XV-F-patch Definitional completions for the frozen joint graph Residual species, correction roles, the amplitude module, and the joint-map clauses: a scope-corrected source patch

We patch the source manifest supplied as XV-F-vav3.md, using only the full supplied texts of XIV-G.tex, XIII-C.tex, XIV-I.tex, and XIV-J.tex. The local amplitude generators, chart-function coefficient class, reconstruction rule, formal target, and five joint-map clauses are now reproduced rather than paraphrased. A polarized stress convention is fixed from the displayed metric–scalar vertex.

Two proposed completions require a narrower release. The metric residual tests in XIV-G do not identify an output bundle over the common graph space. Likewise, generic formulas for a graph differential operator, external weight, and spectator coefficient do not recover their frozen graphwise values. We therefore distinguish source definitions, definitions introduced by the patch, and parameterized formulas. The polynomial weight convention is not attributed to XIII-C, and the normal contact bound six is not used as a bound on the graph differential operator.

The displayed reconstruction is coefficientwise $C^{\infty }$-linear on the transcribed local module. Its chart-function relations are genuine local null presentations; they do not determine the complete six-entry reconstruction kernel. XIV-J's global descent criterion is transcribed, while canonical agreement verifies only its restricted version. The canonical-domain complement equality remains refused.

The four documentary patch ledgers are completed within these boundaries. The stronger assertion that every corresponding frozen-object gap passes XV-F's original F0 is not established. No preliminary lift, null-defect calculation, characteristic prescription, new analytic estimate, or admission of $Q_H{R}_1^{(1)}$ is supplied. Gates F1–F4 remain unentered.

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

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

release{ Completed: the four owned transcription and convention ledgers, including the local amplitude definition, the displayed target, the full joint-map clauses, and the chosen stress polarization.

Explicitly parameterized, not recovered: the residual output fiber and the graphwise correction coefficients.

Still missing: a fully identified frozen codomain, a complete presentation package, the complete reconstruction kernel, the actual presentation-level correction space, and the remaining inherited coefficient definitions.

Ending: the owned documentary patch is complete to its stated scope; the actual graph remains Unresolved at F0. This is not “lift defined and descended.”}

The proposed patch asks whether four source sheets can be completed, definitionally, so that XV-F's Proposition prop:F0 no longer fails on the gaps those sheets own. The question has two distinct readings.

The first is documentary. A formula which is present in a supplied full text, but was previously available only as a paraphrase, can now be reproduced. Its source notation, coefficient class, and scope can be inspected directly. This is the appropriate reading for XIV-I's amplitude definition and XIV-J's target and map-class definition.

The second is mathematical. A role such as “the specified graph differential operator” might be replaced by a generic formula for a finite differential operator. That replacement gives a mathematical template, but it does not identify the specified operator. Similarly, a species name does not by itself construct a vector bundle, and a named spectator does not determine its distributional coefficient. The second reading therefore requires information beyond the existence of a displayed syntactic form.

The patch must not conflate these readings. Its central distinction is between completing a documentary ledger and completing the frozen-object package needed for a lift. The former is accomplished below. The latter is not claimed merely because every row has acquired a formula or an explicit boundary statement.

This distinction is necessary for a proof-ready release. If an unknown frozen coefficient is replaced by an arbitrary parameter and the resulting expression is called the actual coefficient, the model has been changed without a theorem. If instead the parameter remains exposed, the expression is a well-defined family, and its status can be stated honestly. A source patch is useful precisely when it makes that difference visible.

The mathematical manuscript inputs are the full supplied texts of XIV-G, XIII-C, XIV-I, and XIV-J XIVG,XIIIC,XIVI,XIVJ. The supplied XV-F manifest is the record being corrected XVF. It is not treated as an independent verification of all claims in the manuscripts it describes.

The supplied file is named XV-F-vav3.md. The accompanying retrieval account also reports a registration under XV-F-v3.tex. This patch does not assert that those two files are byte-identical or that their line numbers coincide. References to XV-F's labels refer to the supplied manifest and its reported registration, not to an independently inspected missing source file.

XIV-D, XIV-E, XIV-H, XII-A, XII-B, and XI-B are not direct inputs. Their names may occur in copied handoffs, in source bibliographies, or in the list of future dependencies. That occurrence does not import their full contents. A premise explicitly copied into one of the four supplied sheets is usable with the scope of that copied premise; an uncopied predecessor formula is not recovered by a citation to its owner.

The coverage account is documentary context only. In particular, a declared continuation path is not a theorem identifying every object appearing along that path. The patch does not use comparison tracks, classical master-equation statements, or earlier contact certificates to discharge a distributional composition obligation.

Reported line numbers are retained as documentary locators. They have not been independently recounted against a version-controlled source archive. Source labels and the supplied passage contents are therefore reported together. Where an exact label has no separately supplied line range, that absence is stated rather than repaired by an invented number.

The word defines is qualified throughout:

A source definition and a patch convention are not interchangeable. The latter can remove an ambiguity in the notation of this paper, but it cannot retrospectively prove that the source used the same convention.

We also use

A template can define a family perfectly well while remaining insufficient to identify the member frozen in the predecessor. Thus $\mathsf{T}$ is not a weaker kind of ${\mathsf{D}}_{\mathrm{s}}$ which can be promoted by formatting. It records a different mathematical object.

A refusal is also not a definition of an alternative object. The refusal of a particular complement equality closes that proposed identification. It does not provide a set equation for the canonical domain, and it does not prohibit a later independent description of that domain.

The proposed gates are retained, with the distinction between documentary completion and frozen-object identification made explicit.

Under the original stronger reading of P0, every owned core slot would have to be an actual definition or a genuine source refusal. That stronger pass is not established: residual output fibers and frozen graph coefficients are not supplied by the permitted texts. The present P0 is a completed audit, not an assertion that those missing objects have appeared.

The stop rule is accordingly exact. When a completion would require a non-input or an unidentified parameter, the patch stops at that item. It may display the dependence on the missing parameter, but it may not choose the parameter and call the choice inherited. No analytic gate is opened by this documentary work.

The following ledger records ownership without transferring theorems between namespaces. The action column has only the three operations specified in the proposed plan: transcribe, define from stated roles, and leave open. The status of a result of “define from stated roles” is determined separately; it can be a patch convention or a parameterized formula rather than a recovered source object.

