En05-B: An Isotropic Matched Operational Continuum Limit for Partitioned Quantum–Classical Networks

We construct a deterministic family of countable, locally finite metric substrates with one fixed partition schedule at each scale. The construction separates transport lanes from junction interfaces. Every junction is updated in one block, whereas the lane interiors are updated by disjoint nearest-neighbor pairs. Increasing directional resolution therefore does not introduce a direction-by-direction service delay.

Under one physical clock, every dependency edge has Euclidean speed at most a prescribed $c>0$. Uniform estimates on the entire substrate, including intermediate lane vertices, prove convergence of both the metric past envelopes and the scheduled pasts to the Euclidean cone. Routing has vanishing excess time uniformly on bounded journeys and from every starting tick. Classical relays have signalling gap one, and quantum transport is the logical identity channel, including arbitrary untouched references.

Every finite acyclic causal circuit of finite-dimensional complete hybrid instruments has an exact logical realization on sufficiently fine substrates. Local replacements admit matched realizations with the same schedule, preparations, routes, and unchanged channels. A fixed-skeleton operational completion is constructed; evaluation, finite composition, and matched locality extend continuously to it. Its compatibility notion retains local interfaces and intervention slots and is stronger than convergence of one terminal probability law.

The construction proves the isotropic operational existence statement for this explicitly declared finite-experiment class. It does not construct a quantum field theory, a product-preserving spacetime observable algebra, Lorentz covariance, or gravitational dynamics. No uniform degree, block-cardinality, interface-dimension, or computational-efficiency bound is claimed.

The exact causal-cone theorem of En05-A separates three objects: scheduled ancestry, its metric envelope, and actual operational influence CheungEn05A. Its continuum conclusions require additional realization hypotheses. In particular, an upper envelope does not establish boundary transport, and separately convergent experiments need not have locally matched realizations.

The immediate construction problem is therefore:

Construct a family whose Euclidean upper cone and attainable transport cone converge to the same cone, while preserving complete finite experiments and local interventions.

A fixed finite periodic transport model cannot solve the isotropy problem by uniform rescaling alone: its asymptotic velocity set is a polytope Fritz. The cardinal instance is the diamond discussed in En03-A, VII-E, VII-J, and En05-A CheungEn03,CheungVIIE,CheungVIIJ,CheungEn05A. Here the directional set, the quotient size, and the microscopic interfaces are allowed to grow.

scheduling mechanism. A junction is not permitted to participate in several blocks at one tick. Instead, every incident lane has its own endpoint site. All endpoint sites belonging to a junction, together with its memory site, form one small-diameter junction block. The interiors of different lanes are disjoint. Consequently all directions can be served in parallel during a transport frame.

This construction uses the permission in En05-A to admit arbitrary finite complete block channels without a uniform bound on block cardinality or within-block gate depth. It would not establish an existence theorem with those additional bounds.

is completed. The paper supplies:

1. a precise experiment and comparison interface;

2. an increasing-direction geometry with uniform metric estimates;

3. a legal partition schedule and a quantitative routing theorem;

4. direction-uniform classical boundary relays;

5. reference-preserving quantum identity transport;

6. matched realizations of finite adaptive experiments;

7. compositional convergence and a local operational completion.

is not imported. The increasing-stencil and transport proposals mentioned in the supplied research plan are not theorem premises. The geometry, schedule, and charging argument used below are specified and proved here. Companion manuscripts are cited for comparison and documentary alignment, not substituted for missing construction steps.

A causal finite circuit and its local operational completion are defined precisely in Sections sec:experiments and sec:completion.

Fix $c>0$. There are countable substrates $({\Lambda }_n,d_n,1)$, injective location maps $z_n:{\Lambda }_n\to {\mathbb{R}}^2$, physical tick durations $h_n\to 0$, and state-independent periodic partition schedules such that:

1. every block has $d_n$-diameter at most one and a physical spacetime footprint tending to zero;

2. every scheduled dependency path satisfies the exact endpoint bound

   $$\Vert z_n(y)-z_n(x){\Vert }_2\le c(T-s)h_n;$$

3. the metric past envelopes and scheduled pasts converge to the Euclidean past cones, in the uniform local distance-function sense;

4. for every Euclidean unit vector $v$, every $u>0$, and every $x\in {\mathbb{R}}^2$, matched classical relay endpoints converge to

   $$(0,x),(u,x+cuv),$$

   and their readout laws have total-variation distance one;

5. transport of every fixed finite-dimensional quantum register is the identity after logical identification, including every untouched reference;

6. every causal finite circuit in the declared class has a realizing family whose logical complete channels equal its circuit channel exactly, on all sufficiently fine scales;

7. local replacements have matched realizing families with common schedule, resource preparation, routing, unchanged block channels, and common readouts;

8. evaluation, admitted finite composition, and the matched-locality relation extend to the fixed-skeleton operational completion.

The limit operational maximal signalling speed is $c$.

The completion statement concerns the compatibility class explicitly defined here. It is not a completeness theorem for arbitrary growing discrete networks known only through one convergent terminal law.

Proof. Items (i)–(iii) are proved in Theorems thm:metric, thm:routing, and thm:cones. Items (iv)–(v) are proved in Theorems thm:identity and thm:relay. Items (vi)–(vii) are proved in Theorems thm:realization and thm:matched. Item (viii) is proved in Theorems thm:completion and thm:completionmatch. The speed conclusion is Corollary cor:maxspeed.

A finite local interface has observable algebra

where $L$ is finite and every ${\mathcal{H}}_{\lambda }$ is finite dimensional. Its state space is represented by unnormalized positive blocks:

The blocks include classical outcomes, occupation flags, and controller records. Tensor products give multi-site interfaces.

