Landmark XI-D Admission of the fixed-width graviton tree and one ghost-number-zero identity: the admission interface and a free-order closure obstruction

Landmark XI-C constructed a fixed-width classical graviton-tree symbol, not an admitted quantum insertion. We formulate the additional admission interface without replacing XI-B's finite-generator algebra by an unnamed Wick algebra. An essential distinction is made between exact off-shell representatives and classical symbols of an algebra that already imposes the free equations. Requiring the latter to reproduce every linear generator on all off-shell configurations is inconsistent.

A separate calculation decides a necessary condition for the proposed closure identity. The bare dressing ${\mathcal{B}}_f=-\int f{\mathsf{X}}_1[\gamma ]\cdot \partial \varphi$ has ghost number zero, but its free variation is generally nonzero. An explicit witness satisfies all the free multiplet equations and uses a ghost with admissible, unrestricted free Cauchy data. Consequently, for the observation tests in that witness, no admission respecting the classical free differential can make this bare insertion closed by adding only positive coupling-order corrections. The inherited second-order held-fixed-metric defect is also evaluated directly.

No quantum admission map is constructed, and no general nonexistence theorem for an appropriately formulated quantum extension is claimed. The overall ending is therefore Unresolved, with a proved conditional obstruction to the particular bare-insertion closure request. No antibracket, interacting differential, state, mixed loop, cubic Einstein contraction, or sharp-width limit is introduced.

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

Figure fig:scope-XI-D records that separation. The drawing adds no hypothesis.

and legal ending{ Overall ending: Unresolved. The quantum extension, its product, and its admission map are not constructed. The classical tree remains the XI-C symbol.

Two precise results are proved: an exact off-shell-shadow interpretation of the unchanged free-shell algebra is incompatible with its equations; and the bare insertion fails a necessary free-order closure condition for explicit admissible observation tests.

The latter is a conditional obstruction to closure, not a theorem excluding admission alone. No quantum class is released.}

The proposed implication is

None of its arrows is supplied merely by the preceding arrow. In particular, ghost number zero and a compact microcausal kernel are necessary bookkeeping facts, not closure theorems.

The production sheet permits three endings.

1. Constructed. A named quantum extension admits the required symbols, a differential is defined and proved nilpotent through the declared order, and the admitted insertion is closed.

2. Proved obstructed. A stated interface is excluded by an explicit incompatibility. Its quantifiers must identify whether admission, nilpotence, or closure is excluded.

3. Unresolved. An indispensable extension, product, antibracket, or identity remains unconstructed.

Here admission remains unresolved, so the overall ending is the third. Nevertheless, one can prove a necessary-condition obstruction without pretending that an interacting differential has been constructed. Such a result excludes a specified closure request if admission respects its required classical limit.

This distinction matters because the production sheet asks for the closure of the bare dressing ${\mathcal{B}}_f$. That is not the same as closure of a complete perturbative observable whose leading term is $\int f\varphi$. The two coefficient problems have different possible cancellations. Section sec:freeclosure makes this difference explicit.

The standard separation between regular polynomial generators, microcausal functionals, and their quantization is also relevant FR. It does not itself supply the extension required here. Likewise, a BV construction requires more than names for antifields Rejzner. We use neither reference as a substitute for a missing construction.

Retain the supplied conventions:

All coefficient fields are smooth on a neighbourhood of the closed slab. Parentheses on indices include the factor $1/2$.

Fix $0<\epsilon <T$ and

Fix

and an open relatively compact $U$ with

The observation test is a real $f\in {C_c}^{\infty }({M_T}^{\circ })$. The width is never sent to zero.

For a symmetric tensor $r$ define

The independent multiplet is $(\gamma ,\varphi ,c,\bar{c},b)$ with parities $(0,0,1,1,0)$ and ghost numbers $(0,0,1,-1,0)$.

