VII-E-ST: Observation boundaries and finite hybrid feedback on a prescribed conical spacetime Record compatibility, residual separation, certified markings, and causal transport

We construct a supported local spacetime adaptation of the observation-boundary contract of VII-E and the finite hybrid-channel contract of VII-G. The two contracts are proved independently within one laboratory on I-B's prescribed conical Lorentzian product. For the observation boundary, we prove the effect-pullback criterion, its reference-stable form, the exact sharp-measurement coherence criterion, and a qubit separation of trace distance $1/2$ between ideal residual targets with identical records. For hybrid evolution, we prove a complete branch representation, history-sensitive observation closure, spectator consistency, and the exact recovery–constant-complement criterion. An informative classical record consequently excludes recovery of every unknown quantum input, even when the declared residual is also available.

The additional contribution is an end-to-end finite observation package that keeps these distinctions visible in supplied spacetime. We provide a compactly supported $3+1$-dimensional lift of a four-face transverse adjacency exchange, an exact guarded reversible encoding, and a separately constructed qubit instrument selecting strictly ordered slices without changing the conical metric. A two-step specialization has two histories with the same terminal slice but different conditional residuals. After coarse aggregation, the record remains unchanged while two complete residual targets have the exact half-diamond gap $4(1-1/\sqrt{2})/25$. An explicit residual audit enlarges the required input observation space from the diagonal algebra to the full qubit algebra.

A destructive finite clock, complete-channel approximation bounds, and a causal route for both payload and classical control complete the local Grade S construction. The package may serve as a precise testbed for distinguishing record-level claims from instrument-level claims in geometric feedback models. The channel and recovery principles themselves are established finite-dimensional facts; the claimed contribution is their supported assembly and the explicit certificates and separations. No metric emergence, gravitational dynamics, field instrument, radiative coefficient, or quantum Ward identity is constructed.

contracts, one laboratory{ The VII-E contract concerns what a declared observation boundary can distinguish. The VII-G contract concerns a state-independent finite completely positive hybrid evolution. Neither contract derives the cone, and neither is a premise supplying the other.

The four-face exchange is an illustration, not an assumption of the general channel theorems. Matter-controlled slicing is not matter-driven gravitational dynamics.}

A probability law for a classical record does not specify the selective transformation of the surviving quantum system. Likewise, preservation of one named expectation does not imply recovery of every input. These distinctions are basic to the instrument formulation of quantum measurement DaviesLewis,Choi,Watrous.

This paper places two source contracts in a common supported spacetime presentation:

1. VII-E's observation boundary, residual separation, spectator condition, and four-face certificate CheungVIIE;

2. VII-G's finite direct-sum channel, branchwise observation criterion, recovery obstruction, and ordered-slice model CheungVIIG.

Their mathematical statements are reproved below. Citation does not replace a proof, identify two models, or provide missing geometric dynamics.

In Cheung's Enter calculus, runtime quantum names are access rights to tensor factors. Destructive computational-basis readout consumes the measured names and returns classical records. Unmeasured factors can remain available CheungEnter,CheungI. The characterization of the Born strategy in Manuscript I concerns its explicitly specified class and five requirements. We use a specified Born instrument and prove its channel formula; we do not rederive that characterization or extend it to quantum fields.

The accompanying philosophy distinguishes:

An output type does not determine the fate of all physical degrees of freedom used by an apparatus, and a channel does not specify a thermal machine CheungPhilosophy.

I-B supplies a finite operational interface on a prescribed conical spacetime CheungIB. A finite clock can be calibrated to its time coordinate, destructively read, and used for causal control. The geometric time axis, defect worldsheet, and future cone remain after that clock is consumed.

A local Grade S observation package specifies and proves:

1. a prescribed Lorentzian background and a supported regular laboratory;

2. named observation and evolution contracts, with their mathematical domains;

3. a complete finite destructive-clock instrument;

4. a declared finite encoding of labels and selected residual observations;

5. reference-preserving transport with complete dependencies embedded in the supplied causal relation.

Grade E would additionally require a construction deriving the geometric data from operational records. It is not claimed.

boundary. Direct-sum state spaces, Kraus representations, deferred readout, and exact quantum error correction are not new general theorems Choi,GurevichBlass,KnillLaflamme. The contribution here is a supported assembly with explicit geometric and finite-channel witnesses, including the equal-slice residual separation proved in Section sec:two-step.

Fix

The physical smooth background is

where $r>0$ and $\varphi$ is periodic with period $2\pi$. Its completion has the conical worldsheet

No smooth Einstein tensor is assigned on $\mathcal{W}$.

