XXII-F: A conditional operational extraction of Lorentzian causal structure from named quantum data Stage 5 of a limited kinematic emergence programme

This note assembles a conditional Lorentz-cone representation theorem and a separate operational approximation corollary on a named tripartite qubit process-matrix interface. The theorem assumes a predeclared nonempty admissible domain, a single-valued faithful generated affine representation rule, its matched covariance, and homogeneity and cone strict convexity. Published classification then gives a Lorentz cone and a conformal quadratic form. The implication is proved; no operational package satisfying all its hypotheses is constructed. A further symmetry obstruction shows that infinitely repeated indistinguishable experiment blocks cannot admit an injective finite-dimensional embedding covariant under every matched permutation. Conditional on supplied compatible transport protocols, complete-channel telescoping yields endpoint, reference, residual, and arrival-window bounds. Execution resources, classical extraction costs, estimation shots, and retained outputs are separated. An optional finite full-output validation certificate is proved under explicit sampling and calibration assumptions. Finite generation, finite futures, nonzero tolerances, and dynamics limitations remain visible. The release is a self-contained conditional assembly and audit, not an existence theorem for operational spacetime emergence, a metric-field construction, or Einstein dynamics.

XXII-A fixes the faithful-generation target and its non-circularity audit CheungXXIIA. XXII-B distinguishes operational languages and hypothesis statuses CheungXXIIB. XXII-C fixes the process family and the component order CheungXXIIC. XXII-D supplies the separate transport contract and scoped approximation certificates CheungXXIID. XXII-E identifies the additional structures required for dynamics comparison CheungXXIIE.

The assertion under investigation is not

The prospective existence claim concerns a nonempty operational package producing an affine cone representation, followed by a separately justified approximation layer.

Two statements must therefore be separated:

1. Conditional implication. If an admissible domain, representation rule, and the stated selecting properties are supplied, the represented cone is Lorentzian. If compatible finite protocols and estimates are additionally supplied, their operational approximation obeys the displayed certificate.

2. Existence and construction. There actually is a predeclared nonempty operational package on the frozen class satisfying all those hypotheses, with a successful representation rule and compatible protocols.

Theorem thm:assembly and Corollary cor:approximation prove the first statement. Research target target:existence records the second. It remains open in this note.

In particular, the statement “the theorem is unproved because its hypotheses have not been constructed” is inappropriate for the conditional implication. The implication has a proof below. What is unproved is its non-vacuous operational instantiation.

boundary. The Cheung references identify supplied manuscripts or supplied scope passages, not independently verified publications. No documentary relationship is treated as a proof-transfer theorem. The process and channel conventions are supported by OCB,Watrous; the cone classification is attributed directly to Ito and Lourenço ItoLourenco. Local proofs are included for the specific auxiliary statements used here.

No claim about the full series is inferred from an incomplete retrieval inventory. In particular, the identifier En10-A@1 has no mathematical passage supplied for use here. It is not a theorem premise, and no bibliographic metadata or result is invented for it.

Fix

Thus $\dim \mathcal{H}=64$. The tensor-factor order is bookkeeping, not a temporal order.

For a local completely positive map $\mathcal{M}$, use

with the full transpose in fixed bases. Define

These represent deterministic channels and trace-nonincreasing instrument elements, respectively.

Let ${\mathsf{W}}_3$ consist of matrices $W$ satisfying

A finite local instrument has positive elements ${M^a}_{x_a|s_a}$ whose sum belongs to ${\mathsf{C}}_a$. Its complete joint law is

The family class is

Its members $\mathbf{W}=(W_e{)}_{e\in \mathbb{N}}$ are separately tested experiment families.

This is the specific branch fixed in XXII-C CheungXXIIC (def:process, def:class), using the process-matrix probability framework of OCB. Positivity gives nonnegative probabilities; multilinearity and (eq:validity) give normalization.

The access ${\mathsf{A}}_0$ admits every finite-outcome local qubit instrument, with one use of each process port in a selected experiment. Settings are independent of the process resource. The receiver output used for extraction is its local classical record. Sender and other-party outcomes are summed. No postselection, extra shared ancillary resource, or undeclared interparty feed-forward enters the signalling tests.

Local auxiliaries are permitted only insofar as they implement the declared local maps. There is no spatial position, distance, physical speed, or geometric causal relation in this declaration CheungXXIIC (spec:access).

An identifier $e$ is not time or a refinement scale. There is no joint process on all identifiers and no admitted inter-experiment composition. Unrestricted tensor products of valid processes need not be valid processes; a composition requires its own interface and normalization proof JiaSakharwade.

The set ${\mathsf{W}}_3$ is nonempty and compact, and every member has trace eight. In particular,

Proof. The completely depolarizing local channel has Choi matrix $I/2$. Substitution into (eq:validity) gives $1=\mathrm{Tr}W/8$. Positivity and the normalization equations define a closed subset of the bounded finite-dimensional set of positive matrices of trace eight.

For $W_0$, every deterministic local tester has trace two, so

This proves the base class is inhabited, not that a future cone-selecting subclass is inhabited.

For distinct $a,b\in \mathsf{J}$, let $c$ be the remaining party. The sender uses deterministic testers $S,T\in {\mathsf{C}}_a$. The receiver uses an instrument $\{B_z\}$, and $c$ uses $M_c\in {\mathsf{C}}_c$. Define

where $d_{\mathrm{TV}}(P,Q)=\frac{1}{2}{\sum }_z|P(z)-Q(z)|$. Tensor factors always occupy their named slots.

The strength is the attained maximum

and belongs to $[0,1]$.

Proof. Total variation of finite laws equals the maximum probability difference over outcome subsets. Coarse-graining such a subset gives one element $B\in {\mathsf{B}}_b$.

Conversely, every such $B$ extends to a binary instrument by adding

This is positive and makes the sum deterministic. The binary total variation equals the displayed absolute probability difference.

