Landmark X-D Convergence of the completed ramp relational scalar as the regulator width tends to zero

We decide the sharp limit of the completed scalar-only observable of Landmark X-C-v3 for fixed observation test, interaction switch, classical metric coefficients, and plateau gauge algebra. The observation and interaction supports remain separated from the initial surface.

Both completed coefficient tests converge in the smooth topology of one regulator-independent compact test space, with rate $O(\epsilon )$ in every seminorm, and hence in the quotient test topology of the free scalar algebra. The proof integrates the coordinate normal flux before estimating the boundary layer. The limiting positive-order initial positions vanish. The limiting velocities are $-{a_1}^{0\rho }[h]{|}_{\Sigma }$ and $({a_1}^{00}[h]{a_1}^{0\rho }[h]-{A_2}^{0\rho }[h,k]){|}_{\Sigma }$. The quadratic contribution is retained; profile-dependent position moments vanish at order $\epsilon$.

These data identify the sharp action-level transmission problem: continuous coordinates and continuous normal flux. Its smeared linear background kernel is $\theta C^{\ast }({{\mathsf{E}}_0}^{\mathrm{A}}a)$, with a possible jump but no hypersurface delta. It differs from the multiplied-equation prescription with zero positive-order Cauchy velocities.

The fixed-algebra Ward identity passes to the limit. The supplied VII-C hypotheses are copied and audited: its quantum metric block, mixed insertion, normal-topology approximants, and mandatory triple product are absent here. VII-C therefore does not apply and is not contradicted. No new product, quantum metric, or subtraction is constructed. Neither the moving edge witness nor a crude inverse-regulator upper bound is used as the decision.

Figure fig:split-X-D records one view of the same separation. Figure fig:chain-X-D records one view of the same separation. Figure fig:pair-X-D records one view of the same separation. Figure fig:gate-X-D records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-X-D records that separation. The drawing adds no hypothesis.

{Release: Constructed, at fixed input through ${\kappa }^2$}{ Fix $f,g,h,k$ and the plateau algebra as in Section sec:frozen. For one compact set

the completed X-C-v3 tests satisfy

Consequently,

converges coefficientwise in the original quotient test topology.

The explicit limits are built from the sharp action-level transmission coordinates, not the multiplied-equation variant. The fixed-algebra identity passes to the limit. VII-C's named normal-topology extension is not constructed or contradicted.

Not opened: a new $R_n$, a quantum metric, a triangle subtraction, a convolution repair, or joint sharp admission.}

Three endings are available:

1. Constructed. Explicit limiting tests exist in a named topology and identify the limiting prescription.

2. Proved obstructed. For fixed input a named seminorm diverges, or the proposed representative limit leaves the declared test space.

3. Unresolved. The topology, boundary-layer estimate, propagation, or identification is unfinished.

The ending reached is (1).

The question is formal in $\kappa$. It concerns convergence of coefficient tests, not convergence of a numerical perturbation series. A crude ${\epsilon }^{-r}$ upper bound is not a divergence theorem. The moving edge witness is not fixed input. Neither argument is used to decide the limit.

The classical coordinate prescription is copied from X-A.tex XA. The bare insertion representatives and free-field quotient topology are copied from X-B-v3.tex XB. The completed tests, Yang–Feldman recursion, and fixed-regulator Ward identity are copied from X-C-v3.tex XC. Their covariance and product proofs are not repeated.

Let

The quantum scalar is defined on $M$. The slab $M_T=[0,T]\times {\mathbb{R}}^3$ and $\Sigma =\{t=0\}$ belong to the coordinate prescription. Write ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ for the Klein–Gordon Green operators and ${{\mathsf{E}}_0}^{\mathrm{R}},{{\mathsf{E}}_0}^{\mathrm{A}}$ for the massless ones.

The backgrounds $h,k$ are fixed smooth real symmetric tensors on $M$, without field equations or decay assumptions. Only their restrictions to finite causal regions will be used. Set

With $H={\mathrm{tr}}_{\eta }h$, abbreviate

Indices are moved with $\eta$.

