Em01-B Intrinsic Regularization of the Joint Characteristic Map Descent, complete-contraction growth estimates, conditional limits, and output geometry

We study the complete retained XIV Einstein–scalar contraction on the space of actual reconstructed internal kernels. An explicit mixed-vertex formula and finite branch–vertex inventory define its smooth-cross-line regularization on arbitrary joint tests. Every regularized member respects every reconstruction identity.

The internal retarded-frequency estimates are extended from nonzero pole charts to compact frequency sets containing the origin. Localized microlocal approximation proves convergence to ordinary products on their product-admissible domains. A separate estimate controls the complete regularized contraction, including tests meeting the characteristic locus: its distributional order is at most five and its smoothing growth is polynomial, with an explicit exponent in terms of the crossed distribution's local order. This is a moderate-growth estimate, not a smoothing-uniform bound or a characteristic-limit theorem.

A degree-four null coefficient array excludes a specified highest-degree reconstruction rule, while preserving its associated-graded class. Fixed-kernel and continuous-map limit criteria distinguish convergence of the named sum from a construction on the whole reconstructed space. Auxiliary cutoff locality, regularization comparisons, characteristic-conormal projection, and extension ambiguity are treated separately.

No characteristic limit, normalized quantum coefficient, or quantum admission for the complete frozen graph is asserted. The submission consists of the stated partial construction and estimates.

Figure fig:split-Em01-B records one view of the same separation. Figure fig:chain-Em01-B records one view of the same separation. Figure fig:pair-Em01-B records one view of the same separation. Figure fig:gate-Em01-B records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-Em01-B records that separation. The drawing adds no hypothesis.

fixed for this paper{ All retained internal branches and vertex terms are included. The construction is intrinsic to the reconstructed kernel, not to an amplitude presentation.

The complete-family estimate allows polynomial growth as the smoothing parameter tends to zero. It does not establish convergence through the characteristic locus.

Existence for fixed auxiliary choices, independence of those choices, global compatibility, normalization, and quantum admission are separate obligations.}

The immediate object is a local family

obtained by smoothing the grouped crossed distribution and keeping the internal kernel exact. Its value on the named kernel $K_{\mathrm{ret}}$ is the complete retained contraction, not the full globally assembled quantum coefficient.

The new conclusions are summarized below.

The complete supplied XIV-H/I/J sources fix the retained instance XIVH,XIVI,XIVJ. Their internal formulas are reproduced below. Their identification with the designated observable Hessian, global contact inventory, and global canonical-domain construction remain inherited inputs. The new distributional theorems are valid for every fixed smooth source-coefficient array satisfying the stated finite differential form.

The internal-module conclusions are checked within XIV-I's named finite module. No equivalence with a general intersecting-Lagrangian calculus is asserted. Source-history details are placed in Appendix app:evidence.

Distributions and tests use a complex-bilinear pairing. Tensor components are taken in fixed finite-dimensional bases. Local coordinate-density conversions are included in smooth coefficients.

For a compact $L$ and a test supported in $L$, write

For tensor-valued tests take the maximum over components.

The strong distribution topology is generated by

where $\mathcal{B}$ is a bounded test set. Such a set has common compact support and bounded derivatives of every order.

For a closed cone $\Gamma \subset T^{\ast }U\setminus 0$, the normal topology on ${{\mathcal{D}}^{\prime }}_{\Gamma }(U)$ additionally uses

where $(\mathrm{supp}\chi \times V)\cap \Gamma =\varnothing$. The Fourier convention is

with inverse factor $(2\pi {)}^{-\dim x}$.

We use normal-topology pullback continuity, fixed-factor product continuity under the nonopposition condition, and proper-support pushforward continuity BDH (Proposition 5.1 and Theorems 6.1, 6.3). Their wavefront criteria are the standard distributional calculus Hormander (Section 8.2).

Let

Indices are raised and lowered with $\eta$. Trace reversal is

Retain

The frozen pair is

where

The inverse and insertion coefficients remain $-1$, and the scalar-sourced slot occurs once.

