Exact Causal Cones for Partitioned Enter Dynamics: Snapshot–Channel Comparison, Sharp Relay Speed, and Conditional Continuum Limits

We study complete quantum–classical instruments executed on a locally finite metric substrate under partitioned updates of diameter at most $r$. We give a self-contained comparison between snapshot factorization, Heisenberg support containment, and reference-preserving channel no-signalling. The schedule past is an upper bound on operational influence, not an assertion that every scheduled ancestor affects a readout. An explicit relay attains the metric bound, giving $c_{\mathrm{lat}}=r$ on the stated lattice substrates. Arbitrary block instruments and calculus-generated instruments are distinguished; finite-program soundness requires a compositional interface, and simulation requires routing from every starting tick.

Destructive Enter readout, coherent recording, operational discard, and complete residual response are kept separate. Transmissive exceptional blocks violate the original-metric cone, whereas inert exceptional blocks admit a local refinement. Long-range completely positive channels remain affine and cannot universally clone unknown states. Entanglement-assisted state transfer is stated with its communication and timing resources.

The comparison with periodic dependency graphs supplies a sharp isotropy obstruction: a fixed finite periodic transport model has a velocity polytope, and the cardinal model has an ${\ell }^1$ velocity ball rather than a Euclidean ball. We also identify a limitation of the earlier boost-closure formulation: together with unrestricted composition and well-founded executability, it excludes ordinary bidirectional one-tick relays as well as super-cone devices.

Finally, continuum domain of dependence, operational no-signalling, algebraic microcausality, and frame-invariant maximal speed are proved only under explicit geometric, operational, observable-algebra, and frame-covariance hypotheses. The operational hypotheses include matched realizations of local modifications; the algebraic hypotheses concern observables, not merely commuting superoperators. Existence of a family satisfying the full package remains open. No physical metric, Einstein dynamics, or invariant physical speed is derived.

Partitioned finite-range dynamics have an exact finite domain of dependence. Their geometry supplies the propagation bound; the semantics of a programming calculus supplies a separate interface between programs and complete instruments. These two statements should not be conflated.

This manuscript develops the comparison proposed for En05-A and revises the supplied manuscript 085VvAv3. The discrete cone, its relay attainment, and the distinction between semantic and syntactic admissibility are retained. The additional development is a precise alignment with channel-locality interfaces and a velocity-polytope obstruction to the simplest isotropic scaling proposal.

Cheung's En03-A gives one-layer adjoint support, schedule locality, local-record laws, and invariance under legal reorderings of fixed labelled channels CheungEn03. VII-J separates scheduled reachability, operational influence, and an independently declared physical causal relation, and gives a phase-explicit cycle-velocity description CheungVIIJ. Psy04-B emphasizes that scheduled ancestry need not equal influence CheungPsy04B. We prove the comparison statements needed here rather than importing those manuscripts as theorem premises.

The measurement accounting follows the distinctions in Landmark I, En04-A, and Hawk04-B: destructive readout returns a classical record; discarding that record gives a partial trace; and recovery from a complete output is not recovery from a selected residual CheungI,CheungEn04,CheungHawk04B. The syntax-specific results of Landmark I are optional semantic input once the carrier is a qubit interface. They are not needed to prove the geometric cone.

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1. We formulate snapshots as unnormalized hybrid states and prove their equivalence with reduced-channel factorization and adjoint support containment.

2. We prove exact schedule locality and its metric envelope, with complete classical controls and untouched references included.

3. We retain the sharp relay witness $c_{\mathrm{lat}}=r$, while making finite-program soundness and tail-routing simulation precise.

4. We distinguish readout laws, residual response, classical-record reconstruction, and entanglement-assisted transport.

5. We state and prove the timed velocity-polytope comparison and the fixed-periodic-model isotropy obstruction.

6. We correct the boost-closure and continuum interfaces. In particular, we do not retain a physical retiming-stability conclusion from an axiom that is incompatible with ordinary bidirectional relays.

boundary. The metric, schedule, register interfaces, and update rules are declared inputs. The results do not derive a metric from entanglement, Einstein dynamics, a Page curve, an island, a horizon, or a holographic identification. Results about reversible quantum cellular automata are related but have additional hypotheses; no classification theorem for arbitrary instruments is asserted SchumacherWerner,Arrighi. The present exact support bound is structurally different from the exponentially suppressed propagation outside an effective cone in Hamiltonian Lieb–Robinson estimates LiebRobinson.

All statements concern finite time windows and finite experiments. At each site and tick, reserve a finite interface large enough for that experiment. An interface may contain quantum registers, classical records, occupation and placement labels, and controller memory.

A local observable algebra has the form

where $L_{x,s}$ is a finite classical label set and each ${\mathcal{H}}_{x,s,\lambda }$ is finite dimensional. Different occupation patterns may have different quantum dimensions. Inactive reserved slots are not accessible surviving quantum systems.

For a finite region $I$, write

A state is represented by positive blocks

The trace is the ordinary sum of matrix traces.

On a countable substrate we require a compatible family of states on finite local algebras. No density operator on an unspecified infinite tensor product is required. All calculations below reduce to finite regions.

A finite-outcome instrument is a family $\{{\Phi }_k{\}}_k$ of completely positive maps such that ${\sum }_k{\Phi }_k$ is trace preserving. Its recorded channel is

A complete channel includes every reported outcome and every failure outcome. Classical discard sums blocks without conditional normalization.

The channel language is the standard finite-dimensional completely positive map language Choi,Watrous. A local adaptive instruction is one complete channel on its quantum registers and its classical controller blocks.

Every state-dependent input to an update is present in its declared local interface. This includes records, program counters, random seeds, controller memory, and resource registers.

An update may depend on its own classical input block, but not on an unrepresented global history, an unknown quantum-state description, or a remote controller. Its complete map is fixed before the unknown input state is supplied.

Shared randomness is allowed by preparing correlated classical registers at the relevant sites. It is not an unaccounted global scheduler input. This support convention is essential for adaptive locality; compare Cheung VII-G CheungVIIG.

Let $R_{I,s}$ denote restriction to the interface in $I$: quantum factors outside $I$ are traced out and external classical labels are summed over.

The snapshot at tick $s$ in $I$ is

Its classical labels include the identities and placements of live names, record values, and all retained controller data.

If $p_{\lambda }=\mathrm{Tr}{\omega }_{I,\lambda }>0$, its normalized conditional quantum state is ${\rho }_{I\mid \lambda }={\omega }_{I,\lambda }/p_{\lambda }$. At $p_{\lambda }=0$, no conditional state is assigned.

Thus the snapshot determines both the local classical law and the conditional reduced quantum states after non-selectively averaging external records. Conversely, those data determine the snapshot by ${\omega }_{I,\lambda }=p_{\lambda }{\rho }_{I\mid \lambda }$.

The affine object is the array of unnormalized blocks $({\omega }_{I,\lambda }{)}_{\lambda }$, not the normalized conditional states separately. Conditional normalization generally introduces nonlinear ratios. All factorization identities in this paper are identities of complete hybrid states.

For a measured register $S$ and a surviving register $T$, set $M_x=⟨x{|}_S\otimes I_T$. Destructive computational readout is

The measured names in $S$ are consumed; $K$ is classical. The surviving quantum output is $T$, not a renamed copy of $S$. This is the readout accounting used in CheungI,CheungEn03,CheungEn04,CheungHawk04B.

Define the coherent recording isometry

Then

and

Both identities hold with an arbitrary untouched reference.

Proof. Expand ${\rho }_{ST}={\sum }_{x,y}|x⟩⟨y{|}_S\otimes {\rho }_{xy}$. Tracing the recorded $S$ factor kills the terms with $x\ne y$ and leaves ${\sum }_x|x⟩⟨x{|}_K\otimes {\rho }_{xx}$, which is (eq:enter). Tracing $K$ gives ${\sum }_x{\rho }_{xx}$, the partial trace of the original state. Tensoring the calculation with a reference identity changes neither equality.

Equation (eq:recordequivalence) permits deferred canonical readout of the record after the measured factor has been made inaccessible and the record is used only through basis-diagonal classical controls and readout. It is not an equivalence with an unrestricted coherent context that may access $S$, erase $K$, or interfere different record branches.

