X-D-ST: The sharp-width limit of the completed scalar tests Normal-flux transmission and fixed-support convergence on the inherited X-A-ST/X-C-ST host

We establish the sharp-width limit of the completed scalar-linear observable of X-C-ST on the supported host already fixed by X-A-ST. The calculation takes place on the declared Minkowski completion; the conical host, its worldsheet, signature, and future relation remain prescribed inputs.

For fixed smooth classical backgrounds, a fixed interaction switch, and observation and interaction supports separated from the initial surface, both completed coefficient tests converge at rate $O(\epsilon )$ in every seminorm of one regulator-independent compact-test space. Their classes therefore converge in the original free-scalar quotient topology. The proof first integrates the coordinate normal flux and then estimates the shrinking layer. It derives zero limiting positive-order positions and the velocities

The quadratic velocity survives, whereas profile-dependent position moments vanish. A conformal example shows that the $O(\epsilon )$ rate cannot generally be improved to $o(\epsilon )$.

The limit is the continuous-coordinate, continuous-normal-flux transmission prescription. Its observation-smeared linear background kernel can jump at the initial surface but contains no hypersurface delta. It is generally different from the unchanged zero-velocity multiplied-equation prescription. The completed background identity passes to the limit on the gauge algebra frozen before the calculation.

The principal limit theorem is credited to Cheung's X-D. The contribution here is its proved supported transfer, an explicit topology and defect audit, and a finite complete-channel encoding of supplied evaluations. Its prospective usefulness is a reproducible sharp scalar baseline that does not confuse test convergence with quantum metric admission or characteristic composition. No time-ordered hierarchy, loop regulator, continuum-field acquisition, or spacetime-emergence construction is supplied.

limit, not another dynamical completion{ X-B-ST supplies one first-vertex scalar map. X-C-ST supplies one second-vertex map and one iterated first-vertex coefficient. Their sum represents the second scalar response once.

X-D-ST takes the sharp-width limit of the resulting completed tests. It does not add another copy of that response, define another $R_2$, or install a product hierarchy. The metric coefficients remain classical.}

The analytical source is Cheung's X-D, version 3, Convergence of the completed ramp relational scalar as the regulator width tends to zero XD. The predecessor presentation is supplied by X-A-ST, X-B-ST, and X-C-ST XAST,XBST,XCST. The present manuscript states the data needed for the limit and proves its layer, propagation, transmission, and topology assertions. The parent limit theorem is not assumed as a premise.

The contribution is not a new priority claim for the parent limit. It is a supported realization of that result with explicit completion-normal data, preserved counting, and narrowly stated operational consequences.

The identifier X-D-ST is used here for this proposed manuscript. No external registration, submission, acceptance, or registry update is asserted.

Enter distinguishes destructive classical readout from quantum return. The measured quantum names are consumed from the returned quantum interface; the declared output contains a classical record and the retained residual EnterI,EnterPhilosophy.

The philosophy relevant here is the separation of:

1. a mathematically defined evaluation from a measurement that acquires unknown field data;

2. a complete channel from its record marginal;

3. a channel identity from a thermodynamic implementation;

4. a finite operational model on a supplied spacetime from a derivation of that spacetime.

The spacetime Enter model I-B supplies the fixed-host clock and transport interpretation IB. It does not supply the sharp-width limit.

Only ordinary finite Born instruments are used below. Their complete channel identities are proved directly. No abbreviated reconstruction of the Enter characterization theorem is attempted.

Retain exactly the X-A-ST/X-C-ST host instance:

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

2. the supported injective developing chart

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

   whose retained compact reserve stays off $\mathcal{W}$;

3. the separately declared analytical completion

   $$\hat{M}={\mathbb{R}}^{1+3},\eta =\mathrm{diag}(1,-1,-1,-1),M_{T_{\ast }}=[0,T_{\ast }]\times {\mathbb{R}}^3,\Sigma =\{t=0\};$$

4. the inherited signature and time ledger

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

   The host keeps signature $(-,+,+,+)$, while the analytical equations use $(+,-,-,-)$;

5. a retained observation region $\mathcal{O}⋐(0,T_{\ast })\times {\mathbb{R}}^3$, with

   $$D_{\ast }:=J_{\eta }^{-}(\overline{\mathcal{O}})\cap M_{T_{\ast }}⋐\mathcal{L}.$$

   Compact laboratory tests are transported to $\hat{M}$ and extended by zero.