The metric has signature $(-,+,+,+)$. Choose ${\partial }_T$ as future directed. A nonzero future-causal vector $a{\partial }_T+w$ satisfies

Static observers have proper time $\mathrm{d}\tau =\mathrm{d}T$, so the proper-time lapse is one.

On a sufficiently small lifted angular interval,

is a local isometry to Minkowski space. Its spatial coordinates give three local rods.

In the inverse-deck convention the peripheral affine gluing is

It has neither time nor axial translation.

Proof. The signature and causal inequality follow from the diagonal metric. Setting $w=0$ proves the proper-time statement. The local-isometry identity is

A positive angular lift around the apex advances through $2\pi -\mu$; the inverse deck map closes it by rotation through $\mu$. Its time and axial coordinates do not change.

The regular product $M$ is not globally hyperbolic.

Proof. Fix $r_0>0$, an angle, and $z=0$. Let $p,q$ have that spatial position and times $-2r_0/c$ and $2r_0/c$. For $r_n\downarrow 0$, the events $(0,r_n,\varphi ,0)$ lie in $J^{+}(p)\cap J^{-}(q)$, by radial causal travel. They have no convergent subsequence in $M$. Thus this causal diamond is not compact.

A supported presentation consists of:

1. a regular developing chart and a relatively compact laboratory region with closure disjoint from $\mathcal{W}$;

2. compact spatial collars, finite marked cells, and compactly supported tests used by the declared observations;

3. finite quantum and classical registers, their locations, and every controller variable inspected by an operation;

4. a finite schedule with explicit input and output ports and no undeclared incoming boundary inputs.

Any background exterior labels are fixed metadata. No global Cauchy theorem, Green operator, or intrinsic conical field theory is part of this presentation.

All matrix theorems below are independent of a field equation. Developed charts, polygonal markings, and collars pull back through the stated local isometry. They are not declarations that the physical spacetime is global Minkowski space.

For a finite-dimensional Hilbert space $H$, write $\mathcal{B}(H)$ for its operators. A channel is a completely positive, trace-preserving linear map. Adjoint maps use the pairing

We use

A classical register is jointly block diagonal in its designated basis, including its correlations with all other registers. A diagonal marginal alone does not establish classicality.

Let $S$ have computational basis $\{|z⟩\}$, with unmeasured complement $B$. Put $D_z=⟨z{|}_S\otimes I_B$. Destructive readout is

Its quantum output contains $B$, not $S$. The measured names are consumed.

A fresh quantum carrier $A$ can record the computational basis through

The channel $X\mapsto VX{V}^{†}$ returns $S,A,B$. It is not destructive readout and not cloning: $V|+⟩=(|00⟩+|11⟩)/\sqrt{2}$.

With $P_z=|z⟩⟨z{|}_S\otimes I_B$, the represented nondestructive sharp instrument is

It can be implemented using a fresh pointer and destructive readout of that pointer. The original $S$ is returned.

An external geometric label is classical metadata describing a marking, chart, certificate, or location. A dictionary from measurement outcomes to such labels is an additional input. It is not one of the preceding quantum maps.

These distinctions prevent a represented residual qubit from being described as the return of a consumed quantum name.

Let $\{P_z\}$ be a finite projective resolution of $I_H$, let $q:Z\to W$ aggregate fine outcomes, and put

Let $T:\mathcal{B}(H)\to \mathcal{B}(K)$ be a channel and $\{{\mathcal{I}}_w{\}}_{w\in W}$ an instrument on $K$, with $E_w={\mathcal{I}}_w^{\ast }(I_K)$. Compare

This comparison uses represented fine projections on a common input space. For a genuinely destructive protocol, any subsequent removal or identification of $S$ must be included explicitly in $T$; it is not hidden in ${\Delta }_w$.

The record marginals of $\mathcal{F}$ and $\mathcal{G}$ agree for every input state if and only if

For an observation space $\mathcal{O}\subseteq \mathcal{B}(K)$, agreement on every observable ${\sum }_w|w⟩⟨w{|}_R\otimes B_w$, $B_w\in \mathcal{O}$, is equivalent to

Both are equalities of linear maps on the declared observation spaces and remain valid after tensoring with any untouched reference identity.

Proof. The fine aggregated probability is

while the coarse probability is $\mathrm{Tr}(\rho T^{\ast }(E_w))$. Equality against all states is equality of Hermitian effects.

The two $w$-conditioned expectations of $B$ pull back to the two sides of (eq:residual-pullback). States separate Hermitian matrices, and complex linearity gives the general equality. Tensoring an equality of linear maps with a reference identity preserves it.