A substrate is $(\Lambda ,d,r)$, where $\Lambda$ is countable, $d$ is an integer-valued metric with finite balls, and $r\ge 1$ is an integer.

A block schedule is a sequence $({\mathcal{P}}_s{)}_s$ of partitions of $\Lambda$ into finite blocks $b$ satisfying ${\mathrm{diam}}_{d}b\le r$. Tick $s$ maps the tick-$s$ interface to the tick-$(s+1)$ interface.

A semantically admissible update is

where each ${\Phi }_{s,b}$ is a complete channel between the declared input and output interfaces in $b$, satisfying Assumption ass:completecontrol.

Allocation, destructive readout, classical record creation and discard, and relocation of names and records are permitted only within $b$. No output of the block is placed outside $b$.

Relocation changes the spatial address of a wire and is part of the local channel interface. It need not be a reduction step of the calculus. The propagation clock counts block updates, not the number of gates evaluated inside a block. There is no additional gate-depth or per-site capacity bound unless explicitly imposed.

For an observable with finite output support, only finitely many block adjoints are relevant. Their product defines its pullback, including on a countable substrate.

An observable is supported in $I$ when it is the identity on the complementary interface. We identify local observables with their extensions by identity.

Let $\Phi$ be a complete channel between finite tensor-product hybrid interfaces. Fix an input region $I$ and output region $Y$. The following are equivalent:

1. There is a complete channel $K$ such that

   $$R_Y\Phi =K{R}_I.$$

2. Equal input snapshots in $I$ give equal output snapshots in $Y$, for all input states.

3. $${\Phi }^{\ast }({\mathcal{A}}_{\mathrm{out}}(Y))\subseteq {\mathcal{A}}_{\mathrm{in}}(I).$$

When they hold, factorization is stable under every untouched reference: equal input reductions on $I$ together with that reference give equal output reductions on $Y$ together with it.

Proof. (i) implies (ii). For (ii) implies (i), choose a fixed complementary state $\sigma$ and define

The map $K$ is complete and trace preserving. By (ii), $R_Y\Phi (\omega )=K{R}_I(\omega )$ on every state. States linearly span the hybrid input space, so the equality holds as a linear-map identity.

Taking adjoints in (i) gives

which is (iii). Conversely, (iii) implies that expectations of every output observable depend only on the input snapshot in $I$, giving (ii).

Finally, tensor the linear-map identity in (i) with a reference identity. This proves the reference statement. Equality on $I$ alone is not enough to conclude equality jointly with a reference unless the corresponding joint input reductions also agree.

This is the factorization form of the no-signalling interface in Cheung VII-J CheungVIIJ (thm:nosignal). Its adjoint form matches En03-A CheungEn03 (lem:one-layer, thm:locality). The proof above is independent of those source records.

For finite $Y$, put

Then

The reduced channel on $Y$ is determined by the channels of the blocks meeting $Y$.

Proof. The complement of $U_s(Y)$ is a union of blocks. An observable on $Y$ is the identity on their output interfaces. Since the adjoint of a trace-preserving channel is unital, each complementary block adjoint contributes an identity on its input interface. The remaining adjoints act only in $U_s(Y)$. Their composition determines the reduced output channel.

For a finite terminal region $Y$ and terminal tick $T$, set

The scheduled past between $t_0$ and $T$ is

Each $N_s(Y)$ is finite. Moreover,

Indeed, each backward layer adds at most one block diameter.

For a terminal site $A$, let

A block is inner at tick $s$ if it is contained in $I_s$.

If $b\in {\mathcal{P}}_s$ meets $I_{s+1}$, then $b\subseteq I_s$. Consequently, no block that is not inner can place an output into $I_{s+1}$.

Proof. For $z\in b\cap I_{s+1}$ and $w\in b$,

Every output remains in its block.

There is a complete affine channel $K_s$, determined only by the schedule and the inner block channels at tick $s$, such that

Proof. The blocks meeting $I_{s+1}$ are contained in $I_s$ by Lemma lem:geometry. Restrict the input snapshot to their union, apply their product channel, and restrict its output to $I_{s+1}$. This is a complete channel $K_s$.

All other block adjoints act on identity factors when testing an observable supported in $I_{s+1}$; they are therefore invisible by Lemma lem:onelayer. Lemma lem:comparison gives (eq:onesnapshot). External outcomes are summed, not postselected.

Equation (eq:onesnapshot) is the snapshot form of one-layer adjoint locality. The normalized conditional states and their probabilities are recovered from its unnormalized blocks. This formulation does not add an assumption that the initial state is unentangled across the cone boundary.

A readout at $(Y,T)$ is a prescribed complete instrument on the tick-$T$ interface in $Y$. Its outcome law is obtained from ${\Sigma }_T(Y)$. A readout may return both a classical record and a quantum residual; the readout law means its classical marginal.

Let two semantically admissible runs have the same schedule over $[t_0,T]$.

If their initial snapshots on $N_{t_0}(Y)$ agree and their block channels agree whenever $b\cap N_{s+1}(Y)\ne \varnothing$, then their terminal snapshots on $Y$ agree. Equivalently,

In particular, for $Y=\{A\}$, agreement of the initial snapshots on $B(A,r(T-t_0))$ and agreement of every inner block channel imply

Every common readout at $(A,T)$ then has the same outcome law. These conclusions are reference preserving in the sense of Lemma lem:comparison.

Proof. Apply Lemma lem:onelayer backward through the ticks. Definition def:schedulepast gives (eq:adjointcone). At each step the reduced channel depends only on the blocks meeting the next scheduled slice, so the two runs have the same reduced evolution through the scheduled past.

For the metric statement, use (eq:metricenvelope) and induct forward using Lemma lem:dod. The initial snapshots agree on $I_{t_0}$, and the common inner channels give the same $K_s$ at every step. Thus the snapshots agree on $I_T=\{A\}$. Readout laws are functions of that terminal snapshot.

For a fixed complete network and terminal output $Y$, an external input port is influential if two trace-preserving local interventions there, for some admitted initial state, produce different unconditioned output laws on $Y$. Denote the set of such ports by ${\mathsf{Inf}}^{-}(Y)$.

For the complete dependency network,

The inclusion can be strict.

Proof. A trace-preserving intervention outside the scheduled ancestors preserves their reduced state, including when the initial state is entangled. Theorem thm:cone excludes an output change. For strict inclusion, give a channel a declared memory input and let it return the constant classical answer $0$. The memory port is a scheduled ancestor but is not influential.

This is the distinction made explicitly in Cheung Psy04-B CheungPsy04B (thm:schedule). Selectively normalized branches and correlations caused by common ancestors are not counterexamples to this unconditioned intervention statement.

A selected calculus has well-typed finite first-order quantum and classical interfaces. It supports sequential composition, tensor composition on disjoint owned interfaces, finite classical control, allocation, its declared gate library, and destructive readout.

The denotation of a composite program is the corresponding composition of complete instruments. Every selected local generating program terminates and has a strategy-independent denotation. Programs and controllers do not inspect a description of the unknown quantum state.

For a concrete strongly normalizing calculus, the relevant metatheoretic clauses are recorded in Appendix app:interface. Assumption ass:programinterface is explicit because separate evaluability of block terms does not by itself imply that arbitrarily concatenated block evaluations are reductions of one fixed original term.

A finite block-channel network is syntactically admissible when every block channel is the denotation of a selected generating program and the network is well typed under Assumption ass:programinterface. Interfaces, record dependencies, ownership, and branch continuations must match.

The geometric theorem applies to arbitrary semantically admissible channels. Syntactic admissibility is a stronger implementation condition; it does not assert that every arbitrary instrument is expressible over the available gate library.

Every finite acyclic program built from the specified primitives has linear path maps $K_h$ satisfying

For reported outcome $y$, its selective map is

The sum over reported outcomes is a complete channel, including every untouched reference.

Proof. A complete history fixes the classical choices. Compose allocation isometries, gates, wiring maps, and destructive measurement bras along that history to obtain $K_h$. Successive branch normalizations cancel against their probabilities.

