V: Admissible currents, retarded response, and finite Enter readout A local adaptation on a prescribed conical spacetime

We develop a local spacetime-Enter adaptation of the fixed-time scalar-response problem of Manuscript V. The central analytical finding is an exact separation between two obstructions: the complete distributional Ward defect controls raw compact-current representability, whereas its causal propagation controls representability after a harmonic prescription. Initial-surface terms are retained explicitly. A compact-Cauchy residual wave makes the obstruction nonzero on an open dense family of smearings, while compact curvature currents nevertheless interpolate the same response on every declared two-dimensional transverse-traceless plane whose separating region is admitted.

A separate calculation constructs the first retarded coefficient of the compactly switched minimal scalar vertex. Its external-metric version is linear in the scalar field, has coefficientwise scaling degree below four, and has no minimal-extension ambiguity. The one-oscillator version is a mixed quadratic operator with a proved essentially self-adjoint real polynomial realization. A distinct Newton–wave composition has a symmetric cubic Wick form with a joint ultraviolet, transfer, and initial-edge cutoff limit. A regular-region witness distinguishes states agreeing on all polynomial forms of degree at most two.

The operational contribution is an exact, reference-preserving finite instrument obtained by compressing the represented responses and cubic forms before destructive clock readout. For the cubic witness this gives an explicit qubit experiment. This connects admissibility and form-level distinctions to a complete finite readout without identifying them with a continuum field apparatus. Its potential value is methodological: it prevents classical descent, operator representation, and recorded response from being conflated.

The cone, external schedule, flat continuation, vacuum, and Newton prescription are declared inputs. No global conical Green theorem, intrinsic Newton inverse, derived spacetime, Einstein dynamics, or inherited characteristic quantum coefficient is claimed.

of the adaptation{ The positive release is local Grade S in the explicit sense of Definition def:grade. It is not a global intrinsic reformulation of every Fourier or Newton construction on the cone. The analytical results are proved on a declared flat presentation; their admitted local restrictions and finite operational interfaces are placed in the prescribed conical laboratory.}

The problem has three logically distinct stages:

A result at one stage does not construct the next.

The analytical mechanisms studied here originate in Cheung's Manuscript V CheungV. We reproduce the required arguments with explicit domains and attach a finite operational interface. The claims of this adaptation are not assertions that the underlying analytical findings are new relative to that source.

The five analytical objects remain separate:

1. a fixed-time classical response, with the scalar held fixed;

2. its restriction to a specified finite-dimensional TT plane;

3. a first retarded scalar coefficient with an external metric;

4. a first retarded coefficient with one metric oscillator;

5. a Newton–wave cubic scalar form.

The last object is not obtained by identifying it with either retarded coefficient.

In the Enter calculus, destructive readout removes the measured quantum names from the live quantum context. A classical record and unmeasured quantum factors may survive CheungEnter,CheungPhilosophy,CheungEn09. We use the explicitly specified finite Born instrument below; we do not rederive Cheung's characterization theorem from abbreviated axiom names.

The distinctions are important:

• a classical record is not a surviving clock;

• agreement of outcome probabilities is weaker than agreement of complete residual instruments;

• a finite encoding is not automatically a field apparatus;

• a supplied causal geometry is not reconstructed by assigning locations to quantum names.

The geometric setting follows I-B CheungIB. The operational proofs use complete channels, including untouched references, in the usual finite-dimensional sense Watrous.

A local Grade S construction specifies:

1. a prescribed spacetime and regular laboratory;

2. the analytical equations, continuation prescriptions, and domains used to define the responses;

3. an explicit finite response encoding and complete clock instrument;

4. time, rod, and causal dictionaries for the admitted events;

5. a reference-preserving finite transport protocol and a separate resource ledger.

No clause requires the geometry or equations to be inferred from records.

Fix $c>0$, $0<\mu <2\pi$, and $\alpha =1-\mu /(2\pi )$. Prescribe

where $r>0$ and $\varphi \in \mathbb{R}/(2\pi \mathbb{Z})$. The missing apex gives the worldsheet

in the metric completion. It is not a worldline. Signature, lapse, future orientation, and deficit are inputs.