Fix

and the same relatively compact plateau $U$ as upstream:

Choose $\delta >0$ such that

Take $0<\epsilon <\delta$. Thus $g=0$ on $0\le t\le \delta$, and the fixed algebra lies above every ramp in this tail of the family.

The frozen ramp is

It is smooth, takes values in $[0,1]$, and is flat at the endpoints.

The coordinate coefficients have zero Cauchy position and velocity at $\Sigma$, and satisfy

These are the coefficient equations of the divergence-form coordinate problem with density

They agree with retarded propagation of the copied coordinate sources, locally by finite propagation.

Set

The bare tests are

They are actual off-shell tests supported in $K$.

The positive Euler coefficients are

They contain $g,h,k$, never ${\sigma }_{\epsilon }$. The notation ${\mathsf{L}}_{\epsilon },{\mathsf{M}}_{\epsilon }$ in the limit plan therefore reduces to the fixed operators ${\mathsf{L}}_{1,g},{\mathsf{L}}_{2,g}$.

The completed tests are

Equivalently,

This is X-C's test-level recursion, not a new map.

The free-field output is

in the scalar algebra ${\mathfrak{A}}_0={\mathcal{A}}_{\mathrm{off}}/{\mathfrak{J}}_0$, where $\hat{\varphi }(P_{m}v)=0$ for compact smooth $v$. The off-shell warning remains: $R_1(V;F_{P_{m}v})=\hat{\varphi }(-\mathsf{L}v)$ need not vanish. Intermediate tests are not replaced by unspecified on-shell representatives.

The operators ${\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$, ${\mathsf{L}}_{2,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$, and ${\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}{\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ are continuous from ${\mathcal{D}}_K(M)$ to ${\mathcal{D}}_C(M)$, where

If $c_{j,\epsilon }\to c_{j,0}$ in ${\mathcal{D}}_K(M)$, then $d_{j,\epsilon }\to d_{j,0}$ in ${\mathcal{D}}_C(M)$, with

For each output seminorm $p_N$, there are an integer $N^{\prime }$ and a constant $C_N$ such that

Proof. The advanced Green operator is continuous on compact smooth tests. Applying the smooth compact coefficients of the Euler operators gives the stated continuous maps and supports BGP (Chapter 3). For the iteration, the intermediate test is supported in $\mathrm{supp}g\cap J^{-}(K)$, and a second causal past remains in $J^{-}(K)$.

The differences are

Continuity on the fixed Fréchet test spaces gives the seminorm estimate. No unproved fixed derivative-loss count for the Green operator is needed.

The convolution coefficient and X-Enter are not inputs. X-Enter's Option A concerns readout of the convolution coefficient alone; the ramp edge integral is not that coefficient. Encoding the limit below would report it, not prove it.

No triangle residue is subtracted. No gravitational action, graviton propagator, or quantum metric is introduced. The scalar-only field content is not enlarged to meet another manuscript's hypotheses.

VII-C is now available and will be audited in Section sec:viic. Its theorem is a comparison, not a premise of the limit proof. Shared vocabulary, a signature dictionary, or a boundary kernel does not transfer its quantum product problem to this sheet.

The fixed-$\epsilon$ Ward identity is an input, but not a limit theorem. It is used only after convergence. Fixed-$\epsilon$ microcausal admission is likewise not a topology and does not decide the limit.

The compact $C$ above is independent of $\epsilon$ and lies above $t=2\delta$. Define

with seminorms

These are ambient smooth derivatives. No topology on a nonsmooth causal boundary is silently substituted.

The linear output topology is the quotient topology of

The inclusion ${\mathcal{D}}_C(M)\to \mathcal{D}(M)$ and the quotient map are continuous. An individual representative seminorm need not descend to the quotient; convergence passes through the continuous quotient map.

The equation subspace is closed:

by the Green-operator exact sequence BGP (Theorem 3.4.7). Hence $Q_m$ is Hausdorff. No additional topology on the full Wick algebra is required for the linear coefficients considered here.

{}1.12

There will generally be no smooth convergence of coordinate velocities on the entire closed slab: the ramp velocities at $t=0$ are zero, while their limiting right-hand values need not be. This is compatible with convergence of the completed tests, whose supports remain above the surface.

