Landmark IX-D Causal-support interfaces for one sharp ordered word after construction of the declared partial free product

We revise the causal-support interface for the ordered word $B_f,W_g,B_{f^{\prime }}$ using the constructed free sector of IX-Free-2 as an explicit inherited premise. Relative to its stated prerequisites, IX-Free-2 supplies degree-four descent, simultaneous overlap agreement, a truncated partial product ${\star }_c$ on a named tensor domain, and agreement of the two associations through order ${\hslash }^3$. It does not supply a time-ordered or retarded product.

We establish configuration-support bounds for the constructed outputs, including all one-edge terms and the classical word. We identify the calibrated scalar as the constant projection of an existing free word, while proving that this projection still does not define a causal map. An explicit algebraic separation argument shows that wordwise free coherence, even together with an admitted symmetric auxiliary multiplication, does not imply causal factorization.

Separate time-ordered and retarded interfaces are stated with actual tensor domains, old-product overlaps, order-reversing involution, and continuity boundaries. We retain the conditional source-contact incompatibility and prove the conditional retarded-subtraction implication on a domain where every required free product exists. The normalization audit shows that numerator data through ${\hslash }^3$ determine a quotient by $\hslash$ only through ${\hslash }^2$.

The interface and separation analysis is complete. The sharp causal-product release remains unresolved at the unconstructed time-ordered extension and its causal compatibility proofs. The free gate for this word is no longer the live stop. Neither a general free-product enlargement nor a G2-compatible interacting construction is asserted.

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

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

and updated prerequisite{ Causal-product outcome: Unresolved.

IX-Free-2 is inherited as a relative construction of one truncated partial free product ${\star }_c$ on $D_{\mathrm{decl}}$, including both associations of the selected word through ${\hslash }^3$. Its degree-four descent and declared overlap tests are not reopened here.

The live missing construction is a separate time-ordered extension and the resulting causal retarded maps. The named free sector is not the complete free-product domain of an unspecified general IX-D construction. No sharp ${\mathsf{T}}_2$, normalized $R_1$, or $R_2$ is released by this manuscript.}

The selected word is

Product-release outcomes are:

1. Constructed: named maps exist and satisfy every declared interface.

2. Proved obstructed: an explicit contradiction excludes every construction satisfying the specified interfaces.

3. Unresolved: a required construction or compatibility proof is unfinished.

The present causal-product release is (iii). A contradiction for one simultaneous prescription excludes that prescription only, unless all alternatives are separately excluded.

The supplied IX-Free-2 manuscript closes the declared free sector relative to its premises IXFreeTwo. Its result replaces the earlier degree-four descent and overlap stop in IX-Free IXFree. It does not remove the additional causal-extension obligations.

Let

with

Set

The operator ${\mathsf{E}}_0$ is the zero-value, zero-first-time-derivative Cauchy inverse at $\Sigma$. Configurations $\psi =(h,\varphi )$ and configuration directions are smooth and off shell.

For

the generators are

Complex tests define complex-valued actual functionals on real smooth configurations.

The mixed covariance $K(x,y)$ and scalar covariance ${\Delta }_m^{+}(x-x^{\prime })$ are fixed flat free kernels. They are not configuration-dependent metric propagators. The separate notation $\mathcal{K}\xi$ below denotes a metric gauge variation, not the mixed kernel $K$.

The carrier ${\mathcal{P}}_{\Sigma }$ consists of actual finite polynomial functionals with compact distributional kernels satisfying

for every smooth configuration and finite smooth direction list. Here

and the intrinsic charge is

It is well defined on actual linear kernels and vanishes on ${\mathcal{M}}_1$.

The locally convex topology ${\tau }_{\Sigma }$ records strong compact-distribution coefficients and all intrinsic trace coordinates. Classical multiplication, directional differentiation, and conjugation preserve the carrier. Classical multiplication is separately continuous.

The sharp adjoint retains the surface term

For nonzero real $f$, the inherited sharp approximants satisfy

This nonconvergence is not used as a substitute for a charge witness.

