Em02-A Exact Sector Reductions and Local Scale Calculations Retarded seed transitions, complete translation pairings, and the unresolved inherited physical specialization

We reorganize the local fixed-kernel problem around the integrals that must actually be calculated. The complete retarded inventory is retained, while the crossed array is smoothed in eight flat relative variables. Exact adjoints and the compact translation representation are collected in supporting appendices.

The main text gives a row–sector contribution ledger, exact non-tip integration formulas, and a cone-radius treatment of the coincidence transition. For a fixed smooth localization profile, the scalar seed measures supported by $R(b-a)R(a-z)$ and $R(b-z)R(z-a)$ have explicitly calculated order-two coincidence-localization coefficients. The seed supported by $R(b-a)R(b-z)$ has an order-three coefficient. Uniform remainders are proved directly. These powers concern an independent internal localization parameter, not the physical smoothing scale.

The same transition calculations are checked through the translation-distribution construction. A complete derivative certificate, if obtained for identified crossed data, determines the singular primitive under an explicit scalar absolute continuity hypothesis. The smooth-data expansion is retained only as a consistency benchmark.

The supplied excerpts do not determine an executable inherited physical input package. Accordingly, no physical singular-data sector is evaluated, no physical complete coefficient or integrable smoothing remainder is obtained, and no physical limit, authorized correction, or quantum admission is asserted. A completion ledger distinguishes these unresolved tasks from the mathematical calculations proved here.

Figure fig:scope-Em02-A, Figure fig:split-Em02-A, Figure fig:chain-Em02-A, Figure fig:pair-Em02-A, and Figure fig:gate-Em02-A name the stop. Not a proof, and not an admission of a quantum coefficient.

Scope of this revision. The requested physical completion cannot be produced from the supplied input descriptions alone. This manuscript records that stop at the beginning, rather than replacing the missing family with diagnostic data. Its new calculations concern the displayed retarded seed geometry.

The target is one local family

A physical calculation requires the values or complete symbolic definitions of

in one component and dual-pairing convention.

The supplied excerpts provide an internal formula inventory and several conditional interfaces. They do not instantiate (eq:required-inputs) for the inherited weighted observable. In particular, a name for a propagator, source, or root operation does not determine the component operation needed below.

The following is an identification ledger, not an assumption that the entries have been supplied.

Finite symbolic definitions can fill this ledger without a printed component table. The present manuscript does not fill it by copying the arrays of a different application.

The documentary locations for the internal formulas are XIV-H and XIV-I XIVH,XIVI. XIV-J distinguishes reconstruction, the supported-sector reference, and the missing full characteristic map XIVJ. These attributions concern the supplied excerpts only.

If an actual local identity $k=\tilde{k}$ has been proved on a neighborhood of $\mathrm{supp}\zeta$, then

for every smooth crossed coefficient $c$. The vertex test and all its derivatives remain supported there.

The supplied Em01-C plateau statement is such an internal identity on its stated retained patch EmC. Applying it requires identifying the chosen patch, switch, and weighted source with its hypotheses. It does not determine the advanced root from local constancy of $f$, and it does not provide a smoothing gain by itself.

A missing frozen presentation space is a separate operator obligation. It does not prevent a fixed-kernel calculation once the actual values have been identified PatchII.

Target I: non-tip isolation.

A compact patch on which support excludes every row except one. This may decide failure for an identified family and a specified correction class.

Target II: coincidence.

A compact patch meeting $r=z-a=0$. All potentially contributing rows and the cone-tip transition must be retained.

Target I does not solve Target II. The support-isolated Em01-G failure concerns its own identified application EmG; its coefficient is not imported here.

Use $M={\mathbb{R}}^4$, $W={M^4}_{(a,b,z,y)}$, and $\eta =\mathrm{diag}(1,-1,-1,-1)$. The defining kernels and complete adjoints are displayed and proved in Appendix app:adjoints. Their inventory is

There are six operational entries, since $K_9=K_{9d}+K_{9m}$.

The mathematical crossed operation uses

and

Relative derivatives are taken before pullback and raising uses $\eta$. Define

This fixes a mathematical operation. Its physical identification is an unresolved entry of the ledger.

Fix $\zeta \in {C_c}^{\infty }(W)$, a relative cutoff $\chi =1$ near $\pi (\mathrm{supp}\zeta )$, and one compact profile $\rho \in {C_c}^{\infty }({\mathbb{R}}^8)$ with $\int \rho =1$. Put

The differentiated profile is

It satisfies

These identities follow by direct differentiation and integration by parts.

