XXII-D: Operational approximation of an extracted Lorentz cone Stage 3 of a limited kinematic emergence programme

This note specifies an operational approximation layer that could be attached to a successful Lorentz-cone extraction. Extraction alone does not supply the required transport resources, quantum endpoint access, composition law, or calibration. For the unchanged carrier of XXII-C, every component has a finite future; consequently the entire null boundary cannot be approximated by comparable event differences from one fixed component. A separate example exhibits a faithful generated Lorentz representation of a classical signalling order without reference-preserving quantum transmission through its designated channel. These results identify substantive missing interfaces, rather than merely missing estimates. Conditional on a compatible transport realization, we prove complete-channel telescoping, an endpoint certificate, an arrival-window bound, and model-dependent locality and resource lower bounds. Boundary attainment, finite-stencil coverage, moving arrival events, and noise accumulation receive explicit failure witnesses. No successful extraction or transport realization on the frozen process family is constructed, and no metric calibration or invariant physical speed is derived.

XXII-A specifies a nonempty operational Lorentz-cone selection target and a faithful generated affine representation CheungXXIIA (target:selection, def:representation). XXII-B separates the candidate data, additional hypotheses, and failure tests CheungXXIIB. XXII-C fixes a tripartite qubit process-matrix family, its one-use instrument access, and its component-order extraction CheungXXIIC.

The present note does not assert either implication

or

Both would omit substantive hypotheses.

En08-B constructs finite transport protocols for a prescribed geometric boundary and speed CheungEn08B (thm:main, cor:operational-limit). Its result is an approximation theorem on its stated model. It is not a derivation of those prescribed inputs.

The narrower objective considered here is:

After a successful extraction and a separately justified compatible transport realization, construct finite protocols approximating a declared nonempty family of future-null targets, with reference-stable channel error, endpoint and arrival control, and a model-specific resource ledger.

This is a research objective, not an existence theorem of XXII-D.

is proved. The proofs below establish scoped obstructions and conditional certificates on explicitly stated mathematical interfaces. They do not construct the missing realization. The approximation layer is therefore not absent only because extraction remains open: even a successful extraction would leave the transport and calibration obligations to be discharged.

discipline. Cheung's manuscripts are identified by supplied filename, version where available, and internal label. Documentary comparisons are not proof-transfer premises. Published literature supplies the process-matrix, quantum-channel, and quantum-network frameworks OCB,Watrous,CDP. No predecessor proof, empirical implementation, or numerical artifact is represented as independently verified here.

For completeness, the relevant frozen objects are stated without using predecessor auxiliary files.

Let

Use the transposed unnormalized Choi convention

The set ${\mathsf{W}}_3$ consists of positive operators $W$ on $\mathcal{H}$ satisfying

Local instrument insertions give the complete probability law

This is the process-matrix language of Oreshkov, Costa, and Brukner OCB, with the exact access restrictions of XXII-C CheungXXIIC (spec:access).

The frozen family is

Each experiment is tested separately. Its identifier is not time, a refinement scale, or spatial position.

The access ${\mathsf{A}}_0$ permits one use of each party's process interface in a selected experiment. The receiver datum is its local classical instrument record. Other records are summed; postselection and undeclared interparty feed-forward are excluded. Local auxiliary implementations are allowed only insofar as they implement the declared local instrument on that one-use interface.

Let ${\Delta }_e(a,b)$ be the supremum of the total-variation distance between $b$'s unconditional laws when a deterministic insertion at $a$ is replaced, with all other insertions fixed. The direct relation on $\mathbb{N}\times \mathsf{J}$ is

There are no cross-experiment edges. Taking strongly connected components and then reachability gives

For every datum of ${\mathcal{P}}_{3q}$, the component order is a partial order. Each experiment contributes at most three components, and

Proof. Mutual reachability is an equivalence relation. Reachability between its equivalence classes is well defined: changing either representative adds paths within a component. It is reflexive and transitive, and mutual comparison of two classes means that they are the same class.

Every path remains within its experiment identifier. The corresponding directed graph has three vertices, so its component quotient has at most three elements. The future of a component is contained in that quotient.

This reproduces the relevant content of XXII-C's order theorem CheungXXIIC (thm:order). The components are not automatically individual source and receiver laboratories.

An arbitrary sequence of repeated interparty uses, an external unknown quantum message, and a retained quantum receiver port are not specified by ${\mathsf{A}}_0$. A local gate sequence implementing one admitted instrument can be compiled into that instrument; this does not create additional process uses or new event coordinates.

Thus “finite local protocols on the same interface” cannot mean an undeclared enlargement of ${\mathsf{A}}_0$. A transport theorem must state its compatible enlargement and prove the required complete-channel validity.

Nor does (eq:process-family) authorize unrestricted products or feedback compositions of its members. Such products can fail process normalization under unrestricted joint operations JiaSakharwade. A declared composition must have its own validity proof.