The regular conical product is not globally hyperbolic. All global Green operators and the free scalar field used below belong to $\hat{M}$, not to that regular product.

The non-global-hyperbolicity diagnosis and the chart contract are retained from X-A-ST. No conical geometry is reconstructed. The worldsheet is not replaced by a worldline, and no global conical inverse is introduced.

For a laboratory test $f$, define

The supported calculations below use only $D_f$, or a fixed slightly larger retained shadow when neighbourhood seminorms are needed.

A local Grade S release consists of:

1. the inherited regular supported host;

2. explicitly defined limiting scalar tests, topology, normalization, and background identity;

3. a finite installed dictionary of supplied evaluations;

4. a complete destructive-clock instrument, including untouched references;

5. scheduled locality and a nonempty regular, reference-preserving route under declared finite operational permissions.

No metric emergence, Einstein equation, or Grade E requirement is included.

Let $m\ge 0$, and fix smooth real symmetric tensors $h,k$ on a neighbourhood of the closed slab. They have no imposed field equations or spatial decay. Set

The coupling is formal. The limit concerns its coefficient tests, not a numerical perturbation series.

Fix

Retain the same nonempty relatively compact open plateau $U$ and algebra

The group ${\mathcal{G}}_p$ is generated through order ${\kappa }^2$ by flows of $\kappa \xi +{\kappa }^2\zeta$, with $\xi ,\zeta \in {\mathfrak{g}}_p$.

Choose $\delta >0$, once and for all, such that

Only $0<\epsilon <\delta$ will be considered. Thus the same group lies above every ramp in this tail of the family. It is not enlarged as the width shrinks.

Use the predecessor profile

The proof also applies to any other fixed smooth profile with the same constant extensions and endpoint values. Constants may depend on that profile. No uniform theorem for arbitrary $\epsilon$-dependent profiles is asserted.

Indices are raised with $\eta$. Put $H={\eta }^{\mu \nu }{h}_{\mu \nu }$, and abbreviate

These are the coefficients of

Only formal density coefficients are used; a globally nondegenerate numerical metric at finite $\kappa$ is not required.

Let

The inherited divergence-form prescription is

with flat coordinate Cauchy data at $t=0$. Equivalently,

Here

Existence and uniqueness follow successively from the smooth flat Cauchy problem, locally by finite propagation when the backgrounds do not decay BGP. The switch $g$ does not occur in (eq:coordinates).

The Klein–Gordon operator and its positive Green operators are

with

for compact smooth tests. Their supports are respectively advanced and retarded, and their test-to-smooth maps are continuous BGP.

The regulator-independent Euler coefficients are

In particular, $L_1,L_2$ do not depend on $\epsilon$. Their action-Hessian signs are $-L_1,-L_2$, as fixed in X-B-ST/X-C-ST.

Set

and define the actual bare representatives

They are supported in $K$.

The completed tests inherited from X-C-ST are

Write

Then

This formula recalls the existing test recursion. It does not reconstruct its response maps. In particular, $-L_2{\mathsf{E}}_{\mathrm{A}}f+L_1{\mathsf{E}}_{\mathrm{A}}{L}_1{\mathsf{E}}_{\mathrm{A}}f$ is the existing second-order pair, counted once.

All intermediate insertions remain actual off-shell tests. Replacing a supplied insertion by its free class before applying a response rule is not allowed:

need not vanish XBST,XCST.

Let

This compact set is independent of the ramp width and lies above $t=2\delta$.

For a compact $C\subset \hat{M}$, let

The derivatives are ambient smooth derivatives. No smooth structure is assigned to the possibly nonsmooth boundary of $C$.

The inherited degree-one free-field sector has topology

It is the linear sector of the scalar free algebra ${\mathfrak{A}}_0$. Its equation relation is $\mathbf{\varphi }(P_{m}a)=0$, and its commutator is

No topology on a newly enlarged interacting algebra is introduced FR.

