X-B-ST: One scalar retarded map on the X-A-ST host Fixed-ramp admission, first-order background covariance, and the one-contraction boundary

We construct a local Grade S realization of Cheung's fixed-regulator scalar-only X-B prescription on the supported host presentation already fixed by X-A-ST. The quantum field is the free Klein–Gordon scalar on the declared Minkowski completion; the metric coefficients remain classical and off shell. The host geometry is not reconstructed or evolved.

At every fixed positive ramp width, the bare relational dressing is admitted through second background order, both by a joint polynomial wavefront test and, separately, as a scalar-linear functional with a smooth compact test. For every specified off-shell linear representative $F_c$, the normalized response is

We prove continuity, retarded support, independence from the Hadamard contraction kernel, and the absence of positive-$\hslash$ terms in its output test. The completed first coefficient is invariant in the unchanged free algebra on the gauge algebra frozen before the calculation. An explicit equation-ideal representative separates this identity from the retained edge and switch defects.

The four X-B-ii scope closures are incorporated into the same file. Classical normalization on the entire insertion domain determines the degree-one output class; compatible contact modifications are therefore algebraically zero. Conversely, the one-contraction verification does not construct a time-ordered family: the Feynman kernel's square fails the ordinary wavefront product criterion on the diagonal. This is a limitation of the present construction, not a nonexistence theorem for renormalized products.

The contribution is a supported transfer, normalization audit, and explicit scope separation of the source results, not a new priority claim for them. Its potential value is a reproducible starting domain for later response analysis without conflating free-field admission, classical covariance, singular-product construction, and measurement. A finite destructive-clock interface encodes supplied evaluations; it does not acquire a continuum field.

orders, one host, and one map{ Bare dressing admission is established through ${\kappa }^2$. The completed background identity is established through ${\kappa }^1$, namely modulo ${\kappa }^2$. These are different statements.

Only the single scalar-only map $R_1(V_{1,g}[h];\cdot )$ is released. The displayed classical ${\rho }_{2,g}$ is not promoted to a constructed second-order quantum map. X-B-ii supplies scope closures inside this file, not another spacetime or another interaction programme.}

The analytical source records are Cheung's X-B, version 3, and X-B-ii, version 1 XB,XBii. The first constructs one normalized scalar response; the second clarifies its scope, one-contraction boundary, and normalization. The present manuscript transfers that package to the supported presentation supplied in X-A-ST XAST. It does not enlarge the source's quantum field content.

The distinction between a classical relational coefficient and its quantum admission is essential. A classical identity does not define a distributional product, while an admitted free-field expression does not automatically define an interacting response. The standard perturbative algebraic framework makes these domain distinctions explicit FR,DFreference,BF.

The Enter background distinguishes destructive readout from quantum return. A measured quantum name is consumed; the declared output is a classical record together with the explicitly retained residual. Recovery from an accessible subsystem is a different task EnterI,EnterPhilosophy.

We use this discipline only after constructing the field-theoretic objects. In particular:

1. an encoded number is not the continuum field from which that number was mathematically evaluated;

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

3. a finite dependency schedule is not a derivation of a light cone;

4. a prescribed spacetime supporting finite operations is not spacetime emerging from those operations.

The spacetime Enter model of I-B supplies the relevant fixed-host interpretation IB. The finite instruments needed here are proved directly below.

Retain exactly the host instance fixed by X-A-ST:

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

2. its supported injective developing chart

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

   whose retained compact subsets stay off $\mathcal{W}$;

3. the declared flat 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 ledger

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

   The host has signature $(-,+,+,+)$, while all analytical equations in this manuscript retain $(+,-,-,-)$;

5. a retained 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 regular host is not assumed globally hyperbolic. The global free field and all Green distributions below live on $\hat{M}$, not on that regular product.

These are the X-A-ST host data, not new geometric conclusions. The worldsheet, signature, speed, lapse interpretation, chart, and future relation remain inputs.

A local developing chart is not a global Minkowski representation, an intrinsic conical Green operator, or a global field-controlled unitary. We make none of those identifications.

The algebraic construction is defined on the flat completion. Its supported host presentation consists of the same pairings and test actions, restricted to retained compact causal calculations and transported through $\iota$.

For $c\in {C_c}^{\infty }(\mathcal{O})$, set

The support contract is that every coordinate calculation needed by the chosen dressing test lies in $D_{\ast }$; interaction contributions outside $D_c$ do not enter its response.

The map will also be defined for arbitrary compact completion tests. This larger analytical domain is not a claim that every such test has a supported conical presentation.