{@{}P{.22} P{.15} P{.25} P{.29}@{}} and permitted patch actions.
Object & Owner & XV-F record & Patch action
Object & Owner & XV-F record & Patch action
$W$ and $N$ & XIV-G; copied local geometry & Dimensions named; characteristic incidence incompletely recorded in the manifest & Transcribe the common space and the selected locus with their copied scope.
Smooth metric residual test $G$ & XIV-G & Residual testing discussed; output fiber absent & Transcribe its actual domain and tensor type.
$E_{\mathrm{res}}$ & XV-F codomain slot & Unidentified & Define from stated roles a parameterized species-bundle notation; leave open its identification with the actual output fiber.
Collision sets ${\Delta }_I$ & XIII-C & Transcribed in graph variables & Transcribe their definition and choose local relative coordinates without asserting a lift to $W$.
$D_{\Gamma },w_{\Gamma },S_{\Gamma /I}$ & XIII-C roles & Roles, not actual coefficient formulas & Define from stated roles the parameterized contact syntax; leave open the frozen graphwise parameters.
${\mathfrak{C}}_{\mathrm{part}}$ & XIII-C & Class of contact values & Transcribe; do not identify it with a space of maps on presentations.
$T_0$ and the vertex & XIII-C & Stress displayed; polarization ambiguous & Transcribe; define from stated roles the unique symmetric polarization used in this patch.
Adapted charts and ${{\mathfrak{A}}_{\mathrm{R}}}^m(V)$ & XIV-I & Chart and rule transcribed; generators and coefficient class paraphrased & Transcribe all three parts of the local definition.
Local reconstruction relations & XIV-I rule; XIV-J null identity & One relation known; chart-function action incompletely recorded & Transcribe the null identity and unpack the coefficientwise module action.
Complete ${\mathcal{P}}_{\mathrm{int}}$, ${\mathcal{K}}_{\mathrm{int}}$, and $\ker \mathsf{Rec}$ & XIV-J common presentation interface & Incomplete enumeration and kernel & Leave open the complete six-entry specification; do not substitute one local amplitude module.
$R$, $R\ast R$, and kernel support & XIV-I copied normalization & Displays available; a cross-sheet support identification requested & Transcribe the displays and their internal kernel identification; do not transfer to an external namespace.
$K_8$ and $K_9$ & Copied in XIV-I and XIV-J & Displays available; coefficient ingredients incomplete & {Transcribe

} the displays; leave open their unrecovered coefficient ingredients.
$m^{\circ }$ and the target & XIV-J & Target paraphrased & Transcribe the contraction recipe on its proved canonical domain.
Grouped blocks & XIV-J & Replacement symbol disputed & Transcribe $H_T$ and $H$; do not replace $H_T$ by a hatted block or an admissible Hadamard replacement.
Joint-map class & XIV-J & Four clauses paraphrased & Transcribe all five actual clauses, including the complete-graph wavefront requirement.
Descent criterion & XIV-J & Necessary interface & Transcribe the global criterion and the limitation of its verification by canonical agreement.
$U_{\mathrm{can}}$ & XIV-I canonical proposition & Named domain; complement equality refused & Transcribe the named domain and retain the refusal.
Presentation-level $\mathcal{A}$ & XV-F interface; not supplied by XIII-C & Unavailable & Leave open. Contact values do not determine a correction map on every presentation.

Three non-transfers deserve emphasis. First, XIV-G remains unresolved at G2. Its source differential symbol is not the joint principal symbol of a composed response. Second, XIV-J's descent criterion is a global algebraic criterion; only the information obtained from agreement on $U_{\mathrm{can}}$ is restricted to that domain. Third, XIII-C's class remains a class of contact changes in graph coefficients. It is not, without another statement, a vector space $\mathcal{A}\subseteq \mathrm{Hom}({\mathcal{P}}_{\mathrm{int}},Y)$.

These distinctions are not merely terminological. They concern different domains, different codomains, and different identifications. A kernel can be tested against a smooth residual direction without having a pointwise slice. A coefficient change can be supported on a collision without specifying its value on every null presentation. A formal target can be written without defining the characteristic multiplication occurring in that target.

No row in Table tab:nontransfer asserts that an XIV theorem proves an unstated XV identity. Where the patch adopts a convention, the adoption is visible. Where the source has a formula, the formula is reproduced. Where the source has only a role, that role remains a parameter until its actual member is recovered.

{Defines/source statement. XIV-G.tex; thm:release(i),(iii),(iv); reported lines 1710–1751; detailed source typing also appears in prop:sourceoperator.}

For the frozen retained instance, the chosen common space and characteristic test locus are $W$ and $N={\tilde{\mathcal{N}}}_{\mathrm{att}}$, of dimensions sixteen and eight, respectively. For a smooth residual test $G$, the full differentiated source ${\mathcal{J}}_{\mathrm{D}}^{\prime }[G]$ is a differential operator of order at most two. Smooth residual testing can hide nonzero attachment-diagonal wavefront directions even for compact balanced kernels.

XIV-G supplies no complete internal symbol relation, joint section, controlled remainder, or characteristic prescription for its grouped target. Its ending remains Unresolved at G2.

The common space is

where the four factors are open spacetime neighborhoods. The supplied XIV-G text copies the retained characteristic configuration in the form

This is the selected retained locus, with its copied neighborhood restrictions. It is not the assertion that every characteristic configuration of the global selected pair is represented by this local formula.

The dimensions have the stated bookkeeping interpretation: $a$ contributes four parameters, $\mathbf{n}$ contributes two, and $\alpha ,\beta$ contribute two. The attachment equality determines $z$. Thus the copied local locus has eight parameters in the sixteen-dimensional common space. No transverse degree follows from this count, and no amplitude order is assigned to a future extension by subtracting the dimensions.

XIV-G's tested source uses

Its local operator representation is

The coefficients are smooth and supported in $\mathrm{supp}G$. These are the source's local differential-operator statements. They retain the actual source coefficients and ramp derivatives within that copied operator statement.

The test in (eq:Gtest) is a metric direction on $Z$. It is not a compact test section on all of $W$. Pulling it back by the projection to $Z$ gives a smooth object depending only on $z$, but does not generally give compact support in $a,b,y$. Moreover, the mixed residual output also retains a scalar slot at $y$. The metric direction alone therefore cannot be declared to be the full test space for the output coefficient.