The subspace $P_m\mathcal{D}$ is closed in $\mathcal{D}$. Consequently $Q_m$ is Hausdorff. Convergence in ${\mathcal{D}}_C$ passes through the continuous quotient map.

Proof. The continuous map ${\mathsf{E}}_{\mathrm{A}}-{\mathsf{E}}_{\mathrm{R}}:\mathcal{D}\to C^{\infty }(\hat{M})$ has kernel $P_m\mathcal{D}$. One inclusion follows from (eq:green). Conversely, if ${\mathsf{E}}_{\mathrm{A}}d={\mathsf{E}}_{\mathrm{R}}d=:u$, then

a compact causal diamond. Thus $u\in \mathcal{D}$ and $P_{m}u=d$. The kernel is closed, and the remaining assertions follow from the quotient topology.

An individual representative seminorm $p_N$ need not descend to the quotient. The theorem below proves the rate for representatives and then uses the continuous quotient map.

For tests supported in $K$, the operators

are continuous into ${\mathcal{D}}_C$, with the causal support specified by (eq:C).

If $c_{j,\epsilon }\to c_{j,0}$ in ${\mathcal{D}}_K$, then the corresponding completed tests converge in ${\mathcal{D}}_C$. More precisely,

For every $N$, there exist $M_N$ and $A_N$ such that

Proof. Advanced propagation sends a compact smooth test to a smooth function supported in its causal past. Each $L_j$ has compact smooth coefficients supported in $\mathrm{supp}g$. For the iteration, the second causal past remains in $J_{\eta }^{-}(K)$. Green continuity, multiplication, and differentiation give continuous maps between the stated fixed-support Fréchet spaces. Continuity supplies some finite seminorm loss; no universal derivative-loss count is assumed. Subtraction of (eq:completed) gives (eq:reduction).

Choose a compact neighbourhood of $K$ still inside the retained observation region. Its truncated causal past is compact. Multiply $h,k$ by one fixed spatial cutoff equal to one near that entire past.

Finite propagation leaves $w_{\epsilon },v_{\epsilon }$ unchanged on the observation neighbourhood, including their nested second-order dependence. After this localization all sources and solutions have uniformly bounded spatial support on the fixed slab.

We may therefore use

and return afterwards to the original smooth, not necessarily decaying backgrounds. For spatial smooth control we use

This follows directly from Fourier inversion and Cauchy–Schwarz, since $N+2>N+3/2$.

Define the free spatial wave operators

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

Suppose

For smooth spatially localized data,

If

and likewise for $H$, then, for $0\le t\le \epsilon$,

No time derivative of $F$ occurs in these bounds.

Proof. The zero-data wave formula is

Integrating its first integral by parts, using $\mathsf{S}(0)=0$, $F(0)=0$, and ${\partial }_s\mathsf{S}(t-s)=-\mathsf{C}(t-s)$, gives the first line of (eq:flux-form). Differentiation gives the second line.

The Fourier multiplier bounds are

and

Integrating them proves (eq:flux-bounds).

This elementary estimate is the decisive ordering of operations: integrate the normal derivative before estimating the layer. It avoids replacing a bounded integrated effect by a crude inverse-width upper bound. The parent argument is attributed to X-D XD (:flux).

For $\rho =0,1,2,3$, define

Spatial indices run from $1$ to $3$. Both equations (eq:coordinates) have the form in Lemma lem:flux, and each $F_{j,\epsilon }(0)$ vanishes.

For every spatial Sobolev order, uniformly on $0\le t\le \epsilon$,

The corresponding local spatial smooth estimates also hold.

Proof. The first flux pair is bounded in every spatial Sobolev norm independently of $\epsilon$. Lemma lem:flux therefore gives $w_{\epsilon }=O(\epsilon )$ and ${\partial }_t{w}_{\epsilon }=-{\sigma }_{\epsilon }{a}^{0\cdot }+O(\epsilon )$.

These bounds, in sufficiently high spatial orders, make both $F_{2,\epsilon }$ and $H_{2,\epsilon }$ uniformly bounded in any prescribed spatial Sobolev order. In particular, no estimate of ${{\partial }_t}^2{w}_{\epsilon }$ is needed to establish their boundedness.

