IX-D-ST: Causal interfaces and a conditional functional-support bridge on a prescribed cone Ordered local Grade S adaptations of IX-D, IX-E, IX-E-2, and IX-E-3

We organize the causal-interface programme following the constructed sharp free partial product as four ordered, supported stages on the regular part of the prescribed I-B conical spacetime. The adapted IX-A-ST carrier and the IX-Free-2 coefficient assignment are explicit inherited inputs; neither is reinterpreted as time ordering.

The principal analytical result is a complete common-regulator evaluation of the imaginary part of the locked mixed kernel. A direct null-cone distribution is accompanied by a nonzero observation-spacelike contribution between the direct and reflected cones; the two cosine tails cancel beyond the reflected cone. Consequently the commutator vanishes on the entire strongly separated two-way class, while a nonzero observation-spacelike control is not, by itself, a causal obstruction.

We prove exact functional support for the quadratic slot and show that a boundary-trace certificate implies a null-sphere lower-support theorem for the sharp insertion. Those spheres suffice for the full retained two-way commutator bridge; a solid-ball lower-support assertion is unnecessary and is false in the specified flat realization. We also prove the scalar non-lift, an actual-tensor algebraic consistency criterion, conditional causal subtraction, and the one-order loss under normalization.

The potential usefulness is methodological: the results separate a genuine fixed-domain obstruction from an uncertified support inference and identify a precise operator certificate for the next analytical step. The underlying algebraic and kernel findings are attributed to the cited Cheung manuscripts; the present contribution is their supported conical organization, explicit hypothesis audit, and finite certificate interface. Verification of the trace certificate for an otherwise unspecified inherited adjoint remains open. No sharp time-ordered product, normalized retarded product, interacting quantum coefficient, or spacetime-emergence theorem is constructed.

file, four ordered releases{ The requested one-file delivery is followed. The four parts retain distinct stage boundaries and do not assert that registry entries have been installed.

“Complete” means complete proofs of the stated interface, kernel, algebraic, and conditional support results. It does not mean a solution of the explicitly retained time-ordering problem. The final causal-product verdict is Unresolved. No proof of universal nonexistence is claimed.}

The programme begins after the construction of one free partial product, not before it. IX-D separates that product from causal interfaces; IX-E formulates the binary extension problem; IX-E-2 evaluates the missing kernel difference; and IX-E-3 identifies the functional-support certificate needed to use that evaluation on all retained two-way pairs IXD,IXE,IXE2,IXE3.

The finite operational background is Enter's separation between a consumed quantum input name, a classical record, and an unmeasured residual. Its operational characterization has its own structural assumptions EnterArticle,EnterI. We adopt finite Born instruments and prove the complete channel identities needed here; we do not re-prove that characterization.

The philosophical background is a discipline of non-identification: destructive readout is not quantum recovery, a channel is not a thermodynamic machine, and an operational analogy does not supply a geometric dictionary EnterPhil. In I-B, geometry, locations, operational permissions, and clock calibration are supplied separately. The finite clock is not the geometric time axis IB.

A local Grade S package consists of:

1. a prescribed host and a supported regular laboratory;

2. mathematical objects with explicit domains and inputs;

3. complete finite channels, including untouched references;

4. one finite destructive-clock factor with stated calibration;

5. computational dependency bounds, distinguished from physical causal containment;

6. a nonempty regular transport realization with durations and resource counts.

These clauses do not assert emergence of the metric. Grade E is not constructed.

and novelty. The source-level mathematical findings are attributed to N. Cheung. No priority claim over IXD,IXE,IXE2,IXE3 is made. The supported adaptation and its hypothesis audit are different from an independent verification of the whole source corpus.

kinds of missing result. Failure to construct a map is not proof that every such map is impossible. Conversely, a conditional theorem is a proved implication, not a discharge of its hypothesis. Both distinctions are maintained in every release below.

Fix $0<{\delta }_{\mathrm{c}}<2\pi$, $\alpha =1-{\delta }_{\mathrm{c}}/(2\pi )$, and $c_{\mathrm{light}}>0$. The prescribed regular host is

Its stipulated completion has defect

Thus the defect is a worldsheet.

The spacetime (eq:host) has a noncompact causal diamond.