XI-B constructs the algebra ${\mathcal{A}}_0$ generated by compact interior linear smearings and their finite products, with the free equations and graded causal commutation relations imposed XIB. Its reduced metric operator and inverse are

The scalar equation inverse is ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$, whereas its action-Hessian inverse is $-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$.

The copied free differential is

It is nilpotent with $b$ retained and preserves XI-B's free equations and causal relations. These facts do not admit a new composite.

Write ${\mathcal{C}}_{\mathrm{sol}}$ for smooth multiplet configurations satisfying

These equations are equivalent to XI-B's metric–auxiliary equations after using $b=D\gamma$. Ghost Cauchy data are unrestricted. They are not required to vanish at $\Sigma$.

The coordinate operator is

where

The bare insertion and the classical tree are

with

The physical one-graviton response of ${\mathcal{B}}_f$ is $-\mathcal{T}$, at coupling degree ${\kappa }^2$.

XI-C proves that these are compact distributional polynomial symbols satisfying its microcausal exclusion XIC. It does not construct their quantum admission. Its mixed insertion kernel is not smooth finite-rank. The noncharacteristic diagonal symbol has a nonzero $00$ component, with the stated numerator $-1/2$ at $p=\mathrm{d}t$.

Above the ramp, for the named input

the copied response is

XI-A's empty Einstein jet remains empty XIA. That source obstruction is not a quantum-admission obstruction.

X-A supplies plateau covariance of the classical coordinate and switched scalar prescriptions XA. It does not supply quantum closure. X-D is not used.

VII-C's four extension requirements remain unmet: its unchanged Hadamard Wick embedding, exact sharp admission, specified normal-topology convergence, and the multiplication/coefficient continuity package. We neither import its covariance matrix nor claim that the fixed-width symbols meet those separate interfaces.

Let ${\mathcal{P}}_{\mathrm{fin}}$ be the off-shell polynomial algebra of finite products of compact smooth linear smearings. Its degree-$n$ kernels are finite sums of smooth tensor products.

Let ${\mathcal{P}}_{\mathrm{mc}}$ be the classical space declared in XI-C: finite polynomials with compact distributional kernels $u_n$ obeying

where the causal cones include zero covectors and the total zero covector is excluded from a wavefront set. Include graded field slots when discussing free variations.

Then

are classical input statements. Neither ${\mathcal{P}}_{\mathrm{mc}}[[\hslash ]]$ nor its notation defines a quantum product.

For a possible on-shell formulation define

Here equality on free configurations is understood in the graded sense, including evaluations in finite Grassmann algebras. We do not identify ${\mathcal{N}}_{\mathrm{sol}}$ with a generated equation ideal without a separate proof.

A prospective admission consists of:

1. a unital quantum algebra $\mathcal{E}$ over $\mathbb{C}[[\hslash ,\kappa ]]$ and an injective homomorphism $j:{\mathcal{A}}_0[[\kappa ]]\to \mathcal{E}$;

2. a classical-symbol interpretation of the admitted elements on ${\mathcal{C}}_{\mathrm{sol}}$, compatible with the old finite-generator symbols;

3. lifts $Q({\mathcal{B}}_f)$ and $Q(\mathcal{T})$ with leading symbols $[{\mathcal{B}}_f]$ and $[\mathcal{T}]$ in ${{\mathcal{P}}_{\mathrm{mc}}}^{\mathrm{sol}}$;

4. agreement with the old inclusion whenever the symbol is represented by an old finite product;

5. an explicitly stated topology and continuity requirement if admission is to be obtained by kernel limits.

This definition names required data; it does not assert their existence.

An off-shell version must instead start with an off-shell quantum presentation and subsequently specify its equation quotient. Calling the same algebra simultaneously an unchanged free-shell algebra and an exact algebra of functions on all off-shell configurations is not a permissible shortcut.

There is no classical-symbol map on an extension of ${\mathcal{A}}_0$ which both preserves its zero elements and sends every old scalar linear generator $\varphi (u)$ to the off-shell functional $\int u\varphi$ on all smooth scalar configurations.

