Landmark IX-E-3 Functional support of the sharp insertion and the two-way commutator bridge

We isolate the actual functional-support calculation needed to apply IX-E-2's evaluated commutator to the retained two-way sector. The local quadratic functional $W_g$ has functional support exactly $\mathrm{supp}g$. For the sharp insertion, a nonzero smooth surface coefficient cannot be cancelled by the locally bounded bulk term in its distributional metric derivative.

A correction is necessary to the proposed solid-ball argument. For the explicitly specified flat massless zero-Cauchy realization, the initial-surface support is the union of the time-zero null spheres generated by $\mathrm{supp}f$, not necessarily the union of their solid balls. We prove this support statement, exhibit an open missed region, and show that the retained null spheres nevertheless suffice to exclude the commutator slab. Thus a gap inside a solid ball does not by itself provide an obstruction witness.

The bridge proof is complete under a displayed boundary-trace certificate. The supplied prerequisite statements, however, do not specify that certificate for the inherited operator ${\mathsf{E}}_0$. We do not identify a standard Green operator with an unspecified inherited operator by notation alone. The supplied-data release is therefore Unresolved at that specific identification. If the certificate is supplied, the bridge follows without another support calculation. No time-ordered product is constructed.

Figure fig:split-IX-E-3 records one view of the same separation. Figure fig:chain-IX-E-3 records one view of the same separation. Figure fig:pair-IX-E-3 records one view of the same separation. Figure fig:gate-IX-E-3 records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-IX-E-3 records that separation. The drawing adds no hypothesis.

The legal outcomes for this support problem are:

1. Bridge proved. The implication (SB) holds for every retained first-stage pair, including the declared conjugate sector. The double-ordering commutator gate clears, but the overall causal-product release remains Unresolved.

2. Proved obstructed. A retained pair has both actual support orderings certified and a nonzero commutator. Every symmetric causally factorizing map on the fixed domain is then excluded.

3. Unresolved. A named support calculation, cancellation check, or ordering certificate remains unfinished. Absence of a proof is not an obstruction.

There is no Constructed outcome for ${\mathsf{T}}_2$ within this paper.

Let $\sigma$ be complex-linear flip and $\gamma$ componentwise antilinear conjugation. Retain exactly

Here

is the inherited space of actual first-stage tensors. No tensor is removed, including the strongly separated pairs whose commutator was evaluated as zero in IX-E-2.

{{0.92} Release box. On the supplied information, the bridge release is Unresolved at the inherited boundary-trace certificate (TC) stated below.

A complete conditional support proof is given. For the explicitly specified flat massless zero-Cauchy realization, the bridge is proved, although the proposed solid-ball lower-support assertion is false.

That realization is not silently substituted for the inherited $L={\mathsf{E}}_{0}C$. Verification that the inherited operator has the displayed trace remains necessary.

No certified nonzero two-way pair is produced. No time-ordered replacement or retarded construction is begun. }

The notation $L={\mathsf{E}}_{0}C$, the sharp adjoint identity, and an upper causal support bound do not, by themselves, specify the action of ${\mathsf{E}}_0^{\ast }$ on arbitrary compact sources.

In particular, the statement that advanced propagation is supported in a ball is an upper-support statement. It does not say whether propagation has a tail inside the ball, is supported on its boundary, or has a nonzero coefficient at a particular point.

We therefore distinguish:

• results following from the explicitly copied inputs;

• a sufficient boundary-trace certificate for the inherited operator;

• a fully specified flat realization for which that certificate, the support alternative, and the bridge are proved.

This separation is essential to avoid changing the fixed functionals while ostensibly keeping their names.

Work on

Configurations are unrestricted smooth off-shell scalar and symmetric metric configurations. Complex test functions are allowed; detection by real configurations is addressed explicitly.

The inherited sharp insertion and local quadratic functional are

Write

IX-E-2 supplies, on $t,s>0$ and $R>0$,

Away from the cone boundaries it is zero for $R<|\tau |$, equals $1/(32\pi R)$ for $|\tau |<R<u$, and is zero for $R>u$. The direct delta contribution is separate; the reflected cone is a jump, not another delta. The Fourier definition governs the spatial apex.

The same input supplies

These identities are not re-evaluated here IXE2.

The compact-detection lemma is also imported: a nonzero compactly supported distribution $fg\mathrm{Im}K$ in the open half-space gives a nonzero actual commutator on the unrestricted configuration space. No transfer to a constrained quotient is asserted.

The strongly separated witness class has exact zero commutator. The observation-spacelike control strictly inside

has, for ${\partial }_t\varphi =1$ and $h_{00}=\alpha >0$,

for the nonnegative tests specified in IX-E-2. That control is not an obstruction witness without both actual ordering certificates.

