En09-A Realization of Relational Time by Destructive Enter on a Clock Register

We construct a finite operational realization of Page–Wootters clock conditioning using destructive Enter on a named clock register. The complete output consists of a classical clock code and the surviving system; the measured clock names are consumed. An explicit channel identity proves equality of the Born record law and the unnormalized conditional system states with their Page–Wootters counterparts, including arbitrary untouched references. Under a separately stated covariance condition for the clock basis, stationarity of the joint preparation implies unitary relational evolution at the calibrated readings. An exact finite spectral construction establishes that these hypotheses are jointly inhabited.

The probability and update formulas have two distinct justifications: they follow algebraically from the specified instrument, and, when the full Enter characterization hypotheses are satisfied, they are the measurement semantics selected by the previously established operational characterization. We do not rederive that characterization from abbreviated axiom names.

We also give a typed hybrid propagator, complete-channel stability bounds, and refinement independence within an explicitly matched class. The philosophy of Enter is expressed as an interface discipline: a destructive record is not a surviving clock, a decoder, a thermodynamic ledger, or a spacetime geometry. The construction realizes relational extraction, not clock-free quantum gravity, a dynamical causal cone, or spacetime emergence.

. Enter calculus; Page–Wootters conditioning; quantum clocks; history states; destructive instruments; linear resources; reference-stable channels.

The Page–Wootters proposal describes system dynamics through correlations with an internal clock in a stationary joint description PageWootters. Finite clock history states and their circuit realizations provide a concrete setting in which this conditioning can be studied BoetteRossignoli,McClean,Diaz. The present manuscript supplies a particular operational extraction interface for such correlations: destructive Enter on the clock.

Enter is not an alternative name for an arbitrary measurement. Its defining access discipline is that the measured quantum names do not survive as quantum outputs. The readout of those names is classical. Unmeasured system names can nevertheless survive in the same complete process. This distinction is central to Cheung's Enter calculus and to the accompanying philosophy of destructive readout and refusal of identification CheungEnter,CheungI,CheungPhilosophy.

Our claim is therefore an interface realization, not an identification of all relational-time formalisms with a programming primitive. Mathematical conditioning on a clock projector need not itself consume a clock. The construction below chooses a destructive implementation, proves its complete channel, and accounts for the resource that this implementation consumes.

claims that must remain separate.

1. Extraction: a clock-conditioned state is returned as the residual system accompanying a classical record.

2. Dynamics: those conditional states satisfy a specified evolution law. This requires clock covariance in addition to stationarity.

3. Foundational semantics: the measurement rule is selected by an operational characterization inside the full strategy class of the Enter theorem.

The first is an explicit channel identity. The second is proved under stated mathematical hypotheses. The third invokes an identified predecessor result and is not inferred merely by writing down a Born-form instrument.

-claims installed at the outset. No theorem below constructs a residual matter clock, returns the consumed clock as an unknown quantum state, produces an exact all-real reading register, derives a metric or Einstein dynamics, or supplies a Page curve. External calibration, preparation, scheduling, and any subsequently supplied physical embedding remain declared inputs. In particular, finite operational consistency does not determine a Lorentz cone CheungEn08A (Theorem thm:main).

The manuscript is independent of the unresolved characteristic quantum-product programme described in the supplied XIV–XVI records. A finite readout cannot supply an absent field-theoretic coefficient, its representation, or its admission.

Fix disjoint finite sets of quantum names $C$ and $S$, with

Their Hilbert spaces are

A register $Q$ denotes an arbitrary finite untouched reference. It may be entangled with $C\otimes S$. It is not available to the clock-reading program.

Write

The output record algebra is the diagonal algebra in the distinguished basis $\{|k{⟩}_R{\}}_{k\in \mathsf{Z}}$. Thus a recorded output has the form

This is a classical–quantum state, not a coherent quantum return of the clock.

The frozen operational interfaces are:

1. a preparation of ${\rho }_{CS}$, or a declared preparation channel with this output;

2. the accessible clock input $C$;

3. the surviving system output $S$;

4. the retained classical record $R$;

5. all untouched references, acted on by ${\mathrm{id}}_Q$.

If a purification, auxiliary token, encoding apparatus, or fresh clock supply is used, its ownership and access are declared separately. Nothing in the readout theorem acquires a state-dependent expectation token.

Choose an orthonormal clock basis

Choose an injective calibration

The stored object is the finite code $k$. The real number $t_k$ is its declared interpretation; no finite register is asserted to contain every real number exactly.

Let $B_C$ be the calibrated basis-change unitary satisfying

Reading the clock means applying $B_C$, then destructive computational-basis Enter to every name in $C$. If the clock basis is already the computational basis, $B_C=I_C$.

The logical clock Hamiltonian need not be diagonal in the computational presentation. Indeed, confusing an energy basis with a time-reading basis can destroy the desired dynamical interpretation; see Example ex:energy-basis.

Fix self-adjoint finite matrices $H_C$ and $H_S$, and put

A density operator is stationary when

A stronger zero-constraint preparation satisfies

For a positive density operator, (eq:strong-constraint) means that its support lies in $\ker K$, and implies (eq:stationarity). For a pure preparation the stronger condition is $K|\Psi ⟩=0$.

