En06-A: Causal Cones, Increasing Stencils, and Operational Transport A Corrected Layer Account of Partitioned Quantum Dynamics

We combine three supplied manuscripts on partitioned quantum dynamics with a preceding snapshot–channel comparison, separating proved discrete results, geometric scaling results, and additional operational and algebraic requirements. Complete hybrid instruments admit equivalent snapshot-factorization and Heisenberg-support descriptions. This gives an order-theoretic domain of dependence, its exact metric envelope, and invariance under legal reorderings of one finite causal network. Scheduled ancestry bounds influence but need not equal it.

We give a self-contained increasing-stencil construction and a uniform shape estimate

The changing word metrics therefore have Euclidean limiting cone envelopes, while admissible relays have limiting speed $c$ in every direction. For an explicitly defined, full-tail, fixed-wire outcome-law theory, matched modifications satisfy continuum domain of dependence, operational no-signalling, and sharp maximal speed $c$.

The supplied transport manuscript is not treated as an unchecked premise. Its common-final-tick readout convention is incompatible with its asserted boost action, and its spatial routing estimates do not establish the required spacetime support estimates. Additional normalization, scheduling, and algebraic issues are isolated. A corrected transport contract yields, conditionally, a Poincare action on the coarse relational theory in every $D\ge 1$. Observable-algebra microcausality requires a separate localized, product-preserving realization; commuting channels alone do not supply it. Unrestricted device resynchronization closure remains withdrawn because it excludes ordinary bidirectional relays. No instrument-level scaling limit, relativistic field theory, gravitational dynamics, or invariant physical speed is derived.

The supplied manuscripts address different objects:

CONE.

085VvBp1.tex studies finite instrument networks, the order-theoretic and metric cone, semantic refinement, foliation independence, and conditional continuum and dynamical statements CheungCONE.

STEN.

085VvBp2.tex proposes changing word metrics with increasingly dense directional stencils and supplies the main geometric construction CheungSTEN.

TRANSPORT.

085VvBp3.tex proposes a five-part transport argument, a relational outcome-law limit, a tail-limit algebra, and a concrete Poincare action CheungTRANSPORT.

En05-A.

The supplied comparison separates complete snapshots, channels, adjoint support, residual response, and device closure CheungEn05A.

These are source records, not automatic theorem premises. The results asserted here are either proved below or explicitly conditional on displayed hypotheses. In particular, a missing mathematical argument is not repaired by describing it as awaiting machine verification.

The designation of this synthesis is En06-A, superseding the working designation En05-B used in the supplied combination plan.

The following table is normative for the conclusions of this paper. “Conditional” means a mathematical hypothesis is still needed; it does not mean merely that a proof has not been machine checked.

{}1.12

The locality comparisons are with Cheung En03-A, VII-E, VII-G, VII-J, and Psy04-B CheungEn03A,CheungVIIE,CheungVIIG,CheungVIIJ,CheungPsy04B. They concern complete interfaces, prescribed supports, and unconditioned laws.

The optional program semantics are compared with Landmark I, En04-A, Hawk04-B, and En02-C CheungI,CheungEn04A,CheungHawk04B,CheungEn02C. Cheung's publisher record describes the Enter calculus as a linear calculus with an operational characterization of destructive measurement, separating the Born-equipped calculus from the measurement-characterization layer CheungEC. No syntax-specific metatheorem is needed to prove the geometric cone.

The finite-speed geometry is related to periodic-graph velocity polytopes Fritz, partitioned cellular automata ToffoliMargolus, and reversible quantum cellular automata SchumacherWerner. The present class includes irreversible instruments and does not inherit a reversible-QCA classification theorem.

We work with finite time windows and finitely many active registers. A countable site set may have an unlimited reserve of names, but every experiment declares the finite registers it actually uses. Multiple declared registers may occupy one site.

This is an explicit capacity convention. It is not the injective one-qubit-per-site convention of one supplied draft. Arbitrarily many names cannot be gathered into a uniformly bounded-size block under that injective convention.

At a site and tick, a finite hybrid observable algebra has the form

where $L_{x,s}$ contains classical record, controller, occupation, and placement labels. Different labels may have different quantum dimensions. An inactive reserved slot is not an accessible surviving measured system.

For finite $I\subseteq \Lambda$, set ${\mathcal{A}}_s(I)={⨂}_{x\in I}{\mathcal{A}}_{x,s}$. States are positive arrays

Compatible finite-region states suffice on a countable substrate; an unspecified infinite tensor-product density operator is not used.

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

Every outcome, including failure, belongs to this complete channel. Discard of a classical output sums blocks without normalization.

We use the usual finite-dimensional CP-map and channel formalism Watrous.

Every input used by a block is in its declared local interface: quantum systems, classical records, controller memory, random seeds, occupancy flags, and resource registers.

The complete map is fixed independently of the unknown input state. Adaptivity is classical blockwise control on those local inputs. An unrepresented global scheduler or remote controller is not admitted.

This is the controller-support discipline emphasized in VII-G CheungVIIG. Initially correlated resources are permissible in the general channel model only when their interfaces are declared. The product-default coarse model introduced later has a narrower preparation convention.

Let $R_{I,s}$ be restriction to the tick-$s$ interface in $I$: trace out external quantum systems and sum external classical labels. Define

For $p_{\lambda }=\mathrm{Tr}{\omega }_{I,\lambda }>0$, the conditional quantum state is ${\rho }_{I\mid \lambda }={\omega }_{I,\lambda }/p_{\lambda }$. No normalized state is assigned when $p_{\lambda }=0$.

The affine snapshot is the unnormalized block array $({\omega }_{I,\lambda }{)}_{\lambda }$. The normalized conditional states and their probabilities determine it, but conditional normalization is not itself affine. This corrects the snapshot wording in the supplied predecessors.