Fix $m\ge 3$. For a datum $D$, suppose a predeclared rule has supplied

with $\dim V_D=m$, ${\iota }_D:E_D\to V_D$ injective, $K_D$ proper, and $T_D:V_D\to {\mathbb{R}}^m$ invertible, such that

The construction must satisfy the non-circularity and admissibility audits of XXII-A and XXII-C. Neither its existence nor a nonempty admissible domain is asserted in this specification.

If a revised carrier is used, the carrier, order, and representation equations must be restated for that carrier. It is not an unannounced replacement of $E_D$.

The homogeneous strictly convex cone classification of Ito and Lourenço is one conditional route to (eq:lorentz) ItoLourenco (Theorem 7) CheungXXIIA (thm:classification). This note assumes (eq:lorentz) when discussing the approximation interface; it does not derive or apply that classification to process data.

In coordinates supplied by $T_D$, write

The full cone $K_D$ determines the Lorentzian quadratic form (eq:quadratic) up to a positive scalar. It does not select a preferred Lorentz coordinate frame, a normalization of that scalar, or physical length and time units. Consequently it does not determine a numerical speed in physical units.

Proof. In Lorentz coordinates, any other quadratic form vanishing on the same null boundary and positive in the future interior has the form

Vanishing on $(1,u)$ and $(1,-u)$ for every unit $u$ gives $b=0$ and

Thus $C=-aI$, and positivity on $(1,0)$ gives $a>0$. This is the uniqueness calculation recorded in XXII-A CheungXXIIA (rem:conformal).

For every $s>0$, the map $s{T}_D$ still maps $K_D$ onto ${\mathcal{L}}_m$, but replaces $q_D$ by $s^2{q}_D$. Lorentz boosts also preserve ${\mathcal{L}}_m$. For example, on one temporal and one spatial coordinate,

does not select the same observer when $\eta \ne 0$.

Finally, if coordinate units are converted by

then the null equation becomes

The constants ${\ell }_0,{\tau }_0$ are not fixed by cone membership.

convention of this note. All endpoint estimates below are in one declared coordinate frame $T_D$. The normalized null slope is one in those coordinates. A positive comparison unit $\ell$ is used to make endpoint errors dimensionless. Its numerical normalization is declared, not derived.

If clocks, rods, tick durations, or physical units are introduced, their calibration and correspondence to these coordinates belong to the transport realization. Classification does not fix them. A Lorentzian quadratic form on one affine carrier is also not a constructed metric field on a manifold.

Define the formal future-null set

A formal anchored target is

Its target endpoint is $a+v$, which need not lie in ${\iota }_D(E_D)$.

The represented null-pair set is instead

It may be empty. Even a member of (eq:null-pairs) is a represented comparison, not an identity-transmission protocol.

For a fixed source $p$, put

Under Specification spec:extraction, $F_p$ has at most two elements. If

then $v\in F_p$, and the differences are eventually equal to $v$.

In particular, for $m\ge 3$, the entire nonzero future-null boundary cannot be approximated from one fixed component by comparable differences in the unchanged carrier.

Proof. Equation (eq:finite-future) gives the cardinality bound. A finite subset of a finite-dimensional vector space is closed, so the limit lies in $F_p$. Distinct points of a finite set have a positive minimum separation. Convergence therefore implies eventual equality to the limiting point.

The boundary $\partial {\mathcal{L}}_m\setminus \{0\}$ has infinitely many vectors and infinitely many rays when $m\ge 3$. An invertible map preserves that fact, whereas $F_p$ is finite.

The theorem does not deny approximation on a revised operational carrier or with moving source labels whose correspondence is proved. It says that either change is substantive. It also does not obstruct a theorem restricted to a named subset of already represented null pairs.

-vacuous target declaration. A later approximation theorem must state its target set $\mathcal{T}$ and prove that it is nonempty. A claim uniform over ${{\mathcal{N}}_{\mathcal{D}}}^{\mathrm{pair}}$ could otherwise be vacuous.

For all-direction approximation, a typical coordinate target set has

Its endpoints require an operational realization or extension; (eq:generated) alone does not provide them.

Let $M\cong {\mathbb{C}}^d$, $d\ge 2$. If a receiver channel $\Lambda :L(M)\to L(M)$ is measure-and-prepare, then

Proof. Write

For a maximally entangled input

the receiver–reference output is separable. This is the measure-and-prepare characterization of entanglement-breaking channels HSR.

A pure product vector has squared overlap with $|{\Omega }_d⟩$ at most $1/d$, by Cauchy–Schwarz. Convexity gives the same bound for a separable state. Testing the projector $|{\Omega }_d⟩⟨{\Omega }_d|$ therefore gives trace distance at least $1-1/d$ from the ideal Bell output. This input is allowed in the diamond norm, proving (eq:eb-floor).

The Bell-witness argument is also used in Cheung's finite recovery and streaming obstructions CheungEn02C. Its use here does not identify those recovery models with the process family.