Proof. Choose a regular spatial point of radius $r_{\ast }>0$, and two events there separated by more than $2r_{\ast }/c_{\mathrm{light}}$. At an intermediate time, points on the same radial ray with radii tending to zero lie in their causal diamond. They have no convergent subsequence in the regular product.

Choose a product-adapted developing chart

whose closure over the retained laboratory stays off $\mathcal{W}$. The developed metric is $-\mathrm{d}{t}^2+\mathrm{d}{\mathbf{x}}^2$. For analytical comparison formulae use

Changing the overall signature convention does not change the causal relation.

Fix

and retain the entire compact shadow

All observation tests used for the free sector are in $\mathcal{D}(V;\mathbb{C})$. Their complete truncated chart pasts, including their initial sphere points, lie in the retained laboratory.

For developed points in the laboratory,

Both null boundaries and coincidence are included. All occurrences of $J^{\pm }$ below mean these chart sets. They are not assertions about all causal curves of the global conical host. For a compact observation set $F$, put

Every $P_F$ used here is compact and retained in $K_{\ast }$. No finite-propagation assertion crosses the worldsheet.

The carrier is IX-A-ST's adapted $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$, with the following explicitly used properties IXAST.

1. Elements are actual off-shell polynomial functionals with compact distributional configuration kernels. Equality means equality on all smooth configurations, not equality modulo equations of motion.

2. Smooth configurations and compactly supported smooth variations are unrestricted in the retained chart.

3. Classical pointwise multiplication, conjugation, and fixed smooth directional differentiation preserve the carrier. Multiplication by a fixed classical factor is continuous.

4. Every coefficient must satisfy the inherited metric admission condition

   $$({\mathrm{D}}_{\mathbf{\eta }}F{)}_h^{\prime }(\Psi )\in {\mathcal{S}}_1={\mathcal{M}}_1+{\mathcal{Q}}_{\Sigma }$$

   for every configuration and required finite direction list. Existence of a distributional pairing alone is not admission.

5. The supplied local response $L={\mathsf{E}}_{0}C$ has the retained support property: a metric variation outside $P_F$ cannot change $Lh$ on $F$. The IX-A-ST saturation statement makes its retained response independent of auxiliary extensions. No global conical inverse is supplied.

6. The sharp metric adjoint, on its supplied source domain, retains the actual identity

   $$L^{\ast }a=\theta (t)C^{\ast }b+\delta (t)\mathcal{B}(b{|}_{\Sigma }),b={\mathsf{E}}_0^{\ast }a.$$

   The source domain and the actual adjoint must be matched before a formula for $b{|}_{\Sigma }$ is imported.

7. Unchanged regular approximants do not converge:

   $$B_{f,\epsilon }/⟶B_f\text{in }{\tau }_{\Sigma }(0\ne f\text{ real}).$$

The boundary-trace certificate (TC) of Part IV is not included in this assumption.

Saturation is a local response-independence statement, not a global Green-operator theorem. In a fully specified flat zero-Cauchy realization, an adjoint trace can be proved by transposition. Part IV does this for that realization. The weak inherited interface above does not silently identify its complete adjoint source domain with that realization. Global Green-operator results require their own hypotheses BGP.

Changing the parent parameter, regulator description, stage tag, or a proposed time-ordering token leaves (eq:host), $\Sigma$, and $\mathcal{W}$ unchanged.

Proof. They are description-register entries. No metric update is in the command's output type.

Configurations are $\Psi =(h,\varphi )$. Set

Pairings are complex linear in their tests.

Use $x=(t,\mathbf{x})$, $y=(s,\mathbf{y})$, $x^{\prime }=(t^{\prime },{\mathbf{x}}^{\prime })$, and

The locked mixed comparison kernel is

It is a chart comparison formula, not a covariance derived from the conical host.

The first stages are locked as

The minus sign and the factor $2$ are fixed.