The regular product (eq:cone) has a noncompact causal diamond.

Proof. Fix $r_0>0$, one angular coordinate, and $z=0$. Take events $p,q$ at that spatial point with times $-2r_0/c$ and $2r_0/c$. For $r_n\downarrow 0$, the events $(0,r_n,\varphi ,0)$ lie in $J^{+}(p)\cap J^{-}(q)$, by radial timelike travel. They have no convergent subsequence in $M$, because the apex is excluded.

Thus a global wave theorem cannot be transferred merely from local flatness. The standard global Green theory requires an appropriate globally hyperbolic domain BGP.

Choose a regular developing chart

and a reference time $T_0$. Write

The local metric becomes

The analytical presentation is the auxiliary Minkowski space

Its Cauchy evolutions, Fourier transforms, vacuum representation, and Newton inverse are specified parts of the model.

A finite admitted packet consists of finitely many response settings, compact measurement and curvature-test supports, their finite-time causal shadows, and a finite operational schedule. These compact regions must lie inside

with a positive margin from its boundary. They are interpreted in the conical laboratory through the developing chart.

Noncompact auxiliary waves and Newton potentials are restricted to that laboratory; they are not extended by zero across its boundary and are not claimed to define global conical solutions.

For a fixed packet, the laboratory can be chosen after the compact regions have been selected. Regular conical charts of any prescribed finite size can be installed sufficiently far from the apex.

All analytical spaces below refer to $\hat{M}$. Their restriction supplies one declared model on the cone, not a classification of arbitrary conical fields.

A developed straight displacement with

is future causal. At a static port, $\mathrm{d}\tau =\mathrm{d}T$.

Proof. The tangent inequality is $-c^2{\dot{T}}^2+|\dot{x}{|}^2\le 0$. The proper-time statement follows by setting $\dot{x}=0$.

The Newton part will retain nonlocal initial-data dependence. The finite schedule locality theorem will not be applied to that elliptic construction.

Fix $m>0$, and set

For a covector ${\xi }_{\nu }$, define

For a symmetric tensor $h$, write

Trace reversal is an involution in four dimensions.

The linearized Einstein and scalar-response operators are

The following identities hold:

The operator $G^{(1)}$ is formally self-adjoint.

Proof. In $DK\xi$, the derivatives of ${\partial }_{\mu }{\xi }^{\mu }$ cancel. Substitution in (eq:Einstein) proves $G^{(1)}K=0$, and taking a divergence proves the Bianchi identity. Twice applying the product rule gives

which is $-L_{K\xi }\varphi$. Integration by parts in the constant-coefficient expression (eq:Einstein) proves formal self-adjointness.

Let $G_{m,R},G_{m,A}$ be the scalar Green operators for $P$. Their spatial Fourier time kernels are

For componentwise massless waves write $G_{r,R},G_{r,A}$, where $r$ indicates tensor rank, and use

Its time kernel is $S_w(t)=\sin (t|D|)/|D|$.

These formulas define the inverses on compact distributions by transposition. Smooth Cauchy solutions follow from Fourier evolution. Finite propagation follows from the wave energy identity. On finite time intervals, all these maps are continuous in the relevant compact-data smooth topologies.

Fix a real smooth free scalar

with compact smooth Cauchy data, and fix $f\in {C_c}^{\infty }((0,b)\times {\mathbb{R}}^3;\mathbb{R})$. Define the response by

and

The scalar is held fixed in this metric functional.

Put

Then $Pu=-f$, and self-adjointness of the spatial sine propagator gives

The relevant source region is compact:

For every smooth $h$,

where

This is a compact order-zero tensor distribution. It has the form

with

Proof. Integrate the metric derivative in $⟨V,L_h\varphi ⟩$ by parts. The terms containing $V{h}^{\mu \nu }{\partial }_{\mu }{\partial }_{\nu }\varphi$ cancel, leaving (eq:Jraw). A cutoff equal to one near $K_f$ extends the calculation from compact to arbitrary smooth $h$.

Since

the singular contribution is exactly (eq:surface). The remaining density is bounded and piecewise smooth on a compact set.

Pair vector distributions against covectors and define

Direct differentiation gives $K^{\ast }{D}^{\ast }a=\square a$.