An observable supported in $I$ is extended by identity on the complement.

Let $\Phi$ be a complete channel between finite tensor-product hybrid interfaces. Fix 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 on $I$ give equal output snapshots on $Y$, for all input states.

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

The factorization is stable under tensoring with an untouched reference. To infer equality of joint output–reference states, the two inputs must agree on their joint $I$–reference reductions.

Proof. (i) implies (ii). For (ii) implies (i), choose a fixed complementary state $\sigma$, put $E_{\sigma }({\omega }_I)={\omega }_I\otimes \sigma$, and define $K=R_Y\Phi E_{\sigma }$. This is a complete channel. Condition (ii) identifies $R_Y\Phi (\omega )$ with $K{R}_I(\omega )$ on all states, and therefore on the whole linear input space.

Taking adjoints gives (iii). Conversely, (iii) makes the expectation of every output observable on $Y$ a function of the snapshot on $I$; this is (ii).

Tensor the linear channel identity in (i) with a reference identity. The resulting factorization involves $I$ together with the reference, which proves the final statement.

This is the common mathematical form of En05-A lem:comparison, the factorization interface in VII-J thm:nosignal, and the adjoint interface in En03-A lem:one-layer CheungEn05A,CheungVIIJ,CheungEn03A. The proof here does not import any of those manuscripts.

A substrate is $(\Lambda ,d,r)$, where $\Lambda$ is countable, $d$ is integer valued with finite balls, and $r\ge 1$. A schedule $\mathcal{P}=({\mathcal{P}}_s{)}_s$ partitions $\Lambda$ at each tick into finite blocks $b$ with ${\mathrm{diam}}_{d}b\le r$. Tick $s$ maps its input interface to the tick-$(s+1)$ interface.

A semantically admissible update is

where every factor is a complete local channel satisfying Assumption ass:control. Names and records may be relocated only within the block. Allocation and record creation have declared local output interfaces.

The radius bounds one block update, not the number of gates evaluated inside it. No hidden gate-depth or uniform storage bound is imposed.

For finite $Y$, set

The reduced output channel on $Y$ depends only on the block channels meeting $Y$.

Proof. The complement of $U_s(Y)$ is a union of blocks. The output observable is the identity on those blocks. Their adjoints are unital, since their complete channels are trace preserving. They contribute only input identities. The remaining factors act inside $U_s(Y)$.

Define backward sets by

For a finite complete wired instrument network, the output reduction on terminal ports $Y$ factors through the initial reduction on their scheduled ancestors:

Changing complete gates outside the ancestor subnetwork does not change that reduction, provided the wiring and the included gates agree.

For a partitioned schedule, the initial ancestor region is $N_{t_0}(Y)$ from (eq:scheduledpast). No metric is required for the network statement.

Proof. Pull an output observable backward in reverse topological order. If it is the identity on all outputs of a gate, unitality makes it the identity on that gate's inputs. Otherwise its possible support expands only to the declared input predecessors. Include all classical control, outcome, and resource ports in this graph.

At the input cut the support lies in the scheduled ancestors. Lemma lem:comparison gives the channel factorization. Excluded gates act on identity factors, so replacing them by any complete gates on the same interfaces does not change the pullback. For a partitioned schedule the same induction is precisely (eq:scheduledpast).

Multiple readout ports, including ports acquired at different ticks, are covered by this finite-network statement. Report ports may be retained as formal outputs for calculating their joint law; this does not make their information available to a physical remote controller.

If operational influence means a change of unconditioned output law under a trace-preserving local intervention, then

The inclusion need not be equality.

Proof. An intervention outside the ancestors preserves their reduced state, even for entangled initial states. Apply Theorem thm:order. For strict inclusion, declare a memory input but return the constant answer $0$. The memory is scheduled but not influential.

This is the distinction in Psy04-B and VII-J CheungPsy04B,CheungVIIJ. Selective normalization and common-cause correlations are not the interventions quantified here.

The triangle inequality gives

For terminal site $A$, let $I_s=B_d(A,r(T-s))$. If a block meets $I_{s+1}$, then it lies in $I_s$: for $z\in b\cap I_{s+1}$ and $w\in b$,

Two runs on the same schedule have equal terminal snapshots at $(A,T)$ if:

1. their initial snapshots agree on $B_d(A,r(T-t_0))$;

2. their block channels agree at each tick $s<T$ on every block contained in $I_s$.

Every common terminal readout then has the same law. The result is independent of all data and gates outside the cone, including gates on straddling blocks.

Proof. Every ancestor block lies in the corresponding $I_s$, by backward induction using the preceding block geometry. Equation (eq:metricenvelope) places the initial ancestors in the initial ball. Restrict the equal snapshots to that ancestor region and apply Theorem thm:order.

Equivalently, there are complete affine maps

determined only by the relevant inner block channels. This is the snapshot version of Lemma lem:onelayer.

On a fixed schedule, an $x$-perturbation pair has equal initial snapshots on all finite regions excluding $x$, and equal block channels on all blocks excluding $x$. Let $c_{\mathrm{lat}}(\mathcal{P})$ be the supremum of $d(x,A)/\tau$, $\tau \ge 1$, over pairs that change a common readout at $(A,t_0+\tau )$. For a schedule class $\mathfrak{S}$, take the supremum over $\mathcal{P}\in \mathfrak{S}$. An empty positive-distance supremum is $0$.

Every range-$r$ schedule satisfies $c_{\mathrm{lat}}(\mathcal{P})\le r$.

Proof. If $d(x,A)>r\tau$, the initial cone and every inner block exclude $x$. Theorem thm:metric forces equal readout laws.

Every linear extension of the dependency partial order of a finite complete wired network gives the same complete instrument and the same joint recorded-history law.