Smooth Cauchy and Green-operator mapping properties and finite propagation are available for the fixed operators in (eq:L).

These are inherited carrier premises reported in the supplied sources, not an independent verification of the complete IX-A manuscript IXA.

For the prescribed regulator placement,

There are ${u_{\mu }}^{\rho },r\in {\mathcal{D}}^{\prime }(X)$ with

For each compact $K_0\subset X$,

The estimate includes compacts meeting $\mathrm{Rel}\cap {\Delta }_{13}$, where $\mathrm{Rel}=\mathrm{supp}r$ and ${\Delta }_{13}=\{x=x^{\prime }\}$. Parent conversion is exactly $c_{ab}r$, with no additional normal derivative, nonconstant weight, or intersection term. The parent distributions are dagger symmetric.

These are the analytical premises reported through IX-Free and IX-Free-2, including their stated dependence on IX-B and VII-D IXFree,IXFreeTwo,IXB,VIID.

Fix the regulator, scale, and one real parameter $c$ throughout:

No new parameter is selected by association, presentation, causal argument ordering, or observation descendant.

The inherited free sector is subject to:

1. Admission of every output and required directional descendant.

2. Classical multiplication at order zero.

3. Exact agreement with fixed old products on declared overlaps.

4. Retention of the locked first-stage coefficients.

5. The one common calibrated scalar parent $U_c$, with its declared observation compatibility.

6. No subtraction changing a fixed old product.

7. No requirement that unchanged approximants converge sharply in ${\tau }_{\Sigma }$.

8. Respect for all finite actual-presentation relations.

9. Named continuity domains and proofs, without promoting test-family continuity to actual-stage continuity or hypocontinuity.

These are the IX-C interfaces as inherited in IX-Free-2 IXC,IXFreeTwo. The present revision uses their completed sector realization; it does not impose causal factorization by silently adding it to that free contract.

Define the actual spaces

and

The last sums are direct by actual field degree, four and two.

The locked contractions are

The algebraic presentation maps are

Inside ${\mathcal{P}}_{\Sigma }{\otimes }_{\mathbb{C}}{\mathcal{P}}_{\Sigma }$, put

All these tensor products are algebraic.

Let

Relative to its explicitly stated inherited premises, IX-Free-2 constructs coefficient maps

and one truncated partial product

extended over $\mathbb{C}[\hslash ]/({\hslash }^4)$. Its restriction to $D_{\mathrm{old}}$ is the unchanged old product modulo ${\hslash }^4$.

The following results are inherited, not reproved:

1. $q_L,q_R,p_L,p_R$ are injective, and

   $$\ker p_L=\ker p_R=\{0\}.$$

   The degree-four proof uses restricted sharp separation on $k_{00}=0$ and one-scalar, three-metric slot isolation.

2. All four prescribed degree-four coefficients descend to actual functionals.

3. Every nonzero sharp generator has nonzero advanced-trace charge at some smooth configuration. The partial contractions and admitted microcausal comparison kernels have zero charge. In particular,

   $$\mathcal{B}\cap {\mathcal{L}}_2=\mathcal{B}\cap {\mathcal{R}}_2=\mathcal{B}\cap {\mathcal{P}}_{\mathrm{mc}}=\{0\}.$$

4. The new domain is a direct sum of its six prescribed blocks. Its whole-sum old overlap vanishes:

   $$D_{\mathrm{new}}\cap D_{\mathrm{old}}=\{0\}.$$

   All coefficients glue, including the locked first stages.

5. $${\mathcal{C}}_{\mathrm{desc}}={\mathcal{C}}_{\mathrm{glue}}=\mathbb{R}.$$

   For each fixed real $c$, the product satisfies C-I1–C-I9 on $D_{\mathrm{decl}}$.

6. The two associations of the selected word exist and agree through ${\hslash }^3$. The product has actual-functional presentation independence and order-reversing involution.

