IX-B-ST: Source-locked triangle calibration on a prescribed conical Enter spacetime Observation descendants, finite calibration records, and reference-preserving transport

We give a local Grade S adaptation of the IX-B sharp-insertion triangle calibration to a causally saturated regular laboratory on the prescribed I-B conical product. The host metric and its worldsheet defect are inputs; the laboratory closure avoids the defect. The covariance blocks, regulator placement, and VII-D parent finite part are supplied comparison data in the developing chart, not an intrinsic conical field theory.

Relative to the stated parent estimate, the calibrated numerical distribution transfers to every fixed observation differential operator $D$ and smooth multiplier $a$, with the quantitative test-order loss $64\to 64+\mathrm{ord}D$, uniformly on bounded coefficient families and through the reflected-diagonal intersection. We retain the eight contraction skeletons, the unique fully contracted triangle, its assignment factor $2$, and the admitted first-stage coefficient with sign $-2$. The residue is a positive measure on a reflected observation relation, not on an observation equality diagonal. Its observation-derivative orbit is infinite-dimensional, although all compatible descendants can inherit one common parent conversion parameter.

The additional contribution is a supported transfer and finite certificate interface: explicit numerical enclosure errors become fixed-interface complete-channel bounds, and an unknown payload together with its classical calibration history admits exact reference-preserving buffered transport entirely in the regular laboratory. This supplies a controlled benchmark for comparing future product constructions without confusing numerical calibration, continuum acquisition, and quantum multiplication. The analytic calibration results are attributed to IX-B and VII-D; their ultraviolet analysis is not re-proved here. Equality of the two regularized parenthesizations constructs neither sharp second multiplication. The VII-C obstruction remains under its original hypotheses. No renormalized interacting product, quantum Ward identity, Einstein dynamics, or spacetime emergence is established.

scope{ This sheet is placed after IX-B and before the IX-C interface in the present adaptation programme. It does not replace IX-B and does not begin IX-C-ST or a unified IX-ST.

The principal analytic release is conditional on the explicitly stated inherited parent estimate. The admission carrier is the already adapted IX-A-ST carrier. Neither is reconstructed by citation or by a finite table.

The host, initial prescription, covariance, regulator profile, and calibration are fixed inputs. Changing a regulator or subtraction-scale label changes comparison data, not the host metric. The quantum certificate remains -open.}

IX-B calibrates a particular numerical coefficient of the ordered word

on the actual-functional admission carrier of IX-A CheungIXB,CheungIXA. Its numerical finite part is inherited from VII-D, while the normal-topology incompatibility is retained from VII-C CheungVIID,CheungVIIC. IX-B does not introduce a conical construction. The purpose of IX-B-ST is to add a prescribed conical host and a complete finite operational interface without changing that analytic target.

The Enter background is an interface discipline. A quantum name denotes declared access to a tensor factor. Destructive readout consumes the measured names and returns a classical record, together with the specified unmeasured residual. It is not a quantum-return decoder for an unknown input. Cheung's operational characterization has its own structural hypotheses; they are not re-proved or enlarged here CheungEnter. The philosophical distinctions between readout, recovery, implementation, and geometric identification are useful precisely because none is inferred merely from another CheungPhilosophy.

In spacetime Enter, a finite clock encoding is calibrated against an already supplied geometry. Consuming that encoding does not consume the geometric time axis, change the event of readout, or derive a metric CheungIB. We use finite Born instruments directly and prove their complete channel identities.

A local Grade S package here consists of:

1. a prescribed spacetime and a supported regular laboratory;

2. explicit classical or numerical objects with stated domains and proof dependencies;

3. complete finite channels, including untouched references;

4. one destructive clock and an explicit calibration contract;

5. proved computational support and separately certified physical causal containment;

6. one nonempty finite transport realization and its resource ledger.

Grade E, understood here as a claim to derive spacetime or its metric from records or quantum data, is not constructed.

proof versus documentary citation. A cited manuscript is a source record, not a blanket axiom. The particular mathematical inputs used below are restated as Assumptions ass:carrier, ass:tensor, and ass:VIID. The conditional conclusions genuinely depend on those assumptions. No claim of an assumption-free reconstruction of the VII-D analysis is made.

boundary. “Neither sharp second multiplication is constructed” means that no such map is defined by this sheet. It does not deny separately versioned later partial-product constructions. IX-C, IX-Free, and IX-Free-2 are documentary successors, not premises, and their releases are not reversed by this adaptation CheungSuccessors.

