Landmark VII-G Completely positive observation boundaries for finite matter–geometry feedback

We replace an unrestricted deterministic state-functional coupling by a finite classical–quantum transition instrument. Classical labels contain the geometric interface and every controller variable used to select future operations. Matter occupies a fixed finite-dimensional Hilbert space, and the unconditional evolution is linear, completely positive, and trace preserving.

We give an exact observation criterion in the direct-sum state algebra and strengthen it to agreement of stochastic history laws and history-conditioned accessible residuals. A finite backward construction identifies the smallest observable linear spaces needed for a declared continuation. Arbitrary entangled spectator extensions preserve every unnormalized geometric-history block after restriction, provided the complete controller and its certificates are spectator blind. Implementation equivalence is characterized by selective adjoint maps and is stable under compatible continuations.

Two limitations are proved independently: a nonconstant deterministic packet selection cannot be the classical output of an affine channel on a convex state domain, and an informative classical record is incompatible with exact recovery of every input state. Resource and energy balance are stated as operator identities, with resource accounting kept distinct from physical energy. Choi bounds yield checkable assisted stability estimates for both terminal states and complete histories.

A resource-budgeted unsharp qubit instrument selects strictly ordered horizontal slices of a fixed spinless cone. Every branch has an explicit certificate; its classical sector description and a quantitative dephasing approximation are calculated exactly. This example is matter-controlled slicing, not matter-driven gravitational dynamics. No quantum-field source, changing deficit, moving particle, or gravitational field equation is constructed.

The observation interface of VII-E allows a coupling rule of the form

while leaving the matter state unchanged. VII-E explicitly notes that this need not be affine and is not thereby a quantum channel CheungVIIE (Definition def:rules and Remark rem:nonlinear).

A concrete benchmark is the rule called Option N in VIII-B:

Its domain requires the selected deficit to stay in the declared interval. The rule selects one definite geometric packet and does not acquire a stochastic measurement record. The finite source letter of VIII-C uses a supplied source token in a prescribed table-selection command, including the word

A promised token is not an instrument that extracts an expectation from unknown matter CheungVIIIB,CheungVIIIC.

We take these as named comparison targets, not as premises providing a joint completely positive evolution. Proposition prop:affine below identifies exactly why a nonconstant deterministic selection cannot simply be promoted to an unconditional affine channel.

Quantum instruments distinguish outcome probabilities from their selective residual maps DaviesLewis1970. Finite-dimensional complete positivity and its matrix certificates are standard Choi1975,Watrous2018. Linear completely positive classical–quantum dynamics is also an established framework, including stochastic trajectories and broader hybrid generators Oppenheim2023,OppenheimTrajectories2023, OppenheimClasses.

Accordingly, we do not claim novelty for the direct-sum state space, the Kraus construction, the exact recovery–complement criterion, or the general channel observation principle. In particular, VII-E already proves a composition criterion for block-diagonal state spaces CheungVIIE (Proposition prop:classical-composition). Our direct-sum criterion is its intrinsic specialization.

The purpose of VII-G is to organize these facts into a complete finite feedback specification, with:

1. a state-independent transition table and certified geometric output graph;

2. branch-sensitive observation boundaries and stochastic histories;

3. exact consistency with entangled spectators;

4. selective-map implementation tests;

5. resource accounting and assisted error certificates;

6. an explicit finite example satisfying the specification.

All results used in the assembly are proved below.

A geometric label can describe a slice, a germ, a controller state, or a spacetime parameter. Updating such a label is not, by itself, a gravitational evolution law. The distinction is

VII-F supplies a demanding fixed-cone incidence certificate, including full affine holonomy and exact event control, but keeps its spacetime, deficit, and worldline fixed CheungVIIF. We use its certificate discipline, not an unproved extension of its geometry. The example in Section sec:example is simpler: its slices are horizontal and its particle link does not change.

The field-theoretic manuscripts VII-A–D are not prerequisites of the finite results. Their relevance to a later source construction is discussed in Section sec:limits.

Fix a nonempty finite label set $\mathsf{G}$ and a Hilbert space $\mathcal{H}$ of dimension $d<\infty$. Set

For $X=(X_g{)}_g$ and $B=(B_g{)}_g$, use

The identity is $1=(I{)}_g$. The relevant norms are

A hybrid state is a positive element $\rho =({\rho }_g{)}_g$ with ${\mathrm{Tr}}_{\mathsf{G}}\rho =1$. Equivalently,

The blocks are subnormalized:

Conditional normalization is not used in the unconditional evolution law.

We use

The classical-input meaning of the diamond norm is made precise in Section sec:stability.

The label $g$ contains every classical variable used to select subsequent operations. A typical label is

where $\gamma$ denotes geometric data.

