X-C-ST: Second-order scalar completion on the inherited X-A-ST/X-B-ST host Direct linear response maps, completed background covariance, and the propagated edge defect

We complete the fixed-width scalar-only response sector of X-B-ST on the supported host already prescribed by X-A-ST. The new constructions are a second-vertex map and an iterated first-vertex map on specified off-shell linear insertions. Both are defined directly by a finite formal test-level Møller expansion, without a time-ordered product hierarchy. Their smooth compact output tests satisfy retarded support and continuity bounds, and their sum reproduces the inherited second scalar response.

The completed relational field is background invariant through order ${\kappa }^2$ in the unchanged free scalar algebra on the gauge algebra frozen before the calculation. The proof gives an explicit three-term variation: a compact free-equation term, a dressed coordinate defect, and a dressed switch defect. In particular, it retains the propagation of the first-order edge gap into the second coefficient. A complete pure-gauge two-jet closes the remainder left in X-B-ST, while a massless conformal example checks the iteration and its signs.

These coefficients contain no positive power of $\hslash$ in their tests and no Hadamard contraction kernel. This tree-exactness concerns the specified linear rooted prescription, not arbitrary interacting products. The diagonal product boundary of X-B-ST remains unchanged.

The contribution is a supported realization and normalization, sign, and defect audit of Cheung's X-C construction, rather than a priority claim for the underlying quadratic-field method. Its prospective value is an explicit second-order baseline that separates background covariance from singular-product admission and finite readout. A complete destructive-clock instrument encodes supplied evaluations and admits reference-preserving regular transport. It does not acquire a continuum field, evolve the host metric, or establish spacetime emergence.

rule: one first-order map and one second-order pair{ X-B-ST released the single first-vertex map $R_1(V_{1,g};\cdot )$, including its use on the first bare dressing, and proved the completed identity through ${\kappa }^1$.

X-C-ST releases $R_1(V_{2,g};\cdot )$ and $\frac{1}{2}{R}_2(V_{1,g},V_{1,g};\cdot )$. Their sum on $F_f$ represents the one inherited term $\int f{\rho }_{2,g}$. A successor citing both files receives one first-order map and one second-order pair, not two copies of the second scalar completion.}

The analytical source is Cheung's X-C, version 3 XC. The supplied X-A-ST and X-B-ST manuscripts fix the supported classical prescription and its first scalar response XAST,XBST. The first response is recalled as an inherited input, not reconstructed as a new achievement of this file.

Quadratic scalar perturbations admit particularly direct linear-field constructions. Their relation to perturbative agreement has a broader established literature HW,DHP. That context does not supply the particular regulated coordinate prescription or the supported host contract used here. Nor is a general renormalized product family imported from that literature.

All coupling expansions below are finite formal expansions over

with complexification for tests and field coefficients. An equality modulo ${\kappa }^3$ means equality of the three displayed coefficient slots. It is not a numerical error estimate at a nonzero real coupling.

The Enter distinction used here is operational: destructive readout consumes the measured quantum names and returns a classical record together with the explicitly retained residual. A quantum-return decoder is a different interface EnterI,EnterPhilosophy.

This distinction is applied only after the analytical objects have been defined. It prevents the following identifications:

1. a finite table of evaluated numbers is not the continuum field;

2. a complete channel identity is not a field detector or a thermodynamic implementation;

3. a prescribed dependency schedule does not derive a Lorentzian metric;

4. finite operations placed on a supplied spacetime do not, merely by that placement, produce spacetime emergence.

I-B supplies the fixed-spacetime interpretation of the finite clock and regular transport pattern IB. The finite channel identities actually used below are proved in this file.

Retain the X-A-ST instance without modification:

1. the regular time-oriented conical product $({\mathcal{M}}^{\circ },G)$, its prescribed worldsheet $\mathcal{W}$, and its supplied future relation;

2. the supported developing embedding

   $$\iota :\mathcal{L}⟶{\mathcal{M}}^{\circ }$$

   and the regular compact reserve used by the predecessors;

3. the separate analytical completion

   $$\hat{M}={\mathbb{R}}^{1+3},\eta =\mathrm{diag}(1,-1,-1,-1),M_{T_{\ast }}=[0,T_{\ast }]\times {\mathbb{R}}^3;$$

4. the signature and time ledger

   $${\iota }^{\ast }G=-\eta ,t=c_{\mathrm{host}}(T-T_{\Sigma }).$$

   The host signature remains $(-,+,+,+)$; the analytical equations retain $(+,-,-,-)$;

5. the observation region $\mathcal{O}⋐(0,T_{\ast })\times {\mathbb{R}}^3$, with

   $$D_{\ast }=J_{\eta }^{-}(\overline{\mathcal{O}})\cap M_{T_{\ast }}⋐\mathcal{L}.$$

   Compact laboratory tests are transported to $\hat{M}$ and extended by zero.

