Landmark XV-A A Model-and-Domain Audit of the Ramp, Graviton, and Joint-Extension Tracks

This paper audits the dependency and scope of the supplied manuscript record. It distinguishes the convolution-regulated classical prescription, the ramp scalar prescription with classical metric, the quantum-graviton progression, the relative classical BV branch, and the joint characteristic-extension branch. The record does not supply the identifications needed to regard these tracks as one unified successor model.

The mathematical results proved here are audit-level results: a domain-sensitive transfer criterion, reconstruction descent, a presentation-null identity, a finite normal-jet interface, and the logical distinction between partial constructions and admission. Results attributed to earlier manuscripts remain predecessor inputs. In particular, the supplied XIV-J release leaves the full joint characteristic map unresolved at J2. Neither the internal amplitude constructed in XIV-I nor the supported-sector reference constructed in XIV-J admits the required quantum coefficient. The recommended next research target is that unresolved full map, not an interacting Ward identity and not an opening of XV-B.

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

Figure fig:scope-XV-A records that separation. The drawing adds no hypothesis.

Scope and evidence contract. XV-A is a dependency and scope audit, not a new retarded-product construction and not a proof of a one-loop coefficient. Its evidence is exactly the supplied record described in Section sec:evidence. It does not certify manuscripts whose complete texts were not supplied. It catalogues the available model data, domains, orders, and open gates; separates copied inputs from proofs given here; and identifies the next admissible research question. It does not construct a characteristic extension, declare an interacting BRST/BV differential, identify classical evaluation with a quantum quotient, infer nonexistence from a failed sufficient wavefront criterion, or transfer a fixed-regulator result to a changed regulator or sharp-width limit.

The ledger precedes the explanatory narrative. “Copied input” describes the role of a predecessor result in XV-A; the parenthetical status describes what the supplied source says it constructed or left open. “Reported” means that only the supplied coverage account or source locator supports the entry, not a reproduced proof. An omitted formula, carrier definition, or perturbative order is recorded as missing rather than reconstructed.

{}1.15 {@{} p3.45cm p2.65cm p3.70cm p3.10cm p2.15cm p3.00cm p4.25cm @{}} , domain, order, provenance, and non-implication ledger.
Claim & Source & Model & Domain & Order & Status in XV-A & Does not imply

Claim & Source & Model & Domain & Order & Status in XV-A & Does not imply

7rContinued on next page

A fixed witness gives a first-order defect for the convolution prescription. & Supplied audit account; convolution manuscript not otherwise identified. & Spacetime convolution on metric arguments of an auxiliary coordinate problem. Exact action, convolution kernel, and inverse normalization unavailable. & The frozen gauge algebra declared by that manuscript. & Classical first order; exact coupling convention unavailable. & Reported\ obstruction on that algebra. & An obstruction to every coordinate prescription; an identification with a ramp edge integral; or repair by a ramp theorem.

Fixed-width ramp coordinate and scalar prescription has plateau covariance. & X-A@1, abstract, lines 112–147; supplied account. & Ramp prescription at fixed $\epsilon$. Full action and inverse formula not reproduced. & Classical fields, with the stated plateau restrictions. & Classical; exact truncation not reproduced. & Reported\ construction. & Unrestricted fixed-background covariance, sharp-width covariance, or quantum-graviton admission.

Scalar-only free-field admission and one normalized retarded correction complete the first-order identity. & X-B@3, field-content passage, lines 202–236; supplied account. & Ramp scalar quantum field; metric classical. One normalized correction; normalization formula unavailable. & Declared scalar carrier; insertion tests are actual off-shell representatives. & Through the stated first order in $\kappa$; exact $\hslash$ order unavailable. & Reported\ construction. & A quantum metric kernel or replacement of insertion representatives by unspecified shell classes.

Second scalar response maps give the classical completed identity through ${\kappa }^2$. & X-C@3, frozen-sheet passage, lines 488–587; supplied account. & Same reported ramp scalar/classical-metric lane, with second response maps. & Classical identity; off-shell insertion representatives retained. & Classical identity through ${\kappa }^2$. & Reported\ construction. & An all-species quantum Ward identity or admission of a joint graviton coefficient.

Completed scalar tests have a fixed-input sharp-width limit. & X-D@3, fixed-input release, lines 1445–1478. & The X-D fixed-input ramp family. & Completed scalar tests with the inputs held as declared. & Sharp-width limit; further order details unavailable. & Reported\ construction. & Joint sharp admission of a quantum metric kernel or a changed-input limit.

The held-fixed graviton defect of the one-tree observable is nonzero at ${\kappa }^2$. & X-E@1, release, lines 201–230; supplied account. & One matter tree; flat gauge fixing. Exact full action and inverse formula unavailable. & One-tree observable and held-fixed graviton comparison. & Tree calculation; defect at ${\kappa }^2$. & Reported\ nonzero defect. & Failure of every all-tree completion or an obstruction to every gauge choice.

Three trees in background de Donder gauge close the stated all-tree identity. & X-F@1, F6, lines 1269–1318; supplied account. & Background de Donder gauge; three trees. & Declared X-F all-tree domain. & ${\hslash }^0$; precise coupling boundary is not recoverable from the truncated supplied sentence. & Reported\ closure with an unresolved order citation. & An unrestricted BRST theorem, higher-order closure, or a one-loop coefficient.