The width $\epsilon >0$, observations, switches, masses, component pairings, and partial-diagonal class ${\mathfrak{C}}_{\mathrm{part}}$ remain fixed. The smooth switch ${\sigma }_{\epsilon }$ equals one after the frozen onset region.

Work on a sufficiently small patch

with separated closures of $A,B,Y$, $\overline{Z}\subset A$, and

The outer separations $b-a,b-z$ are nonzero and future directed. The residual variable $z$ is retained.

Put

and

This is a smooth submanifold of dimension eight in the sixteen-dimensional $W$.

The inherited canonical domain $U_{\mathrm{can}}$ is not identified with $W\setminus N$ or any larger characteristic complement.

Let $I,J,L,M$ denote component indices in the fixed symmetric-tensor bases. Define constant maps by

and

Write

The designated local source derivative has coefficients ${q_{\alpha \beta }}^{L,I,J}\in C^{\infty }$ with $|\alpha |+|\beta |\le 2$, and kernel

All coefficients of the designated source are retained. The analytic results below require its finite local form, not an unprinted numerical simplification.

The fixed retarded kernels are

Set

The complete retained sum is

with

These formulas, with the fixed coefficient arrays, make the retained input independently specified as a distributional expression. Their physical identification remains the designated inherited identity.

Choose a basis $e_I$ of covariant symmetric tensors and its dual basis. All component contractions below use this basis–dual-basis pairing. Off-diagonal symmetric components therefore introduce no additional ad hoc factor of two.

For signature $+---$, define

Then $T_0[\varphi ]=T(\varphi ,\varphi )$, and differentiation of $T_0$ gives $2T(\varphi ,\psi )$. Hence

The grouped crossed distribution has components

Both terms are defined tensor products. Their difference is retained as a single array. No mixed product or alternative covariance is introduced.

Define the linear submersion

The symbol ${\partial }_{2,\mu }$ differentiates only the second relative variable; ${{\partial }_2}^{\nu }={\eta }^{\nu \lambda }{\partial }_{2,\lambda }$.

For smooth $C$, put

For $K=(K^{I,J})\in {\mathcal{D}}^{\prime }(W;V_{\mathrm{int}})$ and smooth $C$, define

For every compact joint dual test $\Phi =({\Phi }_J)$, this means

Equation (eq:vertex) is the full contraction of (eq:Hessian), with the remaining scalar derivative transferred to the output kernel. It includes the kinetic, trace-kinetic, and mass terms.

Proof. Symmetry of the metric tensor converts the symmetrized kinetic term into $h^{\mu \nu }{\partial }_{\nu }\varphi {\partial }_{\mu }\psi$. Contracting the crossed $\psi$ produces ${\partial }_{2,\mu }{C_I}^{\mu \nu }$. The trace kinetic term produces $-{\eta }_{\mu \lambda }{{\partial }_2}^{\nu }{C_I}^{\mu \lambda }/2$. These are the two contributions to ${Z_I}^{\nu }$.

The polarized mass term is $m^2{\eta }_{\mu \nu }\varphi \psi /2$. The leading minus sign of the Hessian gives the negative mass coefficient in (eq:vertex). Integration by parts in the remaining scalar slot gives the output divergence and (eq:vertex-test). The differentiation of the quadratic stress has already used the factor $1/2$ in the interaction, so there is no additional polarization factor.

The output divergence is a total distributional derivative. It retains derivatives of $g(y)$, both relative-variable contributions to a total $y$-derivative of the cross-lines, and any $y$-dependence of $K$. Appendix app:inventory lists every vertex operation and every internal placement.

The seeds underlying $K_{47}$, $K_{5,+}$, and $K_{5,-}$ are defined tensor-product pullbacks, including at $a=z$. The source incidence (eq:Q) is defined.

On the retained patch, the supported sector

is a finite sum of at most three derivatives of smooth coefficients times $\delta (q_R){\delta }^{(4)}(a-z)$.

