IV-B Represented Records with a Spacetime Enter Interface Three separate scalar-field instruments on a prescribed conical background, with an explicit completion and transfer boundary

We prove an exact interface adaptation of three separately represented binary measurements to a prescribed conical spacetime. A finite consumed clock context produces a classical time code and Ramsey outcome while retaining the complete conditional matter state, including arbitrary untouched references. The adaptation preserves the positive record contrasts and exact interpolation-off controls of the three represented models. Spectral-mask convergence at fixed particle-number cutoff is strengthened to convergence of the complete recorded channels in diamond norm; initial-edge removal in the unbounded model instead gives pointwise trace-norm convergence, without an unsupported uniform channel-norm claim. A separate target-adapted SWAP construction transports finite payloads exactly along prescribed local null segments, with an explicit register ledger.

Two corrections are essential. Local flatness of a cone does not identify its global field representation or Newton inverse with the corresponding Minkowski objects. We therefore state the flat-completion prescription explicitly and prove its local pullback identities, rather than claiming an intrinsic quantum field theory on the entire cone. Likewise, assigning a time to a global controlled unitary does not establish a compact causal implementation. The causal theorem applies to the certified finite transport and control network, not to an unconstructed local realization of the global field apparatus.

The contribution is thus an exact, reference-stable Grade-S interface construction with a precisely delimited geometric interpretation. Its potential usefulness is methodological: represented signals, conditional residuals, regulator errors, geometric calibration, and physical implementation can be compared without identifying their distinct mathematical objects. No priority claim is made for the underlying measurement identities. Records do not determine the metric, lapse, or deficit angle. No spacetime-emergence theorem, Einstein dynamics, gravity-generated-entanglement test, inherited gravitational quantum coefficient, or finite-qubit compilation is asserted.

The operational philosophy of Enter distinguishes a consumed quantum name from a returned quantum system. Destructive readout returns a classical record and whatever other quantum factors the interface explicitly retains. Consumption is a statement about access and output type, not physical annihilation of every implementing degree of freedom. Quantum recovery is a different task: its output must return the specified quantum information, including the required reference correlations. Equality of channels also does not specify a thermal machine or a heat ledger.

These distinctions are central to Cheung's account of Enter CheungEnter,CheungPhil,CheungI. We use specified Born semantics and prove the channel identities needed here. We do not import a measurement-uniqueness theorem from a shortened list of its hypotheses. The spacetime interface follows the prescribed-background interpretation of I-B CheungIB, not a reconstruction of geometry from records.

The phrase

will have only the following typed meaning: a designated finite clock encoding is consumed, whereas the three named spatial-carrier factors survive. It does not mean that the coordinate $T$, the manifold, or a geometric dimension is destroyed.

A Grade-S interface adaptation consists of:

1. a declared spacetime and local rods;

2. separately specified represented measurement models;

3. a finite destructive clock interface;

4. explicit time, location, and causal-wire dictionaries;

5. complete-channel identities and the stated regulator estimates.

An intrinsic-cone field adaptation additionally requires a global field representation, inverse prescriptions, domains, and measurement implementation on that same conical spacetime.

Grade E would require the relevant geometric structure and its calibration to be determined by operational data rather than supplied as inputs.

The construction below achieves Grade S in the first, interface sense, using an explicit flat completion for the represented fields. It does not establish the additional intrinsic-cone field adaptation. This subdivision is necessary: the two meanings are not interchangeable.

has been debugged{ The literal inference

is invalid. So is

We prove the corrected interface result and leave these stronger connecting constructions explicitly unclaimed.}

The coefficient conventions of the represented models are those of IV, version 1 CheungIV. They are restated below, and the operator, record, and interface arguments used in this paper are supplied here. I-B, En09-A, En03-A, En05-A, and En08-B are cited for provenance and comparison; their theorems are not silently substituted for the proofs below. Other companion manuscripts are documentary separations only.

