Landmark XVI-A Fixed-width retarded coefficient obligations Revised proof-obligation manuscript — coefficient status: unresolved

This note fixes the declared normalization and source notation for Landmark XVI-A, separates those declarations from predecessor results, and records the missing data required to formulate and discharge three retarded-slot wavefront obligations. It also corrects a potentially invalid inference: a changed source distribution need not determine a different free-shell class. The X-G variational obstruction and the XI-G frozen-class obstruction remain in force on their respective hypotheses. Standard distributional criteria are supplied as a conditional verification interface, not as estimates for the unidentified XVI-A kernels. No retarded coefficient is constructed, and no theorem resolving obligations 4–6 is claimed.

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

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

{ {0.92} Scope and release status. Throughout, the regulator width satisfies $\epsilon >0$ and is held fixed. Neither X-G nor XI-G is repaired here. The new retarded coefficient remains Unresolved. The displayed coefficient slots are schematic and are not an identity. A missing estimate is not a proof of obstruction. A verified estimate for one operation is not, by itself, construction of a retarded quantum product. }

The document identifier is XVI-A, corresponding to the supplied filename XVI-A.tex. The alternative spelling “X-VI-A” in the request is treated as an editorial spelling of this identifier, not as a separately established landmark.

The available material consists of the supplied XVI-A draft, selected predecessor excerpts, and a supplied coverage summary. It does not include the complete predecessor manuscripts or the later XVI sheets. Consequently, references to predecessor labels below identify the supplied passages; they do not certify a full-series consistency check.

We distinguish four kinds of statement:

1. Declared input: a relation retained from the supplied XVI-A draft. Retention is not an independent derivation.

2. Reported predecessor result: a result stated in a supplied predecessor excerpt or coverage summary, confined to its stated hypotheses.

3. Audit deduction: an elementary consequence whose justification is given here.

4. Open obligation: a proposed analytic task for which the defining data or proof has not been supplied.

A named source dependency, conceptual connection, or inherited interface does not transfer every theorem of the named manuscript. In particular, a free Green operator or free nilpotence statement does not discharge an interacting quantum admission obligation.

The following relations are retained exactly at the level of the supplied draft. No new coefficient, source formula, or wavefront estimate is inferred from them:

The action–equation sign convention is

These formulas remain in the inherited integral notation. No unprovided measure, boundary condition, tensor contraction, or integration-by-parts prescription is silently reconstructed.

The register specifies the roles of the symbols, not unprovided local formulas.

The notation $J_D$ and $D_{\sigma }$ is used consistently. No identification with ${\mathcal{J}}_D$ or ${\mathcal{D}}_{\sigma }$ in unprovided manuscripts is assumed. The symbol $D_{\sigma }$ denotes the correction in (eq:preparedsource); it is not shorthand for applying $D_{\nu }$ to ${\sigma }_{\epsilon }$.

In particular, the displayed action and source relations do not constitute a proof that varying the complete prepared action yields $J_D$. That assertion would require the complete action, the definition of ${\ell }_3$, the switch dependence, and the variational conventions.

The following exclusions remain part of the declared interface:

• Full Einstein–Hilbert weighting is not substituted for (eq:preparedaction); the draft excludes it in order to retain the selected XI-B free operator.

• The X-sheet inverse $-4P{E}_0^R$ is not used. The X-F normalization reported in the supplied material is separately written $-4\mathrm{Ptr}\otimes {E_0}^R$ XF. No identification of $P$ with $\mathrm{Ptr}$ is needed here.

• The factors $-1$ and $-4$ are not interchangeable. A transfer between the corresponding conventions would require an explicit normalization map.

The supplied X-G material reports an obstruction to an exact source-preserving variational upgrade and the strictly positive classical witness

The symbols in this witness belong to the X-G data and hypotheses XG. They are not silently identified with the XVI-A preparation. The strict inequality is retained as a reported predecessor fact, not derived from unspecified assumptions on $M_{\chi }$, $\zeta$, or $\sigma$.

deduction. Corrections beginning at strictly positive formal powers of $\hslash$ cannot cancel this nonzero classical coefficient.