Proof. The coordinate maps

are invertible linear maps. They identify the three seeds with tensor products of existing retarded distributions.

The map $(v,a,z)\mapsto (v-z,v-a)$ is a submersion. Its pullback of the delta at the origin defines the source incidence before differentiation.

The convolution in (eq:R) is support-proper in the intermediate variable on compact endpoint sets. Its spatial transform is

with value $\Theta (t)t^3/6$ at $\rho =0$, obtained by causal time convolution. The displayed formula for $B$ has the same spatial transform.

On the separated chronological patch,

The source contributes at most two derivatives and the output divergence at most one. Thus $K_8,K_9$ have the asserted form; $K_6$ needs at most two derivatives.

Choose adapted charts for the two $K_5$ placements,

with

Here $u_{+}=a,u_{-}=z$, and ${\zeta }_{\pm }$ complete the outer coordinates. Also use response coordinates

For $0\le d\le 4$, the $K_5$ array spaces are

with reconstruction

For $K_{47}$, use arrays multiplying derivatives of order at most four of $R(b-a)R(b-z)$. For $K_6$, use arrays multiplying ${{\partial }_n}^{\alpha }{\delta }^{(5)}(n)$, $|\alpha |\le 2$. For $K_8,K_9$, use separate such arrays with $|\alpha |\le 3$.

Their finite direct sum is ${\mathcal{P}}_{\mathrm{int}}$. The reconstruction map $\mathsf{Rec}$ sums the blocks, and

carries the quotient topology transported from ${\mathcal{P}}_{\mathrm{int}}/\ker \mathsf{Rec}$. All array spaces use compact-set $C^{\infty }$ seminorms.

The fixed kernels are elements of these spaces after finite chain-rule and Leibniz expansion. The spaces permit arbitrary smooth arrays in the specified module; this does not make every array a new physical source.

For every compact $L⋐W$, there are a continuous coefficient seminorm $p_{L,4}$ and a constant $C_L$ such that

Consequently reconstruction is strongly continuous, $\ker \mathsf{Rec}$ is closed, and the quotient is Hausdorff.

Proof. The retarded kernel $R$ is a locally finite positive measure. Indeed its integral on a compact future cone segment has the radial form

Thus the tensor-product seeds are locally order zero. The normal-delta seeds are likewise locally finite measures. Smooth changes of coordinates preserve order-zero estimates on compact subsets, with their density factors included.

Every reconstructed generator is a derivative of order at most four of one of these measures, multiplied by a smooth coefficient. Transferring derivatives to the product of the coefficient and test proves (eq:internal-order). The seminorm may be taken as the maximum of coefficient derivatives through order four on a slightly enlarged fixed compact chart set.

A bounded test set has common support and bounded derivatives, so the estimate proves strong continuity. Closedness of the kernel and Hausdorffness of the quotient follow.

The Fourier transform of the fixed kernel is

For $\rho >0$,

Near a nonzero characteristic sheet ${\lambda }_s=\omega -s\rho =0$, with $|{\lambda }_s|<c\rho$, $0<c<1$,

The full multiplier is homogeneous of degree $-2$.

Proof. The spatial Fourier transform of $R$ is $\Theta (t)\sin (\rho t)/\rho$. Temporal damping gives

The damped existing kernels converge in tempered distributions. Partial fractions give the two identities.

The physical kernel has homogeneity degree $-2$ in dimension four, so its Fourier transform has degree $-2$. Alternatively positive dilation in the damped identity gives the same distributional conclusion.

The temporal inverse transform of the complete (eq:partial-fractions) is $\Theta (t)\sin (\rho t)/\rho$. A restriction of (eq:pole) to a pole chart is an identity on that chart; after a frequency cutoff, its inverse transform is the corresponding microlocalized kernel, not the whole uncut kernel.

For every $R>0$, there is $C_R$ such that

This includes $\kappa =0$. Every frequency derivative of order $k$ has a corresponding bound using $\Vert \psi {\Vert }_{k+1}$.