The external analytic-vector theorem of Nelson Nelson is used, with its hypotheses checked. Page–Wootters conditioning PageWootters is discussed only through the precise finite conditional-evolution statement proved in Section sec:clock. It is not a claim of externally unscheduled quantum evolution.

Fix

Let

and prescribe

The time orientation is that of ${\partial }_T$.

The metric has signature $(-,+,+,+)$. Static observers have proper-time increment $\mathrm{d}\tau =\mathrm{d}T$. For a nonzero vector $v=a{\partial }_T+w$, future causality is equivalent to

On an angular interval with a fixed real lift of $\varphi$, the map

is a local isometry to $({\mathbb{R}}^{1,3},\eta )$, where $\eta =\mathrm{diag}(-1,1,1,1)$.

Proof. The metric matrix is diagonal with one negative and three positive entries. The causal inequality is $-c^2{a}^2+{\Vert w\Vert }_h^2\le 0$; a nonzero causal vector has $a\ne 0$. Along a static curve, $-g/c^2=\mathrm{d}{T}^2$. Finally,

which proves the isometry statement.

The completion adds the singular locus

It is a two-dimensional worldsheet. Fixing $z$ gives one worldline; the full defect is not a worldline. No smooth metric or ordinary pointwise Einstein tensor at the apex is used.

The geometry and its interpretation agree with the prescribed branch of I-B CheungIB. The conical and lapse comparisons in VIII-B and Em02-E remain separate source records CheungVIIIB,CheungEm02E.

A local developing isometry, by itself, supplies neither:

1. a global Minkowski plane-wave or TT representation on $M$;

2. an intrinsic Newton inverse on a conical spatial slice;

3. a local implementation of a globally specified controlled unitary.

Proof. On the angular cover, changing $\varphi$ by $2\pi$ rotates the developed spatial point by $2\pi \alpha$. A generic function $e^{ik\cdot y}$ is not invariant under that rotation. Its pullback therefore does not descend to a single-valued function on the cone. The same issue applies to tensor modes, with the additional tensor rotation. A global mode representation needs compatible gluing data.

For the inverse statement, two smooth flat sources may agree on a developing neighborhood but differ outside it. Their Newton potentials on the neighborhood can differ by a nonconstant harmonic function. For example, a positive source bump outside the neighborhood produces a nonzero potential there. Consequently, restriction to the neighborhood does not determine the Newton response.

For the implementation statement, a unitary on an abstract global Hilbert space has no specified spatial coupling, intermediate dynamics, or accessible local algebra. Adding a time label supplies none of these missing data.

This proposition rules out the stated deductions, not every possible future cone quantization or local apparatus construction.

Fix a regular developing patch $U\subset M$, and write $D=\mathrm{dev}{|}_U$. The analytic field models are defined on a separately declared auxiliary Minkowski spacetime

All global wave evolutions, Fourier modes, TT projectors, Newton inverses, states, and cutoffs are evaluated on $\tilde{M}$.

A compact scalar test $f_U$ with support in $U$ corresponds to $f=D_{\ast }{f}_U$, extended by zero outside $D(U)$. Tensor tests are transported with their tensor type and the volume pairing. A represented field or response $q$ has the patch interpretation $D^{\ast }q$.

This prescription is part of the model. It is not an assertion that $\tilde{M}$ is globally isometric to $M$, or that different completions give identical answers.

Every bounded spatial detector region needed below can be accommodated: a sufficiently distant portion of a developing angular sector contains a Euclidean ball of any prescribed finite radius. Translate the whole flat detector package into such a ball. Simultaneous translation of tests, modes, and coherent labels is unitary in the declared flat representations and does not change the numerical pairings.

Under Definition def:completion, compact pairings satisfy

The corresponding tensor identity also holds. Local differential equations and linearized curvature identities pull back through $D$. A flat zero-data wave calculation agrees with its local calculation whenever the relevant causal domain is contained in $D(U)$.