Proof. Incomparable events have disjoint current wire interfaces. Their extensions by identities commute, including on entangled inputs. Any two linear extensions of a finite partial order are connected by adjacent swaps of incomparable events. Every swap preserves the complete composition.

The statement compares one labelled network under legal orderings. It does not remove dependencies on records, change its metric, or produce boost-related devices. Execution-dependent descriptions of the order of intermediate events should not be identified as the same ordered trajectory; the complete channel and joint labelled records are the invariants.

Local finiteness and acyclicity alone do not give a finite ancestral calculation or a well-founded initial-value problem. The chain $\mathbb{Z}$ with edges $j\to j+1$ is locally finite and acyclic but has infinite past at every vertex. The generalization of Theorem thm:order requires a finite ancestral network relative to an input cut, or a separately proved infinite-network construction and convergence theorem.

For a measured register $S$ and surviving register $T$, destructive computational readout is

The measured names are consumed; $K$ is a classical record.

including every untouched reference.

If $W|x{⟩}_S=|x{⟩}_S|x{⟩}_K$, then

Proof. Expand ${\rho }_{ST}={\sum }_{x,y}|x⟩⟨y{|}_S\otimes {\rho }_{xy}$. The first partial trace gives ${\sum }_x{\rho }_{xx}$. The second gives ${\sum }_x|x⟩⟨x{|}_K\otimes {\rho }_{xx}$. The calculation is unchanged by tensoring with a reference identity.

Coherent recording with an accessible measured factor is not destructive Enter. Deferred readout is valid only for a stated record-compatible context. Operational discard is not physical annihilation. These distinctions agree with Landmark I, En04-A, and Hawk04-B CheungI,CheungEn04A,CheungHawk04B.

A finite acyclic instruction tree built from declared allocations, gates, destructive readouts, classical control, and discard has path maps $K_h$ with

Its reported selective maps are

Its complete denotation is CPTP and reference preserving.

Proof. A complete history fixes the classical choices. Compose the corresponding allocation isometries, gates, and measurement bras. Branch normalizations cancel against branch probabilities. At a readout, ${\sum }_x{M}_x^{†}{M}_x=I$; gates and allocations are isometric. Backward induction over the finite tree proves the completeness identity. Group histories by reported output and tensor the path maps with a reference identity.

This is the comparison with En04-A thm:soundness, Hawk04-B thm:adequacy, and En02-C thm:C1 CheungEn04A,CheungHawk04B,CheungEn02C.

A block is calculus generated only if a well-typed program over the available library denotes its complete instrument. A finite network of such blocks realizes a composite program only when sequential composition, parallel ownership, and input/output wiring match. Those are additional syntactic requirements. Strong normalization concerns reductions of a fixed term, not externally inserted idle ticks or an indefinitely driven substrate.

No exact finite-gate compilation of arbitrary specified instruments, no efficiency claim, and no support-preserving normalization of arbitrary adaptive wiring are asserted here.

A quantum-to-classical channel followed by a decoder whose only nontrivial input is that record is entanglement breaking. It cannot recover every unknown quantum state with its reference.

Proof. The composite has form

With a reference, its output is a sum of positive product operators, hence separable. The identity channel preserves a maximally entangled input, so the channels cannot agree.

Compare Landmark I prop:classical-recovery and the standard measure-and-prepare characterization CheungI,HorodeckiEB. A receiver holding an additional entangled quantum resource has a different interface.

For an isometry $V:C\to R\otimes B$, residual response means non-constancy on the code of $\mathcal{C}(\rho )={\mathrm{Tr}}_R(V\rho V^{†})$. The complement includes all inaccessible systems and preparation purifications of the stated dilation.

A finite isometry has a CPTP left inverse on its complete output. This does not imply recovery from a chosen component or constancy of its complement.

Proof. For a state ${\tau }_C$, define

It is CP and trace preserving, and $\mathcal{L}(V\rho V^{†})=\rho$. For $V|\psi ⟩=|0{⟩}_R\otimes |\psi {⟩}_B$, the $R$-channel is constant and the $B$-channel is the identity.

A responding residual outside the cone of a constant readout is not a signalling path to that readout. The restore map returns a code from the complete output; it does not establish that the code is already in a selected residual. Nor is residual response geometric backreaction; a same-model geometric dictionary would be a separate construction CheungEn04A,CheungHawk04A.

Fix $P,Q$ with $d(P,Q)=\ell >r$. Remove $P,Q$ from their ordinary blocks, discard empty remainders, and insert the exceptional block $\{P,Q\}$. Channels on the shortened blocks must be specified anew; a channel on the original larger block does not automatically restrict to them.

1. If the exceptional channel factorizes as ${\Phi }_P\otimes {\Phi }_Q$, with no cross relocation or classical flow, it can be replaced by singleton channels. The original-metric cone continues to apply.

2. Lossless one-tick bit transport from $P$ to $Q$ violates the original-metric radius-$r$ cone.

3. The geometry change alone neither creates nonlinearity nor permits universal deterministic cloning.

Proof. (a) is equality of the two product updates. For (b), prepare bit $0$ or $1$ at $P$, transport it with the same rule, and read at $Q$. The laws are ${\delta }_0,{\delta }_1$.

For (c), finite compositions of complete instruments are linear on unnormalized hybrid states. They remain CPTP after ignored systems are traced out. Using root fidelity $F(\rho ,\sigma )=\Vert \sqrt{\rho }\sqrt{\sigma }{\Vert }_1$, CPTP maps do not decrease fidelity Watrous. A deterministic cloner of two nonorthogonal pure states of overlap $0<f<1$ would change fidelity from $f$ to $f^2<f$, impossible.

A non-product channel need not be transmissive: a constant Bell-pair preparation has non-product output but no input dependence.