. For arbitrary formal coefficients $a_n$,

because its constant coefficient is $W_{\mathrm{XG}}\ne 0$. Equality of formal series is coefficientwise. This argument concerns formal corrections of order at least $\hslash$; it makes no assertion about numerical cancellation at a chosen nonzero numerical value of $\hslash$.

Changing the action may change the model or the hypotheses of the obstruction. Such a change is not a source-preserving upgrade of the original model merely because it is proposed later. XVI-A supplies neither a counterexample to X-G nor a proof of the excluded upgrade.

The supplied XI-G excerpt defines a free-shell class of a compact cubic kernel $u$ on $M^3$, with two symmetric-tensor slots and one scalar slot XIG. Let $\mathcal{K}$ denote the vector space of compact kernels with this slot structure and tensor-slot symmetry. Write

Here $P_m$ is the scalar free operator used in that excerpt. Compact support allows the displayed pairing with smooth solutions.

The class is

The reported XI-G result is

for the frozen data and hypotheses of its obstruction theorem XIG. The symbol ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ denotes the predecessor's microcausal polynomial target space; no new admission theorem for that space is assumed.

deduction: the exact transfer condition. If a future prepared observable kernel $u_D\in \mathcal{K}$ satisfies

then it belongs to the same class and inherits the class obstruction:

. The set ${\mathcal{I}}_{\mathrm{sol}}$ is a vector subspace by linearity of the pairings. If $u_D-u\in {\mathcal{I}}_{\mathrm{sol}}$, then

Equation (eq:XIGobstruction) then gives (eq:sameclasstransfer).

A nonzero difference $u_D-u$ does not exclude (eq:sameclasscondition). Thus the sentence “the distributions are different, so XI-G does not govern the new distribution” is not justified without a free-shell comparison.

There is a further distinction: $J_D$ is a source label, whereas $u$ in (eq:XIGobstruction) is a compact observable kernel on $M^3$. The supplied material does not define the map from $J_D$ to a corresponding kernel $u_D$. It is therefore not legitimate to identify these objects merely because both are described as distributions.

The corrected conclusion is:

XI-G remains valid for its frozen class. Its applicability to a kernel induced by $J_D$ has not been established or excluded. A same-class proof would transfer the obstruction. A different-class proof would remove that particular transfer, but would not prove admission of the new class.

Even if a kernel induced by ${\sigma }_{\epsilon }{J}_3$ were admitted to some declared class, that would not establish admission of the kernel induced by ${\sigma }_{\epsilon }{J}_3+D_{\sigma }$. The correction and the total kernel must be analyzed.

The supplied XI-F excerpt contains a cubic transformation law with both a gauge-fixing companion and an Einstein-equation contribution. The supplied XI-G coverage also states that its witness lies on the gauge-fixed shell and off the Einstein shell XIF,XIG.

Accordingly:

• the gauge companion is retained;

• the Einstein-shell contribution is retained;

• a gauge-fixed free-shell condition is not used as a substitute for an Einstein-shell hypothesis;

• a transformation law for the unramped response is not asserted for the prepared response without proof.

The relevant prepared terms have not been supplied as explicit kernels. Retaining them is a requirement on a future derivation, not a claim that they have already been constructed.

The supplied target display is retained as a list of visible slots:

Equation (eq:schematic) is schematic, not an identity. It is not assigned a coefficient name, because doing so could misrepresent an undefined sum as a constructed distribution.

The symbol roles are:

• $F_f$: the baseline tested-observable label;

• $O_1,O_2$: correction labels in the proposed observable bookkeeping;

• $V_1,V_2$: interaction-argument labels in that bookkeeping;

• $R_1,R_2$: first and second retarded-product labels, not defined multilinear maps in this note.

No identification of $V_1$ or $V_2$ with a particular density or action is made. No coupling expansion, $\hslash$ expansion, factorial convention, or renormalization prescription is inferred beyond the literal factor $\frac{1}{2}$ in the supplied display.

For tracking purposes only, the obligations are indexed by

This is an editorial correspondence, not a proof that the source $J_D$ induces these slots.

