Landmark IX-E-2 Exact imaginary part of the locked Cauchy kernel and the first-stage commutator on the retained two-way sector

We evaluate the imaginary part of the locked mixed Cauchy kernel and apply it to the first-stage commutator on the fixed comparison domain of IX-E. A common Abel regulator gives an exact distributional decomposition into a direct sine contribution and two cosine contributions. Away from the spatial apex, the imaginary part consists of a direct null-cone delta distribution and a locally integrable contribution in the observation-spacelike region between the direct and reflected cones. The direct and reflected cosine tails cancel outside the reflected cone.

Consequently, the commutator vanishes identically on IX-E's entire strongly separated two-way witness class. An explicit observation-spacelike control has nonzero commutator, but is not certified as a two-way pair of actual functional supports. The remaining domain-wide obligation is therefore an actual functional-support bridge, not another evaluation of the kernel on the settled witness class.

The release remains Unresolved. No time-ordered replacement, contact classification, normalized retarded product, or fixed-domain obstruction is claimed. The inherited free-sector results and external non-identification boundaries are unchanged.

Figure fig:split-IX-E-2 records one view of the same separation. Figure fig:chain-IX-E-2 records one view of the same separation. Figure fig:pair-IX-E-2 records one view of the same separation. Figure fig:gate-IX-E-2 records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-IX-E-2 records that separation. The drawing adds no hypothesis.

This manuscript is a commutator-coverage paper. It does not propose a time-ordered construction.

The target inherited from IX-E is a complex-linear actual tensor map

Here “actual” means that tensors are formed from actual functionals, not from unquotiented test-function presentations. The formal parameter is fixed by conjugation.

We retain IX-E's three legal outcomes unchanged:

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.

The Constructed outcome is retained for series consistency but is unavailable within the deliberately restricted scope of this paper. A vanishing commutator is not a construction.

Let $\sigma$ be complex-linear flip and $\gamma$ componentwise antilinear conjugation:

Thus

The inherited first-stage space and frozen target are

The spaces $\mathcal{B}$ and $\mathcal{W}$ are the inherited spaces of actual first-stage functionals. In particular, every $B_f\otimes W_g$ under consideration is retained.

IX-E proves the stability and containment statements

These are imported results, not new domain claims.

The zero old seed does not evaluate the actual old intersection:

No nonzero old time-ordered map is supplied here. Conjugation saturation cannot be ignored when auditing (eq:old-intersection).

{{0.92} Release box. The outcome is Unresolved on the fixed domain (eq:frozen-domain).

Witness-class result: the commutator vanishes exactly on every pair in IX-E's strongly separated witness class.

Fixed-domain result: no nonzero commutator on a certified retained two-way pair is established. Complete two-way coverage remains unfinished at the actual functional-support bridge.

No retained pair is deleted. No time-ordered replacement is introduced. No implication from witness-class vanishing to a constructed causal product is asserted. }

The internal prerequisites below are the statements supplied for this successor. Their attribution records the supplied provenance; it does not assert independent verification of unprovided source manuscripts. Every imported property used in a proof is stated explicitly. References to other landmarks are not substitute existence theorems Inputs,IXE.

Work in the flat half-space

The configuration space is the unrestricted smooth off-shell scalar and metric configuration space inherited from IX-A. Neither an on-shell quotient nor a gauge-fixed quotient is used.

The sharp insertion is

For $x=(t,\mathbf{x})$, $y=(s,\mathbf{y})$, the locked mixed covariance supplied from VII-C and restated on the VII-D convention sheet is

No graviton positivity is assumed. The inherited statements $H_{ww}=0$ and the characteristic-cone failure of $B_f$ remain unchanged.

The declared free product ${\star }_c$ and its first-stage coefficients are inputs:

These are the inherited exact first-stage formulas in the truncated coefficient space. Their reversal identity is

The released degree-two and degree-four descent, the word through ${\hslash }^3$, the evaluations of $\ker p_L,\ker p_R,{\mathcal{C}}_{\mathrm{desc}},{\mathcal{C}}_{\mathrm{glue}}$, and F9a fixed-support continuity are not reopened. F9a is not promoted to hypocontinuity.