A local Grade S release consists of:

1. the inherited supported host presentation;

2. explicitly admitted field-theoretic objects with their domains and normalization;

3. a 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.

This definition contains no emergence or Einstein-dynamics requirement. Grade E is not claimed.

This section states the chart data supplied by X-A-ST. Its classical covariance proof is not repeated. The new identities proved later are not inferred merely by citing the parent's main theorem.

Let

with the smooth compact-open topology, and let

The backgrounds are real, smooth, classical, and off shell. No Einstein equation, graviton propagator, or quantum metric variable is introduced.

Fix $m\ge 0$, $0<\epsilon <T_{\ast }$, and

The ramp is

It is smooth and flat at both endpoints.

Retain the same nonempty relatively compact open plateau $U$ as in X-A-ST:

The formal group ${\mathcal{G}}_p$ is generated through ${\kappa }^2$ by flows of $\kappa \xi +{\kappa }^2\zeta$, $\xi ,\zeta \in {\mathfrak{g}}_p$. The vector fields are extended by zero. They preserve $g,\sigma$, are initial-trivial, and do not meet the worldsheet.

Write $X_{\xi }={\xi }^{\mu }{\partial }_{\mu }$ on scalars and $D_{\xi }r={\partial }_{\mu }({\xi }^{\mu }r)$ on scalar densities. The formal metric transformation is

The last term is not omitted in a second-order diagnostic.

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

Thus

where

The supplied coordinate coefficients are

The retarded massless inverse on these smooth past-compact sources is understood by finite propagation, equivalently by the zero-data prescription at $t=0$. No spatial decay is assumed on $h,k$.

These are the coefficients of the divergence-form density prescription

They are not coefficients of a convolution regulator or of a wave equation obtained by multiplying a different source equation by $\sigma$.

The handoff includes smoothness, fixed-width continuity, retained-past dependence, and the laws

In particular,

These are supplied classical chart data XA,XAST, not a quantum multiplication theorem.

For later kernel arguments the linear coefficient has the explicit distributional representation

The unsmeared kernel is distributional. Smoothness of its observation-smeared first derivatives does not mean that ${\Lambda }_{\epsilon }(x,z)$ is a smooth two-point function.

Fix $f\in {C_c}^{\infty }(\mathcal{O};\mathbb{R})$, let $K_f=\mathrm{supp}f$, and put

The bare coefficients, on a smooth scalar configuration $\psi$, are

Integration by parts fixes the representatives

Write

The test $f$ is one fixed label-space density. It is not transformed during the background comparison.

The only interaction vertex used in the response construction is

In particular,

The derivative of $g$ is part of the operator.

Formal self-adjointness follows by compactly supported integration by parts. The sign ledger is

The positive Euler operator $P_m$ is not the variational Hessian $S_0^{\prime \prime }$.

Set

The test space has its standard LF topology; on a fixed compact $K$, ${\mathcal{D}}_K$ has all smooth seminorms.

Let

be the positive Klein–Gordon retarded and advanced Green operators. They satisfy

Existence, smoothness, finite propagation, and continuity are the standard Green-operator facts on the globally hyperbolic completion BGP.

For example, after spatial Fourier transformation,

and the advanced formula is the transposed one. The quotient at zero frequency has its continuous value. These equations prescribe Green operators, not quantum Cauchy data.

Put

Then ${{\Delta }_{\mathrm{S}}}^{\mathrm{T}}=-{\Delta }_{\mathrm{S}}$.

Fix one scalar Hadamard bisolution ${\Delta }_{+}$ on $\hat{M}\times \hat{M}$, with

Its wavefront set has the Hadamard null-relation form: both slot covectors are nonzero, and the first lies in one fixed half of the null cone Radzikowski. No state on the conical product is constructed by this choice.

Include zero covectors in the closed Minkowski causal cones ${\overline{V}}_{\pm }$, and define

Use the polynomial microcausal free presentation: a polynomial functional has compact distributional derivative kernels satisfying

for every positive derivative order.

Denote the corresponding off-shell scalar algebra by ${\mathfrak{A}}_{\mathrm{off}}$. Its Wick product is

Each polynomial pair has a finite sum. The usual microcausal contraction theorem supplies this free product and its associativity FR. No interacting product theorem is imported from it.

Let ${\mathfrak{J}}_0$ be the pointwise ideal generated by

Because ${\Delta }_{+}$ is a bisolution,

The same observation applied inside a pointwise multiple shows that ${\mathfrak{J}}_0$ is a star ideal. Define

In particular,

The off-shell insertion domain and the output sector are

Zero is included in this homogeneous degree-one sector. Its topology is the quotient test topology on $\mathcal{D}/P_m\mathcal{D}$. Formal $\hslash$-series are treated coefficientwise.