This matters for duality. A vector-valued distribution with values in a bundle $E$ pairs with compact sections of the dual bundle, with an appropriate density. It does not, without a chosen fiber pairing, pair canonically with sections of the same bundle. A metric on $E$ could identify those two spaces, but no such metric on an output bundle is supplied by XIV-G.

The input metric bundle itself is not rank-mysterious. In four spacetime dimensions a symmetric covariant two-tensor has ten components. What is not identified in XIV-G is a separate bundle called $E_{\mathrm{res}}$ which packages the output residual species with the required value/test duality. The two questions must not be confused.

The phrase residual species in this patch records the field slots left after the specified contractions. In the retained mixed sector those slots are the metric slot at $z$ and the scalar slot at $y$, as recorded in the supplied graph and joint-map statements.

Let ${\mathbb{B}}_{\mathrm{res}}$ be a finite-dimensional complex value space supplied as an additional parameter. Define

and use the flat coordinate density on $W$ to set

For every specified ${\mathbb{B}}_{\mathrm{res}}$, this is a named vector-valued distribution space. The fiber rank is $\dim {\mathbb{B}}_{\mathrm{res}}$.

This convention is introduced here. XIV-G does not identify ${\mathbb{B}}_{\mathrm{res}}$, its rank, a fiber metric, or an output symbol for a bundle named $E_{\mathrm{res}}$. Accordingly, the unparameterized notation $E_{\mathrm{res}}$ is only a species placeholder until such an identification is supplied. It is not an unidentified bundle magically made concrete by writing an arrow to $W$.

The metric tests (eq:Gtest) are part of the inherited testing interface. They are not asserted to be exactly ${C_c}^{\infty }(W;{\mathbb{B}}_{\mathrm{res}}^{\ast })$.

Definition def:Eres is a genuine definition of a family. For each parameter it identifies the total space, projection, dual test space, and scalar pairing. It does not choose the parameter which the frozen joint-map target uses. This is the strongest codomain completion justified without silently inserting an output fiber convention.

One could choose an explicit tensor-product value bundle for the mixed residual slots and compute its rank. That would be a legitimate additional mathematical convention. It would also have to specify the covariant or contravariant value convention, its density, and its identification with the target's contraction indices. This patch does not make that extra choice under the label “from XIV-G.”

The reason for leaving the parameter visible is not that a printed integer is intrinsically necessary for defining a distribution space. A vector bundle defined by explicit bundle operations is an object even if its rank is not separately printed. The missing point here is the identification of the actual output fiber and pairing, not the typography of a dimension statement.

The testing limitation in Input statement input:G remains in force. Smooth testing can remove attachment-diagonal wavefront directions from a tested kernel even when those directions are present in the joint kernel. The source illustrates this with a compactly weighted diagonal delta whose residual test becomes a smooth function.

The present patch neither enlarges that example into a model of the characteristic target nor derives a new estimate from it. Its role is negative and precise: regularity after every smooth metric test cannot be used as a substitute for an untested joint composition theorem. A source symbol of order two is a polynomial on the local operator relation. It is not the symbol of the gravitational response after propagation, and it is not the principal symbol of the full grouped cross-line composition.

Likewise, a definition of $E_{\mathrm{res}}[{\mathbb{B}}_{\mathrm{res}}]$ adds no output wavefront bound. Bundle typing specifies what a value would be if constructed. It does not construct that value, control its characteristic directions, or make it microcausal after pushforward.

Patch G reproduces the common-space and residual-testing statements. It defines a parameterized species-bundle codomain. It does not identify the frozen output fiber, and it does not enter G3–G5 or any of XV-F's analytic gates.

Proof. Equations (eq:W)–(eq:Goperator) reproduce the supplied common-space and testing interface. Equations (eq:Eres)–(eq:vectorD) define a family for a specified parameter. No supplied statement identifies that parameter with an output fiber. The source's unresolved ending is retained in Input statement input:G; none of the definitions here adds a composition or wavefront theorem.

Thus P1 passes as a named-species and conditional-codomain interface. It does not pass as an identification of the actual frozen bundle. This limitation is an owned boundary, not a nonexistence statement about possible residual bundles.

{Defines in the source. XIII-C.tex; sec:class, Collision sets; reported lines 338–349; the definition is unlabelled in the supplied full text.}

For each inventory graph, let $y$ be its interaction vertex and let $x,u,v$ denote its displayed observable vertices when present. For a connected vertex subset $I$ containing $y$, with $|I|\ge 2$,

Intersections and loci obtained after contracting such subsets are included. Diagonals involving only a pre-existing observable subgraph retain their inherited normalization.

The connected subsets are those associated with XIII-C's cycle inventory. They live in the graph variables of that inventory. Nothing in Definition def:collisions identifies every graph variable with one of $a,b,z,y$. In particular, a formula for a collision in a graph with internal vertices is not automatically a formula for its lift or pushforward locus in $W$.

{Defines in the source. XIII-C.tex; eq:vertex; reported lines 416–421.}

The supplied interaction and stress are

Here $\eta =\mathrm{diag}(1,-1,-1,-1)$, consistently with the four supplied sheets. The vertex is a functional, not an Euler operator. Its mixed derivative will be computed algebraically in Patch J; no additional vertex from a non-input is used.

{Defines in the source. XIII-C.tex; def:Cpart, eq:partial-counterterm; reported lines 358–392; the shorter range 358–385 does not include the whole closing qualification.}

The class ${\mathfrak{C}}_{\mathrm{part}}$ consists of finite sums of graphwise contact changes at order ${\hslash }^2$ in the time-ordered product, hence at order $\lambda \hslash$ in the selected retarded expansion. Before pushforward they have the form

Here:

1. $r_I$ is a system of relative coordinates transverse to ${\Delta }_I$;

2. $S_{\Gamma /I}$ is an inherited spectator graph coefficient, with the two remaining field slots retained;

3. $w_{\Gamma }$ is built from $g,\partial g,{\sigma }_{\epsilon },{\sigma }_{\epsilon }^{\prime },{\sigma }_{\epsilon }^{\prime \prime }$ and the smooth external graph weights already present;

4. $D_{\Gamma }$ is the specified finite graph differential operator;

5. the change is even, of ghost number zero and field degree two, and is supported in the existing compact graph closure;

6. it leaves the tree datum and the inherited observable-subgraph normalizations unchanged.

Only well-defined products and pushforwards are members. The bound six is a conservative frozen bound; an established scaling degree can impose a smaller bound on a particular locus. The total-diagonal subclass is ${\mathfrak{C}}_{\mathrm{tot}}\subseteq {\mathfrak{C}}_{\mathrm{part}}$.