For $a$ supported in $M_{+}^{\circ }$, the supplied VII-B identity is

where

There is no ${\delta }^{\prime }(t)$ term. The off-shell surface contribution is retained; an on-shell smoothing does not replace it VIIB.

For $F=\mathrm{supp}f$ and $G=\mathrm{supp}g$, the copied bounds are

The series definition is

Null relation and coincidence are included. Two-way ordering means that both orderings hold.

We use IX-E-2's derivative-support characterization: on the unrestricted affine configuration space, functional support is the closure of the union of the supports of all first-derivative distributions, over all configurations and field components. This characterization is an imported result, not a replacement by the upper set in (eq:upper) IXE2.

The obligation is

The two variations of the sharp insertion are

With $a_{\mu }=-f{\partial }_{\mu }\varphi$, the metric derivative is represented by the full $L^{\ast }a$ in (eq:adjoint). The scalar variation samples $Lh$ on $F$.

For every $g$,

Proof. At the allowed configuration $h_{00}=1$, the metric derivative has kernel $2g$, whose support is $G$; combine this with the upper bound in (eq:upper).

If $n$ is non-null, the map $v\mapsto \mathsf{B}(v)$ is injective.

Proof. Contract (eq:Bsurface) with $n_{\alpha }$:

Thus $\mathsf{B}(v)=0$ implies $v=0$.

Suppose, near a point of $\Sigma$,

where $A$ is locally bounded and $D$ is smooth on $\Sigma$. If $D$ is nonzero at that point, the point belongs to $\mathrm{supp}q$.

Proof. In any sufficiently small spatial neighbourhood choose a compact smooth tensor test $\zeta$ such that $⟨D,\zeta ⟩\ne 0$. Let $\rho$ be a compact smooth temporal function with $\rho (0)=1$, and test against

The surface pairing is the fixed nonzero number $⟨D,\zeta ⟩$. The bulk pairing is $O(\epsilon )$ by local boundedness. Their sum is nonzero for sufficiently small $\epsilon$. Its test support lies in the prescribed neighbourhood.

The convention for $\delta (t)$ here is the inherited surface distribution: it pairs with the trace on $\Sigma$. No additional factor or boundary normalization is introduced.

Lemma lem:no-cancel does not prove that $b{|}_{\Sigma }$ is nonzero. It proves that, once nonzero trace is established with the stated bulk regularity, the bulk term cannot remove its support. These are distinct obligations.

For every compact smooth covector source $a$ in $M_{+}^{\circ }$, the inherited $b={\mathsf{E}}_0^{\ast }a$ is smooth up to $\Sigma$, $C^{\ast }b$ is locally bounded there, and its componentwise trace is

A fixed overall nonzero sign in this formula would give the same support conclusions.

This certificate is more precise than an upper ball bound. Its verification for the inherited operator is not included in the supplied statements. We first prove exactly what it implies.

Assume (TC) and the copied adjoint identity. For every $x=(t,\mathbf{x})\in F$ with $t>0$,

Moreover,

Proof. First suppose $f(x)\ne 0$, and fix $\mathbf{z}$ with $|\mathbf{x}-\mathbf{z}|=t$. Choose a nonnegative compact smooth cutoff $\chi$, positive at $x$, supported in a small interior neighbourhood of $x$ on which

is smooth and nonzero. Define

This is a smooth compact configuration, and its support is away from $\Sigma$. Consequently $a=-f\mathrm{d}\varphi$ is an allowed source for (eq:TC).

On the cone $v=r_{\mathbf{z}}(\mathbf{w})$,

Its temporal trace component is therefore

The integral is finite, and its integrand is positive on a nonempty open spatial set. This construction respects the restriction $a=-f\mathrm{d}\varphi$; it does not treat its components as independently selectable sources.

If configurations are required to be real, write $\varphi ={\varphi }_1+i{\varphi }_2$. Since the trace map is complex-linear and the displayed result is nonzero, at least one real configuration ${\varphi }_j$ gives nonzero trace.

Injectivity of $\mathsf{B}$, followed by Lemma lem:no-cancel, shows that $(0,\mathbf{z})$ lies in the support of the actual metric derivative. The derivative-support characterization places it in ${\mathrm{supp}}_{\mathrm{fun}}{B}_f$.