The local slot $O_2$ is also unconstructed in the supplied material. Discharging obligations 4–6 would not silently settle its status or prove (eq:schematic) to be an expansion identity.

This section states general distributional tests. It does not provide a wavefront estimate for an XVI-A kernel.

Let $X,Y,Z$ be smooth manifolds. Fix compatible smooth density conventions. For $v\in {\mathcal{D}}^{\prime }(X)$, $\mathrm{WF}(v)$ is a conic subset of $T^{\ast }X\setminus 0$. Tensor-valued kernels can be checked in local smooth frames.

For a smooth map $a:X\to Y$, define

For $v\in {\mathcal{D}}^{\prime }(Y)$, a standard sufficient condition for $a^{\ast }v$ is

Under this condition,

See BDHoperations (Proposition 5.1).

For $v,w\in {\mathcal{D}}^{\prime }(X)$, define $-\mathrm{WF}(w)=\{(x,-\xi ):(x,\xi )\in \mathrm{WF}(w)\}$. The standard product criterion is

It suffices for the product $vw$. If

then

See BDHintro (Theorem 13). Pairing two distributions requires such a product criterion, or another explicitly justified construction, together with the appropriate support conditions.

For the projection $\pi :Y\times Z\to Y$, fiber integration of $h\in {\mathcal{D}}^{\prime }(Y\times Z)$ is defined when $\pi {|}_{\mathrm{supp}h}$ is proper. Its wavefront bound is

See BDHoperations (Theorem 6.3). Properness is a support requirement, not a wavefront transversality requirement.

These are sufficient tests for the specified operations. An intersection in an unverified upper bound does not establish an intersection in the actual wavefront set. Failure of a sufficient test does not, by itself, prove that no alternative construction exists BDHintro.

Nor do these tests alone select extensions across singular diagonals or normalize retarded quantum products. Such extension and normalization problems are separate in perturbative constructions BDF.

Suppose a future slot is given by a finite ordered recipe whose inputs are specified distributions. If every operation in that recipe is justified, every integration has the required support control, and any extension step is separately constructed, the recipe defines an output distribution.

The verification proceeds in the order of the recipe: an operation is applied only after its inputs have been defined and its hypotheses checked. Its output estimate is then available for the next operation. This finite induction does not supply the missing input kernels, prove a required final cone condition, or establish a quantum admission theorem.

If the chosen switch is smooth at fixed $\epsilon$, then smooth multiplication is defined and does not enlarge wavefront sets:

The wavefront set of a sum is contained in the union of the wavefront sets of its summands:

These general facts are discussed in BDHintro.

They do not identify $\mathrm{WF}(D_{\sigma })$ or the wavefront sets of the observable kernels derived from $J_D$. In particular, (eq:smoothmult) is not a statement that ${\sigma }_{\epsilon }{J}_3$ is smooth.

deduction. On an open set where a smooth multiplier $a$ is nowhere zero,

after restriction to that open set.

. One inclusion follows from smooth multiplication. Locally, $1/a$ is smooth and $v=(1/a)(av)$, which gives the reverse inclusion.

Thus switching does not remove a singularity at a point where the switch is nonzero merely by being smooth. Possible cancellation in the total source must be proved for the actual distributions; the upper bound (eq:sumbound) neither proves nor excludes such cancellation.

All statements here are at fixed $\epsilon$. No uniform estimate as $\epsilon \downarrow 0$, and no sharp-width limit, is asserted.

An obligation is not mathematically testable until the relevant objects and operations are identified. For each $j\in \{4,5,6\}$, a future proof must provide the following packet.

1. Exact objects. Give the distributions or kernels to be estimated, including their dependence on $J_D$, their tensor or graded slots, symmetry, density conventions, and relevant formal orders. Names such as “the required kernel” are insufficient.

2. Spaces and maps. Give every input and output manifold, bundle, test space, pullback map, diagonal embedding, and projection used by the slot.

3. Operation recipe. Specify the ordered contractions, products, pullbacks, pairings, and integrations. State whether any kernel is initially defined only away from a diagonal.

4. Exact covector statements. State the actual wavefront inclusions to be proved. For each operation, identify the corresponding normal-set, opposite-covector, or projection test. Specify the final allowed cone rather than referring to “the usual estimate.”