7. F9a holds on the named fixed-support test families. F9b, quantum hypocontinuity, associativity for unspecified triples, and closure under unspecified intermediate products are not asserted.

{}1.14

Thus the earlier free-closure clause is discharged only for $D_{\mathrm{decl}}$, and only as the inherited relative premise in Assumption ass:free2. The earlier future-lift obligations concerning time ordering, causal factorization, anti-time ordering, and old retarded overlaps remain undischarged.

The construction does not establish that

for arbitrary declared pairs or arbitrary $H$. An unclaimed enlargement is not a nonexistence theorem.

The inherited first stages are

Write

and

The one-edge outputs are

For $V=f\otimes g\otimes f^{\prime }$, the two-edge outputs are

Their distributional existence, compact kernels, admission, and required descendants are inherited.

The common constructed word is

This is an equality of existing partial-product associations, not an assertion about unspecified triples.

The sign ledger remains the inherited one: each partial has factor $-2$, the complete word has two insertion signs with product $+1$, and the middle-slot assignment factor is $2$, not $4$.

The standard perturbative framework distinguishes a free quantum product, time ordering, and the interacting or retarded construction FR,DF. No existence theorem for local functionals is automatically applied here to the sharp nonlocal carrier.

The support ${\mathrm{supp}}_{\mathrm{fun}}F$ of an actual functional is the complement of the largest open region on which every compactly supported configuration variation leaves $F$ unchanged at every configuration. Metric and scalar support are defined by restricting the variations to the respective field species.

For a finite formal functional, its support is taken to be the union of its coefficient supports when a formal support statement is required.

Observation support and functional support are not interchangeable. The metric dependence of $B_f$ generally extends from $\mathrm{supp}f$ back to the initial surface.

We use $J^{\pm }$ for causal future and past, including null relations, and $I^{\pm }$ for chronological future and past. A binary ordering condition is

Equivalently,

This means that $F$ is not causally earlier than $G$; it also includes spacelike-separated supports.

On the ambient tensor space define

The inherited domain is dagger stable, and

where the pair is declared.

Dagger stability alone does not assert separate flip stability. To use both free orderings without an enlargement, define

The space $D_{\mathrm{cmp}}$ is stable under $\sigma$ and $\gamma$. Both orderings of every pair represented there are declared. In particular, $D_{\mathrm{first}}\subset D_{\mathrm{cmp}}$.

Proof. Flip stability follows directly from ${\sigma }^2=1$. Since $\sigma \gamma (D_{\mathrm{decl}})=D_{\mathrm{decl}}$, one has $\gamma (D_{\mathrm{decl}})=\sigma (D_{\mathrm{decl}})$. Therefore

The first-stage sum contains both orderings, and $\mathcal{B},\mathcal{W}$ are conjugation stable.

The restriction $D_{\mathrm{cmp}}$ is not a closure theorem. A future time-ordering domain must still name the pairs it contains, including the reversed and conjugate pairs needed by its identities.

A proposed bosonic time-ordering system must specify:

1. Actual-functional domains

   $${\mathcal{D}}_2\subset {{\mathcal{P}}_{\Sigma }}^{\otimes 2},{\mathcal{D}}_3\subset {{\mathcal{P}}_{\Sigma }}^{\otimes 3},$$

   the coefficient range and truncation, symmetry, normalization ${\mathsf{T}}_1(F)=F$, and classical order-zero multiplication. Every required reversed, conjugate, or intermediate domain must be named.

2. Admission and actual-presentation descent for every coefficient and required directional descendant.

3. Causal factorization:

   $${\mathsf{T}}_2(F,G)=F{\star }_{c}G\text{if }F\gtrsim G,$$

   where $F\otimes G$ belongs both to the time-ordering domain and to $D_{\mathrm{decl}}$. The comparator is specifically the IX-Free-2 product. Outside $D_{\mathrm{decl}}$, this identity requires a separately constructed free extension. Higher factorization must name every intermediate free product used.

4. Exact declared old time-ordered overlaps and a specified class of permitted contact terms, with admission and descent tests for those terms.