At a readout node, ${\sum }_x{M}_x^{†}{M}_x=I$. Allocation is isometric and gates are unitary. Backward induction over the finite instruction tree gives the completeness identity. Grouping histories gives the reported maps. Tensoring every path map with a reference identity proves the referenced statement.

This is the finite-program accounting in Landmark I, En04-A, Hawk04-B, and En02-C CheungI,CheungEn04,CheungHawk04B,CheungEn02C; the proof is included to make the present fragment self-contained.

Every finite syntactically admissible network denotes the same complete instrument as a well-typed composite program. Legal within-tick orderings of its disjoint block evaluations have the same denotation.

Proof. At a tick, compose the generating programs in parallel on their disjoint owned interfaces. Compose these tick programs sequentially, using finite classical control for the retained histories. Assumption ass:programinterface supplies the composite program and identifies its denotation with the block network.

Channels on disjoint interfaces commute, so legal within-tick orderings give the same channel. Evaluation of the composite program then supplies a valid finite execution in the selected calculus.

The proposition does not turn an indefinitely driven substrate into a reduction sequence of one fixed term. For example, an external driver may request a new gate at every tick forever, although every requested local program terminates. Strong normalization of individual terms does not prohibit that external drive.

A schedule has the tail-routing property if, from every starting tick, every finite collection of names and records at declared initial locations can be brought to any declared target site in finitely many ticks by within-block relocations alone. The finite reserved interface must have sufficient capacity for the routed collection.

On ${\mathbb{Z}}^D$ with the ${\ell }^1$ metric, define, along one coordinate axis, the two partitions with intervals

and singletons at all remaining sites. Lift these partitions along each coordinate line.

A schedule that repeats both phases on every axis is tail routing. Every nearest-neighbor edge lies in an interval of one of the two phases. A name can therefore follow any finite coordinate path by waiting for the required phase. Previously routed names may remain at the target while the remaining names are transported.

Suppose the selected program has a finite input interface and a finite complete evaluation, and its gates belong to the specified library. On a tail-routing schedule with sufficient finite capacity, its instrument can be realized by a syntactically admissible run.

Proof. Choose a target site. Route the program's input names and classical input records to that site, using a state-independent transport plan. Transport is a wire-address change and leaves their joint logical state unchanged.

At a later tick, evaluate the generating program inside the block containing the target site. All required fresh slots are reserved there. The block's denotation is the instrument of the original program. If a prescribed final placement is required, route the returned names and records after evaluation.

This is a simulation of the complete finite instrument. It is not an exact finite-gate compilation theorem for arbitrary specified unitaries, an efficiency theorem, or a claim that each administrative redex is itself the terminal evaluation of an isolated term.

Under the hypotheses of Theorem thm:cone, if a common completed program's reported outcome records are in the terminal region, their law is the same in both runs.

Proof. The reported records are part of the terminal snapshot.

Strong normalization bounds reductions, not externally inserted idle ticks or unsuccessful routing attempts. A finite completion horizon requires a chosen finite scheduled execution, a finite progress condition, or the explicit routing construction above.

For a finite horizon, two runs form an $x$-perturbation pair when they share the schedule, their initial snapshots agree on every finite region not containing $x$, and their channels agree on every block not containing $x$.

The site $x$ signals to $(A,\tau )$, where $\tau \ge 1$, if such a pair produces different laws for a common readout at $(A,t_0+\tau )$. Define

with supremum $0$ if there is no positive-distance signalling.

Every substrate satisfies $c_{\mathrm{lat}}\le r$.

Proof. If $d(x,A)>r\tau$, the initial cone ball excludes $x$, as does every inner block. An $x$-perturbation pair therefore satisfies the metric hypotheses of Theorem thm:cone, so its readout laws agree.

On ${\mathbb{Z}}^D$ with the ${\ell }^1$ metric, for every $\tau \ge 1$ there is an $x$-perturbation pair with $d(x,A)=r\tau$ and readout laws ${\delta }_0,{\delta }_1$. Consequently $c_{\mathrm{lat}}=r$.

Proof. Take $A=0$, $x=r\tau e_1$, and place a classical bit $\beta \in \{0,1\}$ at $x$. At step $j=0,\dots ,\tau -1$, choose the phase containing the interval

Relocate the record from its right endpoint to its left endpoint. All other updates may be identities.

After $\tau$ ticks, the record is at $A$. Reading it gives ${\delta }_{\beta }$. The two runs differ only in the initial bit at $x$ and use identical rules. The lower bound $r$ combines with Proposition prop:upper.

The speed is a supremum over schedules and admitted channels. Attainment does not say that every fixed schedule, every block instrument, or every site in the metric cone transmits information.

Let two initial regions $X_0,Y_0$ satisfy $d(X_0,Y_0)>2r\tau$. Propagate each region forward by adjoining all outputs of every block meeting its current support. Up to tick $t_0+\tau$, the two propagated families are disjoint, and no update before that terminal tick meets both families.

Proof. After $j$ ticks, each family lies within distance $rj$ of its initial region. If a block at step $j<\tau$ met both families, the triangle inequality would give

a contradiction. At the terminal tick, a common site would give $d(X_0,Y_0)\le 2r\tau$, also a contradiction.

This proposition certifies separation of complete propagated interfaces, not merely separation of the original name labels. For syntactic subcomputations with the corresponding ownership interfaces, it supplies the geometric disjointness needed to apply the selected calculus's disjoint-support results.

Let $Q$ be a $d$-dimensional quantum input, $d\ge 2$. Every channel from $Q$ to a finite classical record followed by a channel from that record to a quantum output is entanglement breaking. It cannot equal ${\mathrm{id}}_Q$, including on reference-entangled inputs. Its half-diamond distance from ${\mathrm{id}}_Q$ is at least $1-1/d$.

Proof. The composite has measure-and-prepare form

For an arbitrary reference, its output is a sum of positive product operators and is therefore separable.

Apply the channel to one half of $|{\Phi }_d⟩=d^{-1/2}{{\sum }_{j=1}}^d|j⟩|j⟩$. Every separable state has overlap at most $1/d$ with $|{\Phi }_d⟩⟨{\Phi }_d|$: this follows for pure product states from Cauchy–Schwarz and then for separable mixtures by convexity. Testing the Bell projector gives trace distance at least $1-1/d$ from the ideal output. The half-diamond distance dominates this particular referenced-input distance.

The general measure-and-prepare characterization is standard HorodeckiEB. The present application agrees with Cheung Landmark I prop:classical-recovery, En04-A thm:classical, and Hawk04-B thm:classical CheungI,CheungEn04,CheungHawk04B.

The proposition concerns a decoder whose entire operational input is the classical record and independent fixed ancillas. It does not apply to a receiver that also holds a quantum system entangled with a system at the sender. That additional system is exactly the resource used by teleportation.

For an isometry $V:C\to R\otimes B$, let

The complete residual $B$ responds to the code when $\mathcal{C}$ is not a constant channel on the code. The residual must include all inaccessible systems and preparation purifications belonging to the stated dilation.

Every finite-dimensional isometry $V:C\to R\otimes B$ has a CPTP left inverse on its complete output. This does not imply recovery from $R$, recovery from $B$, or constancy of either complement.

Proof. Choose a state ${\tau }_C$ and define

Both summands are completely positive and their traces sum to $\mathrm{Tr}X$. Thus $\mathcal{L}$ is CPTP and $\mathcal{L}(V\rho V^{†})=\rho$, including references.

For a counterexample to residual constancy, take $V|\psi ⟩=|0{⟩}_R\otimes |\psi {⟩}_B$. The accessible output is constant and the residual is the identity channel. Complete restoration uses $RB$ and returns a new code output; it does not assert that the code was relocated into either chosen output by the restore map.

The distinction is the one emphasized in Cheung En04-A and Hawk04-A CheungEn04,CheungHawk04A.

In the counterexample, place the input and its unchanged residual at a site outside the scheduled past of a readout of $R$. The residual responds to the input while the readout law on $R$ remains constant. The cone theorem does not require all systems outside its readout cone to be input independent. Nor does nonconstant residual response establish geometric backreaction.

Fix sites $P,Q$ with $\ell =d(P,Q)>r$. Their ordinary blocks $b_P,b_Q$ are distinct. Replace them by

discarding empty sets. This is a partition.