The following useful error formula is record-only and unassisted:

Indeed, total variation is the maximum absolute difference on an outcome event. Interchange the finite maximum with the state supremum and use ${\sup }_{\rho }|\mathrm{Tr}(H\rho )|={\Vert H\Vert }_{\infty }$ for Hermitian $H$. No equality with an assisted channel error is inferred.

Record compatibility does not determine a residual instrument. There are ideal targets with identical records on every input whose outputs have trace distance $1/2$ on one qubit input.

If implemented versions have errors at most ${ϵ}_{\mathcal{F}},{ϵ}_{\mathcal{G}}$ on that input, their mutual distance is at least

Proof. Take $H=K={\mathbb{C}}^2$, $T=\mathrm{id}$, the computational projectors, and one coarse outcome $\ast$ with ${\mathcal{I}}_{\ast }=\mathrm{id}$. Both record effects are $I$. For ${\rho }_{+}=|+⟩⟨+|$,

Their difference has residual block

with eigenvalues $1/2,-1/2$. Its trace norm is one. The robust bound is the triangle inequality.

The fine residual is realized by a pointer measurement returning the qubit; it is not primitive destructive readout of that qubit. Improving implementations cannot make distinct ideal target channels equal.

Suppose now that $\{E_w\}$ is projective and that the coarse instrument is Lüders:

Define the two pinching maps

Under (eq:record-pullback), every Kraus operator $A_{\nu }$ of $T$ satisfies

Consequently

Proof. For $\psi \in \mathrm{im}{\Pi }_w$,

Thus $A_{\nu }{\Pi }_w=E_w{A}_{\nu }{\Pi }_w$. For $\psi \in \ker {\Pi }_w$, the same calculation with $E_w$ gives $E_w{A}_{\nu }\psi =0$. Combine the two identities.

Under the sharp hypotheses and record compatibility, the following are equivalent:

1. $\mathcal{F}=\mathcal{G}$ as complete recorded residual channels;

2. $T{\mathcal{D}}_{\Pi }=T{\mathcal{D}}_P$;

3. $$T(P_{z}X{P}_{z^{\prime }})=0\text{whenever }z\ne z^{\prime },q(z)=q(z^{\prime }),X\in \mathcal{B}(H).$$

Thus full compatibility removes fine coherence within each coarse outcome. It need not remove coherence between different coarse outcomes.

Proof. By Lemma lem:sharp, the $w$-block of $\mathcal{G}(X)$ is $T({\Pi }_{w}X{\Pi }_w)$. Its difference from the corresponding $\mathcal{F}$-block consists exactly of the fine off-diagonal corners within ${\Pi }_w$.

Summing the $w$-blocks gives (ii). Conversely, their output supports lie in orthogonal $E_w$-corners, so a zero sum makes each corner zero. Testing a single $P_{z}X{P}_{z^{\prime }}$ proves necessity of (iii), and expansion into corners proves sufficiency.

For injective $q$, record compatibility already suffices. For one coarse outcome, the criterion is $T=T{\mathcal{D}}_P$. These are finite instrument identities, not field-instrument constructions.

An observation space is a complex linear, adjoint-closed subspace containing the identity. It need not be an algebra. Write $\rho {\equiv }_{\mathcal{O}}\sigma$ when the two states agree against every element of $\mathcal{O}$.

For a channel $\Phi$ and observation spaces ${\mathcal{O}}_{\mathrm{in}},{\mathcal{O}}_{\mathrm{out}}$, the following are equivalent:

and

If this inclusion holds successively, compatible continuations preserve the declared agreement.

The reference-stable comparison of two extended inputs requires agreement on ${\mathcal{O}}_{\mathrm{in}}\otimes \mathcal{B}(E)$, not merely agreement of their unreferenced marginals.

Proof. Sufficiency follows by pulling each output observable back. For necessity, take a Hermitian matrix $H$ annihilating ${\mathcal{O}}_{\mathrm{in}}$ under the trace pairing. Since $I\in {\mathcal{O}}_{\mathrm{in}}$, $\mathrm{Tr}H=0$. For a faithful state $\omega$, both $\omega +\epsilon H$ and $\omega -\epsilon H$ are states for sufficiently small $\epsilon >0$. They agree on ${\mathcal{O}}_{\mathrm{in}}$, hence

Finite-dimensional double orthogonal complementation gives the inclusion. Complex linearity completes the proof. Tensoring the inclusion proves the reference statement.

Let $\{K_a\}$ be a complete Kraus family and $\{U_a\}$ unitaries on the surviving system. Immediate pointer readout followed by $U_a$, and coherent controlled $U_a$ followed by pointer readout, implement the same recorded instrument:

This identity is complete under references.

