum is constructed.

For the frozen displayed lifts and fixed regulator width:

1. The vertex ${\mathcal{V}}_g$ is admitted. The tree coefficient (eq:tree-definition) exists and defines one selected causal datum $Q_H{R}_1^{(0)}\in {\mathcal{E}}_H$.

2. Its inherited inverse signs, properness, compact microcausal kernel, and first-completion branch overlap are proved.

3. Causal factorization has the values (eq:ordered-values). Replacing them by $R_1={\mathcal{V}}_g{\star }_H\mathcal{O}$ is Proved obstructed under the printed contraction normalization.

4. The corrected coefficient through $\lambda \hslash$ is Unresolved until the hypotheses of Theorem thm:conditional are established.

5. Classical restriction is applied after admission. The inherited cancellation of ${\Delta }_{\xi }\mathcal{O}$ remains an evaluation identity and does not imply a quantum restriction quotient or a zero interaction response on all of ${\mathcal{C}}_W$.

Proof. Proposition prop:vertex, Theorems thm:literal, thm:tree, and thm:conditional, Propositions prop:overlap, prop:properness, and prop:zeros, together with the stated inherited restriction input.

The affirmative binary release through the requested quantum order is therefore not established. The missing extension is named, its support and normalization obligations are separated, and its absence is not called an anomaly. A manuscript cannot become a proof of that extension merely by labeling the formal expression “renormalized.”

A successor receives:

1. the admitted interaction functional and the fixed displayed composite;

2. the selected retarded convention and its corrected ordered values;

3. the constructed tree datum and its admitted class;

4. the branchwise root-overlap identity;

5. the frozen total-diagonal counterterm class and the explicit internal partial-collision problem;

6. the conditional quantum-admission theorem;

7. the evaluation-only restriction and the two proved tree zero-evaluation subclasses;

8. the two entry differential displays.

It does not receive a complete coefficient ${R_1}^{(1)}$, a representative-independent retarded operation on every quotient class, a reduced algebra, a completed observable, or a binary gauge-consistency result. Opening a subsequent gauge-defect paper through quantum order first requires completion of the present joint-extension gate, or an explicit restriction of that successor's claim to the constructed tree datum.

There is one constructed tree-level binary datum on the unchanged carrier and one later classical evaluation restriction. There is not yet a complete quantum binary datum through the requested order. No gauge-consistency defect has been opened.

correction. The distinction between factorization of ${\mathsf{T}}_2$ and ordered values of $R_1$ follows from the selected retarded definition. The correction changes neither the inherited causal brackets nor the coefficient $-1$ of the metric inverse.

scope. Properness and tree admission are proved. The two-line extension is expressly unresolved. No failed sufficient wavefront condition is promoted to a theorem that every extension fails.

scope. All constructed outputs use the existing polynomial domain and full-shell quotient. No quotient by ${\mathcal{N}}_W$ is taken. The selected-lift construction is not asserted to descend as a binary operation under arbitrary changes of input representatives.

scope. Only the switched metric–scalar vertex is inserted. The Einstein–Hilbert action remains in the model, but none of its interaction vertices is restored in this selected pair. The empty invariant jet is not used as an interaction functional.

. No successful pdflatex execution, proof-assistant verification, independently executed symbolic computation, independent referee review, or journal acceptance is asserted.

XIIIA

N. Cheung, Landmark XIII-A: Classical cancellation on a fixed subclass, and the obstruction to a coefficientwise restriction quotient. Supplied manuscript XIII-A.tex. Evaluation-only handoff, restriction-ideal obstruction, ordering counterexample, and unresolved completion boundary.

XIIF

N. Cheung, {Landmark XII-F: The gauge-fixed remainder of ${{\Phi }^{(2)}}_{\mathrm{D}}$: companion support, the auxiliary pairing, and what is not a Ward identity}. Supplied manuscript XII-F.tex. Copied witness and the exact scope of classical cancellation.

XIIE

N. Cheung, Landmark XII-E: Assembly of the second-order relational coefficient after admission of the density-weighted cubic insertion. Supplied manuscript XII-E.tex. Assembly sign, completion formulas, compact balanced kernels, and scalar-sourced slot.

XIID

N. Cheung, Landmark XII-D: Retarded response and carrier admission for the cubic-only density-weighted Einstein source. Supplied manuscript XII-D.tex. Inherited roots, response normalization, source coefficients, and causal plateau proposition.

XIIA

N. Cheung, Landmark XII-A: One graded polynomial carrier for the finite-ramp multiplet, with ordering-stable composites and a robust observation exclusion. Supplied manuscript XII-A-v2.tex. Polynomial product, action causal matrix, full-shell ideal, free differential, and admissible ordering class.

XIIC

N. Cheung, Landmark XII-C: One declared cubic weighting: actual Euler source, changed Noether identity, and the coupled-jet question. Supplied manuscript XII-C.tex. Boundary only: retained Einstein–Hilbert action and named invariant-jet obstruction.

XIB

N. Cheung, Landmark XI-B: The free de Donder multiplet for the ramp Einstein–scalar sheet: causal propagators, residual gauge, and the layer dressing. Supplied manuscript XI-B.tex. Inverse coefficient $-1$ and free ghost normalization.

FR

K. Fredenhagen and K. Rejzner, Perturbative algebraic quantum field theory, arXiv:1208.1428v2, 2013. Sections 6–7: time ordering, retarded constructions, and the declared domains of local and multilocal renormalization.

FRModels

K. Fredenhagen and K. Rejzner, Perturbative Construction of Models of Algebraic Quantum Field Theory, arXiv:1503.07814, 2015. Comparison framework for the separation of free products and renormalized perturbative constructions.

Hormander

L. Hörmander, The Analysis of Linear Partial Differential Operators I, second edition, Springer, 1990. Distribution products and proper-pushforward wavefront bounds used in the one-line proof.

BGP

C. Bär, N. Ginoux, and F. Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society, 2007, arXiv:0806.1036. Smooth wave Green operators, transposition, causal support, and finite propagation.