An operation table may depend on $g$, but not on an undeclared functional of the current unknown density operator. Adaptive measurement control is represented by recording the outcome in a classical successor label. If a finer outcome affects future choices, it must remain in the controller state even when a coarser geometric report is displayed.

For a fixed finite horizon, any finite instruction tree can be made Markovian by adjoining its node or relevant history to the label. This may require exponentially many labels. No bounded-memory claim for arbitrary horizons is implied.

A certified finite interface consists of:

1. the finite labels $\mathsf{G}$, each with a declared admissible geometric realization and controller interpretation;

2. a directed edge relation $\mathcal{E}\subseteq \mathsf{G}\times \mathsf{G}$;

3. a fixed certificate for each edge $(g,h)\in \mathcal{E}$, specifying all affected output data, matching requirements, controller updates, and any location transport;

4. a declared reserve for implementation history and spectators, not inspected by geometric control.

The certificates are classical data. The definition does not prove a gravitational constraint equation.

For each $g,h\in \mathsf{G}$, let

be completely positive. Require

Nonzero branches are permitted only on certified edges.

Define

Here and below the adjoint is characterized by $\mathrm{Tr}(B\Psi (X))=\mathrm{Tr}({\Psi }^{\ast }(B)X)$.

The map (eq:hybrid-map) is completely positive and trace preserving on ${\mathfrak{X}}_{\mathsf{G}}$. It preserves the classical–quantum state space and has adjoint

Conversely, every completely positive trace-preserving map between two finite direct sums has this branch form. If

then

A canonical extension to the full matrix space on ${\mathbb{C}}^{\mathsf{G}}\otimes \mathcal{H}$ has Kraus operators

This extension discards off-diagonal input label blocks and has block-diagonal output.

Proof. After adjoining any matrix ancilla, a positive direct-sum input has positive blocks. Each amplified branch map is positive, and sums of its outputs are positive. Thus $\mathfrak{F}$ is completely positive. Furthermore,

The block formula proves preservation of classicality, and pairing with $B$ gives (eq:hybrid-adjoint).

For the converse, compose the given map with the completely positive injection into input block $g$ and projection onto output block $h$. These compositions define completely positive ${\Psi }_{h|g}$. Linearity gives the branch formula, and trace preservation on each input block gives (eq:instrument-complete).

Finite-dimensional completely positive maps have Kraus representations: the positive Choi matrix can be decomposed into rank-one positive terms and each vector unvectorized into a Kraus operator Choi1975. Completeness gives (eq:kraus-complete). The operators (eq:lift-kraus) have sum of squared moduli equal to the full identity. Their action reads only the diagonal input blocks and writes only diagonal output blocks, proving the final assertion.

If a second transition has branches ${\Lambda }_{k|h}$, the composed channel has branches

Its certificates are concatenations of the declared edge certificates.

Proof. Substitution into the two block formulas gives (eq:branch-composition). Sums and compositions preserve complete positivity. Moreover,

The certificate assertion means concatenation of already declared certificates; it introduces no new geometric existence claim.

An observation space is a complex linear, adjoint-closed subspace $\mathcal{O}\subseteq {\mathfrak{X}}_{\mathsf{G}}$ containing $1$. Thus it is a finite-dimensional operator system. It need not be closed under multiplication.

Two states agree on $\mathcal{O}$ if

A frequently used choice is

where each ${\mathcal{A}}_g\subseteq L(\mathcal{H})$ is a unital ${}^{\ast }$-subalgebra. This choice observes the complete classical label, because it contains each central block projection.

Use of linear spaces is important: the smallest future-sensitive observation space constructed below need not be an algebra.

Let $\mathfrak{F}:{\mathfrak{X}}_{\mathrm{in}}\to {\mathfrak{X}}_{\mathrm{out}}$ be a channel and let ${\mathcal{O}}_{\mathrm{in}},{\mathcal{O}}_{\mathrm{out}}$ be observation spaces. The following are equivalent:

1. for all hybrid states,

   $$\rho {|}_{{\mathcal{O}}_{\mathrm{in}}}=\sigma {|}_{{\mathcal{O}}_{\mathrm{in}}}⟹\mathfrak{F}(\rho ){|}_{{\mathcal{O}}_{\mathrm{out}}}=\mathfrak{F}(\sigma ){|}_{{\mathcal{O}}_{\mathrm{out}}};$$
2. $${\mathfrak{F}}^{\ast }({\mathcal{O}}_{\mathrm{out}})\subseteq {\mathcal{O}}_{\mathrm{in}}.$$

For the direct-sum spaces ${\mathcal{O}}_{\mathrm{in}}={⨁}_g{\mathcal{A}}_g$ and ${\mathcal{O}}_{\mathrm{out}}={⨁}_h{\mathcal{B}}_h$, this is equivalent to

Proof. The inclusion implies agreement by pulling each output observable back to the input space.