Proof. The first assertions are change of variables and naturality of differential operators under an isometry. The volume form is $\mathrm{d}{\mathrm{vol}}_g=c\mathrm{d}T{\mathrm{d}}^{3}y=\mathrm{d}t{\mathrm{d}}^{3}y$. For the final assertion, subtract the two solutions on their common domain. The difference solves the homogeneous equation with zero Cauchy data; the local energy identity gives uniqueness.

In Model 3, the precise response on $U$ is therefore

The complete source on the right is part of the declared input. Equation (eq:completed-response) is not a response determined by the restricted source alone.

All Hilbert spaces are complex and separable. Instruments act on trace-class operators; their adjoints act normally on bounded operators. Inner products are antilinear in the first entry. We use $\hslash =1$ in the represented models.

For a self-adjoint operator $A$, set

The phase in (eq:kraus) is $e^{-i\theta }{U}_A$, not $e^{-i\theta U_A}$.

Define the complete recorded channel

Equation (eq:ramsey-channel) is CPTP and retains every untouched reference. Its outcome probabilities are

It is implemented by preparing a probe in $|+⟩$, applying

and destructively reading the probe in the basis

Proof. Direct expansion gives ${\sum }_s{K}_s^{\ast }{K}_s=I$, which proves complete positivity and trace preservation. The same expansion gives (eq:ramsey-prob). Moreover,

Tensoring these operator identities with any reference identity proves the complete statements.

Let ${\psi }_0,{\psi }_1\in \mathrm{Dom}A$ be unit vectors, and put

For $\theta =\pi /2$ and $0<\tau \le d/S$,

Proof. The spectral theorem and $\left|e^{ix}-1-ix\right|\le x^2/2$ give

At $\theta =\pi /2$, the record reads the imaginary part. Subtracting the two expectations yields

Use

where the diamond norm is understood on trace-class inputs and arbitrary reference extensions.

For two unitaries $U,V$, let ${\mathcal{I}}_U,{\mathcal{I}}_V$ be the channels defined by (eq:kraus)–(eq:ramsey-channel). Then

For bounded self-adjoint $A,B$,

These estimates survive compatible CPTP pre- and post-composition.

Proof. The probability bound follows from (eq:ramsey-prob). For the complete bound, define isometries

Then

For any reference extension and trace-class input $Z$,

Pinching the first factor produces the recorded channels and cannot increase the norm. This proves (eq:instrument-error). Differentiating $e^{i(\tau -s)A}{e}^{isB}$ and integrating proves (eq:duhamel); negative $\tau$ follows by reversal. CPTP composition is a contraction for the complete comparison norm.

In particular, a two-preparation contrast loses at most $\Vert U-V\Vert$ under this unitary replacement. This is a complete instrument estimate, not merely agreement of two effects.

Work first on the declared completion $\tilde{M}$, with

Fix smooth real spatial TT waves $e,r$ with Schwartz Cauchy data, spanning a nondegenerate real symplectic plane. They have no temporal components. For a free scalar $\phi$, define

This solves $P{S}_{t_0}(a)\phi =-L_a\phi$, with zero correction data, where $L_a\phi =-a^{ij}{\partial }_i{\partial }_j\phi$.

For real $f\in {C_c}^{\infty }((t_0,t_1)\times {\mathbb{R}}^3)$, write

Set

Integration by parts gives

Finite propagation makes the relevant spatial support compact on the finite interval.

Let $u_0,u_e,u_r\in L^2({\mathbb{R}}^3)$ be the one-particle vectors representing $\int f\phi$, ${\ell }_{e,f}$, and ${\ell }_{r,f}$, respectively. Explicitly, if a real linear Cauchy functional is $\int (\alpha {\phi }_0+\beta {\varpi }_0)\mathrm{d}x$, use

For the response functional,

All these vectors are Schwartz.

