IX-A-ST: Boundary-sensitive admission of sharp Cauchy insertions on a prescribed conical Enter spacetime A local Grade S construction with finite evaluation records, a destructive clock, and reference-preserving transport

We construct a supported local adaptation of the IX-A admission domain for the exact off-shell harmonic insertion

on a prescribed $3+1$-dimensional conical spacetime. A causally saturated regular developing presentation contains the entire truncated Cauchy past of each admitted observation. This makes the local zero-Cauchy response independent of auxiliary field extensions without asserting a global Green operator on the non-globally-hyperbolic regular cone.

The resulting space consists of actual compact-distribution polynomial functionals. A canonical trace on their linear metric descendants, taken modulo microcausal linear kernels, defines a Hausdorff locally convex topology. The carrier contains the bulk polynomials, exact sharp insertions, coordinate descendants, and arbitrary finite classical pointwise words. Directional differentiation and conjugation are continuous, and pointwise multiplication is separately continuous. For every nonzero real observation test, an explicit positive advanced-Cauchy moment produces a normalized continuous separator: it equals one on the sharp insertion and zero on every unchanged regular approximant. Thus faithful admission does not identify the sharp element with that regularization limit.

The additional contribution is a supported transfer theorem and a complete finite operational interface around these classical objects. One destructive clock selects installed evaluation tables; their outputs and an arbitrary reference-entangled payload admit exact causal transport through a compact regular laboratory. Finite records can certify the displayed separator but cannot determine the full infinite-dimensional admission topology or the quantum residual. This combination provides a controlled benchmark for later deformation and acquisition proposals while keeping classical admission, statistical readout, and quantum multiplication distinct. The admission construction itself is attributed to IX-A; no priority is claimed for its underlying distributional methods. No renormalized quantum product, quantum associativity, Einstein dynamics, or spacetime emergence is constructed.

boundary{ The metric, developing presentation, initial surface, lapse, calibration, and operational permissions are inputs. The sharp insertion is retained exactly as an off-shell classical functional. Finite tables encode already-defined evaluations; they do not acquire an unknown continuum field.

This is local Grade S, not Grade E. The quantum certificate remains -open. Separate continuity of classical multiplication is not a construction of quantum multiplication, joint continuity, hypocontinuity, or quantum associativity.}

IX-A separates three mathematical tasks: admitting an exact distributional functional, choosing a topology on its carrier, and constructing a quantum product on that carrier CheungIXA. The present paper addresses the first two on a supported regular presentation of the conical product used in I-B CheungIB. It then supplies a finite clock–evaluation–transport interface. The latter does not solve the third task.

The operational background is the Enter discipline. Quantum names specify access to declared tensor factors; destructive readout consumes the measured names and returns a classical record with specified surviving factors CheungEnter,CheungPhilosophy. This interpretation distinguishes a readout from recovery of an unknown quantum state. It also distinguishes a channel identity from a thermodynamic reset law and a mathematical analogy from a physical identification.

We specify finite Born instruments directly. We do not rederive the measurement-characterization theorem of the Enter calculus, enlarge its hypotheses to continuum fields, or infer a spacetime metric from its operational conditions.

A local Grade S package in this paper comprises:

1. a prescribed background and a compact regular support presentation;

2. explicit classical functionals, domains, topologies, and proofs;

3. complete finite instruments with declared input and output interfaces;

4. a finite destructive clock and an explicit calibration contract;

5. scheduled-support and independently certified causal-containment statements;

6. a finite reference-preserving transport realization with a resource ledger.

Grade E, understood here as an emergence claim for spacetime or its metric, is not part of the definition.

of the adaptation. The analytic core follows IX-A's actual-functional construction. The new local work is to state and verify sufficient support saturation, remove any implicit appeal to a global conical inverse, and connect a normalized boundary separator to a complete finite instrument without identifying that instrument with continuum acquisition. General wavefront and Cauchy theory are used in their usual stated domains BGP,BDH,BDHIntro.

sources. The supplied internal sheets are version-qualified source records, not assumptions that substitute for proofs. The propositions needed for the present release are established below. Comparisons with later sheets and with the OpenAI catalogue do not transfer their models, gates, or verification status into this manuscript.

Fix $0<\mu <2\pi$, $c>0$, and

The regular background and its metric are

with $\phi$ of period $2\pi$. The stipulated completion has singular locus

It is a worldsheet. Only its fixed-$z$ sections are worldlines. No smooth metric chart or pointwise Einstein tensor is asserted on $\mathcal{W}$.

The signature is $(-,+,+,+)$. The lapse in $T$ is one, and static observers have proper-time increment $\mathrm{d}T$. These are prescribed I-B data CheungIB.

The regular product $M_{\mu }$ is not globally hyperbolic.

