I-B-v1: A Finite Enter Spacetime Calibrated Destructive Time Readout on a Prescribed Conical Product

We construct a finite operational model on the regular part of an explicit $3+1$-dimensional conical Lorentzian background. Its additional data are finite quantum names, locations, accessible algebras, a complete control and resource interface, and a calibration from a finite clock basis to geometric time labels. Destructive Enter consumes the clock names and returns a classical code together with the surviving matter. We prove the complete instrument identity, the distinction between event-time calibration and history-label calibration, the absence of a universal quantum return, and a finite-dimensional canonical-commutator obstruction. Scheduled influence is contained in the prescribed metric causal relation. A target-adapted finite chain supplies exact reference-preserving SWAP transport for a prescribed local null displacement, with an explicit resource ledger.

Several necessary corrections are incorporated. The conical defect in the product is a worldsheet, not a worldline. An arbitrary clock-basis change is not a spacetime coordinate transformation. A stochastic reading at one fixed event cannot equal several distinct coordinates of that event. A relational lapse cannot replace an entire cosmological metric, and spatial homogeneity does not imply flatness. The construction is a spacetime model with an Enter interface, not emergence of spacetime from records. No inherited quantum response coefficient is admitted and no standing analytic gate is discharged.

The motivating discipline is Cheung's separation of destructive readout, quantum recovery, physical implementation, and geometric identification. Enter removes measured quantum names from the returned quantum interface and supplies a classical record. This is an access statement, not annihilation of every implementing degree of freedom. Likewise, equality of channels is not a thermodynamic account, and an analogy is not a mathematical dictionary CheungPhil,CheungI,CheungEn03.

We adopt specified finite Born semantics. We do not claim to rederive Landmark I's characterization from an abbreviated list of requirements. That characterization has its own structural class and hypotheses CheungI. All mathematical assertions needed for the model below are stated with their hypotheses and proved locally.

The construction follows En03-A's four-part geometric interface: external classical geometry, locations, accessible coupling algebras, and complete control and resource data CheungEn03. Geometry is supplied, not extracted from a count of names or records.

The background is a continuum. An experiment is finite: it has finitely many active names, finite-dimensional quantum factors, finitely many classical outcomes, and finitely many instructions. Classical geometric parameters may be prescribed real numbers. No assertion is made that arbitrary such numbers have been determined by a finite-bit physical calibration procedure.

The main release chooses the conical branch with unit lapse. A cosmological attachment interface is given in Appendix app:lapse; its input geometry is not silently identified with the conical product.

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These corrections restrict the theorem to a coherent, nonempty model; they are not additional emergence assumptions.

boundaries. Throughout this paper, “XV-B is not opened” means that the inherited downstream admission exit is not activated by this construction. It does not deny the existence of a technical manuscript bearing that identifier. XVI-N is not discharged; O.1 remains NOT_CLOSED for the purposes of this release; no inherited quantum coefficient is admitted; and no compilation, numerical execution, physical implementation, or machine verification is asserted. The narrower internal-amplitude advance in XIV-I is not withdrawn, but it is not the full joint characteristic map absent at I3/J2 CheungXIVIJ,CheungStops.

Fix

Let $C_{\mu }$ be the completion of

with spatial line element

Write $o$ for its apex. The circumference of the circle $r=R$ is $(2\pi -\mu )R$, so the opening angle is $2\pi -\mu$.

Set

For any smooth positive lapse $N$ on $M$, prescribe

The main model uses

In geometrized units $c=1$, this is the metric in the proposed conical-product branch. Equivalently, with $\theta =\alpha \varphi$, the angular coordinate has period $2\pi -\mu$ and the angular term is $r^2\mathrm{d}{\theta }^2$.

The singular completion locus is

It is two-dimensional. For fixed $z_0$, the subset ${\mathcal{W}}_{z_0}={\mathbb{R}}_T\times \{o\}\times \{z_0\}$ is a worldline.

The smooth Lorentzian manifold in our theorem is $M$, not $\overline{M}$ equipped with a purported smooth metric at its apex. The completion records the conical defect. No smooth chart or ordinary pointwise Einstein tensor is asserted on $\mathcal{W}$.

For $N>0$, $g_N$ has signature $(-,+,+,+)$, and

Choose the time orientation for which ${\partial }_T$ is future directed. For

a nonzero future-causal vector satisfies

A static observer has proper-time increment

In particular, $T$ is proper time up to the chosen origin for the static observers of (eq:metric).

Proof. In the displayed coordinates the metric matrix is diagonal, with one negative and three positive entries. Its inverse gives the stated norm of $\mathrm{d}T$. The causal inequality is

A nonzero causal vector cannot have $a=0$; the chosen orientation therefore gives $a>0$. Along a static curve, $-g_N/c^2=N^2\mathrm{d}{T}^2$, which proves the proper-time formula.

On a sufficiently small angular interval choose a real lift of $\varphi$. The developing map

is a local isometry from (eq:metric) to Minkowski space with metric $-c^2\mathrm{d}{T}^2+\mathrm{d}{x}^2+\mathrm{d}{y}^2+\mathrm{d}{z}^2$. Its spatial coordinates are three local rods.