The host's regular part is not used as a globally hyperbolic Cauchy spacetime. All global free fields and Green operators in this manuscript belong to $\hat{M}$.

This ledger does not state a third cone. The chart does not supply a global Minkowski representation, an intrinsic Green operator on the conical product, or a global field-controlled unitary.

For $f\in {C_c}^{\infty }(\mathcal{O})$, write

The supported transfer concerns calculations retained in $D_f\subset D_{\ast }$. The new maps are additionally defined on arbitrary compact completion tests, without claiming that every such test has a supported host interpretation.

The local Grade S package in this file consists of:

1. the inherited regular supported host;

2. explicit response maps with domains, normalization, support, and background transformation laws;

3. one finite installed evaluation dictionary;

4. a complete destructive-clock instrument, including untouched references;

5. scheduled locality and a nonempty regular reference-preserving route under declared finite operational permissions.

Grade E, metric emergence, and Einstein dynamics are not included.

Only the explicitly stated host and handoff data are inherited assumptions. A bare citation is not an additional hypothesis. In particular, the comparison sheets listed later do not prove either Theorem thm:maps or Theorem thm:ward. The latter is proved by the three-term variation, not by declaring that every identity in a predecessor automatically transfers.

Let $h,k$ be arbitrary smooth real symmetric tensors on the completion, with no spatial decay or metric field equation imposed, and set

Their topology is the smooth compact-open topology.

Retain the same fixed $\epsilon >0$, smooth ramp $\sigma (t)=\chi (t/\epsilon )$, and compact switch

as in the predecessors, where

The plateau and algebra are unchanged:

For $\xi ,\zeta \in {\mathfrak{g}}_p$, the formal pullback generated by $\kappa \xi +{\kappa }^2\zeta$ belongs to ${\mathcal{G}}_p$. The full background transformation is

Outside the plateau, the same formula will be used only to calculate defects for arbitrary compact interior generators. Those generators are not added to ${\mathfrak{g}}_p$.

Indices are raised with $\eta$. Put $H={\mathrm{tr}}_{\eta }h$ and define

Set

Thus

The switched scalar density is

It is switched before variation. For $P_m=\square +m^2$,

its positive Euler operator is

The two local differential coefficients are

Every displayed derivative of the switch is retained.

The corresponding vertices are

Both are quadratic in the scalar and have compact coefficient support. Compact integration by parts gives

The positive Euler coefficients are not the action Hessians.

Write

It has its standard LF topology, with smooth seminorms on each fixed compact-support stage.

The positive free Green operators satisfy

They act continuously from compact smooth sources to smooth solutions. These are completion-space Green-operator statements BGP; no conical Cauchy theorem is used.

Retain X-B-ST's scalar free Wick algebra and quotient

where ${\mathfrak{J}}_0$ is generated by $F_{P_{m}a}$, $a\in \mathcal{D}$. For

the degree-one relations are

The degree-one topology is the quotient test topology on $\mathcal{D}/P_m\mathcal{D}$.

The insertion domain is instead

It consists of actual off-shell representatives. Output classes are not used as unspecified new input representatives.

The inherited degree-one separation property is

This is X-B-ST's normalization standard XBST (Section C3); it is also the Green-operator exact-sequence mechanism BGP (Section 3.4).

Fix $f\in {C_c}^{\infty }(\mathcal{O};\mathbb{R})$. The coordinate handoff is

The massless inverse here is the supplied zero-data coordinate prescription at $t=0$. Smoothness without spatial decay, fixed-width continuity, and retained-past dependence are inherited chart data.

On ${\mathcal{G}}_p$, the supplied laws are

No proof of X-A-ST's coordinate covariance is claimed again here.

Let

The inherited compact bare tests are

They are smooth, supported in $K$, and continuous in the smooth backgrounds at fixed $\epsilon$. Their scalar-only and joint admissions were established in X-B-ST XBST. The label test $f$ remains fixed.

Define

For every smooth scalar $\psi$,

This fixes the insertion representatives, including their signs.