Conversely, let $H=H^{†}\in {\mathfrak{X}}_{\mathrm{in}}$ annihilate the Hermitian part of ${\mathcal{O}}_{\mathrm{in}}$ under the trace pairing. Because $1\in {\mathcal{O}}_{\mathrm{in}}$, $\mathrm{Tr}H=0$. Choose a faithful hybrid state $\omega$; a multiple of the block identity suffices. For sufficiently small $a>0$, $\omega \pm aH$ are states agreeing on ${\mathcal{O}}_{\mathrm{in}}$. The assumed output agreement implies

Finite-dimensional double orthogonal complementation places ${\mathfrak{F}}^{\ast }B$ in the Hermitian part of ${\mathcal{O}}_{\mathrm{in}}$. Complex linearity gives (eq:observation-inclusion).

If the spaces are direct sums, an output observable may have one nonzero block $B_h$. Formula (eq:hybrid-adjoint) then makes (eq:observation-inclusion) imply every (eq:branch-inclusion). The reverse implication follows by summing those branch images.

This is the direct-sum specialization of VII-E's composition principle and block-diagonal refinement, proved here without an ambient pinching notation CheungVIIE (Theorem thm:composition and Proposition prop:classical-composition).

Define

For a supported input block,

For a fixed $g$, all successor probabilities depend only on the state functional on ${\mathcal{A}}_g$ if and only if

Proof. Sufficiency is (eq:transition-probability). For necessity, apply the faithful-state perturbation proof of Theorem thm:observation to each effect.

This is the effect-pullback observation of VII-E with the successor geometry as the classical outcome CheungVIIE (Theorem thm:pullback). It says nothing by itself about the residual matter state.

Allow a finite sequence of channels with branches ${{\Psi }^{(t)}}_{h|g}$, $t=1,\dots ,T$. All tables are fixed; the current label already contains the controller memory.

For a history $\gamma =(g_0,\dots ,g_n)$, define

Its probability is $p_{\gamma }=\mathrm{Tr}{\rho }_{\gamma }$. The complete history-retaining output is

Forbidden paths have zero blocks.

The map ${\mathcal{H}}_n$ is a channel. Summing its blocks with a common terminal label gives the ordinary terminal hybrid state.

Proof. Each path map is completely positive. Summing probabilities over the final label uses (eq:instrument-complete); repeating this sum backwards over the path leaves ${\sum }_{g_0}\mathrm{Tr}{\rho }_{g_0}=1$. This proves trace preservation. The terminal summation is precisely repeated substitution in (eq:hybrid-map).

Suppose that, for each $t,g,h$,

where each ${\mathcal{O}}_{t,g}\subseteq L(\mathcal{H})$ contains $I$. If two initial hybrid states agree on ${⨁}_g{\mathcal{O}}_{0,g}$, then:

1. all geometric history probabilities through time $T$ agree;

2. for every history ending at $g_n$, the corresponding unnormalized blocks agree on ${\mathcal{O}}_{n,g_n}$;

3. on every positive-probability history, the normalized conditional states agree on the same space.

Proof. For $B\in {\mathcal{O}}_{n,g_n}$, its history-conditioned expectation is

Successive applications of (eq:time-branch-inclusion) put the bracket in ${\mathcal{O}}_{0,g_0}$. Initial agreement therefore proves (ii). Taking $B=I$ proves (i). Equal positive probabilities permit division by the same scalar, proving (iii).

The theorem compares correlated stochastic histories, not merely one-step probabilities or terminal marginals.

Fix terminal observation spaces ${\mathcal{O}}_{T,g}$. Define backwards

These are observation spaces. They are the smallest family containing the terminal spaces and satisfying (eq:time-branch-inclusion).

Agreement on the initial direct sum is equivalent to agreement of all complete length-$T$ history-conditioned terminal expectations. It consequently implies agreement of all shorter history laws.

Proof. Adjoints of completely positive maps preserve adjoints. Hence the displayed span is adjoint closed and contains $I$. Any family satisfying the inclusions must contain the indicated span at each backward step, proving minimality by induction.

The pullbacks of all length-$T$ history-conditioned terminal observables span the recursively constructed spaces. The inserted identity adds no independent expectation: at each step it equals ${\sum }_h{{\Psi }^{(t)\ast }}_{h|g}(I)$. Thus agreement of all those expectations is exactly agreement on the initial spaces. Summing extensions of a shorter history gives its probability.

For a time-homogeneous table and geometry-only terminal observations, one can also construct the boundary for every finite history.

Set ${{\mathcal{O}}_g}^{(0)}=\mathbb{C}I$ and

The family stabilizes after at most $|\mathsf{G}|(d^2-1)$ strict enlargements. Its stable value is the smallest branch-invariant observation family containing the classical effects. Agreement on its direct sum is equivalent to agreement of all finite geometric history laws.