Use ${\delta }_{\mathrm{c}}$ for the conical deficit and $\mu$ for the numerical subtraction scale. Fix

The host is

The angular coordinate has period $2\pi$. The stipulated completion has defect

This is a worldsheet; only fixed-$z$ sections are worldlines. The signature is $(-,+,+,+)$, the lapse in $T$ is one, and static proper time is $T$. These are I-B inputs, not outputs of the calibration CheungIB.

The regular spacetime in (eq:host) is not globally hyperbolic.

Proof. Fix a radius $r_0>0$, an angle, and an axial coordinate. Choose events at that spatial point and times $T_{\ast }\mp 2r_0/c$. Points at time $T_{\ast }$, the same angle and axial coordinate, and radii $r_n\downarrow 0$, belong to the causal diamond between those events by radial causal curves. They have no convergent subsequence in the regular product. That diamond is noncompact.

Fix a product-adapted isometric developing embedding

with developed metric

Choose $T_0,r_0>0$ and $R>r_0+T_0$, and set

All observation tests are supported in $V$. The compact analytic and operational laboratory is contained in $\iota (U)$, has closure disjoint from $\mathcal{W}$, and contains a neighbourhood of $\iota (K_{\ast })$.

For $f\in \mathcal{D}(V)$, the full truncated developed past

is retained. It is contained in $K_{\ast }$. The response prescription has zero value and zero first $t$-derivative at $t=0$, with no additional incoming homogeneous or lateral-boundary data.

For each fixed compact test family, the permitted positive smoothing widths are small enough that the smoothed field kernels remain inside $U$.

The positive support margin is necessary: a chart containing only the observation points would not determine the prescribed zero-Cauchy response. Such laboratories are nonempty by choosing a regular developed ball and a sufficiently short observation interval, sufficiently far from the defect.

For the supported response already supplied by IX-A-ST, two auxiliary field extensions equal near $K_{\ast }$ give the same response on $V$, with the same smooth derivatives and observation pairings. The comparison covariances and the residue are not thereby constructed as host Green functions.

Proof. The local uniqueness and finite-propagation argument in the IX-A-ST saturation contract applies because every $D_f$ lies inside the region on which the extensions agree IXAST ({:saturation}). Their difference has zero source in the relevant truncated past and zero initial response data, so it vanishes on the observation region. Only this supported Cauchy statement is used; it supplies neither a global inverse on $M_{{\delta }_{\mathrm{c}}}$ nor a conical quantum state. The ordinary local Cauchy theory is distinct from global hyperbolicity BGP.

Replacing a regulator label ${\rho }_a$ by ${\rho }_b$, or a scale label ${\mu }_1$ by ${\mu }_2$, in the comparison catalogue leaves $M_{{\delta }_{\mathrm{c}}},g_{{\delta }_{\mathrm{c}}},\mathcal{W},\Sigma$, the lapse, and the installed causal relation unchanged.

Proof. A comparison command acts on the catalogue and finite programme registers. The host data in (eq:host) are held fixed and are not outputs of that command. No metric-update map is included.

This is the fixed-host distinction also documented in VIII-ST STInterfaces.

Write $x=(t,\mathbf{x})$, $y=(s,\mathbf{y})$, and $x^{\prime }=(t^{\prime },{\mathbf{x}}^{\prime })$, and set

This is a space of observation triples in one supported chart; it is not the spacetime host.

All comparison quantities are expressed in one fixed system of units. In particular $\mu \epsilon$ and the scale ratios inside logarithms are dimensionless.

The inherited local formulae are

Here ${\mathsf{E}}_0$ denotes the supported comparison prescription, not a newly introduced host inverse. Both input fields are smooth and off shell. The source $j=\square h$ is not set to zero.

The pair $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$ is the IX-A-ST actual-functional carrier and topology, with the following inherited properties IXAST,CheungIXA.