The first map is recalled, not rebuilt:

X-B-ST supplies its support, continuity, tree-exactness, and first-order completed identity, as well as

The classical target for the additional second-order slot is the supplied expression

For free $u$, it is the second scalar Cauchy coefficient of X-A-ST. For arbitrary smooth $u$, the same formula is a well-defined smooth linear extension, because every source operator has compact coefficients.

The predecessor's incomplete second coefficient was

It did not include (eq:rho2). The new pair below fills that one slot.

On actual formal compact tests define

The inverse is purely formal over ${\mathbb{D}}_2$.

The second-vertex map is

The iterated first-vertex coefficient is

The symbol $\frac{1}{2}{R}_2$ denotes this second Taylor coefficient on the diagonal first-vertex family. It does not name a general retarded product.

The inner expression

is an actual smooth compact test. It is formed before the next Green operation. In terms of the inherited rule, (eq:iteration) can be written as applying that rule to $F_{-a_c}$, not to an unspecified representative of $\mathbf{\varphi }(-a_c)$.

The two maps in (eq:second-map)–(eq:iteration) are defined on every $F_c\in {\mathcal{L}}_{\mathrm{off}}$. Their output tests are smooth and compact, and

They are continuous in the compact-test topology and the smooth-background topology.

For every smooth scalar $u$,

Consequently

No singular product, diagonal extension, or subtraction is used.

Proof. The function ${\mathsf{E}}_{\mathrm{A}}c$ is smooth and supported in $J_{\eta }^{-}(\mathrm{supp}c)$. The coefficients of each $L_{j,g}$ are smooth and supported in the fixed compact set $S=\mathrm{supp}g$. This proves the assertion for $s_c$ and for the intermediate test $a_c$.

The function ${\mathsf{E}}_{\mathrm{A}}{a}_c$ is smooth and has support in $J_{\eta }^{-}(\mathrm{supp}{a}_c)$. Applying $L_{1,g}[h]$ gives

by transitivity of the completion causal relation.

For continuity, the maps

are continuous into their respective smooth spaces; the last two have range in the fixed test space ${\mathcal{D}}_S$. The claim follows by composition, first on each compact-support test stage and then on the LF test space. The dependence on $h,k$ is polynomial.

For normalization, formal self-adjointness and ${{\mathsf{E}}_{\mathrm{R}}}^{\mathrm{T}}={\mathsf{E}}_{\mathrm{A}}$ give

Transposing the two Green operators in the iteration successively gives

At every step, the differential source factor is compactly supported. All propagations are applied to smooth tests or smooth compact sources.

This is the direct scalar-map construction of X-C in the inherited supported presentation XC (:moller, :maps). The equality to ${\rho }_{2,g}$ is its normalization against the classical handoff, not another copy of that response.

If

both new maps vanish.

For $c$ supported in $\mathcal{O}$, all effective interaction vertices lie in $D_c\subset D_{\ast }$. Two extensions of the background data on the same declared flat completion that agree near $D_c$ give the same new output tests. The analogous coordinate-data agreement for $c[q]$ is the inherited X-A-ST completion contract.

Proof. The vanishing follows from (eq:new-support). The second-vertex expression is local in its background coefficients on $S\cap J_{\eta }^{-}(\mathrm{supp}c)$. For the iterated expression, every contributing chain has

Since $S$ lies in the open slab, these vertices and their relevant causal intervals lie in $D_c$. The Green kernels are the same fixed completion kernels. Thus agreement of the coefficients near $D_c$ gives agreement of the resulting tests.

This is not independence from arbitrary global Green realizations on different curved completions. Transport through $\iota$ is the rewriting of these same retained compact pairings, not a construction across the worldsheet.

For arbitrary smooth $u$, define

The transpose of this scalar response is precisely (eq:moller). The second vertex therefore enters with the same negative response sign as the first vertex, while two first responses have positive sign.

For the first vertex alone,

This explains the factor $1/2$ in (eq:iteration). No factorial is assigned after the gauge calculation.

The diagonal coefficient is quadratic, not additive, in its Euler operator. For two specified admissible operators $L,M$, the direct identity is

These mixed terms may not be discarded under the name of polarization or diagonal additivity. Equation (eq:cross-terms) is ordinary operator bookkeeping; it does not install a general mixed retarded-product family.

For $a\in \mathcal{D}$,

Their classes need not vanish. At the full formal level,

Thus neither the formal recursion nor tree-exactness changes the insertion domain into an on-shell quotient.