All tester sets are closed and consist of positive matrices of trace at most two. They are compact. The continuous objective therefore attains its maximum, and the binary probability difference is at most one.

The exact data assignment is

Probabilities and strengths are derived entries, not independent adjustable data. This is exact mathematical data, not a finite statistical record. Equations (eq:strength)–(eq:binary) give a mathematical optimization problem, not a reported numerical execution or a claimed single semidefinite program CheungXXIIC (prop:binary).

On $\tilde{E}=\mathbb{N}\times \mathsf{J}$, set

There are no cross-identifier edges. Their absence describes the separately tested interface, not a measurement of an unspecified global causal relation.

Let $p{⇝}_{D}q$ mean equality or finite directed reachability. Set

The relation ${⪯}_D$ is a well-defined partial order. Its carrier is countably infinite. Each experiment contributes between one and three components, and

Every chain has at most three distinct components.

Proof. Reflexive reachability is transitive, and mutual reachability is an equivalence relation. Changing representatives only adds paths within the relevant components, so the quotient order is well defined. Reflexivity and transitivity descend. Comparison in both directions puts two representatives in the same component, giving antisymmetry.

A path stays within one identifier. Each three-vertex graph has between one and three components, so there are countably many nonempty finite blocks. Every future and chain lies in one block.

This reproduces the relevant order result of XXII-C CheungXXIIC (thm:order). It is an order of components, not an injective order of raw cyclic events. Direct influence is not thereby proved transitive.

For example, a convex mixture of the circuits $A\to B$ and $B\to A$, with independent $C$, admits the law

Both unconditional influences are nonzero: $P(v=1|x,y)=\lambda x$ and $P(u=1|x,y)=(1-\lambda )y$. The quotient merges $A,B$. This is not a closed causal loop in either definite-order branch CheungXXIIB (prop:signalling-cycle).

A matched operational equivalence consists of a bijection of experiment identifiers, a party relabelling in each experiment, and bijections of admitted instruments and outcomes preserving deterministic insertions and coarse-graining, with equality of matched complete probability tables.

It induces a raw-label bijection $f$. Such a self-equivalence is allowed when it preserves a datum, not merely when it maps between two distinct data presentations.

Every matched equivalence preserves strengths, directed edges, reachability, and components. It induces an order isomorphism $\bar{f}:E_D\to E_{D^{\prime }}$.

Proof. Matched laws have the same total-variation distances. The bijection of tests gives identical suprema. Bijections preserve finite paths in both directions, hence mutual reachability and the quotient order.

The permitted equivalences and domain invariance must be fixed before an extraction is applied. Restricting them after encountering a symmetry obstruction changes the claim.

Scheduled ancestry $R_{\mathrm{sch}}$, unconditional influence $R_{\inf }$, and a later independently supplied geometric relation $R_{\sup }$ remain distinct. In a complete definite-order network, $R_{\inf }\subseteq R_{\mathrm{sch}}$; containment in $R_{\sup }$ requires a separate embedding certificate CheungVIIJ,CheungEn06A,CheungXXIIA. The frozen extraction here represents the component order of ${R_{\inf }}_D$, not all three relations at once.

A proper cone is closed, convex, pointed, and has nonempty interior in its finite-dimensional ambient space. The conical hull consists of finite nonnegative linear combinations, including zero.

For an ordered carrier $(E,⪯)$, a representation $(V,\iota ,K)$ has $\iota :E\to V$ injective, $K$ proper, and

Closure is ordinary finite-dimensional closure.

These are the equations of XXII-A CheungXXIIA (def:representation). Generation excludes an independently appended supercone; faithfulness excludes extra comparisons introduced by convexification or closure.

For a proper cone,

Homogeneity means that this group acts transitively on $\mathrm{int}K$. Cone strict convexity means

These are properties of the constructed event-difference cone, not identities with operator positivity or scheduler interleavings.

Fix $m\ge 3$, a finite actual predicate list $\mathsf{H}=(H_1,\dots ,H_r)$, and a predeclared single-valued rule ${\mathfrak{F}}_m$. Let

The conditional theorem requires the following certificates:

1. Non-vacuity. A specified witness $D_{\star }\in \mathcal{A}$, or a valid existence proof, is supplied.

2. Representation rule. For every $D\in \mathcal{A}$, the rule supplies $(V_D,{\iota }_D,K_D)$, with $\dim V_D=m$, satisfying Definition def:representation for $(E_D,{⪯}_D)$. A rule returning only a feasibility set does not satisfy this clause.

3. Selecting properties. Each constructed $K_D$ is homogeneous and strictly convex. These properties are either proved from the actual operational premises or explicitly listed as additional assumptions on the rule's output.

4. Matched affine covariance. The domain is invariant under the declared matched equivalences. For each such equivalence with induced $\bar{f}$, an invertible affine map $v\mapsto A_{f}v+b_f$ satisfies

   $${\iota }_{D^{\prime }}(\bar{f}(p))=A_f{\iota }_D(p)+b_f(p\in E_D).$$

   This includes self-equivalences. Mere existence of an unrelated unlabelled affine isomorphism is not this matched covariance.

5. Input and status audit. The actual rule and predicates use only named operational data and the declared non-geometric interface. No independently supplied spacetime metric, light cone, geometric causal relation, or invariant speed is an input. Every assumed property and class restriction is disclosed. The input does not prescribe the cone claimed to be extracted.

6. Additional-content and countermodel audit. The claimed comparison with the scheduler package is justified in a common operational model universe. Relevant countermodel interpretations and exclusions are identified and proved for this rule and carrier.

This is a certificate template. No actual successful $\mathsf{H},{\mathfrak{F}}_m,D_{\star }$ meeting all clauses is furnished here.

The rule in this specification is a parameter of the implication, not a construction supplied by the note. Properties (K2)–(K3) can make a legitimate conditional theorem; assuming them does not establish their operational origin.