Channels on the shrunken blocks must be specified anew; a channel on $b_P$ does not automatically restrict to one on $b_P\setminus \{P\}$. Channels of unaffected blocks are retained.

The exceptional block may have a complete instrument supported on its two-site interface. It is record transmissive or name transmissive when it admits the corresponding lossless one-tick transport. It is inert when its complete channel is ${\Phi }_P\otimes {\Phi }_Q$, with no cross-site relocation or classical control.

An inert exceptional block can be refined into the two singletons $\{P\},\{Q\}$ with the same complete update. The refined dynamics is admissible in the original metric $d$.

Proof. The factorization identifies the exceptional update with the product of the two singleton channels. All remaining blocks have diameter at most $r$.

The constant channel $\rho \mapsto |{\Phi }^{+}⟩⟨{\Phi }^{+}{|}_{PQ}\mathrm{Tr}\rho$ has a non-product output but transmits no input dependence in either direction. Transmission must be established by an operational witness or a failure of reduced-channel factorization, not by block diameter or non-product structure alone.

Define

This is the shortest-path metric obtained by adjoining an undirected edge of length $r$ between $P,Q$. It is integer valued, has finite balls, and all repartitioned blocks have $d_W$-diameter at most $r$.

Proof. Represent the original metric by the complete weighted graph with edge lengths $d(x,y)$. The triangle inequality allows every path not using the added edge to be shortened to one original edge. A shortest path uses the new edge at most once, since a repeated use contains a positive-length loop that can be removed. This gives the displayed formula and proves that it is a metric.

A $d_W$-ball is contained in a union of three finite $d$-balls: one about its center and ones about $P,Q$, with radius no larger than the specified ball radius. Finally $d_W\le d$ and $d_W(P,Q)\le r$.

All necessity statements below concern the original metric. The cone theorem still holds for the modified metric $d_W$.

A lossless one-tick record transport from $P$ to $Q$ gives a $P$-perturbation pair with readout laws ${\delta }_0,{\delta }_1$ at $(Q,t_0+1)$. Hence the original-metric signalling speed of the extension is at least $\ell >r$.

A lossless one-tick name transport has the same conclusion by transporting a computational-basis bit and reading it at $Q$.

Proof. Prepare the two possible bit values at $P$, use the same transport channel in both runs, and apply the common terminal readout at $Q$. Initial snapshots agree away from $P$. The signal arrives in one tick over original-metric distance $\ell$.

In the exceptional-block model there are runs with identical initial states and identical channels on all original-metric inner blocks for $(Q,t_0+1)$, but different readout laws at that event. In both runs the non-selective quantum marginal at $Q$ is $I/2$.

Proof. Prepare a Bell pair with halves at $P,Q$, using a declared prehistory before the exceptional update is activated. In the exceptional block, measure the $P$-half destructively and deliver its record $b$ to $Q$. Use the $Z$ basis in the first run and the $X$ basis in the second. These are admissible choices of complete exceptional-block instruments.

The resulting record–qubit states at $Q$ are

Their quantum marginals are both $I/2$, but ${\omega }_Z\ne {\omega }_X$.

Read $b$, measure the qubit in the $Z$ basis with outcome $a$, and output $a\oplus b$. The law is ${\delta }_0$ on ${\omega }_Z$ and uniform on $\{0,1\}$ on ${\omega }_X$. The exceptional block is not original-metric inner because $d(P,Q)>r$; all inner channels and all initial data agree.

The theorem violates original-metric autonomy, not locality relative to the complete operational input. The arriving record is part of that input.

For any finite extension made of complete instruments and wire relocations:

1. the input-to-readout-law map is affine;

2. equal complete operational input states give equal laws for a fixed readout;

3. no deterministic program universally clones unknown pure states.

Proof. (a) follows from linearity of complete channels and classical marginalization. For (b), a fixed readout is a channel on its accessed quantum and classical factors.

For (c), let $\mathcal{C}$ be a hypothetical CPTP cloner. Use root fidelity $F(\rho ,\sigma )=\Vert \sqrt{\rho }\sqrt{\sigma }{\Vert }_1$. CPTP maps do not decrease fidelity Watrous. For two pure states with $0<f=|⟨\psi ,\varphi ⟩|<1$, cloning would give output fidelity $f^2<f$, contradicting monotonicity.

For universal qubit cloning there is also a direct affine proof: the two decompositions of $I/2$ into the $Z$ and $X$ basis ensembles would give different two-copy outputs. Landmark I uses that universal-input argument CheungI (thm:nocloning). The fidelity argument additionally explains why a physical deterministic map cannot exactly clone even a specified pair of nonorthogonal pure states WoottersZurek,Dieks.

Suppose the receiver at $Q$ holds one half of a Bell pair whose preparation and distribution were completed before the unknown input is supplied. Suppose also that:

1. a Bell-basis measurement at $P$ is available;

2. its two outcome bits can be delivered from $P$ to $Q$ in $m$ ticks;

3. the receiver can apply the corresponding Pauli correction.

If measurement and correction are each charged one separate tick, the unknown qubit is transported in $m+2$ ticks, including when it is entangled with an untouched reference. The transport is outside the original-metric cone whenever $\ell >r(m+2)$.

Proof. The Bell measurement has four equally weighted corrected branches. In each branch, the $Q$-half is $X^b{Z}^a|\chi ⟩$, up to the fixed Bell-basis phase convention. Delivery of $(a,b)$, followed by $Z^a{X}^b$, leaves $|\chi ⟩$ at $Q$. The corrected branch maps from input to output are $I/2$, up to phase; summing their four CP maps gives the identity channel. Therefore the calculation preserves every reference.

The charged stages are one measurement tick, $m$ communication ticks, and one correction tick. Comparison with the cone requires the stated distance inequality.

This is the standard teleportation resource identity BennettTeleport. If two bits can be batched, $m=1$; if only one bit per tick is available, $m=2$. If local wrappers and transport are admitted as one compound exceptional-block instrument, a different timing convention may charge fewer ticks, but that closure must be stated.

Bell-pair distribution requires a certified ordinary route. On a suitable ${\mathbb{Z}}^D$ schedule it can take $\lceil \ell /r\rceil$ ticks; an arbitrary metric substrate need not admit such a route. A constant-time transfer protocol is not automatically super-cone at every distance $\ell >r$ when its local processing stages are charged separately.

Let sites be embedded by $z:\Lambda \to {\mathbb{R}}^D$, and let $K\subset {\mathbb{R}}^D$ be a declared compact convex velocity set containing $0$. A dependency edge of displacement $\Delta$ and duration $u>0$ is physically contained when $\Delta \in uK$. A nonzero displacement at duration $0$ is not contained in this future cone.

If every edge of a complete dependency network is physically contained, every dependency path is contained in the same convex velocity cone. Operational influence obeys that containment.

Proof. For a path with edges $({\Delta }_j,u_j)$,

Operational influence is contained in scheduled ancestry by Corollary cor:influence.

This is the local-velocity and physical-containment language of Cheung VII-J CheungVIIJ. A transmissive exceptional block supplies an actual edge outside the declared physical cone when its displacement exceeds the admitted one-tick velocity set. An inert exceptional block instead admits the singleton refinement of Proposition prop:inert. It is not correct to assign an inert operation a physical transport velocity merely from the diameter of its conservative two-site envelope.

Let $Q$ be a finite strongly connected directed quotient graph. Vertices include register type and schedule phase. Every edge $e$ has a displacement ${\Delta }_e\in {\mathbb{R}}^D$ and duration $u_e\ge u_{\min }>0$. Its periodic lift permits translated copies of the transitions. Waiting without displacement is allowed.

For a simple directed cycle $c$, set

Define

Let $q=|V(Q)|$ and $D_{\max }={\max }_e\Vert {\Delta }_e{\Vert }_2$. Every dependency path completed within time $u$ satisfies

The set of limiting path velocities as elapsed time tends to infinity is exactly $K_{\mathsf{Sch}}$. This is a statement about potential scheduled reachability, not transmission by every path.

Proof. Remove directed cycles from a finite walk until the remainder has no repeated vertex. Decompose the removed cycles into simple cycles. The remainder has at most $q-1$ edges, so its displacement has norm at most $(q-1)D_{\max }$.