A complete instrument is a finite family of CP maps $\{{\Phi }_k{\}}_k$ whose sum is trace preserving. Its recorded channel is

Failure outcomes are included. Classical discard sums blocks without conditional normalization. We use the standard finite-dimensional CP-map framework Choi,Watrous.

Every quantum or classical factor read or written by a block channel belongs to that block's declared interface. This includes controller memory, records, random seeds, program counters, and resource registers. The channel is fixed before the unknown input is supplied.

This is the controller-support convention of VII-G and En05-A CheungVIIG,CheungEn05A. A schedule or routing plan chosen from an unknown state description is not admitted.

For finite region $I$, write $R_I$ for restriction to its complete interface, tracing quantum factors and summing classical blocks outside $I$. The reduced state $R_I\omega$ is its snapshot.

We use

For finite hybrid interfaces the diamond norm is the corresponding completely bounded trace norm, allowing arbitrary reference extensions.

For a complete finite channel $\Phi$, input region $I$, and output region $Y$, the following are equivalent:

1. $R_Y\Phi =K{R}_I$ for a complete channel $K$;

2. equal input snapshots on $I$ give equal output snapshots on $Y$, for every pair of input states;

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

The factorization is stable under tensoring with an untouched reference identity.

Proof. The first assertion implies the second. Conversely choose a fixed complementary state $\sigma$, put $E_{\sigma }({\rho }_I)={\rho }_I\otimes \sigma$, and define $K=R_Y\Phi E_{\sigma }$. The second assertion gives $R_Y\Phi (\rho )=K{R}_I(\rho )$ for every state. States linearly span the finite hybrid operator space, so this is a linear-map identity. Taking adjoints proves the third assertion. Conversely, the third assertion says that every output expectation depends only on $R_I\rho$, which proves the second.

Tensoring the linear-map identity with ${\mathrm{id}}_R$ proves the reference claim. Equality on $I$ alone does not imply equality jointly with a reference; the appropriate joint reductions on $IR$ must agree.

Let $d$ be an integer-valued graph metric with finite balls. At tick $s$, a partition ${\mathcal{P}}_s$ consists of finite blocks of diameter at most one. The update is

All outputs remain inside their block. On a countable substrate this product is understood through its action on finite observables and compatible finite-region states.

For finite terminal region $Y$ at tick $T$, set

The terminal reduced channel factors through the initial snapshot on $N_s(Y)$, and

If two runs have the same schedule, equal initial snapshots on $N_s(Y)$, and equal channels on the blocks entering this backward recursion, their terminal snapshots agree.

All statements are reference preserving when the corresponding joint initial reductions agree.

Proof. An output observable on $Y$ is the identity on blocks disjoint from $Y$. Their channel adjoints are unital, so they contribute only identities. The remaining adjoints act on the union of blocks meeting $Y$. Iterating proves the support inclusion and the dependence only on the indicated channels. Each backward layer adds distance at most one. Lemma lem:comparison gives factorization and the reference statement.

An unconditioned trace-preserving local intervention outside the scheduled ancestors of $Y$ cannot change its output law. Thus

Equality need not hold.

Proof. Such an intervention preserves the ancestors' reduced state, also jointly with an untouched reference. Apply Lemma lem:locality. For strict inclusion, a channel may have a declared input port but return a constant classical answer independently of that input.

These proofs reproduce the comparison needed from En03-A, VII-J, Psy04-B, and En05-A CheungEn03,CheungVIIJ,CheungPsy04B,CheungEn05A. No absence of entanglement across a cone boundary is assumed.

Work in calibrated length units. For integers $n\ge 4$, define

Here ${\delta }_n$ is the hub mesh, ${\ell }_n$ the maximum nominal lane length, $a_n$ the physical allowance for one graph edge, and ${\xi }_n$ a small separation allowance.

Let

Both orientations occur. The vectors $e_1,e_2$ occur, ensuring connectivity.

The normalized directions $p/\Vert p{\Vert }_2$, $p\in P_n$, have angular covering error tending to zero. One sufficient bound is

Proof. For unit $v$, round $(K_n-1)v$ to a nearest integer vector $p$. The rounding error has Euclidean norm at most $\sqrt{2}/2<1$. Thus $p\ne 0$, $\Vert p{\Vert }_2\le K_n$, and its component in direction $v$ is at least $K_n-2$. The transverse component has norm less than one. Its angular error is therefore at most $\arctan (1/(K_n-2))$, which is bounded by the displayed quantity.

For every $k\in {\mathbb{Z}}^2$, introduce a hub site $H_k$. For every $k\in {\mathbb{Z}}^2$, $p\in P_n$, and $i=0,\dots ,N_n$, introduce a lane site

All these labels denote distinct sites.

The lane indexed by $(k,p)$ has graph edges

Its nominal endpoints are the hubs $H_k$ and $H_{k+p}$, but its endpoint sites are not identified with the hubs.

At hub $H_k$, define its junction set

Add all graph edges between distinct sites of $C_k$, making it a clique. Different junction sets are disjoint.

Let ${\Lambda }_n$ be this graph's vertex set and $d_n$ its unweighted shortest-path metric.

The graph is countable and connected. Its degree is finite and bounded at each fixed $n$, and its metric balls are finite. Every junction clique is finite and has diameter one.

Proof. There are countably many labels. An interior lane site has degree two. A junction has $1+2S_n$ sites, so its sites have at most $2S_n+1$ neighbors, including a possible lane neighbor. The presence of $e_1,e_2$ connects all hubs, and every lane site connects to its endpoint hubs.

A bounded-degree graph has finite balls by the finite branching bound on walks of any fixed length. The clique assertion follows from its definition.

