X-A-conv-ST: A fixed-width convolution obstruction on a prescribed conical Enter spacetime Supported analytical transfer, complete destructive reports, and the separation from the ramp prescription

We transfer the analytical obstruction of Cheung's X-A–Convolution to the supported I-B/X-A-ST setting without changing its regulator, gauge algebra, Cauchy data, or observation test. All wave operators remain on the declared Minkowski completion; the regular conical host is prescribed rather than reconstructed. The complete convolution data are stated explicitly, so the obstruction is proved rather than hidden in a positivity assumption.

The decisive identity is ${\mathsf{E}}_0\mathsf{C}{T}_{g,\epsilon }(K\xi )=S_{\epsilon }\xi$, not the identity required for coordinate covariance. For the parent compact witness we obtain a strictly positive translation-energy formula for $c_{\epsilon }$ at every allowed fixed width. The same coefficient survives evaluation on the scalar solution of the switched action, despite that action's plateau Noether identity. We retain the second-order defects and the genuine pure-gauge two-jet without promoting either calculation to an all-orders statement.

The supported transfer identifies precisely which compact sets must lie in the inherited developing chart. A change of variables, including its density Jacobian and transported two-point kernel, preserves the positive defect; it does not make convolution diffeomorphism natural. Laboratory field interpretation is conditional on this explicit support certificate, which is not implied by the ramp's causal reserve.

We then connect the analytical result to the proposed X-Enter-ST operational package: scalar-only invariant readout is necessarily uninformative, while a complete destructive-clock instrument can report supplied finite evaluations and preserve every unmeasured finite residual. An independently charged second coefficient would permit a conditional relational circuit, but is not constructed. Scheduled locality and reference-preserving SWAP transport concern these finite reports, not acquisition of the continuum coefficient.

The contribution is a controlled analytical–operational non-repair theorem on one retained host. The parent obstruction is credited to X-A–Convolution; the new supported audit makes its domain and interpretation explicit. No convolution repair, quantum characteristic coefficient, interacting differential, ultraviolet completion, or spacetime emergence is established.

in one sentence{ The frozen convolution prescription is proved obstructed at first classical order. Its positive gap survives supported transport and cannot be made into an informative invariant scalar-only output by changing readout.

The finite reporting interface is constructed under declared permissions. A laboratory interpretation of the continuum calculation additionally requires the source-specific chart certificate stated below.}

The principal analytical source is N. Cheung's X-A-conv, version 1, here called X-A–Convolution XConv. It fixes a spacetime convolution of switched metric arguments and proves that its coordinate prescription fails covariance on the gauge algebra declared before the calculation.

The principal operational comparison is Cheung's X-Enter, version 1 XEnter. Its proposed supported companion, X-Enter-ST, is the source supplied in the preceding development XEnterST. That source already separates the coefficient orbit, the complete readout, and the supplied-report instrument.

The present manuscript supplies the analytical half of this supported comparison. In particular, it states the actual mollifier and witness, proves the Cauchy and preflight identities, derives strict positivity, and retains the second-order defects. It does not infer these facts from an unspecified smoothing operator.

The ramp construction of X-A and X-A-ST is a separate prescription XA,XAST. No ramp identity is an input to the convolution obstruction.

Enter is a destructive readout interface. Its measured quantum names are consumed from the continuation's quantum context; its declared output includes the classical outcome and any unmeasured surviving quantum system. Consumption is an access statement, not annihilation of every physical degree of freedom used by an apparatus.

Cheung's operational characterization concerns a specified strategy class. The journal article and Manuscript I distinguish that characterization from a field instrument or a derivation of qubit kinematics EnterArticle,EnterI. The philosophical account emphasizes that readout, quantum recovery, thermodynamic implementation, and gravitational identification are different constructions EnterPhil.

We use explicit finite Born instruments and prove their channel identities directly. No abbreviated reconstruction of the Enter characterization is a premise of the classical argument.