For later support bookkeeping write

Define

The inherited two-edge values have residual forms

The IX-Free-2 record, or the supported IX-C-ST predecessor when accepted with its stated inputs, supplies IXFreeTwo,IXCST:

1. The displayed free coefficients exist in ${\mathcal{P}}_{\Sigma }$, including required directional descendants.

2. With

   $$\mathcal{B}=\{B_f\},\mathcal{W}=\{W_g\},\mathcal{L}=\mathrm{span}\{B_f{W}_g,Q_{f,g}\},\mathcal{R}=\mathrm{span}\{W_g{B}_f,{Q^R}_{g,f}\},$$

   the actual algebraic tensor domains are

   $$D_{\mathrm{first}}=(\mathcal{B}\otimes \mathcal{W})+(\mathcal{W}\otimes \mathcal{B}),$$
   $$D_{\mathrm{second}}=(\mathcal{L}\otimes \mathcal{B})+(\mathcal{B}\otimes \mathcal{R}),D_{\mathrm{new}}=D_{\mathrm{first}}+D_{\mathrm{second}},$$
   $$D_{\mathrm{old}}={\mathcal{P}}_{\mathrm{mc}}\otimes {\mathcal{P}}_{\mathrm{mc}},D_{\mathrm{decl}}=D_{\mathrm{new}}+D_{\mathrm{old}},D_{\mathrm{new}}\cap D_{\mathrm{old}}=0.$$

   The old free coefficients are unchanged.

3. There is one linear coefficient assignment

   $$S={\star }_c:D_{\mathrm{decl}}⟶{\mathcal{P}}_{\Sigma }{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4).$$

   It is extended linearly to formal tensors. No arbitrary intermediate closure is supplied.

4. Both degree-two presentation maps and both degree-four presentation maps are injective. In particular, ${Q^R}_{g,f}\ne 0$ for nonzero $g\otimes f$.

5. For one common real parent $c$,

   $$\begin{array}{rl}{A}_c(f,g,f^{\prime })=& (B_f{\star }_c{W}_g){\star }_c{B}_{f^{\prime }}=B_f{\star }_c(W_g{\star }_c{B}_{f^{\prime }})\\ =& B_f{W}_g{B}_{f^{\prime }}+\hslash (G_{12}+G_{13}+G_{23})\\ & +{\hslash }^2(G_{12,23}+G_{12,13}+G_{13,23})+{\hslash }^3{\mathcal{N}}_c(f,g,f^{\prime })1(\mathrm{mod}{\hslash }^4).\end{array}$$

6. If $\sigma (F\otimes G)=G\otimes F$ and $\gamma (F\otimes G)=F^{\ast }\otimes G^{\ast }$, then

   $$d^{†}=\sigma \gamma d,{D_{\mathrm{decl}}}^{†}=D_{\mathrm{decl}},S(d{)}^{\ast }=S(d^{†}).$$

   Conjugation fixes formal $\hslash$.

7. Fixed-support continuity holds on the named generating families, denoted F9a. Actual-stage continuity F9b and quantum hypocontinuity are not supplied.

No conclusion below reopens degree-four free descent.

The scalar in (eq:word) is

with the source-locked VII-D/IX-B normalization

Its supplied strong numerical estimate is

on arbitrary joint tests in a fixed retained compact set. The parent conversion class is $\mathbb{R}r$, not an evaluated binary time-ordering ambiguity class VIID,IXB.

The numerical facts ${\mathcal{C}}_{\mathrm{desc}}={\mathcal{C}}_{\mathrm{glue}}=\mathbb{R}$ are inherited free-sector facts. They impose no causal normalization and do not run the host curvature.

-D-ST: causal interfaces after the free product

A point belongs to ${\mathrm{supp}}_{\mathrm{fun}}F$ if every neighbourhood admits a compactly supported smooth perturbation that changes $F$ at some smooth configuration. For formal polynomials, support means the union of the supports of their finitely many coefficients. Species supports ${\mathrm{supp}}_{h}F,{\mathrm{supp}}_{\varphi }F$ use only the corresponding variations.

Define

On the unrestricted affine configuration space, ${\mathrm{supp}}_{\mathrm{fun}}F$ is the closure of the union of the supports of all first functional derivatives, over all smooth configurations and species.

Proof. If every derivative vanishes on perturbations in an open set, integration along $\Psi +\lambda k$, $0\le \lambda \le 1$, shows that every such perturbation leaves $F$ unchanged. Conversely, if a derivative pairs nontrivially with a variation, the directional function has nonzero derivative and is not locally constant. Taking all configurations, local variations, and closures gives the assertion.