The extension has an exact cone for the shortened metric

This is the shortest-path metric obtained by adding an edge of length $r$. A shortest path need use that edge at most once; finite $d_W$-balls are contained in a finite union of finite $d$-balls.

Start with a Bell pair at $P,Q$. In one exceptional update, measure the $P$-half in $Z$ or $X$ and deliver its bit $b$ to $Q$. The joint record–qubit states are

Both quantum marginals are $I/2$, but the joint states differ. Measure the qubit in $Z$, obtaining $a$, and report $a\oplus b$. The laws are respectively ${\delta }_0$ and the uniform law. This violates original-metric autonomy, not affinity or equality of laws on equal complete inputs.

A bit-only long-range primitive can also assist quantum teleportation when the receiver holds a previously distributed Bell half. With one measurement tick, $m$ bit-communication ticks, and one correction tick, the charged transport duration is $m+2$. It is outside the original cone only if $\ell >r(m+2)$. A Bell-distribution prehistory, channel capacity, and local gate availability must be declared. An $O(1)$ protocol overhead does not prove super-cone transport at every distance $\ell >r$.

No metatheorem about an arbitrary new syntax is inferred merely from these physical-channel facts.

Two obstructions concern different objects and should be stated separately.

Let $S\subset {\mathbb{Z}}^D\setminus \{0\}$ be finite, symmetric, and generating. Write

If $L={\max }_{s\in S}\Vert s{\Vert }_2$, then

for every integer $k\ge 0$.

Proof. Symmetry gives $0\in C_S$, proving the first inclusion. For $z\in k{C}_S$, Caratheodory's theorem gives $z/k={{\sum }_{i=1}}^{D+1}{\lambda }_i{s}_i$, with nonnegative weights summing to one. This representation follows by eliminating positive weights along affine dependences until at most $D+1$ remain. Set $m_i=\lfloor k{\lambda }_i\rfloor$ and $y={\sum }_i{m}_i{s}_i$. Then ${\sum }_i{m}_i\le k$, so $y\in B_S(k)$, and

The case $k=0$ is immediate.

Under a fixed positive physical length-to-time normalization, the large-scale word-metric cone of a fixed finite $S$ has a cross-section proportional to $C_S$. For $D\ge 2$, it is not a nonzero Euclidean ball.

Proof. Divide Lemma lem:filling by $k$ and let $k\to \infty$. The error tends to zero. The limiting body is a finite polytope; a Euclidean ball in $D\ge 2$ has infinitely many extreme points.

A common input-independent mixture of admissible schedules retains the metric upper envelope, provided the two compared runs agree for each schedule realization and include the relevant control data.

This does not prove that the actual influence region of every averaged dynamics equals the envelope. The identity dynamics has no positive-distance influence. Randomization may suppress signals or alter typical propagation; it does not change a separately defined word-metric envelope. This is the corrected reading of STEN's schedule-averaging claim.

Let $Q$ be a finite strongly connected quotient graph. Each edge has displacement ${\Delta }_e$ and duration $u_e>0$; translated copies of its lift and waiting are allowed. For a simple directed cycle $a$, put

With $q=|V(Q)|$ and $D_{\max }={\max }_e\Vert {\Delta }_e{\Vert }_2$, every path completed within time $u$ satisfies

Its limiting velocity set is $K_{\mathsf{Sch}}$.

Proof. Remove cycles from a walk until the remainder has no repeated vertex. It has at most $q-1$ edges. The removed cycles decompose into simple cycles, and their total displacement is a duration-weighted sum of their velocities. Since $0\in K_{\mathsf{Sch}}$, that contribution belongs to $u{K}_{\mathsf{Sch}}$.

Conversely, repeat cycles in desired time proportions and use fixed connector walks, available by strong connectivity. Connector costs are negligible along increasingly long repetitions; waiting realizes the zero contribution. This approximates every convex combination.

This is the comparison with Fritz and Cheung VII-J Fritz,CheungVIIJ. A finite-time jump is not bounded by the asymptotic polytope alone.

For the cardinal model with velocities $\pm c{e}_i$,

A Euclidean unit direction $u$ has radial speed $c/\Vert u{\Vert }_1$: in two dimensions it is $c$ on axes but $c/\sqrt{2}$ on diagonals. Compare En03-A prop:directional and VII-E's directional comparison CheungEn03A,CheungVIIE.

A fixed finite periodic model therefore has its own non-Euclidean attainability obstruction. It is not identical to an arbitrary schedule's metric envelope. If quotient complexity grows, the finite-time offset must be controlled along with the polytope shape.

We now prove the geometric component proposed in STEN. The metric changes with the mesh; the theorem is not about one fixed substrate.

For $0<\delta \le 1/2$ and $L\ge 4\sqrt{D}/\delta$, a $(\delta ,L)$-stencil is a finite set $S\subset {\mathbb{Z}}^D\setminus \{0\}$ such that:

1. $S=-S$ and $S$ contains $\pm e_i$;

2. $\Vert s{\Vert }_2\le L$ for every $s\in S$;

3. directions of generators with $\Vert s{\Vert }_2\ge (1-\delta )L$ form a $\delta$-net of the unit sphere.

Such stencils exist, with at most $C_D{\delta }^{-(D-1)}+2D$ generators.

Proof. Choose a symmetric $\delta /2$-net $\{u_i\}$ of the sphere. For $D\ge 2$, a maximal separated set gives the stated count by packing disjoint spherical caps; for $D=1$, the sphere has two points.

Round $(L-\sqrt{D})u_i$ to an integer vector $s_i$. The error is at most $\sqrt{D}/2$, so

These vectors are long and nonzero. For $\rho =L-\sqrt{D}$, normalization changes direction by at most $\sqrt{D}/\rho \le \delta /2$. Together with the net error this gives a $\delta$-net. Adjoin negatives and the unit generators.