For coherent copying of an original computational register, agreement with destructive readout holds only after both outputs are restricted to the same record–matter interface. A retained coherent carrier is not an additional common output.

Proof. The isometry

satisfies $W^{†}W=I$. Controlled feedback changes it to ${\sum }_a|a{⟩}_A\otimes U_a{K}_a\psi$. Destructive readout of $A$ gives (eq:deferred). The immediate implementation has the same selective maps.

For computational copying, write $X={\sum }_{z,z^{\prime }}|z⟩⟨z^{\prime }|\otimes X_{z{z}^{\prime }}$. After coherent copying and label-preserving controlled operations, final readout and restriction to the common interface retain precisely the diagonal blocks $X_{zz}$. If both copied measured registers are consumed, the duplicate classical outcome can be explicitly forgotten. No quantum return of a consumed register is involved.

A continuation that reads a phase-sensitive observable of the coherent carrier, or rewrites the controller using that carrier, is outside this equivalence. Deferred-measurement principles require the corresponding interface and control hypotheses GurevichBlass. They do not make remote control instantaneous or preserve an implementation's storage cost automatically.

For Contract E, allow a classical configuration

where $g$ is a geometric label, $\Gamma$ the original accessible interface, $h$ a controller state, and $\mathrm{loc}$ assigns abstract cell labels to names. A geometric rule changes declared geometric data and location labels. A matter rule applies a declared channel. A coupling rule may inspect a declared state functional $\rho {|}_{{\mathcal{A}}_v}$ and update a classical label.

Such a coupling function need not be affine. Contract E's equal-input comparison does not by itself make it a quantum channel.

Adding inaccessible spectator names does not alter: the chosen rule or channel, exact guards, capacity checks, certificates, successor labels, controller updates, or transport of original names. Spectator storage and any required spectator location history are reserved separately. The controller does not inspect spectator outcomes or spectator cardinality.

Under Assumption ass:blind, compare a finite execution with an extension satisfying ${\mathrm{Tr}}_N\tilde{\rho }=\rho$ on the original accessible interface. Use the same initial geometric and control labels and original-name locations.

At every step the executions have the same geometric trace, the same accessible reduced matter state, and the same original-name locations. The spectator extension may initially be entangled.

Proof. For any linear map $\Phi$,

This follows by expansion in matrix units on $N$. A matter step therefore preserves equality of accessible reductions. Equal reductions give equal declared state functionals, so geometric and coupling steps choose the same successors. Blindness makes guards, certificates, and resource checks identical, and the same fixed location maps apply. Induction proves the assertion.

This compares an extension with its restriction. It does not assert that all extensions with one marginal have identical joint reference correlations, or that selective conditioning cannot steer an untouched spectator.

This section supplies an illustration independently of the general observation proofs. The numerical transverse family is the one recorded in VII-E CheungVIIE; its required properties are proved here.

Specialize the prescribed deficit to $\mu =\pi /2$. Use a regular developed patch in the first quadrant, centered at developed spatial coordinates $(20,20)$, and write its local coordinates as $(x,y,z)$. Set $t=c(T-T_{\ast })$. Numerical lengths below use one fixed length unit.

Put

For $-1\le s\le 1$, define

Label the four affine dominance regions $E,W,N,S$.

The graph $t=F_s(x,y)$ is piecewise spacelike and $p$-Lipschitz. For $s_2>s_1$,

Its central edge is

At $s=0$ the central edge becomes one four-face junction. No transverse face disappears there.

Proof. Each affine spatial gradient has norm at most $p<1$. Their maximum has the same Lipschitz bound. A tangent to a face of slope $v$ has squared norm

All four functions increase with $s$ at rate at least $a$.

For $E=W$, equality forces $x=0$, and dominance over $N,S$ is $q|y|\le -ds$, possible precisely on the incoming side. For $N=S$, equality forces $y=0$, with $p|x|\le ds$. The edge tangents have constant $t$, so their lengths are the displayed Euclidean coordinate intervals. At $s=0$, each strict-dominance region contains an open cone.

The rates

show transversality. The four mixed adjacencies persist and satisfy $p|x|-q|y|=ds$ in their respective quadrants.

Use the following $2+1$-dimensional Lorentz matrices, extended by the identity on $z$:

Direct multiplication verifies Lorentz orthogonality, determinant one, and future orientation. On each corresponding plane the transformed time coordinate is $s$.

Set ${\Lambda }_{i\leftarrow j}=C_i{C}_j^{-1}$. Every closed ordered core word is the identity, since

This proves the incoming triple, outgoing triple, and event word identities without interpreting the nontrivial frame changes as curvature.