Proof. Take real $w\in {C_c}^{\infty }({M_T}^{\circ })$ with $P_{m}w\ne 0$. Such tests exist. The equation relation in ${\mathcal{A}}_0$ is

Every algebra extension preserves this equality, and every symbol map preserving zero must assign it the zero functional.

The required off-shell interpretation would instead assign

Evaluate at the smooth configuration $\varphi =P_{m}w$. Its value is

This contradicts the zero relation.

This proposition has deliberately narrow quantifiers. It excludes an exact off-shell shadow of all old generators. It does not exclude an off-shell representative chosen before an on-shell quotient, nor admission formulated as Definition def:interface.

Thus the phrase “$Q({\mathcal{B}}_f)={\mathcal{B}}_f$ as a classical functional” must specify its configuration space. An exact off-shell representative is meaningful; it cannot be the unrestricted classical-symbol interpretation of XI-B's already-quotiented algebra.

A possible source topology can be specified on kernel stages. For compact $K_n$ and a closed cone ${\Gamma }_n$ satisfying (eq:mc), use compactly supported distributions with $\mathrm{WF}(u_n)\subset {\Gamma }_n$ and the normal distribution topology. Besides strong distribution seminorms, its microlocal seminorms have the form

with $(\mathrm{supp}\psi \times V)\cap {\Gamma }_n=\varnothing$. For the insertion, the flat ramp permits the kernel to be viewed as a compact distribution on the surrounding Minkowski space, including its possible closed support at $t=0$.

This names a source-side continuity question. It does not name a quantum target topology or prove continuity of $Q$. The distributional operation results of BDH concern specified wavefront spaces and specified operations; they are not an automatic completion theorem for a quantum algebra.

Likewise, the causal commutator on linear fields does not uniquely define multiplication of all distributional polynomial kernels. The formal causal-Weyl expansion would require repeated contractions. Even for local quadratic symbols, its second contraction involves products whose usual wavefront multiplication criterion is not satisfied. That observation calls for additional product data; it is not a universal nonexistence theorem.

No state is required merely to ask this admission question. Conversely, the absence of a selected state supplies neither the missing product nor an obstruction to every possible state-independent extension.

The finite-rank mismatch proves that XI-B's existing operations alone do not perform admission. Proposition prop:shadow settles an inconsistent off-shell interpretation. Neither fact constructs the on-shell admission of Definition def:interface, and neither excludes every such admission.

Accordingly, D1–D2 end at Unresolved. This is the governing release status.

Since admission has not been constructed, no antibracket on admitted quantum composites is defined here. XI-B's antifields remain external slots. In particular, there is no claimed Jacobi identity for an unnamed bracket.

There is also no constructed $s_1$ or $s_2$. Saying that $s_1$ is “the variation of $V_{1,g}$” leaves an essential ambiguity. In a canonical BV description, an antifield-free interaction changes the antifield equations through its Hamiltonian vector field; it does not, by itself, specify new transformations of the fields. The necessary antifield-dependent terms and their master identity must be supplied separately.

The same issue applies more strongly to the observable-dependent tree $\mathcal{T}(f,g)$. It is an output associated with an observation test. It is not automatically an interaction Hamiltonian defining one universal differential.

Consequently no assertion of interacting nilpotence is made. The gauge-fixed bosonic action's breaking, the auxiliary–ghost completion, and the invariant action's Noether identity remain distinct. Omitting the Einstein cubic vertex is still a scope restriction, not a proof that the missing terms of a master identity vanish.

The next section only tests a necessary leading coefficient of the proposed closure equation. It does not bypass these gates.

The bare insertion has ghost number zero. Its copied free variation, defined on smooth graded configurations, is

It has ghost number one. Moreover,

as a classical identity.

Proof. The coordinate operator is continuous and linear on smooth tensors at fixed width. Apply the chain rule to (eq:B), using (eq:sfree). The scalar has zero free variation. This gives (eq:sB) and the stated ghost numbers. The tree depends only on $\varphi$, so (eq:sT) follows.