The source additionally warns that a spectator can remain singular in its tangential variables. Smoothness concerns the external weights, not the entire spectator distribution. It also warns that differentiating a diagonal distribution can differentiate its multiplying weights upon pushforward. This is not permission to add independent arbitrary higher-jet coefficient functions.

These qualifications are retained. In particular, a polynomial weight convention introduced below is not inferred from the words “built from.” Those words do not specify a polynomial degree, a coefficient field beyond the ordinary constants, or a complete list of admissible graphwise combinations.

Let a graph have vertex-coordinate space $X_{\Gamma }$ and fix $I=\{i_0,i_1,\dots ,i_k\}$. In a coordinate neighborhood choose

Let $s_I$ consist of $x_{i_0}$ and the coordinates of vertices outside $I$. Then

The inverse recovers $x_{i_j}=x_{i_0}+r_{I,j}$ and leaves the other vertices unchanged. This is an explicit coordinate completion of role (i), not a new collision prescription.

The codimension is $4k=4(|I|-1)$ in the graph coordinate space. Coordinate-density conversions, if needed for a different local chart, are smooth coefficient factors. They do not alter which collision is being represented and do not identify this graph codimension with the codimension-eight locus $N$ in $W$.

The following formulas expose every parameter that the source role leaves unspecified.

Fix a graph and a collision chart (eq:collisionchart). Choose the following data.

First, choose a finite-order differential operator

Its actual order and coefficient functions are parameters. The bound $|\alpha |\le 6$ in (eq:Cpartsource) does not bound $d_{\Gamma }$.

Second, list the fixed smooth weight generators already present for that graph, denoted

They are placeholders for inherited external weights, not freely chosen additional functions. Include the graphwise evaluations of the source-named functions $g,\partial g,{\sigma }_{\epsilon },{\sigma }_{\epsilon }^{\prime },{\sigma }_{\epsilon }^{\prime \prime }$ in this list. No formula for ${\sigma }_{\epsilon }$ is introduced.

For the polynomial realization convention of this patch, choose a finite set $F_{\Gamma }\subset {\mathbb{N}}^{{\ell }_{\Gamma }}$ and constant coefficients $a_{\Gamma ,\kappa }$, and define

This is an explicit polynomial family. The source does not state that this family equals its entire admissible weight class.

Third, let

be the spectator parameter. Its name records the contracted graph and retained slots. Equation (eq:Stemplate) does not determine the inherited spectator distribution.

For $|\alpha |\le 6$, define the local template value

Only choices satisfying the source's parity, ghost-number, field-degree, support, normalization, product, and pushforward requirements are candidates for membership in ${\mathfrak{C}}_{\mathrm{part}}$.

The tensor product in (eq:contacttemplate) is meaningful for a singular tangential spectator: it combines independent $s_I$ and $r_I$ variables. Smooth multiplication and the displayed finite differential operator are then meaningful local distributional operations. This supplies a coherent local formula without pretending that a singular spectator is a smooth function.

A spectator whose inherited realization requires further products or pushforwards is not recovered merely by this local notation. Those products remain subject to the source membership condition. Nor is a coordinate expression on one collision chart a proof of compatibility on intersections of several collision charts.

Suppose the parameters in Definition def:contacttemplate are chosen so that the seed in (eq:contacttemplate) is supported in the existing compact graph closure ${\mathcal{K}}_{\mathrm{gr}}$. Then

A well-defined pushforward has support in the image of that set. If this value is used as a contact change at order ${\hslash }^2$ in the time-ordered product, it does not alter the selected tree coefficient.

Proof. The tensor factor ${\partial }^{\alpha }\delta (r_I)$ is supported on $r_I=0$. Multiplication by a smooth function and application of a finite differential operator do not enlarge distributional support. The support hypothesis supplies the compact graph closure. A pushforward, when defined, has support contained in the image of the original support.

For the last assertion, a change $\delta {\mathsf{T}}_2={\hslash }^{2}C$ changes the retarded expression $(i/\hslash )({\mathsf{T}}_2-{\mathcal{V}}_g{\star }_H\mathcal{O})$ by $i\hslash C$. It has no coefficient at ${\hslash }^0$. The inherited tree datum is consequently unchanged.

The compact support in this lemma is a hypothesis inherited from the class, not a consequence of writing a polynomial in smooth weights. A polynomial can have noncompact support. The patch therefore does not insert a new cutoff to manufacture membership. Similarly, preservation of an observable-subgraph normalization is a retained admissibility requirement, not a consequence of differentiation alone.

Equations (eq:Dtemplate)–(eq:contacttemplate) define a parameterized local contact realization. They do not identify the actual frozen $D_{\Gamma }$, $w_{\Gamma }$, or $S_{\Gamma /I}$. They prove neither exhaustiveness of the frozen class nor an identity between that class and a presentation-level correction space $\mathcal{A}$.

Proof. The source roles specify that there is a graph operator, an admissible external weight, and an inherited spectator. They do not supply the parameters in (eq:Dtemplate), (eq:wtemplate), and (eq:Stemplate). A family indexed by unspecified parameters does not identify one prescribed member.

Exhaustiveness would require a theorem that every permitted frozen change is represented by the displayed parameter family, together with a statement of its admissible parameters. Neither is supplied. Finally, the values in (eq:contacttemplate) are distributions. A map in $\mathcal{A}$ would require a value for each presentation, and possibly additional linearity, continuity, covariance, and tensor restrictions. No such assignment follows from a class of values.

P2 is therefore complete as an explicit role-formula ledger. It does not pass the stronger frozen-parameter recovery condition. Calling the displayed parameters “specified” would not change this conclusion. The actual values must be supplied.

A notation correction is needed before transcription. The supplied XIV-I full text defines ${{\mathfrak{A}}_{\mathrm{R}}}^m(V)$, not a module written ${\mathcal{A}}^m(V)$. The latter notation occurs in the XV-F retrieval account. This patch uses the actual source notation and does not postulate an additional equality between two differently named modules.

{Defines in the source. XIV-I.tex; sec:amplitudespace, Actual coordinates for the two placements; reported lines 828–855; the final chart is reported at lines 852–855.}

The two placements use

The maps to $(a,d_{+},r_{+},y)$ and $(z,d_{-},r_{-},y)$ are diffeomorphisms onto their images. On the retained nonzero future cone patches, $(q_{\pm },{\zeta }_{\pm })$ are ordinary regular coordinates for $d_{\pm }$. The residual separation $r_{\pm }$ is retained rather than tested away.