The metric (eq:metric) is locally flat on $M$. In the convention

the gluing map $J_{\mu }$ fixes $T,z$ and rotates the developed $(x,y)$-plane by $\mu$. Thus local flatness does not remove the global conical gluing.

Proof. Differentiating (eq:development) gives

For the deck transformation, the developed spatial angle changes by $-2\pi \alpha =\mu -2\pi$, which is the same rotation as $\mu$.

This geometry is motivated by the conical constructions in VII-E, VII-F, and VIII-B CheungVIIEF,CheungVIIIB. The additional $z$ direction and the lapse interface are declared here; they are not conclusions transferred from a $2+1$-dimensional theorem.

VII-F's particle-supported cellulation event is a geometric construction in its own declared family CheungVIIEF (Theorem thm:event). We do not import its cell counts or uniqueness conclusion into a three-dimensional spatial cellulation. Our operational events below lie in the regular part. Attaching that separate slicing event would require its full geometric data and a clock-and-record dictionary. A slicing event is not already an Enter event.

Choose an open spatial set $U\subset {C_{\mu }}^{\circ }\times \mathbb{R}$ whose development is an open convex Euclidean set, and write

for its spatial developing chart. All finite protocols in the main construction remain inside $U$.

Let $x,y\in U$ and $T_1\ge T_0$. If

then $(T_1,y)\in J_g^{+}((T_0,x))$. If equality holds and $x\ne y$, the straight developed segment, traversed at constant speed, is null.

Proof. Convexity keeps the straight spatial segment in $V$. Its lift has constant coordinate velocity of norm at most $c$, so (eq:tangent-cone) applies with $N=1$. Equality makes its tangent null.

No global formula for $J^{\pm }$ across the singular completion is needed.

An experiment consists of the following data.

1. External geometry. The fixed configuration

   $$(\overline{M},M,g,N=1,\mathcal{W},U,\chi ,c).$$

2. Names and locations. At each semantic cut there is a finite set of live quantum names. Their Hilbert spaces are finite tensor products of qubits. A location map assigns each accessible port to a point of $M$. Several names may be co-located.

3. Accessible algebras. For a declared local collection $A$ of live names, the accessible algebra is the full matrix algebra on those names, tensored with the identity on other live names. A smaller algebra may be declared, but no subsequent operation may silently use an observable outside it.

4. Complete control and resources. A finite instruction tree specifies allocations, permitted unitaries, destructive readouts, classical controls, input and output ports, schedules, location data, retained storage, and all auxiliary factors. Every classical control dependency is either preinstalled task data or a declared causal record wire.

The background metric has no state-dependent update rule in this model.

A fixed classical history has one normalized pure state on its complete live quantum names. Unitary gates and pure allocations preserve this property. At a rank-one destructive readout, each supported outcome has a normalized pure residual. Density operators and recorded channels describe ensembles, inaccessible restrictions, and complete input–output comparisons; they do not assert that a classical mixture is one pure branch.

An untouched reference $E$ may be included in a complete input. It has no available control or communication wire. Its storage is listed separately from the laboratory storage.

Fix disjoint clock and matter names, with

and a finite-dimensional surviving space $H_X$. We consistently order the input factors as $C\otimes X$.

Choose an orthonormal clock basis

and the unitary

The program applies $B_C$, then computational-basis Enter to all clock names. It returns the $q$-bit code $k$, not the names $C$.

Define

The record has diagonal observable algebra

The specified readout has channel

It is completely positive and trace preserving. Its $k$-th subnormalized matter block is

For $p_k>0$, the conditional matter state is ${\sigma }_k/p_k$. No normalized state is assigned to a zero-probability outcome. Every identity holds after tensoring with ${\mathrm{id}}_E$.

Proof. After applying $B_C$, the destructive measurement bra for outcome $k$ is

The Kraus operator including the classical output is

Consequently the channel is (eq:instrument), and

This proves complete positivity and trace preservation. The partial-trace expression follows by expanding in the clock basis. The block probabilities and conditional states are the specified Born semantics. Tensoring each operator identity with $I_E$ proves the reference versions.

Theorem thm:instrument is the finite interface used in En09-A; its conditioning block is not, merely by its form, a metric or a stress tensor CheungEn09.

For comparison, define the coherent recording isometry

where the output retains $C$, adds a quantum carrier $A$, and retains $X$. Define also the represented nondestructive instrument

Among these three interfaces, only ${\mathcal{M}}_C$ removes $C$ from the returned quantum context. Moreover,

Coherent recording returns a quantum carrier and is not itself a classical recorded channel.

Proof. The assertions about returned names follow from the displayed domains and codomains. The partial-trace identity is Theorem thm:instrument. For a superposition of two clock basis states, coherent recording contains off-diagonal terms between the corresponding $C\otimes A$ states. Hence it is not generally block diagonal in $A$.

The equality after discarding $C$ does not identify the larger nondestructive interface with destructive Enter CheungI,CheungEn03.