The integrals exist by smooth Cauchy theory and compact observation support. This is a classical calculation, not application of an unconstructed quantum derivation to an admitted element.

For a free ghost, $D\mathcal{R}c=\frac{1}{2}\square c=0$, so

The ramp layer therefore remains visible to free ghosts with nontrivial Cauchy data.

For every $m\ge 0$ and every $0<\epsilon <T$, there are admissible nonnegative observation tests $f$ and a smooth free multiplet configuration for which

In particular, this variation is not generally a functional vanishing on all free configurations.

Proof. Work over the Grassmann algebra $\mathbb{C}[\vartheta ]/({\vartheta }^2)$ with $\vartheta$ odd and nonzero. Set

Every equation in (eq:freeshell) holds. The ghost is an affine wave solution, and no ghost initial-data restriction in XI-B excludes it.

Here

so

The spatially constant zero-data wave response is therefore

The spatial components vanish. Ramp symmetry gives

It follows that

where $\cos (mt)=1$ when $m=0$.

Choose $t_{\ast }\in (\epsilon ,T)$ with $\cos (m{t}_{\ast })\ne 0$, and a small time interval on which it has a fixed nonzero sign. Choose $f\ge 0$, nonzero, compactly supported in that interval and in a spatial ball. Since $S_{\epsilon }>0$ there, the ordinary coefficient of $\vartheta$ in (eq:witness) is nonzero.

The switch and plateau may be chosen to contain this observation region while satisfying the frozen support requirements. They do not enter (eq:witness).

The theorem supplies explicit allowed tests; it does not assert nonvanishing for every possible fixed $f$. For a particular test, (eq:sB) is the necessary condition to examine. A smearing can annihilate a nonzero field.

Noncompact smooth ghost data are legitimate here, just as in XI-B. If compact ghost Cauchy data are desired, choose them equal to the affine data on a ball containing the initial causal shadow of $\mathrm{supp}f$. Finite propagation makes the resulting ghost identical to the displayed one throughout the dependence region, leaving (eq:witness) unchanged.

Fix an observation test from Theorem thm:freewitness. Suppose a future admission has the following properties:

1. the leading classical symbol of $Q({\mathcal{B}}_f)$ is ${\mathcal{B}}_f{|}_{{\mathcal{C}}_{\mathrm{sol}}}$;

2. the leading differential on that symbol is the copied free action (eq:sB);

3. all further corrections have nonnegative formal powers of $\hslash$ and strictly positive powers of $\kappa$ in the differential beyond $s_0$.

Then it cannot satisfy

through coupling order ${\kappa }^2$ and quantum order ${\hslash }^0$. The same conclusion holds for the physically weighted bare dressing $\kappa Q({\mathcal{B}}_f)$ through coupling order ${\kappa }^2$.

Proof. For the unweighted insertion, the coefficient at ${\hslash }^0{\kappa }^0$ of the proposed identity is $s_0{\mathcal{B}}_f{|}_{{\mathcal{C}}_{\mathrm{sol}}}=0$. Theorem thm:freewitness contradicts it.

For the weighted insertion, the first coefficient is instead ${\hslash }^0{\kappa }^1$, and it is the same nonzero variation. Positive-order corrections to the differential cannot alter this coefficient. Nor can an order-${\kappa }^2$ tree correction alter it; its copied free variation is zero by (eq:sT).

This theorem excludes every prospective admission combined with these closure interfaces for the witness tests. It does not exclude admission alone. It also does not construct an interacting differential or claim a failure of its nilpotence.

Let $F_f=\int f\varphi$. For a complete candidate beginning $F_f+\kappa {\mathcal{B}}_f$, the first closure coefficient could involve both

Theorem thm:conditional does not exclude such a cancellation. The production sheet's equation for ${\mathcal{B}}_f$ alone has no leading $F_f$ term on which that correction can act.