Proof. The dimensions are nondecreasing and their sum is at most $|\mathsf{G}|d^2$, starting at $|\mathsf{G}|$. If one iteration does not enlarge the family, all subsequent iterations are identical. This proves the bound and branch invariance.

The recursively generated elements are spans of finite path effects. Conversely, every finite path effect appears at some iteration. The equivalence follows from the trace pairing with those effects.

No algebra closure is asserted for these minimal spaces. Replacing them by generated ${}^{\ast }$-algebras is permissible but may declare more information accessible than the histories actually require.

If two admissible inputs agree on the declared observation space but a target requires different accessible output expectations, no channel satisfying the corresponding adjoint inclusion realizes that target on both inputs.

Proof. Theorem thm:observation forces equality of the output expectations.

This is the linear-channel form of the observation-factorization obstruction in VII-F CheungVIIF (Proposition prop:obs-obstruction). A capacity check or rounded report is an additional observation, not a consequence of a declared matter source algebra.

Let $N$ be a finite untouched spectator. An initial extension satisfies

It may be entangled within every block.

Adding the spectator does not change:

1. the label dictionary or any input-label interpretation;

2. the selected branch maps, except for tensoring with ${\mathrm{id}}_N$;

3. guard results, successor labels, certificates, or transport of original names;

4. controller updates or the declared resource reserve.

Spectator storage and any spectator location history are separately reserved. Spectator outcomes are not read by the controller.

The assumption concerns the whole transition specification. Equality of one displayed source functional is not sufficient.

For every linear map $\Psi$ on the original matter,

No trace-preservation or product-state assumption is needed.

Proof. Pair both sides with an arbitrary output operator $B$. Their pairings equal

Equality against all $B$ proves the statement.

Under Assumption ass:blind and (eq:block-extension), extend every branch by ${{\Psi }^{(t)}}_{h|g}\otimes {\mathrm{id}}_N$. Then, for every finite history,

Consequently:

1. the complete geometric history distributions agree;

2. the reduced history–matter outputs agree;

3. on every supported history, the conditional original-matter reductions agree;

4. the nonselective spectator marginal is unchanged.

Proof. Apply Lemma lem:partial-trace to each map in (eq:path-block). This proves (eq:path-spectator) without normalizing any branch. Taking traces proves (i); taking the direct sum proves (ii); division by a common positive trace proves (iii).

For (iv), the sum of all branches of a one-step transition is trace preserving on each input label. Pair the spectator reduction with $B_N$. Its expectation after the step is

Iteration proves nonselective invariance.

This extends the partial-trace mechanism of VII-E to selective branches and their stochastic correlations CheungVIIE (Lemma lem:partial-trace, Theorem thm:spectators, and Assumption ass:blind). It does not identify two different spectator extensions as joint outputs.

Suppose an update is allowed only when the total number of matter names is at most $K$. An original interface with $K$ names passes. Adding one product spectator makes the same geometric update fail, even if every source algebra is $\mathbb{C}I$ and no quantum operation touches the spectator.

This violates Assumption ass:blind. It is the capacity leak identified in VII-E CheungVIIE (Example ex:capacity).

For an entangled pair $(|00⟩+|11⟩)/\sqrt{2}$, local projective readout leaves the remote marginal $I/2$ nonselectively, but its conditional states are $|0⟩⟨0|$ and $|1⟩⟨1|$. Thus (iv) is not a claim that the spectator state is unchanged on each outcome. Nor may a subsequent controller inspect an otherwise excluded spectator record without changing the hypothesis.

Let ${\Psi }_{h|g}$ and ${\tilde{\Psi }}_{h|g}$ be two implementations with the same input and output label dictionaries.

The implementations agree on ${\mathcal{O}}_{\mathrm{out}}={⨁}_h{\mathcal{B}}_h$ for every hybrid input if and only if

The equality also holds on ${\mathcal{O}}_{\mathrm{out}}\otimes L(N)$ for every untouched reference $N$, using the same extended input in both implementations.

Proof. Choose an input supported in block $g$ and an output observable supported in block $h$. Equality of all state expectations is equality of the Hermitian adjoint images. Complex linearity gives (eq:implementation-adjoint). Conversely, summing these identities proves output agreement.

For a reference observable $B\otimes C$, the two adjoint images are equal by tensoring (eq:implementation-adjoint) with $C$. Such tensors span the reference-extended observation space.

Suppose both implementations satisfy the same branchwise inclusions (eq:time-branch-inclusion), and their branch adjoints agree on each ${\mathcal{O}}_{t,h}$. Starting from the same input, their complete histories have equal probabilities and equal accessible history-conditioned residual expectations.

Proof. In the backward expectation formula from Theorem thm:histories, replace the final adjoint by the other implementation's adjoint. The inclusion keeps its image in the previous observation space, where the next replacement is valid. Continue backwards to the initial state.