A revised carrier is a separate predeclared branch. It must have a stated order, a map from $E_D$, and a proof of any claimed preservation

Definitions and certificates must then be restated on that carrier. There is no post hoc switch of carrier to obtain classification.

Suppose the objects and certificates of Specification spec:certificates are supplied. Then, for every $D\in \mathcal{A}$, there is an invertible linear map

The represented comparisons satisfy

The representation spans its affine carrier, is generated as in (eq:generated), and is matched-affine-covariant.

The full cone determines the class, up to positive multiplication, of the Lorentzian quadratic form

Under (eq:affine-covariance),

Nonemptiness is inherited from (K1), not proved by classification. If homogeneity and strict convexity are assumed rather than operationally derived, the conclusion retains precisely that conditional status. No transport, calibration, metric field, or dynamics conclusion is included.

Proof. By (K2), $K_D$ is a proper cone in a space of dimension $m\ge 3$. By (K3), it is homogeneous and strictly convex. Ito and Lourenço's Theorem 7 classifies such cones: the lower-dimensional alternatives are excluded, leaving a Lorentz cone ItoLourenco (Theorem 7). This gives (eq:lorentz). Applying its linear bijection to the faithfulness equation proves (eq:lorentz-order). Spanning and generation are the supplied (K2) identities.

For covariance, comparable differences in $D^{\prime }$ are exactly the images under $A_f$ of those in $D$. An invertible linear map commutes with conical generation and closure, so

The marked embeddings are matched by (K4).

To prove quadratic uniqueness, work in Lorentz coordinates and write another quadratic form vanishing on the same nonzero null boundary as

Its vanishing on $(1,u)$ and $(1,-u)$ for all unit $u$ gives $b=0$ and $u^{\mathsf{T}}Cu=-a$. Thus $C=-aI$. Positivity on the future interior gives $a>0$. This proves the uniqueness in (eq:q). Apply it to $q_{D^{\prime }}\circ A_f$, whose null boundary and future interior agree with those of $q_D$, to obtain (eq:q-covariance).

The proof uses the supplied witness only to ensure the domain is nonempty. Clauses (K5)–(K6) state the scope and admission of the operational interpretation; classification does not prove them.

The classification route is the one recorded in Cheung's En08-A and XXII-A CheungEn08A (thm:weaker) CheungXXIIA (thm:classification). Self-duality and irreducibility are not additional requirements of this particular route.

An isomorphism $T_D$ exists; this theorem does not supply a canonical Lorentz frame or an implemented algorithm for computing it. A later numerical or constructive claim must specify its algorithm, input precision, termination, and error certificate.

A finite event carrier cannot generate a cone linearly isomorphic to ${\mathcal{L}}_m$, $m\ge 3$, under (eq:generated).

Proof. There are finitely many nonzero comparable differences $G=\{g_1,\dots ,g_s\}$. If they generate a proper cone, pointedness gives $0\notin \mathrm{conv}G$. Otherwise a nontrivial nonnegative sum equal to zero would put the negative of a nonzero generator in the cone.

Choose an auxiliary Euclidean inner product and the closest point $z\ne 0$ of $\mathrm{conv}G$ to zero. Then

In a convergent sequence ${\sum }_j{\lambda }_{n,j}{g}_j$, ${\lambda }_{n,j}\ge 0$, these inequalities bound all coefficients. A coefficient subsequence proves $\mathrm{cone}G$ is closed.

Every extreme ray of $\mathrm{cone}G$ contains a generator, hence there are finitely many extreme rays. The Lorentz cone has an extreme ray ${\mathbb{R}}_{\ge 0}(1,u)$ for every unit $u$; extremality follows from equality in the triangle inequality. There are infinitely many such rays for $m\ge 3$, and linear isomorphisms preserve extreme rays.

This is the finite-generation check of XXII-A–C CheungXXIIA,CheungXXIIB,CheungXXIIC. The auxiliary inner product is a proof device, not spacetime geometry.

The constant family $W_e=W_0$ has zero strengths:

independent of $S$. Its component order is an infinite antichain, so every comparable difference generating set is empty and $K=\{0\}$. Thus it fails (K2), despite base-class nonemptiness CheungXXIIC (prop:empty-query).

Suppose a datum contains infinitely many identical labelled experiment blocks, and the declared matched equivalences include all finite permutations of those blocks.

There is no injective embedding of its unchanged component carrier into a finite-dimensional affine space satisfying (eq:affine-covariance) for every such self-equivalence.

In particular, no constant process family can belong to a domain satisfying both (K2) and this matched version of (K4).

Proof. Choose the corresponding component containing party $A$ in each identical block, and denote these distinct components by $p_1,p_2,\dots$. Let $v_i=\iota (p_i)$. Every transposition of two selected blocks is a matched self-equivalence, so its affine map transposes the corresponding $v_i,v_j$ and fixes the other selected points.

If the affine space has dimension $m$, choose $m+2$ of these points. There is an affine dependence

with not all coefficients zero. Apply the affine map implementing the transposition $i,j$. Because the coefficients sum to zero, its translation term cancels. Subtract the original dependence to obtain

Injectivity gives $v_i\ne v_j$, hence $a_i=a_j$. This holds for every pair, so all coefficients are equal. Their sum is zero, so all are zero, a contradiction.

This is an obstruction to a marked covariance requirement, not merely to cone shape. It uses no continuity of the extraction rule. An unlabelled affine-isomorphism statement without the commuting condition (eq:affine-covariance) is weaker and is not covered by the proposition.

Distinguishable operational blocks, a justified revised carrier, or an explicitly restricted equivalence convention may avoid this particular premise. Numerical placements based only on identifier order do not repair matched covariance under arbitrary renaming.