1. Its elements are finite polynomial functionals with compact symmetric distributional coefficient kernels. Their admission criterion is

   $$({\mathrm{D}}_{\mathbf{\eta }}F{)}_h^{\prime }(\Psi )\in {\mathcal{S}}_1\text{for every smooth }\Psi \text{ and finite smooth direction list }\mathbf{\eta }.$$

   Here

   $${\mathcal{S}}_1={\mathcal{M}}_1+{\mathcal{Q}}_{\Sigma },{\mathcal{Q}}_{\Sigma }=\{L^{\ast }a:a\in \mathcal{D}(V;T^{\ast }U)\},$$

   and ${\mathcal{M}}_1$ consists of compact linear metric kernels with no nonzero nonspacelike wavefront covectors. The trace of a represented linear derivative is the intrinsic advanced-Cauchy trace defined by that carrier.

2. The topology records strong compact-distribution coefficients and all those trace coordinates. It is Hausdorff. Fixed-direction differentiation and conjugation are continuous. Classical pointwise products stay in ${\mathcal{P}}_{\Sigma }$ and are separately continuous. Joint continuity and hypocontinuity are not assumed.

3. The bulk inclusion is faithful. The functionals $B_f,W_g$ and the selected one-contraction coefficient with $W_g$ belong to ${\mathcal{P}}_{\Sigma }$, with all directional descendants required by admission.

4. For every nonzero real $f$, the unchanged compact-kernel approximants satisfy

   $$B_{f,\epsilon }/⟶B_f\text{in }{\tau }_{\Sigma }.$$

   A normalized continuous boundary separator detects this nonconvergence.

No quantum multiplication on all of ${\mathcal{P}}_{\Sigma }$ is part of this assumption.

This recalls the domain and input interface without reconstructing the carrier, its topology, or its trace proof.

The supplied comparison covariance has blocks

where

and

No positivity of the metric block is assumed.

Retain the VII-C tensor identities and old Wick-symbol convention CheungVIIC ({:tensor}, {:K}):

The nonzero mixed component is

The reverse component is $\overline{K(x,y)}$. The covariance selects the $0$-component at a contraction with $h_{00}$.

On the old bulk polynomial domain, retain

Self-contractions within a Wick symbol are excluded. The usual old-domain operations used at positive cutoff are defined. The unchanged mixed-kernel approximants converge in the fixed-support normal stages of VII-C.

In (eq:K), the infrared powers $P^{-1},P^{-2}$ are locally integrable in three momentum dimensions. The high-frequency terms are understood distributionally. The reflected term is retained. In particular, $H_{ww}=0$ is not the statement $K=0$.

These are prescribed chart comparisons. They are neither derived from the conical metric nor claimed to extend to a global conical two-point function. Functional Wick products and their domain restrictions are reviewed in FR.

Let $\rho \in {C_c}^{\infty }({\mathbb{R}}^4)$ be real, nonnegative, even, Euclidean radial, and normalized by $\int \rho =1$. Use the same $\rho$ on both original field slots of each insertion:

The middle observable $W_g$ is not smoothed.

Use the coordinate Fourier pairing $\hat{\rho }(k)=\int e^{-ik\cdot z}\rho (z){\mathrm{d}}^{4}z$. For $\mathbf{n}\in S^2$, put

Radiality makes ${\vartheta }_{\rho }$ independent of $\mathbf{n}$. Let $K_{\epsilon }$ be (eq:K) with multiplier ${\vartheta }_{\rho }(\epsilon P)$, and set

The exact numerical family under this placement is

Its positive-cutoff smoothness and its identification with the old-domain coefficient are included in the following analytic input.

Let

For compact $A⋐X$, write

Distributional pairings are complex linear in the test.

The prescribed family (eq:triangle) is smooth for positive cutoff and agrees with the fully contracted old coefficient on product tests. There is a regulator-independent distribution $U_m\in {\mathcal{D}}^{\prime }(X)$ such that

where

For every compact $A⋐X$,

The range is $0<\epsilon \le 1/2$, further restricted when required by the supported smoothing margin. The estimate includes compact sets meeting $\mathcal{R}\cap {\Delta }_{13}$, where ${\Delta }_{13}=\{x=x^{\prime }\}$.