Non-descent already occurs for the inherited first map. For example, in the massless case with $h=r\eta$ and a real bump $r$ supported where $g=1$, take $a={\partial }_{t}r$. Then

This recalls the predecessor's representative test; it is not a new construction of its map.

The separate pairings in (eq:individual-normalizations), imposed for every $c\in \mathcal{D}$ and every smooth spatially compact free solution, determine the two degree-one output classes uniquely.

If formal $\hslash$-dependent alternatives are allowed, this statement requires equality of the full normalization series, coefficientwise.

Normalization of the sum alone does not uniquely specify a decomposition into two summands.

Proof. For each output, subtract two candidates with the same individual pairing. Their compact test difference annihilates every spatially compact free solution. Equation (eq:separation) makes its class zero. The argument applies to each $\hslash$-coefficient.

If only the sum is fixed, one may formally add $\mathbf{\varphi }(n_c)$ to one candidate and subtract the same class from the other. Individual normalization or the coefficient recursion is needed to exclude such redistribution.

The direct inverse (eq:moller), and not a choice of contact after observing a defect, fixes the present split. X-B-ST's sheet-compatible first-map contacts remain zero classes. This file does not reopen VII-H's different contact classification.

Define, coefficientwise through ${\kappa }^2$,

For instance,

These are finite formal compositions, not numerical inverses of a background-dependent operator at finite coupling.

Let

This is a linear formal field in the inherited free algebra. No state is part of this definition.

The following statements hold modulo ${\kappa }^3$:

1. The test operator is the unique formal series with leading term $1$ satisfying

   $${\mathcal{R}}_q=1-{\mathcal{R}}_{q}L[q]{\mathsf{E}}_{\mathrm{A}}.$$

   Equivalently,

   $${\mathbf{\varphi }}_q(c)=\mathbf{\varphi }(c)-{\mathbf{\varphi }}_q(L[q]{\mathsf{E}}_{\mathrm{A}}c).$$

2. The scalar map in (eq:scalar-map) obeys

   $$(P_m+L[q])r_{q}u=P_{m}u$$

   for arbitrary smooth $u$, and is the unique solution with the same zeroth-order data and zero positive-order data before the interaction.

3. For every compact $a$,

   $${\mathbf{\varphi }}_q(E_g[q]a)=0.$$

4. If $\mathrm{supp}g\cap J_{\eta }^{-}(\mathrm{supp}c)=\varnothing$, then ${\mathbf{\varphi }}_q(c)=\mathbf{\varphi }(c)$.

5. Its algebraic commutator is

   $$[{\mathbf{\varphi }}_q(c),{\mathbf{\varphi }}_q(d)]=i\hslash ⟨c,({{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{A}}-{{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{R}})d⟩1.$$

Proof. The formal inverse identity

proves (i). Coefficient comparison fixes the first operator as $-L_{1,g}{\mathsf{E}}_{\mathrm{A}}$ and the second as $L_{1,g}{\mathsf{E}}_{\mathrm{A}}{L}_{1,g}{\mathsf{E}}_{\mathrm{A}}-L_{2,g}{\mathsf{E}}_{\mathrm{A}}$.

The scalar-side recursion is

Applying $P_m$ gives (ii). Conversely, its positive-order equations are ordinary free inhomogeneous Cauchy problems with smooth compact sources, so free Cauchy uniqueness determines them.

For compact $a$, ${\mathsf{E}}_{\mathrm{A}}{P}_{m}a=a$, and hence

Equation (eq:free) proves (iii). The support condition in (iv) makes $L[q]{\mathsf{E}}_{\mathrm{A}}c=0$, proving that assertion.

For (v), formal self-adjointness gives

Using the inherited free commutator, its kernel between the original tests is

The advanced contribution is

because ${\mathsf{E}}_{\mathrm{A}}(1-L[q]{{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{A}})={{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{A}}$. The retarded contribution is

because $(1-{{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{R}}L[q]){\mathsf{E}}_{\mathrm{R}}={{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{R}}$. Their difference is ${{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{A}}-{{\mathsf{E}}_{\mathsf{q}}}^{\mathrm{R}}$. Every composition is well defined on the stated compact tests.

The theorem characterizes the declared linear construction XC (:yf). The commutator is not an interacting two-point response or a construction of a state. No assertion that ${\mathcal{R}}_q$ is an automorphism of the unchanged free algebra is made.

The tests $s_c,i_c$, and the completed tests constructed below, contain no $\hslash$ and no Hadamard contraction kernel. They have no positive-$\hslash$ remainder in the directly defined linear rooted prescription.