5. The appropriate involution or unitarity identities, including anti-time ordering rather than an unjustified self-adjointness requirement.

6. A continuity statement on a named domain at a specified strength. F9a must not be promoted to F9b or quantum hypocontinuity.

A time-ordered product need not vanish for spacelike-separated arguments. If causal factorization applies in both directions, the two free orderings must agree there; their common value need not be zero.

Even a contracted contribution may be smooth and nonzero away from coincidence. A distribution supported on a coincidence set vanishes on tests supported away from that set, so contact terms cannot account for such an off-diagonal spacelike contribution. This is consistent with the usual free and time-ordered distinction FR.

A proposed retarded system must specify:

1. The vertex and observable roles and the off-shell subtraction convention. These are not inferred from the letters $B,W$.

2. The causal Green operator and kernel convention. A first-order vanishing requirement is

   $$\mathsf{R}(V;O)=0\text{if}{\mathrm{supp}}_{\mathrm{fun}}O\cap J^{+}({\mathrm{supp}}_{\mathrm{fun}}V)=\varnothing .$$

   A stronger distributional argument-slot support statement must name those slots and be separately proved.

3. Exact agreement with declared old retarded maps and specified smooth classical comparison coefficients.

4. Preservation of every fixed old product; no subtraction may change an already fixed regular coefficient.

5. Admission and actual-functional descent of every nonconstant coefficient and required descendant.

6. A named continuity domain and proof, including the additional domain obligations of any higher retarded formula.

The strict-exterior condition involving $J^{+}$ imposes no vanishing assertion when the supports meet its causal boundary. Coincidence contributions require a contact prescription, while noncoincident null-related points may carry propagator singularities and are not thereby diagonal contact terms. Replacing $J^{+}$ by $I^{+}$ would change the interface and is not justified here.

On a domain where both maps exist, a proposed unnormalized comparison is

It is not yet a definition in this manuscript, since ${\mathsf{T}}_2$ has not been constructed. The conditional theorem in Section sec:D5 states precisely when this comparison has the desired properties.

Replacing $V{\star }_{c}O$ by $VO$ is a different prescription. It requires a separate identity eliminating the relevant free contractions. A specialized comparison reported for another sector does not supply that identity here.

For the following sign calculation only, let $P$ be the specified smooth field operator, let $G_R^{\ast }=G_A$ in the chosen pairing, and suppose $\mathsf{Q}[h]$ is formally self-adjoint. All displayed sources and pairings are legitimate with the stated compact supports. The first variation satisfies

Under Assumption ass:smooth, setting $u_f=-G_{A}f$ gives the smeared comparison test $q=+\mathsf{Q}[h]u_f$.

Proof. The adjoint identities give

An additional minus in the last test would contradict this declared Duhamel convention.

This is a smooth classical sign calculation, not a construction of a sharp $R_1(V_1;B_f)$.

The inherited product ${\star }_c$, its wordwise coherence, and its calibrated constant coefficient do not by themselves supply a time-ordered or retarded system satisfying ${\mathcal{I}}_{\mathrm{TO}}$ or ${\mathcal{I}}_{\mathrm{ret}}$.

Proof. Assumption ass:free2 supplies actual free coefficient maps and their declared overlaps. It supplies no separate ${\mathsf{T}}_2$, no old time-ordered overlap equations, and no anti-time-ordering system.

Equation (eq:word) compares two existing free associations. It contains no causal-factorization identity relating a time-ordered map to ${\star }_c$. Section sec:D3 gives an explicit algebraic separation of these requirements.

Without ${\mathsf{T}}_2$, the difference in (eq:subtraction) is unavailable. Nor do the free statements supply the causal Green convention, higher intermediate domains, or argument-slot support proofs. These remain additional obligations. This is not a proof that a later causal extension cannot exist.

For a compact observation set $F⋐M_{+}$, define

This set is compact in Minkowski space: time lies between zero and the largest time in $F$, and finite propagation bounds the spatial coordinates.