Let $H=K^{\ast }{J}_{\mathrm{raw}}$. Then

where, componentwise,

For every smooth descriptor,

Proof. For a scalar distribution $v$, the product rule gives

using $P\varphi =0$. Also

Multiplying the last term by ${\partial }^{\nu }\varphi (t)$ and using $c(t){\delta }^{\prime }=c(0){\delta }^{\prime }-\dot{c}(0)\delta$ proves (eq:H)–(eq:Hab). Equation (eq:gaugedefect) is compact integration by parts.

The initial-data content can also be read directly.

Let

Then

Proof. Both sides have zero Cauchy data and solve $Pq=P(X_{\xi }\varphi )=-L_{K\xi }\varphi$. Cauchy uniqueness applies.

Thus omitting the initial-surface current or the homogeneous subtraction changes the fixed-time observable. These are the mechanisms identified in V@1 CheungV; they are not introduced by the clock interface.

A smooth field on $\hat{M}$ is spacelike compact if its support is contained in $J_{\eta }(C)$ for some compact $C$. Define

and

A functional on a named metric space $\mathcal{V}$ is representable if one $J\in \mathcal{E}$ satisfies

For the local conical interpretation, the support of $J$ and the response packet must satisfy Definition def:presentation.

These currents have the test-tensor type used by Fewster and Hunt FH. We do not use a radical-exhaustion theorem.

For a compact distribution $C$,

Indeed, the common support is contained in the compact causal diamond between the compact source and itself. The converse follows from the Green identities on compact distributions.

The causal propagations of $J_{\mathrm{raw}}$ and $H$ are smooth. If $E_{1}H=0$, the causal propagation of $J_{\mathrm{raw}}-D^{\ast }{G}_{1,A}H$ is smooth as well.

Proof. The sources are finite sums of slice distributions with smooth compact spatial coefficients and piecewise smooth densities on a bounded time interval. For a density $c(s)$, propagation is

All output-time derivatives act on the smooth Cauchy propagator; spatial derivatives commute with it. On compact time intervals, Cauchy estimates control every derivative. A slice derivative gives a derivative of $S_w(t-s_{\ast })c_{\ast }$, also smooth.

If $E_{1}H=0$, its advanced potential is compact by (eq:compactprimitive). It is piecewise smooth with smooth spatial coefficients, so the same argument applies after $D^{\ast }$.

Let $C$ be a compact symmetric contravariant distribution with $K^{\ast }C=0$. Suppose

is smooth. For a smooth temporal cutoff $\chi$, zero in the remote past and one in the remote future, define

Then $J_C\in \mathcal{E}$ and

Proof. Conservation of $C$ implies

Consequently $G^{(1)}(\chi q)$ is supported in the transition slab of $\chi$. Spacelike compactness makes this support compact. The Bianchi identity proves conservation.

First take $Dh=0$. Set

Both $F,d$ are compact, and

The second term pairs to zero with $h$, because ${\partial }^{\mu }(\mathsf{r}h{)}_{\mu \nu }=0$.

Retarded and advanced uniqueness give

Hence $E_{2}F=-E_2{C}^{♭♭}$. By (eq:compactprimitive), $F+C^{♭♭}=\square A_2$ for compact $A_2$. Since $\square h=0$, it follows that $⟨-F,h⟩=⟨C,h⟩$.

For general $h\in \mathcal{S}$, solve $\square w_0=Dh$ with zero data at $0$. Then $h-K{w}_0\in {\mathcal{S}}_{\mathrm{dD}}$. Both compact conserved currents annihilate $K{w}_0$, so the equality extends to all $h$.

The trace reversal in the gauge term is retained in this proof; it cannot be dropped before the harmonic pairing.

Fix a time $t_{-}<0$. On

let $w_R=G_{1,R}Dh$. For $h\in \mathcal{S}$, let $w_0$ solve $\square w_0=Dh$ with zero data at $0$. Define

The fixed $t_{-}$ ensures that the finite-time laboratory interpretation has a common lower time bound.

Put

Then

If $\zeta =0$, $A_H$ and $J_{\sec }$ are compact.