The unrestricted classical master equation is obstructed, while a relative action identity is available. & Coverage account for XIV-A; no complete proof excerpt. & Reported frozen cubic data; fixed backgrounds. & Unrestricted ghosts versus ghosts supported where both backgrounds are trivial. & Classical; precise coupling truncation unavailable. & Reported\ domain-dependent result. & Observable closure, unrestricted fixed-background covariance, or a quantum differential.

The relative complex does not induce the claimed interacting differential on the unchanged free-shell carrier. & Coverage account for XIV-B; no complete proof excerpt. & Relative classical complex; positive-coupling gauge-fixed equations. & Compact relative ghosts and the separate free-shell carrier. & Classical; agreement only at order zero is reported. & Reported\ carrier separation. & That compact gauge parameters are nonzero homogeneous free ghost solutions, or that the free-shell equation ideal is unchanged.

The internal $K_5$ amplitude remains unresolved at H2. & XIV-H@1, thm:H2, lines 1539–1582. & Two internal placements with fixed retarded lines and smooth weights. & Untested internal kernels; internal characteristic transition. & Internal tree data, before quantum cross-line composition. & Copied input\ (source: unresolved at H2). & Nonexistence of every amplitude construction or admission of a quantum coefficient.

Both $K_5$ internal placements are defined by tensor-product pullback. & XIV-H@1, prop:K5exact, lines 1135–1202. & Products $R(b-a)R(a-z)$ and $R(b-z)R(z-a)$, then finite differential operations. & Internal distributions, including $a=z$. & Internal tree kernel; not the contracted quantum target. & Copied input\ (source: constructed). & A product with the quantum cross-lines on the characteristic locus.

The named internal amplitude module is constructed at I2. & XIV-I@1, thm:I2, lines 1288–1378. & Finite fixed retarded-pole module; both placements; exact finite remainder. & Untested reconstruction with retained residual variables. & Internal frequency estimates; no asserted target transverse degree. & Copied input\ (source: constructed at I2). & A single-Lagrangian symbol theorem, a joint characteristic map, or quantum admission.

The fixed retarded pole is an exact presentation of an existing internal line. & XIV-I@1, prop:pole, lines 726–826. & $(\lambda -i0{)}^{-1}$ with its fixed delta contribution. & Nonzero characteristic-sheet charts. & Internal Fourier representation. & Copied input; elementary chart identity also checked in Section sec:pole. & A new graph regulator, adjustable characteristic counterterm, or characteristic prescription for the joint target.

The common precomposition sum exists, but the joint map is unresolved at I3. & XIV-I@1, thm:I3, lines 1459–1548; thm:release, lines 1729–1814. & $K_{47}+K_5+K_6+K_8+K_9$, with grouped cross-lines still to be composed. & Canonical admissible domain versus tests meeting $N$. & Required quantum target not admitted. & Copied input\ (source: unresolved at I3). & That internal estimates prove the final product, or that the canonical domain is automatically the entire complement of $N$.

A presentation-level map descends exactly when it annihilates the reconstruction kernel. & XIV-J@1, prop:descent, lines 813–846. & Linear reconstruction $\mathsf{Rec}$. & Presentations and their reconstructed image. & Order-independent linear algebra. & Proved here; source result recorded as predecessor input. & Uniqueness on a larger carrier than $\mathrm{im}\mathsf{Rec}$, or descent across $N$ from off-characteristic agreement alone.

$q{\delta }^{\prime }(q)+\delta (q)=0$ gives a nontrivial null presentation. & XIV-J@1, lem:null, lines 853–883. & Adapted internal amplitude chart. & Distributional reconstruction. & Presentation-frequency degree. & Proved here. & That the highest polynomial degree is an intrinsic reconstructed symbol, or that XIV-I's estimates are invalid.

The supported-sector reference is constructed away from the characteristic locus. & XIV-J@1, thm:reference, lines 1215–1299. & $K_{\mathrm{supp}}=K_6+K_8+K_9$ only. & $U\setminus N_U$. & Partial grouped quantum reference; no full admission. & Copied input\ (source: constructed on its stated domain). & The full six-branch map, either characteristic $K_5$ composition, or characteristic composition of $K_{47}$.

Supported normal-order-preserving extension has a conditional coefficient interface. & XIV-J@1, thm:conditionalextension, lines 1427–1470. & Finite response-normal jets. & Supported realizations of normal order at most three. & Conditional extension interface. & Copied input; support-explicit interface proved here. & Existence, selected finite part, uniform estimates, or characteristic normalization.

The full joint map remains unresolved at J2; subsequent gates are not discharged. & XIV-J@1, thm:J2, lines 1608–1649; thm:release, lines 1894–1939. & Unchanged retained instance and frozen map class. & Full reconstructed sum on tests meeting $N$. & Required quantum coefficient. & Copied input\ (source: unresolved at J2). & A universal obstruction or admission from the supported-sector reference.

XIV-J admits no ${R_1}^{(1)}$ and does not open XV-B. & XIV-J@1, cor:noadmission, lines 1808–1822. & Required pushed-forward coefficient and stated output space. & Required output admission $Q_H{R}_1^{(1)}\in {\mathcal{E}}_H$. & Quantum admission unasserted. & Copied input\ (source: no admission in this release). & Nonexistence of every possible coefficient or permission to form its interacting Ward defect.

The evidence has two levels.