For admissible regulators ${\rho }_a,{\rho }_b$,

The regulator conversion is complete within this fixed parent class: no extra normal derivative, nonconstant weight, or independent intersection-supported term occurs. These are the specific VII-D inputs CheungVIID ({:finitepart}, {:conversion}, {:placement}, {:positive}).

No profile integral, ultraviolet sector estimate, reflected intersection estimate, or leading-coefficient ledger is re-proved by this assumption.

A contraction graph is a finite loopless multigraph. Its vertices are insertion symbols, its edges are cross-contractions, and no field slot is used twice. Decorations specify the field species, indices, and derivatives.

Every connected loopless multigraph with at least one and at most three edges is isomorphic to exactly one of:

1. a single edge;

2. two parallel edges;

3. a two-edge path;

4. three parallel edges;

5. a two-edge path with one edge doubled;

6. a triangle;

7. a three-edge path;

8. a three-edge star.

Arbitrary graphs with at most three edges are disjoint unions of these with total edge count at most three, together with isolated vertices.

Proof. One edge forces two vertices. For two edges, connectedness permits either two vertices with a double edge or three vertices forming a path.

For three edges, connectedness permits at most four vertices. With two vertices there is a triple edge. With three vertices, either all vertex pairs are joined or exactly two pairs are joined, with multiplicities two and one. With four vertices the graph is a tree, whose degree sequence is either $(2,2,1,1)$ or $(3,1,1,1)$. These are respectively a path and a star. Distinct listed vertex counts, multiplicities, or degree sequences distinguish the cases. Connected-component decomposition proves the last assertion.

This is the IX-B inventory CheungIXB ({:skeletons}). It is combinatorial and independent of the conical host. It is not an existence theorem for all distributionally decorated versions of those graphs.

For the ordered word $B_{f,\epsilon },W_g,B_{f^{\prime },\epsilon }$, the field-independent order-${\hslash }^3$ coefficient has exactly one graph type: a scalar edge from the first to the third vertex and two metric edges, one from each outer vertex to the middle vertex. Its numerical density is (eq:triangle). The factor $2$ is the middle-slot assignment count. No $ww$-edge occurs.

Proof. Each outer insertion has one scalar and one metric-response slot. The middle insertion has two metric slots. Since scalar–metric covariances vanish, full contraction forces the scalar outer slots to pair with one another. The remaining two outer metric slots must pair with the two middle slots, in two assignments.

Pairing the two outer metric slots instead would leave the middle slots unmatched; that edge would also have covariance zero by Assumption ass:tensor. The locked tensor identity selects index $0$. The scalar derivative edge gives $D_{\epsilon }$, and the ordered metric edges give $K_{\epsilon }$ and its reverse conjugate. The two defining insertion minus signs multiply to $+1$. The assigned placement therefore gives (eq:triangle).

There are three single-edge placements and three choices of a two-edge placement in this word. This identifies placements, not their sharp composition domains CheungIXB ({:unique}).

The map $\Psi$ in (eq:residue) is an embedding proper over compact subsets of $X$. The distribution $r$ is a positive locally finite measure of order zero, with support exactly $\mathcal{R}$. For product tests,

where

uses zero extension of $f$ outside $V$.

Proof. The observation times recover $t,s,t^{\prime }$. On the image,

Since $t+s>0$, these are smooth inverse formulae. They also prove immersion and injectivity.

On the inverse image of a compact subset of $X$, all times are bounded and separated from zero, $\mathbf{z}$ is bounded, and $\mathbf{n}$ lies in the compact sphere. The inverse image is closed in the resulting compact parameter region, proving properness there.

The parameter density is positive, so its pushforward is a positive locally finite measure. Every neighbourhood of an image point has inverse image of positive parameter measure; hence its support is the whole image. Changing $\mathbf{z}$ to $\mathbf{y}=\mathbf{z}-s\mathbf{n}$ in the defining integral gives (eq:productresidue).

The reflected relation has dimension eight and codimension four in $X$. It is not an observation equality diagonal. Moreover,

and

where that intersection is present.

Near a point of $\mathcal{R}$, set

Then

In particular, no $\delta (t-t^{\prime })$ occurs.

Proof. The parameter space has dimension $3+2+3=8$. The differences

prove all diagonal statements. Equality $x=x^{\prime }$ is exactly the one additional relation $t=t^{\prime }$, when restricted to $\mathcal{R}$.