The cycle contribution is

Its total duration is no greater than $u$. Since $0\in K_{\mathsf{Sch}}$, the contribution belongs to $u{K}_{\mathsf{Sch}}$, proving (eq:finiteoffset). Dividing by elapsed time proves the upper inclusion for limiting velocities.

Conversely, approximate a convex combination of cycle velocities and $0$ by spending the corresponding time proportions on repeated cycles and waiting. Strong connectivity supplies fixed connector walks between their base vertices. Choose repetition numbers tending to infinity; connector durations and displacements become negligible. Integer repetition numbers approximate the desired time proportions, giving every velocity in $K_{\mathsf{Sch}}$.

The untimed periodic-graph mechanism is studied by Fritz Fritz. The timed finite-offset version agrees with Cheung VII-J thm:polytope CheungVIIJ. Without strong connectivity, the ordering of reachable strongly connected components must be retained; taking all cycle velocities in the graph can overstate attainability from a specified input.

Take two quotient states with displacements $L,-L$, each of duration one, and allow waiting. All cycles have zero displacement, so $K_{\mathsf{Sch}}=\{0\}$. A first transition still moves a dependency by $L$. The finite-time offset cannot be omitted.

In the cardinal transport model with displacements $\pm a{e}_j$, each of duration $a/c$, and waiting,

For a Euclidean unit direction $u$, its maximal scheduled radial speed is

In two dimensions it is $c$ on an axis and $c/\sqrt{2}$ on a diagonal.

Proof. The cardinal edge velocities are $\pm c{e}_j$. Their convex hull with $0$ is the ${\ell }^1$ ball of radius $c$. Time-sharing cardinal movements and waiting attains every point in that hull asymptotically. A vector $\rho u$ lies in the hull exactly when $\rho \Vert u{\Vert }_1\le c$.

For a prescribed partition schedule whose phases restrict the available movements further, its velocity set may be smaller. Equality above concerns the specified cardinal transport model, not every coordinate schedule.

Let $D\ge 2$. A fixed finite periodic transport model, rescaled with a fixed length-to-time ratio, cannot have a nonzero Euclidean ball as its limiting scheduled velocity set.

More generally, a family of compact velocity polytopes with uniformly bounded numbers of vertices cannot converge in Hausdorff distance to a nonzero Euclidean ball.

Proof. For the fixed model, uniform length and time rescaling leaves (eq:velocitypolytope) unchanged. It is a finite polytope. A nonzero Euclidean ball in $D\ge 2$ has infinitely many extreme points and is not a finite polytope.

For the second statement, let the vertex count be bounded by $M$. Pad each vertex list to length $M$ by repetition. Hausdorff convergence to a compact ball bounds all vertices. A subsequence has coordinatewise convergent vertex lists. Its limiting convex hull is a polytope with at most $M$ vertices and also equals the asserted Hausdorff limit, a contradiction.

This is an obstruction within a specified periodic class, not a nonexistence theorem for arbitrary refinements, nonperiodic substrates, or increasing directional complexity.

For a scaling family of periodic models, suppose

Then paths with elapsed physical times tending to $u>0$ have limiting displacements in $uK$.

Proof. Apply (eq:finiteoffset) in physical units and pass to the Hausdorff limit. The offset vanishes.

The corollary is an upper certificate. It does not supply a channel on each path, a lower relay bound, operational convergence, or frame covariance. If the quotient complexity grows, the physical offset must be controlled as well as the polytope shape.

Let a finite collection of fixed labelled channels have declared complete supports. Suppose a strict partial order makes all channels with intersecting supports comparable. Every legal linear extension has the same total channel.

Proof. Incomparable channels have disjoint supports and commute after extension by identities. Any two linear extensions of a finite partial order are connected by adjacent swaps of incomparable elements. Every swap preserves the channel composition.

Records and resource wires belong to the supports. The proposition does not move a gate before a gate that creates a record it reads, and it does not identify channels obtained by changing a geometric parameter. This is the fixed-channel retiming statement in Cheung VII-E and En03-A CheungVIIE,CheungEn03.

For $\phi :\Lambda \to \mathbb{Z}$, let

An edge from $(x,s)$ to $(y,s+1)$ has retimed duration $1+\phi (y)-\phi (x)$. The retiming is $r$-compatible if

Applying one bijection ${\beta }_{\phi }$ to a complete network does not create a directed graph cycle. For a spatial round trip, the sum of retimed durations equals the original total duration.

Proof. A graph isomorphism preserves directed cycles. On a spatial round trip, the correction terms $\phi (y)-\phi (x)$ telescope.

An $r$-compatible retiming can assign duration $0$ to an ordinary edge. Therefore it does not preserve a numerical bound of slope $r$ when one retains the old spatial metric and uses the new tick coordinate. The causal relation must be transformed with the coordinates. Negative or zero coordinate duration is not, by itself, a realizable feedback device.

We record the following hypothesis for analysis, not as a standing assumption of the discrete or continuum theorems.

Every $r$-compatible retiming image of every realizable transport step type is realizable as a device in one common substrate. Devices may be translated in tick time and finitely composed without additional compatibility restrictions. Ordinary waiting steps are available.

1. Every admitted finite wired device network has a well-founded linearization consistent with its input–output dependencies. Composition along any legal linearization gives the same complete outcome law.

2. If the network is calculus generated and well typed under Assumption ass:programinterface, its linearization is realized by a finite composite program with that denotation.

Every finite semantically admissible block network satisfies E7(a). Every finite syntactically admissible network satisfies E7(b).

Proof. Dependencies increase the original tick, so the finite graph is acyclic. Topological orderings exist. Within a tick, disjoint complete channels commute, giving ordering-independent composition. The syntactic statement is Proposition prop:sound.

Assume Hypothesis hyp:boost. Suppose lossless classical-bit transport is available in both directions between $P,Q$, in one original tick, with $\rho =d(P,Q)>r$. Then an admitted composition has a directed closed chain through one event.

Proof. Put $G=\lfloor \rho /r\rfloor \ge 1$ and define

If $d(x,y)\le r$, the two ceiling terms differ by at most one. Taking the positive part preserves that bound, so $\phi$ is compatible. Also $\phi (P)=G$, $\phi (Q)=0$.

The ${\beta }_{\phi }$-image of the forward transport and the ${\beta }_{-\phi }$-image of the reverse transport both have duration $1-G$. By the closure hypothesis, they are independently realizable in the common substrate.

Starting at $(P,s_0)$, use the forward image, wait at $Q$ for $2(G-1)$ ticks, and use the reverse image. The final event is $(P,s_0)$, because the total duration is

The devices are composed from two different retiming images; this is not the application of one coordinate change to an original network.

The new edges have nonpositive duration and nonzero displacement. They lie outside a future velocity cone; they are not inputs to the positive-duration velocity-polytope theorem.

A directed dependency cycle has no well-founded linearization. For lossless classical-bit legs, a NOT rule on one leg gives no pathwise consistent bit assignment; identity rules allow multiple assignments with the same external initial data.

Proof. A topological ordering of a directed cycle would require an event to precede itself. For the classical examples, the consistency equations are respectively $x=1-x$ and $x=x$. The first has no bit-valued solution; the second has both constant bit solutions. These are statements about pathwise assignments, not about selecting a stationary marginal as a substitute for an acyclic instrument execution.

Hypothesis hyp:boost and E7(a) are incompatible with any bidirectional lossless one-tick transport between distinct sites, even when their distance is at most $r$.

Proof. Let $P\ne Q$ be such sites. Define $\phi (P)=1$ and $\phi (x)=0$ for $x\ne P$. This function is $r$-compatible for every metric because every difference of values is at most one.

The ${\beta }_{\phi }$-image of the $P$-to-$Q$ step has duration zero. The ${\beta }_{-\phi }$-image of the $Q$-to-$P$ step also has duration zero. Unrestricted composition makes $(P,s)\to (Q,s)\to (P,s)$ an admitted dependency cycle. Proposition prop:cyclecontradiction contradicts E7(a).

The formal implication “super-cone bidirectional transport plus strong closure contradicts E7(a)” is valid, but it is not a selective physical exclusion: the same assumptions exclude the ordinary relay used for sharp attainment. Accordingly, unrestricted device retiming closure is not used as an additional axiom of an admissible relay substrate and is not included in the continuum existence conjectures.