Full support of $K$, where inherited, is not used to determine its imaginary part. A real distribution may have full support.

For the adjoint variable $b$ associated with a test covector $a$, the supplied VII-B identity is

Equivalently, in the inherited convention,

There is no ${\delta }^{\prime }(t)$ term. The supplied on-shell smooth representative does not replace this off-shell derivative.

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

These are upper inclusions. They are not lower support theorems.

Under VII-C's joint hypotheses (E1)–(E4), namely the old Wick product on the polynomial domain, off-shell admission of the sharp insertion, sequential normal-topology admission, and hypocontinuous multiplication with a continuous ${\hslash }^3$ coefficient map, the mandatory word has no sequentially continuous extension. Its supplied separated-support residue is

where $A_f$ denotes the inherited reflected light-ray transform. This residue is not subtracted here.

For the supplied VII-D regularization family,

is a strong distributional limit, and admissible mollifiers differ by $c_{ab}r$. This is numerical finite-part data, not an algebra construction. Neither ${u_{\mu }}^{\rho }+cr$ nor the triangle scalar is renamed as ${\star }_c$ or ${\mathsf{T}}_2$.

V's externally retarded object

belongs to its compactly switched minimal vertex problem. Its linear coefficient, finite-mode oscillator coefficient, and cubic Newton–wave Wick form remain different objects. In particular, V does not supply $T_2(V,V)$ or the sharp map sought here.

VI's first-order compact-switch defect and VII-A's restricted plateau intertwining statement are retained. The expression

is an obligation list, not a construction imported into this paper.

VII-E/F and VIII-A–C supply their stated observation, exchange, particle-link, affine-matching, packet, and scheduling certificates. No supplied map identifies those outputs with $B_f$, $W_g$, a triangle finite part, or an element of ${\mathcal{D}}_2$. VIII-C's token policy supplies a semantic promise, not an acquired expectation.

Set

Distributional pairing is complex-bilinear: no conjugation of the test argument is implicit. Distributional conjugation is defined by

Kernel conjugation is at fixed $(x,y)$. It is not the operation $K(x,y)\mapsto K(y,x)$.

The inherited coefficients (eq:ordered-coefficients) give

Proof. The classical terms cancel. The scalar tests and smooth configuration factors in ${{\mathcal{Q}}^{\mathcal{R}}}_{g,f}$ commute, so their order may be changed without exchanging singular distributions. Subtracting the two order-$\hslash$ coefficients gives the factor $-2$ multiplying $K-\overline{K}$. The definition of the imaginary part then gives the factor $-4i$.

For $\epsilon >0$, insert the same factor $e^{-\epsilon |p|}$ into every term of (eq:locked-kernel); call the result $K_{\epsilon }$. This Abel parameter is not the VII-C mollifier.

The locked formula defines a tempered distribution in $(t,s,\mathbf{z})$. Moreover,

The distribution is smooth in $(t,s)$ with values in ${\mathcal{S}}^{\prime }({\mathbb{R}}^3)$.

Proof. Let $\Xi (t,s,\mathbf{z})$ be a compactly supported smooth test function and take its partial spatial Fourier transform. On its time support the absolute value of the momentum coefficient is bounded by

with $C$ independent of $0<\epsilon \le 1$. The partial Fourier transform decreases faster than any inverse power of $|p|$, uniformly on that support.

Near $p=0$, spherical measure gives

At infinity, the Fourier decay supplies an integrable majorant. Dominated convergence therefore defines the unregulated pairing and proves convergence on compactly supported tests.

For Schwartz tests, the same estimate is combined with temporal Schwartz weights sufficient to absorb the explicit factor $t$. This gives the tempered estimate and convergence in ${\mathcal{S}}^{\prime }$. Temporal differentiation of the momentum coefficients introduces only additional nonnegative powers of $|p|$ and polynomial temporal factors. Spatial Fourier decay again dominates them. Differentiation under the paired integral therefore proves smooth dependence on the time parameters.

The kernel in $(x,y)$ is obtained by the invertible coordinate change to $(t,s,\mathbf{z},\mathbf{y})$, with constant dependence on the last variable. Consequently the compactly supported pairings in (eq:commutator) are meaningful for arbitrary smooth configuration factors.