The dimension is $4+3+1+4+4=16$. The charts are local adapted charts on their stated domains. They are not an identification of a local chart with every point of the global graph space, and they do not transport the collision inventory of XIII-C into $W$ without further maps.

{Defines in the source. XIV-I.tex; input:convolution, eq:R; reported display lines 329–335; the preceding subsection identifies $R$ as the kernel of ${E_0}^{\mathrm{R}}$.}

Within the copied XIV-I namespace, $R$ is the kernel of ${E_0}^{\mathrm{R}}$, and the displayed normalization is

The source records $\square {E_0}^{\mathrm{R}}=\mathrm{id}$. The convolution display is a premise concerning existing internal kernels, not a new product prescription for the quantum graph.

Consequently, the support of this copied kernel is not a missing sentence:

Indeed its cone formula has the measure pairing

It vanishes off the forward cone boundary and is nonzero on every open neighborhood meeting that boundary. This elementary support reading introduces no estimate and no uniqueness theorem.

The identification is internal to the supplied XIV-I statement. It does not identify the same printed symbol in an excluded manuscript. Thus a request for the support of the copied kernel is resolved; a request for a separate cross-sheet identification can still be open.

{Defines in the source. XIV-I.tex; eq:generators, def:amodule, eq:amodule, eq:reconstruction; the registered definition is reported at lines 873–901; the generator display is adjacent to that definition.}

Use the source Fourier convention

The inverse transform has factor $(2\pi {)}^{-4}$. Let $M_{\mathrm{R}}$ denote the Fourier transform of the existing kernel $R$. Its source pole representation is

No new choice of boundary value is made here.

For an adapted chart $V$ and an integer $m\ge -2$, define

The module ${{\mathfrak{A}}_{\mathrm{R}}}^m(V)$ consists of finite expressions

with the finite inherited tensor components understood. The coefficients carry their usual compact-set smooth seminorms. The frequency factors are distributions, not ordinary smooth symbols at the characteristic pole.

The module is filtered by the polynomial degree $j+|\beta |$. Its reconstruction is

Normal inverse Fourier transformation is performed on each tensor factor, and coefficient multiplication is performed afterward.

The integer $m$ in this definition is the amplitude-filtration parameter. It is not the scalar mass appearing in (eq:vertex). The source uses the same letter in different contexts; the distinction is maintained here.

The coefficient condition is $C^{\infty }(V)$, with $w=(u,\zeta ,q,r,y)$. It permits dependence on $q$ and $r$. It does not require such dependence for every coefficient, but it does not exclude it. There is no later restriction in the supplied definition to coefficients independent of these coordinates.

Thus the proposed free-family interpretation over chart functions cannot be retained. Physical coefficient multiplication takes place after inverse transformation. It can annihilate or relate normal distributions even though the corresponding frequency expressions have distinct polynomial degrees.

Let

The displayed coefficient module has the action $(hc{)}_{j\beta }=h{c}_{j\beta }$.

For the transcribed local coefficient module,

Consequently the local reconstruction kernel is a $C^{\infty }(V)$-submodule.

Proof. Substitute $(hc{)}_{j\beta }=h{c}_{j\beta }$ in (eq:reconstruction). Associativity of multiplication of a distribution by smooth functions gives

If the second sum is zero, multiplication by $h$ remains zero.

XIV-I does not state this as a separate $C^{\infty }$-linearity theorem. That bibliographic observation is true, but it does not make (eq:modulelinearity) mathematically undecidable for the displayed coefficientwise rule. The rule and action already imply it.

For an enlarged or globally glued presentation space, the action must again be specified. Conditional statements about that larger space remain conditional. The local proof does not manufacture coefficient actions on presentations which the supplied package has not enumerated.

{Defines/source identity. XIV-J.tex; lem:null, eq:null; reported lines 853–883.}

XIV-J gives the null presentation

with separately nonzero reconstructed summands

The distributions in the local reconstruction obey

More generally,

When the indicated generators occur in ${\mathcal{I}}_m$, these give local coefficient null presentations. For a larger presentation space they give null presentations provided that its reconstruction uses the same rule and coefficientwise smooth action.

Proof. For a test $\psi$,

which equals the pairing with $-j{\delta }^{(j-1)}$. The case $j=0$ gives $q\delta =0$.

For the third identity, insert $r^2\phi$ in (eq:Rpair). On the cone points $(t,t\omega )$, $r^2=t^2-t^2|\omega {|}^2=0$. Thus the integral vanishes.

Tensoring the one-dimensional identities with the inner distribution and using Lemma lem:linearity yields the corresponding coefficient relations. The conditional assertion for a larger space follows from the stated rule and action there.

For example, the coefficient array with $c_{10}=q$, $c_{00}=1$, and all other entries zero is a nonzero array with zero reconstruction whenever both indices are permitted. Similarly, the array with $c_{00}=r^2$ has zero reconstruction. These statements exhibit dependence of the reconstructed family over chart functions.

They do not say that the frequency amplitude module itself has the same relation. In particular, $(qi\theta +1)M_{\mathrm{R}}$ is a nonzero frequency presentation whose physical reconstruction vanishes. This is exactly why polynomial degree is a degree of a presentation and cannot by itself define an intrinsic filtration of a reconstructed joint symbol.

The finite Taylor multiplication identity gives

where $\xi$ denotes the passive chart variables. It follows directly by applying the Leibniz rule to a test pairing.

On a product subchart around $q=0$, define

Then

The independence used here is that of the finitely many one-dimensional delta derivatives after the coefficients have been reduced to $q=0$. Testing against a compact function with one prescribed finite $q$ jet extracts each $B_i$.

Equation (eq:localcriterion) is an exact criterion for the transcribed local module. It still contains distributional relations in $r$. It does not claim that the few relations in Lemma lem:relations generate every such relation. It also does not determine cancellations between different placements or between different operational entries in a common reconstruction.

The complete six-entry presentation space is a different object. XIV-J calls it a finite direct sum of the internal coefficient presentation spaces, with summation into the common tensor-valued kernel. That statement does not enumerate every coefficient space and every cross-entry identity in this patch. The $K_{47}$ entry, in particular, is not identified with one instance of the ${\delta }^{(j)}(q){{\partial }_r}^{\beta }R(r)$ local family.