A useful physical boost interface would need frame-labelled devices, admissible inter-frame compositions, and a causal-order-preserving notion of frame change. The fixed-channel ordering proposition supplies none of those data. The construction above also does not prove a general evasion theorem for one-way channels under unrestricted closure of an entire device class.

A scaling family consists of admissible substrates $({\Lambda }_{\epsilon },d_{\epsilon },r)$, site embeddings $z_{\epsilon }:{\Lambda }_{\epsilon }\to {\mathbb{R}}^D$, and physical tick durations $h_{\epsilon }\to 0$. Events embed as

Assume block footprints in physical spacetime have diameter tending to zero on bounded windows. This includes both input and output interfaces of each block firing.

For the cubic example,

Its metric envelope has physical slope $c$ in the ${\ell }^1$ norm.

Let $E_{\epsilon }$ be the embedded event set. For every compact $K\subset {\mathbb{R}}^{1,D}$,

This distance-function formulation avoids requiring $E_{\epsilon }\cap K$ to approximate every compact set, which would fail, for example, for a singleton not lying on a particular mesh.

For an embedded terminal event $q=(T{h}_{\epsilon },z_{\epsilon }(A))$, define the full metric past envelope

A finite experiment uses its restriction to the experiment's starting time.

There is a norm $\nu$ on ${\mathbb{R}}^D$ and $c>0$ such that, for

the distance functions to $J_{\epsilon }^{-}(q)$ converge to those of $J_{\nu }^{-}(q)$, uniformly for sources in a fixed compact set and embedded terminal events $q$ in a fixed compact set.

The uniformity is needed when terminal events or operation supports move within compact sets. It gives the positive-margin exclusion used below. The cubic example has $\nu =\Vert \cdot {\Vert }_1$, not the Euclidean norm.

C2 concerns a metric envelope. It does not assert that a given schedule realizes every direction on its boundary. Theorem thm:isotropyobstruction constrains actual scheduled reachability. The lower operational hypothesis C5 is therefore separate from C2.

Every reported finite experiment has a fixed finite outcome alphabet and includes its complete attempted-run law. Probabilities are compared in total variation on that common alphabet.

A modification footprint includes changed complete local rules and changed initial input sites. An initial modification confined to a region means that the two initial snapshots agree on every finite region disjoint from it. It does not mean that arbitrary correlated initial states with the same site labels are interchangeable.

There is a class $\mathcal{X}$ of limit experiments with declared compact operation and readout supports, satisfying:

1. Every experiment has a realizing family of finite discrete experiments whose complete outcome laws converge in total variation and whose operation footprints approach the declared supports.

2. Realization is compatible with finite disjoint composition and with the declared recorded continuations.

3. If two limit experiments differ only by a modification in a compact set $K^{\prime }$, they admit matched realizing families: the schedule and unchanged background are common, all discrete modification footprints approach $K^{\prime }$, and their initial snapshots agree outside the declared discrete initial modifications. Common readouts have common discrete realizations.

The same clauses may be required with an untouched reference as an additional operational interface.

Clause (iii) is essential. Separate convergence of two experiment laws, without preservation of their local agreement, is not enough to apply a discrete domain-of-dependence theorem.

A compatible family of finite discrete experiments with convergent declared supports in a fixed compact set and total-variation-convergent complete outcome laws determines an experiment in $\mathcal{X}$. Membership and identifications use the stated operational realization, not support alone.

C4 supplies membership of newly constructed convergent families. It is not a compactness theorem ensuring that arbitrary experiment families converge.

Assume C1–C3. Suppose two limit experiments have a common readout at $a$ and differ only by a compact modification footprint $K^{\prime }$ with

Then their readout laws agree.

Proof. Use C3(iii) to choose matched discrete realizations with terminal events $a_{\epsilon }\to a$. Uniform cone convergence and convergence of the modification footprints imply that, for sufficiently small $\epsilon$, every modification lies outside $J_{\epsilon }^{-}(a_{\epsilon })$, with a positive margin.

Because complete block footprints shrink, no changed block meeting the next cone slice is hidden outside the declared modification footprint. The discrete initial cone snapshots agree and the required inner block channels agree. Theorem thm:cone gives equal discrete readout laws. Their total-variation limits are equal by C3(i).

Assume C1–C3. Let two compact operation/readout regions be spacelike separated at positive margin:

Replacing one experiment by another in its region does not change the unconditioned joint record law of the other.

Proof. By compactness and continuity, every point of either region is at positive distance from the causal past of every point in the other. For sufficiently small meshes, uniform C2 excludes all dependency paths in either direction between the two complete discrete regions. The discrete schedule-locality argument applies to the full set of the other experiment's reported records, not only to each marginal record separately. C3(ii)–(iii) supplies matched composites. Pass to the total-variation limit.

Conditioning on a record in the remote region may change a conditional law; that does not contradict this unconditioned statement.

For every $\nu$-unit vector $v$ and duration $u>0$, there are matched discrete relay pairs whose sender modifications approach $A=(0,x)$, whose readouts approach $B=(u,x+cuv)$, and whose limits belong to $\mathcal{X}$. Their two complete readout laws converge in total variation and satisfy

for all sufficiently small meshes. The sender modifications can be localized in arbitrarily small neighborhoods of $A$.

Assume C1–C5. Define $c_{\mathrm{cont}}$ as the supremum of

over positive-duration signalling pairs of limit events, with sender modifications supported arbitrarily close to $A$. Then $c_{\mathrm{cont}}=c$.

Proof. If the ratio exceeds $c$, a sufficiently small neighborhood of $A$ is at positive distance from $J_{\nu }^{-}(B)$. Theorem thm:contDOD excludes signalling, so $c_{\mathrm{cont}}\le c$.

C5 gives pairs on the boundary. Total-variation convergence and the triangle inequality preserve a gap at least $\gamma$ in their limit laws. Thus $c_{\mathrm{cont}}\ge c$.

Suppose an ideal discrete relay pair has laws ${\delta }_0,{\delta }_1$. If its two implemented laws are each within total variation ${\eta }_{\epsilon }$ of the corresponding ideal law, then

For a finite composition of complete channels with local half-diamond errors ${\delta }_{\epsilon ,j}$, one may take ${\eta }_{\epsilon }\le {\sum }_j{\delta }_{\epsilon ,j}$.

Proof. The first claim is the triangle inequality. For the second, telescope the channel compositions one factor at a time. CPTP pre- and postcomposition contract diamond distance. The resulting total channel error is at most the sum of local errors; a terminal readout cannot increase it.

This is the complete-channel error accounting in Cheung En02-B thm:B6 CheungEn02B. For a fixed physical journey requiring $O({\epsilon }^{-1})$ ticks, per-tick error $o(\epsilon )$ is one sufficient condition for a vanishing total allowance. It is not necessary in every error model.

Ideal classical relays already have gap $1$ at every discrete scale. The open task is to realize the required directional endpoint families and compatible operational limits, or to control errors uniformly in an implementation. Discrete relay attainment alone does not establish those combined requirements.

For each scale there is a represented observable algebra ${\mathfrak{B}}_{\epsilon }$ containing local subalgebras associated with bounded spacetime operation regions. It has the following properties:

1. Generators assigned to two regions with no scheduled dependency path in either direction commute in the chosen complete observable realization.

2. There are linear comparison maps $j_{\epsilon }:{\mathfrak{B}}_{\epsilon }\to \mathfrak{B}$ into a common $C^{\ast }$-algebra. For the admitted uniformly bounded generator sequences and their finite products, these maps are asymptotically product preserving:

   $$\Vert j_{\epsilon }(a_{\epsilon }{b}_{\epsilon })-j_{\epsilon }(a_{\epsilon })j_{\epsilon }(b_{\epsilon })\Vert ⟶0.$$

   They also preserve the adjoint asymptotically. Admitted generator images converge in norm.

3. The norm limits generate isotone $C^{\ast }$-subalgebras $\mathfrak{A}(O)\subseteq \mathfrak{B}$. The operational laws extend to states on their generated algebra.

The realization must specify how allocation, consumed wires, discarded systems, and dilation ancillas are represented.