For a two-line row define its smooth ambient amplitude by

The inverse exists for these independent seed charts. For response rows, retain the smooth restricted amplitude

Thus the derivatives have already been applied before any geometric restriction below.

Let ${\mu }_e$ denote the scalar ambient seed measure:

These expressions mean the independent tensor-product measures of Appendix app:adjoints, not products formed after coincident-variable restriction. Write ${\mu }_0$ for the response measure

Status: Proved for displayed mathematical data.

The raw and differentiated complete pairings are

Every fixed-test pairing is smooth for $\delta >0$.

Proof. Apply the adjoint identities of Theorem thm:adjoints. Distributional convolution may be differentiated on compact positive-scale intervals because the profiles form smooth families with bounded test seminorms. Linearity of the vertex then gives (eq:exact-derivative).

Choose a local sphere frame

with $\ell \cdot k=2$ and $E_A\cdot E_B=-{\delta }_{AB}$. Use

Fix a compact regular chart with positive lower bounds for $\alpha$, $\beta -\alpha$, $\beta$, and $\beta -u$. Where a non-tip formula requires $\beta -\gamma >0$, that restriction is imposed separately.

Put

Then

The density is

and ${\mathrm{d}}^{4}r=2\mathrm{d}\gamma \mathrm{d}\upsilon {\mathrm{d}}^2\omega$. The inverse chart and density verification are given in Appendix app:geometry.

On the scalar supports:

For $5a$, directed nullness gives $\gamma ,\upsilon \le 0$. For $5z$, it gives $u\ge 0$. The relations follow by equating the two expressions for $|\omega {|}^2$ in (eq:internal-cones).

Status: Proved for displayed mathematical data.

1. At

   $$b-a=\beta \ell ,r=-ϵ(1,\mathbf{m}),ϵ>0,\mathbf{m}\ne \mathbf{n},$$

   a sufficiently small neighborhood excludes every row except $5a$.

2. If $b-a$ and $b-z$ are future null and nonparallel, a sufficiently small neighborhood excludes every row except $47$.

The complete contraction on tests in either neighborhood equals the surviving row contraction for every smooth crossed array. No nonzero singular coefficient follows from isolation alone.

Proof. In (i),

This excludes the $47$ and $5z$ seed supports; $r\ne 0$ excludes the response rows.

In (ii), write $s=b-a$, $t=b-z$. Then $r=s-t$ and $r^2=-2s\cdot t<0$. Both $5$ supports require $r^2=0$, and the response rows require $r=0$. Differentiation and smooth multiplication preserve these local support exclusions.

Choose $b_0\in C^{\infty }([0,\infty );[0,1])$ with $b_0=1$ on $[0,1]$ and $b_0=0$ on $[4,\infty )$. A fixed coincidence partition is

where ${ϵ}_0$ is independent of $\delta$.

For $L>0$, let

Use the exact five-sector partition

Their sum is one.

Throughout the main calculation, these functions multiply already differentiated seed amplitudes. Thus

Here ${\vartheta }_{e,j,\delta }$ denotes the partition pulled back to the appropriate seed chart, including the fixed tip/away partition if used.

Individual $v_{e,j,\delta }$ are not declared to be derivatives of raw sectors. No motion term is needed in (eq:postadjoint-partition). The different raw-sector convention is recorded in Appendix app:partitions.

In the next table, $P$ means “not excluded by scalar support.” It does not mean a proved nonzero contribution. A zero is an exact support zero in the post-adjoint convention.

Proof. On the $47$, $5a$, and response scalar seeds, $q_R=0$, so $a_{R,\delta }=1$ and ${\theta }_{0R,\delta }=0$. The response seeds have $r=0$, where ${\theta }_{\mathrm{away}}=0$. Because the partition is inserted after the adjoints, no derivative of these vanishing factors has been generated.

The remaining calculation information is:

Every crossed factor in this table is currently unidentified for the inherited family. Consequently no physical scaling exponent is entered.

The identity

implies, on the fixed compact chart,

Thus the $00$ sector permits $(u,v,\xi )=(\delta u_{\ast },\delta v_{\ast },\sqrt{\delta }X)$. The translated equations, including their higher-order terms, are retained in Appendix app:geometry.

Neither this implication nor simultaneous crossed cones forces $r=O(\delta )$. Moreover, the $5z$ support admits $q_1=q_2=0$ with $q_R>0$ fixed; see Proposition prop:counterexample. The $0R$ sector therefore cannot be removed by the core argument.

This section identifies the actual integrals remaining after the adjoints. It does not insert an unspecified physical boundary value.

Let

expressed in the characteristic coordinates; component sums are suppressed temporarily. Let ${\Theta }_{j,\delta }$ be a chosen post-adjoint sector factor.