Proof. Choose a regular spatial point of radius $r_0$, one angle, and $z=0$. Let $p,q$ have that spatial position and times $T_{\ast }\mp 2r_0/c$. Points at time $T_{\ast }$, the same angle and axial coordinate, and radii $r_n\downarrow 0$ lie in $J^{+}(p)\cap J^{-}(q)$ by radial causal curves. They have no convergent subsequence in the regular product. The causal diamond is noncompact.

Let $t=c(T-T_{\Sigma })$, and use developed spatial coordinates $\mathbf{x}$. The local metric is

The choice of units and the normalization of $t$ are fixed inputs.

Choose positive numbers $\delta ,T_0,r_0,R,H$, with $R>r_0+T_0$ and $H>T_0$, and an isometric developing identification of

with a regular open region of $M_{\mu }$, whose compact subregions used below have closures disjoint from $\mathcal{W}$. Set

Observation and coordinate-source tests belong to $\mathcal{D}(V)={C_c}^{\infty }(V)$, with the appropriate tensor indices.

The analytic prescription has zero homogeneous Cauchy data at $t=0$. All field dependence needed by these tests lies in $K_{\ast }$. No undeclared lateral boundary data or incoming homogeneous solution is added.

The strict margin $R>r_0+T_0$ is essential. A chart containing only the observation support is not sufficient.

Such presentations exist: place a sufficiently small developed ball around a regular point and choose a sufficiently short observation interval. Larger finite operational windows can be accommodated by first fixing their size and choosing the regular point sufficiently far from the defect.

For $f\in \mathcal{D}(V)$, define its auxiliary truncated past

This is compact and contained in $K_{\ast }$. The symbol $J_{\eta }$ refers to the developed Minkowski presentation. No equality with a globally minimal causal past on the cone is claimed.

Let $h\in C^{\infty }(U;S^2{T}^{\ast }U)$. Choose a compact cutoff in $U$ equal to one near $K_{\ast }$, extend the cutoff field by zero to auxiliary Minkowski space, and solve the zero-data wave problem there. Its value on $V$, when used in the response below, is independent of the cutoff.

The same statement holds for its smooth derivatives and its pairings with observation tests supported in $V$.

Proof. Two such extensions agree on a neighbourhood of $K_{\ast }$, hence their first differential expressions agree on every $D_f$. The difference of their zero-data solutions has zero source in the truncated past of $\mathrm{supp}f$ and zero Cauchy data there. Finite propagation and local uniqueness give zero difference near $\mathrm{supp}f$. The strict support margin gives the same conclusion for derivatives. These are local consequences of ordinary Minkowski Cauchy theory BGP (Chapter 3).

This use of auxiliary Minkowski space constructs no intrinsic global conical Green operator.

Raise indices with $\eta$, and write

For smooth sources, define the zero-Cauchy response on the supported region by

At zero spatial frequency the quotient equals $t-u$. Thus

For fields without spatial decay, Lemma lem:saturation defines the local interpretation.

Set

and, for a smooth compact covector source $a$ in $V$,

These definitions retain the exact IX-A normalization CheungIXA. No equations are imposed on $h$ or $\varphi$.

Let

A configuration is $\Psi =(h,\varphi )$. The compact-distribution polynomial carrier consists of finite sums

The degree-zero coefficient is a complex number. A compact distribution acts on a smooth function by inserting any cutoff equal to one near its support.

Conjugation means the coefficient involution

On real configurations this is ordinary complex conjugation. This convention preserves complex polynomial dependence.

The symmetric kernels in (eq:polynomial) are uniquely determined by the functional on all smooth off-shell configurations.

Proof. Scaling separates homogeneous degrees. Polarization recovers the symmetric multilinear form in each degree. If that form vanishes on all smooth directions, its kernel annihilates all finite sums of elementary tensor tests. Those sums are dense in the smooth test space on each compact product region. Distributional continuity implies that the kernel is zero.

Consequently ${\mathcal{P}}_{\mathrm{dist}}$ is a space of actual functionals, not a free symbol algebra. Give its coefficient carrier the locally convex direct-sum topology

The strong topology is uniform convergence on bounded sets of $C^{\infty }(U^n)$, componentwise in the bundles.

Let ${\overline{V}}_{\pm }$ be the closed nonspacelike cotangent cones, including zero in individual slots. Define ${\mathcal{P}}_{\mathrm{mc}}$ by

where the total zero covector is omitted. Let ${\mathcal{P}}_{\mathrm{reg}}$ consist of polynomials with smooth compact kernels.

For a closed wavefront cone $\Gamma$, the normal topology used here combines strong distribution seminorms with the localized Fourier seminorms

All assertions involving that topology specify degree, compact support, and wavefront stage. We do not confuse the normal topology with weaker distribution topologies BDH.

Every fixed smooth directional derivative of a bulk polynomial is bulk. At each smooth configuration, its first metric derivative has no nonzero nonspacelike wavefront covector.

The classical pointwise product of two bulk polynomials is bulk.

Proof. Directional differentiation and evaluation contract slots with smooth fields. A forbidden nonspacelike tuple in the remaining slots would give a forbidden tuple of the original kernel by adjoining zero covectors in contracted slots. The convention in (eq:microcausal) therefore gives the descendant statement.