Fix distinct ordered real numbers

The stored object is $k$. Its interpretation as $t_k$ is classical metadata. The finite operator

is a pointer observable, not the continuum coordinate function $T$.

There are two different admissible calibrations.

1. Event-time calibration. For a reading at the fixed event $e$, an input is exactly calibrated when

   $$p_k>0⟹t_k=T(e).$$

   Thus the returned label is the coordinate of the reading event.

2. History-label calibration. Choose a finite event map

   $$\kappa :\{0,\dots ,D-1\}⟶M,T(\kappa (k))=t_k.$$

   A readout at an independently specified event $e_{\mathrm{r}}$ returns the label of $\kappa (k)$. It need not return $T(e_{\mathrm{r}})$. In the examples below all $\kappa (k)$ are earlier points on the sender's static worldline.

Under distinct-label calibration, (eq:event-calibration) holds if and only if there is a unique $j$ with $t_j=T(e)$ and

The readout then returns $j$ with probability one. If $T(e)\notin \{t_k\}$, no normalized exactly calibrated input exists.

Proof. Distinctness permits at most one index $j$ satisfying $t_j=T(e)$. Calibration and normalization give $p_j=1$. Put $Q=I-P_j\otimes I_X$. Positivity and $\mathrm{Tr}(Q\rho )=0$ imply

Therefore $Q\rho =\rho Q=0$, proving the support statement. The converse follows immediately from (eq:instrument).

A coherent history with several supported readings cannot make the single classical number $T(e_{\mathrm{r}})$ equal several distinct $t_k$. Calling $t_k$ a geometric label is legitimate after $\kappa$ is supplied. Calling every such outcome the actual time of one fixed event would be false. Random classical selection among different reading events is another valid construction, but the event selection must then be included as classical input data.

The dictionary is the following explicit collection of maps.

1. Clock map. The basis calibration $B_C$, the finite map $k\mapsto t_k$, and either the event support condition (eq:event-calibration) or the history-event map $\kappa$.

2. Rod map. At each cut, a surviving carrier name $a$ has a prescribed spatial location $x(a)\in U$, with rod label $\chi (x(a))$. In polar notation the corresponding local labels are $(r,\alpha \varphi ,z)$.

3. Causal map. The complete scheduled port graph is mapped to $M$ so that every directed dependency edge maps to a future-causal pair. Ancestors of an output therefore map into its metric past.

These maps are model data. They are not inferred from output probabilities alone.

A pre-read and post-read germ $s_{\ast }^{-}\prec s_{\ast }^{+}$ describes semantic order at one reading event. It need not denote two distinct geometric times. In particular, strict semantic sequencing does not imply a positive proper-time duration for an idealized local instruction.

The distinctions in XIX-A motivate the following ledger CheungXIX.

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In the present model class:

1. even a harmonic scalar profile does not determine the lapse;

2. a finite pointer is not the all-real coordinate $T$;

3. an externally specified density ramp does not, by its specification alone, supply an independent dynamical clock;

4. a slicing-family parameter is not identified with $T$ without an additional calibration.

Proof. For (i), take any two positive constant lapses $N=a$ and $N=b$. The scalar $\xi =T$ is harmonic for both metrics: the coefficients and volume density are independent of $T$, so ${\square }_{g_N}T=0$. The scalar profile is identical, but

depends on the lapse.

For (ii), the pointer has at most $D$ distinct eigenvalues, whereas $T$ takes every real value on $M$.

For (iii), a fixed regulator $r_{\epsilon }(T)=r(T/\epsilon )$ adds a prescribed function of external labels, not a new state variable or evolution equation. It can be used in a calibration if further data are supplied; that does not make an independent dynamical degree of freedom appear.

For (iv), a family of slices may be reindexed by any bijection of its parameter interval without changing the collection of slices. Neither the background metric nor the finite clock instrument selects one such indexing.

The original-versus-modified rod distinction is stronger than an indexing issue: Appendix app:rod-witness gives a compact perturbative witness separating their quadratic lapses.

Finite record laws and complete matter channels in this construction do not determine the rod frame, the lapse, or the global deficit.

Proof. First, apply a spatial Euclidean isometry to the chosen developing chart and all prescribed carrier locations. The abstract name permutations and channels are unchanged, while their numerical rod coordinates change.

Second, retain the same event labels, sites, and scheduled channels, but compare the constant lapses $N=1$ and $N=2$. Every edge satisfying $\Vert \Delta x\Vert \le c\Delta T$ is causal for both. The same matrices produce exactly the same complete channels and record probabilities. Nevertheless, a static interval of coordinate duration $\Delta T$ has proper duration $\Delta T$ in the first calibration and $2\Delta T$ in the second.

Third, any two deficits in $(0,2\pi )$ admit regular developing patches containing isometric sufficiently small Euclidean balls. Install the same finite experiment in such balls. The complete channels coincide, although the global circumference-to-radius ratios and conical gluing angles differ.

This is nonuniqueness relative to the supplied finite operational data. It does not assert that all the displayed geometric configurations are inequivalent under every possible change of coordinates.