The spacetime interface follows I-B: a classical conical background, finite quantum names, accessible algebras, locations, schedules, controls, and resources are separately declared IB. The readout does not select or evolve that metric.

The following distinctions remain binding throughout:

1. Existence and smoothness of a classical coefficient do not prove its gauge invariance.

2. A change of coordinates transporting a frozen regulator is not active covariance of that regulator.

3. A supplied finite report is not acquisition of an unknown continuum functional.

4. Equality of classical outcome laws is weaker than equality of complete instruments with residual systems and references.

Relational-observable and perturbative-gravity frameworks provide context, but do not prove covariance of this particular switched convolution prescription BFR.

Retain the I-B/X-A-ST instance, including its deficit, time orientation, regular laboratory, developing chart, time calibration, operational permissions, and reserved finite transport tube.

Write its regular conical product as

where $c_{\mathrm{h}}>0$ and $0<\alpha <1$ retain their inherited values. The completion defect is

It is a worldsheet.

The already declared analytical completion and chart are

with

The chart is injective on its declared domain and the laboratory does not meet $\mathcal{W}$. No host-update map is added.

The constant $c_{\mathrm{h}}$ is a host calibration speed. It is not the coefficient $c_{\epsilon }$.

The parent calculation below uses its original completion slab

This does not reset the inherited host time reserve to a new value. If the fixed parent data do not fit the retained chart, the corresponding laboratory interpretation requires the additional certificate in Section sec:support. No translation, rescaling, chart enlargement, or change of deficit is silently performed to force a fit.

The regular product $({\mathcal{M}}^{\circ },G)$ is not globally hyperbolic. Consequently, the global Cauchy and Green operators used below are operators on $\hat{M}$, not global inverses on the regular cone.

Proof. Fix a radial ray, $r_0>0$, and $z=z_0$. Choose $a>r_0/c_{\mathrm{h}}$ and set

The points

lie in $J^{+}(p)\cap J^{-}(q)$: the radial trip can be made within the available time. They have no convergent subsequence in the regular product, because their only completion limit has $r=0$. Thus this causal diamond is not compact. Global hyperbolicity would require compact causal diamonds.

The formal metric tensors used in the completion calculation are analytical inputs. Transporting them does not replace the fixed host metric $G$. In particular, the transported analytical background is $-G$, in agreement with (eq:signature), not an unnoticed substitution of the opposite signature.

This section retains the definitions of X-A–Convolution XConv (convention sheet). All subsequent conclusions refer to this frozen choice.

The smooth off-shell inputs are

No decay, equation-of-motion quotient, gravitational constraint, or gauge-fixed quotient is imposed. The scalar is massless.

The formal metric jet is

There is no factor $1/2$ in the definition of $k$. Statements modulo ${\kappa }^3$ mean coefficient identities, not convergence of a numerical perturbation series.

Set $x_{\ast }=(4,0,0,0)$ and $r(x)=|x-x_{\ast }{|}_E$, where the Euclidean norm is used only for supports and convolution. Define

Freeze

Then $g=1$ on $r\le 2$, and $\mathrm{supp}g\subset \{r\le 3\}⋐(0,8)\times {\mathbb{R}}^3$.

For a vector field $\xi$, write

The algebra fixed before the calculation is

Indices are raised and lowered with $\eta$. The switch is fixed background data.

For $\xi ,\zeta \in {\mathfrak{g}}_C$,

gives

The scalar transforms by the same formal pullback.

Let

Freeze one width

Convolution is componentwise in these completion coordinates.

The auxiliary metric is

The regulator acts on the metric arguments of the coordinate functional. It is not inserted into the action, the scalar argument, or the Cauchy inverse. It is neither Lorentz invariant nor diffeomorphism natural.

Define

Freeze

The free equation ${\square }_{\eta }{\rho }_0=0$ holds. Its unboundedness at spatial infinity is allowed by the declared smooth domain; no Hilbert-space normalizability is being asserted.