All distribution identities below are first proved with ambient time variables in $\mathbb{R}$. Interior restrictions require no boundary choice. For inherited tests meeting $\Sigma$, smooth dependence on the time parameters supplies the corresponding half-space pairing. No new boundary contact is introduced into $K$.

Define the Fourier distributions

The same argument as above defines these distributions and their Abel-regulated versions. The letter $H$ here denotes only this cosine distribution; it is not a new Wick covariance.

With the convention (eq:variables),

Equivalently,

Proof. Conjugate (eq:locked-kernel) at fixed observation arguments. The spatial phase becomes $e^{-ip\cdot \mathbf{z}}$. Replacing $p$ by $-p$ restores the original spatial phase without changing $|p|$.

The direct difference is

The reflected difference is

These identities hold with the common Abel regulator, and hence after its distributional limit. Division by $2i$ gives (eq:imag-decomposition).

Thus the correctly named pieces are

The direct contribution is not a pure function of $\tau$, and is not a pure Pauli–Jordan contribution. Distinct Fourier phases do not prevent their integrated distributions from cancelling on an open region.

The substitutions $v=\tau$ and $v=u$ are distributionally unambiguous. For example, $(t,s)\mapsto (\tau ,s)$ is an invertible linear coordinate change, after which $D(\tau ,\mathbf{z})$ is $D$ tensored with the constant distribution in $s$. The same argument applies to $u$. No time-ordering prescription or wavefront extension is hidden in these substitutions.

For $R>0$, angular integration gives

For $\epsilon >0$ and $a\in \mathbb{R}$,

Proof. The integral vanishes at $a=0$. Differentiation with respect to $a$ is justified by the exponential regulator and gives

Integration from $0$ to $a$ proves the assertion.

Using the sine-cosine product identity in (eq:H-radial-regulated) yields

On $R>0$,

Here $\Theta$ is the locally integrable Heaviside distribution; its assigned point value at zero is irrelevant.

Proof. For $R>|v|$, both arctangent arguments tend to positive infinity, so their sum tends to $\pi$. For $R<|v|$, one tends to positive infinity and the other to negative infinity, so the sum tends to zero.

For all $R>0$,

The majorant is locally integrable in $(v,\mathbf{z})\in \mathbb{R}\times {\mathbb{R}}^3$, since spatial polar measure is $R^2\mathrm{d}R\mathrm{d}\Omega$. Thus dominated convergence proves local distributional convergence, including neighbourhoods of the spatial apex. Its limit agrees with the independently defined Fourier distribution.

The radial display is used on $R>0$. At the apex the Fourier definition remains authoritative. The global convergence argument does not leave freedom to add an apex contact.

The companion calculation is

On $R>0$,

Globally, with the Fourier definitions,

Proof. For a one-dimensional compactly supported smooth test function $\zeta$,

The integrable majorant $\Vert \zeta {\Vert }_{\infty }/(1+a^2)$ proves this delta-sequence limit. It is the standard distributional interpretation, not a pointwise limit on the cone NIST.

Applying it to the two translated terms of (eq:D-regulated) proves (eq:D-evaluated). The argument also holds under joint pairing in $(v,\mathbf{z})$ on compact subsets of $R>0$, where $1/R$ is smooth and bounded.

At finite regulator,

both by Fourier differentiation and by (eq:H-arctan). Distributional differentiation is continuous, so the global convergence of $H_{\epsilon }$ proves (eq:D-H-check), including the apex in the Fourier sense.

The momentum singularities have therefore required no infrared finite part. The common regulator evaluates the locked distribution; it does not select a new normalization.

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

Away from the direct and reflected cone boundaries,

Proof. Substitute (eq:H-evaluated) and (eq:D-evaluated) into (eq:imag-decomposition). Since $t,s>0$,

The Heaviside difference is therefore one precisely in the open region $|\tau |<R<u$, and zero in the other two open regions. The direct-cone delta contribution is kept separately.

In the open set $t,s>0,R>0$, the support of $\mathrm{Im}K$ is exactly

The direct cone carries the displayed delta contribution. The reflected cone is a jump locus of the locally integrable contribution, not an additional delta distribution.