The domain

is stable under $\sigma ,\gamma ,†$. It contains $D_{\mathrm{first}}$.

Proof. Flip stability is immediate. Dagger stability of $D_{\mathrm{decl}}$ gives $\gamma (D_{\mathrm{decl}})=\sigma (D_{\mathrm{decl}})$. Thus

The first-stage domain contains both orderings, so is contained in $D_{\mathrm{cmp}}$.

A proposed bosonic causal construction must provide:

1. actual tensor domains $D_2$, and any required $D_3$, with all necessary flips, conjugates, and intermediate domains specified;

2. a symmetric linear map

   $${\mathsf{T}}_2:D_2⟶{\mathcal{P}}_{\Sigma }{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4),T_0(F,G)=FG,{T_1}^{\mathrm{arity}}(F)=F;$$

   the last normalization concerns arity one, not $\hslash$-order one;

3. admitted coefficients and all required descendants, with actual-presentation independence;

4. causal factorization on every retained comparison pair:

   $${\mathsf{T}}_2(F,G)=S(F\otimes G)\text{if }F\gtrsim G;$$

5. exact values on every separately supplied old time-ordered overlap, and a specified admissible contact class;

6. anti-time ordering and involution:

   $${\overline{\mathsf{T}}}_2(d)=S(d)+S(\sigma d)-{\mathsf{T}}_2(d),{\mathsf{T}}_2(d{)}^{\ast }={\overline{\mathsf{T}}}_2(\gamma d);$$

7. a continuity theorem on named domains at a specified strength.

Outside $D_{\mathrm{cmp}}$, every additional free comparator must be independently constructed. Higher causal factorization must name every product it uses.

In particular, symmetry does not mean ${\mathsf{T}}_2(F,G{)}^{\ast }={\mathsf{T}}_2(F^{\ast },G^{\ast })$. The anti-time-ordered map in (eq:antitime) is essential.

A retarded construction must specify its actual argument domain, normalization convention, admitted coefficients and descendants, old retarded overlaps, and a named continuity statement.

For independently unsmeared source or observation slots, a distributional retarded-support assertion additionally requires a specified multilinear distribution and a proof of its support in the declared causal relation. Vanishing for one functional-support ordering is not that slot-support theorem.

A second retarded order requires higher time ordering and all intermediate products in its defining identity. None is supplied by writing an arity-two subtraction.

Assumption ass:free2 supplies neither a system satisfying ${\mathcal{I}}_{\mathrm{TO}}$ nor one satisfying ${\mathcal{I}}_{\mathrm{ret}}$. In particular, (eq:word) is not causal factorization.

Proof. The free assignment has a specified free tensor domain and coefficient values. The time-ordered maps, ordered restrictions, anti-time-ordering equations, old time-ordered overlaps, and retarded slot distributions in the requirements are additional objects. They are not among its outputs. The explicit nonimplication witness in Lemma lem:separation below shows that even adjoining a symmetric admitted auxiliary multiplication does not supply the missing causal axiom.

Let $F=\mathrm{supp}f$, $G=\mathrm{supp}g$, and $F^{\prime }=\mathrm{supp}{f}^{\prime }$. The inherited outputs have the following bounds:

{}1.15

All inclusions are relative to the supported laboratory.

Proof. The supplied response-support property gives the metric bound for $B_f$; its scalar slot is integrated only on $F$. The assertion for $W_g$ is immediate.

The kernels in the displayed free outputs are fixed comparison distributions, not varying configurations. Only their residual fields contribute configuration dependence. Each residual $w$-slot on an observation set contributes its corresponding $P_F$; each local $h_{00}$-slot contributes $G$; each scalar derivative slot contributes its observation support. A classical product depends only on the union of its factors' dependence sets. These observations prove every row directly.

These are the supported versions of IX-D's support ledger IXD ({:support}). They do not assert support of an unconstructed retarded argument-slot distribution.

Let ${\mathrm{ev}}_{0}F=F[0,0]$, coefficientwise. Define

The constructed free word satisfies

Neither projection defines ${\mathsf{TO}}_2$, ${\mathsf{T}}_2$, ${\mathsf{Ret}}_1$, or a higher retarded map.