Pointwise products use symmetrized exterior tensor products. An all-future or all-past nonzero wavefront tuple of such a tensor product would already violate the condition in one nonzero factor.

Use the auxiliary advanced and retarded inverses of ${\square }_{\eta }$, denoted by ${\mathsf{G}}_{\mathrm{A}}$ and ${\mathsf{G}}_{\mathrm{R}}$. They are employed only in the supported prescription.

For a covector source $a\in \mathcal{D}(V;T^{\ast }U)$, set

The formal adjoint of $C$, under the stated tensor pairing, is

Let $n_{\mu }={\partial }_{\mu }t$, and define

The compact kernel of $J(a)$ is

There is no ${\delta }^{\prime }(t)$ term, and

Proof. The zero-data prescription gives $Lh={\mathsf{G}}_{\mathrm{R}}(\Theta Ch)$ on the observation region. Transpose the retarded kernel against the compact source:

All relevant intersections are contained in $K_{\ast }$. The product rule gives (eq:qa). The adjoint has differential order one, so it differentiates $\Theta$ at most once. The advanced response to a smooth compact source is smooth; only the surface factor can contribute wavefront. The support statement follows from finite propagation.

The conventions and the boundary mechanism agree with the IX-A sharp-adjoint construction CheungIXA (:adjoint).

The map $v\mapsto \mathcal{B}(v)$ is injective. If $b(0,{\mathbf{x}}_{\ast })\ne 0$, at least one component of $q_a$ has both nonzero normal directions in its wavefront set at $(0,{\mathbf{x}}_{\ast })$.

Proof. With lower components $v_{\mu }$, direct substitution gives

These identities prove injectivity.

Choose a component with nonzero delta coefficient near the point. After localization it has the form

with smooth coefficients and $v({\mathbf{x}}_{\ast })\ne 0$. The Fourier transform of its delta term is independent of the normal frequency and nonzero at some fixed tangential frequency. Integration by parts in $t$ makes the step term $O(|\tau {|}^{-1})$. It cannot cancel the nonzero order-zero term as $\tau \to +\infty$ or $\tau \to -\infty$. These frequency rays lie in arbitrarily narrow cones around the two normals, proving the assertion.

Define

There is a well-defined linear map

It vanishes on ${\mathcal{M}}_1$ and commutes with conjugation.

Proof. If $m+q_a=m^{\prime }+q_{a^{\prime }}$, then $q_{a-a^{\prime }}=m^{\prime }-m\in {\mathcal{M}}_1$. If its advanced trace were nonzero, Lemma lem:noncancellation would give a timelike conormal wavefront covector, forbidden for ${\mathcal{M}}_1$. Thus the trace is independent of the decomposition. Compact trace support follows from finite propagation. Reality of the differential and Green operators proves the conjugation statement.

The trace is computed from an actual metric derivative. It is not an independently adjoined boundary variable. It need not detect a step singularity whose delta coefficient vanishes. No complete classification of boundary microlocal singularities, and no identity $\ker \mathfrak{b}={\mathcal{M}}_1$, is asserted.

For a smooth direction $\eta \in \mathcal{E}$, put

For a finite list $\mathbf{\eta }=({\eta }_1,\dots ,{\eta }_k)$, write $D_{\mathbf{\eta }}=D_{{\eta }_1}\cdots D_{{\eta }_k}$; the empty list is the identity.

The space ${\mathcal{P}}_{\Sigma }\subseteq {\mathcal{P}}_{\mathrm{dist}}$ consists of all $F$ such that

Its boundary coordinates are

This quantifies over actual distributions, not over chosen representatives of equation-of-motion classes.

The domain contains ${\mathcal{P}}_{\mathrm{mc}}$, all $J(a)$, and all $B_f$. For $\eta =(k,\sigma )$,

Third derivatives vanish. The mixed and linear kernels are compact distributions.

Proof. Bulk admission is Lemma lem:bulk. The first metric derivative of $J(a)$ is $q_a$; its higher derivatives vanish. Differentiate (eq:B) and integrate the scalar variation by parts to obtain (eq:descendants). For a fixed smooth $k$, $Lk$ is smooth on $V$, so the first term in $D_{\eta }{B}_f$ is a regular scalar linear functional. Every source in the remaining metric derivatives is smooth and compactly supported in $V$.

For compact distribution kernels, the bilinear expression in (eq:B) is continuous in smooth fields restricted to a fixed compact neighbourhood of $K_{\ast }$. The kernel theorem gives its compact mixed kernel. Alternatively, multiply the retarded kernel by $\Theta (z^0)$ and transfer the finite $C$ and scalar derivatives to it. The product exists because a nonzero wavefront pair of the wave kernel has nonzero observation-slot covector, whereas a wavefront covector of the step factor has zero observation-slot component. The operations are therefore legitimate BDHIntro.