For a fixed local module, the inherited coefficient seminorms can be written

These are the usual compact-set smooth seminorms named by XIV-I. The source also states continuity of its reconstruction into ${\mathcal{D}}^{\prime }(V)$.

If a topology on the local image is required, this patch adopts the quotient-topology convention

This convention pushes the named coefficient topology to the image. It is not attributed to a separately stated source definition of the topology of the complete ${\mathcal{K}}_{\mathrm{int}}$.

No assertion is made that this quotient topology equals the subspace topology inherited from ${\mathcal{D}}^{\prime }(V)$. No global Fréchet structure, choice of representatives, or continuous right inverse is introduced. A topology for the common graph image still requires its full presentation and gluing conventions.

The canonical proposition of XIV-I is reproduced in Section sec:domain. Its refusal of the complement equation is retained there as a single displayed refusal. It is not reopened as a search problem.

Patch I closes the documentary absence of the local generators, coefficient class, reconstruction rule, and coefficient seminorms. It also removes the artificial uncertainty about local coefficientwise linearity. It does not complete the global presentation domain or classify the full graph reconstruction kernel.

P3 therefore passes for the actual transcribed local amplitude module. A pass for that module is not a pass for every presentation of the six-entry graph.

{Defines the formal target in the source. XIV-J.tex; eq:target; reported lines 719–730.}

The source target is

Its retained internal sum is

Thus there are six operational entries: $K_{47}$, the two $K_5$ placements, $K_6$, $K_8$, and $K_9$. Grouping $K_4+K_7$ does not remove one of the other entries. The supported sector

is a proper sub-sector, not another name for the complete target.

On the inherited proved domain $O=U_{\mathrm{can}}$, define the notation

with the source's tensor pairings, vertex derivatives, and grouped line variables.

This is the displayed canonical-domain recipe from (eq:target). It is not a definition of the characteristic multiplication on all of $W$. It does not give a preliminary lift on presentations.

The source declares a canonical contraction map on its reconstructed kernels on $O$. Definition def:mcirc supplies the missing display for that recipe. It does not independently recover every coefficient hidden in the inherited tensor pairings or prove that the expression can be evaluated on a test crossing the retained characteristic locus.

This is the appropriate correction to D0.10: the recipe is no longer only a paraphrase. Its domain limitation remains part of the definition rather than an omitted qualification.

{Defines the grouped distribution in the source. XIV-J.tex; sec:reference, The grouped cross-line distribution; reported lines 1112–1116; the submersion description follows in the same subsection.}

The source uses the macro H_T, whose expansion is $H_T$. The block tensor is

There is no hatted replacement block in this formula. The symbol $H_T$ is retained rather than rewritten as $H^{\prime }$.

The supplied XIII-C text explicitly states

Thus $H_T$ is the time-ordered line in the supplied product convention. It is not, by that equation, an admissible Hadamard replacement with smooth difference. No definition from XII-A is imported to change its meaning.

In independent relative variables, $D_T$ is a difference of two defined tensor products. The grouped pullback is

The source constructs this pullback by its stated submersion operation. That construction does not multiply ${\mathcal{C}}_{\mathrm{cross}}$ by an internal kernel at $N$.

For the finite vertex differential operators, the source applies derivatives in the two independent relative variables before pullback, and denotes the resulting components by ${{\mathcal{C}}_{\mathrm{cross}}}_D$. The patch keeps that order of operations. It does not expand the grouped difference into separately undeclared mixed products of $H$ and ${G_{\mathrm{act}}}^{\mathrm{A}}$.

The smooth-difference branch of XV-F's cross-line lemma remains conditional. More explicitly, any statement requiring

to be smooth requires an additional hypothesis. Such smoothness is not supplied by (eq:HT), and it is not used here. Keeping a conditional lemma is different from asserting that its hypothesis holds for the frozen time-ordered blocks.

This corrects the disputed replacement notation without changing the grouped target. It also prevents the source's time-ordered difference from being silently replaced by a different smooth-difference problem.

For two scalar directions ${\psi }_1,{\psi }_2$, define

This is the convention used in the patch's mixed Hessian. It satisfies $T(\varphi ,\varphi )=T_0[\varphi ]$. It is introduced explicitly here from the displayed $T_0$ in (eq:vertex), not attributed to an expanded tensor formula in XIV-J.

The stress in Definition def:polarization is the unique symmetric bilinear polarization of $T_0$. For the displayed vertex (eq:vertex),

Proof. Expanding the quadratic expression gives

The coefficient of $s$ is therefore $2T(\varphi ,\psi )$. The metric derivative of $-\frac{1}{2}\int g{\gamma }_{\mu \nu }{T_0}^{\mu \nu }[\varphi ]$ in direction $h$ is $-\frac{1}{2}\int g{h}_{\mu \nu }{T_0}^{\mu \nu }[\varphi ]$. Differentiating this in the scalar direction $\psi$ cancels the factor $1/2$, yielding (eq:Hessian).

Uniqueness follows from the polarization identity for a quadratic map over a field of characteristic zero:

The source mixed Hessian is written exactly in the form (eq:Hessian), with the words “inherited polarized stress tensor” in place of an expansion. The patch fixes that expansion by the explicit vertex calculation. There is now one convention in this paper, not two competing normalizations.

This closes the factor ambiguity for the patch's tensor notation. It does not assert that a located XIV-J theorem had already printed the expansion. Nor does it import a convention from XIV-D or another excluded sheet. The factor is fixed by the displayed functional whose derivative is being taken.

{Defines in the source. XIV-J.tex; def:mapclass; reported lines 732–762.}

Let ${\mathcal{K}}_{\mathrm{int}}$ be the space of actual reconstructed internal kernels obtained from the finite presentations in the copied precomposition package, with tensor indices retained.

The class consists of linear operator maps

which:

1. on the named reconstructed sum give the specified contraction with the grouped cross-lines and mixed vertex Hessian;

2. retain both $K_5$ placements and the residual variable $z$;

3. agree with the canonical composition on its proved domain;

4. are defined on arbitrary compact tests meeting $N$;

5. have a stated and proved complete-graph wavefront bound appropriate to their composition.

The product data and ${\mathfrak{C}}_{\mathrm{part}}$ are unchanged.

The domain consists of reconstructed kernels, not arbitrary amplitude arrays. No representative-independent operation on all free-shell quotient classes is asserted. A generalized composition belongs to the class only after its relation, amplitude, and remainder hypotheses are supplied for the present target. No common product phase is introduced.