The same $g,f,{\xi }_{\ast },\varrho$ are used for every allowed fixed width. No witness is removed after its defect is found.

For a smooth source $s$, define the zero-data inverse on $M_8$:

The multiplier has its continuous value $t-u$ at zero spatial frequency.

The zero-data wave problem has a unique smooth solution for every smooth source on a neighborhood of $M_8$, locally interpreted by finite propagation. It satisfies

For smooth $a$,

where ${\mathsf{H}}_0[a]$ is the homogeneous solution with the Cauchy data of $a$.

Proof. For spatially compact smooth sources, differentiating (eq:Ezero) proves the equation and initial data. The wave energy estimate, after spatial differentiation, bounds the solution on finite slabs by the source and initial-data Sobolev norms. Applied to a difference, it gives uniqueness. Sobolev embedding gives smoothness and continuous dependence.

On a compact observation region, choose compact spatial cutoffs equal to one on its truncated causal past. Finite propagation makes the resulting solutions agree on overlaps. This defines the solution for general smooth sources and preserves uniqueness. These are the standard completion Cauchy arguments BGP.

Finally, $a-{\mathsf{E}}_0{\square }_{\eta }a$ is homogeneous with the Cauchy data of $a$, proving (eq:leftinverse).

For a symmetric tensor $A$, set

Thus

The scalar coordinate functions, with external label $\mu$, are defined by

with the coordinate Cauchy data of $x^{\mu }$.

The unique coefficients of (eq:harmonic) are

Their values and first time derivatives vanish at $\Sigma$.

Proof. Insert the displayed density expansion into (eq:harmonic). The coefficients of $\kappa$ and ${\kappa }^2$ give respectively

and the source displayed in (eq:v). Lemma lem:Cauchy gives the formulas and successive uniqueness.

These are the parent Cauchy formulas XConv (:coordcoeff). In particular,

Switch derivatives are retained. The nonlinear expression is $a_2(S_{\epsilon }(gh))$, not $g{a}_2(h)$ or $S_{\epsilon }(g{a}_2(h))$.

For every $\xi \in {\mathfrak{g}}_C$,

Proof. Because $g=1$ on a neighborhood of $\mathrm{supp}\xi$, $gK\xi =K\xi$. Constant-coefficient derivatives commute with convolution, and direct contraction gives

Consequently,

The support of $S_{\epsilon }\xi$ lies in the Euclidean $\epsilon$-neighborhood of $\mathrm{supp}\xi$, separated from $t=0$. Both Cauchy data therefore vanish. Equation (eq:leftinverse) proves the result.

Coordinate covariance would require $w[K\xi ]=\xi$. The actual value is $S_{\epsilon }\xi$ XConv (:preflight, :preflightstop). This calculation makes no global factorization of $w[h]$ through $\mathsf{C}h$; the switch terms prevent that simplification.

The formal inverse of the coordinate map is

Define

Here $f$ is fixed in relational-label space. No numerical global inverse is assumed.

Set

Taylor expansion gives

The negative first insertion sign is essential XConv (:Ocoeff).

With

the coefficient of $\kappa$ in the variation is

Proof. The zeroth coefficient varies by $\kappa \int f{X}_{\xi }\varphi$. The leading variation of $w$ is $w[K\xi ]=S_{\epsilon }\xi$, so the first insertion contributes $-\kappa \int f{X}_{S_{\epsilon }\xi }\varphi$. Their sum is (eq:firstdefect).

For the frozen data (eq:witness),

Therefore the frozen coordinate and relational prescriptions fail first-order covariance on ${\mathfrak{g}}_C$.

Proof. Expanding the squared difference and using translation invariance of Lebesgue measure gives

Integration against the normalized ${\varrho }_{\epsilon }$, followed by division by $2N$, gives (eq:ceps).