Proof. Every nonconstant term in (eq:word) has positive field degree and vanishes at zero configuration. The displayed identities follow.

The projection has a large kernel: ${\Pi }_{3,0}(\hslash F)=0$ for every $\hslash$-independent $F$, and ${\mathrm{ev}}_0({\hslash }^{j}F)=0$ whenever $F[0,0]=0$. It loses nonconstant coefficient information and has no supplied rule assigning missing binary coefficients, ordered restrictions, or old causal overlaps. A change in a nonconstant coefficient may leave the projection unchanged. This is information loss, not permission to change any locked free coefficient.

The inherited free coherence is compatible with the symmetric, admitted auxiliary map $M_2(F,G)=FG$, although that map fails causal factorization on a retained ordered first-stage pair.

Proof. Choose nonzero tests with

The support bounds imply $W_g\gtrsim B_f$. Both free orderings belong to the first-stage domain.

Classical closure makes $M_2$ admitted and symmetric. It changes none of the inherited free data. But

by the inherited injective presentation map. Thus $M_2(W_g,B_f)\ne S(W_g\otimes B_f)$.

The slabs prove the displayed ordering, not necessarily its uniqueness: different time slabs may still be spacelike separated. The lemma excludes the classical candidate, not every causal extension.

For this subsection only, let $P_{\mathrm{f}}$ be a specified formally self-adjoint field operator and $H_{\mathrm{F}}$ a specified fundamental solution with

The actual source presentation is $j=P_{\mathrm{f}}h$. Differential compatibility requires ${H^{\mathrm{TO}}}_{jh}=(P_{\mathrm{f}}{)}_x{H}_{\mathrm{F}}$. The declared test domain detects the diagonal contact.

Under Assumption ass:source, the additional prescription ${H^{\mathrm{TO}}}_{jh}=0$ is inconsistent.

Proof. Compatibility forces ${H^{\mathrm{TO}}}_{jh}=c_{\mathrm{F}}\delta$. A nonzero real compact test $q$ gives

For a bundle-valued identity select a nonzero diagonal component and matching bundle tests.

The statement excludes only those simultaneous assumptions. An independent central source need not be identified with $P_{\mathrm{f}}h$. A free bisolution equation $P_{\mathrm{f}}H=0$ is a different input. The two-slot contact is not the three-observation residue $r$, and cannot be absorbed merely by changing $c$ IXD ({:contact}).

For the implication below only, suppose:

1. a symmetric actual map ${\mathsf{T}}_2:D_2\to {\mathcal{P}}_{\Sigma }{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$ is constructed, with $\sigma ,\gamma$-stable comparison domain $E_2=D_2\cap D_{\mathrm{cmp}}$;

2. all its coefficients and required descendants are admitted, and actual-presentation descent holds;

3. its classical coefficient is multiplication, and ${\mathsf{T}}_2(F,G)=S(F\otimes G)$ whenever $F\gtrsim G$;

4. (eq:antitime) holds;

5. all claimed old time-ordered and retarded overlaps are supplied and agree exactly, with old unnormalized retarded convention $i({\mathsf{T}}_2-S)$;

6. any claimed continuity is proved on the same named comparison interfaces.

None of these new time-ordering hypotheses is discharged by this assumption's declaration.

Under Assumption ass:future,

has admitted coefficients and descendants, zero classical coefficient, and exact supplied old unnormalized overlaps. It satisfies

and

A common named fixed-support continuity property passes to this difference.

Proof. Admission, descendant admission, and old overlap follow by linear subtraction. The order-zero coefficients cancel.

The support condition is $V\gtrsim O$, so factorization makes the two terms equal. For involution use free reversal and (eq:antitime):

Continuity follows from continuity of addition and scalar multiplication in the named locally convex range.

The vanishing is an ordered-functional-pair assertion, not an independently unsmeared slot-support theorem.

There is a well-defined coefficient shift

Numerator data through order ${\hslash }^3$ determine a normalized quotient only through order ${\hslash }^2$.

