Landmark XII-A One graded polynomial carrier for the finite-ramp multiplet, with ordering-stable composites and a robust observation exclusion

We construct a polynomial quantum carrier for the finite-ramp Einstein–scalar multiplet $(\gamma ,\varphi ,c,\bar{c},b)$ at fixed $\epsilon >0$. The equation operator, the action Hessian, and the Hadamard product matrix have distinct roles. Reduced solution coordinates provide a componentwise wave-propagator separation lemma. An oriented Hadamard smoothing lemma then proves that polynomials vanishing on the complete gauge-fixed free shell form a two-sided quantum ideal, without requiring microcausal equation primitives or a mollification argument.

The regular quotient recovers the causal free-field algebra. Smooth admissible changes of product data yield continuous ordering isomorphisms. Their corrections on the quadratic insertion and the scalar-cubic tree are computed explicitly and lie in the regular subalgebra; consequently the named, formally completed composite sector is ordering-stable. The frozen free differential is checked in both ghost orderings and transported by conjugation.

Continuity is stated at the established level: fixed-support kernel continuity, stagewise product hypocontinuity, a Hausdorff shell quotient, and global separate continuity. Global hypocontinuity and completeness are not released. A regular time-slice theorem identifies ambient and interior-slab free algebras.

Finally, taking XI-G's observation-qualified obstruction as an explicit input theorem, we exclude the detected cubic shell classes for every admissible ordering. Continuous detection functionals make the detected class open and dense on every fixed-support test stage on which a detector is nonzero. The undetected set, the Einstein subshell, and any enlargement remain undecided.

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

Figure fig:scope-XII-A records that separation. The drawing adds no hypothesis.

and proof boundary{ Constructed: the graded polynomial product, regular presentation, free-shell descent, frozen free differential, ordering transport, ordering-stable composite sector, the delimited continuity statements, observation robustness, and the regular ambient–slab identification.

Proved obstructed, conditional on the stated XI-G input: admission of the detected cubic classes into this carrier.

Unresolved or not opened: undetected observations, restriction to the Einstein subshell, global product hypocontinuity, completeness, and enlargement. Absence of an enlargement construction is not a no-go theorem.

The predecessor obstruction is copied, not reopened. No state, interacting differential, quantum retarded product, time-ordered product, loop construction, or sharp-width limit is introduced.}

Let $M={\mathbb{R}}^4$ with $\eta =\mathrm{diag}(1,-1,-1,-1)$, $\square ={{\partial }_t}^2-{\Delta }_{\mathbf{x}}$, and $P_m=\square +m^2$, $m\ge 0$. Fix $T>0$, $M_T=[0,T]\times {\mathbb{R}}^3$, and ${M_T}^{\circ }=(0,T)\times {\mathbb{R}}^3$. The ramp ${\sigma }_{\epsilon }$ is smooth, has fixed width $\epsilon >0$, and vanishes for $t\le 0$. The nonzero switch $g$ belongs to the frozen admissible class of XI-G; in particular its compact support lies in ${M_T}^{\circ }$. Additional plateau hypotheses used by XI-G are part of that input, not inferred from compact support alone.

There is one metric field and no second classical metric fluctuation. The species, parities, and ghost numbers are

The symbols $f,g,{\sigma }_{\epsilon },\eta$ are external data, not field generators. Both $\hslash$ and $\kappa$ are even, ghost-number-zero formal parameters.

Trace reversal, divergence, and the symmetrized gradient are

Parentheses include the factor $\frac{1}{2}$. With $A=2G_{\mathrm{lin}}$ and formal adjoint $D^{\ast }$, the frozen identities are

They fix the normalization ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{E_0}^{\mathrm{R}}$.

The off-shell configuration space $\mathcal{C}$ consists of smooth multiplets $\Phi =(\gamma ,\varphi ,c,\bar{c},b)$. Define the equation operator by

It is not asserted to be formally self-adjoint. Its kernel is the complete gauge-fixed free shell

The action Hessian is a different operator. In the reordered blocks $(\gamma ,b)\oplus \varphi \oplus (c,\bar{c})$,

The ghost block is interpreted with the graded variational convention. Ordinary transpose identities for bosonic matrices are not substituted for graded transpose identities.