Let $F=\mathrm{supp}f$, $G=\mathrm{supp}g$, and $F^{\prime }=\mathrm{supp}{f}^{\prime }$. The constructed functionals have the following separate metric and scalar support bounds:

{}1.14

Proof. A metric variation supported outside $P_F$ cannot change $Lh$ on $F$. This follows from locality of $C$, the fixed zero Cauchy data, and finite propagation. A scalar variation outside $F$ cannot change $\partial \varphi$ on $F$. These facts prove the bounds for $B_f$. The bound for $W_g$ is immediate.

In (eq:Q)–(eq:QR), the covariance is fixed; only the displayed residual fields vary. This gives the partial-contraction bounds. The residual fields in ${\mathcal{T}}_{f,f^{\prime }}$ are $w$ on $F,F^{\prime }$, giving the two past sets.

The support of a classical product is contained in the union of the supports of its factors: a variation outside both supports changes neither factor. Apply this to (eq:oneedges) and the classical word.

For (eq:G1223), the fixed kernels $K,\overline{K}$ introduce no metric configuration dependence. Hence its metric support is empty, while its scalar dependence is confined to $F\cup F^{\prime }$. The other two two-edge outputs contain only $h_{00}$ on $G$ and one residual $w$ slot. This gives their displayed metric bounds and empty scalar support.

The inherited distributional pairings introduce no additional configuration variables.

The test $g$ still enters the coefficient of ${\mathsf{G}}_{12,23}$, even though its residual metric support is empty. Dependence on an observation test is not dependence on the metric configuration.

The bounds are inclusions, not exact-support assertions. They are properties of the constructed free outputs as configuration functionals. They are not retarded argument-slot theorems and do not reconstruct the inherited singular products.

For a smooth compact source $s$, the defining retarded Green property gives

this is the usual Green-operator support convention FR. A specified smooth comparison

is therefore legitimate when its source is smooth and compact and its chosen inverse has the stated Cauchy or retarded property. It remains a smooth classical composition.

Zero-Cauchy inversion agrees with the corresponding retarded solution only under the appropriate zero past-data hypothesis. It is not a Feynman inverse, and arbitrary homogeneous data cannot be discarded.

For $H\in {{\mathcal{V}}_{\Sigma }}^{[3]}$, let

where ${\mathrm{ev}}_{\mathrm{const}}$ extracts the field-degree-zero part of the polynomial coefficient, equivalently its value at the zero configuration.

The constructed free word satisfies

This scalar assignment does not by itself define ${\mathsf{T}}_2$, ${\mathsf{R}}_1$, or a higher retarded map on the intermediate actual-functional stages.

Proof. The first assertion is the constant coefficient of the inherited equality (eq:word). Thus the scalar is now a projection of an existing word, not a projection of an unconstructed association.

Nevertheless, $\Pi$ has a large kernel. For example, $\Pi (\hslash F)=0$ for every $F\in {\mathcal{P}}_{\Sigma }$. Changing a lower-order nonconstant coefficient leaves the displayed constant projection fixed. This is an information-loss statement; it is not permission to change the locked coefficients of ${\star }_c$.

IX-Free-2 supplies the nonconstant free coefficients lost by this projection. It supplies no separate time-ordering coefficients, causal factorization, old time-ordered or retarded overlaps, or anti-time-ordering identities. In particular, the scalar does not select

The existence of the free word therefore does not convert its constant projection into a causal map.

Wordwise free coherence of ${\star }_c$ is compatible with a symmetric admitted auxiliary multiplication ${\mathsf{M}}_2$, with classical term $VO$, that differs from $V{\star }_{c}O$ on a declared first-stage pair satisfying

Proof. Consider the auxiliary map

viewed as a formal map with zero positive-order coefficients. It is symmetric, depends only on actual arguments, and has admitted values by classical carrier closure. Its definition leaves ${\star }_c$ and every inherited wordwise equality unchanged.