Give each hub the location

The nominal location of $L_{k,p,i}$ is

There are offsets $b_{p,i}\in {\mathbb{R}}^2$, with $\Vert b_{p,i}{\Vert }_2<{\xi }_n$, such that

is injective on all sites, remains equivariant under hub translations, and satisfies

Proof. There are finitely many lane types $(p,i)$. Choose their offsets successively in the open ball of radius ${\xi }_n$. For a new type, equality modulo the lattice ${\delta }_n{\mathbb{Z}}^2$ with a hub or a previously chosen type excludes only a countable set of offsets. An open ball is not exhausted by that set. The resulting map is injective and translation equivariant.

A lane edge has length at most

All sites of $C_k$ are within ${\xi }_n$ of ${\delta }_{n}k$, except the hub itself, which is exactly there. A clique edge has length at most $2{\xi }_n\le a_n$.

The physical drawing need not be a planar graph: lane crossings do not create additional graph edges. This is an explicit modelling choice, not an implicit crossing interaction.

Abbreviate

Then

For $k,k^{\prime }\in {\mathbb{Z}}^2$, put $L=\Vert {\delta }_n(k^{\prime }-k){\Vert }_2$. There is a hub route using $m$ stencil steps, with

Every hub on the route is within ${\rho }_n$ of the straight segment joining the two hubs.

Proof. If $k=k^{\prime }$, use no steps. Otherwise put $w=k^{\prime }-k$ and choose

Round each $jw/m$, $j=0,\dots ,m$, to a nearest integer vector $k_j-k$, retaining the exact endpoints. Consecutive differences satisfy

Delete zero differences. Every remaining difference belongs to $P_n$. The number of steps does not increase. The rounding error at each hub is at most ${\rho }_n$ in physical units.

For hubs $H_k,H_{k^{\prime }}$,

Proof. The lower bound follows by summing (eq:edgeLipschitz). For a route with $m\ge 1$ lanes, use one clique edge to enter the first lane, $N_n$ edges on each lane, one clique edge between consecutive lanes, and one clique edge to finish at the last hub. This uses

graph edges. Apply Lemma lem:hubroute. The equal-hub case is immediate.

For every pair of sites $x,y\in {\Lambda }_n$,

Consequently, for displacement at most $R$,

uniformly over all such sites.

Proof. The lower bound again follows from (eq:edgeLipschitz). Every site can be joined to one endpoint hub using at most $N_n+1=F_n$ edges. Choose such hubs $H,H^{\prime }$ for $x,y$. Then

and the corresponding physical distances are also at most $s_n$. The hub distance is therefore at most $\Vert z_n(x)-z_n(y){\Vert }_2+2s_n$. Lemma lem:hubmetric gives

which is (eq:allmetric).

Theorem thm:metric is not restricted to hubs. An intermediate vertex may be physically close to a crossing lane to which it has no direct graph edge. The detour to hubs costs $O({\ell }_n)$, which vanishes. This resolves the intermediate-vertex obligation needed for uniform cone convergence.

The schedule is defined for every integer tick. Its phase is $s\mathrm{mod}{F}_n$.

At phase zero, the blocks are the junction cliques $C_k$; every interior lane site is a singleton.

At phase $j=1,\dots ,N_n$, every hub is a singleton. On each lane $(k,p)$, form the pair blocks

All unmatched lane sites are singletons.

Every phase is a genuine partition into finite blocks of graph diameter at most one. Every block's spatial diameter is at most $a_n$. In the calibrated spacetime norm

the input–output footprint of a firing has diameter at most $\sqrt{2}{a}_n$.

Proof. The junction cliques are disjoint and contain all endpoint sites. At transport phases, the lane chains have disjoint site labels, and the parity pairs on each chain are disjoint. All remaining sites are explicitly assigned singleton blocks.

Clique and pair blocks have graph diameter one. Their physical diameters satisfy Lemma lem:embedding. The tick duration is $h_n=a_n/c$, giving the spacetime bound.

A register at the first endpoint of a lane immediately after its phase-zero firing can advance from index zero to index $N_n$ during the following $N_n$ ticks, moving one graph edge at each tick. It arrives at the next phase-zero boundary.

The same assertion holds in reverse along a lane, because $N_n$ is odd.

Proof. At transport phase $j$, the pair containing indices $j-1,j$ is present. Induction moves the register to index $j$. The last transport tick ends at the next phase-zero boundary.

For reverse travel, the pair containing indices $N_n-j+1,N_n-j$ is present: its lower index $N_n-j$ has parity $j-1$ when $N_n$ is odd.

For any starting tick and hubs at physical separation $L$, a logical register can be delivered to the target hub in elapsed physical time at most

For arbitrary sites at physical separation $L$, one may use

All waiting, junction transitions, initialization into a lane, and terminal delivery are included.

In particular, on journeys with $L\le R$,

and the excess tends to zero uniformly.

Proof. For hubs, wait less than one frame, hence at most $U_n$, for a phase-zero firing. Use the hub route of Lemma lem:hubroute. At the initial phase-zero firing, move the register from hub memory into the first outgoing port. Lemma lem:frame traverses the lane. At each intermediate phase-zero firing, move directly from the incoming port into the next outgoing port, inside the one junction block. Each complete lane consumes one frame of duration $U_n$. After the last lane, the next phase-zero firing moves the register into target hub memory in one further tick. Thus

If the hubs coincide, waiting or local storage satisfies the same bound.

For an arbitrary starting site on a lane, wait until a transport frame starts. If its index is even, start forward movement at the first transport phase; if odd, start at the second. Advance to the last port, retain the register there, and move it into that hub at the next phase-zero firing. This costs at most $2U_n+h_n$. A register at a hub can similarly reach any specified site on one of its outgoing lanes in at most $2U_n+h_n$.