5. Support and extension. Prove the support condition for each integration and the declared retarded support condition. Construct any needed extension and state its normalization. Compact tests do not automatically prove every internal projection to be proper.

6. Terms retained. Show how the full correction $D_{\sigma }$ enters. Retain the gauge companion and the Einstein-shell term whenever the relevant derivation produces them. A cancellation may be used only after it has been proved.

7. Parameters and scope. State that $\epsilon$ is fixed. Specify any other parameter dependence and any topology needed for a claimed functional or parameter-family result. Pointwise estimates are not silently promoted to uniform estimates or algebraic closure.

8. Discharge statement. State exactly which operation and output estimate the proof establishes. Distinguish that result from construction, normalization, covariance, variational compatibility, and quantum admission.

The supplied material does not contain a completed packet for any of obligations 4–6. Accordingly, no slot-specific covector estimate is fabricated here.

The obligations below preserve the original numbering. The indexing is that of (eq:index4)–(eq:index6).

Specify the kernel realization of $R_1(V_1;O_1)$ associated with the prepared source $J_D$, and complete the packet of Section sec:packet for that realization.

Prove the identified wavefront estimates and support conditions needed for every operation in that slot, with $\epsilon$ fixed. State the final output distribution and its required cone bound.

Present gap. The kernel realization, its input and output manifolds, its maps, and its slot-specific discharge criterion have not been supplied.

Status: Unresolved.

Stop. No first retarded slot is constructed by this note.

Specify the kernel realization of $R_1(V_2;F_f)$ associated with $J_D$, and complete the packet of Section sec:packet.

Retain the gauge companion in the complete kernel derivation. Prove the identified wavefront estimates and support conditions for the full expression, with $\epsilon$ fixed. Do not replace the complete expression by one obtained by setting the gauge companion to zero.

Present gap. Neither the complete prepared kernel nor the estimate controlling its gauge-companion contribution has been supplied.

Status: Unresolved.

Stop. No second retarded slot is constructed by this note.

Specify the kernel realization of $\frac{1}{2}{R}_2(V_1,V_1;F_f)$ associated with $J_D$, and complete the packet of Section sec:packet.

Retain the Einstein-shell term in the complete derivation. Prove the identified estimates for repeated contractions, pullbacks, pairings, integrations, and any required diagonal extensions, with $\epsilon$ fixed. Do not substitute a gauge-fixed free-shell condition for an unproved Einstein-shell hypothesis.

Present gap. The quadratic kernel, its repeated-operation recipe, its extension data, and its slot-specific discharge criterion have not been supplied.

Status: Unresolved.

Stop. No quadratic retarded slot is constructed by this note.

Absence of any listed estimate leaves the corresponding obligation unresolved. It is not a proof of nonexistence. Conversely, a later proof of one estimate discharges only the task identified by that proof.

The ledger records the present state of specification and proof. “Not specified” is a defect to be supplied, not a completed discharge criterion.

{}1.15

The generic criteria in Section sec:interface do not fill these cells. An actual kernel and recipe are needed before one can determine which criteria apply.

Even a complete microlocal packet for each retarded slot would not, without further proof:

• construct the local slot $O_2$;

• establish the schematic display as a coefficient identity;

• select or construct renormalized retarded products;

• establish an interacting BRST or BV statement;

• establish covariance or variational compatibility of $J_D$;

• provide a state, positivity result, loop construction, higher-order construction, or sharp-width limit;

• repair the X-G or XI-G obstruction.

A subsequent manuscript seeking an actual coefficient theorem must supply, at minimum:

1. the complete prepared action and its variational conventions, including the formula for ${\ell }_3$;

2. explicit definitions of $J_3$ and $D_{\sigma }$, and a proof relating their sum to that action if such a relation is claimed;

3. the normalized free operator, its field bundles, source domains, and Green kernels;

4. definitions of $F_f$, $O_1$, $O_2$, $V_1$, $V_2$, and the proposed retarded maps;

5. the full gauge-companion and Einstein-shell contributions appearing in the prepared derivation;

6. the compact observable kernel induced by $J_D$, together with its free-shell comparison to the frozen XI-G kernel;

7. completed specification packets and proofs for obligations 4–6;

8. the extension, normalization, and admission results needed to turn the schematic bookkeeping into a coefficient theorem.