1. Direct excerpts. The supplied record reproduces specific propositions, theorems, and release statements from XIV-H, XIV-I, and XIV-J. Their locators are retained in the bibliography.

2. Coverage account and locators. The supplied record also contains a source-based account of the convolution, X-A–X-F, XII–XIII, and XIV-A–XIV-G relationships, together with selected X-series locators. It explicitly disclaims a full-corpus scan and independent verification. The X-F sentence in the proposed outline is truncated. These limitations remain limitations of XV-A.

A result reproduced with its source proof is still a copied predecessor input unless XV-A gives an independent proof of the particular assertion it uses. A locator without the relevant formula does not license fabrication of that formula.

A ledger entry is:

1. constructed if the relevant object is supplied on its declared domain;

2. proved obstructed if a stated theorem excludes the required property in a specified frozen class;

3. unresolved if the required construction or exclusion has not been established in the supplied record;

4. a copied input if XV-A uses it as a predecessor result rather than proving it independently.

The fourth designation is a provenance designation, not a competing mathematical outcome. Thus a copied input may report a constructed, obstructed, or unresolved source outcome.

In particular, “unresolved” is not synonymous with “proved obstructed.” A nonzero defect for one observable is not an obstruction to every completion.

For audit purposes, a model signature is the tuple

Here $S$ and $V$ are the action and designated vertices; $\mathcal{F}$ is the field content and its parity; $\mathcal{B}$ is the fixed background and gauge-fixing data; $\mathcal{R}$ is the switch or regulator family; $G$ is the inverse or response operator with its normalization; $\mathcal{O}$ is the assembled observable; $\mathcal{C}$ is the carrier or field domain; $\mathcal{T}$ is the allowed test and insertion class; and $\nu$ is the perturbative grading and truncation. An unavailable component is marked unavailable, not treated as matching another signature.

This signature distinguishes four domains that must not be silently substituted for one another:

They denote, respectively, off-shell fields, the free shell, a declared classical evaluation set, and a quantum carrier. A restriction to one of them is additional data.

Let $A$ be an identity proved for signature ${\Sigma }_1$ on domain ${\mathcal{C}}_1$. To transfer that identity to an object with signature ${\Sigma }_2$ and domain ${\mathcal{C}}_2$, it suffices to provide maps transporting all operations and arguments used in $A$, together with proofs that:

1. the target arguments belong to the transported domain;

2. the transported operations agree with the target operations;

3. all products, pullbacks, quotients, and limits used in the transport are defined;

4. the normalization and perturbative grading are respected.

Without such a bridge, the identity for ${\Sigma }_1$ supplies no domain-independent transfer theorem for ${\Sigma }_2$.

Proof. Under the four hypotheses, replace each argument and operation in the proved identity by its transported counterpart. The intertwining identities preserve the equality, and the domain hypotheses justify every operation.

For the final assertion, an equality on ${\mathcal{C}}_1$ alone does not specify operations or values on ${\mathcal{C}}_2$. If ${\mathcal{C}}_2$ contains additional arguments, the operations can agree on the transported part and differ elsewhere without contradicting the original equality. If no transport is specified, one may regard the two carriers as independently tagged copies; the original identity then contains no statement identifying their operations. Consequently a transfer conclusion requires additional premises. This is a statement about the scope of the given identity, not a theorem that no bridge can exist.

A bridge need not be an isomorphism of every conceivable structure. It must, however, preserve the structures actually used in the transferred claim. A shared name, an analogous formula, or a declared continuation path is not such a proof.

The supplied account describes a spacetime convolution acting on metric arguments in an auxiliary coordinate problem. It reports a fixed witness producing a first-order defect on that manuscript's declared gauge algebra record.

The bounded audit conclusion is:

The reported convolution prescription is obstructed on the frozen algebra and at the order tested by its witness.

The exact convolution kernel, action, gauge algebra, inverse normalization, and witness formula are not reproduced in the supplied materials. XV-A therefore does not independently verify that obstruction.

The ramp identity concerns another signature. In the absence of an explicit bridge, it neither removes the reported convolution defect nor identifies its witness with a ramp edge integral. No statement about every possible convolution prescription follows.

The reported sequence is

Its scope is recorded as follows XA,XB,XC,XD,record.

1. X-A supplies a fixed-width classical coordinate and scalar prescription with plateau covariance.

2. X-B@3 supplies scalar-only free-field admission and one normalized retarded correction, completing its stated first-order identity.

3. X-C@3 supplies the second scalar response maps and a classical completed identity through ${\kappa }^2$.

4. X-D@3 supplies a fixed-input sharp-width limit of the completed scalar tests.

The lane is scalar-quantum and metric-classical where quantum content is invoked. Its insertion tests remain actual off-shell representatives. Its classical completed identities are not automatically identities on a quotient of quantum metric kernels.

The X-D limit must retain its fixed-input qualification. A limit of completed scalar tests does not, by itself, prove joint sharp admission of a quantum metric kernel. Nor does a covariance identity at fixed $\epsilon$ commute with a sharp-width limit without a theorem establishing the needed convergence and compatibility.

The supplied account distinguishes X-E from X-F XE,XF,record.

X-E uses one matter tree and flat gauge fixing. The reported ${\kappa }^2$ held-fixed graviton defect is nonzero for that one-tree observable. This is an observable-specific boundary, not a universal no-go theorem.