The fifth clause must not be omitted. A map which assigns formal values but lacks the required complete-graph wavefront theorem does not meet the source definition. Displaying the class does not prove that it is nonempty.

The residual-species codomain in (eq:jointmap) has the status specified in Definition def:Eres. Its source name is preserved, while an actual bundle-valued realization still requires the output fiber and pairing. This prevents the map-class display from being used to claim that the unidentified codomain has become a concrete vector bundle.

For scalar tests, set

For a specified value bundle, use compact dual test sections and the same support condition. This is the meaning of “meeting $N$” in clause (iv) of Definition def:mapclass.

This collection is not generally a vector space. Two tests can cancel near $N$, and their sum can fail the meeting condition. Clause (iv) therefore does not define a linear map only on ${\mathcal{D}}_N(W)$. It requires a distributional value which is defined on all compact tests, including those whose supports meet the locus.

Support means the closed support of the test. A meeting test need not have a nonzero pointwise value at a point of $N$. The clause is not an instruction to evaluate a singular kernel at a point. It is also not a wavefront hypothesis on the test: smooth compact tests have no wavefront singularities. The difficulty lies in defining the coefficient which pairs with them.

The source abstract and scope statements say that no full map on these tests was constructed; the reported isolated line locators 203 and 480 point to that boundary. The labelled statements thm:J2 and thm:release give the same boundary unambiguously. This patch does not turn the support definition into a construction on that support class.

{Source proposition, transcribed. XIV-J.tex; prop:descent, eq:descent; reported lines 813–846.}

Let

be the common reconstruction, and let

be linear. Then $\tilde{\mathsf{M}}$ defines a map on reconstructed kernels if and only if

When this holds, the induced map is uniquely given by

Agreement with canonical contraction on $O=U_{\mathrm{can}}$ gives only

where $r_O$ is restriction. It does not prove (eq:descent) across the complement of $O$.

Proof. If $\tilde{\mathsf{M}}=\mathsf{M}\mathsf{Rec}$, a null presentation has image zero. Conversely, two presentations with equal reconstruction differ by a null presentation; condition (eq:descent) makes their proposed images equal. The induced map is therefore well defined and unique.

On $O$, canonical contraction respects equality of reconstructed kernels. A null presentation thus has proposed image restricting to zero there whenever canonical agreement holds. That proves (eq:restricteddescent). A distribution supported in $W\setminus O$ is not detected by this restriction, so the global kernel inclusion does not follow.

The criterion itself is not merely local. It is a global algebraic equivalence for a candidate global value. The source's limitation concerns what canonical agreement verifies. This correction avoids a second ambiguity in the proposed patch: “descent is local” is too imprecise, whereas “the available agreement checks descent only after restriction” is exact.

A local null relation is not coverage of the complete common kernel. Even if a candidate annihilated the presentation (eq:Jnull), it would still have to annihilate every other relation in the complete presentation space, including possible relations between placements and operational entries.

XIV-J constructs a grouped reference for $K_{\mathrm{supp}}$ on $U\setminus N_U$, with an exact normal-jet formula. It also proves a conditional equivalence between a supported normal-order-preserving extension and extensions of its collected coefficient array. Neither result supplies the full $\mathsf{M}$ of Definition def:mapclass.

No new estimate from those results is made here. In particular, the finite normal-jet count is not a transverse degree of the full coefficient, and a conditional coefficient interface is not an extension prescription. The class still includes $K_{47}$ and both $K_5$ placements.

P4 closes the documentary target and clause gaps, and fixes the patch stress convention. It does not close the characteristic-map gate which those clauses define.

{Source proposition, transcribed. XIV-I.tex; prop:canonical; reported lines 1492–1496 within the passage registered at lines 1459–1548.}

On the inherited microlocally admissible domain $U_{\mathrm{can}}$, the reconstructed retained sum, composed with ${\mathcal{C}}_{\mathrm{cross}}$ and the specified vertex operators, equals the canonical grouped expression. The internal reconstruction proves no new normal-set condition for that subsequent product.

Set

the inherited open domain in Input statement input:canonical on which the specified subsequent products and pullbacks have already been proved.

This is a named-domain convention. It does not supply a new list of products beyond the source's grouped target, reconstructed retained sum, cross-line distribution, and specified vertex operations. It does not extend the domain by declaring every collision-free configuration admissible.

XIII-C's graphwise ${\mathcal{U}}_{\Gamma }$ is described by its complete line-wavefront sum condition. The source explicitly does not identify it with the complement of the collision diagonals. This patch also does not identify the collection of those graphwise open sets with $O$ without an inherited graph-to-common-space statement.

A domain can be defined rigorously without one particular complement equation. For example, a complete inventory of proved products and their precise admissibility sets can define it. The refusal below therefore does not mean that an eventual F0 pass is impossible. It means that this proposed shortcut is unavailable.

The refused equation is recorded once:

The supplied XIV-I passage says that no equality (eq:refused) follows from its representation. The supplied XIV-J remark after the supported-sector reference makes the same distinction for the complete six-entry sum. The XV-F manifest additionally records a refusal in XIV-H at reported lines 1661–1667. Since XIV-H is a non-input, that third refusal is retained as a documentary report, not as a newly imported theorem.

Thus the three refusals remain refusals, with their evidentiary levels distinguished:

1. the XIV-I refusal is present in a supplied full text;

2. the XIV-J refusal is present in a supplied full text;

3. the XIV-H refusal is reported by the supplied manifest.

No search for the refused identification is reopened. No alternative equality is inferred from the number of refusals.

The canonical-domain definition permits the expression $m^{\circ }$ to be stated with its correct domain restriction. It also permits the restriction map $r_O$ in Proposition prop:descent to be named. It does not provide the support relation required to regard every contact value as an element of

That relation would require the actual lift or pushforward of the collision supports into $W$ and their position relative to $O$.

In particular, the collision class cannot be inserted into $\mathrm{Hom}({\mathcal{P}}_{\mathrm{int}},Y)$ merely because its members are called contacts. A contact on one graph-coordinate collision can have a nontrivial image under a graph map. Whether that image lies outside the canonical domain is an additional geometric statement.

D0.8 is therefore patched as a named domain with an explicit non-equation. This removes the mistaken requirement that the refused equation itself must be recovered. It does not claim an operationally complete new description of every admissible configuration.