Choose nonzero tests $f,g$ with

Take

Let $F=\mathrm{supp}f$ and $G=\mathrm{supp}g$. Theorem thm:support places the metric support of $B_f$ in $P_F=J^{-}(F)\cap \{t\ge 0\}$ and its scalar support in $F$. Consequently,

Since $F\subset \{0<t<a\}$, every point of $P_F$ has time coordinate between zero and $a$. Thus

The same theorem gives

Causal future-directed curves cannot decrease the time coordinate. Therefore,

Both pair orderings belong to $D_{\mathrm{first}}$.

The locked reverse first stage gives

whereas

The nonzero tensor $g\otimes f$ has a nonzero image under the inherited injective map $q_R$. Therefore ${{\mathcal{Q}}^{\mathcal{R}}}_{g,f}\ne 0$, and the two expressions differ. This proves the asserted compatibility and discrepancy.

The map ${\mathsf{M}}_2$ is not proposed as a time-ordered construction satisfying ${\mathcal{I}}_{\mathrm{TO}}$. Its failure is the explicit witness that a causal axiom cannot be inferred from the free ones. The lemma neither changes the old products nor proves an obstruction to a future causal extension.

Observation operations remain numerical:

The inherited descendant estimate has test-order loss $64\to 64+\mathrm{ord}D$. It does not identify observation differentiation with configuration differentiation or open new functional slots.

Assigning a causal cutoff to $U_c$ would still require new product, descent, admission, and overlap proofs. No such assignment is made here.

For this section only, let $P$ be the declared formally self-adjoint field operator and let $H_F$ be a declared Feynman fundamental solution with

The source presentation is $j=Ph$, and differential compatibility requires the time-ordered contracted $jh$ kernel to be obtained by applying $P_x$ to the contracted $hh$ kernel. The domain includes tests detecting the diagonal contact.

The fundamental-solution, rather than bisolution, character of the time-ordered kernel is an additional differential input; it is not a consequence of free-product coherence FR.

Under Assumption ass:source, the additional prescription ${H^{\mathrm{TO}}}_{jh}=0$ is incompatible with the stated source presentation and differential identity.

Proof. Differential compatibility gives

In the scalar case, choose a nonzero real compact test $q$ and pair with $q\otimes q$. The result is

For a bundle-valued identity, select the nonzero component of the declared diagonal identity and the corresponding bundle test. The zero-kernel prescription gives zero on the same test, which is a contradiction.

This excludes the simultaneous prescription under its explicit hypotheses. An independent central source variable need not be identified with $Ph$. No universal source-extension obstruction is asserted.

If a free contraction is a bisolution, $P_{x}H=0$, its zero source contraction is compatible with that different differential identity. It does not automatically transfer to $H_F$. Applying $P$ again would involve a derivative of a delta distribution; its normalization and bundle factors are not evaluated here.

The two-variable contact $c_F\delta$ is not the three-observation scalar calibration $cr$. IX-Free-2's freedom ${\mathcal{C}}_{\mathrm{glue}}=\mathbb{R}$ supplies no map identifying these objects and cannot absorb the contact by reassignment of the common parent parameter.

The free-closure premise formerly required at this point is now Assumption ass:free2, restricted to $D_{\mathrm{decl}}$. Only the remaining causal-lift hypotheses are assumed below.

For this section only, assume:

1. A separate symmetric time-ordering map

   $${\mathsf{T}}_2:{\mathcal{D}}_2⟶{{\mathcal{V}}_{\Sigma }}^{[3]}$$

   exists on a specified actual tensor domain ${\mathcal{D}}_2\subset {{\mathcal{P}}_{\Sigma }}^{\otimes 2}$ stable under $\sigma$ and $\gamma$. The comparison domain

   $${\mathcal{E}}_2={\mathcal{D}}_2\cap D_{\mathrm{cmp}}$$

   contains every pair used in this section. Thus the two free orderings and their conjugates all belong to $D_{\mathrm{decl}}$.