On

define

The real $\lambda$ defines a numerical model. It is not evaluation of a formal nilpotent parameter at a nonzero real number.

The operator (eq:A1) is essentially self-adjoint on the total finite-particle domain. Its closure contains $\mathrm{Dom}\mathsf{N}$, and

Only the gravitational oscillator and ${\mathrm{Span}}_{\mathbb{C}}\{u_0,u_e,u_r\}$ are active.

Proof. Creation and annihilation satisfy

Products of two such operators obey the stated number bound. Different species commute, so the polynomial is symmetric and changes total number by at most two.

If $\Psi$ has at most $n$ particles, iteration gives

Thus every finite-particle vector is analytic for $A$. Nelson's analytic-vector theorem Nelson proves essential self-adjointness. Approximation by number truncations proves the domain inclusion and bound. Fock factorization over the displayed one-particle span and its orthogonal complement proves the final claim.

Choose smooth $0\le \vartheta \le 1$, with $\vartheta (s)=0$ for $s\le 0$ and $\vartheta (s)=1$ for $s\ge 1$. Replace

and denote the resulting operator by ${A_{\epsilon }}^{(1)}$. Formula (eq:oneparticle) and integration over an interval of length $\epsilon$ give

Consequently,

with uniform sector bounds for ${A_{\epsilon }}^{(1)}$.

For every $\tau \in \mathbb{R}$,

For every trace-class $Z$ on ${\mathcal{H}}_1\otimes {\mathcal{H}}_E$, with any untouched reference $E$,

No diamond-norm convergence is asserted.

Proof. On the sector of at most $n$ particles, telescope the powers in the analytic series. For $2C\left|\tau \right|<1$, the preceding sector estimates give

Finite-particle vectors are dense and all exponentials are unitary, so convergence is strong for these small times. Subdivision of an arbitrary time and finite telescoping prove strong convergence for every $\tau$.

The corresponding Kraus operators and their adjoints converge strongly. For finite-rank $Z$, this implies trace-norm convergence of each Kraus block, also after tensoring with $E$. Approximate a general trace-class $Z$ by finite-rank operators and use the uniform channel bounds to obtain (eq:edge-complete).

For a coherent preparation of total mean number $\nu$, an explicit unitary-vector error is obtained from

At fixed admissible $\tau$, this tends to zero by first fixing a large $n$, then taking $\epsilon \downarrow 0$.

For the simple family $A_{\epsilon }=\epsilon Q$, the unitaries converge strongly to $I$, but $\Vert e^{i\tau \epsilon Q}-I\Vert =2$ whenever $\tau \epsilon \ne 0$. Wave packets concentrated where the phase is near $\pi$ also distinguish the corresponding $\theta =0$ Ramsey records from those of $I$ almost perfectly. This prevents promotion of a strong-limit proof to a uniform instrument-norm claim.

Translate so that $e^{ij}(t_0,0)=E^{ij}\ne 0$. Choose real compact scalar data

where ${\eta }_0=1$ near zero. Then

A small future neighborhood therefore contains a nonnegative compact test $f$ for which

Let $z_{\ast }$ be the scalar coherent label for these data. The states

have identical scalar factors, equal oscillator covariances, and $\Delta ⟨Q⟩=1$, $\Delta ⟨{\Pi }_0⟩=0$. Hence

Their second moments are finite by Proposition prop:A1.

For every fixed $\lambda \ne 0$, some fixed $\tau >0$ and $\theta =\pi /2$ give

The limiting contrast is positive. At $\lambda =0$, the two record laws agree exactly for every $\epsilon ,\tau ,\theta$.

Proof. Apply Lemma lem:contrast with $d=\left|\lambda \right|d_1$. The edge error is at most $\frac{1}{2}[{\mathcal{E}}_{\epsilon }({\nu }_0;\tau )+{\mathcal{E}}_{\epsilon }({\nu }_1;\tau )]$. At $\lambda =0$, the generator acts only on the common scalar factor.