For $n\ge 2$, no rule assigns to each positive-definite inner product on an $n$-dimensional real vector space a Lorentzian bilinear form on that same space, using only that inner product and equivariantly under all linear changes of coordinates.

Proof. At the Euclidean inner product $I$, equivariance makes the assigned form invariant under every orthogonal transformation. Coordinate reflections force its off-diagonal entries to vanish. Coordinate permutations force its diagonal entries to coincide. It is therefore $\lambda I$, which is definite or degenerate, not Lorentzian.

This reproduces the elementary signature refusal recorded in Oai365-A CheungOai365 (Proposition prop:signature). Here we add a time factor, a negative sign, and a time orientation explicitly. A velocity diamond or polytope is not a substitute for that choice.

For $D\ge 2$, ${\mathcal{M}}_C$ has no channel left inverse on its full $C\otimes X$ input space.

Proof. Let

and fix any matter state ${\eta }_X$. The distinct inputs

both have output

A left inverse would send one output to two different inputs.

The proposition concerns the full clock–matter input space. It does not forbid preparing a fresh clock in a known basis state from a classical code. Indeed, on the block-diagonal class

a channel can reconstruct the displayed density operator by preparing $|{\tau }_k⟩$ conditional on $k$. This is recovery of a classical sector, not recovery of arbitrary clock coherence and not return of the consumed names.

Identical clock record laws do not determine the surviving matter state, even on exactly event-calibrated inputs.

Proof. Choose orthogonal matter vectors $|a⟩,|b⟩$. Both inputs

return the deterministic record $0$. Their matter residuals are orthogonal. More generally, postcomposing every selective block with either of two different trace-preserving matter channels preserves the clock probabilities while changing the residual instrument.

Record compatibility and residual compatibility are therefore different assertions, as emphasized in I, VIII-A, and En09-A CheungI,CheungVIIIA,CheungEn09.

For every input operator,

This identity is reference preserving.

Proof. Writing $Z={\sum }_{j,k}|{\tau }_j⟩⟨{\tau }_k|\otimes Z_{jk}$, both sides equal ${\sum }_k{Z}_{kk}$. The same computation retains any reference factor.

The geometric coordinate has not disappeared: for example, an unchanged classical record may remain stored on a static worldline while $T$ increases along that worldline. A fixed record of an earlier reading is not a replacement for all later external time.

For nonzero finite-dimensional $H_C$ and $\hslash >0$, no matrices $A,H$ satisfy

Proof. The trace of a matrix commutator is zero. The trace of the right side is $i\hslash D\ne 0$.

Thus ${\hat{\mathsf{T}}}_C$ is not an exact canonical time operator conjugate to a finite clock Hamiltonian. The continuum coordinate $T$ remains a classical function on the supplied manifold.

Suppose a unitary $U_C$ is applied before the calibrated readout. Its effective pointer observable on the previous input is

Assume the $t_k$ are distinct. If

for a function $f$ on the finite spectrum, then $U_C$ permutes the one-dimensional eigenspaces of ${\hat{\mathsf{T}}}_C$, up to phases. Consequently, a general unitary basis change is not even a finite relabeling of the original pointer, much less a spacetime coordinate transformation.

Proof. The conjugated operator has simple spectrum. Hence $f({\hat{\mathsf{T}}}_C)$ must also have distinct eigenvalues. Its eigenvectors are the original $|{\tau }_k⟩$, whereas those of the conjugated operator are ${U_C}^{†}|{\tau }_k⟩$. Simplicity identifies these two collections of one-dimensional eigenspaces by a permutation.

For a concrete failure, take $D=2$, use the computational reading basis, and apply Hadamard to the input $|0⟩$. The originally deterministic reading becomes a two-outcome distribution. A deterministic relabelling of the original reading cannot do this.

Even an allowed finite relabelling requires more to define a smooth, time-oriented coordinate transformation. If an actual coordinate change $T^{\prime }=f(T)$, with $f^{\prime }>0$, is supplied, tensor transformation gives in the zero-shift case

No such transformation law follows from applying $U_C$ alone. A simultaneous passive transformation of states and observable matrices is instead a change of representation.

The Page–Wootters literature motivates distinguishing extraction of conditional blocks from additional dynamical hypotheses PageWootters,CheungEn09. The following finite example proves exact conditioning without claiming a stationary-universe construction.

Choose any finite-dimensional $H_X$, a self-adjoint ${H_X}^{\mathrm{gen}}$, and

Define the isometry

For every operator $Z$ on $H_X$,

This equality holds with every untouched reference. A record-controlled application of ${U_k}^{†}$ returns the original matter input, including its reference correlations. It does not invert ${\mathcal{M}}_C$ on the full clock–matter input space.

Proof. Orthogonality gives

Substitution into (eq:instrument) proves (eq:history-channel). Controlled inversion gives $D^{-1}{\sum }_k|k⟩⟨k|\otimes Z$. These are operator identities and remain true with a reference. The inverse acts only on the encoded matter task; the phase-erasure counterexample of Proposition prop:no-return still applies to the full input space.