All wavefront calculations, covector sums, diagonal kernels, ${\Delta }_{\mathrm{F}}$, ${\mathsf{E}}_{\mathrm{A}}$, and ${\mathsf{E}}_{\mathrm{R}}$ in this file belong to the completion. A sum of covectors using Minkowski's affine identifications is not, by itself, a cone condition on ${\mathcal{M}}^{\circ }$.

Stage B: admission and the single retarded map B: admission and the single retarded map

A distribution $u$ on ${\hat{M}}^n$ is of split type if

The sum uses the fixed affine cotangent identification on $\hat{M}$.

A split-type distribution satisfies $\mathrm{WF}(u)\subset {\Xi }_n$.

Proof. If all $p_j$ are future causal, their time components are nonnegative. Their sum can vanish only if every time component vanishes, which forces every causal $p_j$ to vanish. The zero total covector is not in a wavefront set. The past case is identical.

The following operations preserve split type:

1. translation kernels $u(x-z)$ and diagonal delta kernels;

2. differentiation, smooth multiplication, and adjoining unused variables;

3. multiplication of two split distributions whose variable sets have exactly one common slot;

4. integration of internal variables when the projection is proper on the relevant support.

The product in (iii) is defined by the ordinary wavefront product criterion.

Proof. For (i), pullback by the difference map gives endpoint covectors $(p,-p)$; the delta cases are analogous. The operations in (ii) do not enlarge wavefront sets, and unused variables have zero covectors.

For (iii), lift both distributions to the union of their variables. An opposing pair would have zero covectors in every unshared slot. Split type would then force the covector in the single shared slot to be zero for each factor as well, contradicting a nonzero wavefront covector. The product criterion therefore holds. Its wavefront bound consists of sums of factor covectors, allowing a zero factor contribution. Those sums remain split.

For (iv), the pushforward bound retains only covectors with zero internal components. The surviving external components must still sum to zero. These are the standard pullback, product, and proper-pushforward rules Hormander (Chapter 8).

Regarded as polynomial functionals of the auxiliary multiplet $(h,k,\psi )$, the three bare coefficients

have compact derivative kernels of split type. They therefore satisfy the joint microcausal cone test on the completion.

This is a mathematical admission test on classical background slots. It does not quantize those slots.

Proof. The zeroth coefficient has a smooth first derivative. The first mixed kernel is a derivative, in its scalar slot, of

It is split by (eq:Lambda) and Lemma lem:split-calculus.

It remains to inspect the highest kernels at second order. With fixed tensor contractions and finite derivatives understood, they are generated by

The $k$-linear term is the corresponding two-slot coordinate kernel.

In the first line the factors share only $x$. In the second line the two deltas share only $w$; their product then meets the Green kernel only at $w$. In the third line attach the delta to the inner coordinate kernel at $w$, then attach the outer Green kernel at that single slot. Every step is covered by Lemma lem:split-calculus. Derivatives do not change this conclusion.

Every retarded path ends in $K_f$, and every coordinate source vertex carries $\sigma$. Consequently all effective background and internal vertices lie in

This set is compact. A smooth compact localization equal to one near it can be inserted before each internal pushforward; it does not change the kernel. The pushforwards are thus proper.

Lower derivative kernels are obtained by multiplying by smooth configurations and integrating the unused slots with that same localization. They remain split. Compactness follows from the same causal support argument. Finally, $\sigma$ is a smooth multiplier, including at $t=0$; no sharp boundary distribution has been inserted. Lemma lem:split-cone completes the proof.

This establishes the supported form of the joint admission reported in X-B XB (:joint). Its proof does not contain the product of a propagator with itself at both identified endpoints.

For fixed $(h,k)$, all tests in (eq:tests) lie in ${\mathcal{D}}_{K_f}$. Their dependence on backgrounds is continuous and polynomial at fixed $\epsilon$: $c_1$ is linear in $h$, and $c_2$ is quadratic in $h$ plus linear in $k$.

Thus each $F_{c_j}$ lies in the specified off-shell linear insertion domain, and $\mathbf{\varphi }(c_j)$ is a well-defined element of ${{\mathfrak{A}}_0}^{(1)}$.

Proof. The supplied coordinate maps are smooth and continuous in the background topology at fixed width. Equations (eq:tests) use their finite products and derivatives multiplied by $f$. This proves smoothness, support, polynomial dependence, and continuity in every ${\mathcal{D}}_{K_f}$ seminorm.