These two conditions are not silently identified. The extraction identity requires neither. Stationarity is used only in the relational-evolution clause.

The calibrated clock basis is covariant for the readings $\{t_k\}$ when

This is an explicit compatibility condition between the clock generator, reading basis, and external calibration.

Covariant clock observables are an established ingredient of relational quantum dynamics Trinity. Here we require only the finite sampled condition (eq:clock-covariance); we do not import an infinite-dimensional physical-inner-product or Dirac-observable construction.

The distinction between a specified instrument and a derived measurement semantics is important enough to make explicit.

Whenever a realization is described as using derived Enter semantics, its measurement strategy belongs to the full structural class of Cheung's characterization theorem and satisfies its composability, no-signaling, nullity, continuity, and typed record requirements in their stated forms.

The imported conclusion is that, for a pure finite configuration $|\Xi ⟩$, a measured subset, and outcome $k$, the destructive component $D_k|\Xi ⟩$ has probability

and, if $p_k>0$, residual ray

No residual is prescribed on a zero-probability branch.

This is the conclusion of Landmark I, version 2, thm:main, with its stated strategy class CheungI; Cheung's journal article distinguishes specified Born semantics from their subsequent operational characterization CheungEnter. Hypothesis hyp:enter retains the full predecessor hypotheses: state sufficiency, coherent dynamics, evaluation conventions, finite configurations, and the precise record condition are not replaced by informal slogans.

We use the characterization as a cited semantic result, not as a new theorem proved from the five abbreviated requirement names. An arbitrary laboratory apparatus is not certified to belong to that class by this manuscript.

Under Hypothesis hyp:enter, destructive readout in the clock basis has unnormalized branch map

The same formula holds with every untouched reference.

Proof. For a pure input, multiplication of the normalized positive-branch residual projector by its probability gives

A zero component gives the zero operator, without assigning a normalized residual.

For a mixed input, purify it with a reference $E$, apply the pure formula with $D_k\otimes I_E$, and take the required reduced view. The result is $D_{k}X{D}_k^{†}$. This expression depends only on $X$, so it is independent of the purification or ensemble presentation. Keeping rather than tracing a reference gives the same formula with its identity factor.

Thus the density-operator channel used below is a consequence of the imported semantics when Hypothesis hyp:enter applies. Independently, it is an explicitly specified finite quantum instrument.

Define the complete destructive clock-readout channel by

Equivalently, in the computational presentation,

For a preparation ${\rho }_{CS}$, define

Equation (eq:pw-normalized) is the clock-conditioned Page–Wootters state for this finite observable.

The appropriate channel target is not the normalized quotient in (eq:pw-normalized). It is the linear recorded map

Keeping subnormalized blocks prevents postselection from being mistaken for a deterministic channel.

For the frozen data of Section sec:data, the following statements hold.

1. Complete instrument identity. The map ${\mathcal{M}}_C$ is completely positive and trace preserving, and

   $${\mathcal{M}}_C={\mathcal{J}}_C$$

   as linear maps on all clock–system operators. This equality remains valid after tensoring with ${\mathrm{id}}_Q$ for every untouched reference.

2. Record law and residual. On input ${\rho }_{CS}$, the probability of code $k$, hence of calibrated reading $t_k$, is

   $$p_k=\mathrm{Tr}\left[(P_k\otimes I_S){\rho }_{CS}\right].$$

   On a positive branch, the surviving system is exactly ${\rho }_S(t_k)$ of (eq:pw-normalized). Under Hypothesis hyp:enter, these are the probabilities and updates selected by the Enter characterization, not an additional measurement postulate introduced here.

3. Linear consumption. The readout consumes every measured quantum name in $C$. Its quantum output context contains $S$, not $C$. A continuation in the same process cannot use a consumed clock name as a quantum input.

4. Relational evolution under covariance. If ${\rho }_{CS}$ satisfies (eq:stationarity) and the reading basis satisfies (eq:clock-covariance), then

   $${\sigma }_{S,k}=U_S(t_k){\sigma }_{S,0}{U}_S(t_k{)}^{†},U_S(t)=e^{-i{H}_{S}t/\hslash }.$$

   In this complete covariant basis, $p_k=1/d$, and therefore

   $${\rho }_S(t_k)=U_S(t_k){\rho }_S(0)U_S(t_k{)}^{†}.$$

5. Complete operational realization. The record, surviving system, and reference extension realize precisely the declared relational extraction. Discarding the record yields

   $${\mathrm{Tr}}_R[({\mathcal{M}}_C\otimes {\mathrm{id}}_Q)(X)]={\mathrm{Tr}}_{C}X.$$

   No quantum clock output or spacetime identification is part of this channel.

Proof. Define Kraus operators from ${\mathcal{H}}_C\otimes {\mathcal{H}}_S$ to ${\mathcal{H}}_R\otimes {\mathcal{H}}_S$ by

Then ${\mathcal{M}}_C(X)={\sum }_k{A}_{k}X{A}_k^{†}$, which is completely positive. Orthonormality of the clock basis gives

Hence ${\mathcal{M}}_C$ is trace preserving.

For an arbitrary operator $X$, expand in the clock basis:

Both

equal $X_{kk}$. This proves (eq:exact-channel-identity) on all operators, not only on the frozen preparation. Replacing $I_S$ by $I_S\otimes I_Q$ proves the reference statement.

