III-B: Classical relational descent in a conical Enter laboratory A local Grade S adaptation with destructive clock readout

We construct a local operational adaptation of the classical descent problem of Manuscript III on the prescribed conical spacetime of I-B. A necessary domain correction is made first: the full regular conical product is not globally hyperbolic. Our evolution and quotient results therefore concern an explicitly supported finite-propagation laboratory inside a regular developing chart, with no incoming boundary data and no interaction with the singular worldsheet.

Within this laboratory we prove the exhibited Poisson-symbol obstruction, the local Killing classification of massive free-shell tangency, first-order jet descent with a past-trivial harmonic section, and a Hausdorff well-posed quotient for the vacuum-metric triangular model. We construct compact coupled Ward pairs with strictly positive on-shell value and distinguish them from an adjoint-exact family that vanishes on shell.

A complete destructive clock instrument is attached through explicit time, rod, and causal dictionaries. We prove its reference-preserving identity and construct a finite evaluation encoding of the descended observables. An explicit three-qubit pilot realizes four compact Ward evaluations with integer values $j(k+1)$, $0\le j\le 7$, $0\le k\le 3$. Target-adapted SWAP transport and causal classical record wires give an exact finite endpoint-channel certificate.

The release is Grade S under the definition given here. The geometry, equations, support restrictions, and finite encoding are inputs. No global conical Cauchy theorem, continuum field instrument, sourced Einstein–scalar completion, derived metric, or inherited quantum coefficient is constructed.

result and its two levels{

The second line does not quantize the first. A register encoding of selected evaluations is not a representation of the entire field quotient.}

correction{ The complete punctured conical spacetime is not assumed to possess a global retarded inverse or a global Cauchy isomorphism. All positive evolution and quotient theorems below have the laboratory support contract of Definition def:laboratory.}

A relational expression must first be a function of the equivalence class that it is intended to describe. A measurement interface cannot repair a failure of that set-theoretic requirement. Conversely, classical descent alone does not construct an operator, a quantum field apparatus, or a record.

Manuscript III separates four classical equation-space questions CheungIII. I-B supplies a finite experiment on a prescribed conical background CheungIB. This paper gives a precise local composition of those structures. Its new interface is deliberately finite: selected values of the proved classical functionals are encoded in finite matrices and read by a complete instrument.

The classical results are reproved for the declared laboratory rather than imported as global statements about the cone. The finite instrument results are also proved directly. Companion citations identify origins and comparisons; no unwritten companion theorem is a premise.

Cheung's Enter calculus treats destructive readout as an operation that consumes the measured quantum names and returns a classical record, with the surviving quantum context explicitly typed CheungEnter,CheungI. The accompanying philosophy distinguishes readout from quantum recovery, a channel from a thermal machine, and an analogy from an identification CheungPhilosophy.

We adopt finite Born and projective-update semantics. We do not claim a new derivation of those semantics outside the structural class of the Enter characterization.

In particular:

1. a consumed clock register is not a destroyed coordinate axis;

2. a classical time label is not a surviving quantum clock;

3. a residual register carrying evaluation data is not automatically a scalar quantum field;

4. a causal schedule on a supplied metric is not a derivation of that metric.

Finite instruments and reference extensions are understood in the usual completely positive framework DaviesLewis,Watrous. Every instrument identity needed for the release is established below.

A local $\mathsf{Grade}\mathsf{S}$ adaptation consists of:

1. a prescribed Lorentzian background, local rods, time orientation, and explicit analytical domain;

2. named classical equations, gauge classes, topologies, and proved descent statements on that domain;

3. a finite named quantum experiment with a complete destructive clock instrument and explicit calibration dictionaries;

4. a declared finite encoding of selected descended evaluations, with a complete instrument proof;

5. a reference-preserving transport and resource certificate whose complete dependencies are causally embedded.

This definition does not require geometry, field equations, or the encoding to be inferred from records.

The phrase $\mathsf{Grade}\mathsf{E}$ will refer only to the stronger, unconstructed emergence programme described in Section sec:emergence. Neither grade is an external accreditation.