For an arbitrary $x\in F$ with $t>0$, take $x_j=(t_j,{\mathbf{x}}_j)\to x$ with $f(x_j)\ne 0$. For a given sphere point write $\mathbf{z}=\mathbf{x}+t\mathbf{n}$, $|\mathbf{n}|=1$, and set ${\mathbf{z}}_j={\mathbf{x}}_j+t_j\mathbf{n}$. The preceding argument applies to each ${\mathbf{z}}_j$. Closedness of functional support proves (eq:sphere-lower).

Finally, any point of $F\cap \Sigma$ is a limit of interior points where $f\ne 0$, unless $f$ is identically zero near it, in which case it is not in $F$. The same construction with $t_j\to 0$ gives sphere points converging to that boundary point. This proves (eq:boundary-F-lower).

The theorem supplies a lower support statement for the fixed test $f$, not merely for a freely chosen source concentrated near one observation.

This section is a diagnostic realization, not a replacement of the inherited operator.

Let

and define

Let ${\mathsf{E}}_0$ be the componentwise solution operator for $P$ with zero displacement and velocity at $t=0$. Its retarded representation is

This is the standard flat three-dimensional wave realization. Its propagation is concentrated on null spheres, rather than throughout the interiors of causal balls Ivrii.

The operator (eq:E-flat) satisfies (TC).

Proof. Pair (eq:E-flat) with a compact source $a$, interchange the compactly supported distributional integrations, and collect the coefficient of $j(v,\mathbf{z})$. This gives

At $v=0$ this is (eq:TC).

Smoothness follows directly by angular integration:

On compact sets in $(v,\mathbf{z})$, the effective $r$-range is bounded. All derivatives can be passed under the integral. Since $C^{\ast }$ is first order with constant coefficients, $C^{\ast }b$ is smooth and locally bounded.

For $f\in {C_c}^{\infty }(M_{+}^{\circ })$, define compact sets

The first is a union of initial null spheres. The second is the union of their truncated backward null cones.

For $f\in {C_c}^{\infty }(M_{+}^{\circ })$,

Proof. The lower inclusion follows from Theorem thm:sphere. For the upper inclusion, every source $a=-f\mathrm{d}\varphi$ has support in $F$. The advanced representation above gives $\mathrm{supp}b\cap M_{+}\subset {\mathcal{N}}_F$. The local differential operator $C^{\ast }$ does not enlarge this support. The surface term has support in $\{0\}\times {\mathcal{S}}_F$.

The scalar derivative has support in $F$, which is separated from $\Sigma$. Thus all first-derivative distributions are supported in the fixed closed set ${\mathcal{N}}_F\cup F$, whose intersection with $\Sigma$ is $\{0\}\times {\mathcal{S}}_F$. Their union and its closure cannot acquire another surface point.

For this realization, the surface-ball alternative is decided exactly by (eq:exact-surface). A ball belongs to functional support precisely when it is contained in ${\mathcal{S}}_F$.

If a point ${\mathbf{z}}_0$ of a proposed ball is outside ${\mathcal{S}}_F$, the continuous function

satisfies $d_F({\mathbf{z}}_0)>0$. Choose $0<\epsilon <d_F({\mathbf{z}}_0)/4$. For $|\mathbf{z}-{\mathbf{z}}_0|<\epsilon$ and $0\le v<\epsilon$, no backward null cone from $F$ meets $(v,\mathbf{z})$. All advanced sources, their local bulk derivatives, and their surface traces vanish there, for every smooth scalar configuration.

This is an explicit missed open region. Its cause is absence of propagation, not cancellation of a nonzero delta by the bulk. The alternative's reference to a “cancelling configuration class” must therefore allow this zero-propagation case.

Let $f$ be a nonzero smooth bump, positive at $(1,0)$, with

Consider the relative open collar

For every $(t,\mathbf{x})\in F$ and $(v,\mathbf{z})\in U$,

No backward null relation is possible. Therefore $b=0$ throughout $U$ for every source $-f\mathrm{d}\varphi$, and the bulk and surface derivative terms both vanish there. The scalar derivative is supported in $F$, disjoint from $U$. Consequently

The time-zero ball associated with $(1,0)$ contains $B(0,1/4)$. Thus the proposed universal solid-ball lower-support assertion is false for the flat realization.

A standard wave support theorem does not repair this assertion: the sphere-versus-ball distinction is exactly the relevant three-dimensional propagation distinction Ivrii.

For the flat realization,

Every nonempty open set meeting $F$ contains scalar dependence at a suitable off-shell metric configuration.

Proof. Choose an interior point $x$ with $f(x)\ne 0$ in the given open set. Take $w$ compactly supported in a smaller interior neighbourhood with

For example, a temporal quadratic multiplied by a cutoff equal to one near $x$ has these properties. Set

Equation (eq:C-flat) gives

Because $w$ and all its derivatives have zero initial data, uniqueness of the zero-Cauchy problem gives

The scalar derivative is therefore

At $x$, its smooth kernel equals $-f(x)Pw(x)\ne 0$. It is detected by a scalar variation in the prescribed open set.