Consequently, existence of a Lorentzian embedding for a poset does not imply existence of a covariant data-derived selector for it. This sharpens the set-valued-versus-single-valued distinction in XXII-C CheungXXIIC (prop:query-covariance).

For a representation of the unchanged carrier, fix $p\in E_D$. There are at most two nonzero comparable differences

If comparable differences from this source converge to $v\ne 0$, they are eventually equal to $v\in F_p$.

Therefore the entire nonzero future-null boundary of a Lorentz cone cannot be approximated from one unchanged component by such differences.

Proof. Equation (eq:future) bounds the nonzero set by two. Allowing $q=p$ adds only zero. A convergent sequence in this finite set has its limit in the set and is eventually constant: take a ball about the limit excluding all other points. A nonzero limit belongs to $F_p$.

The Lorentz cone has infinitely many nonzero null directions when $m\ge 3$. An invertible linear map preserves that fact.

This is the obstruction retained in XXII-D CheungXXIID (thm:anchored-obstruction). An extension or moving-source comparison must be substantive and proved. A restricted nonempty set of represented null pairs is a different, potentially weaker target.

The scheduler package $\mathsf{S}$ records acyclic dependency, finite-experiment termination, commutation of independent operations, schedule-independent complete records, sensitivity equal to strict reachability in its designated models, and confluence. It is not automatically defined for every process matrix CheungXXIIA,CheungXXIIB.

On a declared common model universe $\mathfrak{U}$, let $C$ denote operative class restrictions and $H$ the additional assumptions. The required additional-content audit is

separate from

Where the baseline is not applicable, its comparison dictionary must be supplied instead of assigning $\mathsf{S}$ by name.

En08-A records the scheduler obstruction and its ball-section and orthant realizations CheungEn08A (thm:main). For the standard orthant in dimension at least three, homogeneity holds through positive diagonal automorphisms, but strict convexity fails because $e_1+e_2$ is on the boundary. For the standard diamond cone

the sum of $(1,1,0)$ and $(1,0,1)$ is again on the boundary. These standard cones fail (K3) CheungEn08A (prop:orthant, prop:ball).

Those computations are not a translation of the predecessor models into the frozen component carrier or a new selector. A relevant countermodel requires its process, access, carrier, and rule to be interpreted first. Moreover,

A Euclidean-ball section is not itself a non-Lorentzian cone. Excluding named examples is not classification of every remaining possibility.

Cone extraction does not provide a source injection, unknown-message port, quantum receiver output, valid repeated uses, routing model, or physical calibration. These are not present in ${\mathsf{A}}_0$.

For a datum satisfying the conditional extraction hypotheses, fix:

1. a message space $M\cong {\mathbb{C}}^d$, $d\ge 2$, source injection, quantum receiver output, local memories, and resources independent of the unknown message;

2. the admitted uses, compositions, and access enlargements, with a proof that their complete maps are CPTP, including untouched references;

3. source and receiver maps on the component carrier, or on a predeclared compatible extension, with a proof of every claimed old-order preservation and cone-generation compatibility;

4. one chosen $T_D$, frame, comparison scale $\ell >0$, execution convention, arrival-readout calibration, and their correspondence to the realizing ports;

5. the exact retained quantum residuals, classical histories, failure flags, encoders, decoders, and permitted reductions;

6. a declared nonempty target set and a family of finite protocols, together with their actual error and endpoint estimates;

7. execution, computation, estimation, and residual ledgers, and covariance or evidence statements required by the claim.

No such compatible realization for a successful extraction package is constructed here.

This restates the relevant content of XXII-D's realization contract CheungXXIID (spec:realization). A receiver decoder may not access the source's remaining message-bearing systems or an external reference through an undeclared side channel.

A separately declared finite quantum network is a possible realization language CDP. It is not derived merely by taking a signalling quotient.

If the complete message-to-receiver channel $\Lambda :L(M)\to L(M)$ is measure-and-prepare, then

Local source preprocessing and receiver decoding do not remove this floor when the designated connection remains measure-and-prepare and no additional communication or entanglement resource is admitted.

Proof. For a maximally entangled message-reference input $|{\Omega }_d⟩=d^{-1/2}{\sum }_j|j⟩|j⟩$, a measure-and-prepare channel produces a separable output HSR. Its overlap with $|{\Omega }_d⟩⟨{\Omega }_d|$ is at most $1/d$: Cauchy–Schwarz proves this for pure products, and convexity extends it to separable states. The Bell-projector probability difference is at least $1-1/d$. This is a lower bound on the referenced trace distance and hence on the half-diamond distance.

Preprocessing changes the measured effects, and decoding changes the prepared states. The resulting channel remains measure-and-prepare.

XXII-D supplies a datum with a classical $A\to B$ connection and a separately chosen faithful generated ${\mathcal{L}}_3$ representation CheungXXIID (thm:geometry-not-transport). That representation is a mathematical witness, not a non-circular selector. The channel floor shows why geometry and classical signalling alone do not imply unknown-qubit transport.

Let

A target is $\zeta =(p,v)$, where $p\in E_D$ and $v\in {\text{Null}}_D$. Write

Its desired source is $(s,y)$, its endpoint is $(s+\tau ,y+x)$, and its absolute arrival coordinate is

The endpoint need not be represented by an old event.

Let ${\mathcal{T}}_D$ be a supplied nonempty family of such targets. For protocol $(n,\zeta )$, declare coordinates $a_{n,\zeta }$, $b_{n,\zeta }$ of its realizing source and receiver and write

These coordinate assignments require (R3)–(R4). They do not follow from numbering experiments or circuit instructions.

Use

This is a frame-dependent mathematical comparison norm.