For a concrete preparation, allocate the clock in $|{\tau }_0⟩$, rotate it to $D^{-1/2}{\sum }_k|{\tau }_k⟩$, and apply the controlled unitary ${\sum }_k{P}_k\otimes U_k$. All these operations can be declared at the sender's location before the reading.

Choose a static sender $x_0\in U$, set

and use history-label calibration. The labels refer to the declared points $\kappa (k)$, not to different actual locations of $e_{\mathrm{r}}$. The displayed preparation does not, by itself, prove that a physical matter trajectory visited all the conditional states at those earlier events. It constructs a calibrated finite history encoding.

After consumption, a later reading may allocate a new factor $C^{\prime }$. For example,

preserves the existing record and supplies a new blank. It neither returns the old name nor recovers an unknown phase lost by Enter.

A channel specification does not determine a heat or work bill CheungEn07. For instance, resetting a memory by swapping with a blank ancilla has the same reduced reset channel for every common energy gap $\epsilon >0$. With

an input excited-state probability $p$ produces energy transfers

The blank is consumed and the original information moves to the ancilla. Restoring that ancilla, its preparation resources, and all controls is a separate cyclic-restoration task.

Let $\Lambda$ be a finite set of static sites in $U$, with finite-dimensional site spaces $H_a$. Fix times

At each tick $m$, a partition ${\mathcal{P}}_m$ of $\Lambda$ is supplied. Each block obeys

A tick is one complete product channel

All quantum ports, records, controller memories, and resource wires that are available during execution must be included in the finite network. An external reference has no execution wire.

The access contract is stroboscopic: input and output access is specified at tick boundaries. No continuous-time Hamiltonian or within-tick apparatus dynamics is asserted. A two-site gate is not treated as an instantaneous equal-time interaction. Its dependencies run from input ports at $T_m$ to output ports at $T_{m+1}$.

For $Y\subseteq \Lambda$, write

For a terminal cut $n$, define backward sets by

For the complete scheduled channels,

The terminal reduction to $Y$ factors through the initial reduction to $N_0(Y)$. The factorization is reference preserving. Changing a complete gate outside the ancestor subnetwork does not change the terminal reduction, when wiring and included gates are held fixed.

Proof. At one backward step, an output observable is the identity on every block disjoint from its current support. The adjoint of a trace-preserving channel is unital, so such a block remains an identity. Other blocks can enlarge support only to their union. Induction gives (eq:adjoint-locality).

To obtain a reduced channel, append any fixed state on the initial complement of $N_0(Y)$, apply the complete evolution, and take the output reduction to $Y$. The adjoint inclusion shows that this result is independent of the appended state and agrees with the actual output for every input, including correlated inputs. The same adjoint argument tensored with ${\mathrm{id}}_E$ proves the reference version.

On an excluded gate all pulled-back observables are identities. Any replacement complete channel preserves those identities.

This is the local proof underlying the prescribed-network comparisons in En03-A, En05-A, and En06-A CheungEn03,CheungEn05A,CheungEn06A. It concerns complete unconditioned channels with records retained, not normalized postselection.

Every directed scheduled dependency path in this execution model maps to a future-causal curve in $M$. In particular,

Operational influence under trace-preserving local interventions is contained in the scheduled dependency relation and hence in the prescribed metric causal relation.

Proof. Every elementary dependency connects two sites in one permitted block across duration $\delta$. Condition (eq:block-permission) and Lemma lem:local-causal supply a future-causal segment for that pair. Concatenation supplies a causal curve along a dependency path.

If an intervention acts outside the ancestors, its adjoint acts on an identity in the backward calculation, and therefore cannot change an unconditioned terminal expectation. This proves the influence inclusion.

Declare a two-site scheduled block but choose its channel to be the identity. Its formal block ancestry includes both input sites for each output, although no cross-site influence occurs. Thus the scheduled region can strictly exceed the actual influence region. An entangled input does not invalidate the theorem, but comparing only individual marginals is insufficient for a claim involving joint outputs and references.

Acyclic dependency, finite local updates, and deterministic schedule-independent records do not imply a Lorentzian generated cone.

Proof. Take events indexed by ${\mathbb{N}}^4$, with directed edges $n\to n+e_i$, $i=1,\dots ,4$. Use a deterministic bit relay on any finite predecessor-closed subnetwork: the recorded value is the Boolean OR of incoming values and a local input bit. Every update occurs after its predecessors. The graph is acyclic, has bounded degree, and every finite experiment terminates. Associativity and determinism make records independent of the legal interleaving. A source bit can be detected at every reachable descendant by setting other inputs to zero.

The closed cone generated by edge displacements is

It is not linearly isomorphic to the four-dimensional Lorentz cone. Indeed, $K$ has four extreme rays, whereas the Lorentz cone has infinitely many. An invertible linear map preserves extreme rays.

This is a local counterexample to the proposed inference, not a reproduction of every additional clause of En08-A's stronger countermodel CheungEn08A.