If $f$ is complex, the nonzero complex kernel is still detected by a real test, or by its real and imaginary decomposition. Taking closures gives all of $F$, including its boundary points. Configurations with $Lh=0$ do not defeat this argument: functional support uses all configurations, and the displayed configuration is a detector.

The preceding gap does not provide a two-way pair. Null spheres alone already obstruct the relevant common-past geometry.

Let $t>0$, $s\ge 0$, and $R=|\mathbf{x}-\mathbf{y}|$. Then

Proof. The smallest distance from $\mathbf{y}$ to the sphere centered at $\mathbf{x}$ is $|R-t|$, including $R=0$. The claimed intersection is therefore equivalent to $|R-t|\le s$.

Suppose

and every initial null sphere from each positive-time point of $F$ lies in ${\mathrm{supp}}_{\mathrm{fun}}{B}_f$. If both actual orderings hold, then

Proof. Suppose $R\le t+s$.

If $t=0$, then $R\le s$ makes $x\in J^{-}(y)$. Since $x\in {\mathrm{supp}}_{\mathrm{fun}}{B}_f$ and $y\in {\mathrm{supp}}_{\mathrm{fun}}{W}_g$, $B_f\gtrsim W_g$ fails.

Let $t>0$. If $R<t-s$, then $y\in J^{-}(x)$, and $W_g\gtrsim B_f$ fails. Otherwise $t-s\le R\le t+s$. Lemma (eq:sphere-ball) supplies a point $z=(0,\mathbf{z})$ on the initial null sphere of $x$ with $|\mathbf{z}-\mathbf{y}|\le s$. Thus

and $B_f\gtrsim W_g$ fails.

The argument includes tangencies, $R=0$, $s=0$, and coincidence. Every case contradicts at least one ordering.

For the explicitly specified flat realization, (SB) holds for all retained first-stage pairs.

Proof. The preceding support results supply all hypotheses of Theorem thm:geometric. If $F,G$ are nonempty, compactness and (eq:full-exclusion) give a positive uniform margin

Thus $R$ is bounded away from zero and both direct-cone deltas are disjoint from the observation support.

IX-E-2's Fourier-defined kernel vanishes in the open region $R>t+s$; its common-regulator proof also supplies smooth time dependence with values in spatial distributions. The margin gives the same zero restriction at any initial-time boundary met by these smooth tests. There is no additional apex or boundary contact to assign. Hence $fg\mathrm{Im}K=0$. If either test is zero, the conclusion is immediate.

The missed region (eq:explicit-gap) therefore cannot carry a certified two-way counterexample in the open commutator slab. For any observation pair in that slab, a retained null-sphere point, possibly outside the missed solid-ball interior, lies in the causal past of $y$. This is the required ordering certificate against the proposed control. The control is nonzero, but at least one ordering fails.

The scalar-slot lower theorem is useful for full exclusion (eq:full-exclusion), but it is stronger than needed to annihilate the commutator.

Assume (TC), the copied sharp adjoint, the copied Fourier-defined kernel properties, and unrestricted configurations. Then (SB) holds.

Proof. Theorem thm:sphere gives the initial null spheres and $F\cap \Sigma$ as lower supports. Proposition (eq:Wexact) gives ${\mathrm{supp}}_{\mathrm{fun}}{W}_g=G$.

For $t>0$, every observation pair in

satisfies $t-s\le R\le t+s$. Its initial sphere therefore meets the initial ball of $y$. The corresponding actual support point of $B_f$ lies in $J^{-}(y)$, contradicting $B_f\gtrsim W_g$. For $t=0$, the closed slab reduces to $R=s$; the lower support $x\in F\cap \Sigma$ gives the same contradiction.

Thus $F\times G$ misses the closed commutator slab, including its direct cone, reflected boundary, and spatial apex. The copied Fourier support statement and its boundary continuation imply $fg\mathrm{Im}K=0$. No scalar controllability assumption is needed for this narrower conclusion.

This identifies the exact sufficient certificate whose verification would complete the inherited bridge.

Let $S(d)={\star }_c(d)$ and

The inherited reversal identity gives

Conjugation preserves actual functional support on the unrestricted real configuration space: a dependence vanishes exactly when its conjugate vanishes.