Writing $\rho =b-D\gamma$, the first two action equations become

The scalar sign and the invertible ghost mixing do not change their kernels. Thus

For solution geometry only, use

Then $J:\ker \mathbf{P}\to {\mathcal{C}}_{\mathrm{sol}}$ is an isomorphism. The variable $b$ remains independent off shell; $J$ is not imposed on the off-shell carrier.

Let ${E_{\mu }}^{\mathrm{R}/A}$ be the retarded/advanced inverse of $\square +{\mu }^2$, and put $E_{\mu }={E_{\mu }}^{\mathrm{R}}-{E_{\mu }}^{\mathrm{A}}$. On compact sources,

These inverses have causal support. We use the standard distributional extension and smooth Cauchy theory BGP. The reduced causal operator is

It belongs to $\mathbf{P}$, not to ${\mathsf{K}}_{\mathrm{act}}$.

The retarded/advanced inverses of ${\mathsf{K}}_{\mathrm{act}}$ are

For example, for source $(U,v)$ the auxiliary solution is

Indeed $b=D\gamma -v/2$ because $D\mathcal{R}{E_0}^{\mathrm{R}/A}v=v/2$. Substitution verifies both inverse identities. The scalar and ghost inverse identities follow by direct multiplication.

The action causal matrix $\Delta ={G_{\mathrm{act}}}^{\mathrm{R}}-{G_{\mathrm{act}}}^{\mathrm{A}}$ therefore has blocks

All other blocks vanish. The causal algebra ${\mathcal{A}}_0(M)$ is the completed graded algebra generated by compact linear smearings, modulo the equations and

Odd fields are evaluated in arbitrary finite Grassmann algebras. An identity means equality of every Grassmann coefficient for every such evaluation. This convention applies to restriction, vanishing, polarization, and all evaluation maps below.

A polynomial has the form

where $u_n$ is a compactly supported, graded-symmetric distribution on $M^n$, with finitely many species components. Graded symmetrization is normalized:

The sign $ϵ(\pi )$ is the Koszul sign. Our kernels contain no additional $1/n!$. Consequently

All derivative and polarization identities are understood coefficientwise in the Grassmann algebra.

Let ${\tilde{V}}_{+},{\tilde{V}}_{-}$ be the closed causal covector cones, including zero. For $n\ge 1$, define

The polynomial microcausal carrier ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ consists of the above polynomials with

Degree zero consists of scalars.

The regular classical algebra ${{\mathcal{A}}^{\mathrm{cl}}}_{\mathrm{off}}$ is the subalgebra generated by compact smooth linear smearings; its kernels are finite sums of tensor products of tests. “Regular” here means finite-rank regular, not every smooth compact kernel.

Define the vector spaces

A product on ${\mathcal{E}}_H$ will be justified in Section sec:quotients; it is not assumed in this definition.

Put $h_{\mu }=-W_{\mu }$, where

These distributions are bisolutions and satisfy $h_{\mu }-{h_{\mu }}^{\mathsf{T}}=i{E}_{\mu }$. Their wavefront relation has one fixed orientation:

The relation includes the null-geodesic condition. The same orientation is used for every species. This is product data; no state or positivity condition is selected.

The full matrix is

All other blocks vanish; the ghost blocks include the vector identity. Slot differentiation includes the corresponding tensor index contraction.

For the translation-invariant frozen data,

These identities follow from ${\partial }_y=-{\partial }_x$, ${\mathcal{P}}^2=\mathrm{id}$, and $D\mathcal{R}=\frac{1}{2}\square$. A replacement matrix need not have zero $bb$ block.

The matrix $H$ solves ${\mathsf{K}}_{\mathrm{eq}}$ in both slots and

It is even and has ghost number zero.

Proof. The auxiliary identities in (eq:H) impose $b=D\gamma$ in each slot; the other equations are the scalar wave equations. The causal identity follows from $h_{\mu }-{h_{\mu }}^{\mathsf{T}}=i{E}_{\mu }$ and slot differentiation. For the odd pair,

The reversed pair gives $-E_0$. The displayed nonzero blocks have the stated parity and ghost number.