The $R$-marginal is

Cyclicity of the full trace and ${P_k}^2=P_k$ give

proving (eq:born-clock). Dividing the positive $k$-block by its trace gives exactly (eq:pw-normalized). Lemma lem:density-semantics supplies the semantic attribution under Hypothesis hyp:enter.

For consumption, the register-level readout has the first-order interface

It is implemented by the calibrated basis change and destructive Enter on the $m$ clock names. Each elementary readout removes its measured name from the quantum context. After all $m$ readouts, the surviving quantum context is $S$. Tensor permutation and classical packaging do not recreate a quantum name. This proves the linear-resource assertion. The system output in (eq:linear-interface) is unmeasured residual output; it is not a quantum return of any name in $C$.

To prove the dynamical clause, fix $t=t_k$, write $U_C=U_C(t)$, $U_S=U_S(t)$, and use stationarity:

Consequently,

For partial trace over $C$, the two-projector expression in (eq:pw-subnormalized) equals the corresponding single-projector expression. Using covariance and unitary invariance of the partial trace,

Thus all $p_k$ coincide. Their sum is one and there are $d$ outcomes, so $p_k=1/d$. Normalization proves (eq:relational-evolution).

Finally, the clock-basis expansion gives

The same calculation includes $Q$. The complete channel therefore returns exactly the record and residual specified above. Its codomain has no quantum $C$ factor, and no geometric object occurs in its definition.

The operation acts as ${\mathrm{id}}_Q$, but the conditional reference state may change when a clock outcome is selected. Such steering is compatible with (eq:drop-record): the unconditional reference marginal is unchanged.

The relational-evolution assertion on $S\otimes Q$ additionally requires stationarity of the full preparation under $K\otimes I_Q$. Stationarity of the $CS$-marginal alone does not imply stationarity of every extension.

The channel identity holds for every preparation and every calibrated orthonormal basis. Stationarity alone does not make an arbitrary basis a time observable. Covariance is the extra hypothesis that justifies the Schrödinger-type interpretation in (eq:relational-evolution).

Take one clock qubit and one system qubit, with

Then $K|\Psi ⟩=0$. Enter in the clock energy basis produces system states $|0⟩$ and $|1⟩$, each with probability $1/2$. However, $U_S(t)|0⟩⟨0|U_S(t{)}^{†}=|0⟩⟨0|$ for every $t$; it cannot turn the first conditional state into the second.

The problem is not failure of the extraction identity. It is failure of the proposed basis to satisfy clock covariance for two distinct readings. Changing to the clock basis $|{\tau }_0⟩=(|0⟩+|1⟩)/\sqrt{2}$, $|{\tau }_1⟩=(|0⟩-|1⟩)/\sqrt{2}$, with $t_1=\pi /\omega$, produces the conditional system states $|+⟩$ and $|-⟩$, which do satisfy the asserted evolution.

The theorem's covariance and constraint assumptions are compatible, but not automatic. We now construct a nonempty exact class.

Let $d=2^m$, $\omega >0$, and let $\{{\Pi }_a{\}}_{a=0}^{d-1}$ be mutually orthogonal projections on ${\mathcal{H}}_S$ with ${\sum }_a{\Pi }_a=I_S$; zero projections are allowed. Set

Choose a clock energy basis $\{|e_a⟩{\}}_{a=0}^{d-1}$ and define

Then $W:{\mathcal{H}}_S\to {\mathcal{H}}_C\otimes {\mathcal{H}}_S$ is an isometry, $KW=0$, and the clock basis is covariant at the readings $t_k$. Moreover,

and, for every operator $X$,

All identities hold with every reference.

Proof. Orthogonality and completeness of the projections imply

On the term $|e_a⟩\otimes {\Pi }_a|\varphi ⟩$, the clock and system energies sum to zero, so $KW=0$.

The finite Fourier identity

proves orthonormality of $\{|{\tau }_k⟩\}$. The sign in (eq:clock-spectrum) gives

hence covariance.

Expanding the right side of (eq:history-encoding), using

and summing over $k$, leaves precisely (eq:stationary-encoding). Applying $D_k$ to (eq:history-encoding) gives $D_{k}W=d^{-1/2}{U}_S(t_k)$. Substitution in (eq:readout) proves (eq:spectral-recorded-channel). Tensoring the operator identities with an identity proves the reference versions.

This construction is exact for the stated commensurate spectrum. It does not encode an arbitrary prescribed Hamiltonian into the same finite additive constraint without an additional approximation or spectral-matching theorem. Its readings are periodic with period $2\pi /\omega$; an absolute winding number would require an additional classical interface.

The negative clock generator in (eq:clock-spectrum) is a mathematical constraint convention, not a thermodynamic claim. One may shift its spectrum upward and replace $K=0$ by a fixed total-energy constraint without changing the covariance or readout calculation.

For nonzero finite-dimensional ${\mathcal{H}}_C$, no matrices $T_C,H_C$ satisfy

Proof. The trace of a matrix commutator is zero, whereas the trace of $i\hslash I_C$ is $i\hslash \dim {\mathcal{H}}_C\ne 0$.

This obstruction does not contradict Proposition prop:spectral-realization. Sampled covariance of a finite periodic reading basis is not an exact canonical commutation relation.