X-F changes the gauge-fixing data to background de Donder gauge and uses three ${\hslash }^0$ trees. The supplied account reports closure of the all-tree identity through its stated order. The proposed outline truncates the sentence specifying that order, and the relevant formula is not reproduced. XV-A therefore records the tree grading but does not invent the coupling boundary.

These are distinguishable signatures even when they belong to a research progression. The one-tree defect and the all-tree closure need not contradict one another: they refer to different assembled observables and gauge-fixing data. Neither statement, as supplied, constructs an interacting quantum BRST/BV differential or a one-loop coefficient. No later X-series BRST release is added to this ledger without its source statement.

The coverage account identifies XIV-A and XIV-B as a relative classical branch of the XII–XIII programme record. It reports that the unrestricted antibracket square is nonzero in the frozen kinematic class, whereas the action identity holds for ghosts supported where both backgrounds are trivial.

The observable ledger is separate from the action ledger. The same account reports a shell observable variation equal to an auxiliary pairing under a uniform causal plateau. An action master equation does not, merely by holding, set that pairing to zero.

XIV-B is reported to continue the relative complex. Its compact relative ghosts are not the unrestricted homogeneous free ghosts of the quantum carrier. The relative complex is reported to meet the relevant free shell only at vanishing ghosts. The positive-coupling gauge-fixed equations are also reported to differ from the unchanged free-shell equations.

These are reported source distinctions, not independently verified BV theorems of XV-A. In particular:

1. identifying antifield images with Euler equations is not, without grading and acyclicity hypotheses, a full Koszul–Tate resolution theorem;

2. compact gauge parameters are not thereby nonzero homogeneous free ghost configurations;

3. a Cauchy statement about causal field variations is not a statement about compact relative ghosts;

4. relative action symmetry is not observable closure.

The reported analytic path is

It does not run through XIV-A, XIV-B, XIII-D, or XIII-E. This path records declared continuation, not automatic import of all earlier theorems record.

The coverage account reports the following inherited freeze: XII-C's cubic action and inverse coefficient $-1$; both completion legs from XII-D and XII-E; XII-E's insertion coefficient $-1$; the full observable including the scalar-sourced slot; XII-A's full-shell carrier; and XIII-C's partial-diagonal ambiguity class. The complete formulas defining those data are not supplied. Their signs are therefore recorded as inherited signs, not derived here, and the frozen class is not expanded.

The direct excerpts establish the narrower chain of source outcomes used below:

The first box is a historical stop, not the current first missing gate after XIV-I has supplied its named amplitude.

XIV-H gives the copied formula

Here $R$, $L$, and $\mathsf{A}$ are the inherited retarded kernel, coordinate-response kernel, and tensor map. Their complete definitions are not recreated by this audit.

The source proposition says that each placement is a finite differential expression, of total kernel derivative order at most four, in

That derivative-order statement is a copied input Hkernel.

The elementary existence mechanism for the displayed products can be checked independently.

Let $R\in {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$. Both products in (eq:internalproducts), understood as distributions in the independent variables $(a,b,z)$, are defined by tensor-product pullback, including at $a=z$.

Proof. The first linear coordinate change is

Its inverse is

Thus $F_1$ is a diffeomorphism. The first product means

The tensor product exists, and pullback by a diffeomorphism is defined.

For the second placement use

with inverse $b=z+r$ and $a=z-u$. The same argument applies. Additional residual variables may be carried along by the identity map. Finite differentiation and multiplication by fixed smooth weights preserve existence.

This proposition does not define multiplication by an additional singular cross-line factor in the original attachment variables. That is a further operation.

The copied XIV-I pole chart uses

and

On this chart the source states

For an explicit convention check, take

Then the partial fraction expression is

For $s=+1$, the second denominator is smooth on the selected chart; for $s=-1$, the first is smooth there. This proves (eq:pole) algebraically. The first coefficient has homogeneous degree $-1$, and the regular remainder has degree $-2$.

For $\eta >0$,

Passing to the distributional boundary value yields

Consequently

This checks the temporal sign convention in the supplied pole formula. It does not independently prove the source's complete amplitude estimates or spatial wavefront theorem Ipole.

The delta contribution is fixed by this retarded boundary value. Discarding or adjusting it changes the represented internal line. The damping above computes the Fourier transform of an existing kernel; it is not a regulator for the unresolved graph product.

The copied common presentation is

Here $K_{47}$ groups branches $B_4+B_7$. Thus five named summands represent the retained six-branch sum $B_4,\dots ,B_9$; this notation does not delete a branch. The supported sector is only

The source formulas include

with the inherited bilinear tensor map determined by

These are copied internal formulas, not new joint composition formulas Iamp.

The unresolved operation has the form

where $E_{\mathrm{res}}$ denotes the residual-species bundle. Its required value on (eq:sum) must retain the prescribed grouped quantum cross-lines, vertex Hessian, and tensor coefficients.

The supplied I3 statement requires: agreement with canonical composition on its proved domain; definition on tests meeting $N$; retention of the grouped factors and coefficients; and an appropriate complete-graph wavefront bound Ijoint. Internal reconstruction alone proves none of these requirements for the new cross-line operation.

In particular, XV-A does not identify $U_{\mathrm{can}}$ with $W\setminus \overline{N}$, or with another entire complement of an attachment locus. The direct I3 excerpt explicitly warns that an internal representation does not establish such a domain equality.