An admissible $H^{\prime }$ is a complex-valued block bidistribution with:

1. the same shell equations in both slots;

2. the same graded antisymmetric part $i\Delta$;

3. smooth difference $d=H^{\prime }-H$;

4. ${H^{\prime }}_{AB}=0$ unless $|A|+|B|=0(\mathrm{mod}2)$ and $\mathrm{gh}(A)+\mathrm{gh}(B)=0$.

Thus $d$ is a smooth graded-symmetric bisolution. Smoothness does not mean compact support. Compact kernels make all its contractions proper on their supports. The wavefront sets of $H^{\prime }$ and $H$ agree.

Use graded left derivatives and the graded tensor-product rule. Normalize the even contraction operator ${\Gamma }_H$ by

Define

For homogeneous kernels of degrees $n,m$, its $k$-contraction is

Here $\pi$ drops the paired variables; indices and Koszul signs are contracted as prescribed above.

The orientation convention is fixed once:

for covectors of $u,v$ canceled by that line. Zero entries occur when individual line factors are smooth. This convention is also used for the continuity stages.

Every contraction is defined. Its support is compact, and its positive-degree kernel satisfies (eq:microcausal). A complete contraction is a scalar. The product is associative, unital, even, and ghost-number preserving. It preserves the regular subalgebra and extends coefficientwise to formal series.

Proof. A forbidden product cancellation would require zero covectors in all unpaired variables and $q\in {V_{-\sigma }}^k$, $q^{\prime }=-q\in {V_{\sigma }}^k$ in the paired variables. The corresponding nonzero input wavefront tuple would lie wholly in a forbidden causal cone, contrary to (eq:microcausal). Hence the distribution product exists.

The projection is proper on the compact support of $u\otimes v$. Suppose a nonzero output tuple were wholly future causal. If $\sigma =+$, the second input tuple, consisting of its future external entries and $q^{\prime }$, would be wholly future causal. It must therefore be zero. This forces $q^{\prime }=q=0$ and the second external tuple to vanish. The first tuple is now wholly future causal and must also vanish, a contradiction. For $\sigma =-$, start with the first input. Past-causal output tuples are excluded by the reversed argument. Symmetrization preserves the exclusion. When no variables remain, proper pushforward gives a scalar; there is no degree-zero wavefront claim.

On three factors the even contraction operators ${\Gamma }_{12},{\Gamma }_{13},{\Gamma }_{23}$ commute: their coefficients are independent of the configuration variables and the graded signs are included. Both bracketings equal

The required multi-contractions satisfy the same wavefront cancellation argument, so this identity involves defined distributions, not merely formal symbols. Constants give the unit. Finite-rank kernels contract to finite-rank kernels. The grading assertions follow from Definition def:admissible. For a fixed formal coefficient only finitely many orders of contraction and input coefficients contribute.

The wavefront calculus underlying these constructions is the one used for extended Wick algebras HW,Hormander. In particular, graded signs change coefficients, not the wavefront-existence conditions.

Let $w$ be a compact distribution on $M^n$ with reduced species indices. The following are equivalent:

1. $w$ annihilates every tensor tuple of smooth solutions of $\mathbf{P}$;

2. ${\mathbf{E}}_1\cdots {\mathbf{E}}_{n}w=0$;

3. $w={{\sum }_{j=1}}^n{\mathbf{P}}_j{r}_j$ with compact distributions $r_j$.

If $w$ is smooth, the $r_j$ can be smooth. Every such $w$ annihilates every smooth multisolution, not only factorized solution tuples.

Proof. All assertions reduce componentwise to $P=\square +{\mu }^2$. The wave causal operator maps compact smooth tests to smooth spatially compact solutions. Pairing (i) against these solutions and using $E^{\mathsf{T}}=-E$ proves (ii); tensor products of tests separate distributions.

Choose a smooth time function $\chi$ equal to zero sufficiently early and one sufficiently late. For compact $w$, define

Causal support and the bounded transition interval make $r$ compact. Direct differentiation, using the inverses of $P$, gives

Every operator here belongs to $P$; the action causal matrix $\Delta$ is absent.