Every compact-distribution polynomial depending only on $\varphi$ also belongs to ${\mathcal{P}}_{\Sigma }$, because its metric derivatives vanish. That observation is not a Wick-product theorem for arbitrary scalar distributions.

The domain ${\mathcal{P}}_{\Sigma }$ is a unital commutative associative algebra under classical pointwise multiplication. It is stable under smooth directional differentiation and conjugation.

The subalgebra ${\mathcal{P}}_{\mathrm{gen}}$ generated by ${\mathcal{P}}_{\mathrm{mc}}$, all $B_f$, and all $J(a)$ is contained in ${\mathcal{P}}_{\Sigma }$ and has the same differential and conjugation stability. There is no bound on the finite number of factors in a pointwise word.

Proof. The coefficients of a product are finite sums of symmetrized exterior tensor products, so they remain compact distributions. For a direction list, the Leibniz rule expresses $D_{\mathbf{\eta }}(FG)$ as a finite sum $(D_{I}F)(D_{J}G)$. Its metric derivative at a fixed configuration is

The coefficients here are complex numbers, and the distributional factors belong to ${\mathcal{S}}_1$. Thus (eq:admission) holds for the product.

An additional direction adds an entry to the defining list. Conjugation preserves ${\mathcal{M}}_1$, ${\mathcal{Q}}_{\Sigma }$, and their trace. Finally, (eq:descendants) expresses derivatives of the sharp generators through coordinate generators, regular scalar linear functionals, and constants. The finite Leibniz rule therefore also preserves ${\mathcal{P}}_{\mathrm{gen}}$.

The classical multiplication in this theorem identifies no singular kernel variables. Its closure is not closure under contractions.

Give $C^{\infty }(\Sigma ;T^{\ast }U{|}_{\Sigma })$ its compact-open smooth Fréchet topology, even though individual traces have compact support.

The topology ${\tau }_{\Sigma }$ on ${\mathcal{P}}_{\Sigma }$ is the initial locally convex topology for the coefficient inclusion into ${\mathcal{T}}_{\mathrm{coeff}}$ and all maps

Equivalently it is the topology of the graph embedding

The added seminorms include, for every compact $A\subset \Sigma$ and integer $j\ge 0$,

No completion, infinite polynomial degree, or new product is introduced by this definition.

The topology ${\tau }_{\Sigma }$ is Hausdorff. Every fixed-direction operator $D_{\eta }$ and conjugation are continuous. Evaluation of $F$, or any fixed directional descendant, at any fixed smooth configuration is continuous.

Proof. Lemma lem:unique makes the coefficient inclusion injective into a Hausdorff space, so (eq:graph) is Hausdorff.

Contraction of a compact distribution with a fixed smooth field is strongly continuous: insertion of that field maps bounded sets of smooth target tests to bounded sets of source tests. The finite degree shifts are continuous on the locally convex direct sum. The same argument handles fixed evaluations.

For the boundary coordinates,

Thus differentiation shifts coordinates already present in the topology. Conjugation replaces the configuration and direction list by their conjugates and conjugates the trace.

For each fixed $G\in {\mathcal{P}}_{\Sigma }$, multiplication by $G$ is continuous on $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$. No joint continuity or hypocontinuity is inferred.

Proof. At coefficient level, multiplication by $G$ is a finite sum of tensor products with fixed compact distributions, followed by symmetrization and degree shifts. For a fixed compact distribution $v$, partial application of $v$ maps a bounded set of smooth joint tests to a bounded set of tests in the remaining variables. This proves strong continuity of $u\mapsto u\otimes v$.

For a boundary coordinate, the differentiated Leibniz formula gives a finite sum of terms

The $G$-values and $G$-traces are fixed. The remaining maps are continuous evaluations or defining boundary coordinates. Every coordinate of multiplication by $G$ is continuous.

The proof supplies no uniform estimate for arbitrary bounded sets of varying multipliers. This is the precise distinction between the asserted statement and a hypocontinuity theorem BDH,CheungIXA.

The maps

are continuous and linear. For finitely many fixed compact test supports, every fixed finite pointwise word in these generators depends jointly continuously on those tests in their smooth fixed-support topologies.

Proof. For coordinate functionals, smooth finite-time Cauchy estimates make $a\mapsto {\mathsf{G}}_{\mathrm{A}}a$, its derivatives, and its initial trace continuous on each fixed compact source stage. Formula (eq:qa) then gives strong continuity of $q_a$.

For $B_f$, the compact mixed kernel is a finite derivative of the fixed retarded-step distribution multiplied by $f(x)$. On a common compact support stage, strong seminorms of this kernel are bounded by finitely many smooth seminorms of $f$. This follows by applying its finite distributional order to the bounded family of joint tests multiplied by $f$.

The only nonzero types of boundary coordinates are

Higher direction lists give zero. Each map in (eq:Btrace) is continuous on fixed test supports. The test-function LF topology is final over those stages, proving the linear continuity assertions.