There is a datum in the frozen class whose component order has a faithful generated representation with cone ${\mathcal{L}}_3$, but whose designated $A$-to-$B$ connection is classical and cannot transmit an unknown qubit with half-diamond error below $1/2$ using local preprocessing and receiver decoding without an additional communication resource.

The representation in this example is an independently chosen mathematical witness, not an operational extraction satisfying XXII-A's non-circularity requirement.

Proof. For every experiment, use the definite-order circuit that initializes a qubit at $A$, connects $A$'s output to $B$'s input through

and discards $B$'s output. Party $C$ receives an independent initialized qubit and has its output discarded. Definite-order circuits give valid process matrices OCB.

Preparing $|0⟩$ or $|1⟩$ at $A$ and measuring at $B$ gives ${\Delta }_e(A,B)=1$. There is no reverse edge and no edge incident to $C$. The extracted order is therefore the disjoint union of

Let $u_e$ be a countable dense sequence in the unit circle of ${\mathbb{R}}^2$. Define, only for this mathematical example,

The comparable differences are exactly

The $A_e,B_e$ pair is future-null. Within one experiment, the spatial distance from $C_e$ to $B_e$ is at least three, exceeding the temporal difference one. Across experiments, the spatial separation of events at different time levels is at least nine, again exceeding their temporal difference. Distinct events at the same time level are also incomparable. The embedding is therefore injective and faithful for ${\mathcal{L}}_3$.

The closure of the cone generated by $(1,u_e)$ is ${\mathcal{L}}_3$: density supplies every unit-circle boundary generator, and the closed unit disk is the convex hull of that circle. These differences span ${\mathbb{R}}^3$. Thus all three representation equations hold.

Now introduce a qubit message at $A$ and permit a quantum receiver output at $B$, but add no new communication channel or shared entanglement resource. Every influence reaching the receiver through (eq:classical-channel) is a classical measurement outcome. Local source processing and receiver decoding leave the resulting message-to-receiver map in measure-and-prepare form. Proposition prop:eb-floor with $d=2$ gives the error floor $1/2$.

The spatial placements and dense directions above were chosen independently of the operational data. They prove a nonimplication from representation equations and cone shape to transport capacity. They do not define an admissible data-derived selector.

The theorem does not refute a future extraction package that includes a separately proved quantum transmission condition. It identifies why that condition cannot be replaced by classical signalling, component reachability, or Lorentz geometry alone.

For a datum with the conditional output of Specification spec:extraction, a transport theorem must supply:

1. Quantum ports and resources. A named message space $M$, with $d=\dim M\ge 2$, source injection, receiver quantum output, admissible local memories, and initial resources independent of the unknown message. The qubit case is the minimal extension of the frozen wire dimensions. Larger messages require their own capacity and implementation statement.

2. Validity and composition. The admissible uses of the $W_e$, their auxiliary extensions, and any classical or quantum wiring between uses. Prove positivity and normalization of the resulting complete maps, including untouched references. Reproducing the frozen classical tests after the declared reductions must be proved, not assumed from similar notation.

3. Carrier correspondence. Source and receiver ports associated with components, or a declared extended event carrier ${\hat{E}}_D$, together with its comparison map. A component containing several laboratories does not choose one of them as a source by itself.

4. Geometric compatibility. Coordinates of the realized ports in the extracted affine carrier, and the relation between realized dependency or influence comparisons and the extracted order. If an extension is used, prove the claimed preservation of old comparisons and the claimed compatibility of its generated cone. Additional ports do not automatically preserve (eq:generated).

5. Execution and calibration. An actual finite execution model for the protocols, with elementary communications, local gates, guards, records, tick conventions, and arrival readings. Its calibration to the chosen extracted coordinates is stated separately from cone selection.

6. Complete comparison interfaces. The declared retained residual systems, classical histories, failure flags, encoders, decoders, and permitted terminal reductions. The receiver decoder may not access the source's remaining message-bearing systems through an undeclared side channel.

7. Evidence and invariance. Exact mathematical premises or actual implementation allowances, their quantifiers, and any statistical coverage. State covariance under permitted operational relabellings and coordinate changes.

None of these clauses is certified for a cone-selecting datum here.

A finite definite-order channel network is a possible realization language, but not a consequence of a process matrix's signalling graph. Combs and quantum networks supply frameworks for such declared sequential interfaces CDP. The scheduler of a realizing protocol is an execution input; it is not retrospectively claimed as the order derived from the raw data.

If all these structures are additionally assumed, the resulting approximation theorem remains conditional on them. To claim a realization from the frozen data, they must instead be constructed from those data under named additional premises.

All channels in this section are completely positive and trace preserving on their stated finite-dimensional interfaces. Use

The diamond norm compares channels completely, including an untouched reference Watrous (Chapter 3). It is not assigned to an arbitrary process-matrix probability functional.

For every channel $\Lambda :L(M)\to L(M^{\prime })$,

This follows from trace preservation. It does not establish preservation of the message–reference correlations.