Let

The nonhorizontal region is

Take

The collar margins are

They are strictly positive throughout the parameter window.

The northeast resting collar polygon has spatial vertices

Reflections give the other three. They partition $K\setminus R_s$ and lie at $t=2+s$. The collar chart is

and the core charts are $A_i=C_i$. Thus all transitions are $A_i{A}_j^{-1}$, including their translations. Closed mixed words telescope in the full affine group.

The capped transverse patch has eight two-dimensional faces and 25 actual edges on either regular epoch. Its noncentral edge classes are:

Every noncentral edge has length greater than $1/2$. The only transverse incidence change is the central exchange.

Proof. The vertices are the two central endpoints, four inner corners, four inner side midpoints, four outer corners, and four outer side midpoints. The central edge and four mixed arms, followed by the listed collar edges, exhaust their incidences.

For the incoming northeast arm, the displacement is $X_s(p,1,p/q)$ in $(t,x,y)$. Its squared norm is ${X_s}^{2}1921/625$. For the outgoing arm use $Y_s(q,q/p,1)$, with squared norm ${Y_s}^{2}1921/1521$. These give the arm formulas, which agree at zero. All remaining edges are horizontal and have the listed Euclidean lengths.

The smallest collar-join length is $3/5$; the arms exceed $3\sqrt{1921}/25$. All faces have open interiors: core faces near their inner side midpoints and collar faces by the strict margins. The central dominance calculation then isolates the only transverse incidence change.

A transverse $2+1$ picture alone is not a compactly supported hypersurface marking in $3+1$ dimensions. Define

Use the spatial box

The graphs $t={\hat{f}}_s$ are continuous piecewise spacelike hypersurface markings with spatial Lipschitz constant at most $3/5$. They are strictly ordered:

They equal $t=2+s$ near $\partial \hat{K}$, and coincide with the transverse capped family for $|z|\le 1$.

They extend by the horizontal marking outside $\hat{K}$ on the regular conical product. The modification has compact spatial support disjoint from $\mathcal{W}$.

Proof. The affine pieces have spatial gradients among

Minima and maxima preserve the common Lipschitz bound. Their parameter slopes are at least $a=4/5$, proving strict ordering.

For $|z|\le 1$, $L_s=s-4<-3$, while $F_s\ge -4/5$. Thus the lift agrees with (eq:transverse-cap). For $|z|\ge 13$, $L_s\ge 2+s$, and the graph is horizontal. Near the lateral boundary of $K$, the transverse collar already has height $2+s$. This proves boundary matching and compact support.

Translation by $(20,20)$ places the entire box in a regular first-quadrant chart of the $\mu =\pi /2$ cone. The local isometry pulls the construction back. Outside the box the marking is unchanged.

For an affine piece $t=v\cdot x+\beta (s)$, a fully specified adapted chart is the standard Lorentz boost

followed by time translation $(s-{\gamma }_v\beta (s))e_0$. For $v=0$, use its continuous identity-boost value. Thus every lifted piece has an explicit affine chart, and all closed transition products still telescope.

The four-face statement concerns the transverse section $|z|\le 1$. In the full lift its junction is extended in the axial direction. We do not relabel it as an isolated $3+1$-dimensional particle event, nor claim a catalogue of all additional taper incidences.

Let

At the transverse central event, dot products of the two boundary rays within an $E$-face give cosine $-1121/1921$; within an $N$-face they give $671/1921$. The other two faces are reflections. Hence

To check the sign, $A>\pi /2$, and

Thus $B>\pi -A$.

At an incoming triple junction, the two equal face angles have sum $\pi +A$, and the remaining angle is $B$. For example the equal-angle cosine is $-39/\sqrt{1921}$, and the double-angle identity gives that sum. The defect is $\pi -A-B={\kappa }_0/2$. The outgoing calculation interchanges the two pairs and has the same result.

The marking therefore redistributes a transverse intrinsic angle defect while the ambient metric remains fixed and locally flat. This is not a bounce or the creation of spacetime curvature.

Let $Q_{-}$, $Q_{+}$ denote the two regular transverse epochs and $Q_{0^{-}},Q_{0^{+}}$ their event germs. At the event, retained central edge slots have zero length and realize the same junction; they are metadata, not regular embedded one-cells.

The forward event rule has guard

It changes the mode to one, spends one permission, and chooses the outgoing germ without advancing $s$. A fixed abstract location map sends the incoming central slot and its endpoint labels to the outgoing event-junction slot; persistent labels are unchanged. No physical matter is transported through the patch by this metadata rule. If a spectator has an abstract location there, its location history is separately reserved.