Choose these initial and final hubs. Each is within physical distance $s_n=c{U}_n$ of its respective site. The hub-to-hub separation is at most $L+2s_n$. Applying (eq:hubtime) and adding the two endpoint costs gives

Storage is achieved by leaving the relevant memory factor unchanged, not by changing the partition schedule.

Each tick lasts $a_n/c$, and every graph edge has physical length at most $a_n$. Junction ticks are charged even though their physical displacement is very small. The efficiency loss appears explicitly in

Thus the upper cone and the lower transport bound use the same clock.

Any fixed finite collection of planned logical routes can be run on the same schedule with the preceding per-route timing bounds, provided finite reserved interfaces are large enough. Routes and storage plans are chosen before the unknown input is supplied.

Proof. Assign a distinct register colour to each logical connection. Reserve a slot of the appropriate type for that colour at the sites it may use, together with hub-memory slots. During transport phases, swap the corresponding slots across each pair block. During junction phases, perform all prescribed colour permutations and memory operations inside the one junction block.

Different colours are distinct tensor factors. An arbitrary finite permutation of them is a legal local block channel; its internal gate depth is not separately charged in this model. No hub site belongs to more than one block. A finite collection uses only finitely many colours and slots at each site.

This is not a bounded-throughput theorem. It is an existence statement with the interface-growth permission explicitly retained.

Events are embedded by

For embedded terminal event $q=(T{h}_n,z_n(A))$, define

The continuum past is

Every scheduled path from $(x,s)$ to $(y,T)$ satisfies

Consequently

Proof. Every transition stays in one block and crosses graph distance at most one. Its physical displacement is at most $a_n=c{h}_n$. Sum the displacement bounds. The metric inclusion follows from the same graph-distance bound.

The embedded events are locally dense in ${\mathbb{R}}^{1,2}$. For every compact source set and compact terminal set, the distance functions to both $J_{n,\mathrm{met}}^{-}(q)$ and $J_{n,\mathrm{sch}}^{-}(q)$ converge uniformly to the distance function to $J_c^{-}(q)$, uniformly over embedded terminal events in that terminal set.

The assertion uses full past sets and distance functions, not intersections with arbitrary compact sets.

Proof. A spatial point has a hub within ${\rho }_n$, and a time has a tick within $h_n$. This proves event approximation.

Fix an embedded terminal $q$, and a point $p=(t,x)\in J_c^{-}(q)$ with $0\le t_q-t\le H$. Choose a hub $H_y$ within ${\rho }_n$ of $x$.

For the metric past, Theorem thm:metric gives

Put

Then $a_n{d}_n(H_y,A)\le c(t_q-s{h}_n)$, so $(s{h}_n,z_n(H_y))\in J_{n,\mathrm{met}}^{-}(q)$. Its distance from $p$ is at most

which tends to zero uniformly for bounded $H$.

For the scheduled past, Theorem thm:routing gives a scheduled route of duration at most

Choose an earlier source tick using

in the preceding floor formula. The route then finishes no later than tick $T$. Waiting at the terminal supplies a path ending exactly there. The resulting approximation distance is at most

By Lemma lem:physicalcone, both discrete sets are contained in the continuum cone. Their distance functions are therefore no smaller than the continuum distance function.

For source and terminal points in fixed compact sets, a nearest point on the closed continuum cone lies in a uniformly bounded region: the apex itself gives a uniform distance bound. The elapsed-time parameter $H$ for those nearest points is therefore uniformly bounded. Apply the preceding approximations to those points. This supplies the reverse distance-function inequality with an error tending uniformly to zero.

Although the proof of Theorem thm:cones does not require an asymptotic velocity theorem, the finite-periodic certificate in the plan is also satisfied.

The phase-explicit periodic dependency quotient has at most

vertices. Every quotient edge has physical displacement at most $a_n$ and duration $h_n$. Hence

Its limiting scheduled velocity polytope ${K_n}^{\mathrm{vel}}$ satisfies

and therefore converges in Hausdorff distance to $c{\overline{B}}_2$.

Proof. Modulo translations there are one hub type and $S_n{F}_n$ lane types. Including $F_n$ schedule phases gives the vertex bound. The edge displacement bound is Lemma lem:partition. Since $F_n=O(n)$, $S_n=O(n^2)$, and $a_n=O(n^{-9})$, the offset is $O(n^{-5})$.

The quotient is strongly connected. Every graph edge is exposed at some phase, waiting is always possible, and the substrate graph is connected. One can therefore traverse a finite graph route and adjust the terminal phase by waiting.

For completeness, the cycle argument is as follows. Removing cycles from a quotient walk leaves at most $q_n-1$ edges. The removed cycles decompose into simple directed cycles. Their displacement divided by duration is a time-weighted convex combination of simple-cycle velocities. Conversely, strong connectivity supplies fixed connectors between cycle bases, whose cost becomes negligible under repeated cycles. Waiting supplies zero velocity. Thus the limiting velocities form the convex hull of zero and the simple-cycle velocities, and finite-time displacement has offset at most $(q_n-1)D_{\max ,n}$.

The outer inclusion follows from the exact edgewise speed bound. For the inner inclusion, fix $n$ and a unit vector $v$. Choose increasingly distant hubs in direction $v$. Theorem thm:routing reaches them in time at most ${\lambda }_{n}L/c+2U_n+h_n$. Add waiting to make elapsed times asymptotic to ${\lambda }_{n}L/c$. The limiting velocity is $(c/{\lambda }_n)v$. Convexity and waiting give the entire inner ball.

This reproduces the relevant finite-offset mechanism rather than using a velocity diagram as a substitute for the uniform cone proof. It aligns with Fritz and VII-J Fritz,CheungVIIJ. The very small choice ${\ell }_n=n^{-8}$ is deliberately inefficient: it makes the growing quotient transient vanish as well.