Choose sequences

and corresponding stencils $S_n$. Set

The word metric is

and the combinatorial block radius is $r=1$.

The physical word ball is

It is not $h_n{B}_{d_n}(0,k)$; the latter would rescale twice.

The one-tick reach is at most ${\epsilon }_n$, not necessarily exactly ${\epsilon }_n$, since the stencil need not contain a vector of length exactly $L_n$. Accordingly, $c$ is an exact upper normalization and an asymptotically attained speed.

Proof. The upper inclusion follows from the length bound. For unit $u$, choose a long generator of direction within ${\delta }_n$ of $u$. Then $\Vert s/L_n-u{\Vert }_2\le 2{\delta }_n$, so the support function of $P_n$ in direction $u$ is at least $1-2{\delta }_n$. The supporting-half-space description of a convex body gives the lower inclusion. Radial projection gives the Hausdorff bound.

For every $n$, $k\ge 0$, and $x\in {\Lambda }_n$,

In particular, the second estimate tends uniformly to zero for $k{\epsilon }_n\le R<\infty$.

Proof. Apply Lemma lem:filling with maximum generator length at most $L_n$, then multiply by $h_n$. This gives (eq:shapepoly). Translation invariance and Lemma lem:polyapprox give

Use the triangle inequality.

The additive rounding error is independent of $k$. The anisotropy term grows as $k{\epsilon }_n$; the whole error is not independent of $k$.

One tick reaches $h_n(S_n\cup \{0\})$, not every lattice point in ${\epsilon }_n{P}_n$. For example, with $S=\{\pm 1,\pm 3\}$, the point $2$ lies in $\mathrm{conv}(S)$ but is not reachable in one word step.

Every generator edge extends to a partition into generator pairs. Every finite word path can therefore be relayed one edge per tick by admissible partitions.

The schedule class has $c_{\mathrm{lat}}=1$. Its maximal Euclidean signalling speed is

Proof. For a generator $s\ne 0$, the graph of edges $z\leftrightarrow z+s$ is a disjoint union of bi-infinite chains, the cosets of $\mathbb{Z}s$. Choose one representative for each coset and write its sites as $z_0+ks$. Pair $2k+p$ with $2k+p+1$, $p=0,1$. Either phase is a full partition.

Choose the phase containing each desired path edge and let other blocks act identically. Connectedness and finite balls on the infinite word graph give a vertex at every integer distance, so geodesic bit relays attain the combinatorial upper bound $1$.

Every block has physical diameter at most $h_n{\max }_s\Vert s{\Vert }_2$. The same one-layer argument bounds influence by that reach per tick. A record transport along a maximum-length edge attains the corresponding one-tick ratio. Long generators give the displayed lower bound.

For a finite declared register collection, choose sufficient local capacity and route names one at a time using unit-generator edges, keeping already delivered names stationary. This gives a routing construction in the schedule class. It does not show that every arbitrary fixed schedule routes.

A fair fixed schedule that repeats both matching phases of every generator is tail routing, but may have a different physical throughput. The distinction between schedule-class attainment and a fixed schedule's speed remains essential.

A useful explicit nested family is, for sufficiently large $n$,

Then $h_n=8^{-n}$, the spatial and tick grids are nested, and $L_n{\epsilon }_n\to 0$. Nesting the site sets does not nest the generators or identify their dynamical channels.

Use the spacetime comparison metric

This is a topology and error metric, not a Lorentz-invariant metric.

Events embed as $(s\Delta t_n,x)$. Define the full discrete metric past of an apex $a_n=(T_n\Delta t_n,A_n)$ by

Finite experiments use its truncation below their starting tick.

Local convergence means that, on every compact comparison set, points of either set have distance tending to zero from the other full set. It does not require approximating points to remain inside the same compact set.

The event grids become locally dense. If $a_n\to a=(t_a,x_a)$, then

locally, uniformly for apexes in bounded sets.

If $e_n=d_c(a_n,a)$, then the excess of $J_n^{-}(a_n)\cap K$ over $J_2^{-}(a)$ is at most $e_n$. For a bounded $K$, the reverse excess is at most

for a fixed sufficiently large $R_K$ and all large $n$.

Proof. The grid $\Delta t_n\mathbb{Z}\times h_n{\mathbb{Z}}^D$ approximates every point with error at most $\max ({\epsilon }_n/2,\sqrt{D}{h}_n/2)$ in $d_c$.

The exact upper word-ball containment gives $J_n^{-}(a_n)\subseteq J_2^{-}(a_n)$. Translate a point by $a-a_n$; it then lies in $J_2^{-}(a)$, at distance at most $e_n$.

Conversely, translate $p\in J_2^{-}(a)\cap K$ by $a_n-a$ to a point $p^{\prime }\in J_2^{-}(a_n)$. Round its time downward to the tick grid. The time error is less than $\Delta t_n$, and its spatial coordinate lies in the Euclidean ball of radius $k{\epsilon }_n=c(T_n\Delta t_n-s\Delta t_n)$. For bounded $K$ and bounded apexes, these radii are bounded by a fixed $R_K$. Theorem thm:shape gives a word-ball point with spatial error at most $(D+1){\epsilon }_n+2{\delta }_n{R}_K$. Together with the translation and time rounding this gives the bound.

Thus the corrected geometric requirements C1 and C2 are theorems. The condition involving event grids intersected with every compact set in CONE is too strong: a singleton absent from a given grid cannot be approximated by that grid's intersection with the singleton.

For the increasing-stencil family,

Every scheduled dependency path of $k$ ticks has Euclidean displacement at most $ck\Delta t_n$.

Proof. A distance-one displacement is a generator displacement. Sum the one-step bounds and use the triangle inequality.