For a word with fixed supports, its coefficient kernel is an exterior tensor product of fixed distributions multiplied by products of the varying smooth test functions. Finite-order estimates on the common compact region prove joint continuity of this restricted parameterization. Its boundary coordinates follow from the finite Leibniz formula. This restricted statement does not establish joint continuity of multiplication on all of ${\mathcal{P}}_{\Sigma }$.

The inclusion ${\mathcal{P}}_{\mathrm{mc}}\hookrightarrow {\mathcal{P}}_{\Sigma }$ is injective and continuous on each fixed-degree, fixed-support, fixed-wavefront normal-topology bulk stage. All its boundary coordinates vanish.

The topology induced on ${\mathcal{P}}_{\mathrm{mc}}$ is its strong coefficient subspace topology, not an asserted identification with the finer normal topology.

Proof. Injectivity follows from uniqueness of actual kernels. Lemma lem:bulk puts every relevant linear metric descendant in ${\mathcal{M}}_1$, where $\mathfrak{b}$ vanishes. Only the coefficient topology remains on the bulk. Normal stages contain those strong seminorms, giving the stated continuous inclusion.

The following signature-sensitive identity is the explicit separation mechanism of IX-A CheungIXA (:witness, :IX1).

Let $f\in \mathcal{D}(V;\mathbb{R})$ be nonzero. Define

Then

The initial trace is compactly supported inside $\Sigma$.

Proof. All spatial supports are uniformly bounded on the finite time interval in question. Put

Integration of ${\square }_{\eta }{b}_0=a_0$ gives $-B_0^{\prime \prime }=A_0$. The advanced solution has zero data at sufficiently late times. Consequently,

There are no integration boundary terms because $f$ is compact inside $V$. If $u_f=0$, then $tf$ is independent of $t$; compact time support implies $f=0$, a contradiction. Support saturation places the entire trace support inside the initial ball.

For every nonzero real $f$,

The linear map $f\mapsto B_f$ is injective, also after complexification.

Proof. At ${\Psi }_f=(0,{\varphi }_f)$, the metric derivative is $-q_a$. Lemmas lem:positive and lem:noncancellation give a nonzero timelike wavefront component, forbidden for a bulk linear metric descendant. The derivative is nonzero, so $B_f\ne 0$. For a complex source, separate real and imaginary parts on real configurations.

Choose a real even compact mollifier $\rho$ of integral one. Extend the compact mixed kernel $K_f$ by zero in the developing coordinates and define

For sufficiently small $\epsilon >0$, all supports remain in a common compact subset of $U^2$. Let $B_{f,\epsilon }$ be the resulting actual polynomial. Equivalently, using compatible local extensions, $B_{f,\epsilon }[h,\varphi ]=B_f[{\rho }_{\epsilon }\ast h,{\rho }_{\epsilon }\ast \varphi ]$. Each approximant has smooth compact kernel and belongs to ${\mathcal{P}}_{\mathrm{reg}}$.

For every nonzero real $f$, there is a continuous linear functional ${\mathcal{Q}}_f:{\mathcal{P}}_{\Sigma }\to \mathbb{C}$ such that

In particular,

although their coefficient kernels converge strongly.

If two supplied numerical evaluations of ${\mathcal{Q}}_f$ have absolute errors at most ${ϵ}_0,{ϵ}_1$ on a bulk element and on $B_f$, their reported separation is at least $1-{ϵ}_0-{ϵ}_1$.

Proof. Choose $\chi \in {C_c}^{\infty }(\Sigma )$ equal to one on the trace support in Lemma lem:positive, and set

This is a continuous coordinate of the graph topology. Equations (eq:Btrace) and (eq:positive) give its value one on $B_f$; Proposition prop:bulkinclusion gives zero on the bulk. No net of bulk elements can converge to $B_f$ in this topology.

Strong convergence of the mollified compact kernel follows by transposition: convolution of a bounded set of smooth tests converges uniformly with all finitely required derivatives on the common compact set. The error assertion is the scalar triangle inequality.

Thus the closure of the bulk in ${\mathcal{P}}_{\Sigma }$ is contained in the closed subspace ${\bigcap }_{\Psi ,\mathbf{\eta }}\ker {\mathfrak{b}}_{\Psi ;\mathbf{\eta }}$, and excludes every nonzero real sharp insertion. Equality with that closed subspace is not claimed.

In the mixed variables $(x,z)$, define the translation-invariant angular cone

The second set includes $\xi =0$.

The mixed kernel satisfies $\mathrm{WF}(K_f)\subseteq {\Gamma }_B$. On a sufficiently large common compact support stage,

The identity from these admitted mixed kernels to their actual functionals in $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$ is not sequentially continuous at $K_f$ for nonzero real $f$.

Proof. The retarded kernel has diagonal conormals and null-propagation pairs $(\xi ,-\xi )$. Multiplication by the step factor in the $z$-slot adds

At this boundary branch $x^0>0=z^0$, so a diagonal contribution cannot meet the boundary there. Finite differentiation and smooth coefficients preserve the upper cone (eq:GammaB).