Proof. On fixed compact intervals, the principal-value subtraction and the delta term give

The same bound holds uniformly for shifts by $\pm \rho$, $0\le \rho \le R$, after enlarging the fixed interval.

The global distributional interpretation of (eq:partial-fractions) is therefore

The shifted tests have a common compact temporal support. The integrand is bounded by $C_R\Vert \psi {\Vert }_1/|\xi |$. Its spatial integral is finite:

The expression is the Fourier transform of the retarded kernel, as follows by the temporal transform of its spatial sine representation. It thus defines the same distribution at the origin, rather than an extra extension chosen there. This proves (eq:MR-order). For derivatives, transfer the derivatives to the test.

Define five-frequency generators

Fix $j+|\beta |\le 4$, a frequency multiindex $\nu$, and a compact set $B\subset {\mathbb{R}}^5$. There is $C_{B,j,\beta ,\nu }$ such that

In particular, the estimate covers every compact normalized annulus

including its part where $\kappa =0$ and $\theta \ne 0$. It also gives a fixed finite-order bound on low-frequency compact sets.

For finite smooth coefficient arrays, coefficient derivatives through order $r$ multiply the right-hand side by their order-$r$ compact-set seminorm. The degree-at-most-three part satisfies

uniformly for $w$ in a fixed compact parameter set $H$.

Proof. Lemma lem:origin, applied after integration in the compact $\theta$-variable, bounds $A_{j\beta }$ by a first-derivative test seminorm. The polynomial factors are bounded with their needed derivatives on $B$. Transferring $\nu$ derivatives gives the order $1+|\nu |$ bound.

Homogeneity is exact:

This proves (eq:frequency-bound).

For clarity, a compact normalized annulus has a finite cover consisting of a neighborhood of $\kappa =0$, the two nonzero-sheet neighborhoods, and a noncharacteristic remainder. For example, first separate $|\kappa |<2a$ with a fixed $a>0$. On its complement, use sheet neighborhoods $|\omega -s\rho |<c\rho$; outside smaller such neighborhoods the quantity $|p(\kappa )|/|\kappa {|}^2$ has a positive lower bound. The pole formula controls the sheet pieces, the smooth multiplier $-1/p$ controls the remainder, and Lemma lem:origin controls the first region. Thus no nonzero-sheet formula is substituted for the origin region. The global estimate already controls the whole cover with one finite test order.

Smooth parameter differentiation acts only on coefficients. Finally, degree $d\le 3$ scales as $L^{d-2}$, and division by $L^2$ costs at most $L^{-1}$. The finite sum gives the last bound.

Separate inverse Fourier transforms reconstruct ${\delta }^{(j)}(q){{\partial }_r}^{\beta }R(r)$, followed by smooth coefficient multiplication. These presentation estimates are not transverse scaling estimates of the crossed graph.

Fix $U⋐W$. Choose $\chi \in {C_c}^{\infty }({\mathbb{R}}^8)$ equal to one near $\mathfrak{f}(\overline{U})$, and $\rho \in {C_c}^{\infty }({\mathbb{R}}^8)$ with $\int \rho =1$. Put

Define

The smoothing parameter $\delta$ is not the frozen switch width $\epsilon$.

For each $\delta >0$, (eq:F) is a continuous linear map from the quotient space ${\mathcal{K}}_{\mathrm{int}}$ to strong distributions. Its lift satisfies

It is defined on every compact joint test, including tests meeting $N$. Its value on $K_{\mathrm{ret}}$ uses the complete branch and vertex inventory.

Proof. The array $C_{\delta }$ is smooth. The test formula (eq:vertex-test) uses smooth multiplication, finite component contractions, and output differentiation. These operations, for fixed smooth coefficients, are continuous on strong distributions.

Composing with Lemma lem:reconstruction gives a continuous map on presentations. It depends only on their actual reconstructed distribution and therefore descends continuously for the quotient topology. A null reconstruction pairs to zero with every smooth test in (eq:vertex-test). This proves all null identities simultaneously.