A transport slot has the same finite hybrid type at its sender and receiver. Its physical motion is a sequence of tensor-factor permutations or swaps. Quantum information is not measured during this motion.

For every fixed finite-dimensional register $Q$ and every planned route of Theorem thm:routing, there is a complete implementation whose logical sender-to-receiver channel is

For every untouched reference $R$ and every state ${\rho }_{QR}$,

The same assertion holds for a finite hybrid register.

Proof. Reserve one transport colour for $Q$, and prepare all its unused slots in a fixed blank state. At each pair firing, swap the two slots. At each planned junction transition, permute the arriving slot into the next outgoing slot or into hub memory. The timetable in Lemma lem:frame sends the unique logical factor to the next endpoint. Induct through the route.

The total transformation on that logical factor and the blank slots is a tensor-factor permutation. After identifying the returned factor as $Q$, tracing the other blank factors leaves ${\rho }_Q$. The identical permutation calculation with ${\mathrm{id}}_R$ leaves ${\rho }_{QR}$. Classical block labels are relocated with their quantum blocks, without normalization or readout.

Destructive Enter on $S$, with surviving factor $T$, is

Dropping $K$ gives ${\mathrm{Tr}}_S{\rho }_{ST}$, not ${\mathrm{id}}_S$. The transport construction uses no such operation on the transported unknown register. This preserves the distinctions of Landmark I, En04-A, and Hawk04-B CheungI,CheungEn04,CheungHawk04B. No teleportation resource is used.

For every $x\in {\mathbb{R}}^2$, Euclidean unit vector $v$, and $u>0$, there are matched discrete relay pairs with sender events tending to $A=(0,x)$ and readout events tending to $B=(u,x+cuv)$. Their complete readout laws are ${\delta }_0,{\delta }_1$ at every sufficiently fine scale. The sender modification footprints shrink to $A$.

Proof. Choose hubs within ${\rho }_n$ of $x$ and $x+cuv$, and supply the bit at the first hub at tick zero. Their separation $L_n$ satisfies $L_n\to cu$. Use the common route and channels of Theorem thm:identity. The two runs differ only in the bit supplied at the sender.

Let the readout occur at the first delivered hub-memory event. Its elapsed time is bounded above by ${\lambda }_n{L}_n/c+2U_n+h_n$, which tends to $u$. The exact physical cone gives the lower bound $L_n/c$, which also tends to $u$. Thus the readout events converge to $B$. Reading the transported bit gives ${\delta }_0,{\delta }_1$. Preparation and any charged local readout occupy shrinking blocks, so the sender modification footprint tends to $A$.

The classical relay establishes the lower signalling bound. Theorem thm:identity is the separate quantum statement. A record-only measurement followed by preparation is not substituted for an unknown-state identity channel.

A finite typed skeleton $\Gamma$ consists of:

1. a finite directed acyclic graph of labelled vertices and labelled quantum or hybrid wire factors;

2. fixed finite input and output types at each vertex;

3. external input and output ports, including complete records;

4. a point $p_v=(t_v,x_v)\in {\mathbb{R}}^{1,2}$ at each vertex;

5. for every edge $v\to w$, the causal constraint

   $$t_w-t_v\ge \Vert x_w-x_v{\Vert }_2/c.$$

An edge is the identity connection on its labelled factor. Additional piecewise-linear causal connections are represented by finitely many labelled identity waypoints.

A vertex may act on several owned factors, but no quantum factor is duplicated by wiring. A reported instrument includes every outcome, including inactivity and failure flags.

A causal finite circuit assigns a complete finite hybrid channel ${\Phi }_v$ to each vertex of a typed skeleton. Initial fixed resources are prepared at declared source vertices. The unknown external input may be entangled across its ports and with an untouched reference.

Every classical control read at a vertex is part of an incident or locally retained wire factor. The channel at that vertex is fixed before the unknown input is supplied.

Its complete channel ${\Phi }_{\Gamma }$ is evaluated in a topological order. Independent operations on disjoint owned factors commute, so the channel does not depend on the chosen legal order.

Operation support means the labelled vertex points, or their declared compact neighborhoods. Wire routes and retained memories are also part of the skeleton, but are not reselected when a local instrument is replaced.

The class is closed under finite disjoint composition and under well-typed recorded continuations whenever their gluing is acyclic and obeys the displayed causal constraints.

A finite adaptive instruction tree with fixed finite outcome sets, declared finite interfaces, and causal branch connections can be represented by a finite typed skeleton of complete classically controlled instruments.

Proof. Unroll the finite tree. Reserve fixed output slots and an activity flag for every possible continuation. At a branching vertex, use classical control to place the active quantum factor into its selected continuation slot by a permutation; fill inactive slots with fixed blank states. The quantum factor is not copied. Classical outcome and activity flags may be copied locally and transported along their declared edges.

At a merge, the local flag selects the active input slot by a complete controlled channel. Every inactive or failure branch is retained in the full interface. The tree is finite, so this uses finitely many vertices and factors. All controls are represented locally.

This is complete-channel accounting, not postselection. It is the finite-interface setting compared with En04-A and En02-C CheungEn04,CheungEn02C. A syntactic Enter-calculus implementation would require the additional compositional and gate-library hypotheses of En05-A; no exact compilation of arbitrary instruments is asserted here.

For a realization, let $E_n$ place the external logical inputs into their designated physical slots and append the common prepared background. Let $D_n$ restrict the final physical output to the designated logical output slots and complete records, then apply their fixed type identifications.

Both maps are CPTP. They are not physical nonlocal transport operations: they are comparison maps identifying specified external interfaces.