Equality of effects is only the case $B=I$. It cannot replace the selective-map criterion.

Fix an input label $g$. Choose a Kraus representation for every nonzero branch and write $a=(h,\alpha )$. Define

Completeness implies ${V_g}^{†}{V}_g=I$.

Destructive computational readout of $A$, followed by reporting only $h$, gives

The matter register is returned. The carrier is consumed. This is a represented instrument, not destructive readout of the matter source itself.

The Born readout interface is the specified semantic primitive of the Enter calculus CheungEnter. No measurement-strategy uniqueness theorem is used as a premise here.

Let

where $J$ is the unnormalized Choi matrix. Among dilations with orthogonal pointer sectors for the reported outcomes $h$, the minimum environment dimension is $R_g$.

A common pointer reserve of

qubits suffices. If matter is encoded in $s$ qubits, a unitary extension of (eq:dilation) uses those $s$ matter qubits and $e$ fresh pointer qubits. Full fine readout produces at most $e$ classical bits per step.

Proof. A Choi matrix of rank $r_{h|g}$ has a decomposition into precisely that many rank-one positive terms, giving a Kraus family of that size. Conversely, a family with $q$ operators gives a Choi sum of $q$ rank-one terms, so $q\ge r_{h|g}$.

An environment sector of dimension $q_h$ gives at most $q_h$ Kraus operators when expanded in a sector basis. Thus $q_h\ge r_{h|g}$, and summing sectors gives the lower bound $R_g$. Minimal Kraus families in (eq:dilation) attain it.

Encode the pointer in $e$ qubits. The isometry on the subspace with pointer input $|0{⟩}^{\otimes e}$ extends to a unitary by completing orthonormal bases. Destructive readout consumes the pointer and returns a finite classical fine index.

This is an existence statement for a unitary dilation. It is not an exact-synthesis claim for arbitrary matrix entries in a fixed discrete gate library. After readout, any retained fine index is classical implementation history. It must either be explicitly available to the controller or remain outside its observation boundary.

For a horizon $T$, retaining all fine indices requires at most $Te$ bits. Storing all geometry labels additionally requires at most $(T+1)\lceil {\log }_2|\mathsf{G}|\rceil$ bits. These are sufficient reserves, not optimal adaptive-memory bounds. No spectator is charged against them.

Suppose a source instrument has Kraus operators $M_a$, with ${\sum }_a{M}_a^{†}{M}_a=I$, and its outcome selects a successor ${\Gamma }_a(g)$ and unitary $U_{{\Gamma }_a(g)}$.

The immediate implementation reads out the pointer of

and then applies $U_{{\Gamma }_a(g)}$. The coherent implementation first applies

and then reads out $A$.

Both implementations have selective maps

They therefore agree on the full classical-record–matter output, including every entangled reference.

Proof. Projection onto pointer outcome $a$ gives

This is the same branch operator as immediate readout followed by feedback. Equality of branch maps, and then tensoring with a reference, proves the conclusion.

Before readout, the coherent pointer is quantum, not an already existing classical geometry. The proposition compares the completed outputs. It does not equate arbitrary intermediate geometric traces or authorize an intervening controller to inspect pointer coherences. This is consistent with the restricted deferred-readout boundary of VII-E and with precise circuit formulations of deferred measurement CheungVIIE, GurevichBlass2022 (Theorem thm:deferred).

For a qubit, consider two fair two-outcome instruments:

where

Both have effects $I/2$. On input $|+⟩⟨+|$, their normalized residuals are respectively $|+⟩⟨+|$ and $I/2$, at trace distance $1/2$.

If the next geometric output records an $X$-basis measurement, the first implementation gives

whereas the second gives $p(a,\pm )=1/4$. The complete two-step geometric laws have total-variation distance $1/2$.

The continuation observes a residual direction not determined by the first record effects. Its permission must be reflected in the backward observation space. This is the same record–residual separation emphasized by VII-E CheungVIIE (Theorem thm:residual-obstruction and Theorem thm:luders). Identical accessible geometry does not certify quantum recovery.

Let $\mathcal{S}$ be a convex set of quantum states. Suppose an affine map assigns to every $\rho \in \mathcal{S}$ a probability distribution concentrated at one classical label:

Then $\gamma$ is constant on $\mathcal{S}$.

The same conclusion holds for packets in any classical outcome space in which distinct packets can be separated by a measurable event.

Proof. If $\gamma (\rho )\ne \gamma (\sigma )$, then for $0<t<1$, affinity gives

This distribution is not concentrated at one label. The mixture belongs to $\mathcal{S}$, giving a contradiction. In a general outcome space, an event separating the two packets has probability strictly between zero and one on that mixture, with the same contradiction.