History-state algorithms are established quantum-computing constructions McClean,BoetteRossignoli,Diaz. Their connection to Enter can be expressed without presuming that every history state satisfies the additive constraint (eq:constraint).

Let $V_0,\dots ,V_{d-1}$ be unitaries on ${\mathcal{H}}_S$, and define

Then $W_V$ is an isometry, and computational-basis Enter on the clock realizes the channel

The clock names are consumed and the system names survive.

Proof. Orthogonality of the clock basis gives

The $k$-branch destructive component is

Equation (eq:circuit-extraction) follows from Theorem thm:main. Its resource assertion is the same typing calculation.

A preparation interface for $W_V$ consists of

The controlled unitaries are declared operations. Exact synthesis from a particular finite gate library, its approximation error, and its storage cost are separate implementation obligations.

If $V_{k+1}=G_k{V}_k$, define maps

Then the positive propagation Hamiltonian

annihilates $W_V$, because $L_k{W}_V=0$. This is a legitimate history-state constraint, but generally not the noninteracting sum $H_C+H_S$. A propagation constraint is not silently substituted for the frozen additive constraint.

{Analogs grounded in Theorem thm:main and Corollary cor:circuit-history. A correspondence between interfaces is not an identity between physical theories.}

Let $q:\mathsf{Z}\to \mathsf{Y}$ be a declared finite coarse-graining. Reporting only $y=q(k)$, while retaining $S$, gives branch maps

Their sum is trace preserving. On a positive branch,

This is generally a mixture of fine conditional states, not the state at a single exact time.

Proof. Classical relabeling adds the corresponding diagonal output blocks. Completeness follows by summing over all $y$. The normalized formula is division of the resulting subnormalized block by its trace.

Failure or rejection is treated in the same way: its flag is part of the complete output alphabet. If the rejected residual is discarded or replaced, that action is an additional declared channel. A successful normalized branch alone is not the complete experiment.

For a local clock instrument with Kraus operators $\{F_{y,a}:{\mathcal{H}}_C\to {\mathcal{H}}_{C^{\prime }}{\}}_a$, suppose the entire output $C^{\prime }$ is also discarded or consumed, while $S$ is untouched. Its residual system branch is

The equality follows by pairing both sides with every system observable and using trace cyclicity. Thus a represented local POVM can provide a further destructive conditioning interface.

Equation (eq:destructive-povm) does not say that effects determine arbitrary residual instruments. It uses the special assumptions that only the clock is acted on and that all quantum clock output is removed. Measuring an ancilla pointer while returning a quantum clock is a different interface.

The supplied X-Enter selective-readout theorem is an appropriate comparison for retaining failure branches and proving selective-map, rather than effect-only, agreement CheungXEnter (Theorem thm:relational). Its encoded phase model is not a premise or a specialization of the clock Hamiltonian construction here.

The philosophical contribution is not an extra dynamical axiom. It is the requirement that an interpretation remain attached to the input, access rights, outputs, and continuation that actually support it CheungPhilosophy. For the present construction this discipline has five precise consequences.

Before the readout, the preparation contains quantum correlations between a clock and a system. After the readout, the accessible interface contains a finite classical label and a conditional system. The correlations supply the relational description; Enter supplies the chosen operation that makes a classical label available.

The operation does not create the family $U_S(t_k)$ from nothing. Its dynamical content was already fixed by the preparation, constraint, and clock calibration. Enter realizes extraction of that content.

A consumed quantum name is absent from the continuation's quantum context. This does not assert annihilation of the laboratory device, of its environment, or of every physical degree of freedom that implemented the channel.

A reversible dilation may retain information in inaccessible degrees of freedom. Such degrees of freedom are not thereby returned through the declared clock-reading interface.

For $d\ge 2$, ${\mathcal{M}}_C$ has no channel left inverse on its full clock–system input space. In particular, with a fixed system preparation, its output cannot universally return an arbitrary unknown quantum clock state.

Proof. Let

and let ${\eta }_S$ be any fixed system state. The distinct inputs $|{\chi }_{+}⟩⟨{\chi }_{+}|\otimes {\eta }_S$ and $|{\chi }_{-}⟩⟨{\chi }_{-}|\otimes {\eta }_S$ have the same output:

A left inverse would have to map this single output to both inputs, which is impossible.

This does not exclude recovery of information from a separately declared code in the surviving system or another accessible factor. For example, a record-controlled inverse of (eq:spectral-recorded-channel) recovers its encoded system input. That is not a return of the consumed clock names, and it does not invert ${\mathcal{M}}_C$ on the full input space.

The classical marginal contains $\{p_k\}$. The complete output also contains the blocks $\{{\sigma }_{S,k}\}$. Agreement of the first does not imply agreement of the second.

There are recorded channels with the same clock outcome law on every input and different conditional system states.

Proof. Let $S$ contain a qubit, let $Z_S$ be its Pauli $Z$, and compare

Unitary conjugation preserves each block trace, so their record laws agree on every input. For input $|{\tau }_0⟩⟨{\tau }_0|\otimes |+⟩⟨+|$, the conditional qubit output is $|+⟩$ for $\mathcal{F}$ and $|-⟩$ for $\mathcal{G}$. The residual states are orthogonal.