Strong convergence was proved above. For a localized Fourier seminorm on a closed cone separated from ${\Gamma }_B$, use a slightly larger separated angular cone. The localized kernel has arbitrary polynomial decay in that larger cone and polynomial growth globally. Convolution regularization introduces bounded smooth multipliers converging to one on fixed frequency balls. Split the Fourier integral into a fixed ball, a near-frequency region in the larger separated cone, and its angularly separated complement. The first part tends uniformly to zero. Arbitrary rapid decay controls the second, and rapid decay of the localization transform controls the complement. Choosing the decay order larger than the requested seminorm order and the distribution growth order gives a tail bound uniform in $\epsilon$. This proves normal convergence, the standard compact regularization argument for the stated constant angular cone BDH.

Theorem thm:separator gives nonconvergence of the image functionals and hence the last assertion.

, not evasion by subtraction. VII-C records a particular logarithmic Wick obstruction under normal-admission continuity and old-product agreement CheungVIIC. The present topology explicitly drops the regular-to-sharp admission-continuity hypothesis. It does not subtract the old regular products or show that a replacement quantum algebra exists. VII-D's numerical finite part and IX-B's subsequent bookkeeping are separate records, not premises of this argument CheungVIID,CheungIXB.

If two admissible presentations agree on every smooth off-shell configuration, they define the same element of ${\mathcal{P}}_{\Sigma }$ and the same topology coordinates.

Kernel permutations, tensor identities, compact integration by parts with all surface terms retained, and valid bulk–surface decompositions preserve the admitted element. No equation-of-motion quotient is used.

Proof. Lemma lem:unique identifies their symmetric coefficient kernels. Equality of actual polynomials identifies every directional and metric derivative. Proposition prop:trace therefore identifies every boundary coordinate.

The stated operations preserve actual distributional pairings. In particular (eq:qa) is an equality, but deleting its delta term is not an allowed rewriting.

For the selected coefficient comparisons only, fix in the auxiliary developed coordinates

where

The graviton block is not asserted positive. This is neither a global cone Hadamard state nor a represented interacting gravitational field.

On its existing bulk domain, retain the usual contraction formula

For polynomials the sum is finite. The distinction between this old domain and a quantum extension is standard in the functional formulation FR.

For $a\in \mathcal{D}(V;T^{\ast }U)$, smearing a massless metric covariance against $q_a$ in one slot gives a smooth function of the other slot. Reverse ordering has the same property.

Proof. The kernel $q_a$ has only timelike conormal wavefront covectors. A nonzero covariance wavefront covector is null in each slot and nonzero in both. The distributional composition criterion therefore has no opposing pair in the contracted variable. Its output wavefront can contain a covector only if such a matching pair survives, which is impossible. Compactness of $q_a$ supplies properness for the integration. The output is smooth BDHIntro.

The following already-defined local contraction outputs are actual elements of ${\mathcal{P}}_{\Sigma }$:

1. coefficients of $B_f$ with a smooth compact linear metric or scalar functional, in either order;

2. the scalar-contracted metric quadratic functional

   $$T_{f,f^{\prime }}[h]=\int f(x)f^{\prime }(x^{\prime })w^{\mu }[h](x)w^{\nu }[h](x^{\prime }){{\partial }_{\mu }}^x{{\partial }_{\nu }}^{x^{\prime }}{\Delta }_m^{+}(x-x^{\prime }){\mathrm{d}}^{4}x{\mathrm{d}}^4{x}^{\prime };$$

3. the single-contraction coefficient with $W_g=\int g{h_{00}}^2$, $g\in \mathcal{D}(V)$, in either order.

These are selected coefficient-membership statements. They do not define a quantum multiplication on ${\mathcal{P}}_{\Sigma }$.

Proof. A smooth linear metric smearing makes the contracted metric covariance smooth. Its image under the local $L$ is smooth on $V$. The remaining output is therefore a regular scalar linear functional. A smooth scalar linear smearing gives a smooth scalar function $v$, and the output is, up to the fixed contraction sign, $J(f\mathrm{d}v)$.

For (eq:selfcoefficient), pair the differentiated scalar covariance against the smooth compact two-variable coefficient at each fixed $h$. This defines a continuous quadratic functional with compact metric kernel. Varying one $w$-factor leaves the other factor smoothly smeared against the scalar covariance. The resulting source for the varied $w$ is smooth and supported in $\mathrm{supp}f$ or $\mathrm{supp}{f}^{\prime }$. Its first metric derivative is therefore a finite sum of $q_a$'s. Every further directional descendant has the same property.