Consequently any proved original-sector bridge clears the double-ordering commutator test on the declared conjugates and flips. For the real-coefficient flat realization, $B_f^{\ast }=B_{\overline{f}}$ and $W_g^{\ast }=W_{\overline{g}}$; the distributional statement itself also applies directly to those tests.

This transfer does not prove the bridge on the original pairs. It also does not evaluate relations among sums of actual tensors.

If a retained pair has both actual orderings and ${\mathcal{C}}_{f,g}\ne 0$, no symmetric causally factorizing map ${\mathsf{T}}_2:{\mathcal{D}}_2\to {\mathcal{V}}^{[3]}$ exists.

Proof. Factorization and symmetry would require

This contradicts (eq:commutator). A higher coefficient or contact cannot equalize unequal prescribed free values.

No pair satisfying this proposition's premises is established on the supplied inherited data.

The logical status is now precise.

The copied adjoint specifies how an already determined $b={\mathsf{E}}_0^{\ast }a$ enters the derivative. It does not give $b{|}_{\Sigma }$. The copied upper bound likewise does not give that trace. Finally, a mixed covariance formula is not, without a stated covariance-to-Green-operator identification, a formula for ${\mathsf{E}}_0^{\ast }$ on every source $-f\mathrm{d}\varphi$.

The flat realization proves that the proposed universal solid-ball assertion is not a safe substitute for this missing information. It also proves that solid-ball support is unnecessary: the null-sphere certificate is sufficient.

For the explicitly specified flat massless zero-Cauchy realization, the bridge is proved.

For the inherited operator on the supplied prerequisite information alone, the release is Unresolved at verification of (TC). Neither a proved fixed-domain obstruction nor a constructed time-ordered product is released.

Proof. The first statement follows from the verified trace, Theorem thm:minimal, and conjugate transfer.

For the inherited operator, the supplied statements do not include the defining trace required by that theorem. The conditional calculation cannot identify two operators merely because both are denoted by ${\mathsf{E}}_0$. No alternative complete lower-support proof or certified two-way counterexample is supplied. This is the named Unresolved outcome.

The outstanding request is specific: provide the defining Cauchy equation and initial conditions, or the equivalent boundary-trace formula, for the inherited ${\mathsf{E}}_0$. If it is (eq:E-flat), the proof above completes the bridge without reopening the kernel evaluation.

Even a proved bridge leaves the actual-tensor relation test, coefficient and descendant admission, named fixed-support continuity, exact old overlap, contact equations, analytic involution, and conditional subtraction obligations unchanged. No replacement distribution is declared to be the sole remaining task.

V's external $R_{1,1}$ and Newton–wave cubic, VII-D's numerical finite part, VI's compact-switch defect, VII-A's plateau intertwiner, and the VII-E/F and VIII certificates remain non-identifications. No conical current or token is identified with the present commutator.

The free-sector descent and word coherence remain closed. F9a is not promoted to hypocontinuity. VII-C remains in force under its joint hypotheses (E1)–(E4). G2 remains unopened. No finite-part subtraction is renamed as a product, and no $R_1$, $R_2$, ${\mathsf{T}}_3$, or Feynman replacement is begun.

The central correction is mathematical rather than terminological: in the flat massless realization, actual initial support is generated by null spheres, and solid-ball interiors can be invisible to every configuration. Nevertheless, those spheres meet the initial ball of every observation pair in the closed commutator slab. They therefore prohibit the relevant ordering and prove the bridge.

For the inherited operator, the remaining burden is not another witness or kernel calculation. It is verification of the displayed boundary-trace certificate. Keeping that identification explicit is necessary to preserve the fixed target and the honest release status.

IXE

Landmark IX-E: Binary time ordering and causal subtraction on the declared sharp comparison sector. Supplied manuscript IX-E.tex. Used for the fixed-domain obstruction and the distinction between the commutator gate and the complete causal interface.

IXE2

Landmark IX-E-2: Exact imaginary part of the locked Cauchy kernel and the first-stage commutator on the retained two-way sector. Supplied manuscript. Used for the evaluated kernel, commutator, compact detection, witness zero, observation-spacelike control, derivative-support characterization, and conjugate reversal identity.

VIIB

Landmark VII-B. Sharp adjoint statement as quoted in the supplied IX-E-3 specification. Only the displayed adjoint and surface coefficient are imported. An unprovided defining kernel for the inherited ${\mathsf{E}}_0$ is not reconstructed.

Ivrii

V. Ivrii, Partial Differential Equations, Chapter 9, Section 9.1, “Wave equation in dimensions $3$ and $2$.” University of Toronto online lecture text. Used for the standard flat Kirchhoff realization and its sphere-supported propagation, not to identify an unspecified internal operator.