For the density, the spatial change to $(\mathbf{y},\mathbf{x}-\mathbf{y},\mathbf{w})$, at fixed times, has determinant of absolute value one. Write $\mathbf{x}-\mathbf{y}=(a+\sigma )\mathbf{n}$, producing the polar factor $(a+\sigma {)}^2$. The replacement of $\mathbf{y}$ by $\mathbf{z}$ is triangular with unit determinant. The time block is unchanged. The inverse coordinate formulae are smooth near the relation. Restricting the density to $\sigma =\mathbf{w}=0$ and comparing with $\mathrm{d}\nu$ gives the coefficient $c_0{a}^{-2}$.

The density and its normalization agree with the supplied VII-D comparison CheungVIID ({:residue}). The positivity argument is made in parameter measure, not by assigning a sign to a formal delta expression.

There exist nonnegative $f,g,f^{\prime }\in \mathcal{D}(V;\mathbb{R})$ with pairwise disjoint supports such that

For those tests,

No family of counterterms supported on ${\Delta }_{12}\cup {\Delta }_{23}\cup {\Delta }_{13}$ removes this divergence on all joint tests. The same conclusion holds for observation faces avoided by the chosen compact supports.

Proof. Choose

and one unit vector ${\mathbf{n}}_0$. The relation contains the triple

All three points lie in $V$ and are distinct. Take nonnegative bumps with mutually disjoint sufficiently small supports, positive near these points. Their product is positive on an open parameter neighbourhood of this triple. Lemma lem:measure gives the positive residue pairing.

Assumption ass:VIID gives the logarithmic term plus a finite distributional pairing and a vanishing remainder. The contribution $-\log \mu \mathcal{C}$ is constant at fixed $\mu$, proving (eq:log).

The product-test support avoids every observation equality diagonal. Every distribution supported on their union pairs to zero with it, even when the distribution depends on $\epsilon$. Such counterterms cannot cancel the nonzero logarithm.

This retains the VII-C separated-test obstruction locally CheungVIIC ({:log}, {:separated}). It does not exclude the supplied subtraction on $\mathcal{R}$, and it is not a theorem about the conical worldsheet or a curvature singularity.

Let

be a fixed differential operator on $X$, with smooth coefficients, and let $a\in C^{\infty }(X)$. The ordered operation is

Its transpose, relative to the displayed coordinate density, is

There is no coefficient conjugation in this transpose.

For $\Phi \in {\mathcal{D}}_A(X)$,

with a combinatorial constant depending only on dimension and $d$. Differential operators and multiplication do not enlarge the test support.

Proof. Apply every derivative of order at most $64$ to the finite sum defining $D^{\mathrm{t}}(a\Phi )$, then use the product rule on $b_{\beta }a\Phi$. The total number of derivatives distributed among the three factors is at most $64+|\beta |\le 64+d$. Taking suprema and summing the finitely many multinomial coefficients gives (eq:leibniz).

Under Assumption ass:VIID,

More precisely,

where $C_{A,a,D}$ can be chosen as the coefficient expression in (eq:leibniz). The estimate is uniform on families with those coefficient seminorms bounded. It includes compact sets crossing $\mathcal{R}\cap {\Delta }_{13}$.

Proof. Apply (eq:parentestimate) to $D^{\mathrm{t}}(a\Phi )$, whose support remains in $A$, and use Lemma lem:leibniz. A bounded subset of $\mathcal{D}(X)$ has a common compact support and bounded smooth seminorms of every fixed order. Thus the right side tends to zero uniformly on each bounded test set, which is the asserted strong convergence.

The parent estimate already holds at the reflected-diagonal intersection. Applying a support-preserving transpose does not remove that intersection or introduce a new exclusion. No new ultraviolet argument is required.

This is the local transport of the IX-B derivative theorem CheungIXB ({:descendants}), not a re-proof of VII-D's ultraviolet analysis.

For the same operation,

The last convergence has the summed order-$64+d$ remainder bound of the two parent estimates.

Proof. The first two identities follow by applying the linear operation $aD$ to (eq:parentconversion)–(eq:parentscale). Subtracting the two instances of (eq:descendantrate) proves the last assertion, since the leading logarithmic terms agree.