Proof. The formula vanishes in the complementary open regions. It is a strictly positive smooth function on $|t-s|<R<t+s$. The closure of that region within $t,s>0,R>0$ is the displayed set, so every point of that set lies in the support. The formula itself contains no reflected delta term.

Thus observation-spacelike pairs can have nonzero imaginary part. The relevant spacelike region is the region between the two cones, not the region beyond the reflected cone.

Two checks follow directly from the Fourier identity. The functions $H$ and $D$ are respectively even and odd in $v$. At $t=0$, the locked kernel vanishes and (eq:imag-decomposition) gives zero because $H(-s)=H(s)$. At $s=0$, the two cosine terms cancel and

These are identities in the spatial distribution variable, not assignments of pointwise cone values.

For actual functionals, retain IX-E's definition

Causal relations include null relation and coincidence.

Both orderings hold exactly when no point of one actual functional support is causally related in either direction to a point of the other. Indeed, the first ordering excludes one direction and the second excludes the reverse. In particular, double ordering excludes intersection of the actual supports.

Suppose $f,g$ have time supports in $[0,T]$ and spatial supports in

respectively. If

then both $B_f\gtrsim W_g$ and $W_g\gtrsim B_f$ hold.

Proof. A point in $P_F\cup F$ has time in $[0,T]$ and spatial distance at most $T+r$ from $x_0$. A point in $G$ has time in $[0,T]$ and spatial distance at most $s_0$ from $y_0$. Their spatial distance is greater than $T$, whereas their time difference is at most $T$. The upper support sets are causally disjoint. The actual supports are subsets of them by (eq:support-upper).

For every $f,g$ satisfying (eq:strong-separation),

The first equality is distributional and the second is an identity of actual functionals on every allowed configuration.

Proof. On $\mathrm{supp}f\times \mathrm{supp}g$,

Hence neither direct-cone delta meets the observation support. Moreover,

there, so the cosine contributions cancel.

The inequalities have a positive uniform margin on the compact observation support. Even if a time support meets $\Sigma$, there is an ambient open neighbourhood on which

On that neighbourhood the global Fourier identity gives $\mathrm{Im}K=0$. Multiplication by $f(x)g(y)$ therefore gives the zero distribution, with no boundary ambiguity. Equation (eq:commutator) then vanishes for every smooth configuration.

This result needs no detection lemma: there is no nonzero distribution to detect. The strong separation is precisely why this witness class does not meet the reflected cone.

Define smooth nonnegative bumps

Let ${\mathbf{e}}_1=(1,0,0)$, and take

Their time supports lie in $[3/8,5/8]\subset (1/4,3/4)$, their spatial radii are $1/10$, and their center distance is $3$. Taking $T=1$ gives

This is a retained two-way pair with exact zero commutator. No numerical integration is involved.

A zero on the settled witness class must not be confused with zero on all observation-spacelike pairs.

There are nonnegative compactly supported tests $f,g$ whose observation supports are strictly spacelike separated and for which ${\mathcal{C}}_{f,g}\ne 0$. The calculation alone does not certify double ordering of their actual functional supports.

Proof. Use the bumps (eq:bumps), set $\eta =1/20$, and choose

On their product support,

Thus the whole product support lies strictly inside

No cone distribution meets it.

Choose the unrestricted smooth off-shell configuration

Then (eq:commutator) and (eq:imag-regions) give

The integral is finite and strictly positive before multiplication by the nonzero prefactor: the denominator is bounded away from zero and both bumps are nonnegative and nonzero. More explicitly, writing $I$ for the integral,

This is an exact pairing with explicit tests, not numerical sampling.

The tests (eq:control-tests) are observation-spacelike. The support of $B_f$, however, can extend into $P_F$. No lower support calculation establishing both actual orderings has been supplied for this pair. Accordingly, (eq:control-pairing) is a nonzero observation-spacelike control, not an obstruction witness.

The following elementary fact makes the remaining bridge a precise commutator obligation.

Let

be a nonzero compactly supported distribution with support in M_+^\circ^2. On the unrestricted smooth off-shell configuration space, ${\mathcal{C}}_{f,g}\ne 0$ as an actual functional.