The following statements are independently proved at the elementary linear and distributional level. They also identify the precise scope of the corresponding copied XIV-J results.

Let ${\mathcal{P}}_{\mathrm{int}}$ be a vector space of presentations, let

be linear, and let

be linear. There is a unique linear map

satisfying $\tilde{\mathsf{M}}=\mathsf{M}\mathsf{Rec}$ if and only if

If ${\mathcal{K}}_{\mathrm{int}}=\mathrm{im}\mathsf{Rec}$, this is descent onto ${\mathcal{K}}_{\mathrm{int}}$. If not, uniqueness is asserted only on the reconstructed image.

Proof. If the factorization exists and $\mathsf{Rec}A=0$, then $\tilde{\mathsf{M}}(A)=\mathsf{M}(0)=0$.

Conversely, assume (eq:descent). Define

If $\mathsf{Rec}A=\mathsf{Rec}B$, then $A-B\in \ker \mathsf{Rec}$, so their proposed images agree. The map is therefore well defined, linear, and uniquely determined on $\mathrm{im}\mathsf{Rec}$.

Suppose a proposed presentation map agrees with canonical contraction on an open set $U_{\mathrm{can}}$. For $A\in \ker \mathsf{Rec}$, this proves only

It does not prove $\tilde{\mathsf{M}}(A)=0$ on $W$.

Proof. Restriction can fail to be injective. If $p\in W\setminus U_{\mathrm{can}}$, a nonzero delta distribution at $p$ restricts to zero on $U_{\mathrm{can}}$. Thus a possible distribution supported outside the canonical domain is not detected by canonical-domain agreement. This demonstrates the missing implication; it does not assert that such a term satisfies every other condition of the frozen map class.

In one normal variable $q$,

Accordingly, in an adapted reconstruction convention satisfying

one has

If $R\ne 0$, the two summands reconstruct separately to nonzero distributions with opposite signs.

Proof. For $\psi \in \mathcal{D}(\mathbb{R})$,

Hence $q{\delta }^{\prime }=-\delta$. Tensoring this identity with $R(r)$ gives (eq:null). The tensor $\delta (q)\otimes R(r)$ is nonzero when $R$ is nonzero: choose separated tests whose two pairings are nonzero.

A polynomial-degree splitting of presentations does not, merely by being defined on ${\mathcal{P}}_{\mathrm{int}}$, define a splitting on reconstructed kernels. Any proposed degree-based joint operation must satisfy (eq:descent).

Proof. The null presentation (eq:null) relates summands of different polynomial degrees whose reconstructions are nonzero. A rule on presentations that separates them must still respect their sum's zero reconstruction. Proposition prop:descent is exactly the condition ensuring that it does.

This qualification does not invalidate XIV-I's estimates on its named presentation module. It prevents those estimates from being relabelled, without further work, as an intrinsic transverse-degree theorem for the quantum target Jdescent,Jnull.

XIV-J supplies a reference only for (eq:supported) on $U\setminus N_U$. For a vertex component $D$, the source writes its normal form as

The insertion maps, coefficients, and trace operations are those of the source chart.

The copied wavefront bound is

This bound, including the source traceability argument for the cross-lines, is a copied input, not a new wavefront theorem proved by XV-A Jreference.

The reference is not a value on tests crossing $N_U$. It omits the characteristic joint compositions of $K_{47}$ and both $K_5$ placements. Consequently it is not the full map (eq:map).

The following local theorem makes the conditional extension interface precise and supplies the support qualification needed for its coefficient characterization.

Work in a product normal chart

with coordinates $(x,q)$. Tests and cutoffs are taken within this chart. For a multi-index $\gamma$, define

A distribution $t\in {\mathcal{D}}^{\prime }(U)$ has supported normal order at most $r$ along $S$ if it is supported in $S$ and annihilates every test $\varphi$ satisfying

This is the finite normal-jet condition used below.

A distribution has supported normal order at most $r$ if and only if it has a unique representation

Proof. Choose $\chi \in \mathcal{D}(V)$ equal to one near $0$ and set

For $|\beta |,|\gamma |\le r$,

If $t$ has the stated normal order, define

The lift is continuous on test-function spaces, so $b_{\gamma }$ is a distribution.

For $\varphi \in \mathcal{D}(U)$, put

The difference

has all normal jets through order $r$ equal to zero. Applying $t$ therefore gives

which is (eq:normal).

Conversely, each insertion in (eq:insert) is supported in $S$ and depends only on the indicated jet, proving the stated normal order. Testing (eq:normal) on ${\mathsf{J}}_{\gamma }\psi$ extracts $b_{\gamma }$ and proves uniqueness.

For a smooth multiplier $c$, the Leibniz rule gives

Indeed, apply the definition to $c\varphi$ and expand ${{\partial }_q}^{\alpha }(c\varphi )$. This checks the signs in the smooth version of (eq:reference). The distributional cross-line trace version remains the source theorem, with its own microlocal hypotheses.

Let $A\subset U$ be relatively closed and let $t^{\circ }\in {\mathcal{D}}^{\prime }(U\setminus A)$ have supported normal order at most $r$ along $S\setminus A$. Write its unique coefficients as

on that open set.