For every finite protocol record the actual finite vector

where:

• $U$: number and types of process-resource uses;

• $Q_{\mathrm{peak}}$: peak live quantum qubits under the stated storage and apparatus convention;

• $G,D_{\mathrm{gate}}$: elementary gate count and gate depth;

• $C_{\mathrm{q}},C_{\mathrm{cl}}$: quantum and classical communication, in stated units and with multiplicities;

• $E_{\mathrm{share}}$: admitted shared-entanglement resources;

• $A_{\mathrm{tot}}$: total auxiliary allocation, not peak use;

• $N_{\mathrm{shots}}$: attempted validation or estimation runs, not uses of the unknown payload.

Separately record:

1. Classical computation. Input precision, finite data queried, extraction or optimization steps, arithmetic model, classical memory, and termination.

2. Estimation. Settings, shots, scores, calibration, confidence allocation, and transfer allowances.

3. Residual and output interface. Every retained quantum residual and classical register, its dimension or alphabet, availability, and declared discard or consumption.

No system is counted twice merely because a classical register is represented mathematically by a diagonal quantum register. An actual coherent register must be charged to its actual live quantum use.

An external untouched reference is unavailable to the decoder. If a validation procedure prepares one, its preparation, storage, and measurements belong to the validation ledger.

A count of compatible channel-comparison factors is not automatically an elementary gate depth. Footprint sites are not automatically peak memory. A retained coherent dilation and an adaptive implementation of the same channel can have different memory costs CheungEn02A,CheungEn02B.

No numerical or asymptotic resource bound follows from the cone. A uniform approximation claim needs supplied finite functions

componentwise, or a clearly weaker per-target bound. These functions and their proofs are additional protocol data. The ledger definition does not construct them.

All maps in this section are CPTP on their stated finite-dimensional interfaces. Set

The diamond norm includes untouched references and contracts under channel pre- and postcomposition Watrous (Chapter 3). No diamond norm is assigned to an unspecified process-matrix probability functional.

Suppose

are compatible compositions with matched intermediate interfaces, and

Then

Proof. Insert and subtract intermediate compositions replacing one factor at a time. Each difference contains one changed factor, with channels before and after it. Their diamond norms are one, so each contribution is bounded by its factor difference. The triangle inequality gives the sum. The half-diamond distance between two channels is at most one.

Histories, quantum memories, and retained residuals belong in the step interfaces. Hidden correlated memory is not an independent fresh error allowance. The same stability architecture is printed in En08-B and in complete-history comparisons CheungEn08B (lem:telescoping) CheungHawk03A (lem:telescoping).

For a retained output system $F_{n,\zeta }$, define the full target

where ${\sigma }_{n,\zeta }$ is a specified input-independent state. It can include an arrival register, history, residual, and failures. The comparison is on this entire declared retained interface, not automatically on every unretained physical environment.

Let

After logical identification, its target is ${\mathrm{id}}_M$. The unconditional reference marginal is unchanged by every channel, including a replacement channel. This fact alone does not establish identity transport or preservation of message-reference correlations CheungXXIIB,CheungXXIID.

Define

The source term excludes comparison with an incorrectly anchored protocol.

Fix a datum and a representation supplied under Theorem thm:assembly. Suppose a compatible realization of Specification spec:transport supplies, for every $\zeta \in {\mathcal{T}}_D$ and $n$:

1. a finite complete protocol ${\Gamma }_{n,\zeta }:L(M)\to L(M\otimes F_{n,\zeta })$ and target (eq:full-target);

2. matched ideal and implemented factorizations with at most $H_n$ comparison steps, encoder and decoder allowances ${\alpha }_n,{\beta }_n$, complete step allowances at most ${ϵ}_n$, and

   $$d_{\diamond }({\overline{\Gamma }}_{n,\zeta },{\Theta }_{n,\zeta })\le {\mu }_n;$$

3. uniform endpoint estimates

   $$\Vert a_{n,\zeta }-T_D{\iota }_D(p){\Vert }_{\ast }+\Vert x_{n,\zeta }-x{\Vert }_2\le {\delta }_n,|{\tau }_{n,\zeta }-\tau |\le {\omega }_n;$$

4. the actual ledgers and proved bounds (eq:resource-bound);

5. a retained arrival register with calibrated absolute reading $z\mapsto {a^{\mathrm{arr}}}_{n,\zeta }(z)$. Its ideal marginal is supported on nonfailure codes obeying

   $$|{a^{\mathrm{arr}}}_{n,\zeta }(z)-t_{\zeta }|\le w_n.$$

All allowances are nonnegative. Uniformity is over the declared target set for this datum, not automatically over every admissible datum.

Set

Then

For every message-reference input, the complete actual output is within trace distance $e_n$ of the ideal referenced output. The same allowance bounds the logical referenced transport error and the residual reduction's distance from the input-independent ideal residual.

For every input, the probability of a failure code or an arrival reading outside the window in (P5) is at most $e_n$. The supplied protocol resource bounds remain ${\mathbf{r}}_{n,\zeta }\le {\mathbf{B}}_n(D)$.

If

the full and logical anchored endpoint errors converge uniformly to zero. If additionally $w_n\to 0$, the arrival readings concentrate in shrinking target windows with failure probability tending to zero.

These are endpoint conclusions. They assert neither exact finite-scale null attainment nor a global evolution at all intermediate times.

Proof. Lemma lem:telescoping bounds the implementation difference from the ideal factorization. The design allowance ${\mu }_n$ and the triangle inequality give (eq:channel-certificate).

Tensoring with an untouched reference and evaluating on a state gives the complete trace-distance bound. Partial traces are channels, so they give the logical and residual reduction bounds. In particular,

Insert the endpoint estimates into (eq:full-error)–(eq:logical-error) to prove (eq:endpoint-certificate).

The bad-code projector is an effect on the complete output. Its ideal probability is zero. Its actual probability differs from zero by at most the trace-distance allowance $e_n$. This proves the arrival statement.

Resource bounds are the supplied execution estimates, not a consequence of telescoping. The convergence assertions follow from the displayed inequalities.