If an interface is padded into a larger reserved space, choose a CPTP decoder that inverts the padding on its code subspace and sends the complementary weight to a fixed admitted output state. Thus padding has a complete deterministic extension. The comparison is

Convergence means half-diamond convergence of these maps. It includes every untouched reference and implies total-variation convergence of every fixed complete finite readout.

convention. One may reserve interfaces separately for each finite experiment, as in En05-A. Alternatively, fix in advance a number of slots and a slot dimension tending to infinity with $n$. Every fixed finite skeleton then fits on all sufficiently fine scales. The substrate and partition schedule do not depend on its input state or its local instrument choices.

For every causal finite skeleton, there are phase-zero firing times $t_{v,n}\to t_v$ and approximating hubs $H_{v,n}\to x_v$ such that every prescribed wire can be delivered after its source operation and before its target operation. The choices depend only on the skeleton, not on the channel assignments.

Proof. Choose each approximating hub by a fixed nearest-hub rule. Its error is at most ${\rho }_n$.

Choose a topological rank $r_v\in \{0,\dots ,M\}$, strictly increasing on every edge. Let $L_{\ast }$ bound all edge lengths in the finite skeleton, and put

These are phase-zero times and converge to $t_v$. For an edge $v\to w$,

The approximating hub distance is at most $\Vert x_w-x_v{\Vert }_2+2{\rho }_n$. Charge one tick for the source operation and use Theorem thm:routing from its output tick. Operation plus delivery costs at most

The causal inequality and the definition of $e_n$ make this no larger than $t_{w,n}-t_{v,n}$. All arrivals may be retained in hub memory until the target firing.

The same argument covers null connections and zero-duration co-located edges. Their added rank-dependent delays vanish.

Restricting the realization theorem to strictly timelike wires would not establish the directional boundary relays. Lemma lem:timing instead delays finitely many vertex times by vanishing amounts, preserving their declared limiting locations.

Every causal finite circuit has a realizing family on the substrates and schedules above such that

for every sufficiently large $n$. Its vertex-operation footprints converge to their declared supports.

The identity holds after tensoring with every untouched reference.

Proof. Use Lemma lem:timing to choose locations, firing times, and state-independent routes. Reserve one transport colour per connection and the required memory factors. Lemma lem:traffic implements the finite traffic concurrently.

At each vertex firing, all input factors and local controls are present in its junction block or hub memory. Apply the specified complete instrument on those factors and return every declared output, including records and failure flags. Unused factors receive identity channels. Every output wire follows its prescribed route. Incoming early factors remain in memory.

In a topological induction, every transport connection has logical channel equal to the identity by Theorem thm:identity. Each vertex therefore receives precisely its logical circuit inputs and applies precisely ${\Phi }_v$. After the output restrictions, the total logical channel is ${\Phi }_{\Gamma }$.

The operation is confined to a junction block whose spatial diameter and firing duration tend to zero. Its hub and firing time converge to the declared vertex point. This proves support convergence.

The induction is an equality of complete linear maps. Tensoring it with ${\mathrm{id}}_R$ proves the reference assertion.

Two circuits form a $K$-local replacement pair if they have the same typed skeleton, source and output placements, transport connections, and unchanged resources, and their vertex channels differ only at vertices supported in the compact set $K$.

An initial-input modification confined to $K$ means equality of the joint reductions on all external input factors outside $K$, including any untouched reference being compared.

A $K$-local replacement pair admits realizing families with:

1. the same substrate and partition schedule;

2. the same reserved interfaces, resource preparation, firing times, transport routes, and unchanged block channels;

3. different block channels only at the corresponding local vertex firings, whose complete footprints approach $K$;

4. the same discrete implementation of every common readout;

5. equal initial snapshots outside the declared discrete initial modifications, also jointly with a reference when that is required.

These conclusions persist under admitted finite compositions and recorded continuations.

Proof. The realization choices in Lemma lem:timing and Theorem thm:realization depend only on the common skeleton and its finite interface types. Choose them once for both circuits.

At an unchanged vertex, use the same complete local channel. At a replaced vertex, replace only its assigned complete instrument inside the same junction firing. Its input and output types are unchanged, so all transport permutations and storage rules remain the same. A different classical outcome changes the state of a represented controller; it does not change an undeclared scheduler.

Resource registers and route colours are initialized identically. A changed external input is placed only in its assigned local slots. The comparison encoding is tensorial on those slots with a common background; exterior reductions therefore agree whenever the logical exterior reductions agree.

A finite composite is realized by the same procedure on its common composite skeleton. Lemma lem:adaptive provides the fixed complete interface for recorded continuations. References are not read or modified.

A replaced instrument may ignore some scheduled inputs. Theorem thm:matched does not identify every ancestor with an influential port. It preserves the local replacement relation needed to apply Lemma lem:locality.

For compatible complete channels ${\Phi }_j,{\tilde{\Phi }}_j$,

Tensor composition satisfies the analogous sum bound.

Proof. Subtract the products by replacing one factor at a time. CPTP pre- and postcomposition contract diamond distance, so each term is bounded by its local error. For tensor products, replace one tensor factor at a time and use that a channel has diamond norm one. Sum the resulting bounds.

This is the compatible half-diamond accounting compared with En02-B and En05-A CheungEn02B,CheungEn05A; diamond distance is the reference-stable comparison used throughout Watrous.

Suppose the logical implementation of vertex $v$ has error $e_{v,n}$, and the complete logical implementation of connection $e$ has identity error $r_{e,n}$. Then

If the right-hand side tends to zero, all complete finite readout laws converge in total variation, uniformly over logical inputs and untouched references.