Apply this construction in each slot. The $T_j$ commute, and

If (ii) holds, this expression vanishes. Telescoping $w-T_1\cdots T_{n}w$ with $\mathrm{id}-T_j={\mathbf{P}}_j{R}_j$ gives (iii), with compact primitives. The construction preserves smoothness. Finally (iii) annihilates every smooth multisolution by transposition. It also proves (i).

A compact full-multiplet kernel $w$ annihilates all smooth shell tuples if and only if

with compact distributional primitives. Smooth $w$ admits smooth primitives. Such a kernel annihilates every smooth full-shell multisolution.

Proof. Use the invertible differential change of variables $\Phi \mapsto (Y,\rho )$, $\rho =b-D\gamma$. Expand each slot using $\Phi =JY+e_b\rho$. Terms containing a $\rho$ slot already factor through that slot's equation $b-D\gamma$. The remaining reduced kernel is $(J^{\mathsf{T}}{)}^{\otimes n}w$ and satisfies Lemma lem:reducedseparation. Integration by parts expresses the resulting terms as (eq:fullseparation). The transformations are differential and preserve compact support and smoothness. The converse and the multisolution statement follow by transposition.

If $z\in {\mathcal{E}}^{\prime }(M^k)$ is microcausal, then

is a smooth full-shell multisolution. The statement holds for $H^{\prime }$ and for the transposed line orientation as well.

Proof. The input projection is proper on $\mathrm{supp}z$ over every compact output set. The tensor wavefront bound for $H^{\otimes k}$ has entries $(p_j,-p_j)$ with all nonzero $p_j$ in $V_{\sigma }$; zero pairs cover smooth factors. An undefined product would require a nonzero input tuple of $z$ entirely in ${V_{\sigma }}^k$, which microcausality excludes. An output wavefront point would require the same type of input tuple when the input covectors are canceled for pushforward. It is therefore excluded too: $\mathrm{WF}(S)=\varnothing$. The bisolution equations imply the shell equations in each output slot. Adding smooth $d$ does not alter the argument. Transposition reverses the common orientation, and microcausality excludes both orientations.

Smearing some slots of a microcausal kernel against a smooth configuration preserves microcausality: a forbidden remaining tuple would lift to a forbidden original tuple with zero covectors in the smeared slots. Thus the lemma applies to $G^{(k)}(\Phi )$.

There is no parity-preserving unital algebra homomorphism from ${{\mathcal{A}}^{\mathrm{cl}}}_{\mathrm{off}}[[\hslash ]]$ to ${\mathcal{A}}_0(M)$ sending every linear smearing to its corresponding causal generator.

Proof. The image of a graded-commutative source is graded commutative. But the scalar commutator is $-i\hslash ⟨u,E_{m}v⟩1$, nonzero for suitable tests.

Define

Let ${\mathcal{N}}_{\mathrm{reg}}$ be the ordinary graded ideal generated by

One has ${\mathcal{N}}_{\mathrm{reg}}={\mathcal{N}}_{\mathrm{sol}}\cap {{\mathcal{A}}^{\mathrm{cl}}}_{\mathrm{off}}$. Its formal extension is also the two-sided ${\star }_H$-ideal generated by the equation functionals.

Proof. A regular polynomial uses finitely many linear coordinates. By Corollary cor:fullseparation, the linear coordinates vanishing on the shell are precisely compact smooth equation functionals. Choose a homogeneous basis separating these coordinates from those whose restrictions are independent. On the independent even coordinates, the finite coordinate map on the solution space is surjective and ordinary polynomial evaluation is faithful. On the odd coordinates, Grassmann evaluation is faithful on the exterior algebra. Thus a polynomial vanishing on the shell belongs to the ideal generated by its vanishing linear coordinates. This proves the equality.

A contraction hitting $E_r$ vanishes because ${{\mathsf{K}}_{\mathrm{eq}}}_{x}H={{\mathsf{K}}_{\mathrm{eq}}}_{y}H=0$. Therefore an equation factor passes through a quantum product, with its Koszul sign:

Ordinary and quantum multiplication generate the same ideal.