There is a further Cauchy-data distinction. For a compact plateau parameter $c=2\xi$, X-A's preflight gives $L_{\epsilon }\mathcal{R}c=c/2$. A proposed scalar co-transformation proportional to $c\cdot \partial \varphi /2$ can then participate in a complete observable's coefficient cancellation.

The free witness above is not such a plateau parameter. Its response is $S_{\epsilon }(t)$, not $t$, because the coordinate data are frozen while the free ghost has nonzero initial time derivative. X-A never asserted covariance for those ghost data. Thus neither its classical theorem nor ghost number bookkeeping proves the requested bare closure.

The preceding obstruction is at free order. It must not be confused with XI-C's second-order background defect. We retain the latter and reconstruct its classical notation from X-A, rather than assuming that an unsupplied quantum coefficient exists.

For a smooth tensor $r$ put

For a free scalar $\varphi$ define

This is the scalar-only first coefficient of X-A.

Let $H$ and $k$ be the first and second metric coefficients. Use X-A's density coefficient $a_2[H]$ and define

Let ${\mathsf{V}}_{2,g}[H,k]$ be X-A's switched second scalar operator and set

The completed classical coefficient is

All these expressions are classical evaluations on actual smooth configurations.

For every smooth $r$,

Proof. The change in ${\mathsf{X}}_2$ is ${\mathsf{X}}_1[r]$. The change in ${\rho }_2$ is $-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{V}}_{1,g}[r]\varphi$. No other displayed term depends on $k$.

Now set $H=h+\gamma$ and

Under the admitted classical transformation

hold $\gamma$ fixed. Full transformation of $H$ would require the additional second coefficient ${\mathcal{L}}_{\xi }\gamma$. X-A's complete classical intertwiner and Lemma lem:affine therefore give

The tree has no external $h,k,\gamma$ argument and contributes no variation under this operation. The backward difference has the opposite sign.

Take $m=0$ and the free configurations

They satisfy the reduced metric and scalar wave equations. Set $b=D\gamma$, with ghosts and antighost zero.

Choose nonnegative nonzero $a\in {C_c}^{\infty }((\epsilon ,T))$ and, on a sufficiently large causal cylinder, $\xi =a(t){\partial }_t$. Multiply it by a spatial cutoff whose derivatives lie outside the observation dependence region. Choose the plateau to contain its compact support, and observe after the support of $a$. Finite propagation permits the following calculation in that cylinder.

For $r={\mathcal{L}}_{\xi }\gamma$,

There is no ramp-layer contribution. Direct zero-data integration gives

Also,

Consequently

for nonnegative nonzero observation tests in the stated region.

This is an evaluation on a free multiplet configuration. The defect is therefore not generally an equation zero. A cancellation of (eq:forward) must supply the omitted co-transformation, not replace ${\mathcal{L}}_{\xi }\gamma$ by $\mathcal{R}c$. Those transformations have different dependence on the metric.

Equation (eq:backgroundwitness) concerns the completed second-order classical candidate. Equation (eq:witness) concerns the free variation of the bare insertion. Neither is a quantum Ward identity, and neither can replace the other.

reduction. Replace the metric by a prescribed classical background $r$ before taking the scalar response. Its first response is $-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{V}}_{1,g}[r]\varphi$, and its completed second coefficient is (eq:A2). Setting that background to zero gives the zero specialization of those response maps. Setting an external quantum metric configuration to zero after the graviton response does not remove $\mathcal{T}$, which has no such external argument.

gauge. The correct identities are

whereas

need not vanish. A pure-gauge background is not a vanishing test for a tree sourced by scalar stress.

empty Einstein jet. The response $r_g$ and its divergence exist. Above the ramp,

for a free scalar. The named input gives a nonzero source. The tree propagates this de Donder violation; it does not set it to zero or restore the ungauge-fixed Einstein equation.

loop. No scalar Hadamard bisolution, covariance matrix, or retarded quantum product is selected. No mixed-loop expression is multiplied, extended, or subtracted. Missing loop data are an unresolved stop, not a no-go theorem for the classical tree.

gravitational interaction. No cubic Einstein vertex is contracted. Its classical role in nonlinear identities has not disappeared. A future interacting master identity must account for the appropriate gravitational, scalar, ghost, and antifield terms.

and width. The initial coordinate data remain frozen at $\Sigma$. No boundary algebra, boundary charge, reflected-residue subtraction, or $\epsilon \to 0$ limit is constructed.