2. Every time-ordered coefficient is admitted, respects actual-presentation relations, and agrees with the declared old time-ordered coefficients. Normalization, permitted contact terms, and the claimed continuity strength on named domains are specified. No actual-stage continuity is inferred from inherited F9a.

3. Both ${\mathsf{T}}_2(V,O)$ and $V{\star }_{c}O$ have classical term $VO$. On comparison pairs with

   $${\mathrm{supp}}_{\mathrm{fun}}O\cap J^{+}({\mathrm{supp}}_{\mathrm{fun}}V)=\varnothing ,$$

   causal factorization gives

   $${\mathsf{T}}_2(V,O)=V{\star }_{c}O.$$

4. Free involution reverses order:

   $$(V{\star }_{c}O{)}^{\ast }=O^{\ast }{\star }_c{V}^{\ast }.$$

   The binary anti-time-ordering identity on ${\mathcal{E}}_2$ is

   $${\overline{\mathsf{T}}}_2(F,G)=F{\star }_{c}G+G{\star }_{c}F-{\mathsf{T}}_2(F,G),$$

   and the time-ordering involution is

   $${\mathsf{T}}_2(F,G{)}^{\ast }={\overline{\mathsf{T}}}_2(F^{\ast },G^{\ast }).$$

   Every expression in these equations is defined on the specified domains. Equation (eq:futurefreeadjoint) is inherited; the additional time-ordering identities are not.

5. On a specified old retarded overlap

   $${\mathcal{E}}_{\mathrm{old}}\subset {\mathcal{E}}_2\cap D_{\mathrm{old}},$$

   the old unnormalized retarded convention is exactly $i({\mathsf{T}}_2-V{\star }_{c}O)$, using the unchanged old time-ordered and free coefficients.

The time-ordered existence, factorization, anti-time-ordering, and retarded-overlap hypotheses are not discharged by this manuscript.

The domain condition in part (a) is substantive: if a desired pair or reversed free product lies outside $D_{\mathrm{decl}}$, a free enlargement must be proved before these formulas can be applied to it.

Under Assumption ass:future, define on ${\mathcal{E}}_2$

This is an actual tensor map, equivalently a bilinear map on its declared pairs, with admitted coefficients and zero classical term. It has the exact declared old unnormalized retarded overlaps and satisfies

Furthermore,

If both terms have a proved continuity property at the same strength on the same named domain, their difference has that property.

Proof. Both summands are linear maps on the actual tensor domain ${\mathcal{E}}_2$, so their difference respects every actual relation. Its coefficients belong to the vector space ${\mathcal{P}}_{\Sigma }$, with the required descendants, by admission of the two coefficient systems. Equal classical terms cancel.

On an excluded causal pair, (eq:causalfuture) makes the difference zero. On ${\mathcal{E}}_{\mathrm{old}}$, the assumed old values and convention give the required old overlap.

For involution, use both order reversal and anti-time ordering:

The conjugate pair is in ${\mathcal{E}}_2$, by the stated domain stability.

Finally, subtraction is continuous in the coefficient topologies. Hence any common proved continuity statement is preserved.

The proposition proves a conditional vanishing property for declared pairs. It does not prove a stronger retarded distributional slot-support theorem. Nor does it construct its time-ordering hypotheses by assuming only free coherence.

Suppose a formal numerator in ${{\mathcal{V}}_{\Sigma }}^{[3]}$ has zero classical term:

Its normalized quotient by $\hslash$ is determined only modulo ${\hslash }^3$:

It is not uniquely determined as an element of ${{\mathcal{V}}_{\Sigma }}^{[3]}$.

Proof. Multiplication by $\hslash$ in ${{\mathcal{V}}_{\Sigma }}^{[3]}$ annihilates ${\hslash }^{3}F$ for every $F\in {\mathcal{P}}_{\Sigma }$. Thus two quotients differing by such a term have the same numerator modulo ${\hslash }^4$.

Conversely, comparison of coefficients shows that the constant, linear, and quadratic quotient coefficients are exactly $N_1,N_2,N_3$. These are therefore uniquely determined in ${\mathcal{P}}_{\Sigma }{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^3)$.