After a declared identification $M^{\prime }\cong M$, identity transport is the stronger channel equality

A replacement channel satisfies (eq:reference-marginal) and generally destroys those correlations. This is the distinction of XXII-B's Candidate D CheungXXIIB and the reference-stable transport statements of En08-B CheungEn08B (lem:identity).

For a declared residual and record system $F_n$, a complete ideal transport target is

where ${\sigma }_n$ is input independent. It may include a declared arrival-record law. The logical target is ${\mathrm{id}}_M$, obtained by tracing out $F_n$.

If

then the logical reduction ${\Lambda }_n={\mathrm{Tr}}_{F_n}{\Gamma }_n$ satisfies

For every state ${\rho }_{MR}$,

Proof. Partial trace is a channel, so postcomposition contracts the diamond norm. Tensoring with the reference and evaluating on a state of trace norm one gives (eq:reference-error).

A logical-output estimate alone is not the same numerical estimate for the entire retained record–residual channel. A full comparison must be supplied if that stronger conclusion is claimed.

Let

be compatible channel compositions on matching step interfaces. Suppose

Then

Proof. Insert intermediate compositions replacing one factor at a time. Each difference contains one changed factor, preceded and followed by channels. Since channels have diamond norm one, each term is bounded by the diamond norm of that factor difference. The triangle inequality and division by two give the sum. The half-diamond distance between two channels is at most one.

The same argument is printed in En08-B CheungEn08B (lem:telescoping). Here it applies only after the realization contract supplies actual compatible channels. Memory, classical control histories, and retained residuals must be present in the step interfaces; an unmodelled correlated memory is not an independent fresh error term.

If also

then

If $k_n\le H_n$ and $e_{n,j}\le {ϵ}_n$, the sum is at most $H_n{ϵ}_n$.

Proof. Apply Lemma lem:telescoping, then the triangle inequality with ${\Theta }_n$, and cap the result at one.

The allowance ${\mu }_n$ is a design comparison, not a geometric endpoint error. Geometric and channel errors are kept separate.

Let $Z_n$ be a retained finite classical arrival register with a declared calibration $z\mapsto a_n(z)\in \mathbb{R}$. A failure code is included in its alphabet. For a target arrival coordinate $\tau$, let the good codes satisfy

and exclude failure codes.

If the ideal record marginal of ${\Theta }_n$ is supported entirely on the good codes and $d_{\diamond }({\Gamma }_n,{\Theta }_n)\le e_n$, then, for every input state, including a referenced input,

Proof. The bad-code projector is an effect on the complete output. Its ideal probability is zero. Its actual probability differs from that value by at most the output trace distance, bounded by the half-diamond error.

This is an unconditional complete-attempt bound. It is not a normalized postselected channel guarantee on accepted trials. The instrument and availability of the decoded output at the claimed arrival must themselves be supplied by the realization.

Choose a nonempty declared target set $\mathcal{T}$ of anchored future-null targets. In the selected coordinates write

For each $\zeta =(a,v)\in \mathcal{T}$ and refinement $n$, a realized protocol has declared source coordinates $a_{n,\zeta }$ and receiver coordinates $b_{n,\zeta }$ in the same coordinate space. Write

The coordinate assignments are part of R3–R5; they are not inferred from experiment numbering.

Set

This is a comparison norm in a chosen frame, not a newly derived physical spatial metric.

For a decoded logical channel ${\Lambda }_{n,\zeta }$, define

The source term matters: displacement accuracy alone could compare a protocol beginning at the wrong operational source. The identity target requires a declared logical identification; the cone does not assign a quantum transport channel to a vector.

Suppose a compatible transport realization supplies, for every $\zeta \in \mathcal{T}$ and $n$:

1. a finite protocol and complete decoded channel ${\Gamma }_{n,\zeta }:L(M)\to L(M\otimes F_n)$;

2. an ideal complete target ${\Theta }_{n,\zeta }(X)=X\otimes {\sigma }_{n,\zeta }$;

3. matching ideal and implemented channel factorizations with at most $H_n$ steps, uniform design allowance ${\mu }_n$, encoder allowance ${\alpha }_n$, decoder allowance ${\beta }_n$, and complete step allowance ${ϵ}_n$;

4. uniform endpoint estimates

   $$\Vert a_{n,\zeta }-T_{D}a{\Vert }_{\ast }+\Vert x_{n,\zeta }-x{\Vert }_2\le {\delta }_n,|{\tau }_{n,\zeta }-\tau |\le {\omega }_n.$$

Then, for the logical reduction ${\Lambda }_{n,\zeta }={\mathrm{Tr}}_{F_n}{\Gamma }_{n,\zeta }$,

In particular, uniform anchored endpoint approximation holds if

The same channel allowance bounds all message–reference output errors. If the ideal arrival records satisfy the window condition, it also bounds the failure and out-of-window probability.