Proof. The coefficient expansion gives the shift uniquely. In ${\mathcal{P}}_{\Sigma }{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$, multiplication by $\hslash$ annihilates every ${\hslash }^{3}C$. Hence that final coefficient of a putative quotient is undetermined. Equivalently, $\hslash$ is not invertible in $\mathbb{C}[\hslash ]/({\hslash }^4)$.

Since ${\mathsf{T}}_2$ is not constructed, no normalized $R_1=\mathcal{L}({\widehat{\mathsf{Ret}}}_1)$ is released, even at the lower determined order. Neither $R_2$ nor ${\mathsf{T}}_3$ is defined.

Retain the copied first-order witness, on its original hypotheses,

In a coupling expansion, a correction beginning at ${\kappa }^2$ cannot cancel this nonzero order-$\kappa$ variation.

Proof. The order-$\kappa$ coefficient of a correction divisible by ${\kappa }^2$ is zero. The copied nonzero coefficient therefore remains unchanged.

This preserves the witness conditionally; it does not construct its switched successor, a new gauge action, or G2.

Part I proves the causal-interface specification, laboratory-local configuration-support bounds, scalar non-lift, free/causal separation, conditional source-contact incompatibility, conditional subtraction, and normalization audit.

The sharp causal-product release remains Unresolved. No sharp ${\mathsf{TO}}_2$, ${\mathsf{T}}_2$, normalized $R_1$, or $R_2$ is constructed.

Proof. The preceding results establish the stated claims. The future time-ordered hypotheses have not been supplied as a construction, so the subtraction implication cannot be instantiated as an unconditional causal product.

-E-ST: binary consistency on a fixed actual domain

No nonzero old time-ordered seed is supplied by the input. Set

This target retains every first-stage tensor, its flip, and its conjugates.

The domain (eq:Dtwo) is contained in $D_{\mathrm{cmp}}$ and is stable under $\sigma ,\gamma ,†$. An old time-ordered map on a nonzero comparison space is not supplied by declaring $O_2=0$.

Proof. Lemma lem:comparison puts every generator in $D_{\mathrm{cmp}}$. The involutions permute the generators and preserve their complex span. The zero seed defines only the zero map on the zero space.

Write

No old time-ordered value is inferred merely from the existence of the old free product. Under additional conjugation-stability information the intersection may be evaluated, but that requires its own proof. No retained tensor is deleted after a failed test.

Both $F\gtrsim G$ and $G\gtrsim F$ hold exactly when their actual functional supports have no causally related pair in either direction. In particular those supports are disjoint, with neither null relation nor coincidence.

Proof. The two equations respectively exclude $a\in J^{-}(b)$ and $b\in J^{-}(a)$ for support points. Reflexivity excludes coincidence.

A null-related support pair is outside at least one ordered region. A limit from ordered pairs does not define its missing value without a separately proved topology and extension.

Suppose $f,g$ have times in $[0,T]$ and spatial supports in $B({\mathbf{x}}_0,r)$ and $B({\mathbf{y}}_0,s_0)$. If

and all retained shadows fit in the laboratory, then both actual orderings hold.

Proof. A point of $P_F\cup F$ has time in $[0,T]$ and spatial distance at most $T+r$ from ${\mathbf{x}}_0$. Its distance from any point of $G$ is therefore greater than $T$, while their time difference is at most $T$. The upper support sets are strictly spacelike, and actual supports are their subsets.

This constructs chart comparison regions, not a Lorentz cone from operational data.

On the first-stage sector,

Proof. Subtract (eq:firststages). Their classical terms coincide. Only smooth configuration factors are reordered. Distributional linearity and $K-\overline{K}=2i\mathrm{Im}K$ give the result.

Neither full support of $K$ nor injectivity of either presentation map computes this difference. A real distribution may have full support.

If $fg(K-\overline{K})$ is a nonzero compact distribution in the interior observation chart, then ${\mathcal{C}}_{f,g}\ne 0$ as an actual functional.

Proof. Density of finite sums of product tests gives smooth compact $a(x),b(y)$ with a nonzero pairing. Choose a smooth scalar with

and choose $h_{00}=b$, all other components zero. These are allowed off-shell configurations. For real-only configurations, real and imaginary decomposition supplies a real detecting pair.