For every $a\ne 0$, the inner integral is positive. If it vanished, continuity would imply $\psi (x)=\psi (x-a)$ everywhere. Iteration would make a nonzero compactly supported function invariant under arbitrarily large translations, which is impossible. The mollifier is strictly positive on $0<|a{|}_E<\epsilon$, a set of positive measure. This proves strict positivity.

For $\xi ={\xi }_{\ast }$ and $\varphi =x^1$, (eq:firstdefect) is exactly $c_{\epsilon }$.

This is the positive-energy mechanism of the parent XConv (:positivegap). It concerns every allowed fixed positive width. No width-removal argument is used.

A modification starting at order ${\kappa }^2$, or at a positive quantum order while preserving the classical first coefficient, cannot restore the failed formal identity.

Proof. Formal equality requires equality of each coefficient. Such a modification leaves the nonzero classical coefficient $c_{\epsilon }$ unchanged.

This excludes only the stated order-preserving repair. It does not exclude a changed classical prescription.

For completeness, retain the parent's first-derivative gravitational density representative

and

The Christoffel symbols and curvature use the convention

Write

using $q=\eta +\kappa h$, and freeze

The free integral is interpreted variationally with compact variations. The chosen density is not replaced by another representative after switching XConv.

For a metric jet, its scalar equation through the retained order is

Thus the second vertex has one factor $g$, not $g^2$.

Define

With free input ${\rho }_0$ and zero positive-order Cauchy data, the selected scalar is

where

These formulas follow by expanding (eq:scalarequation) and applying Lemma lem:Cauchy. In particular,

No switch derivative has been discarded.

For the formal transformations generated by ${\mathfrak{g}}_C$,

Proof. Before truncation the scalar density coefficient can be written

with $\eta$ interpreted here as a contravariant density. The curved density is natural under pullback. The only possible mismatch is $(1-g)(F_{\kappa }^{\ast }\eta -\eta )$. It vanishes because $g=1$ wherever the transformation acts, and the transformation is the identity elsewhere. Also $F_{\kappa }^{\ast }g=g$.

Divergence of a vector density is natural. The pulled-back solution has the same Cauchy data because the transformation is the identity near $\Sigma$. Successive Cauchy uniqueness proves the assertion.

Let

and use the fixed free scalar ${\rho }_0=x^1$. For

the first coefficient difference is

Proof. Lemma lem:scalarcov gives ${\rho }_{1,g}[q_{\lambda }]=\lambda X_{{\xi }_{\ast }}{\rho }_0=\lambda \psi$. Lemma lem:preflight gives $w[q_{\lambda }]=\lambda S_{\epsilon }{\xi }_{\ast }$. The evaluated first relational coefficient is therefore

Smearing gives (eq:orbit).

This is the parent's evaluated obstruction XConv (:evaluatedgap). The scalar correction has been derived from the switched action, not appended as an external force or assumed to be a symmetry of an unchanged free scalar shell.

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

as a formal variational identity. This is compatible with $c_{\epsilon }>0$.

Proof. The unswitched first-derivative gravitational representative differs from the Hilbert density by a divergence. Together with the natural scalar density, this gives local current coefficients ${n_j}^{\mu }$ satisfying

The currents are local in a finite jet of $\xi$. Their support is compact in the gauge-support region. These are the coefficient Noether identities for the fixed representative XConv (:actionidentity).

Varying (eq:action) and integrating by parts gives

The boundary terms vanish by compact support. Every remaining factor involving $1-g$ or $\mathrm{d}g$ vanishes on the gauge support.

The convolution regulator appears nowhere in the action. It appears in (eq:harmonic), so the independent action identity cannot change (eq:preflight).

For the scalar sign check alone,

The calculation is off shell and retains its switch terms.

Let

and let $a_2(A,B)$ denote the symmetric polarization

Define

Then

For the bare functional $k=0$, write ${\delta }_{\xi }={{\delta }_{\xi }}^{(0)}+\kappa {{\delta }_{\xi }}^{(1)}$, where

Set

The variation is

Proof. The leading variation of $w$ is $b$, while

Thus their sum is $d_2$. Differentiating $\mathcal{J}$ gives

The second-order integrand from (eq:Ocoefficients) is consequently

Using $d_2=R_{2,\xi }+X_{\xi }w$ yields (eq:seconddefect).