This elementary example expresses the same interface separation emphasized by VIII-A's residual-obstruction theorem and XIX-C's record–residual analysis CheungVIII,CheungXIXC. It explains why complete recorded channels, rather than record frequencies alone, appear in the stability statements below.

Let $\mathcal{S}$ be a convex state set. If an affine channel assigns each $\rho \in \mathcal{S}$ a classical distribution concentrated at one label $\ell (\rho )$, and distinct labels are distinguishable, then $\ell$ is constant.

Proof. If $\ell (\rho )\ne \ell (\sigma )$, affinity forces the classical output on $a\rho +(1-a)\sigma$, $0<a<1$, to be

This distribution is not concentrated at one label, a contradiction.

Accordingly, the record $k$ is a sample from a law determined by the clock preparation. It is not an exact single-run label of an arbitrary unknown expectation value. Finite supplied tokens, repeated-preparation estimation, and approximate acquisition are different tasks, as stressed in XIX-C's prop:token and prop:affine CheungXIXC.

Equation (eq:drop-record) is a channel equality. It does not assign measurement work, heat, reset cost, an energy budget, or environmental leakage. A thermodynamic implementation requires its own apparatus and resource specification.

Likewise, the code $k$, calibrated scalar $t_k$, and program schedule are not a metric tensor or a matter-defined relational lapse. Em02-E's classical lapse construction uses its own background and relational-map hypotheses CheungEm02E. No comparison map identifying that lapse with the present clock code has been constructed.

Thus the analog in Table tab:analogs is useful precisely because it does not promote a finite interface correspondence into an identity with gravitational spacetime.

The relational label $t_k$ and the external operational schedule parameter $s$ have different roles. The former labels a conditional state extracted from one preparation. The latter locates preparation, readout, and later operations in a declared experiment.

Before readout, use the live operator space

After readout, use

The space ${\mathcal{A}}_{+}$ is a finite classical–quantum algebra. There is no surviving clock factor in it.

Choose a readout event with ordered germs

They denote the interfaces immediately before and after the event. They may share a numerical scheduling coordinate with distinct event-order tags; no positive physical duration is inferred from this notation.

Let

be the declared pre-event propagator. On a stationary preparation it leaves the preparation unchanged.

For the post-event interface, let ${\mathcal{V}}_k(s,u)$ be any declared CPTP propagator satisfying

Define

The minimal construction chooses every ${\mathcal{V}}_k=\mathrm{id}$. Time-dependent Hamiltonians or other admitted post-event dynamics are also allowed. No time-homogeneous semigroup is assumed.

Define the typed operational propagator by

With the event germs included as endpoints, every expression is a CPTP map between its declared interfaces and

for every compatible ordered triple.

Proof. The pre-event and post-event maps have their stipulated composition laws. A crossing interval contains exactly one readout. Splitting such an interval before the readout composes two pre-event factors; splitting it after the readout composes two post-event factors. In either case the factors reduce to the middle line of (eq:hybrid-T). No composition introduces a second ${\mathcal{M}}_C$. Complete positivity and trace preservation are preserved under composition.

The type distinction is necessary. Writing ${\mathcal{M}}_C{\mathcal{M}}_C$ would be meaningless on these interfaces: the first readout has already consumed its clock input.

Under the covariance hypotheses, the mathematical family

acts on the recorded interface and satisfies a composition law. It transports a conditional system state from the represented reading $u$ to $t$ while leaving the stored record unchanged. It does not create a new observation of the consumed clock or rewrite the original event as if it had occurred at a new reading.

Multiple physical clock readings require additional clock resources or a different nondestructive protocol, with their own complete instruments. One destructive readout of a history state is not, by itself, a multi-time intervention experiment.

For compatible finite linear maps define

For states, $d_{\mathrm{tr}}(\rho ,\sigma )=\frac{1}{2}{\Vert \rho -\sigma \Vert }_1$. The half-diamond normalization gives $0\le d_{\diamond }(\Phi ,\Psi )\le 1$ for channels and includes arbitrary reference-entangled inputs Watrous (Chapter 3).

By (eq:exact-channel-identity),

This is equality on the full fixed interface, not merely agreement at one calibrated preparation.

Suppose $\widetilde{{\mathcal{M}}_C}$ is a channel with the same classical record and system interface, with classical output blocks, and

For every reference-entangled input, write its two outputs as

Then

Every subsequent complete channel preserves the same error upper bound. If $p_k\ge p_{\min }>0$ and $\epsilon <p_{\min }$, then ${\tilde{p}}_k>0$ and

Proof. The diamond bound gives trace distance at most $\epsilon$ for the two complete outputs. Trace norm is additive on their orthogonal classical blocks, proving the first inequality in (eq:record-block-bound). Taking the classical marginal is a channel and proves the second. Channel postprocessing contracts trace distance, including with a reference.

Fix $k$, abbreviate

The classical marginal bound implies $\left|p-q\right|\le \epsilon$; thus $q>0$ under the stated lower bound. The sum of the norms of all other output blocks is at least $\left|p-q\right|$, because their total trace difference is $q-p$. Therefore

Using positivity and $\mathrm{Tr}\tilde{\sigma }=q$,

State trace distance is also at most one.