Proof. Corollary cor:assembly gives the uniform complete-channel allowance. Lemma lem:full-logical transfers it to logical channels and referenced inputs. Substituting the endpoint estimates into (eq:endpoint-error) gives (eq:certificate). Equation (eq:certificate-convergence) makes its right side tend to zero. Proposition prop:arrival gives the last statement.

This is an implication from constructed or explicitly assumed protocol data. It does not assert that those data exist. It also proves no global channel evolution at intermediate times.

A cone contains no elementary gate size, no tick duration, and no storage cost. An unconditional lower bound on gate count or memory as a numerical function of distance therefore cannot be inferred from cone selection alone.

For example, a positive rescaling of the affine embedding rescales all coordinate lengths while leaving its order and conical generation unchanged. It does not change a given gate count. Any numerical distance–resource comparison must fix its scale and allowed elementary operations.

The following lower bounds concern finite acyclic networks of complete channels, not arbitrary process matrices. Assume:

1. an external message $M\cong {\mathbb{C}}^d$ is introduced at a source; other initial resources are fixed independently of that message;

2. all quantum wires, classical controls, routing choices, and retained records occur in the dependency graph;

3. source and receiver interfaces have specified spatial coordinates in the chosen extracted frame;

4. every spatially nonzero dependency hop advances by at most $a_n>0$;

5. execution has $N_n$ rounds and every source–receiver dependency path contains at most $N_n$ such hops;

6. no undeclared communication or decoder access bypasses this graph.

The step bound and round convention are additional realization premises. They are not supplied by (eq:lorentz).

Under Specification spec:bounded-step, if there is no dependency path from the message source to the receiver interface, its logical channel is a replacement channel. Its error from identity satisfies

Proof. Pull a receiver observable backwards through the acyclic network. A channel whose outputs carry only the identity observable contributes the input identity, because its adjoint is unital. Including classical and quantum predecessors gives support only on the receiver's scheduled ancestors. If the message input is absent from those ancestors, the receiver law is independent of every message state. This is the scheduled-support argument of VII-J and En06-A CheungVIIJ CheungEn06A (thm:order).

Linearity then gives

for one state $\sigma$. On a maximally entangled message–reference input, the output is $\sigma \otimes I_R/d$, whose Bell overlap is $1/d^2$. The Bell-projector test gives (eq:constant-floor).

Under Specification spec:bounded-step, suppose

Let $r=\Vert x{\Vert }_2$ be the target spatial displacement and suppose $\Vert x_{n,\zeta }-x{\Vert }_2\le {\delta }_n$. Then

If each round has calibrated duration $h_n>0$, measured from the declared source-input cut, then that execution duration is at least the right side of (eq:depth-lower) multiplied by $h_n$.

Proof. Lemma lem:no-path forces a dependency path. If it contains $k$ spatially nonzero hops, the triangle inequality gives

The reverse triangle inequality gives $\Vert x_{n,\zeta }{\Vert }_2\ge (r-{\delta }_n{)}_{+}$. This proves the integer bound. The duration statement is the declared round-duration convention.

If $a_n=\ell /n$, the bound is linear in $n$ for a fixed nonzero target once ${\delta }_n\to 0$. At a fixed scale it is linear in distance. The choice $a_n=\ell /n$, however, is an additional realization prescription.

The ratio $a_n/h_n$ is a model-dependent propagation ceiling. No invariant physical speed is derived by naming that ratio.

Suppose a realizing network uses $q_n$ fixed spatial sites. All quantum and classical communication edges between distinct sites have length at most $a_n$. If a source–receiver dependency path exists and their displacement is $x_{n,\zeta }$, then

If, additionally, the resource model keeps a $d$-dimensional register at every used site live simultaneously, its live quantum dimension is at least $d^{q_n}$.

Proof. The communication path lies in a connected finite graph on the used sites. Erasing loops gives a simple spatial graph path with at most $q_n-1$ edges. The triangle inequality gives $\Vert x_{n,\zeta }{\Vert }_2\le (q_n-1)a_n$, proving (eq:sites). The simultaneous-register clause gives the product dimension.

Without the simultaneous-register premise, a site-count bound is not a peak-memory bound. Dynamic allocation, retirement, buffering, and reuse require their actual execution ledger. Cheung's one-shot and streaming ledgers illustrate precisely this distinction CheungEn02A,CheungEn02B. No memory optimum for those models is imported here.

A finite protocol records at least

where:

• $U$ is the number and type of process-resource uses;

• $G$ counts gates under a stated elementary gate convention;

• $N$ is execution depth or calibrated rounds;

• $Q$ is peak live quantum memory;

• $q$ is the distinct-site or port footprint;

• $C$ is classical communication, distinguished from storage;

• $A$ is total auxiliary allocation, distinguished from peak use.

The error allowances and endpoint estimates are recorded alongside this vector, with their dependence on $n,\zeta$.

No $O(n)$ gate, footprint, or memory upper bound is established for the frozen process family in this note. En08-B's resource scaling belongs to its constructed static-site transport model CheungEn08B (thm:main).