The second row is not a quotient by arbitrary compact gauges: the proposed transformations generally fail to preserve its equation space.

The notation ${\mathsf{C}}_1$ names a model. It does not mean that the fields have only $C^1$ regularity. All classical fields in this paper are smooth.

Fix

Let

and prescribe

The metric completion has singular worldsheet

where $o$ is the cone apex. We use only the smooth regular part. The background and its unit lapse are supplied data, as in I-B CheungIB.

The regular product $(M,\bar{g})$ is not globally hyperbolic. Consequently, local flatness alone does not justify applying a global globally-hyperbolic Green-operator theorem to all of $M$.

Proof. Fix $r_0>0$, one angular coordinate, and $z=0$. Take

For any sequence $r_n\downarrow 0$, the events

belong to $J^{+}(p)\cap J^{-}(q)$. Radial travel has length $r_0-r_n<r_0$, while each available time interval is $2r_0/c$, so radial timelike routes exist.

The sequence has no convergent subsequence in $M$, because its only possible radial limit is the excluded apex. Thus this causal diamond is not compact. Global hyperbolicity requires compact causal diamonds.

A global theory could supply an additional extension or boundary prescription at the singularity. None is selected here. Standard global wave theorems explicitly require an appropriate globally hyperbolic domain BGP.

Choose a regular spatial neighborhood $U$ with a developing chart

whose image is a Euclidean ball. Locally it is obtained from

followed by a rigid Euclidean coordinate change and restriction.

Fix a reference time $T_0$ and put

Then

Choose $a,r_{\ast }>0$ with

The physical laboratory is

For analysis use the auxiliary flat slab

and the support tube

A supported laboratory field is a smooth field on $\hat{\mathcal{L}}$ supported in ${\mathcal{C}}_{r_{\ast }}$. It is interpreted on $I\times B_R$ through (eq:flat-chart), and is zero outside the spatial laboratory on the conical time slab.

All Cauchy data below are supported in ${\overline{B}}_{r_{\ast }}$. Newton potentials are computed on the auxiliary ${\mathbb{R}}^3$ from those declared compact sources. They are not asserted to be inverses of the global conical Laplacian.

The strict margin in (eq:margin) gives a uniform zero collar. Thus a supported field extends smoothly by zero from the chart to the whole regular conical time slab. Differential equations satisfied in the chart remain satisfied across that collar.

Conversely, uniqueness below is asserted only in the declared supported class. It is not uniqueness among arbitrary conical solutions with unspecified incoming or singularity data.

In the developing chart:

1. a future-causal displacement satisfies

   $$\Delta T>0,|\Delta x|\le c\Delta T;$$

2. a straight segment with this inequality is future causal;

3. a static observer has $\mathrm{d}\tau =\mathrm{d}T$.

Proof. The tangent inequality is $-c^2{\dot{T}}^2+|\dot{x}{|}^2\le 0$, with the future orientation $\dot{T}>0$. Apply it to a constant-velocity segment. For a static curve, $\bar{g}=-c^2\mathrm{d}{T}^2$.

Only containment in the supplied causal relation is needed. No global equality between an auxiliary chart past and the full conical causal past is assumed.

Fix $m>0$, and define on the auxiliary slab

Here $m$ has the inverse-length normalization appropriate to $s=c(T-T_0)$.

For a covector ${\xi }_{\mu }$, set

For symmetric $h$, put

and

Symmetrization includes the factor $1/2$.

For smooth fields,

and

Also,

Proof. In $DK\xi$, the derivatives of ${\partial }_{\mu }{\xi }^{\mu }$ cancel. Substitution into (eq:Glin) gives $G^{(1)}K=0$. Taking a divergence in (eq:Glin) cancels the wave and divergence terms pairwise.

Apply the product rule twice to $\square ({\xi }^{\nu }{\partial }_{\nu }u)$; the mass term commutes with $X_{\xi }$. Substitution of $K\xi$ into (eq:Lh) gives the negative commutator.

Finally use

in the divergence formula for the scalar wave operator.

These calculations fix the signs in all subsequent gauge actions.