There is a canonical surjective homomorphism

sending linear symbols to causal generators, with kernel ${\mathcal{N}}_{\mathrm{reg}}[[\hslash ,\kappa ]]$.

Proof. Let $V=\mathcal{D}/{{\mathsf{K}}_{\mathrm{eq}}}^{\mathsf{T}}\mathcal{D}$, with the species grading. The regular classical quotient is $\mathrm{Sym}(V_{\mathrm{even}})\otimes \bigwedge (V_{\mathrm{odd}})$. The contractions descend to $V$ by the bisolution equations. Their graded antisymmetric part is $i\Delta$, so linear generators satisfy exactly (eq:CCR).

The graded symmetrization map from the causal presentation to this deformed symmetric algebra is bijective: on finite words its highest field-degree part is ordinary graded symmetrization, and all contraction corrections lower degree. The triangular map is invertible degree by degree. Every word uses a finite-dimensional coordinate span; no finite-dimensional assumption on $V$ is needed. Extending coefficientwise gives the formal isomorphism. Composing its inverse with the quotient map defines ${\pi }_{\mathrm{free},H}$ and proves the kernel claim.

The space ${\mathcal{N}}_{\mathrm{sol}}[[\hslash ,\kappa ]]$ is a two-sided quantum ideal. Hence

is a surjective algebra homomorphism. There is an injective homomorphism $j_H:{\mathcal{A}}_0(M)[[\kappa ]]\to {\mathcal{E}}_H$ making

commute.

Proof. Fix $F\in {\mathcal{N}}_{\mathrm{sol}}$, $G\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$, and $\Phi \in {\mathcal{C}}_{\mathrm{sol}}$. Graded polarization of the identically vanishing restriction gives that $w=F^{(k)}(\Phi )$ annihilates every solution tuple. By Corollary cor:fullseparation, $w={\sum }_i{{{\mathsf{K}}_{\mathrm{eq}}}_{\mathsf{i}}}^{\mathsf{T}}{r}_i$.

By Lemma lem:smoothing, $S=H^{\otimes k}{G}^{(k)}(\Phi )$ is a smooth multisolution. Consequently

The primitives need not be microcausal: they are paired with a smooth multisolution, not inserted into a distributional product.

The $k=0$ term is $F(\Phi )G(\Phi )=0$. Every higher contraction vanishes by the preceding argument. For $G{\star }_{H}F$, use the transposed orientation from Lemma lem:smoothing; no symmetry of the noncommutative product is assumed. Kernel evaluation and the displayed derivative contraction are the same proper composition, so no approximation step is needed. This proves the ideal property and formal descent.

Finally the restriction of $Q_H$ to ${{\mathcal{A}}^{\mathrm{reg}}}_{\mathrm{off},H}$ has kernel ${\mathcal{N}}_{\mathrm{reg}}[[\hslash ,\kappa ]]$ by Proposition prop:regularideal. The induced map $j_H$ is therefore injective, and the diagram commutes.

Define the odd, ghost-number-one symbol derivation by

It acts on kernels by differential operators and species replacement, and satisfies ${s_0}^2=0$.

The operator $s_0$ preserves ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ and ${\mathcal{N}}_{\mathrm{sol}}$. It is a derivation of ${\star }_H$ precisely when

The condition holds for the frozen matrix. Thus $s_0$ descends to a nilpotent derivation of ${\mathcal{E}}_H$.

Proof. Differential operators do not enlarge wavefront sets. On the shell, $\mathcal{R}c$ is a wave solution and $D\mathcal{R}c=\frac{1}{2}\square c=0$; also $\square b=0$. Hence the odd linear variation is tangent to the shell and preserves its vanishing ideal.

Commuting the linear symbol variation through a contraction gives $\delta H$. The quantum Leibniz defect vanishes if $\delta H=0$. Conversely, applying the defect to two linear smearings detects every component of $\delta H$ as a scalar, proving necessity.

For the frozen data, both nontrivial ghost orderings cancel:

The pairs $(b,\bar{c})$ and $(\bar{c},b)$ vanish because $H_{bb}=0$. Every remaining pair has only zero contraction blocks.