A later operational approximation theorem must discharge:

1. Finite protocols and quantum transmission. Construct protocols for one declared nonempty target set and supply the compatible access enlargement of Specification spec:realization. Prove transmission of the unknown message with an untouched reference. Do not substitute preservation of the reference marginal for identity transport. Theorem thm:geometry-not-transport is a standing transmission-capability test.

2. Anchored endpoint and complete-channel control. Provide actual source and receiver maps, complete comparison channels, residual and record conventions, and estimates of the form (eq:certificate). State whether the claim is logical-output comparison or a full retained-interface comparison. Use an event-aware topology when arrival events move.

3. Resource upper ledger. Give explicit resource functions under Definition def:ledger, including every operative communication and auxiliary resource. Prove the error scaling needed for (eq:certificate-convergence). No predecessor asymptotic count is transferred without its realizing-model correspondence.

4. Model-dependent locality lower bounds. State the elementary step bounds, execution rules, and cost model under which lower bounds are claimed. Theorems thm:depth and Proposition prop:site-lower apply only under their stated premises. A spatial ceiling does not prove availability of every null direction, nor does a depth bound automatically bound memory.

5. Arrival and calibration. Construct the declared output availability, arrival window or distribution, and calibrated record interface. Account for failed and late attempts without normalized postselection. Frame, scale, units, and clock correspondence must retain the status described in Proposition prop:calibration.

6. Carrier compatibility and failure tests. Retain Theorem thm:anchored-obstruction and the tests of Section sec:tests. If the operational carrier or allowed compositions are enlarged, prove their compatibility with the extracted representation. State separately any exact null-attainment or continuum evolution claim.

These obligations supplement extraction. They do not repair an unsatisfied extraction target or a circular geometric input.

Let

Every nonzero finite conical combination is strictly inside ${\mathcal{L}}_m$: the strict triangle estimate gives

when at least one ${\lambda }_j>0$. Nevertheless,

by density of rational interior points in the closed unit ball.

This is a generator example, not a faithful quantum event representation. It shows why a closure-produced null boundary need not contain an attained generator or protocol. XXII-B gives the same scope test CheungXXIIB (Failure test F3).

Even convergence of implemented endpoints to a null target does not assert equality with that target at any finite refinement. Exact logical transmission and exact null-coordinate attainment are also different predicates.

In a chosen Lorentz coordinate frame, suppose every moving step has nonnegative duration and spatial velocity in a fixed finite set

For $m\ge 3$, choose a unit vector $u\notin S$ and set

If a protocol's endpoint has

and total step duration ${\tau }_n$, then

If $h_S(u)=0$, sufficiently accurate endpoints are impossible. If $h_S(u)>0$ and ${\delta }_n\to 0$, then

Thus this stencil cannot approximate the null target $(r,ru)$ with vanishing arrival delay.

Proof. For a unit $u$, equality $u\cdot s=1$ with $\Vert s{\Vert }_2\le 1$ requires $s=u$. Hence $h_S(u)<1$. Because $0\in S$, it is nonnegative.

Writing step durations as $\Delta t_j\ge 0$ gives

The endpoint estimate gives $u\cdot x_n\ge r-{\delta }_n$. The conclusions follow.

The same reasoning permits velocities in $\mathrm{conv}S$. An increasing stencil may avoid this particular obstruction, but requires an actual routing and resource theorem. Isotropy is not proved by the upper bound $\Vert s{\Vert }_2\le 1$ alone.

The outline's En08-B labels prop:exact-null and prop:fixed-stencil are treated as documentary identifiers. The tests in this section are established by their printed arguments, not by importing unsupplied predecessor proofs.

Let ${\tau }_n>\tau$ and ${\tau }_n\to \tau$. On a common message-plus-flag interface define

Then endpoint arrival times converge and the post-arrival transport channels are identical, but

Proof. For $\tau \le t<{\tau }_n$, the flags of the two outputs are orthogonal. Their trace distance is one for every state input. The half-diamond distance is therefore one. The channel distance cannot exceed one.

Uniform estimates away from the arrival event, or a declared event-aware endpoint topology, can be appropriate. Arrival-time convergence alone is not uniform channel convergence through a discontinuity.

On a qubit let

For an unencoded sequence of $k$ such steps,

Consequently, if $k_n\to \infty$, $e_n\to 0$, and $k_n{e}_n\to a>0$, the endpoint error tends to $1-e^{-a}>0$.

Proof. Since ${{\mathcal{R}}_0}^2={\mathcal{R}}_0$,

The half-diamond distance between ${\mathcal{R}}_0$ and identity is one: it is at most one for channels, and the input $|1⟩⟨1|$ gives orthogonal outputs. Homogeneity of the norm gives (eq:noise).

Finally,

This is a lower-bound witness for the displayed noise model without error correction. It does not prove that every nonzero per-step allowance prevents convergence.

Conversely, if a telescoping upper bound fails to tend to zero, that fact alone proves only failure of that certificate. Error cancellation or a separately constructed correction scheme requires its own analysis.