If a retained pair satisfies both actual orderings and ${\mathcal{C}}_{f,g}\ne 0$, no symmetric causally factorizing map ${\mathsf{T}}_2$ exists on the fixed $D_2$.

Proof. Factorization and symmetry would force

contradicting the nonzero order-$\hslash$ coefficient. Higher orders or a contact added to the proposed time-ordered value cannot satisfy two unequal prescribed equalities.

The quantifier covers every extension on this fixed domain that retains the specified free values. Without a certified nonzero two-way pair, it is a conditional obstruction criterion, not an evaluated no-go theorem IXE.

On $D_2$, both free orders exist. Define

It has classical coefficient multiplication and obeys

Let $(d_i,t_i)$, $i\in I$, be all forced ordered values and any separately supplied old time-ordered values. Set

The index set need not be finite.

A complex-linear map on $D_2$ satisfying the prescribed values, symmetry, classical multiplication, and (eq:antitime) exists algebraically if and only if, for every finite relation,

This conclusion includes no sharp distributional construction or continuity theorem.

Proof. Write ${\mathsf{T}}_2=A+X$. Symmetry requires $X$ to descend to $Q$; (eq:antitime) requires $X(\gamma q)=-X(q{)}^{\ast }$. Thus its forced values are $X([d_i])=x_i$ and $X([\gamma d_i])=-x_i^{\ast }$, giving necessity.

Conversely, (eq:relationtest) defines such a map on the conjugation-stable span of these classes. Multiplying its range by $i$ makes it conjugation-equivariant. Extend the resulting real-linear map from the real fixed-point subspace to a real complement in $Q$, then complexify and multiply by $-i$. Pull back to $D_2$ and add $A$. The extension takes values in $\hslash {\mathcal{P}}_{\Sigma }{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$, so does not change the classical coefficient. All values are finite linear combinations of admitted elements. No continuity of the chosen complement is implied.

This is the actual-tensor criterion of IX-E IXE ({:algebraic}). It tests finite relations among any number of prescribed tensors, not only pairwise double-ordering consistency.

A construction realizing the criterion analytically must add all of the following:

1. Complete consistency of forced ordered values and all actual tensor relations.

2. Exact proposed replacement kernels, their source and differentiation conventions, ordered restrictions, singular sets, and residual configuration functionals.

3. Distributional existence, compact configuration kernels, carrier admission, and every required directional descendant.

4. A specified admissible contact family and simultaneous solution of ordered-restriction, old-overlap, admission, symmetry, and anti-time-ordering equations.

5. A named continuity theorem. One possible weak target uses seminorms

   $$F⟼|{\mathrm{D}}_{{\eta }_1}\cdots {\mathrm{D}}_{{\eta }_r}F(\Psi )|$$

   on fixed-support generating families. This is not actual-stage hypocontinuity.

Observation-time ordering alone is insufficient for (req:analytic): the sharp insertion has nonlocal configuration dependence. One cannot silently commute time ordering with differentiation, the Cauchy response, or its sharp adjoint.

The regular-functional time-ordering formula and a renormalized singular extension are different operations FR. Extension principles require proved hypotheses at the actual extension locus BF. Those references are documentary comparisons here, not existence theorems for this sharp sector. No scaling degree or ambiguity class for its unresolved binary extension is evaluated.

If Requirement req:analytic and the stated time-ordering identities are discharged on $D_2$, Theorem thm:conditionalret gives an unnormalized binary retarded difference through ${\hslash }^3$. Its normalized coefficient shift is determined only through ${\hslash }^2$.

Proof. Apply Theorem thm:conditionalret on the fixed comparison domain and then Lemma lem:hbaraudit.

Part II proves the double-ordering geometry, a nonempty strong-separation criterion, the commutator identity, fixed-domain obstruction criterion, and algebraic extension criterion. Its subtraction theorem is conditional.

At this stage the necessary restriction of $K-\overline{K}$ has not been evaluated on the intended two-way sector, no nonzero old time-ordered map is supplied, and no sharp analytical extension is constructed. The release is Unresolved, not a universal obstruction.

Proof. The positive statements are proved above. Neither an analytical candidate nor a certified violating two-way pair has been produced in Part II. These are distinct missing discharges.