The overall ending is Unresolved, because D2 has not been constructed or excluded in its consistent on-shell formulation. The paper does not promote a classical symbol to a quantum class.

At the same time, the bare-closure request has a concrete lower-order problem: for the diagnostic tests, an admission with the correct free classical action cannot make ${\mathcal{B}}_f$ closed by positive-order coupling corrections. This conclusion is independent of a chosen state and precedes a loop calculation.

A successor therefore receives three distinct tasks:

1. construct a quantum admission with an explicit distinction between off-shell representatives and on-shell symbols;

2. define the antibracket, antifield-dependent interaction, and differential on a named domain;

3. identify the complete observable whose coefficient closure is intended, instead of requiring closure of its bare dressing alone.

It also receives both evaluated witnesses. Neither may be erased by relabelling a background pullback as the free ghost rule.

The justified release is

A constructed quantum class would require an additional admission theorem and a different, fully specified closure problem.

scope. The off-shell-shadow incompatibility, free-shell witness, necessary-condition obstruction, and classical defect evaluation are proved in the displayed conventions. The quantum admission interface is formulated but not solved.

. The supplied X-A is used for classical Cauchy prescriptions and plateau intertwiners. XI-A, XI-B, and XI-C supply the signed action, free multiplet, limited algebraic domain, and classical tree. The scalar completion is written explicitly, so an unsupplied X-B-v3 or X-C-v3 theorem is not required as an additional premise.

quantum certification. No admitted composite, interacting BRST cohomology class, state, renormalized retarded product, or loop extension is claimed.

. No successful pdflatex execution, proof-assistant check, executed symbolic computation, independent referee review, or journal acceptance is asserted. All nonzero witnesses are analytic.

XA

N. Cheung, {Landmark X-A: Classical covariance of one fixed-regulator switched relational scalar through order ${\kappa }^2$}. Supplied manuscript X-A.tex. Used for the classical prescription, Cauchy facts, and complete plateau intertwiners.

XIA

N. Cheung, Landmark XI-A: A switched Einstein–scalar action for the ramp relational scalar: sign conventions, Noether identities, and an empty fixed-input Einstein jet. Supplied manuscript XI-A.tex. Used for the signed action and Einstein-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. Used for the free equations, causal algebra, and differential.

XIC

N. Cheung, Landmark XI-C: One fixed-regulator graviton tree for the ramp relational scalar: the retarded symbol, its normalization, and the boundary of the free-algebra handoff. Supplied manuscript XI-C.tex. Used for the classical tree, kernel bounds, sign, and qualified admission handoff.

FR

K. Fredenhagen and K. Rejzner, Perturbative algebraic quantum field theory. arXiv:1208.1428v2, 2013. Background for the distinction between regular polynomial generators, microcausal functionals, and quantum products.

Rejzner

K. Rejzner, Batalin–Vilkovisky formalism in locally covariant field theory. Ph.D. thesis, arXiv:1111.5130v2, 2013. Background for separating antifield slots, BV structures, and quantum differential constructions; no such construction is imported.

BDH

C. Brouder, N. V. Dang, and F. Hélein, “Continuity of the fundamental operations on distributions having a specified wave front set (with a counter example by Semyon Alesker),” Studia Mathematica 232 (2016), 201–226. arXiv:1409.7662. Used only to identify the separate normal-topology distributional continuity question; not as a quantum-completion theorem.