The layer contains no interaction vertex. All estimates here concern the coordinate coefficients.

For a fixed compact observation region, its backward causal region in the slab is compact. Multiply $h,k$ by a fixed smooth cutoff equal to one on a neighbourhood of that causal region. Finite propagation, including the nested second coordinate equation, leaves the coordinates on the observation region unchanged.

After this localization the backgrounds and coordinate coefficients have uniformly bounded spatial support on $[0,T]$. We may use spatial Sobolev norms and then return to the original backgrounds. For $\Vert \cdot {\Vert }_r=\Vert \cdot {\Vert }_{H^r({\mathbb{R}}^3)}$, Fourier inversion and Cauchy–Schwarz give

the relevant high-frequency integral converges because $2(N+2)-2N=4>3$.

Let

The sine quotient has its continuous value $t$ at frequency zero.

Suppose

Then

If $F,H$ are uniformly bounded in every spatial smooth seminorm on $0\le t\le \epsilon$, then

in every local spatial smooth seminorm, uniformly on that layer. No time derivative of $F$ is needed for these bounds.

Proof. Integrate the zero-data Duhamel formula once by parts in time. The initial term vanishes because $F(0)=0$.

Fourier multipliers give

Consequently, if $M_r(F)$ and $M_r(H)$ denote the corresponding suprema on the layer,

Spatial Sobolev embedding proves the stated smooth estimates. The bounds integrate the flux rather than differentiate the ramp before estimating.

Define

Spatial indices range from $1$ to $3$. Both normal flux terms vanish initially, and

The first application of Lemma lem:flux gives

Thus $F_{2,\epsilon },H_{2,\epsilon }$ are uniformly bounded in every spatial seminorm, without a bound on ${{\partial }_t}^2{\mathsf{X}}_{1,\epsilon }$. The second application gives

These are also exact normal-current estimates. For ${\mathsf{X}}_{\epsilon }=x+\kappa {\mathsf{X}}_{1,\epsilon }+{\kappa }^2{\mathsf{X}}_{2,\epsilon }$, the positive-order normal currents are

They satisfy ${\partial }_t{\Pi }_{j,\epsilon }=-{\partial }_i{J}_{j,\epsilon }^i$ and start at zero. The spatial currents are uniformly bounded, so their layer integrals are $O(\epsilon )$.

Write

Uniformly for $0\le s\le 1$, in every local spatial seminorm,

The exit positions tend to zero, and the exit velocities tend to

The limiting data depend only on the profile endpoints.

Proof. Replace the smooth coefficients at $\epsilon s$ by their initial traces in (eq:first-layer)–(eq:second-layer). The error is $O(\epsilon )$. Integrating the velocities gives the position formulas.

To display every ramp-derivative contribution, the expanded sources are

The first equation implies

The singular part of the second source is therefore

One quadratic contribution comes from $-{\sigma }_{\epsilon }^{\prime }{a}^{00}{\partial }_t{\mathsf{X}}_{1,\epsilon }$; the other comes from $-{\sigma }_{\epsilon }{a}^{00}{{\partial }_t}^2{\mathsf{X}}_{1,\epsilon }$. Their integrals each give half of the quadratic velocity:

Their sum agrees with the exact flux calculation. Discarding either term gives an incorrect exit velocity.

The moments $I_1(1),I_2(1)$ multiply $\epsilon$ in the exit positions. Other profile-dependent moments similarly occur in remainders, not in (eq:velocities).

The same limiting data follow for any fixed smooth profile with the same constant extensions and endpoint values. The proof does not assert uniformity for profiles themselves varying with $\epsilon$.

Retarded propagation has not erased the layer: the finite quadratic contribution in $v_2$ survives. Profile independence concerns that retained datum, not the absence of all profile-dependent corrections.

Define the right-hand limiting coordinates by

These are ordinary smooth Cauchy problems on $t\ge 0$, locally by finite propagation.

For an explicit statement of the data at $t=\delta$, let ${s_j}^{\rho }$ be the right-hand sources in (eq:limit-cauchy). Then

All layer contributions are included through the derived velocities.