Applied to (eq:optionN), this obstruction requires that the selected packet genuinely vary: $\theta \kappa \ne 0$, the source functional is nonconstant on the convex domain, and the packet's declared deficit distinguishes the outputs. It does not prohibit deterministic evolution of an already classical label.

Stochastic output, a nonconvex promised input class, or a controlled approximation is a different target. A nonlinear rule may also be retained explicitly outside the present channel programme.

If completely positive maps satisfy

then

In particular, all record probabilities are state independent.

Proof. The Choi matrix of the identity is

and has rank one. Each positive $J({\Psi }_a)$ lies below $J(\mathrm{id})$, so its range is contained in the one-dimensional range of $J(\mathrm{id})$. Thus $J({\Psi }_a)=p_{a}J(\mathrm{id})$. Choi injectivity gives the result.

Thus an informative instrument cannot retain the matter-unchanged property of (eq:old-coupling) on every quantum state.

The next criterion is standard. It is also recorded in En02-C, but is restated and proved for the present use rather than imported with that manuscript's radiation benchmark CheungEn02C (Lemma lem:decouple). Channel-level continuity versions belong to the information–disturbance literature KSW2008.

A finite-dimensional channel $\mathcal{N}$ has a CPTP left inverse if and only if a complementary channel has the form

for a fixed density operator $\tau$.

Proof. Let $V:{\mathcal{H}}_{\mathrm{in}}\to {\mathcal{H}}_{\mathrm{out}}\otimes E$ be a Stinespring isometry.

Suppose a recovery exists and let $W:{\mathcal{H}}_{\mathrm{out}}\to {\mathcal{H}}_{\mathrm{in}}\otimes F$ dilate it. Then $(W\otimes I_E)V$ dilates the identity channel. Every Kraus operator of the identity is proportional to the identity by the rank-one Choi argument. Consequently this composite isometry has the form

for one fixed unit vector. Tracing over the recovered input and $F$ shows that the original complementary output is constant.

Conversely, diagonalize $\tau ={\sum }_j{\lambda }_j|j⟩⟨j|$, omitting zero eigenvalues. Write $V={\sum }_j{V}_j^{\mathrm{raw}}\otimes |j⟩$. The complementary identity implies

Hence ${V_j}^{\mathrm{raw}}=\sqrt{{\lambda }_j}{V}_j$, where the $V_j$ are isometries with mutually orthogonal ranges. Let $P={\sum }_j{V}_j{V}_j^{†}$, and choose a density operator ${\omega }_{\ast }$ on the input. Define

This is completely positive and trace preserving. Since $\mathcal{N}(X)={\sum }_j{\lambda }_j{V}_{j}X{V}_j^{†}$, orthogonality gives $\mathcal{RN}(X)=X$.

Let $\{{\Psi }_a\}$ be an instrument. If at least one effect ${\Psi }_a^{\ast }(I)$ is not a scalar multiple of $I$, then:

1. its nonselective matter channel ${\sum }_a{\Psi }_a$ has no CPTP left inverse;

2. its complete classical-record–matter channel $\mathcal{I}(X)={⨁}_a{\Psi }_a(X)$ also has no CPTP left inverse.

Both assertions concern recovery of every input state.

Proof. For (i), use the dilation with environment basis $|a,\alpha ⟩$. Measuring its environment gives the instrument probabilities. A constant complement would make every such probability independent of the input, contradicting the nonscalar effect. Apply Theorem thm:decouple.

For (ii), suppose $\mathcal{R}\mathcal{I}=\mathrm{id}$. Restrict the recovery to each classical output block, obtaining a CPTP map ${\mathcal{R}}_a$. The completely positive maps ${\mathcal{R}}_a{\Psi }_a$ sum to the identity, so Lemma lem:identity-branches gives ${\mathcal{R}}_a{\Psi }_a=p_a\mathrm{id}$. Taking traces and using trace preservation of ${\mathcal{R}}_a$ implies ${\Psi }_a^{\ast }(I)=p_{a}I$, again a contradiction.

A nondemolition measurement of one observable is compatible with this corollary: its eigenstates may be preserved while coherence between eigenspaces is disturbed. Recovery on a proper code subspace is also a different task.

For a constant qubit channel $\mathcal{N}(X)=\mathrm{Tr}(X)\tau$,

Proof. Every decoded channel is constant, with some output $\sigma$. On a maximally entangled input its output is $I/2\otimes \sigma$, whose Bell overlap is $1/4$. Testing the Bell projector gives half-diamond error at least $3/4$.

Preparing $I/2$ attains the bound. Indeed, the completely depolarizing qubit channel equals the uniform mixture of the four Pauli conjugations. Subtracting the identity and using the triangle inequality gives half-diamond distance at most $3/4$.

Only this constant-channel calibration is used. No streaming-memory or radiation error from a different channel is transferred.

Let a declared Hermitian hybrid observable be

The reservoir term, if used, is part of the label specification. The evolution parameter and physical interpretation must be declared independently of a circuit tick or slicing parameter.