Applying the lemma again gives $v_{\epsilon }=O(\epsilon )$ and ${\partial }_t{v}_{\epsilon }=-F_{2,\epsilon }+O(\epsilon )$. Substituting the first-coordinate estimates in (eq:fluxes) gives the final expression. Sobolev embedding gives the local spatial smooth version.

The positive-order normal currents are

They obey conservation equations

where

Their layer integrals are $O(\epsilon )$. These are completion-normal currents, not host evolution equations.

Write

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

The limiting positive-order initial positions are zero, and the exit velocities converge to

These limiting data depend on the profile endpoints, not its interior moments.

Proof. The fixed smooth coefficient difference $a(\epsilon s)-a_{\Sigma }$, and similarly for $A$, is $O(\epsilon )$ in every localized spatial norm. Use this in (eq:layer-bounds). The estimates in (eq:flux-bounds) also apply with the current $t$ in place of $\epsilon$; their remainders are $O(t)$. Integrating the velocities from zero therefore gives the position expansions with $O({\epsilon }^2)$ remainders.

At $s=1$, $\chi (1)=1$. The profile moments are multiplied by $\epsilon$ in the positions, whereas the velocities approach (eq:velocities).

The quadratic term in ${\mathsf{v}}_2$ must not be halved. For a direct audit, the expanded second source contains both

The first equation implies

Consequently the singular part of the second source is

Since

the two quadratic contributions sum to the full term in (eq:velocities). This audit is consistent with, but not a substitute for, the integrated flux estimate.

Define $w_0,v_0$, smooth on the right-hand closed slab, by

Their data are determined by Lemma lem:profile, not chosen independently.

The ordinary flat Cauchy problem constructs these solutions successively. For example, if ${s_j}^{\rho }$ denotes the corresponding right-hand source and $u_1=w_0,u_2=v_0$, then

Thus no initial-layer datum is silently lost before propagation to the observation region.

After the fixed spatial localization, for every integer $r\ge 0$,

Every spacetime derivative converges at rate $O(\epsilon )$ on fixed compact subsets of $(0,T_{\ast }]\times {\mathbb{R}}^3$.

Proof. For $t\ge \epsilon$, the ramp is identically one. Set

The exit expansions and smoothness of the right-hand solutions give $O(\epsilon )$ position and velocity data for both $z_j$ at $t=\epsilon$, in every spatial Sobolev order.

The first difference satisfies

For a free wave with initial time $s$, Fourier multipliers give, on a fixed bounded time interval,

Indeed, $\mathsf{C}$ is bounded in each Sobolev norm, $\mathsf{S}:H^r\to H^{r+1}$ is uniformly bounded on bounded time intervals, and ${\mathsf{C}}^{\prime }:H^{r+1}\to H^r$ is uniformly bounded. The constant is independent of $s=\epsilon$.

This proves the estimate for $z_1$, using higher initial norms where needed. The second equation is

Its source is $O(\epsilon )$ in every spatial Sobolev order. Terms with ${{\partial }_t}^2{z}_1$ are controlled by ${{\partial }_t}^2{z}_1=\Delta z_1$; all coefficient derivatives belong to the fixed smooth backgrounds. Equation (eq:wave-energy) proves the estimate for $z_2$.

Higher time derivatives follow inductively from these equations, using higher spatial norms and the fixed coefficient derivatives. Sobolev embedding gives spacetime smooth convergence away from $t=0$. Finally finite propagation removes the auxiliary spatial localization on the retained region.

There is generally no smooth convergence of coordinate velocities on the whole closed slab: the finite-width velocities vanish at $t=0$, while the limiting right-hand velocities need not vanish.

Set

and define

The products with $f$ extend smoothly by zero to the completion, because $f$ is supported away from the initial surface.

For each fixed input satisfying (eq:separation),

For every $N\ge 0$, there is a constant $A_N$ and a sufficiently small positive width bound such that

Consequently,

The limiting tests are independent of the choice of fixed ramp profile with the stated endpoints.