As a functional of the only quantum configuration $\psi$, $F_{c_j}$ has smooth first derivative $c_j$ and vanishing higher derivatives.

The scalar test support lies in $K_f$, while dependence on classical backgrounds can extend through $D_f$. Neither inclusion is asserted to be an equality for every configuration. Joint admission and Proposition prop:linear are separate statements.

For $h$ classical and the specified representative $F_c\in {\mathcal{L}}_{\mathrm{off}}$, define

The notation

means (eq:Rone) by transposition. It is not an application of a Green operator to an unspecified unsmeared quantum field.

For an arbitrary smooth scalar $u$, define the smooth linear response expression

When $P_{m}u=0$, this is the first retarded correction with the inherited free input. For arbitrary $u$, it is simply its displayed smooth linear extension.

Equation (eq:Rone) defines a map

linear in the insertion test and linear in the quadratic vertex. Its output test is smooth and compact, with

For every smooth $u$,

The map is continuous in the smooth background and compact-test topologies. It is additive in the vertex and vanishes when

No diagonal extension or subtraction is used.

Proof. The advanced Green operator sends $c$ to a smooth function supported in its causal past. The differential operator $L_{1,g}[h]$ has smooth coefficients supported in $\mathrm{supp}g$, giving (eq:retardedsupport).

Since $L_{1,g}[h]u$ is compactly supported, transposition in (eq:Green) gives

The final integration by parts is legitimate because the operator coefficients are compactly supported.

On each fixed-support test stage, Green-operator continuity gives smooth seminorm estimates on the compact interaction region. Multiplication by the background coefficients and their finitely many derivatives is continuous. The output lies in the fixed space ${\mathcal{D}}_{\mathrm{supp}g}$. These estimates, and continuity of the Green operator on the LF test space, give the stated continuity and its passage to the quotient. Linearity and the causal vanishing follow directly.

This proves the principal X-B normalization and support statement in the present presentation XB (:retarded, :retarded). The signs agree with (eq:Hessians): the action inverse is $-{\mathsf{E}}_{\mathrm{R}}$, not $+{\mathsf{E}}_{\mathrm{R}}$.

For retained laboratory tests, the response and dressing tests agree for two auxiliary completion extensions that coincide near the complete causal support used by their calculation and have the same inherited coordinate initial data.

The resulting compact pairings are unchanged by transport through $\iota$. This is exact patch agreement, not a global conical representation.

Proof. Coordinate coefficients agree by the supplied X-A-ST completion contract. For the response, ${\mathsf{E}}_{\mathrm{A}}c$ at an interaction point uses only the causal region joining that point to $\mathrm{supp}c$. On a retained calculation this region lies in the declared saturated compact support. The Green equations there have the same source and terminal prescription, so uniqueness gives the same test action. Applying the same local $L_{1,g}[h]$ gives the same $d_c$.

Transport of a compact distributional pairing through a chart is defined by transposition on tests and densities. It rewrites, rather than changes, that pairing.

For arbitrary completion tests outside the support contract, no such conical transfer is asserted.

Although its output is in ${\mathfrak{A}}_0$, the map need not factor through the free class of its input:

need not vanish.

Proof. Use ${\mathsf{E}}_{\mathrm{A}}{P}_{m}a=a$. For a concrete example let $m=0$, choose a real bump $r$ supported where $g=1$, and take $h=r\eta$. Then

Choose $a={\partial }_{t}r$, with ${\partial }_{t}r\ne 0$. Against the smooth free solution $u=t$,

Every element of the equation ideal vanishes on smooth free solutions. Hence the output class is nonzero, although $[F_{P_{0}a}]=0$.

Thus the geometric prescription must supply the representative before the map is applied. This is not an ambiguity of (eq:Rone).

For the prescribed scalar-only pair, the output test in (eq:Rone) has no $\hslash$-dependence:

It is also independent of the choice of Hadamard contraction kernel with the stated commutator. This is tree-exactness of this one map, not an all-orders interacting identity.

Proof. For the verification, introduce only the auxiliary kernel

and the expression

Here $V=-\frac{1}{2}⟨\psi ,L\psi ⟩$, with $L$ formally self-adjoint and compactly supported. Both ${\mathsf{E}}_{\mathrm{A}}c$ and ${\Delta }_{+}c$ are smooth: the Hadamard kernel has no wavefront covector with a zero second slot. Thus the expression is well defined.

Since ${F_c}^{(n)}=0$ for $n\ge 2$,

Consequently

The numerator is exactly divisible by $\hslash$. The zero-contraction term and the Hadamard kernel cancel; a second cross-contraction is absent. Taking $L=L_{1,g}[h]$ proves the assertion.