Expected energy is conserved for every hybrid input if and only if

The branchwise relations

are sufficient.

If the norm of the defect in (eq:energy-identity) is at most $\delta$ for every $g$, the expected one-step error is at most $\delta$, and the accumulated error after $n$ steps is at most $n\delta$ for a common observable and uniform defect.

Proof. Expected conservation is $⟨\mathcal{E},\mathfrak{F}\rho ⟩=⟨\mathcal{E},\rho ⟩$. Testing all states supported in one input block gives the operator identity.

Under (eq:branch-energy),

For a defect $D_g$,

Iteration and the triangle inequality give the accumulated bound.

Condition (eq:branch-energy) is spectral intertwining. It is not the assertion that postselecting an energy-correlated outcome preserves the unconditional input mean on that branch.

Let $r(g)\ge 0$ be a resource budget and let $c_{h|g}\ge 0$ be a declared edge cost satisfying

On every supported history,

Its expectation satisfies

Proof. The history identity telescopes (eq:path-resource). For the expected identity,

and the effects sum to $I$.

These costs may count permissions, attempts, records, or other declared finite resources. They are not automatically physical energies.

Suppose $\{{\Phi }_{a|g}{\}}_a$ is a complete instrument and $a\mapsto {\Gamma }_a(g)$ is a fixed routing table whose outputs all have valid certificates, including any declared no-update output. Then

is a complete hybrid transition.

Observation compatibility holds if each routed map satisfies

Proof. Grouping completely positive maps preserves complete positivity, and summing all groups gives the original trace-preserving instrument. Summing the indicated adjoint inclusions gives the branch inclusions.

Rerouting does not undo the matter disturbance of the rejected proposal. Nor does rerouting automatically preserve an observation boundary established for a different output algebra.

If the only retained branch is $\Phi (X)=P_{0}X{P}_0$, its trace is $\mathrm{Tr}(P_{0}X)$, not $\mathrm{Tr}X$. The normalized successful output

is not a trace-preserving linear map on the full input domain. A complete model must retain an admissible failure branch or another complete fallback.

For a local instrument $\{{{\Psi }_a}^R\}$ on $R$, with ${\sum }_a{{\Psi }_a}^R$ trace preserving,

This holds for entangled inputs. It does not assert equality of conditional remote states.

Proof. Pair the remote output with $B$. Its expectation is

Let a finite site set $X$ carry local algebras

The local classical variables include all controller records read at that site. The global algebra is ${⨂}_{x\in X}{\mathfrak{X}}_x$, equivalently a direct sum over classical configurations.

At instruction $t$, suppose a fixed envelope $R_t\subseteq X$ contains every quantum and classical register read or written by that instruction, and the complete instruction is a channel

Within $R_t$, the channel may select different operations from its classical controller blocks. The envelope must nevertheless include those controls.

For $Y\subseteq X$, define backwards

Every output observable supported on $Y$ pulls back to an input observable supported on $S_0$. Equal input reductions on $S_0$ therefore give equal output reductions on $Y$.

If $X$ has a metric and $\mathrm{diam}{R}_t\le \Delta$, then

Proof. If $R_t$ is disjoint from the current support, the observable is the identity on that entire envelope. The channel adjoint is unital, so the envelope contributes no support. If they meet, support can enlarge only to their union. Induction proves (eq:local-backwards).

A new point lies within distance $\Delta$ of a point in the previous backwards set. The triangle inequality gives the metric bound. Finally, equality of input reductions gives equality of every pulled-back output expectation.

This theorem does not inherit locality from a controller that reads a global geometry label and then selects a remote support. Such a controller must itself be included in the envelopes.

The distinction is the one already present in VII-E's prescribed schedule theorem and VIII-C's prescribed block-channel bound CheungVIIE (Theorem thm:locality and Proposition prop:direction), CheungVIIIC (Theorem thm:C5). The present extension permits feedback inside a declared local hybrid channel. It is still a dependency theorem, not a spacetime light-cone theorem. No physical directional causal dominance is deduced.

For a hybrid linear map $D$, define its diamond norm through the canonical full-matrix extension that first pinches the input label and then applies the direct-sum map. Equivalently, one may amplify the direct-sum map by arbitrary matrix references and use the trace norm (eq:hybrid-norms).

For two transitions, set

The assisted hybrid norm satisfies

Moreover,

Proof. For an amplified block input $X=(X_g{)}_g$,

For the full-matrix extension, pinching is a trace-norm contraction also after amplification. This follows either from duality with a unital completely positive adjoint or by averaging diagonal phase unitaries. Hence the same upper bound holds there.

An input supported in one maximizing block gives the reverse inequality. The branch bound is the triangle inequality in the output direct sum.