There is a relevant positive classification: a proper, full-dimensional, homogeneous, strictly convex cone in dimension at least three is linearly isomorphic to a Lorentz cone ItoLourenco. Here “proper” means closed, convex, pointed, and with nonempty interior; strict convexity is understood in the cone sense. This theorem classifies a cone satisfying its additional hypotheses; it does not construct those hypotheses from a scheduler.

XXII-F records the additional extraction, invariance, approximation, resource, and layer-separation obligations for an operational instantiation CheungXXIIF. We do not claim to discharge that existence target. Declaring (eq:metric) is not a metric-independent operational extraction rule.

Fix $x_0,x_1\in U$, $x_0\ne x_1$, with

Fix a departure time $T_{\mathrm{d}}$. The prescribed target pair is

The developed straight segment is future null.

Choose any integer $m\ge 1$. Install the $m+1$ sites

where the notation $x_m=x_1$ is understood. Each site has a register $A_j\cong {\mathbb{C}}^d$, with $d=2^p\ge 2$. Set

At tick $j=1,\dots ,m$, apply SWAP to $A_{j-1}\otimes A_j$ and identities to all other sites.

The station locations and tick duration are task data. They are not selected by the unknown payload or inferred from its entanglement.

Prepare $A_1,\dots ,A_m$ in the fixed blank $\sigma =|0⟩⟨0|$, independently of the payload and any reference. For every state ${\rho }_{A_{0}E}$, after tick $j$ the state is

under the fixed register identifications. Consequently:

1. the logical source-to-destination channel is exactly ${\mathrm{id}}_d$, including every untouched reference;

2. all non-output route registers end in a jointly payload-independent blank state;

3. the arrival event is exactly $q$;

4. every scheduled dependency is metric causal;

5. the protocol uses $m$ two-register SWAP gates, $m+1$ payload-sized registers, and $m$ ticks.

Proof. At $j=0$, (eq:swap-invariant) is the preparation. At tick $j$, register $A_j$ is blank and independent. SWAP exchanges it with the entire current payload factor $A_{j-1}$, including all correlations with $E$. The previous site becomes blank, proving the invariant by induction.

At $j=m$, the destination-reference state is the original payload state under the fixed identification. The remaining registers are blank, proving the complete-channel assertions.

Each active pair has separation $a=c\delta$. Hence its input-output dependencies satisfy the local causal certificate. The payload positions at successive ticks trace the prescribed null segment. Finally $m\delta =L/c$, and the gate and register counts follow from the explicit schedule.

This theorem uses a target-adapted substrate. It is not a claim that an arbitrary null displacement is exactly attained on one fixed lattice. En08-B instead supplies resource-controlled approximation for a prescribed null boundary on its declared lattice family CheungEn08B. Our construction has the same essential reference-preserving permutation identity, but a different spatial permission contract. No resource formula is transferred between these models.

For increasing $m$, the gate radius and tick duration tend to zero, while total travel time remains $L/c$ and register count grows. This family is not a continuum quantum field theory.

The theorem certifies the complete finite channel and the causal embedding of its tick-boundary dependencies. It supplies no continuous-time Hamiltonian that implements a two-site SWAP inside one tick, no detector model for its intermediate dynamics, and no field-theoretic derivation of the channel. Those are additional physical-realization obligations, not consequences of the endpoint permutation.

Let the surviving clock-conditioned matter be the register $A_0$. Write ${\mathcal{T}}_m$ for the complete route channel with its blank preparation, retaining all route outputs. Let $\iota$ identify $A_0$ with $A_m$.

For every input state on $C\otimes A_0\otimes E$, clock readout at the sender followed by the route has complete output

where

The clock factor $C$ is absent.

Proof. Apply Theorem thm:transport to each subnormalized positive block of Theorem thm:instrument, and sum the blocks. Linearity gives (eq:composed-channel).

The record in (eq:composed-channel) remains at the sender. Writing a joint output state is not instantaneous delivery of that record to the receiver. If a receiver operation is to depend on $k$, a causal classical record wire must be added, or the record-controlled operation must be performed locally before departure.

With a $q$-qubit clock and $p$-qubit payload registers, the explicit laboratory quantum peak is at most

After clock consumption it is $p(m+1)$ until the route residuals are released. The clock record has $q$ classical bits. An untouched $r$-qubit reference contributes $r$ to a separate external-storage ledger, or to a total ledger if that convention is chosen. The preinstalled schedule, site metadata, and blank preparations are declared resources, not outputs supplied by the record.

The peak in (eq:peak) does not include undeclared synthesis ancillas: the named local unitaries are permitted primitives of this model. No compilation into a smaller universal gate library is asserted.

For compatible channels define

If the implemented preparation, readout, $m$ route gates, and output interface differ from their ideal versions by at most ${\epsilon }_{\mathrm{prep}}$, ${\epsilon }_{\mathrm{read}}$, ${\epsilon }_1,\dots ,{\epsilon }_m$, and ${\epsilon }_{\mathrm{out}}$, respectively, then the complete output channel differs by at most

Proof. Insert and subtract hybrid compositions in which one component at a time is replaced by its ideal version. The triangle inequality and contraction of the diamond norm under pre- and postcomposition by channels give the bound.