Changing a quadratic vertex's free Wick ordering adds only a scalar-independent term. Its first functional derivative is zero and does not change this calculation.

The auxiliary expression has not been named a time-ordered family. Its exact boundary is proved in Section sec:oneboundary.

Define the unswitched positive Euler coefficient

For every smooth vector field $\xi$,

This is an identity on arbitrary smooth scalars.

Proof. The density coefficients are

In particular,

Expansion of $Q_{K\xi }u$ gives

Expanding $D_{\xi }{P}_{m}u-P_m({\xi }^{\nu }{\partial }_{\nu }u)$ gives the same expression.

Define the off-shell completed first test and its class by

The free quantum field is held fixed in this comparison.

For an arbitrary compact interior parameter $\xi$, not necessarily in ${\mathfrak{g}}_p$,

All three displayed tests are smooth and compact. The first is explicitly in $P_m\mathcal{D}$.

Proof. The changed test is

Equation (eq:switchterms) gives $L_{1,g}[K\xi ]=Q_{K\xi }+{\Sigma }_{g,\xi }$. By Lemma lem:Q and $P_m{\mathsf{E}}_{\mathrm{A}}f=f$,

Substitute this identity and separate

Compactness of the gauge support makes $X_{\xi }{\mathsf{E}}_{\mathrm{A}}f$ compact. The other supports are compact because $f$, or the coefficients of $\Sigma$, are compact.

The coefficients of ${\Sigma }_{g,\xi }$ are supported in

This uses closed supports; it does not replace them by the open set $\{g\ne 1\}$.

For every smooth classical $h$ and every $\xi \in {\mathfrak{g}}_p$,

Equivalently,

is background invariant modulo ${\kappa }^2$.

Proof. The coordinate term in (eq:variation) vanishes by the supplied preflight identity (eq:preflight). The switch operator vanishes because $g=1$ and $\mathrm{d}g=0$ on a neighbourhood of $\mathrm{supp}\xi$. The remaining difference is

whose field class is zero.

This is the X-B first-order background identity XB (:variation, :ward). Off shell the difference is generally the nonzero equation-ideal representative just displayed. A general scalar pullback is not asserted to be an automorphism of the unchanged free Klein–Gordon shell. No interacting BRST differential or quantum master equation has been defined.

Passing (eq:variation) to the free algebra gives

where the integral denotes its compact test action.

For an initial-trivial compact parameter,

Indeed,

The inherited edge witness is

for which

It is outside ${\mathfrak{g}}_p$.

A coordinate gap does not by itself prove a nonzero total smeared on-shell response for every fixed $f,g$. The test can miss it, a resulting compact test can be an equation zero, and the switch contribution in (eq:defectclass) can also be present. If an excluded edge parameter happens to lie in an already chosen interaction plateau, the switch term vanishes and the coordinate term is isolated. No edge plateau is assumed to exist for every $g$.

Any established nonzero order-$\kappa$, order-${\hslash }^0$ defect cannot be canceled by terms starting at ${\kappa }^2$, or at positive $\hslash$-order, while its lower-order prescription is fixed. This coefficientwise observation does not identify the present defect with the different VI-B defect VIB. The original algebra is neither shrunk nor enlarged after the calculation.

Define

For every smooth $u$,

Thus the map on the first bare insertion is exactly the mixed term of the classical relational expansion, with its minus sign.

Proof. Theorem thm:retarded identifies the left side with $⟨c_1[h],{\rho }_{1,g}[h;u]⟩$. Use $c_1[h]={\partial }_{\mu }(f{w}^{\mu }[h])$ and integrate by parts.

For $\xi \in {\mathfrak{g}}_p$,

This is a transformation law, not an invariance claim.

Proof. The two maps $h\mapsto L_{1,g}[h]$ and $h\mapsto c_1[h]$ are linear, while $c_1[K\xi ]=D_{\xi }f$. Expand

On the plateau $L_{1,g}[K\xi ]=Q_{K\xi }$. For every compact test $a$,

The second term vanishes in ${\mathfrak{A}}_0$. Apply this with $a=c_1[h]$ and $a=D_{\xi }f$.

For comparison, the supplied coordinate laws imply the bare-test identities

To verify the second line algebraically, substitute (eq:coordinatelaw) into $B=w\cdot \partial w-v$. It gives

The change of (eq:barecoeff) is then

Transposition yields (eq:c2change).

The physical first-vertex correction has an additional factor $\kappa$. Its retained expansion is

Only $c_0,c_1$ enter the response part through this physical order.

For the classical second Euler coefficient write

The supplied classical response is