Linearity in $K$, together with (eq:Kret), retains every term. Appendix app:inventory specifies the finite vertex expansion without a leading-term truncation.

Here “intrinsic” means representation-independent on ${\mathcal{K}}_{\mathrm{int}}$. It does not claim independence of the flat relative-variable coordinates or of the mollifier.

Let $u\in {\mathcal{E}}^{\prime }({\mathbb{R}}^d)$ and let $A\subset {\mathbb{R}}^d\setminus 0$ be a closed angular cone with $\mathrm{WF}(u)\subset {\mathbb{R}}^d\times A$. Then

in the normal topology of ${{\mathcal{D}}^{\prime }}_{{\mathbb{R}}^d\times A}({\mathbb{R}}^d)$.

Proof. Strong convergence follows by transposing convolution:

The expression converges in smooth seminorms uniformly on bounded test sets.

Compact support gives polynomial Fourier growth, and the wavefront condition gives rapid decay on every closed cone separated from $A$. Since

a frequency ball–tail split gives weighted Fourier convergence on such cones.

For localized Fourier seminorms, convolve this identity with $(2\pi {)}^{-d}\hat{\chi }$. In a slightly enlarged tested cone, use rapid decrease of $\hat{u}$. In its complement, angular separation forces the Schwartz argument of $\hat{\chi }$ to control the total frequency, dominating the polynomial bound of $\hat{u}$. This gives uniformly small weighted tails. Strong convergence controls the bounded frequency region. All normal seminorms therefore converge.

Let $f:\Omega \subset {\mathbb{R}}^n\to {\mathbb{R}}^d$ be a linear submersion, $u\in {\mathcal{D}}^{\prime }(\Omega )$, $v\in {\mathcal{D}}^{\prime }({\mathbb{R}}^d)$, and $D$ a constant-coefficient differential operator. Let $L⋐\Omega$, and let $\chi =1$ near $f(L)$. If on a neighborhood of $L$,

then

strongly in distributions locally near $L$. A finite list of component pairings can be treated simultaneously.

Proof. At a fixed point of $L$, compactness of normalized directions and closedness of wavefront sets give small base neighborhoods and closed angular cones containing the relevant wavefronts, with a strict nonopposition margin after pullback by $d{f}^T$.

Differentiate before smoothing:

Choose a relative cutoff $\sigma$ equal to one on a smaller neighborhood of the image and supported within the chosen relative neighborhood. Compact support of $\rho$ ensures that, for sufficiently small $\delta$, the smoothed factor on that smaller neighborhood equals ${\rho }_{\delta }\ast [\sigma D(\chi v)]$.

Lemma lem:mollification applies to this compact distribution in its constant angular cone. Submersion pullback continuity and fixed-factor product continuity then give the local convergence. A finite partition covers $L$.

For a finite list of pairings, make the neighborhood small enough for each pairing and apply the argument with its own cones. There is no need to take a union of incompatible component cones.

The crossed operators are exactly $\mathrm{id}$ and the four ${\partial }_{2,\mu }$; raised derivatives are fixed linear combinations. They do not enlarge wavefront sets. The map $\mathfrak{f}$ is a linear submersion.

For a fixed $K$, define ${\mathcal{O}}_K$ as the open set where, componentwise for every vertex pairing,

On compact subsets of ${\mathcal{O}}_K$,

strongly in distributions, independently of the normalized compact mollifier and of a cutoff equal to one near the relative image.

The limit agrees with the inherited canonical value where that value is the ordinary contraction satisfying (eq:OP).

Proof. Apply Lemma lem:local-product to every pairing in Appendix app:inventory. Smooth weights, finite sums, and output derivatives preserve convergence. Each admissible mollifier has the same ordinary product as its limit.

This does not identify ${\mathcal{O}}_K$ with an entire characteristic complement. A generalized canonical composition needs its own approximation theorem.