Proof. There exist compactly supported smooth $a(x),b(y)$ such that

To justify this reduction, choose a compact product rectangle containing the relevant supports. A compactly supported smooth test on the rectangle can be periodized inside a larger rectangle and expanded in its smooth Fourier series. Multiplication by separate compact cutoffs in the two variables gives approximating finite sums of tensor-product tests, converging with every derivative on a fixed compact set. If a distribution vanished on all tensor-product tests, continuity would therefore make it vanish on every test.

Choose a smooth scalar configuration satisfying

and choose $h_{00}=b$, with the other metric components zero. Equation (eq:commutator) detects $T_{f,g}$. If real configurations are required, decomposing $a,b$ into real and imaginary parts gives a real tensor-product detector. Compactness is required of the kernel, not of $\varphi$.

For interior tests, distributional vanishing of $T_{f,g}$ and vanishing of the actual commutator are consequently equivalent. This uses unrestricted off-shell configurations and cannot be transferred without proof to a constrained quotient.

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

Proof. Factorization and symmetry would require

This contradicts (eq:commutator). The nonzero order-$\hslash$ coefficient remains nonzero modulo ${\hslash }^4$. No higher coefficient or contact can make the two prescribed free values equal.

This is IX-E's fixed-domain obstruction criterion, now equipped with the evaluated kernel IXE (Theorem 4.6). Its missing premise is still certification of a nonzero two-way pair.

The remaining coverage obligation is

It concerns every retained first-stage pair, not just the strongly separated class.

A sufficient route is to prove that two-way functional separation excludes the observation region

including the direct-cone contribution, with a separate check of any apex or boundary cases used in that argument. This exclusion would imply (eq:SB). It is sufficient, not asserted necessary: multiplication by tests can also annihilate a distribution through vanishing or cancellation.

For $R\le t+s$, the time-zero balls

intersect. Indeed, along the segment joining the centers one can choose distances $a\le t$ and $R-a\le s$ precisely when $R\le t+s$. The resulting point

lies in $J^{-}(x)\cap J^{-}(y)$.

This explains the reflected geometry. It does not show that $z\in {\mathrm{supp}}_{\mathrm{fun}}{B}_f$. Membership in the upper set $P_F$ is not membership in the actual support. Likewise, a use of $y\in {\mathrm{supp}}_{\mathrm{fun}}{W}_g$ requires more than $y\in G$.

For a differentiable functional on the unrestricted affine smooth configuration space, its functional support is the closure of the union of the supports of its first-derivative distributions over all configurations and all field components.

Proof. If the functional is independent of perturbations supported in an open set $U$, every first derivative annihilates such perturbations. Conversely, if every first derivative annihilates them at every configuration, then for a perturbation $\eta$ supported in $U$,

This is exactly absence of functional dependence in $U$. Taking the complement of the union of such open sets proves the support characterization.

For the sharp insertion, the metric variation is

Set $a_{\mu }=-f{\partial }_{\mu }\varphi$ and $b={\mathsf{E}}_0^{\ast }a$, in the inherited adjoint convention. Its metric derivative is represented by the full distribution

The scalar variation remains

Lemma lem:derivative-support therefore identifies what a lower support proof must examine. It does not evaluate those supports. Such a proof must control the actual bulk distribution, its surface trace, all allowed scalar gradients, and possible cancellations. A nonzero candidate cone, or the support of one formal summand, does not by itself establish nonzero dependence of the complete functional.

The surface term cannot be discarded, and the on-shell smoothing cannot replace it. Actual-presentation cancellations must also be respected: support belongs to the functional, not to a chosen test presentation.

Conjugation preserves functional support on the unrestricted real configuration space, since vanishing of a dependence is equivalent to vanishing of its complex conjugate. The same statement holds for its usual complexification.

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

The inherited reversal identity gives

Indeed,

Thus a complete commutator-vanishing proof on the original first-stage pairs transfers to their declared conjugates. This transfer is explicit; it does not supply the missing proof on the original pairs. Nor does generatorwise vanishing evaluate all actual linear relations among prescribed tensors.

No proof of (eq:SB) is supplied here. No retained counterexample with both actual support orderings is supplied either.

The kernel identity is closed. The strongly separated witness calculation is closed. The nonzero control is closed as an observation-spacelike calculation. The domain-wide stop remains the actual functional-support bridge.