This estimate is not exact causal containment of an erroneous device. Nor does a per-gate error tending to zero imply a vanishing total error for an arbitrarily growing number of gates.

For a declared smooth geometric metric $q$, a declared covariant source tensor ${\mathsf{T}}_{\mu \nu }$, constants $\Lambda$ and ${\kappa }_{\mathrm{grav}}\ne 0$, define the comparison equation

This equation is not a rule already contained in (eq:instrument). A finite conditional density matrix is not a covariant spacetime tensor until a field, observable, state, renormalization prescription, and map to tensor-valued distributions are supplied.

A symmetric, conserved, traceless tensor on a background need not source that background through (eq:einstein).

Proof. In a flat four-dimensional chart with metric $\eta =\mathrm{diag}(-1,1,1,1)$, let

It is constant and symmetric, its covariant divergence vanishes, and ${\eta }^{\mu \nu }{\mathsf{t}}_{\mu \nu }=0$. For $\Lambda =0$, the left side of (eq:einstein) is zero while the right side with $\mathsf{T}=\mathsf{t}$ is nonzero.

This is the four-dimensional version of the elementary refusal in VIII-C; local Ward identities likewise do not, by themselves, supply Einstein dynamics CheungVIIIC,CheungWard.

If the components of a nondegenerate metric are constant in all spacetime coordinates, its connection coefficients and curvature tensors vanish. Positive constant rescaling preserves this conclusion. A family of such constant metrics has zero linearized Einstein tensor for its parameter variation.

Proof. Every spacetime derivative of every component vanishes, so

The curvature formulas then vanish. Constant rescaling and differentiation with respect to an external family parameter retain spacetime-constant components.

The needed statement is the spacetime-constant one CheungXXIIE. The phrase “spatially constant” alone would be insufficient: in units $c=1$,

is spatially homogeneous but has

These follow directly from ${{\Gamma }^0}_{ij}=H{q}_{ij}$ and ${{\Gamma }^i}_{0j}=H{{\delta }^i}_j$.

By Proposition prop:holonomy, the main conical metric has $G_{\mu \nu }=0$ on its regular part. This says nothing about a distributional defect source on $\mathcal{W}$, which is not constructed here. Its conical gluing is prescribed rather than sourced by the Enter residual.

More generally, if a cosmological Einstein–scalar background is adjoined, its equations must be imposed on the geometric side. A quotient or relational scalar construction does not insert a missing metric source equation. This is the equation-space discipline illustrated by III and VI CheungIII,CheungVI.

The experiment has no map

in its specification. Consequently conditioning, changing a reading basis, forgetting a record, or allocating a fresh clock does not change the prescribed metric. Any such update would be an extension of the model with its own consistency and dynamics obligations.

Fix $0<\mu <2\pi$, $c>0$, a regular convex developing patch $U$, a clock dimension $D=2^q\ge 2$, and a payload dimension $d=2^p\ge 2$. Then the following finite operational package exists.

1. Its smooth background is the time-oriented $3+1$-dimensional Lorentzian manifold

   $$M={\mathbb{R}}_T\times {C_{\mu }}^{\circ }\times {\mathbb{R}}_z,g=-c^2\mathrm{d}{T}^2+\mathrm{d}{r}^2+{\alpha }^2{r}^2\mathrm{d}{\varphi }^2+\mathrm{d}{z}^2.$$

   Its completion has the declared conical worldsheet $\mathcal{W}$.

2. It has one designated live clock factor $C$, finite matter names, a location map, accessible algebras, and a complete finite control and resource interface.

3. Destructive clock Enter is exactly (eq:instrument), including every untouched reference. It consumes $C$, writes a classical code, and retains the subnormalized conditional matter block.

4. An explicit dictionary matches the code to geometric time labels and the surviving carrier names to three spatial rods. Exact equality with the actual reading-event coordinate holds on the event-calibrated support class of Proposition prop:fixed-event. Nontrivial multi-outcome history readout instead uses the separate event map $\kappa$.

5. Scheduled dependencies are contained in the prescribed metric causal relation. Actual intervention influence is contained in scheduled dependency.

6. Every prescribed straight future-null displacement inside the patch has the finite target-adapted SWAP protocol of Theorem thm:transport, with exact complete-channel correctness and the resource ledger (eq:peak). Its composition with clock readout is (eq:composed-channel).

7. The clock instrument has no universal left inverse on its full input space, and no finite canonical time commutator is available. The supplied metric and its time coordinate remain after clock consumption.

Signature, spatial chart, speed, lapse, calibration, and transport permissions are inputs. Einstein dynamics are not derived.

Proof. The geometric assertions are Propositions prop:geometry and prop:holonomy. Choose any finite set of sites in $U$, assign qubit registers, and prescribe the finite primitives of Definition def:experiment; this supplies item 2.