A positive lower probability bound is essential for a uniform normalized-branch statement. Complete-channel convergence alone does not control conditioning on arbitrarily rare outcomes. The exact covariant finite model has $p_{\min }=1/d$, but this bound need not remain uniform if the clock dimension grows.

Let $G_1,\dots ,G_N$ and ${\tilde{G}}_1,\dots ,{\tilde{G}}_N$ be compatible channels, with corresponding factors on the same comparison interfaces. Then

Proof. Subtract the two compositions and replace one factor at a time:

Channels have diamond norm one. Submultiplicativity therefore bounds each term by ${\Vert {\tilde{G}}_j-G_j\Vert }_{\diamond }$. The triangle inequality and division by two prove the sum bound. The distance between channels is at most one.

This lemma applies to finite adaptive instruction trees after their classical controls and all reported outcomes are represented on the complete interfaces. A branch deleted by postselection is not such a trace-preserving factor.

Fix a complete finite target channel $\Phi :\mathcal{B}({\mathcal{H}}_{\mathrm{in}})\to {\mathcal{A}}_{\mathrm{out}}$, including its preparation, calibration, clock Enter instrument, all retained records, residual ownership, and any acceptance or rejection rule.

An admissible refinement supplies, for each scale $j$:

1. an implemented channel ${\tilde{\Phi }}_j$ on these same external comparison interfaces;

2. an ideal factor chain on compatible intermediate interfaces whose complete composition is exactly $\Phi$;

3. corresponding implemented factors, compared with those ideal factors on the same intermediate interfaces;

4. an explicit assignment of factor errors to encoding, flow, event, classical record processing, and decoding;

5. any needed padding, identification, or finite classical coarse-graining maps, included as factors rather than suppressed.

Intermediate physical registers may depend on $j$. The external comparison algebra does not.

The existence of the ideal matched factor chains is an assumption of the refinement class, not a consequence of a shared finite diagram. In particular, this definition does not infer a continuum target from the native finite clock.

In Definition def:refinement, suppose the complete non-flow error totals are

Suppose the flow factors correspond to intervals of lengths $h_{j,\ell }$, with

and have half-diamond errors bounded by $A{h_{j,\ell }}^{1+\alpha }$, where $A<\infty$ and $\alpha >0$ are independent of $j,\ell$. Then

If ${\delta }_j\to 0$ and the four non-flow totals tend to zero, the complete channels converge to $\Phi$.

Any two admissible refinement families satisfying these conditions for the same frozen target converge to the same limit.

Proof. The total flow error satisfies

Apply Lemma lem:telescoping to the complete matched factor chain and group the remaining factor errors by their declared roles. This proves (eq:refinement-budget).

For two families with bounds $b_j\to 0$ and ${b^{\prime }}_j\to 0$,

Both families consequently have the same channel limit.

This is the precise error decomposition

A one-step error of order $O(h)$ alone does not force the accumulated error to vanish. Nor does a proposed product formula automatically supply the complete-instrument estimates required here. The supplied En05-A comparison distinguishes geometry, schedule, and channel semantics CheungEn05A.

The XXI-A–C identifiers in the development plan motivate this hybrid and complete-experiment interface CheungXXIOutline. Their full texts were not supplied. The estimates used here are proved in this manuscript rather than attributed to unavailable proposition numbers.

Refinement must distinguish:

1. convergence of externally calibrated classical readings;

2. convergence of the quantum residual channel after forgetting some or all records;

3. convergence of the complete recorded experiment on a common record–residual interface.

The third implies the relevant marginal channel convergences. Neither the first nor the second, by itself, implies the third. Proposition prop:record-not-residual illustrates the residual gap even when the record laws agree exactly.

Finite classical codes also do not converge to an exact all-real record in total variation merely because their calibrated values become dense. For example, let ${\mu }_j$ be any probability measure with finite support in $[0,1]$, and let $\lambda$ be uniform Lebesgue probability. If $F_j=\mathrm{supp}{\mu }_j$, then

Hence their total-variation distance is one at every scale. A weak limit can still exist, but it is a different convergence claim. Fixed finite bins or a specified bounded-statistic interface can be used instead; the binning or testing map must be declared.

Finally, common finite samples do not select a unique interpolation, generator, or continuum limit. XIX-C's prop:two-limits and prop:interpolation record precisely this non-determination CheungXIXC. Theorem thm:refinement avoids it by freezing the target and restricting the comparison class explicitly.

No physical causal geometry is needed for the channel identity. If a realization is assigned locations, a causal embedding can be added as a separate contract.

Let the complete finite dependency graph include every actual quantum input, classical control, preparation, routing operation, and retained record. Assign each event a location $p$ in a set with a supplied transitive causal relation $⪯$. Require every dependency edge $p\to q$ to satisfy $p⪯q$.

Under this edge condition, every dependency path lies in $⪯$. For local channel realizations of the graph, Heisenberg support of a terminal observable is contained in its scheduled ancestor region. Thus operational influence is bounded by that region and, through the certified embedding, by the supplied causal relation.

Proof. Path containment is transitivity. For support containment, pull the terminal observable backward through the circuit. The adjoint of a trace-preserving local channel sends an identity on an untouched factor to the identity, and can enlarge support only to the input factors of that event. Induction through the finite graph places the support in the ancestor region. An intervention on a factor outside that region cannot change the terminal reduced output, because its pulled-back observables act as the identity there.