This is a bound on potential information paths as well as relocations of a named object. A geometric shape theorem and an operational path attainment statement remain logically distinct.

This section provides an unconditional operational result without using the disputed general transport theorem.

At scale $n$, an admitted experiment has:

1. a finite deterministic schedule of active singleton or generator-pair blocks, with identity completion;

2. finite declared quantum and classical interfaces, deterministic wire routing and allocation interfaces, and complete locally controlled payload instruments;

3. a product default state on all declared initial registers, with no external shared random seed or correlated undeclared resource;

4. a fixed finite tuple of labelled report ports, each acquired by a local readout at its own tick and site;

5. a nonempty compact declared support $K_n$ containing all active input/output block footprints and all retained wire paths and dwell segments between their uses.

The readout acquisition times need not coincide. A report retained solely to calculate the joint law is not an operational communication wire. If used downstream, its physical wire must be declared.

The skeleton consists of everything except the locally controlled payload maps. The complete report law is ${\mu }_n$.

This is a deliberately restricted class. Adaptive payloads are included; arbitrary outcome-dependent wiring is not silently assumed to have a support-preserving fixed-wire compilation.

A limit object is $\xi =(K,R,\mu )$, where $K$ is compact, $R$ is a finite tuple of labelled acquisition events in $K$, and $\mu$ is a law on fixed finite alphabets.

It belongs to $\mathcal{X}$ if experiments exist at every sufficiently large scale with

Two objects differ by a modification in compact $K^{\prime }$ when matched full-tail families have the same skeletons and report designations, and payloads differ only on block footprints in $(K^{\prime }{)}_{{\rho }_n}$, with ${\rho }_n\to 0$. The initial default state is common.

The theory $\mathcal{T}$ consists of these objects, this witnessed modification relation, admitted disjoint composites, and conditioning of laws at positive-probability events.

The relation is not determined by schedule membership alone. It requires matched local interventions.

If a matched modification pair has

then its two complete report laws agree.

Proof. Every discrete dependency path obeys the physical bound in Proposition prop:C2b. Report acquisition events converge to $R$, and modified footprints approach $K^{\prime }$. The positive margin therefore places every modified gate outside the ancestor subnetwork of the report ports for all sufficiently large $n$. Their initial inputs and included gates agree. Apply Theorem thm:order to the full report tuple, then pass to the total-variation limit.

The same proof applies to any subset of report ports. It proves equality of their joint marginal, not merely equality of each one-port marginal.

Let ${\xi }^1,{\xi }^2\in \mathcal{X}$ have compact supports satisfying

for some $\delta >0$. Their independent composite belongs to $\mathcal{X}$, with law ${\mu }^1\otimes {\mu }^2$. Each marginal is independent of the choice of the other experiment. More generally, modifications outside the past of a chosen report subset leave that subset's joint marginal unchanged.

Proof. The inequality gives positive separation of spatial shadows. Hausdorff convergence makes the active site sets of the two families disjoint for all large $n$. Their independent product-default preparations, complete channels, and report ports factor. The product laws converge because

For the marginal statement apply Theorem thm:coarseDOD to that report subset.

Microcausality does not require spacelike outcomes to be independent. Entangled background states can correlate them. The product conclusion above uses the independent product-default experiment class; it is not a derivation of a product vacuum in a quantum field theory. The general channel no-signalling theorem permits entangled inputs.

If a realized support is the union of two positively spacelike separated compact components, with one report in each, its law factorizes in this product-default class. At small mesh, no block or retained wire can connect the components, so the finite circuit splits. This is the restricted product-rigidity comparison with TRANSPORT.

For every Euclidean unit $v$, duration $\sigma >0$, and source position $x$, there are matched full-tail experiments with source modifications approaching $A=(0,x)$, report events approaching $B=(\sigma ,x+c\sigma v)$, and laws exactly ${\delta }_0,{\delta }_1$ at every scale. Their declared supports converge to the closed segment from $A$ to $B$.

Proof. Choose a long generator $s_n$ with direction within ${\delta }_n$ of $v$, a source site $x_n\to x$, and $k_n$ with $k_n\Delta t_n\to \sigma$. A singleton source payload writes bit $0$ or $1$. Relay the bit along $k_n$ successive $s_n$-edges using the matching phases of Proposition prop:matchings, then read it locally. The constant finite number of preparation/readout ticks adds $O(\Delta t_n)$, which vanishes.

The displacement is $k_n{h}_n{s}_n$, with length between $(1-{\delta }_n)k_n{\epsilon }_n$ and $k_n{\epsilon }_n$, and direction tending to $v$. It tends to $c\sigma v$. The footprints and paths lie within vanishing neighborhoods of the segment, and all source differences lie in a shrinking neighborhood of $A$. The report laws remain deterministic and distinct.

Define $c_{\mathrm{coarse}}$ as the supremum of $\Vert x_B-x_A{\Vert }_2/(t_B-t_A)$, $t_B>t_A$, over matched modification signalling pairs localized arbitrarily near $A,B$. Then

Proof. For a ratio greater than $c$, sufficiently small neighborhoods of $A$ have positive distance from the past of neighborhoods of $B$. Theorem thm:coarseDOD excludes a law difference. Theorem thm:relays attains the limiting boundary.

Thus the geometric relay membership problem is solved for this defined full-tail coarse class. This does not prove existence of an instrument-level realization satisfying the stronger diamond-norm C3 of CONE.

For positive-probability $E$, total-variation convergence implies

For example, if $d_{\mathrm{TV}}({\mu }_n,\mu )={\eta }_n<\mu (E)/2$, the conditional distance is at most $4{\eta }_n/\mu (E)$. This is convergence of conditional laws, not a deterministic physical postselection protocol.