Commutation of two CP superoperators does not imply commutation of elements of an observable algebra, and a channel adjoint need not be multiplicative. C6 is an additional observable-level construction hypothesis. A common local unitary dilation can motivate such an interface, but that dilation and its spacetime localization must be provided. They are not constructed by the operational cone theorem.

Assume C1–C3 and C6. For bounded open regions $O_1,O_2$ whose closures are spacelike separated at positive margin,

Their admitted non-selective operations also obey operational no-signalling.

Proof. Uniform C2, the positive margin, and shrinking block footprints exclude scheduled dependency paths between the two regions at sufficiently small scales. By C6(i), corresponding discrete generators $a_{\epsilon },b_{\epsilon }$ commute.

Let their comparison images converge to $a,b$. Asymptotic multiplicativity gives

Finite polynomials in each set of generators commute with finite polynomials in the other. Norm continuity of multiplication extends commutation to the generated $C^{\ast }$-algebras. Operational no-signalling is Corollary cor:contNS.

The theorem is a transfer theorem under C6, not a construction of a quasi-local algebra or a quantum field theory from arbitrary block instruments.

The limit theory has a group $\mathcal{G}$ of bijections of ${\mathbb{R}}^{1,D}$ such that:

1. transformations carry admitted experiments and their supports to admitted experiments and preserve their complete outcome laws;

2. they preserve the order defined by $J_{\nu }^{-}$ in both directions;

3. they preserve calibrated spacetime volume in the fixed time and length units, and hence allow no change of the common scale;

4. they act transitively on future timelike inertial directions, represented by parallel classes of future timelike lines.

Clause (iii) is stronger than saying that no pure dilation is a group element. The latter alone would not exclude transformations that combine a dilation with a Lorentz transformation.

Let $D\ge 2$. Assume C1–C5 and C7, with Euclidean $\nu$. Every $g\in \mathcal{G}$ is an orthochronous Poincare transformation. The operational maximal speed $c_{\mathrm{cont}}=c$ has the same value in every admitted inertial frame.

Proof. Rescale the time coordinate to $ct$. The cone order is the Minkowski causal order. The Alexandrov–Zeeman theorem identifies its bijective order automorphisms, in spacetime dimension at least three, as

where $\lambda >0$ and $L$ is an orthochronous Lorentz transformation, possibly reversing spatial orientation Alexandrov,Zeeman.

Such a map multiplies spacetime volume by ${\lambda }^{D+1}$. C7(iii) therefore forces $\lambda =1$. C7(iv) connects the distinguished inertial direction with every admitted one. By C7(i), signalling pairs and their law gaps transform with the frame. Poincare transformations preserve the boundary cone, and Theorem thm:contspeed gives its attained maximal speed $c$.

For $D=1$, arbitrary increasing bijections of the two null coordinates preserve causal order, so the cited causal-automorphism reconstruction does not give the same rigidity Zeeman. Additional affine and calibration assumptions are required.

C7 is a covariance hypothesis of a limit theory, not unrestricted retiming closure of discrete devices. It is not affected by retaining the original tick grading in admissible discrete networks.

There exists a scaling family of admissible substrates and schedules satisfying C1–C2 with Euclidean limiting norm and admitting the directional matched relay families required by C5.

A family as in Conjecture conj:family can be equipped with realizations satisfying C3–C4 and C6, and its limit theory satisfies C7. Finite experiments in each admitted frame have complete acyclic instrument semantics.

Unrestricted device retiming closure (Hypothesis hyp:boost) is deliberately not included: Theorem thm:boostobstruction makes it incompatible with the ordinary bidirectional relay class.

Record-boost constructions, interleaving arguments, and Gromov–Hausdorff or reachable-set comparisons remain possible research proposals, not proofs of the conjectures. Their logical targets must be specified:

1. Anisotropy alone does not force a positive probability for a Born-null event. A record-boost contradiction needs an independently justified operational covariance interface.

2. Interleaving that remains within the declared block diameter still obeys Theorem thm:cone. A claimed violation must identify which locality or clock hypothesis changes.

3. Persistence of anisotropy can obstruct Euclidean frame covariance. It does not contradict no-signalling relative to the anisotropic cone itself.

Increasing directional or quotient complexity, or abandoning the fixed periodic class, may evade Theorem thm:isotropyobstruction. Neither evasion establishes the full limit package.

The proposal to check finite graphs with $N\le 20$ and one exceptional block remains a programme, not an execution claim. An exhaustive computation must specify:

1. a finite catalogue of graphs, schedules, horizons, and register interfaces;

2. a finite exact gate/instrument library, or a symbolic parameterization with proved algebraic coverage;

3. complete outcomes, failure flags, and controller supports;

4. exact or validated numerical arithmetic and its error bounds.

The bound $N\le 20$ alone does not make the class of all quantum instruments finite or justify an exhaustive enumeration.

For a fixed finite network, the factorization identity can be checked on a basis of matrix units in each classical block. For a chosen complementary state $\sigma$, form $K=R_Y\Phi E_{\sigma }$ and test

That is a channel identity, not a sample check on selected input states.

Suppose a complete implemented channel $\tilde{\Phi }$ satisfies $d_{\diamond }(\tilde{\Phi },\Phi )\le e$, where $\Phi$ factors through the initial cone snapshot. Two inputs with equal cone snapshots have implemented output reductions at trace distance at most $\min \{1,2e\}$. The estimate is reference stable when the corresponding joint cone–reference reductions agree.

Proof. The two ideal outputs agree. Insert that common output between the implemented outputs and use the triangle inequality. Each implemented–ideal discrepancy is at most $e$. Trace distance is at most one. The diamond allowance includes references.

For finite input dimension $d$, a normalized Choi comparison gives

A separate already-certified diamond allowance is added without another factor $d$. This is the certificate accounting emphasized in Cheung Hawk02-E and Hawk04-A CheungHawk02E,CheungHawk04A; see also Watrous.

Flagged truncations must remain deterministic. For a projection $P$, the map

is CPTP. For states commuting with $P$, the trace-distance error in a common enlarged output space is exactly the discarded probability. That equality is not asserted for arbitrary states with low–high coherences. Cheung En02-E gives this accounting for its specified Gaussian packet states CheungEn02E (thm:cutoff).

A finite-graph certificate is neither a continuum limit nor evidence that the operational, algebraic, or covariance construction gaps have been discharged.

Cheung VIII-B constructs a variable-deficit spatial-cone dictionary on its declared background CheungVIIIB (thm:dictionary). That cone is not the scheduled past defined in (eq:schedulepast), and its construction is not an Alexandrov–Zeeman reconstruction.

Cheung Em02-E gives a classical relational lapse on a declared background CheungEm02E (thm:lapse). It does not supply the frame group in C7 or a derivation of the metric from the hybrid input.

The characteristic-product programme in XIV-H–XIV-J is also separate. The earlier internal-amplitude stop in XIV-H is followed by the internal construction reported at I2 in XIV-I; the full joint characteristic map remains unresolved at I3 and J2 in the supplied releases CheungXIVH,CheungXIVI,CheungXIVJ. A finite channel implementation or a causal-cone proof cannot define that missing distributional operation. No inherited quantum coefficient is admitted by this manuscript.

The discrete results establish the following chain:

The explicit relay supplies the separate lower bound and makes the metric lattice speed exactly $r$ on the stated substrates. Neither scheduled ancestry nor the metric envelope is identified with the minimal influence region of every channel.

The continuum problem is not a second proof of the discrete cone. It is a construction problem involving isotropic reachability, matched operational limits, a product-preserving observable realization, and calibrated frame covariance. The velocity-polytope obstruction identifies why fixed finite periodic rescaling does not solve it.

All continuum conclusions remain conditional. The coordinate-retiming calculation does not establish physical boost closure, and the unrestricted closure analyzed here is incompatible with ordinary bidirectional relays. No invariant physical speed is claimed.

The following clauses describe the optional calculus interface. They are not premises of Lemma lem:onelayer, Lemma lem:dod, or Theorem thm:cone, which are channel theorems. A concrete calculus must discharge them on its own stated typing and evaluation domain.

E1: Termination.