The logarithm in (eq:descconversion) is a numerical conversion on supplied chart distributions. It is not a running of the host curvature.

Suppose that, after fixing smooth configurations $\Psi =(h,\varphi )$, a selected numerical coefficient is exactly

with $a[\Psi ]\in C^{\infty }(X)$. If $\Psi \mapsto a[\Psi ]$ is continuous in the compact smooth seminorms through order $64+d$, then its numerical finite part is

Convergence is uniform on configuration sets for which those multiplier seminorms are bounded.

Proof. Insert $a=a[\Psi ]$ in Theorem thm:descendants and use its uniform coefficient bound.

Finite products of smooth field jets satisfy the multiplier condition under the corresponding compact smooth bounds. Smooth Cauchy responses may be used when their required smooth seminorm bounds are supplied. This rule does not multiply by an additional singular spectator distribution, classify arbitrary decorated graphs, or prove carrier admission of every resulting functional.

An observation derivative such as ${\partial }_t$ uses the time coordinate of the developing chart. Thus its coordinate vector is related to ${\partial }_T$ by the fixed calibration. Nevertheless, differentiating a kernel argument is not evolving the host metric, implementing a clock operation, or constructing quantum evolution without external scheduling. This is the precise meaning of the time-axis separation.

On the supported observation space $X$, the distributions

are linearly independent over $\mathbb{C}$. No finite-dimensional subspace containing $r$ is invariant under $-{\partial }_t$.

Proof. Let

On $\mathcal{R}$,

The measure $r$ is nonzero by Theorem thm:diagonal.

Suppose a finite relation ${{\sum }_{j=0}}^N{c}_j{r}_j=0$ has $c_N\ne 0$. For $N\ge 1$, pair the relation with $F^N\Phi$. When $j<N$, every term in ${{\partial }_t}^j(F^N\Phi )$ still contains a factor $F$ and vanishes on $\mathcal{R}$. At order $N$, its restriction is

The relation therefore gives

A nonnegative compact test positive near a point of $\mathcal{R}$ detects this nonzero measure, since $({\partial }_{t}F{)}^N$ has constant nonzero sign there. This is a contradiction. The $N=0$ case follows from $r\ne 0$.

An invariant finite-dimensional space containing $r$ would contain every $r_n$, contradicting independence.

This carries the IX-B orbit statement to the regular chart CheungIXB ({:independence}). It is not a renormalization-group trajectory of the host.

Fix $\rho ,\mu$ and define

If observation compatibility is imposed by the actual operation $aD$, the compatible descendant family is

with the same $c$ for every declared derivative and multiplier. An infinite-dimensional residue orbit therefore does not imply one independent normalization constant per derivative.

Proof. This is the linear image under $aD$ of the one-parameter affine parent family. Independently choosing a new constant for a derivative would change, rather than follow from, the stipulated compatibility identity.

If arbitrary real smooth weights were admitted, the space

would be infinite-dimensional, even among first–third exchange-symmetric weights. This proposition does not declare those weights admissible counterterms.

Proof. Consider $a_n=(t+t^{\prime }{)}^n$, which are exchange symmetric. If $P(t+t^{\prime })r=0$ for a polynomial $P$, positivity of the parameter measure implies that $P(t+t^{\prime })$ vanishes throughout every open parameter patch of $Z_U$. There is such a patch by the explicit interior witness above, and $t+t^{\prime }$ varies over an open interval on that patch. A polynomial vanishing on an open interval is identically zero. Thus the weighted distributions are linearly independent.

The class $\mathbb{R}r$ is the complete parent regulator conversion class in Assumption ass:VIID. It is not the span of all derivatives and is not a classification of arbitrary weighted or decorated extensions.

In the normal coordinates of (eq:density), $r$ has transverse scaling degree four. A nonzero pure normal derivative of order $k$ has degree $4+k$. This follows from the dilation of a delta distribution in four normal variables. It is not the test-order loss $64+d$, and it does not restrict arbitrary tangential coefficient functions to a finite-dimensional space.

For every positive cutoff allowed by the supported prescription, the field-independent order-${\hslash }^3$ coefficients of

are equal and have common value

After the single common subtraction their numerical limit is

The corresponding statement on arbitrary compact joint tests is the strong distributional limit in Assumption ass:VIID.