Proof. Evaluate the finite DAG in a topological order, tensoring each step with identities on the other live logical factors. Apply Lemma lem:telescope. A complete terminal readout is CPTP and cannot increase the error. Its classical trace distance is total variation.

If two implemented relay laws are respectively within ${\eta }_{0,n},{\eta }_{1,n}$ of ${\delta }_0,{\delta }_1$, then

A sum of local half-diamond errors bounds each ${\eta }_{b,n}$.

Proof. Use the triangle inequality and $d_{\mathrm{TV}}({\delta }_0,{\delta }_1)=1$. The second statement follows from Lemma lem:telescope and terminal readout contraction.

For the explicit ideal construction every error above is zero. For an approximate factorized logical implementation with $O({a_n}^{-1})$ charged elementary steps on a fixed physical journey, a uniform per-step error $o(a_n)$ is sufficient.

This last statement requires the specified logical factorization. It is not a claim that arbitrary errors coupling all registers in all blocks of a large conservative cone have only $O({a_n}^{-1})$ contributions. Such a model needs its own complete finite-channel error ledger.

The identity qubit channel and computational-basis dephasing give the same deterministic computational readout on input $|0⟩⟨0|$. They differ on $|+⟩⟨+|$, and on a Bell-entangled input. Consequently convergence of that one terminal law cannot identify a reference-preserving channel, its local slots, or its responses to replacement instruments.

The completion must retain enough information to define composition and matched replacements. This is the substantive issue behind C4 in En05-A CheungEn05A.

Fix a finite labelled DAG and every finite local type. Let ${{\mathcal{X}}_{\Gamma }}^{\circ }$ be the circuits whose wire inequalities are strict:

Include the chosen finite list of identity waypoints in the vertex set. Interface types, vertex labels, and replacement slots are fixed.

For two assignments on this skeleton, define

If a fixed prepared state is retained as separate source data, add its trace distance; equivalently treat its preparation as a one-dimensional-input source channel.

This topology compares complete local channels, not only their effects on one prepared input. Equality in the completion is equality of the labelled local data. Equal terminal channels alone are not used to identify different skeleton-local experiments.

A family is compatible for the completion if its labelled skeleton and all local finite types are fixed, its vertex data converge in the position part of (eq:completionmetric), and its complete local channels are Cauchy in the channel part.

Compatibility of a replacement pair additionally retains the same unchanged coordinates, source data, and local channels outside the declared replacement slots.

The metric completion of ${{\mathcal{X}}_{\Gamma }}^{\circ }$ is precisely the class ${\mathcal{X}}_{\Gamma }$ with the same fixed local types and the closed causal inequalities

It has the following properties:

1. every compatible local family has a unique limit as labelled local data;

2. its complete circuit channels converge in half-diamond distance to the channel of that limit;

3. admitted finite tensor compositions and acyclic typed gluings are continuous and extend to the completion;

4. every limit object has the matched discrete realization constructed in Section sec:realization.

Proof. For fixed finite types, the space of CPTP maps is closed in the finite-dimensional vector space of linear maps: positivity of every finite Choi block and trace preservation are closed conditions. It is bounded and hence compact, and in particular complete in diamond norm Choi,Watrous. The position space is complete. Closed causal inequalities define a closed subset of their finite product. Thus ${\mathcal{X}}_{\Gamma }$ is complete.

Every closed-causal assignment is approximated by strict-causal ones. Choose a topological rank $r_v$, and replace $t_v$ by $t_v+r_v/k$, leaving positions and instruments unchanged. Every edge gains strictly positive time and the assignments converge. This identifies the completion.

Circuit evaluation is a finite tensor-and-composition expression in its local channels. Lemma lem:telescope proves its continuity and the channel limit. Coordinates specify admissibility and supports; they do not alter the logical channel formula.

A typed gluing forms a finite composite skeleton. When it is acyclic and satisfies the causal constraints, its channel is the corresponding tensor-and-composition expression. The same bound proves continuity. Vertex support coordinates and boundary identifications are retained in that expression. Finally, Theorem thm:realization includes all closed-causal assignments, including boundary connections.

Suppose compatible pairs $(E_k,F_k)$ on a common skeleton differ only in fixed local replacement slots, with those slots' supports converging into a compact set $K$. Their completion limits form a $K$-local replacement pair and admit the matched realizing families of Theorem thm:matched.

Equality of exterior initial reductions, including a reference, also survives the limit.

Proof. Every unchanged local channel and coordinate is equal in the two members at each $k$. Their limits are equal by continuity of the coordinate and channel projections. The changed supports lie in $K$ in the limit. The resulting pair therefore meets Definition def:replacement; apply Theorem thm:matched.

Partial trace and classical marginalization are continuous. Equality of exterior initial reductions is a closed condition, also after tensoring the fixed comparison reference.

Consider a family on one fixed finite typed skeleton whose support coordinates converge and whose complete readout laws, for a declared input preparation, converge in total variation. Even if its local channels are not Cauchy, a subsequence of its local assignments converges to an admitted circuit with that limiting law.

For matched pairs on the same fixed finite slots, a common subsequence can be chosen so that the two limiting circuits remain matched.

Proof. The finite product of local CPTP-map spaces is compact. Select a convergent subsequence of all local channels and any separately retained finite preparation states. The causal constraints are closed, so the limiting assignment is admitted. Continuity of circuit evaluation gives its readout law. The original laws have a prescribed limit, so the subsequence has that same limit.

For pairs, use compactness of the joint finite product. Equal unchanged channels and equal exterior reductions remain equal under the common subsequence.

Theorems thm:completion and thm:completionmatch construct a unique local completion for compatible contextual data. Proposition prop:weakC4 supplies law-level membership on the same fixed finite slots, but does not infer a unique instrument from one law.