Nonempty targets and existence of protocols are hypotheses of the corollary. Their construction remains separate from its proof. A nonempty extraction domain is not a nonempty domain of compatible transport realizations.

The coordinate normalization makes null slope one. It determines neither a calibrated physical speed nor clock and rod units. The arrival register uses absolute target coordinate (eq:arrival-target); the endpoint duration uses the displacement coordinate $\tau$. Their connection is supplied by the realization, not assumed from identical notation.

The exact data assignment (eq:data) has no shot count. Finite empirical certification is a different interface. The following proposition gives one explicit finite certificate for complete channels, not for the infinite cone or its exact signalling quotient.

Fix a finite nonempty set of validation tasks $k=1,\dots ,K$. Task $k$ tests a fixed complete channel ${\Gamma }_{0,k}$ from a $d=2^b$-dimensional message to a declared $o_k=2^{q_k}$-dimensional output, possibly after a fixed padding isometry. All retained records, residuals, and failures intended in the claim are included in that output.

Let ${\Gamma }_{\star ,k}$ be the claimed implemented channel on the same interface and ${\Theta }_k$ its specified ideal target. Define normalized channel Choi states

This convention is distinct from the transposed unnormalized process testers in (eq:choi).

The Choi output dimension is $N_k=d{o}_k$. Let $P_{k,\nu }$, $1\le \nu \le {N_k}^2-1$, be its nonidentity Pauli strings, so

Write

Conditionally on frozen training and calibration information:

1. every task, target, decoder, score, and positive shot count $n_{k,\nu }$ is fixed before validation;

2. scores $Z_{k,\nu ,i}\in [-1,1]$ are independent within each task-setting pair and have a common mean $m_{k,\nu }$;

3. all settings for a task test the same complete channel; references are untouched by the decoder, and input preparation and ancillary independence are justified;

4. every attempted trial, including losses and failures, receives its predeclared score;

5. on a joint event $G$, with $\mathrm{Pr}(G^c)\le \beta$,

   $$|m_{k,\nu }-c_{k,\nu }|\le b_{k,\nu },d_{\diamond }({\Gamma }_{\star ,k},{\Gamma }_{0,k})\le {b_k}^{\mathrm{tr}};$$

6. the complete-output access, padding, quantum-reference preparation, and Pauli measurement implementation are admitted and calibrated.

This is additional validation access, not the original classical receiver access ${\mathsf{A}}_0$. No unaccounted adaptive stopping, setting removal, or channel drift is covered.

This follows the scope discipline of Cheung's complete-record certificates CheungOai265B,CheungOai280C. No reported empirical result is copied as evidence.

Under Specification spec:validation, choose $0<\alpha <1$ and let

Set

With probability at least $1-\alpha -\beta$, simultaneously for all tested tasks,

Consequently, acceptance only when $U_k\le {\epsilon }_k$ gives

The shot ledger is exactly

For a prescribed common radius $r>0$, taking every count equal to

gives that radius or smaller and an explicit total of $J_{\mathrm{val}}{n}_{k,\nu }$ attempted shots.

Proof. Hoeffding's inequality for scores in $[-1,1]$ gives

A conditional union bound followed by integration gives simultaneous sampling coverage with failure probability at most $\alpha$. Intersect with $G$. Then

where clipping at two is valid because both exact coefficients belong to $[-1,1]$.

The difference of the two normalized Choi states has Pauli expansion

Its identity coefficient is zero. Orthogonality and $\Vert X{\Vert }_1\le \sqrt{N_k}\Vert X{\Vert }_2$ give

For differences of channels with input dimension $d$,

To see the factor, a pure message-reference input can be written $(I\otimes B)|{\Omega }_d⟩$ with $\Vert B{\Vert }_{\infty }^2\le d$. Its output difference is the corresponding two-sided multiplication of ${\Delta }_k$, whose trace norm is at most $d\Vert {\Delta }_k{\Vert }_1$. The diamond norm of a channel difference is optimized on such referenced states Watrous (Chapter 3). Divide by two and add the transfer allowance to obtain (eq:validation-bound).

The joint failure probability is at most $\alpha +\beta$. Every false acceptance lies outside the simultaneous coverage event. The shot identities and prescribed-radius bound follow directly from the definitions.

This certificate concerns the full declared output because all its Choi coefficients are covered. Three Bell correlations on the logical output alone do not certify arbitrary retained residuals.

The statistical conclusion is joint frequentist coverage, not a posterior probability conditional on acceptance. It covers only the finite tested tasks. Extending it to an infinite target family requires a separate coverage or uniformity theorem.

The certificate does not prove the sampling, CPTP, access, calibration, or transfer assumptions. It does not decide exact zero signalling or certify homogeneity and strict convexity of an infinite generated event cone.

For a measured factor $S$, an untouched residual $B$, and a declared orthonormal readout basis, the deterministic record channel is

Its Kraus operators are $|z{⟩}_R\otimes ⟨z{|}_S\otimes I_B$; their adjoint products sum to $I_{SB}$. Thus it is CPTP, including reference extensions, and

Its outputs are $R,B$, not the consumed quantum factor $S$. This is the supplied finite readout convention and consumption discipline of I, En09-A, and XVIII-A CheungI,CheungEn09A,CheungXVIIIA.

The surviving complement can remain quantum and correlated with a reference. It is not a quantum return of the measured factor. A decoder returning the unknown message is a different map with a different correctness requirement. No normalized postselected branch replaces the deterministic channel in the corollary.

The following remain scoped tests, not sufficient premises for extraction CheungXXIIB,CheungXXIID.

• Boundary generation versus attainment. For

every nonzero finite conical combination is interior to ${\mathcal{L}}_m$, by the strict triangle inequality, but $\overline{\mathrm{cone}G}={\mathcal{L}}_m$ by density. A generated null boundary need not contain an attained generator or implemented protocol.