Fix a finite parameter grid

stored exactly. Store rounded length reports only as reports. A rounded zero length cannot replace the exact guard: if $\ell =v(-s)$, floor quantization at resolution $\epsilon$ reports zero for $0\le -s<\epsilon /v$.

Let $x$ be a tuple of valid location words and $\tau (x)$ its fixed event transport. On valid forward codewords prescribe

At zero, both central reports are zero and all common noncentral formulas agree, so the report words remain valid.

The finite map (eq:event-code) has a permutation extension to the complete binary code space. That permutation has an exact NOT–CNOT–Toffoli implementation with clean work uncomputed. It acts as the identity on internal matter.

Proof. The output history determines $x$, and its permission determines $k$, so the forward map is injective. Input and output sets are disjoint because their modes differ. Exchange each valid input with its designated output, and fix all other strings. The exchanges are disjoint transpositions. The decoder reads exactly the outgoing germ, spent permission, and transported locations.

For completeness, any transposition of two $N$-bit strings can be synthesized along a Gray path between them. The symmetric sequence of adjacent transpositions exchanges the endpoints and fixes intermediate strings. An adjacent transposition is a NOT on the differing bit, controlled on all other bits. Negative controls are converted by NOT gates; the conjunction of positive controls is computed with Toffoli gates into clean work, used once, and uncomputed. Apply this construction to the finite transposition list. History is deliberately retained; it is not clean work.

If the code has $N_E$ bits, at most $\max \{N_E-2,0\}$ clean conjunction bits suffice for this construction. This is a finite reversible existence proof, not a reported compilation.

The preceding family supplies a local certificate containing:

1. piecewise spacelike ordered markings and transverse incoming/outgoing central edge laws;

2. complete transverse edge data, affine charts, and exact ordered products;

3. a compact transverse collar and a compactly supported $3+1$-dimensional lift matching a fixed horizontal exterior;

4. a transition stratum with exact guard and fixed abstract location transport;

5. a decoder-correct reversible finite event implementation, with separately charged history and spectator reserves.

It is a type-A transverse incidence illustration only. It changes neither $\mu$, $\mathcal{W}$, nor $J_{\mu }$.

Proof. Lemma lem:central supplies (i). The explicit charts and Proposition prop:edge-data supply (ii). The collar construction and Proposition prop:lift supply (iii). The event definitions and Lemma lem:event-code supply (iv)–(v). The construction lies in one regular developing patch and is horizontal near its boundary. It changes no global gluing map or worldsheet data.

The connection with polygonal descriptions of $2+1$ gravity is contextual DJH,CheungVIIE. No polygon Hamiltonian, particle hop, field equation, or global transition catalogue is deduced.

Let $\mathsf{G}$ be a finite classical label set. Each label contains every variable used by the controller: geometric marking, permission, relevant records, and control memory. Matter occupies a fixed $d$-dimensional space $H$. Set

States have $X_g\ge 0$ and ${\sum }_g\mathrm{Tr}{X}_g=1$. The blocks are subnormalized; their normalization is not part of the unconditional evolution.

A certified edge $g\to h$ includes all changed classical data, exact guards, geometric matching, and location maps. No edge certificate is inferred from its probability alone.

A linear map between finite direct sums is completely positive and trace preserving exactly when it has branch form

with

Its adjoint is

A canonical channel on the full matrix space has Kraus operators

It discards off-diagonal input label blocks and produces block-diagonal output. It is a specified lift, not a reconstruction of quantum geometric coherences.

Proof. After adjoining any matrix ancilla, a positive direct-sum input has positive blocks. Complete positivity of the branches and summation prove complete positivity of $\mathcal{F}$. Equation (eq:branch-complete) gives trace preservation.

Conversely, compose a given completely positive map with injection into input block $g$ and projection onto output block $h$. These maps give completely positive branches. For a map between matrix algebras, its Choi matrix is positive; diagonalizing that matrix and unvectorizing its rank-one terms gives the displayed Kraus representation Choi. Trace preservation on each input block gives (eq:branch-complete).

The adjoint follows by pairing. Multiplication of the lifted Kraus operators gives the full identity as their sum of squared moduli, and their action uses only diagonal input blocks.

The table ${\Psi }_{h|g}$ is fixed independently of the current unknown density operator. Matter dependence enters through Born probabilities

An undeclared state-dependent compiler is not part of this law.

Composition is explicit:

Completeness follows by summing its adjoint effects. Certificates concatenate along the declared paths; this does not create a certificate for a previously uncertified edge.

For observation spaces in finite direct sums, preservation of agreement for all states is equivalent to

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

In particular, successor probabilities depend only on $\rho {|}_{{\mathcal{A}}_g}$ exactly when $E_{h|g}\in {\mathcal{A}}_g$ for every $h$.