For every spatial Sobolev order $r$,

after the fixed spatial localization. All spacetime derivatives converge at rate $O(\epsilon )$ on compact subsets of $[\delta ,T]\times {\mathbb{R}}^3$. In particular, the ramp data at $t=\delta$ converge to (eq:delta-data).

Proof. For $t\ge \epsilon$, ${\sigma }_{\epsilon }=1$ and every positive-order ramp derivative vanishes. Put $w_j={\mathsf{X}}_{j,\epsilon }-{\mathsf{X}}_{j,0}$. The layer lemma and smoothness of the right-hand solutions give $O(\epsilon )$ Cauchy data for both differences at $t=\epsilon$ in every spatial Sobolev norm.

The first difference satisfies $\square w_1=0$. The homogeneous wave formula bounds $w_1,{\partial }_t{w}_1$ by those data uniformly on the fixed interval. For the second difference,

The source is $O(\epsilon )$ in every spatial Sobolev norm: use the first estimate and ${{\partial }_t}^2{w}_1=\Delta w_1$. The inhomogeneous Duhamel formula gives the second estimate.

Higher time derivatives follow inductively from these equations, using higher spatial norms and derivatives of the fixed smooth coefficients. Sobolev embedding gives the asserted smooth convergence. The constants are independent of the initial time $\epsilon \in (0,\delta )$.

Define

and

They are smooth compact tests on $M$, because $f$ vanishes near $\Sigma$. Define $d_{j,0}$ by (eq:limiting-completed).

For each fixed input satisfying the sheet, $d_{1,0},d_{2,0}\in {\mathcal{D}}_C(M)$, and for every $N$,

Thus $d_{j,\epsilon }\to d_{j,0}$ in ${\mathcal{D}}_C(M)$, and their free-field classes converge in $Q_m$.

Proof. Proposition prop:coordinate-convergence controls every coordinate derivative on a neighbourhood of $K$. The formulas (eq:bare) are finite smooth products and derivatives, multiplied by $f$; their differences are $O(\epsilon )$ in every compact-test seminorm. Lemma lem:reduction gives the same rate for the completed tests. Their supports lie in the fixed compact $C$, and the quotient map is continuous.

The representatives converge before taking equation classes. No quotient cancellation is needed to rescue a divergent representative family.

X-B's crude bound differentiated the ramp before integrating the solution and consequently contained powers of ${\epsilon }^{-1}$. Lemma lem:flux explains why that bound does not describe the present fixed-input limit: ${\partial }_{t}u+F$ is estimated without differentiating $F$. The theorem supplies boundedness and the actual rate on the declared supports.

Let $\theta$ be the characteristic function of $t>0$, and set

Extend the coordinate map by $x$ for $t<0$, and by $x+\kappa {\mathsf{X}}_{1,0}+{\kappa }^2{\mathsf{X}}_{2,0}$ for $t>0$.

The limiting coordinates are the unique formal piecewise smooth continuous coordinates equal to $x$ for $t<0$ and satisfying

distributionally. Their normal flux is continuous across $\Sigma$.

Proof. The flux is a locally integrable piecewise smooth function. Its distributional divergence has a surface delta with coefficient equal to the normal-flux jump.

The negative-side flux is ${{\delta }_0}^{\rho }$. The positive-side initial flux is

Tangential positive-order coordinate derivatives vanish initially because the initial positions vanish. The jump is zero exactly for (eq:velocities). The positive-time bulk equations are (eq:limit-cauchy), so (eq:transmission) holds.

Conversely, continuity and normal-flux matching force the same initial data. Coefficientwise Cauchy uniqueness gives uniqueness.

This argument defines the flux first; it does not multiply a delta by an unspecified discontinuous velocity. The limiting positive-order positions vanish, but the velocities generally do not.

Write

For a compact covector test $a$ supported in the open slab, set $b={{\mathsf{E}}_0}^{\mathrm{A}}a$. X-A's ramp kernel identity and dominated convergence give

The background kernel is therefore

It is smooth on either side, locally bounded, and compactly supported after restriction to $t\ge 0$. It can have a jump, but contains no hypersurface delta.