Proof. Use independent fields at the three vertices. The cross-contraction operators ${\mathcal{P}}_{12},{\mathcal{P}}_{13},{\mathcal{P}}_{23}$ have field-independent coefficients and commute on the positive-cutoff old-domain expressions. Both parenthesizations consequently give the finite polynomial expansion of

followed by field identification. The old products in this calculation are defined under Assumption ass:tensor. Proposition prop:unique identifies their scalar order-three graph and multiplicity. The asserted limit is the parent input. Product-test density identifies the same joint numerical distribution; no independent graph is assigned to one parenthesization.

The theorem carries the IX-B regularized identity CheungIXB ({:regularassociation}). It concerns a schedule of old-domain evaluations, not a physical causal past.

Subtracting $\log ((\mu \epsilon {)}^{-1})r$ from a positive-cutoff coefficient changes a coefficient of an already fixed old product. It is a numerical comparison prescription, not an unchanged old Wick product.

Conversely, assigning only a finite value at the sharp kernel cannot make the unchanged divergent regular sequence sequentially continuous. The finite part does not remove VII-C's incompatibility CheungVIIC,CheungVIID.

The selected one-contraction output for the ordered pair $(B_f,W_g)$ is

It is an actual compact-distribution bilinear functional and belongs to ${\mathcal{P}}_{\Sigma }$. Thus

is a well-defined specified first-stage output. This does not define a quantum product on the full carrier.

Proof. Existence and full carrier admission of the selected contraction output are the input in Assumption ass:carrier, inherited from IX-A-ST. Multiplying by fixed smooth compact tests preserves its compact coefficient support.

For its normalization, differentiation of $W_g$ supplies the factor $2$. The tensor identity in Assumption ass:tensor selects $K(x,y){\partial }_t\varphi (x)$. There is one defining minus sign from $B_f$, giving $-2$. Classical closure supplies $B_f{W}_g$, so the stated formal first-stage polynomial is well defined.

The normalization agrees with IX-B CheungIXB ({:partial}, {:Qpartial}). The positive assignment factor of the full triangle and the negative first-stage coefficient are consistent: the full triangle has two insertion minus signs.

The parent assignment $U_c$, the admitted $Q_{f,g}$, and Theorem thm:regularassociation do not define either

where ${\star }_{\Sigma }$ is only a placeholder for a further construction. In particular, the trilinear numerical assignment

is not by itself a binary map on actual intermediate carrier elements.

Proof. The arguments of $U_c$ are observation tests. The arguments of the missing second-stage map would instead include actual functionals such as $Q_{f,g}$ and $B_f{W}_g$. The parent distribution gives no value on arbitrary such intermediate elements or their finite relations.

A binary factorization must prove descent from all test presentations to actual functionals, admission of all nonconstant outputs and their directional descendants, compatible overlap with old products, and the required coherence identities. None of these maps is defined by the scalar pairing (eq:Nc). Equality of two regularized scalar limits therefore cannot supply either missing composition.

This is a boundary of the present construction, not a theorem excluding every possible binary factorization. The next IX-C interface asks for both locked first stages, actual-functional descent, and coefficientwise associativity conditions. No result from a later IX sheet is used to pass that interface here CheungSuccessors.

Let $\tau (x,y,x^{\prime })=(x^{\prime },y,x)$, and define

The explicit formulae give

Indeed $D_{\epsilon }(-z)=\overline{D_{\epsilon }(z)}$, and exchange swaps the two mixed factors. The strong finite-part limit then gives

For $\mathcal{O}=aD$, put

Then

An $x^0$-derivative is exchanged with an $x^{\prime 0}$-derivative; a descendant need not itself be exchange invariant. These are distributional symmetry identities, not a quantum algebra involution theorem.

The topologies and operations in the following table are different.

There is no sequentially continuous extension of the unchanged regular scalar coefficient to the sharp pair in the normal kernel stages of Assumption ass:tensor, agreeing with that coefficient on regular kernels.

Consequently the VII-C algebra-extension requirements cannot all hold simultaneously: unchanged old-product agreement, exact sharp admission with the specified normal admission continuity, bounded-set hypocontinuous quantum multiplication, and continuous scalar coefficient extraction.