The source stop is unchanged. Canonical agreement is an agreement where the subsequent operations are already defined. It is not the characteristic prescription missing from those operations.

The proposed outline places $K_8$ and $K_9$ wholly outside the input set. That is not correct for the supplied texts. XIV-I and XIV-J copy their formulas explicitly. Their upstream owner may be XIV-H, but the copied displays are available as premises of the four-sheet patch, with their copied scope.

{Defines the copied response formulas. XIV-I.tex; input:Hresponse, eq:Q, eq:T, eq:K8, eq:K9; reported lines 257–321, with formulas reported at lines 285–310.}

Those displays are

and

The source indices and maps are retained. In particular, the copied source uses $\mathcal{P}$ and $\mathsf{b}$ in these displays. This patch does not identify those symbols with differently styled maps in the XV-F retrieval account.

The coefficients ${q_{\alpha \beta }}^{L,I,J}$ are specified by role as the actual differentiated-source coefficients. They are not expanded in these displays. The functions and fiber maps also remain inherited inputs rather than newly chosen coefficients. Thus the formulas are available, while their complete instantiation is still missing.

Likewise, the supplied sheets display

and define the role of the constant bilinear map by

These equations define dependencies. They do not expand $a_1$, $a_2$, ${\dot{a}}_2$, the graph source, or all inherited switches and observations.

The exact correction to D0.13–D0.14 is therefore not “the formulas are unavailable.” It is “the formulas are copied, but the frozen coefficient ingredients are not completely recovered from the permitted inputs.”

XV-F's Remark 21 suggests that the attachment delta forces $K_8$ to remain singular in the timelike interior where $B$ is locally constant. The displayed inner derivative ${\partial }_{b^{\alpha }}$ in (eq:K8) prevents that inference.

For the copied defined incidence operation in (eq:K8),

This support consequence does not prove a conormal order or a characteristic composition theorem.

Proof. The function $B$ in (eq:Rsource) is, almost everywhere, the constant $1/(8\pi )$ on the interior of $J^{+}$ and zero on its exterior. Its distributional first derivatives are therefore supported on $\partial J^{+}$.

The incidence $Q$ is supported on $v=a=z$. In the copied pushforward, differentiation in $b$ acts on the factor $B(b-v)$; $Q$ is independent of $b$. Hence

The copied incidence operation is defined, and its support is contained in the image of $v=a=z$, $b-v\in \partial J^{+}$. The outer $b$ derivative and smooth factors do not enlarge support. This gives (eq:K8support).

This is an algebraic support reading of the copied formula, not a new estimate. It does not settle the source's regular conormal order statement. It does show that the old interior-support diagnostic cannot be used to assert a potential contradiction: the supposed interior contribution has already been killed by the displayed derivative.

The correction also illustrates why a source patch must inspect operators rather than only seed factors. The presence of $B$ before differentiation does not determine the support of the differentiated kernel.

This ledger is intentionally finer than the proposed nine-row affirmative ledger. Several original proposed rows mixed a local definition with a global completion or a generic formula with a frozen coefficient. Those distinctions cannot be suppressed in a proof-ready version.

In particular, the entry “D0.3–D0.4 defines for the chart-function case” must not be read as a determination of the full common kernel. The entry “D0.17 formulas displayed” must not be read as recovery of every specified graph parameter. The entry “species named” must not be read as identification of a value/test duality.

The following documentary gaps in XV-F's Proposition prop:F0 are removed:

1. the missing local amplitude generators and coefficient class;

2. the missing display of the canonical-domain target recipe;

3. the incomplete and disputed description of the grouped blocks;

4. the missing full list of joint-map clauses and the meaning of a compact test meeting $N$;

5. the factor ambiguity in the stress convention adopted by this patch.

The four patch ledgers are completed, including the explicit boundaries on residual-bundle and contact-role completion.

However, the stronger statement that every owned frozen-object gap now passes the original F0 is not established. In particular:

1. the actual residual output bundle is not identified;

2. the complete common presentation space, image topology, and reconstruction kernel are not recovered;

3. the actual graphwise correction parameters and the presentation-level space $\mathcal{A}$ are not supplied;

4. the remaining inherited coefficient definitions are not fully instantiated.

Consequently no preliminary lift for the actual frozen graph is defined by this patch, and no actual null-defect test is available. Gates F1–F4 are not entered.

Proof. Definition def:amodule reproduces the local coefficient class and generators, giving (i). Definition def:mcirc and (eq:target) give (ii). Equations (eq:blocks)–(eq:Cross) give (iii). Definitions def:mapclass and def:meeting give (iv). Definition def:polarization and Lemma lem:Hessian give (v).

Proposition prop:Gstatus states the residual-codomain boundary. The distinction between the local coefficient module and the complete graph presentation is recorded in Section sec:I, and its topology limitation is explicit in (eq:quotienttopology). Proposition prop:Cstatus proves that the contact templates do not recover their frozen parameters or determine $\mathcal{A}$. The copied response displays leave the coefficient ingredients listed in Table tab:assembled open.

A preliminary lift requires the actual presentation domain, the actual value codomain, and values assigned according to the frozen composition rule. Naming these dependencies and exposing missing parameters does not give those assignments. A null defect additionally requires the actual reconstruction kernel and correction map class. They are not supplied. The F0 stop rule therefore applies, without an analytic obstruction being asserted.

The proposition is a statement about what this source patch establishes. It is not a theorem that no lift exists. Likewise, it is not a theorem that the unnamed residual bundle cannot be chosen. It says that the patch has not recovered the frozen choice and has not constructed the lift.

The omission of a separately printed fiber rank would not by itself prevent a definition if the bundle were otherwise specified. The present failure concerns the missing identification of the output object and pairing. More importantly, even a later choice of that bundle would not remove the independent presentation, coefficient, correction-map, and composition obligations.

The requested ending “core package completed for the four owned gaps” is valid only if “completed” means that the four documentary tasks have been disposed of with their scope and unresolved parameters explicitly recorded. It is not valid as an assertion that all corresponding actual objects have been recovered.

This patch adopts the first, defensible meaning and rejects the second. Its release is therefore:

This is a substantive correction to the proposed outline rather than an unsupported affirmative release. It preserves the outline's intended separation between definition work and analytic work, while also preserving the difference between a parameterized definition and recovery of a prescribed frozen member.

No quantum coefficient is constructed or admitted. In particular,