It would be incorrect to move that stop to a replacement distribution, or to describe the control as an evaluated fixed-domain obstruction.

Even a future proof of (eq:SB) would clear only the first-stage double-ordering commutator test. IX-E's actual-tensor relation test is separate.

For clarity, its algebraic structure can be recorded without constructing a new map. With

a candidate ${\mathsf{T}}_2=A+X$ must satisfy

For prescribed tensors $d_i$ with prescribed values $t_i$, put $x_i=t_i-A(d_i)$. In the flip quotient

the inherited finite-relation condition is

where each sum is finite. This condition is not evaluated here. IX-E's algebraic sufficiency proof uses a Hamel complement and does not establish continuity IXE (Theorem 5.1).

The ordered slab also remains relevant: its nonzero ${{\mathcal{Q}}^{\mathcal{R}}}_{g,f}$ excludes classical multiplication as the time-ordered map. A later slab proves an ordering, not necessarily its uniqueness, and is not a substitute for a two-way calculation.

If the commutator gate is eventually clear, the inherited E2–E6 obligations remain. They include a specified replacement distribution, admission of coefficients and descendants, named fixed-support continuity, exact overlap, contact and involution compatibility, and the hypotheses of conditional subtraction. This successor begins none of those constructions.

Relative to the stated inherited inputs and the results proved in this manuscript, the release on (eq:frozen-domain) is Unresolved. Neither a Constructed outcome nor a Proved obstructed outcome is established.

Proof. No time-ordered map satisfying the complete interface is constructed. The explicit certified two-way class has zero commutator. The explicit nonzero control has not been certified as two-way in actual functional support, and no violation of the actual-tensor relation test is evaluated. The support bridge remains unfinished. These establish precisely the third release status, not nonexistence of every possible future construction.

All supplied free coefficients and the triangle parameter remain fixed. No contact is added to repair unequal prescribed values; no ${u_{\mu }}^{\rho }$ is subtracted and renamed as a product. Removing a retained pair would change the target, not solve it.

G2 remains unopened. In particular, the inherited compact-switch witness

retains its nonzero first-order dressing variation under its stated hypotheses. A correction beginning at second order cannot cancel a nonzero first-order coefficient while that prescription is retained.

No Feynman replacement, scaling-degree extension, normalized $R_1$, $R_2$, or ${\mathsf{T}}_3$ is begun. No Einstein-equation or compilation claim is made. No repair of VII-C and no normal-topology algebra claim follows from the present calculation.

The locked kernel has an explicitly evaluated imaginary part, supported away from the spatial apex on the direct null cone and the observation-spacelike region between the direct and reflected cones. Both the direct cosine contribution and the reflected cosine contribution are essential: their tails cancel beyond the reflected cone.

On IX-E's strongly separated two-way witness class the resulting commutator is exactly zero, as a distribution and as an actual functional on every allowed configuration. This closes that class without qualification. The explicit nonzero spacelike control shows why observation separation alone cannot replace actual functional-support ordering.

No fixed-domain obstruction is established. A domain-wide conclusion requires the actual functional-support bridge, with the sharp adjoint surface term retained and actual-presentation cancellations controlled. Even universal commutator vanishing would not construct a time-ordered extension or discharge the remaining actual-relation and analytic obligations.

The release is therefore Unresolved at complete two-way commutator coverage. The evaluated kernel, the settled witness class, and the remaining support question are now separately visible without reconstructing the series.

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, support ordering, first-stage coefficients, retained-pair obstruction, algebraic relation test, and conditional subtraction boundary.

Inputs

Internal prerequisite statements supplied for Landmark IX-E-2. The quoted statements from I–V, VI, VII-A–F, VIII-A–C, IX-A, IX-B, IX-D, IX-Free, and IX-Free-2 are imported only in the explicit forms recorded in this manuscript. No unprovided full-source theorem or nonzero old time-ordered map is reconstructed.

NIST

National Institute of Standards and Technology, Digital Library of Mathematical Functions, Sections 1.16 and 1.17(i). Background for distributional differentiation and delta-sequence interpretation. The regulator identities and pairing limits used here are proved directly.