Proof. Use the faithful-state perturbation argument of Theorem thm:composition within the direct-sum algebra. An output observable can have just one nonzero block, proving necessity of every branch inclusion. Summation proves sufficiency. Apply the same argument to the single effect $E_{h|g}$ for the final statement.

This is the direct-sum specialization of a general adjoint criterion, not a new general channel theorem.

For a finite sequence of fixed tables and a history $\gamma =(g_0,\dots ,g_n)$, define

The history map is a channel: each path map is completely positive, and summing the last successor, then the preceding ones, proves trace preservation.

Suppose observation spaces satisfy the branchwise closure

Initial agreement on ${⨁}_g{\mathcal{O}}_{0,g}$ implies:

1. equality of all geometric history probabilities;

2. agreement of every unnormalized history block on its declared terminal observation space;

3. agreement of normalized conditional expectations on every positive-probability history.

Proof. Pull a terminal $B\in {\mathcal{O}}_{n,g_n}$ backward along its fixed history. Repeated application of (eq:history-closure) places the pullback in ${\mathcal{O}}_{0,g_0}$. Pairing proves (ii); $B=I$ proves (i). Divide by the same positive probability for (iii).

Branch closure is necessary for the corresponding universal one-step comparison at every node, by Theorem thm:observation. It is not asserted necessary for accidental agreement on one input or one terminal test.

Let the input observation algebra be $\mathrm{span}\{I,Z\}$. Let

where $H$ is the Hadamard matrix. The summed channel has zero output $Z$-expectation on every input, so it preserves agreement for that terminal observation. But

The inputs $|+⟩⟨+|$ and $|-⟩⟨-|$ agree on the input diagonal algebra and have the same outcome probabilities, yet their conditioned $Z$-expectations differ.

For a fixed horizon and terminal observation spaces, define

These are the smallest observation spaces satisfying (eq:history-closure). Agreement on their initial direct sum is equivalent to agreement of all complete history-conditioned terminal expectations.

For a stationary table and geometry-only reports, iteration from ${{\mathcal{O}}_g}^{(0)}=\mathbb{C}I$ stabilizes after at most $|\mathsf{G}|(d^2-1)$ strict enlargements. The stable space characterizes agreement of all finite geometric history laws.

Proof. Adjoints preserve adjoints, so the recursively defined spans are observation spaces. Any closed family must contain these spans, proving minimality. The recursively generated elements are precisely spans of pathwise pullbacks. The inserted identity contributes no independent condition because ${\sum }_h{\Psi }_{h|g}^{\ast }(I)=I$.

For stationary iteration, the sum of dimensions starts at $|\mathsf{G}|$ and is at most $|\mathsf{G}|d^2$. An iteration without enlargement is already fixed. Path effects generate exactly the stable space.

No multiplicative closure is required. Replacing the minimal space by its generated algebra may expose more information than the chosen histories need.

Assume complete spectator blindness and an initial extension with ${\mathrm{Tr}}_N{\tilde{X}}_g=X_g$ for every $g$. For every complete history,

Thus all history laws, restricted unnormalized residual blocks, certificates, and original-name locations agree.

Proof. Equation (eq:restriction) applies to each selective branch. Induction along a path proves the block identity. The hypotheses retain the same path labels, exact guards, controller tables, and certificates. The assertion includes zero-probability paths as zero blocks.

The recovery theorem in this section concerns the entire quantum input algebra $\mathcal{B}(H)$, not merely one observable or a classical subset of states.

Let $\mathcal{N}:\mathcal{B}(H)\to \mathcal{B}(K)$ be a finite-dimensional channel. The following are equivalent:

1. a channel $\mathcal{R}$ satisfies $\mathcal{RN}={\mathrm{id}}_{\mathcal{B}(H)}$;

2. a complementary channel is constant: ${\mathcal{N}}^c(X)=\mathrm{Tr}(X)\sigma$;

3. its Kraus operators satisfy

   $${A_{\alpha }}^{†}{A}_{\beta }={\lambda }_{\alpha \beta }{I}_H$$

   for a positive matrix $\lambda$;

4. after unitary mixing of Kraus operators,

   $$\mathcal{N}(X)=\sum _j{p}_j{V}_{j}X{V}_j^{†},{V_i}^{†}{V}_j={\delta }_{ij}{I}_H,p_j>0,\sum _j{p}_j=1.$$

All recovery identities hold completely under references.

Proof. A Kraus dilation gives

so (ii) and (iii) are equivalent.