For $\nu \ge 0$, smooth data supported in ${\overline{B}}_{r_{\ast }}$, and smooth forcing supported in ${\mathcal{C}}_{r_{\ast }}$, the equation

has a unique smooth solution on $\hat{\mathcal{L}}$. It is supported in ${\mathcal{C}}_{r_{\ast }}$. Evolution is continuous in the compact-data smooth topology and the compact-open smooth topology on each closed subinterval of $I$.

If $F=0$ for $s\le 0$, its zero-data solution is zero there.

Proof. Let ${\Omega }_{\nu }=(-\Delta +{\nu }^2{)}^{1/2}$. Spatial Fourier transformation gives

The sine quotient has its continuous value at zero frequency. On compact time intervals the source has uniformly compact spatial support, so the formula and its differentiated versions are well defined in Sobolev spaces of every order.

For integer $n\ge 0$, an energy containing the $L^2$ norms of ${\partial }_s{\partial }^{\beta }u$, $\nabla {\partial }^{\beta }u$, and ${\partial }^{\beta }u$, $|\beta |\le n$, satisfies

in the forward direction, with the corresponding backward estimate. Gronwall's inequality gives uniqueness and continuous dependence. Sobolev embedding and the equation give smooth control.

The local energy identity outside the expanding initial ball has nonnegative outward characteristic flux. The source bound $|x|\le r_{\ast }+|\sigma |$ therefore keeps the solution in ${\mathcal{C}}_{r_{\ast }}$, in either time direction. Zero forcing and zero data on the past side give zero there.

This is a flat supported proof, consistent with the general smooth wave framework BGP. No global conical Green operator is invoked.

Let ${\mathcal{S}}_g$ consist of supported laboratory tensors satisfying $G^{(1)}(h)=0$, and let ${\mathcal{S}}_{\varphi }$ be the solutions of $P\varphi =0$ with smooth data supported in ${\overline{B}}_{r_{\ast }}$. Let

This is a linear subspace of ${\mathcal{S}}_g$. Define the unchanged product map

No topological quotient theorem for this row is needed.

For a compact smooth spatial source $f$, define

Then $\Delta \mathcal{N}f=f$, and $\mathcal{N}\Delta f=f$ for compact smooth $f$. For the latter assertion, the difference is harmonic and decays at infinity.

Apply $\mathcal{N}$ at each time and put

The potentials may have spatial tails. They are evaluated only in the specified chart functional.

For $\xi \in {C_c}^{\infty }(I\times B_{r_{\ast }};T^{\ast })$,

Proof. Direct differentiation gives

Use Newton inversion, ${\xi }^i={\xi }_i$, and ${\xi }^0=-{\xi }_0$.

For real $f\in {C_c}^{\infty }(\mathcal{L})$, define

where $\lambda$ is a real parameter.

There exist fixed real $f,\varphi ,\xi$ in the preceding classes such that

and

For $\lambda \ne 0$, this symbol does not descend through $q_0$. Its first-order coefficient also does not descend.

Proof. Choose a bump $\zeta$ supported in $B_{r_{\ast }}$, equal to one near $0$, and scalar data

Then ${\partial }_1\varphi (0,0)=1$. On a small relatively compact neighborhood of that event, ${\partial }_1\varphi >0$. Choose nonnegative nonzero $f$ supported there and a compact vector ${\xi }^{♯}$ equal to ${\partial }_1$ near $\mathrm{supp}f$. Thus $J>0$.

The metric representatives $0,K\xi$ are equivalent and both solve the vacuum metric equation. Their scalar labels agree. Lemma lem:poisson gives the displayed gap, which no function of their common quotient label can reproduce.

This is the local form of the exhibited-symbol obstruction in III CheungIIIProduct. It is not a statement about every scalar functional. For example, a metric-independent scalar functional already descends through $q_0$.

The elementary two-representative bound gives, for every candidate $G$ on the product quotient,

Replace $\mathcal{N}$ by the Fourier multiplier

where the cutoff removes a neighborhood of $0$, has compact frequency support, and tends pointwise to $1$ as $ϵ\downarrow 0,\Lambda \uparrow \infty$. For the same representatives,

Their regulated gap is $-\lambda J_{ϵ,\Lambda }$, where

Indeed, the spatial Fourier transforms of $\xi$ have a uniform Schwartz bound on the compact time interval of $\mathrm{supp}f$; dominated convergence gives uniform convergence of the cutoff vectors there. The lower bound therefore tends to $|\lambda |J/2>0$.