-E-2-ST: exact imaginary part and witness-class vanishing

Introduce one Abel parameter $ϵ>0$, multiplying every momentum term in (eq:K) by $e^{-ϵ|p|}$. This represents the existing kernel. It is not the VII-C smoothing profile and does not define a new product.

The locked formula defines a tempered distribution in $(t,s,\mathbf{z})$, smooth in $t,s$ with values in ${\mathcal{S}}^{\prime }({\mathbb{R}}^3)$. The common-Abel family converges to it distributionally.

Proof. After spatial Fourier transformation of a joint test, the multiplier is bounded on compact time sets by $C(|p{|}^{-1}+|p{|}^{-2})$. Both powers are locally integrable with three-dimensional momentum measure. At infinity the transformed test is rapidly decreasing. Dominated convergence proves the pairing and the Abel limit. Temporal derivatives introduce only polynomial factors in momenta and times, handled by the same estimates. Temporal Schwartz weights absorb the explicit factor $t$, giving temperedness.

Define

Both have the same common-Abel interpretation. The changes $v=t-s$ and $v=t+s$ are submersion pullbacks, equivalently tensor products followed by invertible linear coordinate changes.

One has

and therefore

Proof. Conjugate at fixed $x,y$ and replace $p$ by $-p$. For $q=|p|$, the direct difference is

The reflected difference is $-i\cos (qu)/(4q^2)$. The identities hold with the same Abel factor, and Lemma lem:Fourier permits its distributional removal.

The cosine distribution is the locally integrable function

Moreover $D=-{\partial }_{v}H$. For $R>0$,

The derivative of (eq:H) specifies the distribution at the spatial apex, with no added point-supported term.

Proof. Angular integration gives

For real $a$,

as follows by differentiating in $a$, integrating the elementary Laplace transform, and using the value zero at $a=0$. Consequently

It converges almost everywhere to (eq:H). For $R>0$, its absolute value is at most $1/(4\pi R)$, which is locally integrable in $(v,\mathbf{z})$. Dominated convergence gives the global distributional limit, including $R=0$.

Fourier differentiation gives $D=-{\partial }_{v}H$. Differentiating the indicator at $R>0$ gives (eq:D). Since the global limit of $H_{ϵ}$ has already been identified as a locally integrable function, its distributional derivative leaves no undetermined apex contact.

For $t,s>0$ and $R>0$,

Away from the two cone boundaries it equals

The reflected cone is a jump locus, not an additional delta distribution.

Proof. Substitute Lemma lem:radial into (eq:imagdecomp), using $t+s>|t-s|$. The direct delta terms are retained separately.

This rederives the locked evaluation of IX-E-2 IXE2; it changes no inherited coefficient.

In the open positive-time chart,

Proof. For $R>0$, (eq:imag) vanishes off the displayed closed slab and is positive and smooth in its interior. Every point of the slab with $R>0$ is a limit of interior points. At $R=0$, the slab requires $t=s$. Those points are also limits of interior points. Lemma lem:radial supplies the global distributions and introduces no support outside the closure.

At $t=0$, the time-slice identity is $K(0,s,\mathbf{z})=0$. At $s=0$,

These are spatial distribution identities, not assigned pointwise values on a null cone. The inherited observation tests here lie at positive times; no enlargement of that test domain is required.

Every pair satisfying Lemma lem:spacelike obeys

Both actual support orderings hold for every such pair.

Proof. On $F\times G$,

There is a positive compact separation from the commutator support (eq:imagsupport). Thus its product with $fg$ is zero distributionally. The ordering statement was already proved by the upper-support argument.

There are nonnegative nonzero compact tests $f,g$ supported in the laboratory with

for which ${\mathcal{C}}_{f,g}\ne 0$. Observation-spacelike separation does not certify both actual functional-support orderings.

Proof. Choose $\lambda >0$ small enough that neighbourhoods of

and their retained shadows fit in the chart. At the central pair, $0<R=\lambda <2\lambda$. The strict inequalities persist on sufficiently small product neighbourhoods.

Take nonnegative bumps there, set ${\partial }_t\varphi =1$ near $F$, and $h_{00}=1$ near $G$. Then