• Finite-stencil delay. If all step velocities lie in a fixed finite $S\subseteq \{\Vert s{\Vert }_2\le 1\}$, with $0\in S$, choose a unit vector $u\notin S$. Then $h_S(u)={\max }_{s\in S}u\cdot s<1$. An endpoint within ${\delta }_n$ of $ru$ satisfies

by projection onto $u$. When $h_S(u)>0$, vanishing spatial error forces an arrival lower bound $r/h_S(u)>r$. When it is zero, sufficiently accurate arrival is impossible.

• Moving arrival jumps. Compare a channel returning a failure flag before $\tau$ and identity with a success flag after $\tau$, with the corresponding jump at ${\tau }_n>\tau$. Even if ${\tau }_n\to \tau$, orthogonal flags between the two arrival times give uniform half-diamond distance one. Endpoint topology does not assert uniform convergence through such an unmatched jump.

• Accumulation is model dependent. For the qubit replacement channel ${\mathcal{R}}_0(X)=\mathrm{Tr}(X)|0⟩⟨0|$, let ${\mathcal{F}}_e=(1-e)\mathrm{id}+e{\mathcal{R}}_0$. Idempotence gives

Thus $ke\to a>0$, with $e\to 0$, can leave error $1-e^{-a}$. Failure of a telescoping upper bound to vanish, by itself, is not a lower bound for every noise or correction model.

• Tolerance versus exact influence. Channels uniformly within $ϵ$ of one intervention-independent channel have signalling strength at most $2ϵ$, not necessarily zero. The family $(1-\delta ){\mathcal{R}}_0+\delta \mathrm{id}$ is at distance $\delta$ from replacement and transmits a bit with distinguishability $\delta >0$. A tolerance graph is not the exact graph (eq:influence).

• Endpoint limits versus continuum evolution. A terminal comparison does not supply intermediate channels, a composition law, or a unique interpolation. For example,

has

uniformly on compact time intervals. Two different $c$'s can be assigned the same coarse $n=1$ channel and have different limits. They choose different missing refinements; no exact subdivision compatibility with that coarse member is asserted. This is the scope of XIX-C's refinement warning CheungXIXC (prop:two-limits, prop:interpolation).

An exact edge can also disappear at a process limit. Mix a nonsignalling $W_0$ with a definite $A\to B$ identity process $W_{\to }$:

The contribution of $W_0$ to each signalling difference is zero, so ${\Delta }_{W_{\delta }}(A,B)=\delta$. The edge is present for every $\delta >0$ and absent at the limit. A convergent sequence of probability tables therefore need not supply compatible component-quotient maps CheungXXIIC (prop:strength-stability).

A full affine Lorentz cone determines $[q_D]$, not a preferred observer, physical normalization, or system of units. If

its normalized null slope becomes ${\ell }_0/{\tau }_0$. Those calibration constants are not selected by cone membership CheungXXIID (prop:calibration).

Translating a constant representative across its affine carrier gives constant metric components. Its Christoffel symbols and curvature are zero. Even after a local differentiable extension is supplied, positive conformal multiplication preserves every pointwise cone. With signature $(+,-,\dots ,-)$, the first variation of $e^{2\lambda f}\eta$ is $2f\eta$, and

The right side can be nonzero while all cone fields are unchanged. XXII-E supplies this calculation and its precise comparator conventions CheungXXIIE (thm:flat, prop:conformal).

Neither a process matrix nor an identity transport channel assigns a matter Hamiltonian or stress-energy tensor. Symmetry and conservation alone also allow nonzero improvements of the displayed derivative form CheungXXIIE (prop:energy, prop:improvement). Consequently, no dynamics matching is hidden in Theorem thm:assembly or its approximation corollary.

An operational instantiation must discharge:

1. Non-vacuous additional package. Name an actual finite hypothesis list, exhibit its admissible witness, and prove the genuine additional-content comparison. Complete the relevant countermodel dictionary and exclusion audit. Base-class nonemptiness is insufficient.

2. Single-valued extraction and invariance. Construct the actual rule and prove faithfulness, generation, properness, spanning, and matched covariance. Respect Proposition prop:symmetry. A feasibility set, a choice justified only by the desired geometry, or a hidden numerical ordering of identifiers is not this construction.

3. Honest selecting hypotheses. Derive homogeneity and strict convexity from operational premises, or explicitly retain them as assumptions on the constructed cone. Classification does not derive them.

4. Compatible approximation. Construct valid transport access, the nonempty target family, finite protocols, source/receiver correspondence, residual outputs, and endpoint and arrival estimates. Respect finite futures and the event-aware topology. The corollary proves estimates for supplied protocols, not their existence.

5. Actual resource and evidence functions. Supply the numerical or symbolic resource functions and their proofs under a fixed execution model. Separate payload execution, classical extraction cost, estimation shots, and residual interfaces. Finite validation covers only its tested tasks unless a further coverage theorem is provided.

6. Layer separation. Keep cone extraction, operational approximation, geometric embedding, calibration, metric extension, stress-energy, and dynamics as distinct claims. No observation boundary, catalogue record, or prescribed inverse is identified with an extracted physical light cone.

No complete operational package satisfying F1–F6 is supplied here.

{}1.12

The component table is a scope ledger, not a proof-dependency identification. In particular, a valid theorem about a prescribed cone, a classical lapse, a free stress class, or an independently appended recovery channel does not construct the present operational bridge.

The completed mathematical release comprises:

1. the self-contained named process and component-order interface;

2. Theorem thm:assembly, a proved conditional implication;

3. the finite-generation, finite-future, and matched-symmetry obstructions;

4. Corollary cor:approximation, a proved conditional complete-output endpoint certificate;

5. the separate ledgers and optional finite validation Proposition prop:validation;

6. obligations F1–F6 and the retained non-identifications.