The propagated defect has explicit Cauchy data:

These follow by convolving (eq:H) with $S_w$.

For the fixed $f,\varphi$:

1. $B_f(\cdot ,\varphi )$ is representable on $\mathcal{S}$ if and only if $H=0$, equivalently $\gamma =a=b_0=0$.

2. Each of the following is equivalent to $E_{1}H=0$: representability of $B_R$ on ${{\mathcal{S}}_{\mathrm{ret}}}^{-}$; representability of $B_0$ on $\mathcal{S}$; and representability of $B_f{|}_{{\mathcal{S}}_{\mathrm{dD}}}$.

3. The latter condition is exactly the vanishing of the two Cauchy data in (eq:zetadata).

When the conditions hold, representatives are obtained from (eq:smoothing), using $C=J_{\mathrm{raw}}$ in (i) and $C=J_{\sec }$ in (ii).

Proof. Every compact conserved current annihilates every smooth pure gauge. In (i), all compact $K\xi$ belong to $\mathcal{S}$, so representability forces $⟨H,\xi ⟩=0$ for every compact $\xi$. This is $H=0$. The bulk and slice supports, and arbitrary first normal jets of slice tests, separate $\gamma ,a,b_0$. The converse follows from Lemma lem:smooth and Theorem thm:smoothing.

For (ii), residual gauges with $\square \xi =0$ belong to all three tested spaces and are fixed by both prescriptions. They must be annihilated. Such annihilation is equivalent to $E_{1}H=0$: necessity follows by testing $\xi =E_{1}v$ and using the skew-adjointness of $E_1$; sufficiency follows from $H=\square A_H$ with compact $A_H$.

Assume $E_{1}H=0$. Compact integration by parts gives

For $B_0$, use

On $Dh=0$, the correction $D^{\ast }{A}_H$ pairs to zero. Smoothing $J_{\sec }$ now gives one representative for all three named functionals. Finally, homogeneous-wave uniqueness proves (iii).

A conserved section current need not be compact when the propagated defect is nonzero. A harmonic prescription is not a section of the complete gauge quotient: a residual wave satisfies $s_{R}K\xi =s_{0}K\xi =K\xi$.

There are compact smooth scalar and vector Cauchy data and a nonzero residual gauge $K\xi$ for which the three functionals $B_f,B_R,B_0$ agree and are positive for some nonnegative compact $f$. For those fixed data, nonzero evaluation is open and dense in a specified fixed-support Fréchet space of real smearings.

Proof. Choose real $F\in {C_c}^{\infty }({\mathbb{R}}^3)$ with $\beta ={\partial }_{1}F\not\equiv 0$, and solve

with other vector components zero. Then $(K\xi {)}_{01}{|}_0=\beta$, and

The scalar $w$ has zero initial data, so Proposition prop:subtraction gives $S_0(K\xi )\varphi =w$. It is positive in a small open future region. Choose a nonnegative nonzero $f$ supported there. Both prescriptions fix this residual gauge.

Fix a compact $Q⋐(0,b)\times {\mathbb{R}}^3$ whose interior meets that positive region, and let

The functional $f\mapsto \int fw$ is continuous and nonzero. Its kernel is a proper closed hyperplane, with empty interior. Its complement is therefore open and dense. All Cauchy data used above are literally compact.

This proves failure on the named spaces, not a dense-TT nonrepresentability theorem.

Choose independent real waves $v_1=e,v_2=r$ satisfying

with Schwartz Cauchy data and a common open Fourier gap around zero momentum. Put ${\mathcal{V}}_2={\mathrm{span}}_{\mathbb{R}}\{e,r\}$.

Such waves are obtained by applying the smooth TT projector away from momentum zero to compact smooth Fourier data and then performing wave evolution. A nonzero Fourier-gap datum is not compactly supported in position: compact support would make its Fourier transform entire, and an open gap would force it to vanish.

For a spatially Schwartz descriptor, $K\xi$ cannot be a nonzero spatial-TT tensor. Indeed, trace and divergence give ${\partial }_i{\xi }_i=0$ and $\Delta {\xi }_j=0$, hence ${\xi }_j=0$; the temporal-spatial component and decay then give ${\xi }_0=0$.