Equation (eq:seconddefect) retains the parent's defects XConv (:infdefects). It is not replaced by zero and is not asserted beyond the displayed order.

For $\xi \in {\mathfrak{g}}_C$, define

Thus

The straight line $\eta +\lambda \kappa K\xi$ is not substituted for this jet.

Set

The coordinate coefficients are

The relational scalar coefficients are

Proof. For the auxiliary comparison metric $e^{\lambda \kappa {\mathcal{L}}_b}\eta$, the scalar functions $e^{\lambda \kappa X_b}{x}^{\mu }$ solve the harmonic equation. They have the frozen data because $b$ vanishes near $\Sigma$. Their coefficients are $\lambda b^{\mu }$ and ${\lambda }^2{b}^{\nu }{\partial }_{\nu }{b}^{\mu }/2$.

The actual auxiliary metric has the same first coefficient and an additional second coefficient ${\lambda }^2{d}_{\xi }$. Linearity of (eq:v) in $J$ adds ${\lambda }^2{\mathsf{E}}_0\mathsf{C}{d}_{\xi }$. This proves the coordinate formulas.

Substitute the scalar pullback expansion and these coordinates into the formal composition. The quadratic operator is

Since $[X_{\xi },X_b]=X_{[\xi ,b]}$, this is (eq:puretwo).

This is the retained parent pure-gauge test XConv (:purecoordinates, :pureobservable). It records the defect, not its cancellation. For the witness (eq:witness), the first smearing is already $\lambda c_{\epsilon }$.

Let

and

These sets are compact.

At fixed $\epsilon >0$, the coefficients (eq:Ocoefficients) are continuous polynomial functionals on the stated smooth spaces. Their first- and second-order joint kernels are smooth and compactly supported. Through order ${\kappa }^2$, dependence on $h,k$ is contained in $Q_{\epsilon }$, and dependence on $\varphi$ is contained in $K_f$.

Proof. For a constant symmetric tensor basis $e_A$, put

This is a jointly smooth parameter family: parameter derivatives commute with the linear Cauchy problem, and differentiated local energy estimates give continuous dependence. The polarized second family

is jointly smooth by the same argument and the finite Leibniz rule.

For example, integrating the scalar derivative by parts gives the first kernel

The second coefficient similarly involves finite derivatives of smooth products of $\Lambda$ and $\Omega$, with factors $f(z)$, $g(y)$, and $g(y^{\prime })$. This proves smooth compact support. Pairings with these kernels are bounded by finite products of compact-open smooth seminorms, proving continuity.

A wave source affecting $z\in K_f$ lies in $D_f$. A convolution input affecting such a source lies within Euclidean distance $\epsilon$ and in $\mathrm{supp}g$. In a nested second-order term the earlier wave source remains in $D_f$ by causal transitivity. Each metric input is thickened only by its own single convolution. The scalar derivatives are evaluated on $K_f$.

This is the parent support mechanism XConv (:smoothkernel, :support). It is an $\epsilon$-thickened dependence bound, not exact metric causality in the original field arguments. The convolution can sample a metric value slightly to the future of an intermediate wave-source point.

These classical statements do not construct the separate interacting characteristic quantum product.

For the retained chart $\mathcal{L}$, distinguish:

1. Witness certificate:

   $$P_{\epsilon }:=K_f+B_{\epsilon }⋐\mathcal{L}.$$

   Because $K_f\subset \{r\le 1/8\}$,

   $$P_{\epsilon }\subset \{r\le 1/4\}⋐U_C.$$

   This suffices to transport the numerical witness pairing, its compact gauge parameter, and its convolution dependence.