Theorem thm:instrument proves item 3. For item 4, choose distinct $t_k$ and a static sender $x_0$. The family

is a nonempty event-calibrated family. A classical task input selects which $e_k$ and preparation are used. For a pure matter input these are pure complete laboratory preparations. Thus actual event-time matching is exhibited, not merely postulated.

For a separate nontrivial history experiment, choose $\kappa (k)=(t_k,x_0)$, read at a later fixed event, and use the finite preparation (eq:history-isometry). Proposition prop:history gives all $D$ outcomes and their complete conditional outputs. The rod map is the explicit developing chart $\chi$.

Theorems thm:locality and thm:metric-containment give item 5. The sites (eq:chain-sites), their partition schedule, Theorem thm:transport, and Corollary cor:clock-transport give item 6. All preparations, clocks, site factors, records, and references have the declared ledger.

Propositions prop:no-return and prop:no-ccr prove item 7. The geometry is fixed metadata throughout the specification, so no instrument operation removes or changes its coordinate function. No source-extraction or metric-update map has been introduced.

In this model, Enter acts on a finite quantum encoding calibrated to the metric's time axis. It can change the selected clock observable through preceding local gates, extract a time label, condition surviving matter, and enable declared causal classical control. It does not consume a spacetime dimension, move the readout event backward in $T$, or derive a new metric.

Proof. The positive statements are the specified unitary and instrument operations. Their outputs and locations are those already proved. The negative statements follow because $M,g,T$ are unchanged classical components of the specification.

The following six stronger implications have not been supplied.

1. A chart reconstructed from records without a predeclared location map.

2. Lapse coefficients reconstructed from records without a geometric clock and background prescription.

3. A Lorentz cone selected by scheduler consistency alone. The conditional classification and XXII-F's operational existence target remain distinct from our prescribed cone.

4. A covariant stress tensor extracted from the residual that sources the same metric through a proved dynamics law.

5. The native quantum-lapse construction and its compatible joint regulator removal.

6. A calibration to proper time derived from records rather than supplied and checked against a declared metric and observer.

The last item does not retract Proposition prop:geometry: for the chosen unit-lapse static observers, equality of coordinate and proper-time increments is proved. What is not derived is the calibration of a physical device to that ideal geometric clock.

. A prescribed spacetime and a finite Enter experiment have been paired by written maps. This is a spacetime model under Enter in the stated operational sense. It is not spacetime emergence, holographic reconstruction, a quantum field theory, or a derivation of gravity. No operational wormhole certificate is identified with a geometric bridge. No catalogue family is a premise.

This appendix gives a self-contained classical composition interface. It does not replace the conical metric by a polynomial lapse or claim to reproduce every native input of XX-A or Em02-E. Those manuscripts supply their own frozen background, clock, rods, and quantum admission contracts CheungXXA,CheungEm02E.

Throughout the appendix set $c=1$, and use $\epsilon$ for a perturbation parameter. It is neither a schedule tick nor a clock reading.

Suppose a smooth Lorentzian metric $q_{\epsilon }$ and a scalar clock ${\mathcal{T}}_{\epsilon }$ are supplied on a region where $\mathrm{d}{\mathcal{T}}_{\epsilon }$ is timelike. Define the positive raw lapse

Supply original slice-harmonic rods and suppose the resulting coordinate map

is locally invertible, with inverse ${\alpha }_{\epsilon }$. Then the relational lapse is

In these relational coordinates the full metric has ADM form

Indeed, writing the metric as $A\mathrm{d}{\tau }^2+2B_i\mathrm{d}\tau \mathrm{d}{y}^i+{\gamma }_{ij}\mathrm{d}{y}^i\mathrm{d}{y}^j$, set ${\beta }^i={\gamma }^{ij}{B}_j$ and $n^2=-A+{\gamma }^{ij}{B}_i{B}_j$; completing the square gives (eq:adm), and inversion gives $q^{\tau \tau }=-n^{-2}$.

Thus a lapse alone does not specify the metric. Along a curve of constant rods, proper-time rate is

when the curve is timelike. Along a normal curve $\mathrm{d}{y}^i=-{{\beta }_{\epsilon }}^i\mathrm{d}\tau$, it is $n_{\epsilon }$. Constant rods and normal observers must not be silently identified.

If $q_{\epsilon },{\phi }_{\epsilon }$ are required to satisfy Einstein–scalar equations, those equations are additional geometric input. Merely supplying $n_{\epsilon }$ does not impose them.

Work in background-adapted coordinates for which the background label map is the identity. Suppose the raw lapse has factorial jets

At fixed clock label, let the forward spatial rod map be

Let the inverse time map at fixed spatial argument be

The full inverse first inverts $R_{\epsilon }$, then substitutes the resulting argument into ${\theta }_{\epsilon }$.

Writing

one obtains

and

Proof. Substitution into $R_{\epsilon }(\tau ,S_{\epsilon }(\tau ,y))=y$ gives

Expanding ${\theta }_{\epsilon }(\tau ,S_{\epsilon }(\tau ,y))$ gives the time component, including $-2{Y_1}^i{\partial }_i{u}_1$.

Under these hypotheses,

where