The actual response coefficients are

The surface term (eq:surface) pairs to zero with a spatial-TT tensor, but that observation alone does not prove conservation.

Define the electric curvature

It annihilates $K\xi$, and $\mathcal{E}(v_a)=-\Delta v_a/2$.

For compact smooth symmetric $C^{ij}$, define

Then

There is $\rho \ge 0$, compactly supported in the open response slab, for which

is positive definite. Define

Then $⟨J_a,v_c⟩={\delta }_{ac}$, and

represents $B_f$ on ${\mathcal{V}}_2$.

Proof. The two curvature fields are independent on the full open slab. Otherwise a nonzero combination $v$ would have $\Delta v=0$ there. Each spatial slice is Schwartz, hence zero, and wave uniqueness would contradict independence.

Consequently finitely many component evaluations separate the two-dimensional coefficient space. Small nonnegative bumps around those points produce a positive definite Gram matrix: its limiting quadratic form is a positive sum of squares of separating evaluations. Fix a sufficiently small choice.

Equation (eq:curvaturecurrent) and matrix inversion give the duality. Evaluating (eq:Jplane) on ${\lambda }_1{v}_1+{\lambda }_2{v}_2$ proves the claim.

For the conical adaptation the selected bumps must fit the laboratory. If the laboratory was fixed earlier, positive Gram rank there is an additional admission condition. Independence on a full auxiliary slab is not silently replaced by independence on an arbitrary small patch.

The finite-plane ambiguity is exactly

On compact distributions,

is an idempotent projection onto $\mathrm{Ann}({\mathcal{V}}_2)$. Thus the interpolation problem has codimension two. It says nothing about uniform bounds or compatibility as the tested plane grows. This reproduces the finite-plane mechanism of V@1, not a dense-space extension CheungV.

Let

Use the auxiliary Minkowski vacuum and field

For a real compact smooth $q$,

where $\Phi (v)=a(v)+a^{†}(v)$, with annihilation antilinear. The map $\mathsf{K}$ takes compact smooth tests continuously to Schwartz vectors.

Let

For real $g,f\in {C_c}^{\infty }(\hat{M})$ and a real smooth external tensor $h$, define

The perturbative convention is interaction $+\lambda \mathcal{V}$ and

For $H_{\mu \nu }=g{h}_{\mu \nu }$, put

It is formally self-adjoint and

The two-insertion normalization is fixed by vacuum Wick expansion and the scalar Feynman distribution

where

In these conventions,

The insertions are formed with off-shell labels before free-field representation, as in the retarded-product framework DF. We construct only this insertion pair.

Every numerical cross-contraction in this insertion pair has relative scaling degree at most three in four dimensions. Its extension preserving that coefficientwise degree is unique. The compact quadratic vertex preserves ${\mathcal{D}}_{\varphi }$, and the required two-insertion expressions define continuous forms in the finite-sector Schwartz topology.

Proof. There is only one scalar field at the second insertion. The cross-contractions are therefore $W_m$, ${W_m}^F$, or one first derivative of either distribution. No double cross-line coefficient occurs.

Rescaling momentum gives $W_m(ϵz)={ϵ}^{-2}{W}_{ϵm}(z)$. After spacetime testing, rapid momentum decay and the locally integrable bound $(|p{|}^2+{\nu }^2{)}^{-1/2}\le |p{|}^{-1}$ prove convergence to the massless distribution as $\nu \downarrow 0$. The same argument on the two time half-spaces applies to ${W_m}^F$. Thus their scaling degrees are two, and a first derivative has degree at most three.

A difference of extensions is supported at the relative origin. Every nonzero such distribution has scaling degree at least four, so it is excluded. This is the subcritical extension mechanism of BF.

For the domain statement, expand the compact quadratic vertex in creation and annihilation operators. Pair kernels contain the Fourier transform of a compact smooth coefficient at a sum of positive energies and decrease rapidly in both momenta. Exchange kernels have polynomial growth along the momentum diagonal and rapid decay in differences; they map Schwartz wavefunctions to Schwartz wavefunctions. Finite particle shifts preserve the algebraic domain.