This is a regularization of the declared auxiliary classical symbol. It is not a cone-wide inverse prescription, an operator regularization, or a field-measurement theorem.

Use the connected open sublaboratory

At every $p\in {\mathcal{D}}_0$, every jet

with symmetric $H$ and

is realized by a member of ${\mathcal{S}}_{\varphi }$.

Proof. At time $s=s(p)$, prescribe compact polynomial-times-bump Cauchy data near $x(p)$ realizing $A,b_{\mu },H_{ij},H_{0i}$. The equation supplies $H_{00}$.

Choose the support ball so small that its backward or forward propagation to time $0$ remains inside $B_{r_{\ast }}$; the strict inequality defining ${\mathcal{D}}_0$ permits this. Cauchy uniqueness then identifies the resulting local solution with the solution evolved from those compact time-zero data.

For smooth $\xi$ on ${\mathcal{D}}_0$, the following are equivalent:

1. $P(X_{\xi }\varphi )=0$ on ${\mathcal{D}}_0$ for every $\varphi \in {\mathcal{S}}_{\varphi }$;

2. $K\xi =0$ on ${\mathcal{D}}_0$;

3. in developing coordinates,

   $${\xi }^{\mu }(x)=a^{\mu }+B^{\mu }{}_{\nu }{x}^{\nu },B_{\mu \nu }=-B_{\nu \mu },$$

   for constants $a,B$.

A nonzero smooth vector compactly supported in ${\mathcal{D}}_0$ therefore fails the tangency condition.

Proof. At a fixed point, Lemma lem:jets and (eq:commutator) imply

for every allowed jet. Take $A=0,H=0$, and arbitrary $b$, obtaining $\square \xi =0$. Next take $A=0,b=0$, and arbitrary trace-free symmetric $H$. The symmetric derivative must be a multiple of ${\eta }^{\mu \nu }$. Finally choose $A=1,b=0$, and $H$ of trace $m^2$. Since $m>0$, that multiple is zero.

Differentiating the Killing equation and permuting indices gives

Connectedness gives the affine form. Conversely such an affine Killing field makes the commutator zero. An affine field vanishing on a nonempty open set is zero.

This is a local classification in a developing chart CheungIIIKilling. It does not assert that every local Poincare generator extends through conical holonomy. No massless version is inferred.

The distinction from Theorem thm:product is essential: that theorem fixes $\varphi$ and changes only its metric representative. The present theorem tests whether moving $\varphi$ preserves its unchanged equation.

Use the dual numbers

Statements in $\mathbb{D}$ are coefficient identities, not numerical remainder estimates.

Let ${\mathcal{E}}_{+,r_{\ast }}$ consist of supported smooth fields on $\hat{\mathcal{L}}$ that vanish for $s\le 0$. For $F$ in this class, let $R_{\square }F$ be the zero-data solution of

at $s=0$, continued as zero for $s<0$. Lemma lem:evolution proves existence, smoothness, and preservation of the support class.

For $a\in {\mathcal{E}}_{+,r_{\ast }}$,

The second identity follows by homogeneous-wave uniqueness.

Define

and

For $\xi \in {\mathcal{G}}_R$,

A homogeneous vector wave in ${\mathcal{G}}_R$ is zero.

Proof. Use $DK\xi =\square \xi$ and (eq:inverse). Then

The other assertions follow from this equation, $G^{(1)}K=0$, and uniqueness for zero wave data.

Notice the sign: $w_R(K\xi )=+{\xi }^{♯}$, whereas $w_{\mathrm{P}}(K\xi )=-{\xi }^{♯}$. The two compensators have different definitions.

Let ${\mathcal{J}}_R$ consist of triples

such that

with the scalar Cauchy data supported in ${\overline{B}}_{r_{\ast }}$. The forced equation and Lemma lem:evolution keep the scalars in ${\mathcal{C}}_{r_{\ast }}$.