Let ${\chi }_1,{\chi }_2$ both equal one on a neighborhood of $\mathfrak{f}(L)$, where $L⋐U$. For a fixed compact mollifier, the two regularized vertex maps have identical values on tests supported in $L$ for every sufficiently small $\delta$.

Proof. The distribution $({\chi }_1-{\chi }_2)C$ vanishes on a relative neighborhood of the compact $\mathfrak{f}(L)$. Its distance from a smaller compact neighborhood is positive. For sufficiently small $\delta$, convolution with ${\rho }_{\delta }$, and all its derivatives, therefore vanish on that smaller neighborhood. The crossed coefficients and the test action (eq:vertex-test) coincide.

This is valid even for tests meeting $N$. It proves locality in the cutoff, not independence of the mollifier at $N$.

The compact array $\chi C$ has finite distributional order. Fix any integer $q\ge 0$ and constant $B$ such that, componentwise,

on a fixed compact neighborhood of its support. A compact distribution always admits such a choice. No universal minimal $q$ is claimed.

For every integer $r\ge 0$,

The constants depend on the fixed compact array and finitely many derivatives of $\rho$.

Proof. Use

If a compact-support cutoff is needed for the pairing, take it equal to one near $\mathrm{supp}(\chi C)$. Each derivative in the $q$-order distribution estimate costs at most ${\delta }^{-1}$, and the kernel contributes ${\delta }^{-8-|\alpha |}$. The cutoff derivatives contribute only lower powers, absorbed for $\delta \le 1$.

For every compact $L⋐U$, there are $C_L>0$ and a continuous coefficient seminorm $p_{L,4}$ such that

Here $q$ is any fixed order in (eq:cross-order).

The estimate applies to arbitrary joint tests meeting $N$, to the complete finite vertex operation, and in particular to the named retained sum $K_{\mathrm{ret}}$. It proves a finite order-five, polynomial-growth bound. It does not prove a bound independent of $\delta$.

Proof. Apply the order-four internal estimate (eq:internal-order) to the smooth test appearing in (eq:vertex-test).

Four derivatives of $gZ(C_{\delta }){\partial }_y\Phi$ require at most five derivatives of $C_{\delta }$ and at most five derivatives of $\Phi$. The mass contribution needs at most four derivatives of $C_{\delta }$, and is bounded by the same right-hand side. The fixed linear map $\mathfrak{f}$, metric coefficients, smooth $g$, and finite component sums contribute fixed constants.

Lemma lem:smooth-bounds, with $r=5$, gives ${\delta }^{-8-q-5}={\delta }^{-q-13}$. All internal blocks are included in the reconstruction estimate, so this is an estimate of the complete regularized expression. No cancellation or branch nonvanishing assumption is used.

For the fixed array $A_{\mathrm{ret}}$ representing $K_{\mathrm{ret}}$, (eq:moderate) is a nonconditional estimate

It establishes moderate growth of the actual retained family. Its power is deliberately coarse. Removing that loss or proving summable differences remains a substantive characteristic problem.

Choose coordinates $(\tau ,\nu )$ with $N=\{\nu =0\}$, $\nu \in {\mathbb{R}}^8$, and fixed compact test supports. Set

Whenever the scaled supports remain in the chart,

Proof. The fixed coordinate transformation and its density conversion have bounded derivatives on the compact chart. The order-five seminorm of the scaled test is bounded by $C{\lambda }^{-8-5}\Vert \Phi {\Vert }_5$. Apply (eq:moderate).

This estimate is uniform after the displayed powers are included. It is neither a smoothing-uniform transverse bound nor an intrinsic transverse degree of the limit.

The proved row is not relabeled as the missing smoothing-uniform estimate. A future improvement must preserve the complete tensor and branch sum before asserting a cancellation or a smaller growth exponent.

A linear map $\tilde{\mathsf{M}}:{\mathcal{P}}_{\mathrm{int}}\to {\mathcal{D}}^{\prime }(U)$ defines a unique map on ${\mathcal{K}}_{\mathrm{int}}$ exactly when

Continuity of the lifted map gives continuity of the descended map for the quotient topology.