Neither the provenance note nor the general analytic interface is a substitute for these materials.

The following corrections are incorporated:

1. Notation is fixed. $J_D$ and $D_{\sigma }$ are used throughout; no unverified calligraphic aliases are introduced.

2. Normalizations remain separate. The selected inverse is (eq:inverse), and the X-sheet factor $-4$ is excluded.

3. The sign is unchanged. The convention remains $r=-u_D$.

4. Class transfer is stated correctly. Different distributions can represent the same free-shell class. The exact conditional transfer argument is given in Section sec:class.

5. Missing estimates are not obstructions. Generic sufficient criteria are not asserted for unspecified kernels.

6. No term is silently deleted. The gauge companion and Einstein-shell term remain explicit requirements.

7. The schematic is not promoted to an identity. No coefficient name or equality is attached to (eq:schematic).

8. Compilation claims are limited. The source is arranged as one pdfLaTeX document with an internal bibliography. A successful compiler execution is not claimed.

{ {0.92} Final release statement. X-G and XI-G remain in force on their respective hypotheses. Their applicability to a changed model or changed observable class must be checked, not assumed. Obligations 4–6 remain open. The retarded coefficient is Unresolved. No coefficient construction, interacting quantum admission, or sharp-width limit is claimed.

The manuscript is complete as an audited statement of scope, conditional verification criteria, and missing proof data. It is not a completed proof of the proposed retarded coefficient. No theorem resolving the XVI-A target is claimed. }

Draft

Landmark XVI-A. User-supplied draft, filename XVI-A.tex. Source of the frozen declarations, the sign convention, obligations 4–6, and the schematic slot display. No external publication status is asserted.

XIB

Manuscript XI-B, version 1, XI-B.tex, supplied identifier 862627cd. Supplied excerpts: “Free-multiplet release,” lines 205–225; “Off-shell nilpotence,” lines 383–396; “Free causal polynomial algebra,” lines 743–762. The associated supplied coverage summary is used only within its stated scope.

XIF

Manuscript XI-F, version 2, XI-F-v2.tex, supplied identifier ef5fb4a8. Supplied excerpt: “The cubic response obeys the complete second-order law,” lines 954–980. This reference does not assert the corresponding law for the prepared XVI-A response.

XIG

Manuscript XI-G, version 1, XI-G.tex, supplied identifier 3cae2be9. Supplied excerpts: “Free-shell class,” lines 494–507; “Separation and the admission criterion,” lines 954–987; “The free-shell class of the cubic kernel is obstructed,” lines 1107–1134. The obstruction is used only for the frozen class and hypotheses reported there.

XF

Manuscript X-F, version 1, X-F.tex, supplied identifier 4c1ccd1e. Supplied coverage summary and source locators: field content and second-order metric, lines 450–486; fixed-regulator release, lines 1425–1469. The complete manuscript was not supplied.

XG

Manuscript X-G, version 1, X-G.tex, supplied identifier d362d49d. Supplied coverage summary of the source-preserving variational obstruction and its positive classical witness. Source locators include the exact-upgrade definition, lines 245–261, and the qualified release, lines 1841–1885. The full proof was not supplied.

BDHintro

C. Brouder, N. V. Dang, and F. Hélein, A smooth introduction to the wavefront set, arXiv:1404.1778 [math-ph] (2014). In particular, Theorem 13 and the discussion of fundamental wavefront-set properties.

BDHoperations

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), arXiv:1409.7662v2 [math.FA] (2016). In particular, Proposition 5.1 and Theorems 6.1 and 6.3.

BDF

R. Brunetti, M. Dütsch, and K. Fredenhagen, Perturbative Algebraic Quantum Field Theory and the Renormalization Groups, arXiv:0901.2038v2 [math-ph] (2009). Used for the distinction between distributional operations and the separate extension and normalization problems of perturbative quantum products.