The following are equivalent:

1. There exists $t\in {\mathcal{D}}^{\prime }(U)$ supported in $S$, of normal order at most $r$, such that

   $$t{|}_{U\setminus A}=t^{\circ }.$$

2. Each ${b_{\gamma }}^{\circ }$ extends to a distribution $b_{\gamma }\in {\mathcal{D}}^{\prime }(S_0)$.

For any such coefficient extensions,

is the corresponding supported realization. The equivalence does not assert that the coefficient extensions exist or obey additional normalization, growth, or wavefront requirements.

Proof. If (i) holds, Proposition prop:normal extracts its unique coefficients. On the open part of $S$ outside $A$, uniqueness of normal-jet coefficients shows that their restrictions are ${b_{\gamma }}^{\circ }$. This argument is local on that open part and hence applies even if $A$ is not a product set.

If (ii) holds, the finite insertion sum is defined on $U$, is supported in $S$, and has normal order at most $r$. Its restriction has the prescribed coefficients and therefore equals $t^{\circ }$ by Proposition prop:normal. No step constructs the coefficient extensions or supplies further bounds on them.

An arbitrary extension of $t^{\circ }$ can contain a distribution supported in $A$ but outside $S$. Such a term need not have a response-normal-jet representation. Thus the coefficient equivalence is asserted here for supported realizations, as indicated by the source theorem's title and context. If “normal order” is used in another convention, this support condition must be retained explicitly.

For the XIV-J interface, $r=3$, $S=S_{\mathrm{resp}}$, and $A=N_U$ in the retained chart. The theorem clarifies the interface but supplies no characteristic coefficients for it Jconditional.

The coverage account distinguishes the codimension-five response incidence, the codimension-eight characteristic locus, and the relative codimension-three flag. Those labels are recorded as source bookkeeping, not independently derived dimensions. They do not replace the extension problem on $N_U$.

Suppose a wavefront condition $P$ is sufficient for the existence of a specified ordinary product. Failure of $P$ alone does not prove nonexistence of every product or generalized composition.

Proof. Sufficiency has the form $P\Rightarrow E$, where $E$ is the existence conclusion under the theorem's hypotheses. It does not imply $\neg P\Rightarrow \neg E$. A converse exclusion theorem would be an additional result.

This proposition does not authorize a particular alternative composition. It explains why an unresolved product gate is neither automatically constructed nor automatically obstructed.

Let $C$ be a closed collision locus and let $O$ be an open set disjoint from $C$. If $\mathrm{supp}c\subset C$, then $c{|}_O=0$. Consequently adding $c$ cannot change a distributional germ on $O$.

Proof. Every test supported in $O$ is supported away from $\mathrm{supp}c$ and therefore pairs to zero with $c$.

This is the limited reason collision contacts cannot, merely by being added, repair a missing characteristic operation on a neighborhood disjoint from those collisions.

The coverage account records further boundaries record:

1. an exact smooth-slot identity is not a joint-symbol theorem;

2. a common phase that fails clean or nondegenerate hypotheses does not become valid by being renamed;

3. a rank discrepancy is not automatically composition excess;

4. individual symbol descriptions do not define a joint product;

5. smooth residual testing can remove attachment-diagonal wavefront information;

6. a branch diagnostic is not a cancellation theorem for the entire six-branch sum.

The detailed phase, rank, and tangent calculations are not reproduced here. XV-A uses these as reported exclusions of particular shortcuts, not as newly proved universal obstructions.

An identity involving only an action and its differential does not by itself establish that a separate observable is closed under that differential.

Proof. The equation for the action contains no condition determining the differential of every other element. Algebraically, consider the graded polynomial algebra generated by an even element $x$ and an odd element $\xi$, with differential

Take the action element to be $S=0$ and the observable to be $O=x$. Then $dS=0$ but $dO=\xi \ne 0$. Thus closure of the action is not a sufficient abstract premise for closure of an independently specified observable. An additional observable theorem is required.

This elementary independence statement is not a new BV construction for the manuscript family.

The next analytic research target continues the XIV-J stop only. It must keep the inherited reconstructed kernels, complete branch content, grouped cross-lines, tensor coefficients, normalization, carrier, and ambiguity class. A change of any of these is a changed problem requiring a new signature and a new statement of dependencies.

The direct source ending is:

XIV-J constructs its supported-sector reference on its stated off-characteristic domain, but constructs no full map, joint section, or admitted quantum coefficient Jverdict,Jrelease,Jadmission.

The following table is a proposed order of research obligations, not a list of completed results and not a claim that the source's gate labels are identical to these six descriptive labels.

The complete-graph control required at stage 1 concerns the joint operation; stage 5 concerns the final output after the remaining section, compatibility, and output operations. Neither is supplied by an internal-line wavefront estimate.

The meanings of $Q_H$ and ${\mathcal{E}}_H$, and the normalization of the pushed-forward coefficient, are inherited interface data whose complete definitions were not supplied. XV-A consequently records the admission line without manufacturing a replacement output space or operator.

For this audit, “XV-B may open” means that the inherited admission conclusion

has been established for the required quantum coefficient, with the preceding construction and output obligations discharged. This is a project dependency contract, not a theorem that no other research project may be pursued.

The supplied XIV-H, XIV-I, and XIV-J releases do not establish (eq:admission). Under the exit contract, they do not open XV-B. This is not a nonexistence theorem for the coefficient.