Proof. All coefficients in (eq:moller) are compositions of ordinary Green and differential operators on tests. The same is true after substituting the bare test polynomial $c[q]$. This proves the assertion algebraically.

The fixed Wick presentation provides a consistency count, not an alternative construction. A quadratic vertex and one linear insertion have three scalar legs, permitting one rooted cross-contraction and leaving one leg. Two quadratic vertices and a linear insertion have five legs. A connected rooted three-entry graph needs two edges, consumes four legs, and leaves one leg. Since the insertion has only one leg, that graph is a chain ending at the insertion. Three contractions would require six legs and are impossible.

The relevant contraction powers are removed by their first- and second-order response normalizations. A vertex-only vacuum component disconnected from the insertion is not part of the rooted prescription; self-contractions are absent in the fixed Wick presentation FR. The count explains the quantum order of the direct formulas, but does not define any missing singular product.

The $\hslash$ in (eq:free) and (eq:perturbed-commutator) is unaffected. Tree-exact tests are not a statement that the algebra is classical or that every quantum correction in a larger sector vanishes.

The inherited X-B-ST nonextension statement remains in force XBST (:nonextension). Its auxiliary kernel

contains the full nonzero diagonal conormal in its wavefront set. Thus at every diagonal point it contains both

The ordinary wavefront product criterion therefore does not define ${\Delta }_{\mathrm{F}}\cdot {\Delta }_{\mathrm{F}}$ there Hormander (Chapter 8).

Equations (eq:second-map) and (eq:iteration) never perform that operation. The second Green propagation in (eq:iteration) acts on the smooth compact test $a_c$. Sequential action on smooth tests is not an extension of an identified-endpoint square of singular kernels.

No $T_2(V,V)$, $T_3$, or general nonlinear-insertion $R_2$ is supplied. Failure of the sufficient product criterion is not a universal nonexistence theorem for renormalized extensions. Such extensions and their normalizations are additional analytical tasks BF,DFreference.

Set

Equivalently,

The completed field is

Its second coefficient contains exactly four terms:

Every $e_j$ is smooth and compact, with

Their background dependence is continuous at fixed $\epsilon$. The retained outputs have field degree one, with zero allowed; $e_1$ is linear in $h$, and $e_2$ is quadratic in $h$ plus linear in $k$.

For every smooth free solution $u$,

The last line is the inherited full classical second relational coefficient, counted once.

Proof. The bare tests are supported in $K$. The support and continuity of every response term follow from Theorem thm:maps and the inherited first-map theorem. Their finite sums have the stated support and topology.

The coordinate coefficients satisfy $w[h]$ linear in $h$ and $v[h,k]$ quadratic in $h$ plus linear in $k$. The same degree statement follows for $c_1,c_2$, and then for (eq:completed). Every output is $\mathbf{\varphi }$ of a smooth compact test, so scalar-only admission needs no further singular-kernel theorem.

Transpose the response terms using (eq:individual-normalizations). For the mixed term,

The bare terms give $O^{(j)}$, and the new sum gives ${\rho }_{2,g}$.

The proposition concerns the retained degree-one outputs. The vertices are quadratic, their auxiliary products need not have degree one, and the finite instruments below are channels rather than field elements. No universal background-degree bound for arbitrary further compositions is asserted.

The variation is calculated before imposing the plateau. Let $\xi ,\zeta$ now be arbitrary smooth compact interior vector fields, and let $q^{\prime }=F^{\ast }q$ have the full law (eq:metriclaw).

Write

The first acts on scalar functions. The second is density pullback, including the Jacobian.

For compact formal tests $a$ and smooth formal scalars $u$,

Proof. Integration by parts gives ${D_{\xi }}^{\mathrm{T}}=-X_{\xi }$. Thus

the inverse of the scalar pullback through this order. Multiplication by $F^{\ast }$ proves the identity.

For arbitrary compact generators in the open slab, define

Write

The ${\theta }_j$ are tests; the $S_j$ are differential operators.

The coordinate defect has compact coefficients with $\mathrm{supp}{\theta }_j\subset K$, and

For $F\in {\mathcal{G}}_p$, ${\Theta }_F=0$.

Proof. Expansion of (eq:defects) gives (eq:theta). Formal pullback coefficients are differential operators and do not enlarge coefficient supports. This proves the compact-support statement.

On ${\mathcal{G}}_p$, substitute the inherited coordinate laws (eq:coordinate-laws). They give