Proof. Choose the separated witness in Theorem thm:diagonal. Normal convergence of the two insertion kernels is part of the inherited VII-C input. Sequential continuity of a scalar extension would make its values on that convergent pair converge. Equation (eq:log) contradicts this.

For the algebra consequence, hypocontinuity gives separate continuity and continuity along convergent pairs of sequences: split a product difference into two terms and use uniformity over the bounded convergent family in the other factor. Apply this first with the fixed middle factor, then with the other convergent insertion. Continuous scalar extraction would force convergence of the same divergent coefficient.

The proposition retains the original quantifiers of VII-C CheungVIIC ({:extension}, {:noalgebra}). It is not a universal obstruction to all carrier topologies or all partial products. No global comparison between every normal topology and ${\tau }_{\Sigma }$ is asserted. The distinction between separate continuity, hypocontinuity, and other distribution topologies matters BDH.

Fix one operation $aD$, one compact test $\Phi$, and one catalogue choice $(\rho ,\mu )$. Set

Suppose supplied numerical approximants ${\hat{A}}_{\epsilon },\hat{b}$ obey

Then

For $\epsilon =2^{-n}$, the analytic term is bounded by the same constant times $2^{-n}(1+n\log 2)p_{A,64+d}(\Phi )$. It tends to zero.

Proof. Subtract $z$, insert the two exact pairings, apply the triangle inequality, and use Theorem thm:descendants. The dyadic statement is substitution.

The theorem is an error-propagation statement, not an executed quadrature or a general computability theorem for arbitrary smooth inputs. The constants and numerical enclosures used by an actual finite certificate must be supplied and justified.

Choose a finite catalogue with classical index $i\in \mathcal{I}$. An entry records:

• the test and source-description identifiers;

• the regulator $\rho$, scale $\mu$, and cutoff;

• the observation operator $D$ and smooth weight $a$;

• the selected signs and slot multiplicities;

• supplied pairings or certified enclosures and the remainder bound in (eq:certificateerror).

Only finitely many choices occur in one experiment. An arbitrary smooth function or real parameter is not claimed to have been acquired or exactly encoded by finitely many bits.

A finite report word $r(k,i)$ stores the installed data selected by a clock code $k$. It may include symbolic identifiers and rational interval endpoints. Its meaning is a certificate about supplied comparison data, not an observation of an unknown field or of the residue itself.

Let $C$ have dimension $2^m$, with orthonormal calibration basis $\{|{\tau }_k⟩\}$. Let $S$ contain all surviving factors and let $E$ be any untouched reference. Put

The calibrated clock readout is

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

It returns subnormalized conditional blocks and consumes the quantum clock names. The geometric coordinates and metric are not consumed.

Proof. The recorded Kraus operators are $|k{⟩}_R\otimes D_k$. Their squared moduli sum to $I_{CS}$. Expanding $X$ in the calibration basis proves the partial-trace expression and the record-discard identity. Tensoring these identities with $I_E$ proves the complete statements. The declared destructive output interface has no quantum $C$-factor, whereas the host is fixed metadata.

This is the I-B instrument interface, not a field measurement CheungIB ({:instrument}). The complete conditioning distinction is also documented in En09-A and the spacetime observation sheets CheungEn09,STInterfaces.

If a fixed readout event $e$ has distinct labels $t_k$ satisfying

at most one label has positive probability. Nontrivial multi-outcome programme or history readout therefore does not assert that all labels equal the time of one event.

The clock channel has no universal left inverse on its full input space.

Proof. Distinct labels prove the first assertion. For the second, the two clock states $(|{\tau }_0⟩\pm |{\tau }_1⟩)/\sqrt{2}$, tensored with the same payload, give identical destructive outputs. A universal left inverse could not recover both.

We impose no stationarity or covariance condition for an unscheduled relational evolution interpretation. External scheduling remains.

Let $G$ be the catalogue register, padded to $2^g$ labels. Its intended inputs are classical in the basis $\{|i⟩\}$. The canonical full-matrix extension pinches that basis. Let $P$ be an arbitrary quantum payload.

For each $(k,i)$, choose an installed real score $v_{ki}\in [-1,1]$, and put

Define