Proof. XIV-H leaves the named uniform internal amplitude unresolved at H2. XIV-I subsequently constructs that amplitude at I2 but explicitly leaves the joint characteristic operator map unresolved at I3. XIV-J constructs the supported-sector reference only on $U\setminus N_U$, while explicitly leaving the full joint map unresolved at J2 and the subsequent gates undischarged.

Thus none of these supplied releases asserts the required full construction and admission. Indeed, the XIV-J no-admission corollary states exactly that release boundary. By the exit contract, a partial internal amplitude or partial reference is insufficient to open XV-B. No source result excludes every future construction in the frozen class, so nonexistence does not follow.

The theorem proves the audit consequence from copied release statements. It does not independently re-prove their analytic estimates or their account of every omitted calculation.

The following repairs are part of the final formulation.

1. Evidence coverage. The supplied relationship account is not promoted to a full-corpus verification. Direct excerpts and reported relationships are distinguished.

2. Truncated X-F order. The missing end of the X-F sentence is not guessed. Any future full-source revision must insert the exact coupling boundary and its theorem locator.

3. Convolution identity. The convolution witness is not identified with the ramp edge integral. Its obstruction remains bounded by its frozen algebra.

4. Historical versus current stop. H2 is the XIV-H stop. After I2 is supplied, the relevant open operation is I3, continued as J2 in XIV-J. Listing H2 does not mean it remains the first missing gate.

5. Reconstruction image. Descent determines a unique map on $\mathrm{im}\mathsf{Rec}$. Uniqueness on a larger ${\mathcal{K}}_{\mathrm{int}}$ requires that this carrier be exactly the reconstructed image or that additional extension data be given.

6. Supported conditional extension. The normal-jet coefficient equivalence explicitly requires support on the response incidence. It is not an equivalence for arbitrary extensions with additional support elsewhere on the characteristic locus.

7. Domain equality. The canonical-domain representation is not treated as proving an equality between $U_{\mathrm{can}}$ and an entire complement of $N$.

8. Degree bookkeeping. The null presentation qualifies the use of polynomial degree as intrinsic data. It does not invalidate the estimates on XIV-I's presentation module.

9. Graph content. $K_{47}$ already groups two branches. $K_{\mathrm{supp}}$ contains only $K_6+K_8+K_9$. Neither notation permits omission of either $K_5$ placement or of the regrouped $K_{47}$ target.

10. Regulator and limit. The internal Fourier damping represents an existing retarded line. It is not a graph-product regulator. The fixed-input scalar sharp-width limit is not joint quantum-graviton sharp admission.

11. Status language. “No coefficient is admitted in this release” is not rewritten as “no coefficient exists.” A failed sufficient product criterion is not a universal exclusion.

12. Algebraic carrier checks. An algebra generated by local functionals need not consist solely of local functionals: multiplying two single-integral functionals generally gives a double-integral expression. A proposed carrier must specify whether it includes the resulting multilocal elements.

13. Tangency and parity checks. If a proposed differential sends a compact antighost to an unrestricted auxiliary field, preservation of the declared compact-antighost domain needs proof. A single coupling convention across even and odd species also requires a generator-level graded-sign check. These are checks to perform, not verified defects of a manuscript whose complete differential was not supplied.

14. Registration language. The supplied account states that XV was not among the registered manuscripts. This document is the proposed XV-A audit. Its preparation neither changes that historical coverage statement nor establishes the admission needed for XV-B.

Relative to the supplied evidence record:

1. the convolution, ramp scalar/classical-metric, quantum-graviton, relative classical, and joint-extension entries have distinguishable model and domain signatures;

2. no supplied identification theorem transports all the necessary structures between those signatures;

3. the direct XIV-H–XIV-I–XIV-J excerpts distinguish defined internal kernels, a constructed internal amplitude, and a partial off-characteristic supported-sector reference from the unresolved full joint characteristic map;

4. reconstruction descent is necessary for any linear presentation-level construction to define a map on reconstructed kernels;

5. the conditional normal-jet interface constructs no coefficient extensions by itself;

6. the supplied releases do not establish the admission (eq:admission), and hence do not open XV-B under the exit contract.

Accordingly, the supplied record does not yet define a unified successor series with transferable results across these tracks. This conclusion is about the supplied definitions and proofs. It neither excludes a future unification theorem nor proves a universal obstruction to the unresolved analytic construction.

Proof. For (i), compare the recorded field content, regulator, gauge-fixing data, assembled observable, and domains in Table tab:ledger and Section sec:lanes. The convolution and ramp prescriptions have different regulated coordinate problems; the ramp quantum content keeps the metric classical; the graviton releases use distinct tree assemblies and gauge fixing; the relative branch restricts ghosts; and the analytic branch retains a separate full-shell quantum target and joint composition problem.

For (ii), the supplied evidence contains no bridge satisfying the operation and domain requirements of Proposition prop:transfer. This is a finite-record statement: it does not claim that no unprovided theorem exists.

For (iii), the cited H2, I2, I3, reference, J2, and release statements explicitly distinguish those objects and their domains. The tensor-product definition in Proposition prop:tensor also shows why existence of an internal kernel is a logically earlier operation than its joint cross-line composition.

Assertion (iv) is Proposition prop:descent. Assertion (v) is Theorem thm:conditional: its hypotheses ask for coefficient extensions and do not produce them. Assertion (vi) is Theorem thm:noadmission.