For field-independent $\xi \in {\mathcal{G}}_R$, set

Equation (eq:jet-action) defines an additive action on ${\mathcal{J}}_R$:

Proof. The support and past-trivial conditions are preserved. The scalar equation transforms to

The metric equation follows from $G^{(1)}K=0$. The group identities are immediate.

The additive action concerns retained coefficients. Commutators of the corresponding formal diffeomorphisms start at the discarded second order.

Each orbit of ${\mathcal{J}}_R/{\mathcal{G}}_R$ has exactly one representative with $Dh=0$, namely

For every real compact $f$ in the laboratory,

descends to ${\mathcal{J}}_R/{\mathcal{G}}_R$. Its scalar coefficient satisfies

Proof. Apply $T_{\xi }$ with ${\xi }^{♯}=-w_R(h)$. Lemma lem:section puts the metric in the required slice, and Proposition prop:jet-action preserves the equations.

The section is constant on each orbit because the two changes $X_{\xi }{\varphi }_0$ and $X_{\xi }{\varphi }_0$ cancel. Two representatives in the slice differ by a past-trivial homogeneous wave, which is zero.

The integrand in each coefficient of (eq:jet-observable) is therefore orbit invariant. For the final identity, writing $w=w_R(h)$,

The proof retains the forced scalar correction CheungIIIJet. Discarding ${\varphi }_1$, or declaring it independently free, would change the equation space and invalidate this conclusion.

The evaluation of a field-dependent section is an orbit calculation. Differentiating that section in a symplectic or Hamiltonian reduction is an additional operation; derivatives of $w_R(h)$ would then have to be retained.

Choose $0<\rho <r_{\ast }$, a real cutoff ${\chi }_0\in {C_c}^{\infty }(B_{r_{\ast }})$ equal to one on ${\overline{B}}_{\rho }$, and $A_{\ast }>0$. Let

Finite propagation gives

The ${\mathsf{C}}_1$ model is

Its metric equation is vacuum linearized gravity. The scalar background stress is not inserted into that equation.

This is not the full Einstein–Klein–Gordon linearization about a nonlinear on-shell background. The latter has different equations and compatibility conditions Hack.

Write

Put $\bar{A}=\mathsf{r}A$, $\bar{B}=\mathsf{r}B$. Let ${\mathcal{X}}_{r_{\ast }}$ be the smooth tuples supported in ${\overline{B}}_{r_{\ast }}$ satisfying

Give ${\mathcal{X}}_{r_{\ast }}$ its inherited fixed-support smooth Fréchet topology.

Let ${\mathcal{S}}_{r_{\ast }}$ be the solutions of (eq:C1) supported in ${\mathcal{C}}_{r_{\ast }}$, with compact-open smooth topology on the auxiliary slab. Support and differential equations are closed conditions.

Evolution and time-zero restriction are continuous inverse linear maps

In particular, every admissible datum determines one supported conical-laboratory solution.

Proof. First evolve $A,B$ componentwise by the wave equation. Then solve the scalar forced equation with data $u,v$. Both are furnished by Lemma lem:evolution.

The field $d_{\nu }=(Dh{)}_{\nu }$ obeys $\square d_{\nu }=0$, with

Both vanish by (eq:constraints), hence $Dh=0$.

Uniqueness holds first for $h$, then for $\phi$. The source $L_h\Phi$ is supported in ${\mathcal{C}}_{r_{\ast }}$. Multiplication by the fixed smooth derivatives of $\Phi$ is continuous in the relevant seminorms. The energy estimates prove continuity of evolution. Restriction and differentiation prove continuity of the inverse.

The zero collar from (eq:margin) gives the claimed conical-laboratory interpretation.

This is a fixed-support local version of the Cauchy construction in III CheungIIICauchy. No inductive limit over supports escaping the developing chart is taken.

Let

be covector initial data $(a_{\mu },b_{\mu })$. Evolve them by

The coupled transformation is

Its induced data map $\mathfrak{g}:{\mathcal{G}}_{r_{\ast }}\to {\mathcal{X}}_{r_{\ast }}$ is

together with

These equations specify the full quotient datum, including the homogeneous scalar data.