For (iii), define the mixed kernel $H_{wh}=(L\otimes I)H_{hh}$ on the supported observation variables. It is a defined distribution: the retarded-step composition has the same transversality used in Proposition prop:generators. The left-ordered coefficient is

At fixed smooth $\varphi$, smearing the $x$-slot is precisely a metric covariance smeared with $q_{f\mathrm{d}\varphi }$. Lemma lem:smoothing makes the result smooth in $y$. Multiplication by $g$ gives a regular first metric derivative. Scalar differentiation replaces $\varphi$ by a fixed smooth direction and preserves the argument. Reverse ordering changes the covariance orientation, not this smoothness conclusion.

The cases above match the selected admission tests identified in IX-A CheungIXA (:knownoutputs). No subsequent multiplication of their outputs by another sharp insertion is assigned here.

One may form

with its finite product topology. Projection to the classical slot and field-independent coefficient evaluations are continuous. The old bulk polynomial space embeds linearly and injectively. This does not supply a deformed product, quantum associativity, a state, or interacting admission.

Transport the tensor fields, densities, kernels, and initial surface through the developing identification of Definition def:support. The supported construction on the cone has exactly the admission, topology, separation, and presentation properties proved above.

It is independent of auxiliary field extensions equal near $K_{\ast }$. A product-adapted change of developed coordinates $t^{\prime }=t$, ${\mathbf{x}}^{\prime }=Q\mathbf{x}+\mathbf{a}$, with $Q$ orthogonal, gives the corresponding isomorphic construction on transformed support data.

No chart through the worldsheet, global conical inverse, or boost-invariant identification of different initial prescriptions is implied.

Proof. The developing map is an isometry on the complete analytic support. Its tensor pullbacks and density transformations carry all finite-order expressions and compact distribution pairings to the same local formulas. Causal support saturation and Lemma lem:saturation prove extension independence.

The displayed product-adapted coordinate changes preserve ${\square }_{\eta }$, the zero-Cauchy surface, and the finite-propagation prescription. Distribution pullback by these diffeomorphisms is continuous, with continuous inverse, in the coefficient and smooth trace spaces. Hence it preserves the graph topology. The final limitations follow from the domains actually used.

The continuum theorem precedes the finite interface. A finite record can report a supplied evaluation without becoming a measurement of the continuum object from which that evaluation was defined.

Choose a finite catalogue

and finitely many real linear diagnostics ${\ell }_j$ on an appropriate real subspace. Allowed examples include:

• the real or imaginary part of a coefficient pairing with a fixed smooth joint test;

• evaluation of a fixed directional descendant at a fixed smooth configuration;

• a boundary pairing ${\int }_{\Sigma }\chi \cdot {\mathfrak{b}}_{\Psi ;\mathbf{\eta }}(F)$;

• the normalized separator ${\mathcal{Q}}_f$.

These maps are continuous by the preceding theorems.

Each finite table entry ${\ell }_j(F_x)$, or a certified enclosure for it, is supplied classical data. No general computability theorem for arbitrary smooth inputs is assumed. In particular, continuity alone is not an algorithm for evaluating an arbitrary real number exactly.

Choose a finite bound $M>0$ such that all selected entries have absolute value at most $M$, and put

for the installed normalized score selected by clock code $a$. A finite report word $r(a,x)$ may additionally store prescribed rounded values and exact catalogue flags. It is a report of installed data, not a universal admission test.

For the catalogue containing $B_f$ and any finitely many regular approximants, ${\mathcal{Q}}_f$ gives exactly the entries one and zero. Its existence and nonzero normalization were proved, not inferred from numerical sampling.

Let $C$ have dimension $2^m$ and an installed orthonormal basis $\{|{\tau }_a⟩\}$. Let $S$ contain every surviving factor and let $E$ be any untouched reference. Put

Calibrated basis change followed by destructive Enter has channel

It is CPTP, including with ${\mathrm{id}}_E$, and

Its quantum output contains $S$, not $C$. On a branch of positive probability one may normalize the returned block; zero branches require no conditional state.

Proof. The Kraus operators are $|a{⟩}_R\otimes D_a$. Their squared moduli sum to ${\sum }_a{P}_a\otimes I_S=I$. Expansion of $X$ in clock matrix units shows that both expressions return its diagonal $aa$-block. Summing the blocks gives the partial-trace identity. The calculation is unchanged with an additional reference.

This is the finite interface compared with I-B, VI-B, En09-A, and the observation sheets CheungIB,CheungVIB,CheungEn09,CheungObservation. The proof is a complete finite-instrument calculation Watrous.

At one fixed actual reading event $e$, distinct calibrated labels $t_a$ satisfying

permit at most one positive-probability outcome. A nontrivial multi-outcome history or programme label therefore requires a separate interpretation, not that event-time equality.

The clock channel has no universal left inverse on its full input space. No finite matrices obey $[T_C,H_C]=i\hslash I$ for $\hslash \ne 0$. Clock consumption does not consume $T$, $t$, or a spacetime dimension.

Proof. Distinct labels prove the first assertion. The two clock states $(|{\tau }_0⟩\pm |{\tau }_1⟩)/\sqrt{2}$, with the same payload, give identical destructive outputs, excluding a universal inverse. Tracing the commutator equation gives a contradiction. The geometric coordinates remain fixed classical inputs.