No assertion is made about unrestricted families with growing numbers of hidden vertices, changing intermediate dimensions, or only one convergent terminal law. Such families are outside the declared compatibility class. This is a restriction of the existence theorem's operational domain, not an unproved promotion of terminal convergence to the broader C4 formulation.

Let $E,F$ be a locally matched pair in the finite-circuit class or its local completion, with a common readout supported in a compact set $Y$. Suppose every changed operation and initial-input modification lies in compact $K$, and

Then the complete readout laws on $Y$ agree. The conclusion is reference preserving in the joint-reduction sense of Lemma lem:comparison.

Proof. Use Theorem thm:matched, or its completed version, to choose common discrete schedules, backgrounds, routes, and readouts. The terminal readout events and changed firing footprints converge to their declared supports.

The positive margin and shrinking footprints imply that, for all sufficiently large $n$, no changed firing can be an ancestor of any common readout. Indeed every such dependency path would obey the exact physical cone of Lemma lem:physicalcone, contradicting the positive limiting separation. Initial changed inputs are excluded by the same argument.

The exterior initial snapshots and the required ancestor channels therefore agree. Lemma lem:locality gives equal discrete joint readout laws. Their logical complete channels converge, in fact are exact in the ideal construction, so their limits agree. The same proof uses joint cone–reference reductions.

For two compact operation/readout regions satisfying

for some $\eta >0$, a matched local replacement in either region does not change the other's unconditioned complete joint record law.

Proof. Compactness and the positive margin separate each region from the causal past of every point in the other. Apply Theorem thm:DOD to the other region's full finite record set.

Conditioning on a remote outcome is a different operation and is not excluded by this statement.

Define continuum event signalling by matched sender modifications localized in arbitrarily small neighborhoods of an event $A$, a common finite readout at $B$, and a nonzero unconditioned readout-law gap. For positive-duration pairs,

Every Euclidean direction attains the boundary speed in this limiting sense.

Proof. If the ratio exceeds $c$, a sufficiently small neighborhood of $A$ is at positive distance from $J_c^{-}(B)$. Theorem thm:DOD excludes a gap. Thus $c_{\mathrm{op}}\le c$.

Theorem thm:relay supplies a matched boundary pair in every direction with gap one. Its two limiting readout laws are still ${\delta }_0,{\delta }_1$, proving the reverse inequality.

The timing estimate can be written

The relative loss separates into

Angular resolution is controlled independently by Lemma lem:angles; hub rounding costs ${\rho }_n$. All these quantities vanish in the estimates relevant to bounded journeys. The periodic transient is separately controlled by Proposition prop:offset.

1. High-degree graph versus partition schedule. A high-degree stencil alone does not provide fast transport. Endpoint separation and one junction block eliminate the proposed direction-by-direction service bottleneck.

2. Hub-only estimates. Metric estimates at hubs alone would not establish C2. Theorem thm:metric controls every intermediate site.

3. An upper cone versus an attained cone. The physical upper bound is exact at every scale, whereas the lower routing loss tends to zero under the same clock.

4. Boundary timing. Null connections cannot in general be realized with zero discrete lateness. Finite topological-rank delays make them realizable while their support locations converge.

5. Separate convergence versus matching. Realization is selected once from the common skeleton. Replacing an instrument does not reselect its remote resources, routes, or schedule.

6. One terminal law versus completion. The local completion retains complete local channels and labelled intervention slots. Example ex:law prevents a law-only identification.

7. Classical signalling versus quantum identity. The former uses a known classical bit; the latter transports a quantum tensor factor without destructive readout.

8. A semantic realization versus compilation. Arbitrary finite complete instruments are admitted as semantic block channels. Exact compilation over a particular gate library is a separate theorem, not a consequence of this construction.

9. Fixed-scale periodicity versus fixed-model rescaling. Each scale is periodic, but its quotient and stencil grow. The family therefore does not contradict the fixed-periodic polytope obstruction.

The geometric and operational package established here is:

In the notation of En05-A, this supplies the geometric C1–C2 requirements, C3 on the declared circuit class and its local completion, and C5. It supplies a proved fixed-interface local completeness construction rather than the unrestricted reading of C4.

algebras. No spacetime observable algebra satisfying En05-A's C6 is constructed. Commuting disjoint channel operations and operational no-signalling are not promoted to algebraic microcausality. Channel adjoints need not be multiplicative.

. The schedules have a preferred discrete time grading. No calibrated frame group satisfying C7 is constructed. Fixed-labelled-channel reorderings are not boosts CheungVIIE,CheungEn03,CheungEn05A. The unrestricted device-retiming closure withdrawn in En05-A is not reinstated.

automaton limits. Reversible quantum cellular automata have their own structural hypotheses SchumacherWerner. Dirac-equation continuum comparisons concern a different dynamical limit Song. Neither supplies, nor is required for, the matched finite-instrument theorem proved here.

geometry. The metric graph, physical locations, speed calibration, and local channels are declared inputs. The construction derives no physical metric from entanglement, no Einstein equation, and no gravitational backreaction from residual response. The readout distinctions of En04-A remain separate from such a dictionary CheungEn04.

analytic programme boundaries. This transport construction does not construct the inherited joint characteristic map. The reported internal amplitude advance at I2 in XIV-I is not confused with its remaining I3 obligation, or with the full J2 obligation in XIV-J CheungXIVI,CheungXIVJ. No inherited quantum coefficient is admitted. The downstream XV-B admission exit is not opened, O.1 is not closed, and XVI-N's separate prepared-insertion obligations are not discharged.

The central scheduling problem is solved here by a specific architecture: distinct lane endpoints, one local junction clique, and synchronized parallel lane frames. Its cost is growing degree, block size, and interfaces. Its benefit is an explicit estimate under one clock: