Landmark IX-E Binary time ordering and causal subtraction on the declared sharp comparison sector

We formulate the binary causal-extension problem on a fixed actual tensor domain inherited from IX-Free-2 and prove its algebraic consistency tests, its first-stage commutator identity, its double-ordering obstruction, an algebraic extension criterion, and the conditional unnormalized subtraction theorem. We correct the double-ordering geometry: two support orderings require causal disjointness, not overlapping functional supports. Neither full support of the locked contraction kernel nor injectivity of the first-stage maps determines the commutator.

The release outcome is Unresolved. The supplied information does not specify the restriction of $K-\overline{K}$ needed to evaluate the decisive causal test, an old time-ordered map on a nonzero comparison subspace, or the sharp distributional extension and continuity data. Consequently this manuscript does not claim a constructed causal product or a universal obstruction. Its proved results identify exactly what a subsequent construction or no-go proof must discharge. Degree-four free descent is not reopened.

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

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

The target is a complex-linear actual tensor map

on one fixed flip-stable comparison domain. An “actual” tensor means an element of the algebraic tensor product of actual functionals, rather than an unquotiented presentation by test functions.

There are three legal outcomes:

1. Constructed: all coefficients exist and every declared interface clause holds on the fixed domain.

2. Proved obstructed: a contradiction excludes every map satisfying that interface on the fixed domain.

3. Unresolved: at least one required consistency, extension, admission, contact, continuity, or overlap obligation remains undischarged.

Failure of one proposed replacement kernel excludes that candidate only. Removing an offending tensor changes the target and does not discharge a claim covering that tensor.

{{0.92} Release box. The domain chosen below contains all of $D_{\mathrm{first}}$ and its conjugates. No offending first-stage pair may subsequently be deleted. The outcome of the supplied-data audit is Unresolved. Free-word coherence does not imply causal factorization. No ${\mathsf{T}}_3$, $R_2$, general free-domain enlargement, G2 release, or repair of VII-C is asserted. }

The inherited hard stop is retained: the causal-product release is Unresolved because the time-ordered extension and its remaining causal, admission, and overlap obligations are unfinished, not because the degree-four free word has failed to descend.

. Injectivity is not an ordering comparison. The inherited equations

and the slab witness ${{\mathcal{Q}}^{\mathcal{R}}}_{g,f}\ne 0$ imply

They do not imply ${\mathcal{Q}}_{f,g}\ne {{\mathcal{Q}}^{\mathcal{R}}}_{g,f}$. For example, two injective linear maps can be the same map. The difference of the two contractions must be computed independently.

. Contacts cannot repair inconsistent prescribed values. If both support orderings hold, symmetry and factorization require the new value to equal both fixed free values. If those values are unequal, no alteration of the new value can satisfy both equations. This argument applies even on the permitted contact set.

. Fixed old coefficients must agree exactly. A contact is normalization freedom only where normalization is unfixed. A contact discrepancy from a supplied, fixed old coefficient remains a failed overlap until it is removed by a permitted normalization preserving all other conditions.

. Double ordering means causal disjointness. The sentence that a two-way pair requires overlapping functional supports is incorrect. With the ordering used here, both orderings exclude every causal relation between the two supports, including coincidence and null relation. Proposition prop:geometry proves the precise statement.

. A time slab proves an ordering, not necessarily its uniqueness. A later slab proves $W_g\gtrsim B_f$. It does not, by itself, rule out $B_f\gtrsim W_g$: supports in different time slabs can still be spacelike separated. Uniqueness of the ordering needs an actual causal relation between actual functional supports.

The following are imported premises. They are not proved anew in IX-E. The actual definitions of the sharp functional carrier, $C$, ${\mathsf{E}}_0$, the locked kernel $K$, and the old products belong to the prerequisite sources.

The labels C-I1–C-I9 are retained as inherited free-sector hypotheses. Their individual wording was not included in the supplied text; no purported verbatim reconstruction of those clauses is made here. The proofs below state the particular inherited properties they use explicitly.

The word “new” distinguishes the absent sharp extension from any old time-ordered product that may separately be supplied.

The inherited algebraic tensor domains are

The two associations of the inherited word agree with

Equation (eq:word) is free-word coherence. It supplies neither a binary time-ordered extension nor causal factorization.

No closure under intermediate multiplication is imported. In particular, an arbitrary expression

need not belong coefficientwise to the declared free domain.

Let $\sigma (F\otimes G)=G\otimes F$ be complex-linear flip. Let

be componentwise antilinear conjugation. Then

Conjugation fixes the formal parameter $\hslash$. The inherited reversal property is

The comparison domain is

Thus both free orderings exist on $D_{\mathrm{cmp}}$. Flip stability of all of $D_{\mathrm{decl}}$ is not assumed.

Admission includes the IX-A requirement

at every smooth configuration and for every finite list of required directions. The admitted coefficient space is treated as a complex vector space stable under conjugation. Admission of required descendants is part of the coefficient requirement, not an automatic consequence of a pairing existing.

The sharp adjoint identity is

Dropping its second term changes the functional; it is not a presentation rewrite.

Functional support is the closed support of configuration dependence. The imported bounds are

These are upper bounds. Their nonempty intersections do not prove nonempty intersections of the actual functional supports.

No nonzero old comparison subspace carrying an explicitly supplied old time-ordered map is present in the supplied data. The only old seed space on which a map is determined without additional information is

with its unique zero map. We therefore freeze the minimal full first-stage target

This choice does not claim a nonzero old time-ordered overlap. Any actual old intersection must still be audited:

One must not infer ${\mathcal{O}}_{\mathrm{act}}=0$ merely from $D_{\mathrm{new}}\cap D_{\mathrm{old}}=0$: conjugation saturation needs its own subspace information.

Under (eq:free-star), the domain (eq:fixed-domain) satisfies

It contains every tensor in $D_{\mathrm{first}}$.

Proof. By definition, $\sigma (D_{\mathrm{first}})=D_{\mathrm{first}}$. Since $D_{\mathrm{first}}\subset D_{\mathrm{decl}}$, this implies $D_{\mathrm{first}}\subset D_{\mathrm{cmp}}$. Also

Because dagger commutes with flip and preserves $D_{\mathrm{decl}}$,

Consequently $\gamma =\sigma †$ preserves $D_{\mathrm{cmp}}$. Every generator in (eq:fixed-domain) therefore belongs to $D_{\mathrm{cmp}}$. Applying either involution permutes these generators; antilinearity of $\gamma$ preserves their complex span.

The interface is classical multiplication, symmetry, coefficient and descendant admission, actual-presentation independence, exact specified old overlaps, causal factorization, anti-time-ordering involution, and one named fixed-support continuity statement. A nonzero old seed may be added only by a separately declared change of target, accompanied by its actual old map.

For actual functionals $F,G$, set

The causal relations include their null boundaries and coincidence.

Let $A={\mathrm{supp}}_{\mathrm{fun}}F$ and $B={\mathrm{supp}}_{\mathrm{fun}}G$. Both $F\gtrsim G$ and $G\gtrsim F$ hold exactly when no point of $A$ is causally related in either direction to a point of $B$. In particular, $A\cap B=\varnothing$.

Proof. The first condition excludes pairs $a\in A,b\in B$ with $a\in J^{-}(b)$. The second excludes pairs with $b\in J^{-}(a)$. These are precisely the two directions of causal relation. If $a=b$, reflexivity gives both relations, so coincidence is excluded as well.

A null-related pair in the actual supports violates at least one ordering condition. An ordered prescription cannot be extended to such a pair merely by taking a limit. Conversely, sufficiently separated spacelike supports satisfy both orderings and are part of the consistency test, not an unordered exceptional region.

In the flat half-space $M_{+}=[0,\infty )\times {\mathbb{R}}^n$, suppose $f,g$ are supported at times in $[0,T]$, with spatial supports in balls $\overline{B}(x_0,r)$, $\overline{B}(y_0,s)$, respectively. If

then the upper support sets in (eq:support-bounds) are causally disjoint. Hence both $B_f\gtrsim W_g$ and $W_g\gtrsim B_f$ hold.

Proof. A point of $P_F\cup F$ has time in $[0,T]$ and spatial distance at most $T+r$ from $x_0$. A point of $G$ has spatial distance at most $s$ from $y_0$. Their spatial separation is therefore greater than $T$, whereas their time separation is at most $T$. They are strictly spacelike. The actual functional supports are subsets of these sets.

Thus two-way first-stage pairs can be specified without knowing exact functional supports. What remains unknown is the locked kernel difference on their observation supports.

For the inherited slab witness, the bounds

give $W_g\gtrsim B_f$: no later point can lie in the causal past of an earlier point. They do not alone decide the reverse ordering.

The locked first-stage coefficients are

The brackets denote the supplied distributional pairing, including its configuration arguments. Classical configuration factors commute; no interchange of singular distributions is being made.

On the declared first-stage sector,

Proof. Subtract the two inherited first-stage free products. Their classical terms agree. Reorder only the smooth configuration factors and the scalar test factors in the second expression of (eq:locked-q). Distributional linearity gives (eq:commutator). Finally, $K-\overline{K}=2i\mathrm{Im}K$.

Equation (eq:commutator) is the calculation determined by the supplied kernel data. A further evaluation requires a formula, conjugation identity, or support theorem for $\mathrm{Im}K$. Full support of $K$ says nothing of this kind: a real distribution can have full support while $K-\overline{K}=0$.

Let

be a nonzero compactly supported distribution whose support is contained in the interior of $M_{+}^2$. Suppose the configuration space permits arbitrary smooth scalar and metric configurations. Then ${\mathcal{C}}_{f,g}\ne 0$ as an actual functional.

Proof. There are compactly supported smooth functions $u(x),v(y)$ with

Indeed, finite sums of tensor-product test functions are dense in the test-function space on a product. This can be seen by a partition of unity into product charts followed by smooth Fourier approximation on compact rectangles. Continuity of a distribution would force it to vanish if it vanished on every such tensor.

Extend $u$ smoothly by zero where necessary and choose

Then ${\partial }_t\varphi =u$. Choose the smooth metric configuration with $h_{00}=v$ and all other components zero. Equation (eq:commutator) gives a nonzero value. If only real configurations are used, decomposition into real and imaginary parts supplies real $u,v$ detecting the complex distribution.

No compactness of $\varphi$ is required: compactness belongs to the functional kernel, while configurations are smooth. If the configuration space is constrained, the realization assumption in this lemma must instead be proved within that space.

If $B_f\otimes W_g\in {\mathcal{D}}_2$, both support orderings hold, and ${\mathcal{C}}_{f,g}\ne 0$, no symmetric causally factorizing map ${\mathsf{T}}_2$ exists on ${\mathcal{D}}_2$.

Proof. Factorization and symmetry would give

This contradicts Proposition prop:commutator. The nonzero coefficient at order $\hslash$ remains nonzero in $\mathbb{C}[\hslash ]/({\hslash }^4)$. The argument is independent of every contact choice and every higher coefficient.

Theorem thm:obstruction is conditional, not an evaluated obstruction. Lemma lem:spacelike supplies geometric test regions; the supplied excerpt does not evaluate $K-\overline{K}$ there.

The symmetric candidate ${{\mathsf{T}}_2}^{\mathrm{cl}}(F,G)=FG$ fails factorization on the inherited ordered slab witness.

Proof. On that witness $W_g\gtrsim B_f$ and

The candidate has no such coefficient.

This proves the copied separation statement: free coherence, including (eq:word), does not itself furnish causal factorization.

Double-ordering tests are necessary but not sufficient. Every actual linear relation among prescribed tensors must also be respected. This obligation is distinct from the already closed degree-four descent.

Write $S(d)={\star }_c(d)$ on ${\mathcal{D}}_2$ and define

The inherited classical coefficient of $A$ is multiplication. Equation (eq:free-star) implies

Let $\{(d_i,t_i){\}}_{i\in I}$ be the collection of all prescribed values: $t_i=S(d_i)$ for every ordered pair, together with $t_i={{\mathsf{T}}_2}^{\mathrm{old}}(d_i)$ for every tensor on a supplied old overlap. The index set need not be finite. Assume these values and their required descendants are admitted, and their classical coefficients equal multiplication. Put

Let

Conjugation induces an antilinear involution on $Q$.

There exists a complex-linear map ${\mathsf{T}}_2:{\mathcal{D}}_2\to {\mathcal{P}}_{\mathrm{adm}}{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$ satisfying all prescribed values, classical multiplication, symmetry, and the binary involution identities if and only if the following finite-relation condition holds:

for every finite $I_0\subset I$ and every complex choice of coefficients.

This is an algebraic theorem. It does not assert fixed-support continuity or identify a distributional time-ordering construction.

Proof. Write ${\mathsf{T}}_2=A+X$. Symmetry is equivalent to $X(\sigma d)=X(d)$, so $X$ factors through $Q$. The binary involution is equivalent to

For prescribed tensors this requires the assignments

Any linear map must respect their finite relations, proving necessity of (eq:relation-test).

Conversely, that condition defines a complex-linear map $x$ on

The space $U$ is conjugation-stable, and the assignments give $x(\gamma u)=-x(u{)}^{\ast }$. Set $y=ix$. Then

On the real fixed-point space $U_{\mathbb{R}}=\{u\in U:\gamma u=u\}$, the map $y$ takes values in the real fixed-point space of $\hslash {\mathcal{P}}_{\mathrm{adm}}{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$.

Choose a real vector-space complement of $U_{\mathbb{R}}$ in $Q_{\mathbb{R}}$, extend $y$ by zero there, and complexify. Every $q\in Q$ has the decomposition

into real fixed-point vectors. The extended $y$ therefore remains conjugation-equivariant. Define $X=-iy$, pulled back to ${\mathcal{D}}_2$, and ${\mathsf{T}}_2=A+X$.

The range of $X$ lies in $\hslash {\mathcal{P}}_{\mathrm{adm}}{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$, so the classical coefficient is unchanged. The construction gives symmetry, prescribed values, and (eq:X-involution). All outputs are finite linear combinations of admitted coefficient functionals. Linearity of the carrier conditions preserves their required descendants.

For a two-way pair the quotient identifies $[d]$ and $[\sigma d]$, while the forced corrections have opposite signs. Condition (eq:relation-test) therefore requires their common correction to vanish, reproducing Theorem thm:obstruction.

The arbitrary complement in the proof is not a continuity argument. It can produce a discontinuous map even when every prescribed coefficient is admitted. The theorem consequently cannot be used to label the complete IX-E interface Constructed without the remaining analytic proof.

Only after consistency of the forced data is established should an unordered extension be proposed. For every contraction, the construction must specify:

1. the exact inherited free distribution and slot convention;

2. the proposed time-ordered distribution;

3. its differential identity and source-contact normalization;

4. its restrictions to every ordered region;

5. the singular set requiring extension;

6. the residual actual functional after contraction.

The flat contractions mentioned in the prerequisite sheet include $K$ and

Their names do not specify a time-ordered replacement. In particular, replacing a differentiated contraction requires deciding whether differentiation is applied to a fundamental solution or whether a distinct differentiated time-ordering convention is used.

The sharp metric slot must be derived from the actual functional derivative through $L={\mathsf{E}}_{0}C$, retaining (eq:sharp). Time ordering by observation times alone is not justified by (eq:support-bounds). Nor may differentiation, sharp Cauchy inversion, or the sharp adjoint be silently commuted with time ordering.

The standard perturbative algebraic framework distinguishes time-ordering on regular functionals from the renormalized extension needed for local functionals FR. Its Feynman-kernel formulas must be used with their distributional hypotheses FR. Those results do not, without a separate verification of hypotheses, construct this nonlocal sharp sector.

A wavefront analysis must determine the actual unresolved set. Observation diagonals, the Cauchy surface, and the reflected branch of $K$ are candidates until excluded. The binary target does not require importing the degree-four triangle solely because that triangle exists in the inherited word.

For every coefficient, the proof must establish existence of its distributional pairings, compact configuration kernels, the metric derivative condition (eq:admission), all required descendants, and annihilation of actual-presentation kernels.

Existence on a test family does not establish admission. In particular, an expression omitting the boundary contribution in (eq:sharp) cannot be justified by referring to a different presentation of the same functional.

For definiteness, a possible weak fixed-support target is the following. Give ${\mathcal{P}}_{\mathrm{adm}}$ the locally convex topology generated by

for all configurations and finite direction lists under consideration. For every named test-function presentation with supports in fixed compact sets, require each $T_j$ to be continuous from the product of the corresponding ${C^{\infty }}_K$ spaces into this topology.

This is one explicit continuity strength; it is not asserted to follow from the supplied F9a statement. It neither claims normal-topology continuity on the full carrier nor hypocontinuity of multiplication. A different topology may be chosen, but it must be named and proved, rather than inferred from the absence of the joint VII-C assumptions.

If a finite-scaling-degree point-extension theorem applies in dimension $n$, uniqueness holds below degree $n$; at higher degree the degree-preserving freedom is a finite sum of derivatives of delta of order at most $\lfloor \mathrm{sd}(u)-n\rfloor$ BF. Application to a submanifold requires the corresponding transverse and microlocal hypotheses BF. No such hypotheses or degrees have been evaluated here for the sharp Cauchy or reflected loci.

The parent identity

classifies its triangle finite part. It does not classify binary time-ordered ambiguities.

On every supplied old overlap one must compute

and solve for permitted contacts so that this difference is exactly zero. A smooth nonzero spacelike discrepancy away from the named contact set is not a contact.

All coefficients of ${\star }_c$, all supplied old free and time-ordered coefficients, and the parent parameter $c$ remain fixed. No subtraction inside ${\star }_H$ is authorized.

If source slots are included, differentiating

shows why the simultaneous prescription

cannot retain that fundamental-solution identity when $c_F\ne 0$. This excludes that simultaneous prescription; it does not exclude every possible sharp causal extension.

Define anti-time-ordering by

The required identity is

Using (eq:A), equations (eq:antitime)– (eq:time-star) are exactly

Symmetry instead requires $X(\sigma d)=X(d)$. The two equations are different.

Contacts must solve the overlap, ordered-restriction, admission, symmetry, and involution equations simultaneously. Averaging a candidate is not proof that its old overlap or causal restrictions survive. The algebraic theorem isolates these constraints, but does not solve the sharp distributional contact problem.

The next result audits directly the binary implication quoted from IX-D. It does not presume that its hypotheses have been discharged.

Suppose ${\mathsf{T}}_2$ is constructed on ${\mathcal{D}}_2$ with admitted coefficients and required descendants, classical multiplication, factorization, exact supplied old overlaps, and (eq:antitime)–(eq:time-star). Suppose the inherited free map has the corresponding admission and old-overlap properties. Define

Then:

1. $\hat{\mathsf{R}}(d)\in {\mathcal{P}}_{\mathrm{adm}}{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$, including required descendants;

2. its classical coefficient vanishes;

3. its supplied old unnormalized retarded overlaps agree exactly;

4. for $d=V\otimes O$, it vanishes if

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

5. $\hat{\mathsf{R}}(d{)}^{\ast }=\hat{\mathsf{R}}(\gamma d)$.

If both input maps satisfy the same named fixed-support continuity statement into a locally convex coefficient space, their difference satisfies that statement.

Proof. Admission and descendant admission follow from linearity of the admitted carrier. The classical coefficients of the two maps are both multiplication, so they cancel. Exact old overlap follows by subtracting the two exact old values.

The stated support condition is equivalent to

Thus $V\gtrsim O$, and factorization makes the two terms in (eq:Rhat) equal.

For conjugation, use (eq:free-star) and (eq:time-star):

Finally, addition and scalar multiplication are continuous in a locally convex vector space. No multiplication-continuity upgrade is used.

The support assertion concerns the stated functional-support ordering of a pair. It is not a distributional support theorem for independently unsmeared argument slots.

The linear coefficient-shift map

is well defined. Order-three numerator data do not determine an order-three normalized coefficient in ${\mathcal{P}}_{\mathrm{adm}}{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$.

Proof. The coefficient expansion in ${\mathcal{P}}_{\mathrm{adm}}{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$ is unique, so $L$ is well defined. Within ${\mathcal{P}}_{\mathrm{adm}}{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$, multiplication by $\hslash$ annihilates every ${\hslash }^{3}C$. Hence, for every $C\in {\mathcal{P}}_{\mathrm{adm}}$,

The coefficient $C$ is undetermined. Equivalently, $\hslash$ has no inverse in $\mathbb{C}[\hslash ]/({\hslash }^4)$.