Conditioning a product law on equality of two independent fair bits creates correlated bits. It need not be realizable with the same disconnected support in the product-default class.

Finite outcome simplices and the hyperspace of nonempty compact subsets of a compact metric space are compact. They give subsequential support–law limits Burago. They do not, without more work:

• fill a subsequence into a family at every fine scale;

• preserve matched modifications and composition;

• provide common quantum input/output embeddings;

• establish diamond-norm convergence.

Full-tail filling is one purpose of the conditional transport contract below. It is not obtained merely by renaming all subsequential limits as experiments.

TRANSPORT proposes the following chain CheungTRANSPORT:

The parts have different mathematical content.

For a fixed-wire experiment at scale $j$, let $e=(s,b)$ and $e^{\prime }=(s^{\prime },b^{\prime })$, $s^{\prime }<s$ excluded, be consecutive participations of one register, with $s^{\prime }>s$. Let $m(b)$ be the midpoint of a pair or the site of a singleton. Then

The midpoint segment is within $O({\epsilon }_j)$ of the declared worldline-closed support.

Proof. The register is static at a site $p$ between its participations. Its deterministic output position from $b$ and input position in $b^{\prime }$ are both $p$. Thus $p\in b\cap b^{\prime }$. Each midpoint is within ${\epsilon }_j/2$ of $p$. The time difference is at least one tick.

The retained dwell segment at $p$ belongs to the declared support. Interpolating the midpoint offsets bounds the midpoint segment's distance from that dwell segment by at most ${\epsilon }_j$.

An orthochronous Poincare transformation preserves the future-causal endpoint displacement. Its Lipschitz constant in $d_c$ bounds the image support error. This proves the geometric comparison, not the scheduler estimate.

Suppose a fixed finite complete payload network is realized on new carriers with the same wire graph and interfaces. Transport blocks act only as wire-address changes. At each payload event the original complete map is applied under the corresponding tensor-factor relabelling.

Then the realized network has the original complete channel under the external-interface relabelling, including every reference.

Proof. Choose a topological ordering of the original network. The new realization respects its dependencies. Initially identify corresponding wire spaces by a tensor relabelling. At every payload event, conjugacy of its local map carries the identified intermediate state to the next identified state. Address changes do not alter internal logical states. Induct over the finite network, retaining every outcome branch. Theorem thm:foliation removes the choice of legal ordering. Tensor the calculation with a reference identity.

This is a channel-network simulation statement. It is comparable with En02-C's reference-preserving soundness CheungEn02C, but does not assert availability or compilation of arbitrary payload instruments over a hardware library.

The limit class defined literally in TRANSPORT, whose designated readouts all occur at one final tick, is not closed under boosts. This fails already for two independent deterministic readouts.

Proof. Every realizing family in that class has one acquisition time $a_n=t_n^{+}\Delta t_n$ for every designated readout. If two readout events converge, their limiting time coordinates must therefore agree.

Two singleton deterministic readouts at $r_1=(0,0)$ and $r_2=(0,L{e}_1)$, $L>0$, are realizable, with law ${\delta }_{(0,0)}$. A boost of nonzero velocity $v$ in direction $e_1$ gives time coordinates $0$ and $-\gamma vL/c^2$, which differ. The transformed object has no realizing family satisfying the common-final-tick definition.

Holding earlier records until a final tick preserves values, but changes the acquisition events and adds physical storage segments. It cannot prove covariance of the originally labelled spacetime data. Definition def:experiment removes this obstruction by allowing distinct actual acquisition times.

A route lying exactly on a spatial segment, with endpoints fired at their correct nominal times and no late arrival, need not stay near the intended spacetime segment.

Proof. In one dimension take $0<v<c$, nominal endpoints $(0,0)$ and $(T,vT)$, and intended path $x=vt$. Travel instead at speed $c$ until $q=(vT/c,vT)$, then wait until $T$. The spatial route is exactly $[0,vT]$, and the destination is reached early, not late.

Write $\ell =vT$. The distance in $d_c$ from $q$ to the intended segment is

Indeed minimizing $\max (c|t^{\prime }-\ell /c|,|\ell -v{t}^{\prime }|)$ balances the two terms at $t^{\prime }=2\ell /(c+v)\in [0,T]$. The positive distance does not disappear when the digital mesh tends to zero.

This example applies to the image of a long idle source worldline. The supplied endpoint-lateness induction does not bound departure times, distributed waiting, or the spacetime location of early arrivals. A pacing estimate must be supplied.

Propositions prop:simultaneous and prop:pacing are not absence-of-machine-checking observations. They identify an incompatible definition and an insufficient estimate. Additional problems with lane separation, charging, and local inner simulation are documented in Appendix app:audit. They do not establish that every alternative transport construction is impossible.

For the fixed-wire experiment class of Definition def:experiment, assume the following.

For each $g\in {\mathcal{P}}^{\uparrow }$ there are numbers $a_j(g)\to 0$, independent of the complexity and payloads of a source skeleton at scale $j$. For every source experiment $\xi$ at that scale, and every sufficiently large target scale $m$, there is a transported experiment with:

1. exactly the same complete report law and labelled alphabets;

2. declared support at Hausdorff distance at most $a_j(g)+b_{\xi ,g,m}$ from $g{K}_{\xi }$, and each acquisition event within that bound of its corresponding $g$-image, where $b_{\xi ,g,m}\to 0$ as $m\to \infty$ at fixed $\xi ,g$;

3. a skeleton and routing plan independent of payloads;

4. matched transports of payload-modification pairs, with differences confined within the same error of the transformed modification footprints.

The spacetime bound covers all paths, waits, allocation interfaces, and retained record storage, not just spatial routes and meeting endpoints. The identity case is required as well.