The multiplied-equation prescription instead solves

If ${\mathsf{H}}_0[u_0,u_1]$ denotes the homogeneous wave solution, then

Thus the prescriptions differ in general, although they can agree for special backgrounds with vanishing relevant trace.

With $n=\mathrm{d}t$, define

The multiplied-equation kernel is

This distinction identifies the candidate; it does not prove joint sharp microcausal admission or define a quantum metric product.

Smooth coordinate convergence through $t=0$ generally fails, as anticipated in the topology sheet. The completed scalar tests nevertheless converge because their supports remain separated from that surface.

The supplied VII-C.tex VIIC proves a continuity obstruction for a particular boundary extension. We now copy its actual assumptions rather than infer them from their labels.

In this subsection only, VII-C uses

Its insertion is

with the sharp zero-Cauchy inverse and unrestricted smooth off-shell configurations.

Its Wick matrix includes a quantum graviton block

and a scalar block $H_{\varphi \varphi }={\Delta }_m^{+}$, with zero cross blocks.

The signature dictionary is ${\eta }_{\mathrm{VII}}=-\eta$, ${\square }_{\mathrm{VII}}=-\square$, and ${\mathsf{E}}_{0,\mathrm{VII}}=-{\mathsf{E}}_0$ for the same source. Raised-index formulas must be recomputed accordingly. This dictionary does not identify either the coordinate prescription or the quantum field content.

The symbols in the next assumption belong to VII-C: ${\mathcal{P}}_{\mathrm{mc}}$ is its multiplet polynomial domain, $\mathcal{A}$ its proposed extension, $b_f$ its degree-$(1,1)$ sharp kernel, and $B_{f,\epsilon }$ its compact-mollified sharp insertion. They are not X-D objects.

A proposed extension consists of a Hausdorff locally convex $\mathbb{C}[[\hslash ]]$-algebra $\mathcal{A}$, with the following properties.

1. ${\mathcal{P}}_{\mathrm{mc}}[[\hslash ]]$ embeds into $\mathcal{A}$, and its product is the fixed Wick product.

2. The algebra admits the sharp insertions $Q_{\Sigma }(B_f)$, with

   $$[Q_{\Sigma }(B_f){]}_{{\hslash }^0}=B_f$$

   as functionals on all smooth configurations.

3. On the degree-$(1,1)$ kernel stages used above, admission is sequentially continuous for the normal topology and agrees with the old inclusion on regular kernels. In particular,

   $$B_{f,\epsilon }⟶Q_{\Sigma }(B_f)in\mathcal{A}.$$

4. Multiplication is hypocontinuous with respect to bounded sets. The field-independent ${\hslash }^3$ coefficient map

   $$a⟼{\mathrm{ev}}_0[{\hslash }^3]a$$

   is continuous and agrees with the usual coefficient extraction on the old polynomial domain.

The cross-references in the original wording refer to VII-C's fixed product and normal-topology stages ${{\mathcal{E}}^{\prime }}_{{\Gamma }_B}(K)$. No such stage is being substituted for ${\mathcal{D}}_C(M)$.

{}1.14

This is failure to match the required construction, not a claim that the scalar Wick product fails continuity. The assumptions are not established as a package on this sheet, and the mandatory objects needed for VII-C's contradiction are absent.

VII-C's named word is

For its mollified sharp kernels, its fully contracted ${\hslash }^3$ coefficient has a nonzero logarithmic term for suitable tests. That theorem is about a quantum multiplet product, not the seminorms of $d_{j,\epsilon }$. X-D neither constructs that word nor subtracts its reflected light-ray coefficient.

VII-C's bulk representation, after integration by parts, and X-A's multiplied-equation kernel carry a boundary delta; X-D's limit kernel does not. This identifies the candidate and does not repair VII-C.

VII-C itself excludes only the named normal-topology extension with the stated admission and product requirements. An algebra omitting $W_g$, or a different construction not meeting those requirements, is outside that conclusion. The present scalar-only limit is such a smaller object. Therefore VII-C does not apply and is not contradicted.

For $F\in {\mathcal{G}}_U$, generated formally by $\kappa \xi +{\kappa }^2\zeta$, $\xi ,\zeta \in {\mathfrak{g}}_U$, the complete metric law is