For comparison channels independent of an intervention, uniform implemented allowances $e$ imply at most a $2e$ intervention difference by the triangle inequality. They do not imply exact absence of influence CheungXXIIB (prop:tolerance).

A protocol can therefore approximate identity on a target while having small additional signals outside an asserted exact support region. Exact support and tolerance support require different guards and different statements.

The separation

holds for the complete definite-order networks used in Section sec:locality. Containment in an independently supplied geometric relation $R_{\sup }$ requires a separate embedding certificate CheungVIIJ. No endpoint norm allowance supplies that certificate.

A family of terminal channels and endpoint comparisons need not define compatible channels at all intermediate times. Any evolution-limit claim must provide its own encoders, decoders, complete interfaces, and statement such as

together with its composition and continuity hypotheses.

XIX-C's two-refinement and interpolation witnesses show the missing information in an inference from one coarse channel or finite samples CheungXIXC (prop:two-limits, prop:interpolation, def:limit). They do not exclude uniqueness after appropriate additional data. Naming a consistency condition is not its construction or a uniqueness proof.

{}1.13

In the XXII-C row, the bibliographic key denotes the supplied XXII-C.tex; no external LaTeX reference file is used.

The completed release consists of:

1. the finite-future and anchored-boundary obstruction;

2. the separation between cone geometry and quantum transmission;

3. the corrected calibration statement;

4. the compatible transport-realization contract;

5. complete-channel, endpoint, arrival, and bounded-step conditional certificates;

6. the six approximation obligations and explicit failure tests.

The abstract conditional certificates are proved. They are not instantiated by a successful cone-selecting datum and a compatible protocol family.

Supply:

1. one nonempty admissible data domain and a predeclared successful extraction meeting the XXII-A and XXII-C target;

2. one separately justified transport realization ${\mathsf{B}}_D$, including every enlarged access or composition interface;

3. one declared nonempty target family and its source/receiver correspondence;

4. finite protocols satisfying A1–A6, with the applicable convergence, resource, locality, and arrival statements;

5. the required relabelling invariance and evidence allowances.

If the target covers the whole null boundary from a fixed source, the unchanged carrier obstruction must be repaired by an explicit, compatible carrier or realization change. This research target has no existence proof here.

Both non-vacuity gates matter: a nonempty extraction domain is not a nonempty domain of compatible transport realizations. Conversely, constructing a transport network for a prescribed Lorentz cone does not meet the operational extraction target.

boundaries. No successful operational extraction, invariant physical speed, metric field, exact null transport event, or geometric causal embedding is established. No Einstein dynamics, holographic reconstruction, Page curve, gravity-generated entanglement, or quantum evolution without external time is proved.

Nothing here changes O.1, opens XV-B, discharges XVI-N or Route B, or alters H2, I3, or J2. No quantum coefficient is admitted. No successful pdfLaTeX compilation, programme execution, laboratory implementation, or proof-assistant verification is asserted CheungXIXE.

gap statement. A successful extraction must first be constructed. A compatible quantum transport realization, calibration, and nonempty target family must then be supplied independently of the mere cone classification. Only after those constructions can Theorem thm:certificate certify operational endpoint approximation.

On the unchanged XXII-C carrier, universal fixed-source null-boundary approximation is obstructed by finite futures. On a compatible revised realization, its existence remains an open construction problem. En08-B remains a theorem about its prescribed boundary, not about a boundary derived by the present programme.

CheungXXIIA

N. Cheung. XXII-A: Freeze the precise claim and the gap. Stage 0 of a limited kinematic emergence programme. Supplied manuscript XXII-A.tex, n.d. Relevant labels: target:selection, def:representation, thm:classification, and rem:conformal; conditions (T1)–(T5), (A1)–(A6).

CheungXXIIB

N. Cheung. XXII-B: Catalogue the additional quantum data and the hypothesis ledger. Stage 1 of a limited kinematic emergence programme. Supplied manuscript XXII-B.tex, n.d. Relevant items: Candidate D, ledger syntax, failure tests F3–F5, prop:tolerance, and prop:two-limits.

CheungXXIIC

N. Cheung. XXII-C: Freeze one quantum process class and the extraction obligations. Stage 2 of a limited kinematic emergence programme. Supplied manuscript XXII-C.tex, n.d. Relevant labels: spec:access, def:class, thm:order, def:feasible, def:extraction, and target:selection.

CheungXXIIIC

N. Cheung. XXII-C, supplied manuscript XXII-C.tex, n.d. Same manuscript as the preceding reference; component-ledger identification only.

OCB

O. Oreshkov, F. Costa, and Č. Brukner. Quantum correlations with no causal order. Nature Communications 3 (2012), 1092. doi:\ 10.1038/ncomms2076. Preprint: https://arxiv.org/abs/1105.4464.

Watrous

J. Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. Chapters 2, 3, and 6. Author's text: https://cs.uwaterloo.ca/ watrous/TQI/.