2. Full analytical certificate:

   $${\mathcal{R}}_{\epsilon }:=D_f\cup Q_{\epsilon }\cup ({\overline{U}}_C+B_{\epsilon })⋐\mathcal{L}.$$

   The required neighborhoods and the original time reserve are also retained. This sufficient certificate covers the causal computation, the convolution inputs, and the whole frozen supported gauge algebra used in its chart representation.

A certificate is a geometric inclusion for the actual retained chart. Compactness of the displayed sets alone does not prove their inclusion in that chart.

The original ramp reserve may cover $D_f$ for its own test without covering $Q_{\epsilon }$, the parent convolution supports, or the parent gauge region. Therefore it does not automatically establish (eq:fullcertificate).

No claim is made that the supplied host description by itself proves either certificate for the numerical placement $x_{\ast }=(4,0,0,0)$. The supported statements below have these explicit hypotheses. This is a restriction on the interpretation claimed, not a reduction of ${\mathfrak{g}}_C$ after finding the defect.

Let $V=\iota (\mathcal{L})$, let $\chi ={\iota }^{-1}:V\to \mathcal{L}$, and let

Here $\chi$ denotes the inverse chart only in this subsection; it is unrelated to the switch profile.

For a completion vector field $v$, denote its transported field by $v^{\iota }={\iota }_{\ast }v$. For a scalar $s$, write $s^{\iota }=s\circ \chi$. Then

whenever the witness certificate holds.

The transported convolution is the two-point operator

For compact chart-supported inputs, their pullbacks are extended by zero outside the chart. For the full calculation, certificate (SC-2) guarantees that the relevant convolution inputs are contained there.

Under (SC-1), the pairing (eq:transportpairing) is well defined and equals the same strictly positive $c_{\epsilon }$.

Under (SC-2), the relevant completion coefficient calculation and its gauge transformations admit the corresponding chart representation. In particular, with transported analytical operators,

and the witness first coefficient is still $\lambda c_{\epsilon }$.

Neither conclusion constructs a global conical Green operator or covariance of an intrinsically redefined cone convolution.

Proof. The vector–covector contraction is preserved by pushforward and pullback. The change-of-variables formula supplies $J_{\chi }$, proving (eq:transportpairing) and positivity.

Substituting the component transformation of $v$ into the original convolution and changing its input variable gives (eq:transportkernel). Thus it is the same operator in the supported presentation.

For (SC-2), finite propagation and Proposition prop:kernels place every required argument in the named reserve. Representing the already defined completion operators in that chart preserves the coefficient identities, including (eq:preflight). No inverse on all of ${\mathcal{M}}^{\circ }$ is needed.

In a developed Cartesian chart the volume factor may be one. It cannot be deleted from an arbitrary host coordinate presentation.

The formula for the two-point kernel also explains why passive coordinate transport is not a new active diffeomorphism identity for the regulator.

This section records the operational half of X-Enter and the supplied proposed X-Enter-ST XEnter,XEnterST. The proofs are included so the present manuscript has no external source-code dependency.

For an encoding $\mathsf{Enc}:\mathbb{R}\to \mathcal{S}$ and a fixed output map $\Phi :\mathcal{S}\to \mathcal{Y}$, the scalar-input output is $\Phi \circ \mathsf{Enc}$. Its declared boundary includes all reported classical words and every returned quantum factor.

It is informative about $a$ if this composite is nonconstant. Orbit invariance means constancy on $\mathsf{Enc}(\lambda c_{\epsilon })$, $\lambda \in \mathbb{R}$.

If

then

Proof. For any $a$, choose $\lambda =a/c_{\epsilon }$. Theorem thm:positive guarantees that this is defined.

No continuity, Born rule, finite dimension, or linear encoding is used. The conclusion concerns the composite on the encoded family, not constancy of $\Phi$ on unrelated states.

The charge $c_{\epsilon }$ is fixed during this comparison; $a$ is the varying coefficient. Reporting fixed calibration metadata is not forbidden.