Selected well-typed programs terminate. A strongly normalizing calculus supplies this at the reduction level; a finite acyclic instruction tree supplies it for the explicit fragment here. To obtain a finite evaluation tree from strong normalization one also uses the finite-branching evaluation interface. External idle ticks are not reductions.

E2: Confluence and disjoint commutation.

The complete denotation is independent of legal evaluation strategy. Disjoint enabled quantum operations commute. Global confluence must be supplied separately: commutation of only disjoint redexes does not logically imply confluence for every possible pair of enabled redexes. The source comparison is Landmark I thm:confluence; fixed-network channel ordering is the different statement in En03-A thm:retiming CheungI,CheungEn03.

E3: Internal no-signalling.

Complete programs on disjoint owned quantum and classical interfaces cannot change each other's unconditioned output law. The source comparison is Landmark I thm:internal-ns CheungI. On the substrate, complete ownership and controller support remain explicit modeling conditions.

E4: Records.

Destructive measurement, coherent recording, and surviving quantum interfaces are distinguished. The exact identity required here is Proposition prop:drop, with its context restriction. Compare Landmark I def:record and En03-A def:recording, def:nondestructive CheungI,CheungEn03. A record-only uniqueness or compatibility theorem is not an unrestricted residual-instrument theorem.

E5: Local complete-instrument semantics.

The program's first-order instrument depends only on its complete input state and is affine on unnormalized hybrid states. No-cloning and ensemble indistinguishability follow at that semantic level. Landmark I's lem:path, thm:adequacy, and thm:nocloning provide the source comparison CheungI. Its prop:classical-recovery is an additional classical-record reconstruction obstruction, not the statement of affinity itself.

E6: Mixed states and admitted adaptive control.

The instrument extends linearly to mixed inputs and tensorially to untouched references. Finite adaptive and randomized strategies are admitted when their complete classical histories and controls are included in the interface. Arbitrary state-inspecting or remote-history-inspecting schedulers are not covered. Proposition prop:path proves this extension for the explicit finite fragment; compare En04-A and Hawk04-B CheungEn04,CheungHawk04B.

The implementation results additionally require sequential and parallel compositionality, well-typed wiring, and sufficient declared routing capacity. These conditions are not consequences of the spatial cone. Landmark I's circuit adequacy concerns its available gate library; it is not an exact compilation theorem for every arbitrary block instrument.

The substantive corrections made here are:

1. Snapshots are unnormalized classical–quantum block states. Affinity is not assigned to normalized conditional states separately. All retained placement and controller information belongs to the snapshot.

2. The schedule past is distinguished from its metric envelope and from actual influence. Reference statements specify agreement of joint input reductions with the reference.

3. Destructive Enter is not identified with unrestricted coherent recording. The inaccessible measured factor and the allowed record contexts are explicit.

4. Soundness is a theorem about finite compositional program networks, not arbitrary driven runs with finitely many live names. Routing is required from every starting tick.

5. Strong normalization alone is not claimed to bound elapsed substrate ticks. The completion and routing hypotheses are retained.

6. Channels on blocks shortened during repartitioning are specified anew. They are not silently inherited by restriction.

7. Non-product exceptional operations are not assumed transmissive. Classical-record reconstruction and entanglement-assisted transport have different receiver interfaces.

8. Teleportation includes its Bell resource, communication capacity, and charged processing ticks. A constant protocol overhead does not prove super-cone transport at every distance greater than $r$.

9. Fixed-gate ordering invariance and coordinate/device retiming are separated. The floor-based cycle calculation is retained as a formal implication, while the stronger ordinary-relay incompatibility is proved and the physical retiming-stability claim is withdrawn.

10. Site homogeneity is not promoted to transitivity on all Euclidean directions. The explicit cardinal diamond supplies the directional comparison.

11. Local mesh approximation uses distance functions rather than intersections with arbitrary compact sets. Cone convergence is uniform on bounded source and target sets.

12. The continuum operational interface includes matched realizations that preserve local modifications and unchanged data.

13. Algebraic microcausality requires a localized observable realization and cross-product-preserving convergence. Commuting superoperators alone do not provide it.

14. Frame calibration excludes the scale factor of every causal automorphism, not merely pure dilation elements.

15. Isotropic and frame-covariant existence remain open. Anisotropic no-signalling is not treated as a contradiction.

Companion manuscripts are cited as source records for comparisons; their model-specific constructions are not transferred as premises. In particular:

• This paper does not alter the supplied characteristic-map, prepared-insertion, or bulk-restoration release boundaries. It does not open the downstream XV-B admission exit or close O.1.

• The internal amplitude construction reported in XIV-I is not the full joint map absent at I3 and J2. The historical H2 stop in XIV-H is not confused with that later, narrower advance.

• The supplied external mathematics catalogue is documentary only. Its labels concerning factoring, circuit complexity, classical capacity, or metric Alexandrov curvature are not a block-instrument compilation theorem, a continuum limit, or the Alexandrov–Zeeman causal-automorphism theorem.

• No proof-assistant verification, newly executed finite-graph enumeration, independently rerun numerical calculation, or successful pdflatex execution is asserted here. The source uses only standard packages and an internal bibliography.

Cheung085

N. Cheung, An Exact Causal Cone for Partitioned Enter-Calculus Dynamics: Domain of Dependence, Necessity, and a Conditional Continuum Limit. Supplied manuscript 085VvAv3 (1).tex, August 2026. Predecessor revised in the present manuscript.

CheungEC

N. Cheung, Enter Calculus. Figshare manuscript, version 2, 2026. DOI supplied with the predecessor: 10.6084/m9.figshare.33087515.v2.

CheungI

N. Cheung, Landmark I. Version 2, supplied companion manuscript I.tex. Finite Enter characterization and metatheory; especially thm:main, thm:confluence, lem:path, thm:adequacy, thm:nocloning, thm:internal-ns, and prop:classical-recovery. Source archive: https://enter.grok.me/.

CheungEn03

N. Cheung, En03-A. Version 1, supplied companion manuscript En03-A.tex. Kinematics and causal cuts on a prescribed schedule; especially lem:one-layer, thm:locality, cor:local-records, thm:retiming, and prop:directional. Source archive: https://enter.grok.me/.

CheungVIIJ

N. Cheung, Landmark VII-J. Version 1, supplied companion manuscript VII-J.tex. Scheduled influence, physical containment, and timed cycle-velocity polytopes; especially thm:nosignal, thm:schedule, thm:physical, thm:local-velocity, and thm:polytope. Source archive: https://enter.grok.me/.

CheungPsy04B

N. Cheung, Psy04-B. Version 1, supplied companion manuscript Psy04-B-v1.tex. Complete finite causal networks and scheduled support; especially thm:schedule and the influence-containment corollary. Source archive: https://enter.grok.me/.

CheungVIIG

N. Cheung, Landmark VII-G. Version 1, supplied companion manuscript VII-G.tex. Operational locality with explicit controller support; especially thm:locality. Source archive: https://enter.grok.me/.

CheungVIIE

N. Cheung, Landmark VII-E. Version 1, supplied companion manuscript VII-E.tex. Prescribed kinematics and fixed labelled-channel retiming; especially thm:retiming. Source archive: https://enter.grok.me/.

CheungEn04

N. Cheung, En04-A. Version 1, supplied companion manuscript En04-A.tex. Typed preparation, complete finite-program soundness, destructive readout, and complete residual response; especially thm:soundness, thm:drop, thm:classical, and def:response. Source archive: https://enter.grok.me/.

CheungHawk04A

N. Cheung, Hawk04-A. Version 1, supplied companion manuscript Hawk04-A.tex. Complete-output preservation, complementary channels, and finite channel certificates. Source archive: https://enter.grok.me/.

CheungHawk04B

N. Cheung, Hawk04-B. Version 2, supplied companion manuscript Hawk04-B.tex. Typed finite-program semantics, destructive readout, discard, and classical-record reconstruction; especially thm:adequacy, thm:drop, and thm:classical. Source archive: https://enter.grok.me/.

CheungEn02B

N. Cheung, En02-B. Version 1, supplied companion manuscript En02-B-v1.tex. Finite decoder accounting and compatible half-diamond error budgets; especially thm:B6. Source archive: https://enter.grok.me/.

CheungEn02C