Set

Define

and

Thus ${\mathcal{B}}_f=\int {b_f}^{ab}(x,z){\gamma }_{ab}(z)\varphi (x)\mathrm{d}x\mathrm{d}z$. The kernel of $s_0{\mathcal{B}}_f$ is obtained by applying ${{\mathcal{R}}_z}^{\mathsf{T}}$ to its metric slot.

For the scalar stress, let ${\tau }^{ab}(y;y_1,y_2)$ be the symmetric bilinear local kernel of the frozen quadratic stress ${T_0}^{ab}$:

Equivalently $\tau$ is a fixed finite sum of products of derivatives of $\delta (y-y_1)\delta (y-y_2)$ with constant coefficients. This convention includes no polarization factor beyond the one displayed.

The coordinate response to the metric Green operator is

For compact external sets the integration is over a compact causal region, because $z\in J^{-}(x)\cap J^{+}(y)$ and $t_z\ge 0$. The retarded wavefront relations imply $\mathrm{WF}(Q)\subset \{(x,p;y,-p)\}$.

The frozen tree and its symmetric kernel are

The factor $\frac{1}{2}$ is retained after integration by parts.

The product defining $\Xi$ exists: a canceled external covector would force $p=0$, hence the $Q$ covector to vanish; the other external covectors then force the $\tau$ covector to vanish. The input relations and pushforward give

A tuple in one closed causal cone cannot have zero sum unless it is zero. These kernels are therefore microcausal.

For tests supported in $K_f,K_g$, valid compact support bounds are

Support and cone bounds for a graded-symmetric quadratic kernel are understood after the corresponding finite permutations.

Let ${\mathcal{W}}_H$ be the algebra of finite ${\star }_H$-words in compact linear fields and

with coefficients in $\mathbb{C}[\hslash ,\kappa ]$. Set $\mathfrak{m}=(\hslash ,\kappa )$. The sector ${\mathcal{S}}_H$ consists of $X\in {\mathcal{E}}_H$ such that, for every $p\ge 1$, some $W_p\in {\mathcal{W}}_H$ satisfies $X-W_p\in {\mathfrak{m}}^p{\mathcal{E}}_H$.

This is the formal coefficientwise completion, not closure in the analytic topology introduced later. For admissible $H^{\prime }$, define safely ${\mathcal{S}}_{H^{\prime }}={\bar{\alpha }}_d({\mathcal{S}}_H)$ once the ordering map is established.

The completion allows coefficients with no uniform word-length bound. It is not identified with finite words having arbitrary formal scalar coefficients. It contains $j_H({\mathcal{A}}_0(M)[[\kappa ]])$.

The sector is graded, unital, closed under ${\star }_H$, and $s_0$-stable. Every coefficient admits a compact microcausal lift.

Proof. These claims first hold for finite words by Theorem thm:product. The differential preserves the linear fields, maps ${\mathcal{B}}_f$ to its included image, annihilates that image, and annihilates the scalar-only tree. It is a quantum derivation. Products and this linear differential preserve every formal order, so the claims pass to the stated completion.

For admissible $H^{\prime }=H+d$, define

For a polynomial, its self-contraction expansion terminates. For formal series it is defined coefficientwise.

The map ${\alpha }_d$ preserves ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}[[\hslash ,\kappa ]]$ and ${\mathcal{N}}_{\mathrm{sol}}[[\hslash ,\kappa ]]$, and

It fixes constants and linear fields, is the identity modulo $\hslash$, has inverse ${\alpha }_{-d}$, and satisfies ${\alpha }_{d_1}{\alpha }_{d_2}={\alpha }_{d_1+d_2}$. It induces ${\bar{\alpha }}_d:{\mathcal{E}}_H\to {\mathcal{E}}_{H^{\prime }}$ with

Under the causal presentation, the regular descended map is the identity of ${\mathcal{A}}_0(M)[[\kappa ]]$.

Proof. A smooth self-contraction is multiplication followed by a proper projection. Output wavefront tuples lift with zero entries in the contracted slots, so microcausality is preserved. The self-contraction operators commute and lower field degree by two, proving the inverse and cocycle identities.