CDP

G. Chiribella, G. M. D'Ariano, and P. Perinotti. Theoretical framework for quantum networks. Physical Review A 80 (2009), 022339. doi:\ 10.1103/PhysRevA.80.022339. Preprint: https://arxiv.org/abs/0904.4483.

JiaSakharwade

D. Jia and N. Sakharwade. Tensor products of process matrices with indefinite causal structure. Physical Review A 97 (2018), 032110. doi:\ 10.1103/PhysRevA.97.032110. Preprint, under the title Process Products Need Qualification: https://arxiv.org/abs/1706.05532.

HSR

M. Horodecki, P. W. Shor, and M. B. Ruskai. Entanglement breaking channels. Reviews in Mathematical Physics 15 (2003), 629–641. doi:\ 10.1142/S0129055X03001709. Preprint: https://arxiv.org/abs/quant-ph/0302031.

ItoLourenco

M. Ito and B. F. Lourenço. The $p$-cones in dimension $n\ge 3$ are not homogeneous when $p\ne 2$. Linear Algebra and its Applications 533 (2017), 326–335. doi:\ 10.1016/j.laa.2017.07.029. Preprint: https://arxiv.org/abs/1701.05801. Theorem 7 is a classification reference, not an operational transport premise.

CheungEn08A

N. Cheung. En08-A, version 1, manuscript En08-A.tex. Enter Calculus Companion, n.d. Relevant supplied labels: thm:main, thm:mainpositive, thm:representation, thm:weaker, and the boundary, tolerance, and finite-sample warnings. Registry: https://enter.grok.me/.

CheungEn08B

N. Cheung. En08-B, version 1, manuscript En08-B.tex. Enter Calculus Companion, n.d. Relevant supplied labels: lem:identity, lem:telescoping, thm:main, def:comparison, and cor:operational-limit. The outline additionally names prop:exact-null and prop:fixed-stencil; no unsupplied proof of those propositions is used here.

CheungVIIJ

N. Cheung. VII-J, version 1, manuscript VII-J.tex. Enter Calculus Companion, n.d. Relevant supplied labels: thm:schedule, thm:nosignal, thm:physical, and thm:release. Scheduled ancestry, operational influence, and supplied geometric causality have separate interfaces.

CheungEn06A

N. Cheung. En06-A, version 1, manuscript En06-A.tex. Enter Calculus Companion, n.d. Relevant supplied labels: thm:order, lem:onelayer, cor:influence, def:experiment, and def:limit. Finite complete-network support and the separate coarse outcome-law completion.

CheungXIXC

N. Cheung. XIX-C, version 1, manuscript XIX-C.tex. Enter Calculus Companion, n.d. Relevant supplied labels: def:limit, prop:two-limits, prop:interpolation, and thm:main. Refinement non-determination and the complete operational-limit interface.

CheungEn05B

N. Cheung. En05-B, version 1, manuscript En05-B.tex. Enter Calculus Companion, n.d. Relevant supplied label: thm:main. Isotropic matched operational existence for a prescribed constant $c>0$; documentary comparison only.

CheungEn06B

N. Cheung. En06-B, manuscript En06-B.tex. Enter Calculus Companion, n.d. Relevant supplied scope record: thm:T. Finite-network transport on an increasing-stencil family with prescribed speed; documentary comparison only.

CheungEn02A

N. Cheung. En02-A, version 1, manuscript En02-A-v1.tex. Enter Calculus Companion, n.d. Relevant supplied label: thm:Dprime. One-shot instrument environment ledger and its separation from adaptive peak memory.

CheungEn02B

N. Cheung. En02-B, version 1, manuscript En02-B-v1.tex. Enter Calculus Companion, n.d. Relevant supplied labels: thm:batch, thm:B5, and prop:observable. Batch, streaming, observable, and subsystem-recovery resource tasks have different scopes.

CheungEn02C

N. Cheung. En02-C, version 1, manuscript En02-C-v1.tex. Enter Calculus Companion, n.d. Relevant supplied labels: thm:C2, thm:C3, lem:firstmeasurement, and cor:robustgap. Bell witnesses and a model-specific streaming recovery obstruction; not a transport realization for the present class.

CheungOai265A

N. Cheung. Oai265-A, manuscript Oai265-A.tex. Enter Calculus Companion, n.d. Relevant supplied label: thm:separation. Static area-law records, recovery, and radiation histories are distinct tasks; documentary negative control.

CheungOai265B

N. Cheung. Oai265-B, version 2, manuscript Oai265-B-v2.tex. Enter Calculus Companion, n.d. Relevant supplied labels: ass:statistics, thm:main, eq:coverage, and eq:falseaccept. Complete-record finite-confidence channel certificate; not evidence for the realization contract in this note.

CheungXIXE

N. Cheung. XIX-E, version 1, manuscript XIX-E.tex. Enter Calculus Companion, n.d. Relevant supplied passage: “What remains outside the release.” Documentary programme boundary record.