Scheduled ancestry, actual operational influence, and $⪯$ remain three different relations. Channels may fail to use a scheduled edge, while entanglement may produce correlations without intervention-induced signalling. The embedding contract does not reconstruct the supplied relation from the program.

These distinctions match VII-J's two targets: complete decoded simulation and physical embedding of the actual elementary operations CheungVIIJ. They do not derive a Lorentz-invariant speed or select a Lorentz cone; En08-A's scheduler-consistency separation remains applicable CheungEn08A.

If a fixed geometric packet or a source-replacement controller is attached, its matching certificates, guards, and residual ownership must also be supplied. The VIII constructions are comparisons for such contracts, not a geometric conclusion of the clock-readout theorem CheungVIII.

For one declared finite clock–system preparation and one complete destructive clock-readout event:

1. the retained classical code supplies the calibrated relational label;

2. the surviving system block is exactly the Page–Wootters conditional state;

3. the measured clock names are consumed;

4. the complete recorded instrument agrees with its relational extraction target on every input and with every reference;

5. sampled covariance and stationarity imply the stated conditional unitary evolution;

6. under the full Enter characterization hypotheses, the Born record law and projective residual update have the inherited derived status;

7. matched refinements obey the complete-channel budget and class-relative uniqueness theorem.

Proof. Items (i)–(vi) are Theorem thm:main, with Lemma lem:density-semantics for the semantic attribution. Proposition prop:spectral-realization gives an exact inhabited finite constraint class. Item (vii) is Theorem thm:refinement.

the release does not do. It does not remove external preparation or scheduling time. It does not construct an internal matter clock that remains available after the declared destructive event. It does not derive a dynamical cone, a continuum metric, Einstein dynamics, a field-theoretic measurement apparatus, or fully quantized gravitational relational observables. A finite history state and its record are not automatically spacetime emergence.

The classical relational-observable and lapse results associated with VI and Em02-E retain their own equation spaces, backgrounds, and quantum obligations CheungBoundaries,CheungEm02E. No Enter readout closes those additional obligations.

non-transfer. No characteristic joint product, quantum coefficient, interacting Ward identity, or downstream admission is constructed. The XIV-I internal amplitude and XIV-J supported-sector reference are not replaced by a finite clock instrument. XV-B version 2 may exist as a technical continuation while the distinct downstream quantum-admission exit remains unopened. This manuscript does not change that status and does not discharge XVI-N CheungBoundaries.

negative controls. The supplied OpenAI catalogue and the identifier “Family 271” remain documentary negative controls only. No claim or proof from that family is used, reproduced, or verified here. No authorship, formalization certificate, or physical interpretation is transferred from an OpenAI document to this construction. The standing documentary entry

is unchanged. These statements record scope; they are not an independent audit or a universal nonexistence theorem.

. The philosophical and mathematical content can be summarized as

The label is a record, not a surviving clock. The record is not a decoder. The channel is not a thermal machine. The analog is not an identification with spacetime.

The following corrections are necessary for the completed statement to be mathematically and operationally sound.

1. A clock basis must be declared. Computational-basis Enter realizes the intended observable only when the clock basis is already computational or a calibrated basis change is included.

2. Stationarity does not imply the desired evolution for an arbitrary observable. Sampled covariance is explicit in Definition def:covariance; Example ex:energy-basis supplies a counterexample without it.

3. Stationarity and a zero-energy constraint differ. Equations (eq:stationarity) and (eq:strong-constraint) are not interchangeable. The exact finite construction satisfies the stronger condition.

4. Uniform readings are a consequence in the exact covariant complete basis. One cannot simultaneously assume arbitrary clock weights and exact covariance of every basis projector for a stationary preparation.

5. Only the measured register's quantum output is absent. The complete process still returns the unmeasured system. Saying that the entire process has “only a classical output” would incorrectly discard its residual.

6. The Born attribution is conditional on the full predecessor class. Writing the instrument in Born form proves its algebraic law, not the predecessor's operational characterization. Hypothesis hyp:enter identifies the imported result and its applicability condition.

7. Normalized conditioning is not a linear channel. The instrument uses subnormalized blocks. Zero-probability branches have no prescribed normalized state.

8. Reference stability is an operator statement. Equality on one frozen state would not establish diamond equality. Equation (eq:exact-channel-identity) holds on every operator. Relational evolution of a reference-inclusive preparation additionally requires its own full stationarity.

9. A consumed clock cannot be measured again in the same continuation. The hybrid composition crosses the readout exactly once. Further physical readings need additional resources or a different instrument.

10. General circuit histories need not satisfy the additive constraint. Their propagation Hamiltonian (eq:propagation-H) is kept distinct from $H_C+H_S$.

11. Refinement uniqueness is class-relative. The ideal target, calibration, complete output interface, and local consistency estimates are frozen. Shared finite samples alone do not prove uniqueness.

12. No unsupplied theorem is silently imported. The XXI-A–C material available here is an outline-level comparison. The hybrid and telescoping results needed for En09-A are stated and proved above.

The bibliography distinguishes journal articles, internal manuscript records, and outline-only material. Internal documents are cited by the supplied identifier and version rather than by an invented publication date, DOI, or formal title.