A unified transferable successor would require compatible signature identifications and the missing domain-sensitive operations. Neither is supplied by the audited record. The last sentence follows because an absent bridge or an unresolved construction is not an exclusion theorem.

If the joint-extension lane is selected, the next research manuscript should be described as:

A continuation of XIV-J addressing the full reconstruction- descending joint characteristic map for the frozen six-branch target.

Its initial deliverable is a construction on tests meeting $N$, or a precise obstruction in the unchanged class. It must address $K_{47}$, both $K_5$ placements, and the supported sector together. It must not rename the supported-sector reference as the full map or import XIII-E's unfinished limit.

The map is the next target; the remaining section, transverse, contact, output, and admission obligations must still be tracked. Until (eq:admission) is actually established, XV-B remains closed under the stated exit contract.

If instead the ramp or graviton lane is selected, the successor must declare that lane's own signature, order boundary, and carrier. It cannot claim to continue the XIV-J analytic stop merely by using related notation.

{{0.91} XV-A establishes a source-bounded model-and-domain audit and the elementary interface results proved above. It constructs no new joint characteristic product, admits no new quantum coefficient, and declares no interacting BRST/BV differential. The full joint map is unresolved in the supplied XIV-J release; the next research gate is that map, and XV-B is not opened. }

record

Supplied coverage account and proposed XV-A crosswalk. Retrieval corpus identifier 3883b893, retrieval 3. The account reports selected relationships and explicitly disclaims a complete corpus scan and independent verification. Used here only as supplied documentary evidence.

XA

Manuscript X-A, version 1. X-A.tex, identifier 31fc6ad1. Abstract, lines 112–147. Locator and scope account supplied; complete text not supplied.

XB

Manuscript X-B, version 3. X-B-v3.tex, identifier 103c6c32. “Field content before any gate,” lines 202–236. Locator and scope account supplied; complete text not supplied.

XC

Manuscript X-C, version 3. X-C-v3.tex, identifier 9dc463c0. “Frozen sheet and C1: the copied second vertex,” lines 488–587. Locator and scope account supplied; complete text not supplied.

XD

Manuscript X-D, version 3. X-D-v3.tex, identifier d0aeb7af. “Fixed-input X-D release,” lines 1445–1478. Locator and scope account supplied; complete text not supplied.

XE

Manuscript X-E, version 1. X-E.tex, identifier 0eb14066. “Release and the three endings,” lines 201–230. Locator and scope account supplied; complete text not supplied.

XF

Manuscript X-F, version 1. X-F.tex, identifier 4c1ccd1e. “F6: VII-C audit with three trees,” lines 1269–1318. Locator and scope account supplied; complete text not supplied. The coupling-order sentence in the proposed outline is truncated.

Hkernel

Manuscript XIV-H, version 1. XIV-H.tex, identifier 05d90ea1. Proposition prop:K5exact, “Exact two-placement kernel formula,” lines 1135–1202. Direct excerpt supplied.

Hverdict

Manuscript XIV-H, version 1. Theorem thm:H2, “Exact H2 verdict,” lines 1539–1582. Direct excerpt supplied. The release theorem at lines 1847–1895 is separately located in the supplied account.

Ipole

Manuscript XIV-I, version 1. XIV-I.tex, identifier 13365486. Proposition prop:pole, “Exact pole decomposition and chart matching,” lines 726–826. Direct excerpt supplied.

Iamp

Manuscript XIV-I, version 1. Theorem thm:I2, “The internal amplitude gate is Constructed,” and adjacent precomposition formulas and proposition, lines 1288–1378. Direct excerpt supplied; the final adjacent proposition excerpt is partial.

Ijoint

Manuscript XIV-I, version 1. Joint-map interface, Proposition prop:canonical, and Theorem thm:I3, lines 1459–1548. Direct excerpt supplied; the ending of the reproduced proof is partial.

Irelease

Manuscript XIV-I, version 1. Theorem thm:release, “XIV-I mathematical release,” lines 1729–1814. Direct excerpt supplied.

Jdescent

Manuscript XIV-J, version 1. XIV-J.tex, identifier 3c48e041. Proposition prop:descent, “Descent criterion for a coefficient-level construction,” lines 813–846. Direct excerpt supplied.

Jnull

Manuscript XIV-J, version 1. Lemma lem:null, “A normal identity invisible to polynomial-degree bookkeeping,” lines 853–883. Direct excerpt supplied.

Jreference

Manuscript XIV-J, version 1. Theorem thm:reference, “Supported-sector reference distribution,” lines 1215–1299. Direct excerpt supplied.

Jconditional

Manuscript XIV-J, version 1. Theorem thm:conditionalextension, “Conditional supported realization,” lines 1427–1470. Direct excerpt supplied. XV-A states the support condition explicitly in its independent normal-jet formulation.

Jverdict

Manuscript XIV-J, version 1. Theorem thm:J2, “Exact J2 verdict,” lines 1608–1649. Direct excerpt supplied.

Jadmission

Manuscript XIV-J, version 1. Corollary cor:noadmission, “No quantum admission in this release,” lines 1808–1822. Direct excerpt supplied. The admission-interface passage is located at lines 1794–1807 in the supplied account.

Jrelease

Manuscript XIV-J, version 1. Theorem thm:release, “XIV-J mathematical release,” lines 1894–1939. Direct excerpt supplied.