No stationarity or covariance hypothesis for a conditional unitary-time interpretation is imposed. The finite clock selects an externally scheduled finite programme; it does not generate unscheduled quantum time CheungEn09.

Let $G$ be the catalogue register, and let $P$ be an arbitrary finite quantum payload. The intended classical catalogue interface is jointly block diagonal in $G$, including its correlations with other systems. Its canonical extension to all matrices first pinches $G$.

For

define the recorded channel

The map (eq:tablechannel) is CPTP and acts as identity on the surviving payload in each subnormalized classical branch. It preserves arbitrary untouched reference correlations in that channel sense. Its conditional sign mean on catalogue entry $x$ and clock code $a$ is $v_{a,x}$.

It is implementable by a controlled finite probe isometry, probe readout, and classical report generation.

Proof. Kraus operators are

Since ${\sum }_y{p}_y(a,x)=1$, their squared moduli sum to $I_{CGP}$. The formula on matrix units proves the reference claim.

For the probe, use the normalized vector

as a controlled preparation, followed by destructive readout. The resulting finite isometry extends to a unitary on a finite larger register. The sign expectation is $p_{+}-p_{-}=v_{a,x}$.

Conditional reference states may change upon selecting a correlated branch. The unconditional reference marginal is unchanged. Reference preservation does not mean absence of conditional steering.

Suppose a second normalized probe vector is installed for each $(a,x)$, with Euclidean distance at most $\delta$ from the first. Keep the same input, output, pinching, and report interfaces. Then

Here $d_{\diamond }(\Phi ,\Psi )=\frac{1}{2}{\Vert \Phi -\Psi \Vert }_{\diamond }$.

Proof. Retain $(a,x)$ as orthogonal flags before common classical postprocessing. The two controlled isometries differ in operator norm by at most $\delta$. For isometries $V,\tilde{V}$, subtraction into two terms gives

Common channel postprocessing contracts the diamond norm, including reference extensions Watrous.

Approximating a real value by a different orthogonal label does not itself give small diamond distance. The proposition approximates normalized amplitudes on one fixed interface.

No finite family of linear scalar diagnostics on ${\mathcal{P}}_{\Sigma }$ determines every admitted functional. In particular, the finite evaluation table does not decide all boundary traces, all wavefront conditions, or convergence of arbitrary families in ${\tau }_{\Sigma }$.

Proof. For $N$ diagnostics choose $N+1$ linearly independent regular scalar linear functionals with disjoint nonzero smearings in $U$. They belong to ${\mathcal{P}}_{\mathrm{reg}}\subset {\mathcal{P}}_{\Sigma }$. The map from their span to ${\mathbb{C}}^N$ has a nonzero kernel. Thus a nonzero admitted functional has all listed readings zero. A finite list cannot replace the defining family of topology coordinates. No universal decision procedure is introduced by storing finitely many values.

A finite table can nevertheless encode the exact separator for a declared finite catalogue. That is a restricted certificate, not a continuum reconstruction.

A record and a surviving quantum interface are different outputs. The following finite comparisons sharpen that distinction around the evaluation channel.

Let $P_z$ be input projections, let $q$ aggregate their fine outcomes, and put

For a channel $\mathcal{T}$ and an output instrument $\{{\mathcal{A}}_w\}$, compare

The record laws agree on every input exactly when

Agreement on a declared recorded residual algebra $\mathcal{O}$ holds exactly when

These identities are stable under untouched references.

Proof. Pair each recorded output block with $I$, or with $B$, and move the maps to the input by adjoints. States separate Hermitian effects. Linear extension gives the residual-algebra identities; reference extension tensors the corresponding maps with identity.

If ${\mathcal{A}}_w(Y)=Q_{w}Y{Q}_w$ and ${\mathcal{T}}^{\ast }(Q_w)=R_w$ for projections $Q_w,R_w$, then

Agreement with the aggregated fine residual instrument further requires, and is equivalent to,

for the declared residual observables.

Proof. Write ${\mathcal{T}}^{\ast }(B)=V^{†}(B\otimes I)V$ with $V$ an isometry obtained from a Kraus family. Expansion shows

Thus $(Q_w\otimes I)V=V{R}_w$, giving the first identity. Insert it into Proposition prop:compatibility; multiplication by distinct $P_z,P_{z^{\prime }}$ proves the remaining equivalence.

Two instruments can have identical classical records on every input while their residual states have trace distance $1/2$ on one qubit input. Their residual source expectations can therefore differ.

Proof. Use one coarse outcome and compare identity with computational dephasing. Both record channels are constant. On $|+⟩⟨+|$, their residuals are $|+⟩⟨+|$ and $I/2$. The difference is $X/2$, with eigenvalues $1/2,-1/2$. The trace distance is $1/2$, and the observable $X$ has expectations one and zero.