If the conditional construction is completed, the clean normalized output is the coefficient shift

with any phase convention stated explicitly. A zero choice for an additional ${\hslash }^3$ coefficient is a truncation convention, not a coefficient derived from order-three numerator data. No normalized product is released here because ${\mathsf{T}}_2$ has not been constructed.

Higher retarded identities require higher time ordering and the relevant intermediate products. Consequently neither $R_2$ nor ${\mathsf{T}}_3$ is defined.

Relative to the supplied prerequisite information, the IX-E release outcome on (eq:fixed-domain) is Unresolved. No Constructed or Proved obstructed outcome is established.

Proof. A Constructed outcome requires a map satisfying every declared clause. No sharp replacement distribution, admitted coefficient construction, complete old-overlap solution, or continuity proof has been supplied.

A Proved obstructed outcome requires an evaluated incompatibility. Theorem thm:obstruction supplies such a criterion, but its nonzero kernel premise has not been established on a two-way pair. Likewise, no violation of (eq:relation-test) has been evaluated. The ordered slab excludes classical multiplication only. Therefore the remaining obligations have precisely the third legal status.

The earliest unevaluated causal calculation is the restriction of

to admitted pairs for which the support geometry supplies both orderings, followed by the full actual-relation test. The exact locked kernel formula or an appropriate conjugation-support theorem is needed to finish it. Independently, a nonzero old-overlap claim requires the actual old map, not merely its name.

The inherited product ${\star }_c$ is not ${\mathsf{T}}_2$ and is not $R_1(V_1;B_f)$. The triangle scalar $U_c$ is not a causal subtraction. V's $R_{1,1}$, the oscillator coefficient, and the Newton–wave cubic remain different objects. VI's relational coefficient and VII-A's plateau intertwiner are not the map sought here. VII-E/F and VIII certificates have no supplied map into ${\mathcal{P}}_{\mathrm{adm}}$.

A cutoff or numerical modification of $U_c$ supplies none of the missing order-$\hslash$ and order-${\hslash }^2$ binary coefficients. The parent finite-part conversion also supplies no binary overlap classification.

The compact-switch witness copied from IX-A and IX-D is

with nonzero dressing variation under its stated hypotheses. In the coupling expansion its nonzero order-$\kappa$ coefficient cannot be canceled by a correction beginning at order ${\kappa }^2$: coefficient extraction at order $\kappa$ annihilates every such correction. Binary time ordering therefore does not open G2.

VII-C remains in force under its original joint hypotheses (E1)–(E4), which are distinct from the gate labels of this paper. Subtracting a scalar finite part does not prove normal-topology admission continuity. F9a and separate pointwise continuity are not promoted to F9b or hypocontinuity.

The first-stage commutator is controlled by the imaginary part of the locked kernel, not by full support or injectivity. Double ordering means causal disjointness of functional supports. A nonzero commutator on any retained two-way pair is a universal obstruction on the fixed domain, and no contact can remove that contradiction without changing prescribed free data.

If the prescribed values instead satisfy all actual relations, Theorem thm:algebraic supplies an algebraic extension criterion. Its arbitrary linear extension is not a substitute for a sharp distributional construction or continuity proof. Admission, contacts, old overlap, and continuity must still be established at the declared analytical strength.

Once those obligations are discharged, Theorem thm:subtraction supplies the unnormalized binary retarded difference through ${\hslash }^3$. Normalization shifts the determined order to ${\hslash }^2$. With the information presently supplied, the honest release remains Unresolved; the free-sector descent and every stated external boundary remain unchanged.

FR

K. Fredenhagen and K. Rejzner, Perturbative algebraic quantum field theory, arXiv:1208.1428 [math-ph], 2012. Used for the distinction between regular time-ordering formulas and renormalized extensions; not invoked as an existence theorem for the sharp sector.

BF

R. Brunetti and K. Fredenhagen, Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds, Communications in Mathematical Physics 208 (2000), 623–661; arXiv:math-ph/9903028. Used only for extension principles under their stated scaling-degree and microlocal hypotheses.

Internal

Internal prerequisite statements: IX-A, IX-Free-2, and IX-D, as quoted in the supplied IX-E specification. The full source manuscripts, the individual wording of C-I1–C-I9, the locked formula for $K$, and a nonzero old time-ordered comparison map were not supplied. Statements imported from these prerequisites are identified as premises rather than reconstructed proofs.