Status: Proved for displayed mathematical data.

On compact supports within the indicated coordinate domains, the two-line differentiated integrals are the following exact scalar integrals.

For $47$, put ${\upsilon }_{47}=-|\omega {|}^2/[4(\beta -\gamma )]$. Then

For $5a$ on $\gamma <0$, put ${\upsilon }_{5a}=|\omega {|}^2/(4\gamma )$. Then

On $\upsilon <0$, the alternative formula eliminates $\gamma =|\omega {|}^2/(4\upsilon )$ and uses density $\beta /(8{\pi }^2|\upsilon |)$.

For $5z$ on $0<\gamma <\beta$, put

Then the integral is

On $\upsilon >0$, an alternative eliminates

and has density

in $\mathrm{d}\tau \mathrm{d}\upsilon {\mathrm{d}}^2\omega \mathrm{d}v{\mathrm{d}}^2\xi$.

The response measure has density

Its amplitude is the already computed response adjoint, not a newly chosen tangential jet.

Proof. Away from a cone vertex, $R(x)=(2\pi {)}^{-1}\Theta (x^0)\delta (x^2)$. The full null-frame density is

For $47$, the constraint Jacobian in $(u,\upsilon )$ is $16\beta (\beta -\gamma )$. For $5a$, it is $16\beta |\gamma |$, or $16\beta |\upsilon |$ in the alternative chart. For $5z$, the two Jacobians are respectively $16\gamma (\beta -\gamma )$ and $16\upsilon (\beta -\gamma )$. Combining these with $(2\pi {)}^{-2}$ proves the formulas. The directed support conditions select the indicated domains.

For the response seed, integrate ${\delta }^{(4)}(r)$ in the original $r$ density, and then $\delta (4\beta u)/(2\pi )$. The result is $\beta /(2\pi )$.

The two $5a$ charts cover compact past-null non-tip regions after a finite partition: at least one of $|\gamma |$ and $|\upsilon |$ is bounded below.

For $5z$, the formulas above cover the non-tip region within the additional domain $\beta -\gamma \ge c_0>0$. Separation of $b-z$ alone does not imply that inequality: a nonzero future-null $b-z$ can be parallel to $k$. Outside this domain one must use another chart or the independent cone-radius formula below. No global coverage is claimed from these two denominators alone.

To finish a singular non-tip calculation, the actual ${\dot{c}}_{\delta }$ must now be inserted into (eq:47-integral)–(eq:5z-integral). Its boundary values, derivatives, contractions, and remainder are not determined by the scalar densities.

The coincidence transition is treated in independent cone coordinates rather than by extrapolating the non-tip denominators.

For $T\ge 0$ and $\mathbf{m}\in S^2$, write $n_{\mathbf{m}}=(1,\mathbf{m})$. The exact cone measure is

For a compact smooth ambient scalar amplitude $H$,

Both radial variables range over $[0,\infty )$. These formulas include $T=0$ without introducing a $1/|\gamma |$ denominator.

Introduce a second parameter $ϵ>0$, independent of $\delta$, and define

Set

Both constants are finite and strictly positive.

In the characteristic chart define the incidence measure ${\nu }_0$ by

Thus, locally,

Status: Proved for displayed mathematical data.

Fix a compact characteristic localization as above. For amplitudes $H$ supported there, and sufficiently small $ϵ$,

Here $K$ is a fixed enlarged compact neighborhood, and the constants are uniform for the stated support class. For the response measure,

exactly.

These are localization asymptotics in $ϵ$. They are not smoothing asymptotics in $\delta$.

Proof. For $5a$, use (eq:radial-5a). On the $r=-T{n}_{\mathbf{m}}$ cone, $|r{|}_{\mathrm{E}}^2=2T^2$. Replace the amplitude by its value at $T=0$. The removed cone factor has total mass

The remaining integral is $⟨{\mu }_0,H⟩$. The amplitude difference is at most $CT\Vert H{\Vert }_{1,K}$, and

The remaining cone and passive measures have bounded mass on a fixed compact set. This proves (eq:tip-5a).

For $5z$, use (eq:radial-5z). At $T=0$, the ambient point is $(z,z+S{n}_{\mathbf{n}},z,y)$, again giving ${\mu }_0$ after renaming the base variable. The same radial mass and Lipschitz estimate prove (eq:tip-5z).