This contract is not asserted as a theorem for the full class. The source's desired form $a_j(g)\le C_g{\epsilon }_j$ is one sufficient choice. The fine error may depend on the finite source complexity, because the diagonal argument fixes a source before refining it.

Assume T. Suppose experiments ${\xi }_j$, on an infinite sequence of source scales, satisfy

in Hausdorff, event, and total-variation topology. For every $g\in {\mathcal{P}}^{\uparrow }$, there is a full-tail realizing family for $(gK,gR,\mu )$. The identity case supplies full-tail filling.

Proof. For each source scale $j$, choose an increasing threshold $m(j)$ so that all target scales $m\ge m(j)$ admit its transport and have $b_{{\xi }_j,g,m}\le 2^{-j}$. For $m(j)\le N<m(j^{\prime })$, with $j^{\prime }$ the next source scale, use the transport of ${\xi }_j$ at scale $N$. As $N\to \infty$, the selected $j\to \infty$. Its support error is $a_j(g)+2^{-j}\to 0$, acquisition events converge to $gR$, and its law is ${\mu }_j\to \mu$.

No uniform bound on the fine error across increasing source complexities is hidden here. Uniform vanishing of the coarse error is the essential separate requirement.

The group ${\mathcal{P}}^{\uparrow }$ consists of affine translations and orthochronous Lorentz transformations for the declared cone of speed $c$, including spatial reflections and excluding time reversal.

Assume T. Then

defines an action of ${\mathcal{P}}^{\uparrow }$ on $\mathcal{T}$, for every $D\ge 1$. It preserves realizability, outcome laws, witnessed modification relations, spacelike independent composites, and conditioning of laws.

Proof. Lemma lem:diagonal gives realizability of $g_{\#}\xi$. Apply it simultaneously to matched families using T3–T4 to preserve the modification relation.

A fixed Poincare map preserves strictly spacelike separation. On compact positively separated supports, continuity gives a positive transformed margin. Independent composites of the transformed objects exist by Proposition prop:coarseNS; their product law is the transformed product law.

Conditioning acts on the unchanged law. On object data, $(gh{)}_{\#}=g_{\#}{h}_{\#}$, and $g_{\#}^{-1}$ supplies the inverse. Preservation of the existential relations in both directions follows by applying T to the inverse map.

Under T, the maximal signalling speed of $\mathcal{T}$, expressed in every admitted inertial coordinate frame, is $c$. This concrete conclusion holds for every $D\ge 1$.

Proof. The action preserves signalling relations and law gaps. It maps future-causal and boundary-null separations to future-causal and boundary-null separations. Positive-duration super-cone signalling is excluded and boundary relays are present in every transformed direction by Corollary cor:coarsespeed and the action.

A limit operational theory admits a group of spacetime bijections that:

1. preserves admitted experiments, their supports, their operational relations, and complete laws;

2. preserves Minkowski causal order in both directions;

3. preserves the calibrated spacetime volume in the fixed time and length units;

4. acts transitively on future timelike inertial directions.

For $D\ge 2$, C7${}^{\mathrm{cal}}$ implies that every group element is an orthochronous Poincare map. If the theory also has the sharp boundary relays, its maximal operational speed is frame invariant.

Proof. The causal-automorphism theorem gives $g(p)=a+\lambda Lp$, $\lambda >0$, with $L$ orthochronous Lorentz Alexandrov,Zeeman. Its spacetime volume multiplier is ${\lambda }^{D+1}$. Calibration forces $\lambda =1$. Operational preservation and transitivity then transfer the upper bound and its boundary attainment between admitted frames.

Merely excluding pure dilations as group elements is not the same as fixing the scale factor of every causal automorphism. The calibrated clause makes the intended requirement explicit.

For $D=1$, causal order alone admits independent increasing reparametrizations of the null coordinates Zeeman. The concrete conditional action theorem does not require that classification, because its group is Poincare by construction.

The conditional action in Theorem thm:action is an action on supports, labelled acquisition events, laws, and the stated relations. From it alone one cannot infer:

1. a limit action on quantum output instruments or residual states;

2. a limit action by automorphisms of a dynamical observable net;

3. isometries between the microscopic word metrics;

4. closure of a physical device class under unrestricted inter-frame composition;

5. an identification of $c$ with a measured physical constant.

Proof. The defined objects contain none of the extra data in (i)–(iv). For (i), the channels $\rho \mapsto |0⟩⟨0{|}_K\otimes \rho$ and $\rho \mapsto |0⟩⟨0{|}_K\otimes \tau \mathrm{Tr}\rho$ have the same record law but different quantum outputs and complements. Thus their equality after forgetting the output data cannot recover those data.

At each finite scale the word-metric stable body is a finite polytope, not the Euclidean ball; a generic rotation does not preserve it. A Lorentz transformation preserves the declared limiting Minkowski quadratic form, but no map on the microscopic metrics has been constructed.

Finally, construction (eq:family) is possible for every chosen $c>0$ by selecting $\Delta t_n={\epsilon }_n/c$. The formalism does not select a physical time calibration or a unique dimensional speed.

At each scale the chosen dynamics has a distinguished tick foliation and generally anisotropic directional reach. This is compatible with a conditional coarse action in the limit. Neither such an action nor the geometric construction is a spacetime emergence theorem.

For a nested family, let

The tensor product is the quasi-local inductive limit over finite register sets BratteliRobinson. The inclusions of the scale storage algebras are unital $\ast$-monomorphisms. This is a valid algebraic construction.

It does not identify Heisenberg observables at different scales, or prove that moving report effects have norm limits.

Choose distinct register factors ${\rho }_n$ and $p_n=|0⟩⟨0{|}_{{\rho }_n}$, extended by identity. The product-default state gives ${\omega }_0(p_n)=1$ for every $n$, but