By Lemma lem:patch, all compact coefficient pairings have the same values in the declared conical patch. This is the precise, completion- dependent adaptation of Model 1, not a global TT field on the cone.

On the flat completion define

and

Let ${\mathsf{P}}_{\mathrm{TT}}(p)$ be the orthogonal TT projection. Use modes

with a factor $2$ in the TT field, as in IV CheungIV.

For $u^{+}=u,u^{-}=\overline{u}$, and similarly for $v$, define

The four operators are annihilation–annihilation, annihilation–creation, creation–annihilation, and creation–creation, with the reality relations supplied by real $f$.

Put

The tensor $P_2$ is not the particle-number projection.

Take smooth real even masks $0\le a_R,b_R\le 1$, equal to one for $R^{-1}\le |p|\le R$ and $|q|\le R$, respectively, and supported in the corresponding doubled regions. Set

The pair coefficient is Hilbert–Schmidt and the mixed coefficient is bounded. There are constants independent of the admitted masks such that

Proof. Write $\ell =t_1-t_0$,

Spatial integration by parts and one temporal integration by parts, retaining the initial endpoint $F(0,k)$, give

For $k=ϵp+\delta q$, transversality gives ${\mathsf{P}}_{\mathrm{TT}}(p)(q\otimes q)={\mathsf{P}}_{\mathrm{TT}}(p)(k\otimes k)$. After absorbing polynomial transfer factors into the arbitrary spatial decay,

For pair signs this implies, after increasing $n$,

The radial integral is finite. The gravitational low and high tails have squared integrals $O(R^{-2})$ and $O(R^{-1})$. The scalar high tail is $O(R^{-1})$, by splitting $|p|<1$ from $|p|\ge 1$ and transferring weights between $p,q,p+q$. This proves the first estimate.

For mixed signs, (eq:master) gives

and, on $|p|\ge 1$, also the bound with $⟨q{⟩}^{-1}$. The low part is Hilbert–Schmidt, with low-tail norm $O(R^{-1})$. The two high-tail Schur integrals are $O(R^{-1})$, using the appropriate $p$- or $q$-weighted bound; the remaining low-$p$, high-$q$ term decays faster. The Schur test proves the second estimate.

The proof does not replace the mixed operator-norm estimate by an unproved Hilbert–Schmidt estimate.

Identify the product Fock space with ${\Gamma }_{\mathrm{s}}({\mathfrak{h}}_g\oplus {\mathfrak{h}}_{\phi })$. Put

and let $G_R$ be the symmetric tensor obtained by placing $P_{2,R}$ in the two cross-species slots and adding its flip. Thus $\Vert G_R\Vert =\sqrt{2}{\Vert P_{2,R}\Vert }_{\mathrm{HS}}$. Define the symmetric quadratic

With $\mathsf{N}={\mathsf{N}}_g+{\mathsf{N}}_{\phi }$ and ${\Pi }_M=1_{\{\mathsf{N}\le M\}}$, sector estimates give

For completeness, on the $n$-particle sector, $\mathrm{d}\Gamma (T)$ has norm at most $n\Vert T\Vert$, while pair annihilation has norm at most $\sqrt{n(n-1)}\Vert G\Vert$. Pair creation is its adjoint. Summing the orthogonal output sectors proves (eq:B2graph); compression and $\Vert \mathrm{Re}C\Vert \le \Vert C\Vert$ prove (eq:B2compression).

Define

Each generator is bounded self-adjoint. Its free scalar term is compressed by ${\mathsf{N}}_{\phi }$, not by total number.

Replacing the first term of (eq:A2) by ${\Pi }_M(I_g\otimes \Phi (u_0)){\Pi }_M$ can make its record depend on gravitational occupation even at $\lambda =0$. The scalar-only compression is required for the exact off-control.

For fixed finite $M\ge 2$, put

Then

and

The estimate is uniform over states and references.