Proof. Two presentations have the same reconstruction exactly when their difference belongs to $\ker \mathsf{Rec}$. Annihilating this kernel makes the proposed image independent of the representative. The quotient universal property proves the continuity statement.

In one $K_5$ block and one tensor component, choose a compact coefficient cutoff $b$, and define

The array (eq:null-array) belongs to the defined degree-at-most-four space and has zero reconstruction:

For a suitable $b$, its degree-four part reconstructs to the nonzero distribution

Thus $A\mapsto \mathsf{Rec}(A_{[4]})$ does not descend.

Proof. The identity $q{\delta }^{(j)}=-j{\delta }^{(j-1)}$ follows by applying the derivative distribution to $q\phi (q)$. Take $j=4$, tensor with $R(r)$, and multiply by $b$.

Choose $b=1$ near a product region meeting the support of $R$. A test with a nonzero third $q$-jet and nonzero retarded-kernel pairing detects the degree-four part. Proposition prop:descent then excludes the rule.

For $F_d=\mathsf{Rec}(P_{5,\pm ,d})$, the quotient class

is nevertheless well-defined: the discarded terms are already in $F_{d-1}$. This algebraic associated-graded statement selects no representative or joint symbol.

A continuous section of a specified quotient $q:B\to S$ requires a complemented kernel. Indeed a section gives the projection $I-sq$; conversely a continuous projection $P$ onto $\ker q$ gives $s(qb)=(I-P)b$. No such actual joint quotient and section are constructed here.

Fix $K_{\ast }\in {\mathcal{K}}_{\mathrm{int}}$. Suppose every test pairing of $t_{\delta }={\mathsf{F}}_{\delta }(K_{\ast })$ has a limit, and for every compact $L⋐U$ there are $C_L,m_L$ such that

uniformly in $\delta$. Then $t_{\delta }$ converges strongly to a distribution $t$.

This conclusion concerns $K_{\ast }$ only, not a continuous map on all of ${\mathcal{K}}_{\mathrm{int}}$.

Proof. The limit inherits the local finite-order estimates, so is a distribution. On a bounded test set supported in $L$, the tests are totally bounded in the $C^{m_L}$ norm. Use a finite net, the uniform estimate, and convergence on its finitely many tests to obtain uniform convergence on the bounded set.

Fix $K_{\ast }$ and put $t_n={\mathsf{F}}_{2^{-n}}(K_{\ast })$. Suppose on every compact $L⋐U$ there are an order $m_L$, summable numbers $a_{n,L}$, and numbers $b_{n,L}\to 0$ such that

and

Then the full family has a strong distributional limit, with remainder bounded by

on that smoothing shell.

Proof. The first estimate makes the telescoping dyadic sequence Cauchy in a fixed finite-order test seminorm. Its limit is a distribution because the initial term and the sum of increments satisfy a common local bound. The second estimate controls every value between dyadic parameters. The displayed tail estimate follows by telescoping and implies strong convergence.

For the named $K_{\ast }=K_{\mathrm{ret}}$, this theorem is a concrete research criterion. No summability is supplied by the polynomial bound of Theorem thm:moderate.

Fix a closed cone $\Gamma \subset T^{\ast }U\setminus 0$. For the complete family suppose:

1. Every pairing ${\mathsf{F}}_{\delta }(\mathsf{Rec}A)(\Phi )$ converges.

2. For each compact $L$, there are $C_L,m_L$ and a continuous input seminorm $p_L$ such that

   $$|{\mathsf{F}}_{\delta }(\mathsf{Rec}A)(\Phi )|\le C_L{p}_L(A)\Vert \Phi {\Vert }_{m_L}$$

   uniformly in $\delta$.

3. Every Fourier seminorm admissible for $\Gamma$ has a bound

   $$q_{r,\chi ,V}({\mathsf{F}}_{\delta }(\mathsf{Rec}A))\le Cp(A)$$

   with a continuous input seminorm, uniformly in $\delta$.