For arbitrary compact metric data define

Call this auxiliary reconstruction $\mathfrak{R}(A,B)$. It may initially be noncompact.

For $x=(A,B,u,v)\in {\mathcal{X}}_{r_{\ast }}$,

When the equality holds, every reconstructed descriptor is supported in ${\overline{B}}_{r_{\ast }}$. The range $\mathfrak{g}({\mathcal{G}}_{r_{\ast }})$ is closed.

Proof. For true compact gauge data, (eq:gauge-data) gives

Newton inversion recovers $a$, and the remaining equations recover $b$.

Conversely suppose the stated equality holds. All Newton potentials in (eq:reconstruction) decay at infinity. Outside ${\overline{B}}_{r_{\ast }}$, $A_{ij}=0$, so

Differentiation and index permutation give

The exterior of a ball is connected. Thus the spatial $a_i$ are affine there, and decay makes them zero.

Also $b_0=0$ outside the ball, and $A_{0i}=0$ gives $b_i=-{\partial }_i{a}_0$. The equation $B_{ij}=0$ then gives ${\partial }_i{\partial }_j{a}_0=0$ there. Again decay forces $a_0=0$, and hence $b_i=0$. The scalar equations in the assumed equality provide exactly (eq:scalar-data).

Finally, Newton convolution from a fixed compact smooth support stage into $C^{\infty }({\mathbb{R}}^3)$, with compact-open topology, is continuous. Derivatives can be transferred to the source; the Newton kernel is locally integrable. Therefore

is a continuous map into a finite product of Hausdorff smooth spaces. Its kernel is precisely the gauge range, so that range is closed.

The exterior argument is an auxiliary membership test CheungIIIClosed. It neither imports noncompact gauge descriptors nor selects a global conical Poisson boundary condition.

Define

Both spaces are Hausdorff locally convex spaces, and evolution induces a topological linear isomorphism

Proof. Theorem thm:Cauchy maps the data gauge range exactly onto the solution gauge range, by covariance and uniqueness. The former is closed by Lemma lem:closed, so both quotients are Hausdorff. The continuous inverse unreduced maps induce continuous inverse quotient maps.

This is a relative theorem for the declared supported representatives. It is not a global TT-section theorem or a classification of all gauge orbits on the cone.

Let $S_0(h)\Phi$ be the zero-scalar-data solution of

and let $H_0(q)$ denote the homogeneous Klein–Gordon solution with the Cauchy data of a smooth scalar $q$.

For every admitted residual wave gauge,

Consequently a zero-scalar-data prescription is preserved only when $X_{\xi }\Phi$ has zero scalar Cauchy data.

Proof. Both sides have zero data and satisfy

Use uniqueness. The last assertion follows by evaluating the transformed scalar data.

A residual wave generally has nonzero initial data and is not past trivial. Its intersection with the past-trivial homogeneous-wave class is zero. The section of Section sec:jet therefore does not remove the residual group of the present quotient.

For compact smooth $J^{\mu \nu }=J^{\nu \mu }$ and scalar $q$ supported in the laboratory, define

If

then ${\mathcal{O}}_{J,q}$ is invariant under the coupled transformation for every smooth descriptor for which that transformation is considered. It descends continuously to ${{\mathcal{S}}_{r_{\ast }}}^{\mathrm{red}}$.

If invariance is required for every compact descriptor, the local identity (eq:Ward) is also necessary.

Proof. Integration by parts, with no boundary term, gives

This proves sufficiency and, by compact test separation, the stated necessity. Compact smooth pairings are continuous in the solution topology, so quotient continuity follows.

The pointwise identity is sufficient for the smaller residual-wave group. Its necessity for that smaller group is not asserted.

For $\psi \in {C_c}^{\infty }(\mathcal{L})$, let

The pair $(J_{\psi },P\psi )$ satisfies (eq:Ward), but

whenever $P\phi =-L_h\Phi$.

Proof. Using $P\Phi =0$, direct differentiation gives

Also integration by parts in (eq:Lh) gives