No successful predicate list, affine selector, or compatible protocol family is constructed. In particular, the symmetry obstruction shows why a mathematical embedding witness for an infinite poset need not become a covariant operational extraction.

Find one actual predeclared package on ${\mathcal{P}}_{3q}$ comprising:

1. a finite hypothesis list and an exhibited nonempty admissible domain satisfying the required additional-content audit;

2. a single-valued rule satisfying the representation, selecting-property, covariance, and input certificates of Specification spec:certificates, or a fully specified justified revised-carrier branch;

3. a nonempty subdomain with compatible transport realizations and declared nonempty null-target families;

4. finite protocols with proved resource functions and allowances satisfying the relevant approximation and evidence conclusions;

5. any required computation, estimation, extension, and relabelling-compatibility theorems.

A full fixed-source null-boundary target cannot use the unchanged finite-future carrier without an explicit compatible change. This existential target is not established in XXII-F.

The logical order is

followed separately by

Neither arrow supplies the hypotheses of the other. Dynamics matching requires further structures identified in XXII-E.

boundaries. No metric field on an operationally identified manifold, preferred frame, physical units, invariant physical speed, attained null transport event, stress-energy extraction, or Einstein dynamics is established.

No holographic reconstruction, Page curve, gravity-generated entanglement, observation-boundary/light-cone identification, or quantum evolution without external time is proved. No result changes O.1, opens XV-B, discharges XVI-N or Route B, or alters H2, I3, or J2. No inherited characteristic quantum coefficient is admitted.

No predecessor proof is represented as machine-checked. No numerical optimization, native programme execution, laboratory implementation, or successful pdfLaTeX build is asserted CheungXIXE.

precise remaining gap. The missing result is the non-vacuous operational construction: an actual additional package, a matched single-valued affine rule, and a separately valid transport realization with finite protocols and proved resource functions.

Until that construction is supplied, the programme has a rigorous conditional assembly theorem, operational error certificates, and scoped obstructions, but not an existence theorem extracting Lorentzian causal structure from the named quantum data. The conditional results cannot be promoted into that existential claim by renaming assumed outputs as derived geometry.

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 items: target:selection, def:representation, thm:classification, rem:conformal, and acceptance conditions (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: ledger syntax, Candidate D, prop:signalling-cycle, prop:finite-generation, 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: def:process, spec:access, def:class, lem:base, prop:binary, thm:order, prop:strength-stability, prop:empty-query, prop:query-covariance, and target:selection.

CheungXXIID

N. Cheung. XXII-D: Operational approximation of an extracted Lorentz cone. Stage 3 of a limited kinematic emergence programme. Supplied manuscript XXII-D.tex, n.d. Relevant labels: thm:anchored-obstruction, prop:calibration, thm:geometry-not-transport, spec:realization, thm:certificate, and failure tests F1–F6. The duplicate XXII-C bibliographic alias is not copied.

CheungXXIIE

N. Cheung. XXII-E: Dynamics matching on an extracted Lorentz cone. Stage 4 of a limited kinematic emergence programme. Supplied manuscript XXII-E.tex, n.d. Relevant labels: thm:flat, prop:conformal, prop:energy, prop:improvement, thm:surjective, and target:dynamics.

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.

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 supplies the classification used in Theorem thm:assembly.

Watrous

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

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, titled Process Products Need Qualification: https://arxiv.org/abs/1706.05532.

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.

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, titled General Entanglement Breaking Channels: https://arxiv.org/abs/quant-ph/0302031.

Hoeffding

W. Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association 58 (1963), 13–30. doi:\ 10.1080/01621459.1963.10500830.

CheungEn08A

N. Cheung. En08-A, version 1, manuscript En08-A.tex. Enter Calculus Companion, n.d. Relevant supplied labels: thm:main, prop:ball, prop:orthant, thm:mainpositive, thm:representation, and thm:weaker. Scheduler obstruction and conditional cone classification. 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. Transport approximation on prescribed geometric and speed inputs.

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. Three-relation discipline. No unsupplied label thm:separation is required for this citation.

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, and def:limit. Complete-network scheduled support and its separate limit interface.

CheungXIXC

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

CheungHawk03A

N. Cheung. Hawk03-A, version 1, manuscript Hawk03-A.tex. Enter Calculus Companion, n.d. Relevant supplied labels: lem:telescoping and ass:physical. Complete-history stability and its separately required physical bridge. No black-hole or gravitational interpretation is transferred here.

CheungEn02A

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

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. Model-dependent batch, streaming, and logical-readout resource tasks.

CheungOai265A

N. Cheung. Oai265-A, manuscript Oai265-A.tex. Enter Calculus Companion, n.d. Relevant supplied labels: def:determination and thm:separation. Static area-law records are not recovery histories or gravitational identification theorems.

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 certification.

CheungOai280C

N. Cheung. Oai280-C, version 1, manuscript Oai280-C.tex. Enter Calculus Companion, n.d. Relevant supplied labels: con:validation, prop:calibration, and thm:simultaneous. Declared complete-channel coverage and validation discipline; not evidence for the present extraction package.

CheungI

N. Cheung. I, version 2, manuscript I.tex. Enter Calculus Companion, n.d. Relevant supplied labels: thm:main, thm:inhabitation, and thm:update. Finite Born-strategy characterization on its stated strategy class. No measurement-uniqueness proof is imported into the present theorem.

CheungEn09A

N. Cheung. En09-A, version 1, manuscript En09-A.tex. Enter Calculus Companion, n.d. Relevant supplied label: thm:main. Complete destructive clock readout, reference extension, and quantum-name consumption.

CheungXVIIIA

N. Cheung. XVIII-A, version 2, manuscript XVIII-A-v2.tex. Enter Calculus Companion, n.d. Relevant supplied item: eq:readout, “Enter is readout, not quantum return.” A finite record–residual convention, not a decoder or geometry theorem.

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 boundaries and verification limits.