Proof. Combine Lemma lem:spectral, (eq:B2compression), and Proposition prop:instrument-error.

Choose a smooth compact-annular TT coherent label whose classical field $k_{\ast }$ is nonzero somewhere. Choose compact smooth scalar data so that $L_{k_{\ast }}{\phi }_{\ast }$ is nonzero there. The same Hessian construction used in Model 1 provides such data. Then ${\psi }_{\ast }=S_{t_0}(k_{\ast }){\phi }_{\ast }$ is not zero. Choose a nonnegative compact bump $b$ supported where it is nonzero and set

Let $z_g,z_{\phi }$ be the corresponding coherent labels and take

Expanding the four legs gives exactly

For total coherent mean number $\nu$, define

Number bandwidth two and (eq:B2graph) imply

Therefore

The same number estimates bound $\Vert {A^{(2)}}_{R,M}{\Psi }_s\Vert$ uniformly in $R,M$.

For fixed $a,\lambda \ne 0$, one can choose a single $\tau >0$, independent of $R,M$, and $\theta =\pi /2$, such that

where

For sufficiently large but finite $M$, the limiting contrast is positive. At $\lambda =0$, the two record laws agree exactly for every $R,M,\tau ,\theta$.

Proof. Let $d=|\lambda a|d_2$, and let $S_2$ be a common bound on the sum of the two generator second moments. Choose $0<\tau \le d/S_2$. The proof of Lemma lem:contrast gives

Take $c_2=\tau d/4$ and $e_M=\tau |\lambda |E_M/2$. The two probability errors from the spectral replacement total at most $e_R(M)$. Finally, ${A^{(2)}}_{R,M}(0)=I_g\otimes F_{0,M}$, and the scalar preparations are identical.

The conical patch interpretation uses Definition def:completion. The masks remain masks of that completion; they are not silently changed into intrinsic cone spectral projectors.

Translate $t_0$ to zero. On the completion, let $Q_{\mu \nu }$ be a smooth conserved symmetric source, Schwartz in space with all seminorms locally uniform in time. Define

and

For trace-reversed data $A_{\mu \nu },B_{\mu \nu }$, prescribe

With $D_0=\sqrt{-\Delta }$, put

where

Equation (eq:NW) defines a smooth response ${\mathcal{R}}^{\mathrm{NW}}Q=j$, continuous from suitable source Schwartz seminorms to compact smooth seminorms. It satisfies

These equations are built into the prescribed response; they are not derived gravitational dynamics.

Proof. The Newton multiplier $-|p{|}^{-2}$ is locally integrable in three dimensions. Its differentiated multipliers and the wave propagators give smoothness and compact seminorm bounds. The Newton data can have Coulomb tails; square integrability of the metric itself is not needed.

The wave formula gives $\square \overline{j}=-2Q$. Set $d_{\nu }=-{\dot{\overline{j}}}_{0\nu }+{\partial }_i{\overline{j}}_{i\nu }$. The prescribed data give $d(0)=0$. Since $\Delta a=-2\rho$ and $\Delta V_i=2J_i$, the wave equation also gives $\dot{d}(0)=0$. Conservation gives $\square d=0$; uniqueness implies $d=0$. Substitution into the linearized Einstein operator gives $G^{(1)}(j)=-\frac{1}{2}\square \overline{j}=Q$.

On

use the massive free scalar and the fixed vacuum-normal-ordered stress

For $\Psi ,\Xi \in {\mathcal{D}}_3$, its matrix element ${t_{\mu \nu }}^{\Psi ,\Xi }(x)$ is a finite sum of products of Schwartz wave packets and derivatives. Polarized Klein–Gordon conservation shows that it is conserved.

For real compact symmetric $J^{\mu \nu }$, define

The preceding seminorm estimates prove that this is a well-defined sesquilinear form, symmetric for real $J$. Its conical patch interpretation is (eq:completed-response) and Lemma lem:patch.