Consequently, a physical convention

would, under the conditional hypotheses, be determined only through ${\hslash }^2$ by the order-three data. A normalized result through ${\hslash }^3$ requires the numerator through ${\hslash }^4$, or a separately specified equivalent convention. There is no inverse of $\hslash$ in $\mathbb{C}[\hslash ]/({\hslash }^4)$.

Higher retarded maps additionally require higher time-ordering and all intermediate free products in their defining identities DF. Neither the binary implication nor the triangle coefficient constructs $R_2$, and the inherited partial domain does not authorize unspecified intermediate products.

Under the stated IX-A/VI switch-gap hypotheses, there is a compact gauge parameter $\xi$, supported strictly after $\Sigma$, with

and

For the specified switch and witness,

The switched forced-response variation is zero, while the dressing variation is $-a_f$.

These are copied witness hypotheses for the specified gauge and switch, not assertions about arbitrary switches or arbitrary gauge parameters.

Merely placing $\xi$ in a switch-transition region does not prove either witness equality.

Under Assumption ass:switch, a modification beginning at coupling order ${\kappa }^2$ cannot remove the order-$\kappa$ switch gap while the first-order prescription is retained.

Proof. The witness compares

If a corrected expression has formal variation

vanishing of the formal series requires the coefficient of $\kappa$ to vanish. It does not. Later coefficients cannot alter that coefficient.

The coupling order $\kappa$ is distinct from the contraction order $\hslash$. IX-Free-2's free construction does not cancel this first-order witness or open G2.

A condition $g=1$ near an observation support does not by itself remove a witness involving the dressing's larger metric-support region. The reported VII-A plateau intertwiner is restricted to its stated plateau-supported gauges and retains the switch derivatives; it is not thereby a quantum Ward identity.

The expression

remains bookkeeping until every displayed map, domain, overlap, and support property has been constructed. Membership of $B_f$ in the carrier does not admit an unspecified correction ${{\mathcal{B}}_f}^{(2)}$. No such correction is admitted or constructed here.

The objects below are not interchangeable. The assertion is that no identification needed by this construction has been supplied, not that special numerical coincidences are universally impossible.

{}1.14

The complete V, VI, VII-A, VII-E/F, and VIII sources are not provided here. Their entries record the stated scope boundaries, not independently certified theorems used in an unconditional causal construction.

The updated dependency chain is

For a general causal construction, any required free-domain enlargement is an additional obligation alongside the time-ordering step. The old degree-four stop is no longer the live stop for this word.

Relative to the explicitly inherited carrier, calibration, and IX-Free-2 premises, this manuscript establishes:

1. Separate time-ordered and retarded interfaces with actual-domain requirements.

2. Configuration-support bounds for all displayed constructed free outputs, including the one-edge terms and the classical word.

3. Identification of ${\mathcal{N}}_c(f,g,f^{\prime })$ as the constant projection of the constructed free association, together with its non-lift to a causal map definition.

4. An explicit separation of free wordwise coherence from causal factorization.

5. The source-contact contradiction under the simultaneous hypotheses of Assumption ass:source.

6. The conditional admitted subtraction, causal vanishing, old-overlap, and involution implication of Proposition prop:conditionalret.

7. The precise normalized $\hslash$-order limitation.

8. Persistence of the copied first-order switch witness against corrections beginning at order ${\kappa }^2$.

It constructs no sharp ${\mathsf{T}}_2$, normalized $R_1$, or $R_2$ on the requested sector.

The causal-product release is Unresolved. The live stop is the time-ordered extension and its remaining causal, admission, and overlap obligations, not degree-four descent of the inherited free word.

Proof. The statements follow from Requirements req:TO–req:ret, Theorem thm:support, Theorem thm:nonlift, Lemma lem:separation, Proposition prop:contact, Proposition prop:conditionalret, Lemma lem:hbaraudit, and Proposition prop:switch.