For $47$, a sufficiently small coincidence neighborhood satisfies $\beta -\gamma \ge c_0>0$. Use (eq:47-integral) without a smoothing amplitude. On its seed,

Set $\gamma =ϵG$, $\omega =ϵW$. Then

and the cutoff confines $(G,W)$ to a fixed bounded set. The density becomes

Its leading factor is ${ϵ}^3/(8{\pi }^2)$. The cutoff differs from $b_0(2G^2+|W{|}^2)$ by $O({ϵ}^2)$, uniformly on that bounded set. The amplitude differs from its incidence value by $O(ϵ)\Vert H{\Vert }_{1,K}$, and the density factor differs from $1/(8{\pi }^2)$ by $O(ϵ)$. Therefore the leading coefficient is

Changing $\sqrt{2}G$ to a Euclidean coordinate and then using three-dimensional polar coordinates identifies this constant as $c_{47}$ in (eq:transition-constants). All errors are $O({ϵ}^4)\Vert H{\Vert }_{1,K}$. Finally ${\theta }_{ϵ}=1$ at $r=0$, proving the response identity.

The coefficients and remainders in Theorem thm:transition are explicit scalar seed calculations through the coincidence transition. The $47$ seed has no internal cone vertex there, whereas the $5$ seeds contain a cone radius tending to zero. Their different localization powers are derived from their measures, not imposed as a common $r$-dilation.

At positive $\delta$, all amplitudes ${H_{e,{\dot{c}}_{\delta }}}^{I,J}[\Phi ]$ are smooth, so the theorem applies to them. Define

Then

The ambient derivatives in these amplitudes precede their incidence evaluation.

If $C^{\chi }$ has distributional order at most $h$, the elementary convolution estimate and the derivative-placement ledger give

Indeed, one additional ambient derivative of each smooth amplitude adds at most one crossed derivative and one test derivative to the bounds in Appendix app:bounds.

These estimates do not provide an integrable complete $\delta$-remainder. Substituting $ϵ=ϵ(\delta )$ would also require control of the complementary region. The incidence expressions in (eq:transition-insertion) still contain the unidentified crossed coefficient and have not been evaluated as physical smoothing coefficients.

For a smooth crossed array $H_0$, let

Resolving this finite functional into crossed components gives compact distributions ${{\Lambda }_{\Phi ,A}}^{\zeta }$ in the relative variables. With $\mathfrak{d}(x,v)=x-v$, define

Status: Proved for displayed mathematical data.

The translation distribution is compact and satisfies

Consequently

Proof. The compact-local internal order-four estimate and the vertex give order at most five for each relative functional. Both tensor factors are compact, so their difference pushforward exists. Tensor evaluation gives $⟨{{\Omega }_{\Phi }}^{\zeta },\psi ⟩={{\Lambda }_{\Phi }}^{\zeta }(\psi \ast C^{\chi })$. The adjoints prove (eq:translation-bridge). A detailed order and support estimate appears in Appendix app:translation.

Choose a translation cutoff $\vartheta =1$ near zero. For sufficiently small $\delta$, both profiles sample only $\vartheta {{\Omega }_{\Phi }}^{\zeta }$ exactly.

The potential row fibers are

where ${\Sigma }_e$ is the corresponding scalar seed support. They are potential contributors only. Classifying their singular, regular, and degenerating pieces requires the actual crossed singular structure. The equation $\pi (w)=v$ alone does not impose $r=0$.

For fixed $ϵ$, define a row functional by inserting ${\theta }_{ϵ}$ after its adjoint, exactly as in $V_{e,\delta ,ϵ}$. Construct its translation distribution ${\Omega }_{e,\Phi ,ϵ}$ by the same component tensor pushforward. Then

To check the coefficients rather than merely the representation, take an arbitrary compact translation profile $\psi$ and put $c=\psi \ast C^{\chi }$. Theorem thm:transition yields

Thus the same constants and incidence functionals occur in the seed and translation descriptions for every such $\psi$, not only the chosen mollifier.

This verifies the calculated internal localization coefficients. It does not supply the missing physical origin-germ expansion.

A full homogeneous germ decomposition is unnecessary if one can directly prove

with evaluated complete coefficients and

For the inherited family, neither the coefficients nor this remainder have been calculated.

If both routes establish (eq:desired-certificate), their complete sums agree by (eq:translation-bridge). Individual fibers need not correspond to individual sectors. After scalar absolute continuity and an anchor are used to integrate the certificate, uniqueness of finite real power–logarithm expansions identifies the singular coefficients and finite part; see Appendix app:decisions.

The remaining physical estimates are not hidden inside the word “complete.”

The mathematical transition remainders in (eq:two-parameter-errors) are proved, but they do not resolve any of these complete physical smoothing estimates.

The status categories used here are:

• proved for displayed mathematical data;

• physically identified;

• physically evaluated;

• conditional;

• unresolved.

No claim in the present revision reaches either of the two physical completion categories.