Suppose $\mathcal{R}$ has Kraus operators $B_{\nu }$. The composed Kraus operators $B_{\nu }{A}_{\alpha }$ represent the identity channel. Its Choi matrix has rank one, so every $B_{\nu }{A}_{\alpha }=c_{\nu \alpha }I$. Consequently

This proves (i)$\Rightarrow$(iii).

Diagonalize the positive matrix $\lambda$ by a unitary mixing of the Kraus operators. Nonzero transformed operators have products ${L_i}^{†}{L}_j=p_i{\delta }_{ij}I$. Set $V_i=L_i/\sqrt{p_i}$, obtaining (iv). Their range projections are orthogonal. With $P={\sum }_i{V}_i{V}_i^{†}$, the map

is completely positive and trace preserving for any fixed state ${\tau }_H$. It recovers every matrix unit and hence every operator. Tensoring the identity proves complete recovery.

This is the full-input form of the standard error-correction and information–disturbance structure KnillLaflamme,KSW; the explicit proof fixes our domain.

For an instrument with recorded channel

if some effect $E_w={\Psi }_w^{\ast }(I)$ is not scalar, no channel recovers every input from the entire recorded output, including its residual.

Proof. Choose branch Kraus operators $K_{w,\nu }$. A dilation can duplicate the classical outcome into its environment:

The environmental marginal has outcome-$w$ probability $\mathrm{Tr}(E_w\rho )$. A constant complement would make this probability independent of every input, requiring $E_w=p_{w}I$. Apply Theorem thm:decouple.

A constant record law is necessary, not sufficient, for full recovery. A replacement channel has a constant record and is not reversible. Conversely, random unitary branches with input-independent recorded probabilities can be corrected using the record.

The corollary does not prohibit copying already classical data. It also does not preclude recovery on a proper code subspace or preservation of a named commutative algebra.

Let a convex domain of states be mapped affinely to classical probability distributions. If every output is a point mass, the output label is constant.

Proof. If ${\rho }_0,{\rho }_1$ have different point-mass outputs, an interior convex combination has the corresponding nontrivial mixture, not a point mass.

Thus a nonconstant deterministic expectation-to-packet rule, such as the comparison called Option N, cannot be promoted unchanged to the unconditional classical output of an affine channel. Such a nonlinear rule may be studied under a different contract. A supplied source token is not a measurement extracting an expectation from unknown matter CheungPackets.

For declared Hermitian observables $H_g$, preservation of their expected value for every hybrid input is equivalent to

The branch intertwinings $H_h{K}_{h|g,\nu }=K_{h|g,\nu }{H}_g$ are sufficient. A uniform operator defect at most $ϵ$ gives a one-step expectation error at most $ϵ$, and at most $nϵ$ after $n$ steps for one common observable.

Separately, if a permission function and edge costs satisfy

on every supported branch, then each history obeys

Proof. Pair ${\mathcal{F}}^{\ast }H-H$ with every state supported in one block. The branch sufficiency follows by summing $K^{†}{H}_{h}K=K^{†}K{H}_g$. For a normalized hybrid state, the absolute defect expectation is bounded by the largest block operator norm. Iteration proves the accumulated bound. The resource identity telescopes pathwise.

Calling $H$ “energy” does not provide a physical Hamiltonian, reservoir, work budget, or Einstein equation. Permission costs are not automatically physical energies.

This construction does not use the four-face family or its event rule. It uses only the prescribed conical product and finite matrices, adapting the ordered-slice example of VII-G CheungVIIG.

Fix an integer $B\ge 1$, physical times $\tau >0$ and $0<\delta <\tau$, and define

A label denotes the horizontal regular marking

The nominal controller tick $k\tau$, selected slice time $T(k,q)$, and physical times of local gate implementations are different declared quantities.

For $k<B$, set

At $k=B$, use an absorbing identity branch.

Every successor is an admitted label and advances the selected slice time by $\tau +a\delta >0$. Each marking is embedded and spacelike on the regular product. The same spatial cells and any fixed compact regular collar are used on every label. Original-name cell transport is the identity, permission decreases by one, and the complete peripheral affine word remains $J_{\mu }$.

Proof. The bounds and parity in (eq:labels) are preserved. A horizontal slice has induced metric equal to the spatial metric of (eq:cone). Formula (eq:slice-time) gives the positive increment. Horizontal time translation commutes with $J_{\mu }$; it creates no time or axial shift in the peripheral word. No spatial cell or conical parameter changes.

Only compact regular portions are used by the operations. The extension of a marking toward the completion is background metadata, not an interaction with $\mathcal{W}$ and not a claim that these slices are global Cauchy surfaces.

Let matter be one qubit. Set

and

For an active input $g$,