Proof. Proposition prop:propagation controls every coordinate derivative on a fixed neighbourhood of $K$. The first bare test is a finite derivative of $f{w}_{\epsilon }$. For the second, write

Uniform boundedness of the coordinate derivatives and their $O(\epsilon )$ differences gives the same rate for every derivative of $B_{\epsilon }-B_0$. The Leibniz rule applied to (eq:bare) proves the bare-test estimates, with support in $K$.

Lemma lem:reduction proves the completed-test estimates and their support in $C$. The inclusion ${\mathcal{D}}_C\to \mathcal{D}$ and quotient map $\mathcal{D}\to Q_m$ are continuous.

The limiting data (eq:velocities) do not depend on interior profile moments. Cauchy uniqueness in (eq:limit-cauchy), followed by (eq:limiting-tests), proves profile independence.

This reproduces the fixed-input conclusion of X-D XD (:convergence); no parent convergence theorem was used in its proof.

The representative tests converge before taking equation classes. No quotient cancellation rescues a divergent representative family. For every continuous seminorm $q$ on $Q_m$, continuity of ${\mathcal{D}}_C\to Q_m$ also gives

This is not an operator-norm estimate in a Hilbert representation or a channel estimate for a field detector.

The coordinate coefficients on $K$, the limiting tests, and their compact pairings depend only on the backgrounds near $D_f$, with the same initial data. Extensions on the declared flat completion agreeing there give the same supported result.

Transport through $\iota$ preserves these compact pairings and their convergence. No global conical propagator or gluing around the worldsheet is obtained.

Proof. The coordinate equations are solved successively by flat finite propagation. Every nested retarded dependence ending in $K$ remains in $D_f$. The limiting initial data are the traces of those same local coefficients on the relevant initial shadow.

For the response terms, all contributing interaction vertices lie in $\mathrm{supp}g\cap J_{\eta }^{-}(K)$. A second advanced test action remains in that causal past. The same local coefficients and fixed completion Green operators therefore give the same tests.

Distribution transport through a chart is defined by transposition on tests and densities. It rewrites the retained pairing. On a fixed compact reserve, the chart and its inverse have bounded derivatives of every required finite order, so the associated fixed-support test topologies are equivalent.

Let $\theta (t)=1_{\{t>0\}}$, and put

Extend the limiting coordinate map by

Its positive-order positions are continuous at $\Sigma$.

The map $X_{\mathrm{sh}}$ is the unique formal piecewise smooth continuous map equal to $x$ for $t<0$ and satisfying

as a distribution, where the flux is formed first as a locally integrable piecewise smooth function. Its completion-normal flux is continuous across $\Sigma$.

Proof. The derivative of a continuous piecewise smooth coordinate function is a piecewise smooth locally integrable function; there is no coordinate-jump delta in that first derivative. Thus the displayed flux is defined coefficientwise by ordinary multiplication of piecewise smooth functions.

Its divergence has a surface delta whose coefficient is the jump of its normal component. On the negative side that component is ${\eta }^{0\rho }={{\delta }^{\rho }}_0$. On the positive side its initial value is

Tangential positive-order derivatives vanish initially, because the positive-order positions are identically zero on $\Sigma$. The jump vanishes exactly for (eq:velocities). The bulk equations are (eq:limit-cauchy), proving (eq:transmission).

Conversely, continuity forces the zero positive-order initial positions. Flux matching determines the two velocities successively. The smooth right-hand Cauchy problems then have unique solutions.

The normal here is the declared completion covector $n=\mathrm{d}t$. These velocities are coordinate-coefficient data, not a newly derived host lapse or a measured observer velocity. The host time conversion remains the inherited one.

The flux-first interpretation is important: it does not multiply a surface delta by an unspecified discontinuous velocity.

Define

Parentheses include the factor $1/2$.

For a compact smooth covector test ${\ell }_{\rho }$ supported in the open slab, set $b_{\rho }={\mathsf{E}}_{0,\mathrm{A}}{\ell }_{\rho }$. Then

Thus the observation-smeared linear background kernel is

It is compactly supported, smooth on each side of $\Sigma$, and locally bounded. It can jump at $\Sigma$, but it has no hypersurface-delta summand.

The finite-width kernels satisfy