Completely normal-ordered finite-sector matrix elements are smooth in the insertion coordinates. In the contracted terms, smearing the numerical line against $f$ first produces a smooth function; compact coefficient multiplication and integration by parts leave a compactly smeared linear field. These operations give the stated finite-sector form bounds.

For the stated normalization,

The test $q_{g,h,f}$ is real, smooth, and compact. The unsmeared coefficient is

For spatial-TT $h$,

Proof. The completely normal-ordered cubic block is identical in the time-ordered and ordinary products and cancels in their difference. Contracting the scalar insertion against either differentiated stress factor, and against the trace and mass terms, gives the Euler operator ${\mathsf{Q}}_{gh}$.

Equation (eq:Feynman), followed by the prefactor $i$, gives $-G_{m,R}{\mathsf{Q}}_{gh}\hat{\varphi }$. Smearing with $f$, transposing the Green operator, and using formal self-adjointness yields

The coefficients of ${\mathsf{Q}}_{gh}$ are compact, so the resulting test is compact smooth. The remaining assertions follow by applying $P$ and using the free equation.

The sign in (eq:external) is positive: the minus sign is already contained in $u_f=-G_{m,A}f$. The first retarded coefficient is not the uncancelled three-scalar normal block CheungV.

On a free scalar,

Thus replacing $L_{gh}$ by $g{L}_h$ is generally incorrect.

For a pure gauge,

Consequently,

This formula retains the trace-reversed switched tensor. Constancy of $g$ near $\mathrm{supp}f$ need not eliminate terms in its causal past.

For every nonzero spatial-TT wave $h$, there are real compact $f,g$ and a one-particle Schwartz vector $v$ such that

Proof. Choose $x_{\ast },p_{\ast }$ with $h^{ij}(x_{\ast })p_{\ast i}{p}_{\ast j}\ne 0$. A small momentum bump, with phase centered at $x_{\ast }$, gives a positive-frequency wave $w_v$ for which $\mathrm{Re}(h^{ij}{\partial }_i{\partial }_j{w}_v)$ has one nonzero sign on a small neighborhood.

Take a nonnegative real $\rho$ supported there, $f=P\rho$, and compact $g=1$ near $\mathrm{supp}\rho$. Then $\hat{\varphi }(f)=0$, $u_f=-\rho$, and $q_{g,h,f}=-{\mathsf{Q}}_h\rho$. Integration by parts gives

There is no contradiction: the retarded construction was defined on off-shell insertion labels before representation. It is not a map on the unchanged free test quotient in the second entry.

Let ${\mathcal{H}}_g={\Gamma }_{\mathrm{s}}(\mathbb{C})$, with

This infinite-dimensional oscillator is distinct from the finite clock introduced later.

Set

where ${\mathcal{D}}_g$ is the finite oscillator-number span. Applying the preceding contraction calculation separately to $e,r$ gives

No metric–metric contraction enters this insertion pair. The oscillator metric is not a local relativistic metric field.

Suppose an initial time $t_{\mathrm{in}}$ precedes the supports of $g$ and $f$. Quantizing the classical bilinear

by replacing $q,\pi$ with $Q,\Pi$ gives exactly ${\mathcal{R}}_{g,f}$ on ${\mathcal{D}}_{\mathrm{mix}}$.

Writing $N=N_g+N_{\varphi }$, there is $C<\infty$ with

Both ${\mathcal{R}}_{g,f}$ and, for every real $\lambda$,

are essentially self-adjoint on ${\mathcal{D}}_{\mathrm{mix}}$.

Proof. On free scalar matrix elements, ${\mathsf{Q}}_{ga}\varphi =L_{ga}\varphi$. The source is zero before $t_{\mathrm{in}}$, so

Smearing proves the switched identity, including derivatives of $g$.

Creation-annihilation estimates give

The two factors commute and

This proves the bound. Schwartz mode vectors and finite particle shifts prove invariance of the common core. Reality proves symmetry.

Each polynomial changes total particle number by at most two. For a vector supported in sectors up to $M$,

After division by $k!$, the resulting analytic-vector series converges for a positive range of its parameter. The core is dense, so Nelson's theorem Nelson proves essential self-adjointness.