For an arbitrary linear map $D:L({\mathbb{C}}^d)\to L(K)$, write

For every such map,

For a column map $D_g$, its Choi matrix is the direct sum of the branch Choi matrices in the output label.

Proof. Apply $D$ to one half of the normalized maximally entangled state. Its output is $\chi (D)$, giving the lower bound.

For the upper bound, adjoin an arbitrary reference space $N$. Any unit vector in $N\otimes {\mathbb{C}}^d$ can be written

with $A:{\mathbb{C}}^d\to N$, and therefore $\Vert A{\Vert }_{\infty }\le 1$. For two unit vectors $u,v$, represented by $A,B$,

Its trace norm is at most $\Vert J(D){\Vert }_1$. A singular-value decomposition of an arbitrary input operator extends the bound to

Taking the supremum proves the upper bound. The direct-sum assertion follows from linearity of the Choi construction.

These are certification bounds, not numerical results inherited from another finite channel. They use the normalization (eq:choi-convention) explicitly.

Suppose two executions use the same label dictionaries and satisfy

Their terminal channels obey

The same bound holds for their complete geometry-history channels.

For a common input, including an arbitrary reference, the trace distance of the corresponding outputs, and the total-variation distance of their geometric history laws, are at most

Proof. Insert and subtract one implemented step at a time. Each term has the difference of one pair of steps sandwiched between channels. Channels have diamond norm one, and the diamond norm is submultiplicative Watrous2018. The triangle inequality gives (eq:telescoping).

For history retention, a current history is a classical input label. Its next-step column is $D_{g_t}$, with the preceding history copied into the output label. Copying that fixed classical prefix does not change the norm of the column. Proposition prop:diamond-column therefore gives the same one-step bound, now on the history algebra. Apply the same telescoping argument.

A common normalized input gives trace-norm difference at most the channel diamond bound. Trace distance contributes the factor $1/2$. Reading only the geometric record is a channel, so it cannot increase trace distance; on diagonal states that distance is total variation. Distances between states are at most one.

For an observation space $\mathcal{O}$, define

Then

Indeed, interchange the suprema and optimize the Hermitian expectation by an eigenstate in a maximizing block. This is the hybrid form of VII-E's accessible-output distance CheungVIIE (Proposition prop:accessible). It is unassisted unless a reference is included explicitly.

An exact clock guard $j=0$ is stable under an exact finite representation, but not under arbitrary perturbation of a real parameter. Similarly, if a length is $c|s|$, a floor report at resolution $h$ is zero whenever $0<c|s|<h$. It cannot serve as an exact event detector.

Theorem thm:stability applies to already specified channels on the same finite dictionaries. It does not convert a discontinuous geometric guard into a uniformly continuous map of real geometric data. Such a claim requires a separate margin or event-resolution theorem, as emphasized in VII-E and VII-F CheungVIIE, CheungVIIF.

This section witnesses the consistency of the finite specification. It is not a replacement for VII-F's particle-link incidence construction.

Fix an integer budget $B\ge 1$, a tick duration $\tau >0$, and $0<\delta <\tau$. Set

The resource is $r(k,q)=B-k$. The controller tick is $k\tau$. The selected slice height is

These two quantities are distinct.

Use the fixed ultrastatic spinless cone

with its wedge boundary rays identified. Each label denotes the horizontal slice $T=T(k,q)$, with the same fixed sector cellulation, stationary axis, and any fixed annular collar.

For $k<B$, define

At $k=B$, the label is absorbing.

Every successor in (eq:example-successors) belongs to ${\mathsf{G}}_B$. Its slice is embedded and spacelike, and advances in spacetime time by $\tau +a\delta >0$.

The complete affine peripheral word is

on every label, with the same marked stationary axis. Location transport is the identity on the persistent cell labels. The resource decreases by exactly one on every active edge.

Proof. The inequalities and parity in (eq:example-labels) are preserved by $k\mapsto k+1$, $q\mapsto q+a$. A horizontal graph has induced metric equal to the spatial cone metric and is embedded. Formula (eq:slice-height) gives the time increment.

The upper seam maps to the lower seam by the rotation $R(\pi /2)$, which fixes the time axis pointwise. A positively oriented primitive meridian crosses that seam once; its angular lift advances by $3\pi /2$. Its affine word is therefore $J$. Moving the adapted horizontal chart by $-T(k,q)e_0$ commutes with $J$, so there is no induced affine time translation. The cellulation and collar are unchanged spatially, giving identity cell-label transport. Finally, $B-(k+1)=B-k-1$.

The cone interpretation and the need to retain full affine data are consistent with the classical point-particle description and the fixed-cone certificate of VII-F DJH1984,CheungVIIF. Here no deficit or worldline changes.

Let $\mathcal{H}={\mathbb{C}}^2$, $P_j=|j⟩⟨j|$, and

Choose $0\le \eta \le 1$ and define

Let