The only imported nonstandard semantic theorem used in the main argument is the Enter characterization identified in Hypothesis hyp:enter. Its full proof is not reconstructed here. The finite channel identity, covariance argument, spectral realization, hybrid composition, stability estimates, and class-relative refinement result have explicit proofs in this manuscript.

The public publisher record for CheungEnter, consulted for bibliographic metadata, gives acceptance on 28 September 2026 and states that the final formatted article is forthcoming. This status is not converted into a publication date for any internal manuscript.

The supplied internal records support the stated comparisons and standing non-claims. They are not treated as independent machine verification, and similarities in notation do not transfer their mathematical hypotheses to En09-A.

No successful pdflatex execution, proof-assistant check, numerical execution, hardware experiment, independent referee review, or journal acceptance is asserted. The source is arranged as a single conventional pdflatex manuscript with its bibliography included.

PageWootters

D. N. Page and W. K. Wootters, “Evolution without evolution: Dynamics described by stationary observables,” Physical Review D 27 (1983), 2885–2892. https://doi.org/10.1103/PhysRevD.27.2885.

BoetteRossignoli

A. Boette and R. Rossignoli, “History states of systems and operators,” Physical Review A 98 (2018), 032108. https://doi.org/10.1103/PhysRevA.98.032108.

McClean

J. R. McClean, J. A. Parkhill, and A. Aspuru-Guzik, “Feynman's clock, a new variational principle, and parallel-in-time quantum dynamics,” Proceedings of the National Academy of Sciences 110 (2013), E3901–E3909. https://doi.org/10.1073/pnas.1308069110.

Diaz

N. L. Diaz, P. Braccia, M. Larocca, J. M. Matera, R. Rossignoli, and M. Cerezo, “Parallel-in-time quantum simulation via Page and Wootters quantum time,” Physical Review Research 7 (2025), 033294. https://doi.org/10.1103/wpnf-4nnn.

CheungEnter

N. Cheung, “The Enter Calculus: A Linear Quantum Lambda Calculus with an Operational Characterization of Measurement,” Frontiers in Physics 14 (2026), article 1954943. Accepted 28 September 2026; final formatted version forthcoming according to the publisher record consulted on 13 October 2026. https://doi.org/10.3389/fphy.2026.1954943.

CheungI

N. Cheung, Manuscript I, I.tex, version 2, n.d. Supplied manuscript record and theorem excerpts. In particular, thm:main characterizes Born probabilities and positive-branch destructive residual rays within its declared strategy class. Enter Calculus Companion, https://enter.grok.me/.

CheungPhilosophy

N. Cheung, The Philosophy of Enter: Destructive Readout, Quantum Recovery, and the Refusal of Identification, EntPhil-v2.tex, version 2, n.d. Introductory philosophical map, not a registered theorem premise. https://enter.grok.me/enter.

CheungEn08A

N. Cheung, Manuscript En08-A, En08-A.tex, version 1, n.d. Supplied manuscript record and excerpts, especially thm:main, “Failure of reconstruction from scheduler consistency.” Enter Calculus Companion, https://enter.grok.me/.

Trinity

P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, “Trinity of relational quantum dynamics,” Physical Review D 104 (2021), 066001. https://doi.org/10.1103/PhysRevD.104.066001.

CheungXEnter

N. Cheung, Manuscript X-Enter, X-Enter.tex, version 1, n.d. Supplied manuscript record and excerpts, especially thm:relational, “Conditional selective relational readout.” Its encoded phase model is cited as a comparison only. Enter Calculus Companion, https://enter.grok.me/.

CheungVIII

N. Cheung, Manuscripts VIII-A, VIII-B, and VIII-C, VIII-A-v3.tex, VIII-B-v3.tex, and VIII-C-v3.tex, versions 3, n.d. Supplied records and excerpts concerning fixed-presentation matching, record–residual separation, affine acquisition obstructions, guarded packet flow, and residual ownership. In VIII-A, see A:thm:residual. These are comparison records, not geometric premises of En09-A. Enter Calculus Companion, https://enter.grok.me/.

CheungXIXC

N. Cheung, Manuscript XIX-C, XIX-C.tex, version 1, n.d. Supplied record and excerpts, especially thm:main, “Finite incidence does not determine continuum evolution,” and the identified prop:token, prop:affine, prop:two-limits, and prop:interpolation. Enter Calculus Companion, https://enter.grok.me/.

CheungEm02E

N. Cheung, Manuscript Em02-E, Em02-E-v1.tex, version 1, n.d. Supplied manuscript record and excerpts, especially thm:lapse, “Complete relational lapse through second order.” Its classical background-dependent lapse is not identified with the present clock record. Enter Calculus Companion, https://enter.grok.me/.

Watrous

J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018. Chapter 3: similarity and distance among states and channels. Author's manuscript and errata interface: https://cs.uwaterloo.ca/ watrous/TQI/.

CheungEn05A

N. Cheung, Manuscript En05-A, En05-A.tex, version 1, n.d. Supplied manuscript record and introductory excerpts. Finite-range schedule geometry, complete channel locality, and continuum-comparison cautions are cited as distinctions, not as automatic transfer theorems. Enter Calculus Companion, https://enter.grok.me/.