Let $D_d=(\hslash /2)⟨d,{\delta }^2⟩$. Graded symmetry of $d$ gives the Leibniz identity

Exponentiating commuting operators proves (eq:intertwining), the smooth-change ordering equivalence FR.

For $F\in {\mathcal{N}}_{\mathrm{sol}}$ and $\Phi \in {\mathcal{C}}_{\mathrm{sol}}$, $F^{(2k)}(\Phi )$ annihilates solution tuples. The kernel made from paired copies of $d$ is a smooth multisolution in all $2k$ slots. Corollary cor:fullseparation therefore makes its pairing with $F^{(2k)}(\Phi )$ zero. Thus ${\alpha }_d{\mathcal{N}}_{\mathrm{sol}}\subset {\mathcal{N}}_{\mathrm{sol}}$; the inverse proves equality. Descent follows. Constants and linear fields have too few slots to self-contract, and every nonidentity term contains $\hslash$. The regular assertion follows because the descended homomorphisms agree on their topological generators.

For the quadratic and cubic seeds,

Here $c_d(f)$ is a scalar and ${\ell }_d(f,g)\in {C_c}^{\infty }(M)$.

Proof. The quadratic formula is the terminating exponential. The image $s_0{\mathcal{B}}_f$ has species $(c,\varphi )$; its only possible self-contraction is forbidden by ghost number.

For the scalar cubic, the second derivative contributes the factor $3\cdot 2$, canceled in part by the $\frac{1}{2}$ in (eq:alpha). No fourth derivative remains. For the projected kernel ${\ell }_d$, a nonzero output wavefront point would come from $(0,0,p)\in \mathrm{WF}(t_{f,g})$. The sum-zero bound (eq:sumzero) forces $p=0$. Thus ${\ell }_d$ is smooth, and proper projection makes it compact.

The transported sector equals the formal coefficientwise completion generated by the unchanged seeds:

Proof. Every ordering correction in Proposition prop:seedcorrections lies in the regular part: constants and compact smooth linear fields. The regular part is transported onto the regular part. Hence each transported seed belongs to the unchanged-seed algebra. Applying the inverse ordering map proves the opposite inclusion. Formal-order preservation gives equality of the completions.

This is a theorem about this named sector, not general invariance of unchanged Wick formulas.

The operator ${s_0}^{H^{\prime }}={\bar{\alpha }}_d{s}_0{\bar{\alpha }}_d^{-1}$ is a nilpotent, ghost-number-one derivation and preserves ${\mathcal{S}}_{H^{\prime }}$. It equals the symbol rule $s_0$ exactly when $\delta d=0$, with $\delta$ defined in (eq:differentialcondition).

Proof. Conjugation gives nilpotence, the derivation property, and sector stability. The symbol rule is a derivation for $H^{\prime }$ exactly when $\delta H^{\prime }=\delta d=0$. In that case its commutator with the self-contraction generator vanishes. Conversely equality on the quotient would make the symbol rule a quantum derivation there; its defect on two regular linear fields is a scalar testing $\delta d$. Scalars are not killed by the shell quotient, so all these tests must vanish.

For a closed cone $\Gamma \subset {\dot{T}}^{\ast }X$, use the normal topology on ${{\mathcal{D}}^{\prime }}_{\Gamma }(X)$: the strong distribution seminorms

and the Fourier seminorms

For compact $K\subset M^n$, set

Finite species components and graded symmetry carry their induced topologies.

The normal-topology facts used here are hypocontinuity of tensor products and compatible distribution products, continuity of transversal pullback and support-proper pushforward, and hypocontinuity of smooth multiplication BDH. Fixing one factor gives a continuous linear map. Constant-coefficient differentiation is continuous, as follows from

On ${C^{\infty }}_{K_f}(M)$, the maps $f\mapsto b_f$ and $f\mapsto {{\mathcal{R}}_z}^{\mathsf{T}}{b}_f$ are continuous into their stages with cone $Z_2$. The bilinear map $(f,g)\mapsto t_{f,g}$ is jointly continuous from ${C^{\infty }}_{K_f}\times {C^{\infty }}_{K_g}$ to the stage ${{\mathcal{E}}^{\prime }}_{K_T,Z_3}$.