En06-B Constructive Spacetime Transport for Finite Instrument Networks and an Exact Bell Instrument–Observable Limit

We construct a payload-independent spacetime realization of every finite future-causal fixed-wire instrument network on an explicitly specified increasing-stencil lattice family. A residue-class construction supplies private transport lanes inside an ordinary lattice, including in one spatial dimension. Rounded affine paths give an every-prefix pacing estimate. Local gathering, unchanged payload instruments, and local dispersal take a vanishing amount of physical time; long routes operate in parallel. The complete instrument is preserved exactly, including every untouched reference.

For a calibrated finite source experiment at scale $j$, and a fixed proper orthochronous Poincare transformation $g$, the construction gives, at every sufficiently fine target scale $n$,

with $C_g$ independent of source-network complexity. Acquisition events, individual retained paths, and matched payload-modification footprints satisfy corresponding estimates. A full-tail diagonal argument follows.

We then construct a Bell-pair measurement network with local setting inputs, retained quantum outputs, and an untouched quantum memory. Its logical instrument is exactly scale-independent. An explicit finite unitary apparatus dilation defines an isotone spacetime observable net with spacelike commutation, product-preserving comparison maps, and an exact state–instrument pairing. The prepared Bell specialization has CHSH value $2\sqrt{2}$, while its unconditioned marginals are independent of the remote setting.

The result is a finite-network continuum realization. It is not a quantum field theory, a derivation of physical light speed, or an exact microscopic boost symmetry.

The fixed-wire interface recorded in Cheung's En06-A distinguishes a finite deterministic schedule, complete payload instruments, locally acquired reports, and a worldline-closed declared support CheungEn06A. Its transport contract requires two different types of error: a coarse source error whose coefficient does not grow with the source graph, and a fine implementation error which may depend on that fixed graph and must vanish.

This paper supplies such a construction for the target family defined in Section sec:substrate. The family is part of the theorem, not an unnamed inherited assumption. Its increasing directional resolution is essential. Fixed-periodic velocity polytopes and their anisotropy are discussed by Fritz Fritz,FritzCorrigendum and, within the supplied manuscript series, by Cheung in En05-A and VII-J CheungEn05A,CheungVIIJ.

The construction uses ordinary lattice sites and singleton or two-site blocks. It does not introduce coincident copies of a site. Instead, finitely many residue classes reserve disjoint site sets for the wires and payload junctions of a fixed network. This gives a particularly direct repair of the lane-collision problem.

1. The target substrate must be specified. A Euclidean upper speed bound alone does not supply isotropic routing. Our target generator set is stated explicitly.

2. Future-directed transport uses a time-orientation-preserving group. We use the proper orthochronous Poincare group ${\mathsf{P}}_{+}^{\uparrow }$. A time reversal does not preserve the directed network class. Spatial parity could be added without changing the proof, but time reversal is not included.

3. Channel comparison and observable comparison are different. An arbitrary channel's Heisenberg adjoint need not preserve products. The observable comparison below is instead an explicit $\ast$-isomorphism on a selected logical dilation algebra.

Cheung's En05-B is a documentary comparison for separated lanes, junction interfaces, and reference-preserving identity transport CheungEn05B. En05-C is a comparison for complete-device specification and product-preserving observable identifications CheungEn05C. None of their unseen proofs is used below.

Discrete Lorentz covariance for particular quantum walks and cellular automata is a distinct construction ArrighiFacchiniForets. Likewise, the scaling-algebra framework of Buchholz and Verch starts with a local quantum field theory and addresses its short-distance limits BuchholzVerch. Neither result is substituted for the finite measurement-and-record construction proved here.

Fix a spatial dimension $D\ge 1$ and a constant $c>0$. On ${\mathbb{R}}^{1+D}$, write points as $z=(t,x)$ and use

All Hausdorff distances and neighborhoods in this paper use $d_c$. For a set $K$, let

An edge from $z_v=(t_v,x_v)$ to $z_w=(t_w,x_w)$ is future causal when

If its duration is zero, its endpoints coincide.

For $g\in {\mathsf{P}}_{+}^{\uparrow }$, let $L_g$ be the operator norm of the linear part of $g$ with respect to ${\Vert \cdot \Vert }_c$. Then

Translations do not enter $L_g$.

For finite-dimensional Hilbert spaces, an instrument is a finite family of completely positive maps

such that ${\sum }_a{\Phi }_a$ is trace preserving. Its complete record-retaining channel is

Classical registers are represented by their diagonal matrix algebras. When a full-matrix extension is needed, input classical registers are dephased in their declared bases.

We compare complete channels using

The diamond norm and finite-dimensional channel representations are standard; see Watrous Watrous. The instrument distinction originates in the operational framework of Davies and Lewis DaviesLewis.

An admitted network consists of the following data.

1. A finite directed acyclic graph, with typed quantum or classical wire ports and explicitly named input and output interfaces.

2. A spacetime event $z_v$ for every vertex. Every directed wire edge has future-causal endpoints and is represented geometrically by the straight segment between them.

3. For every payload vertex, a complete finite-dimensional instrument on its entire declared input interface, with the entire output interface specified. Locally controlled payloads may read declared classical input wires. Their output types and wire routing do not depend on the outcomes.

4. Explicit allocation, preparation, consumption, and terminal interfaces. A default experiment has a product preparation on its declared initial registers and no undeclared shared resource.

5. Labelled report acquisitions. Each label names the local payload readout which first exposes that report. A report used downstream has a declared physical wire. A tuple retained only for statistical bookkeeping is not thereby a communication wire.

6. All retained quantum, classical, apparatus, and environmental storage segments which belong to the chosen physical realization.

7. A nonempty compact declared support $K$ containing the physical network support $F$, and a specified class of matched payload modifications.

The skeleton consists of all these data except the variable payload maps. In particular, skeleton data include register types, routing, acquisitions, and allowed modification locations.

Complete maps, rather than restrictions to a smaller regional algebra, are specified. This is the distinction emphasized in the extension-gap comparison in En05-C CheungEn05C.

The external channel is the contraction of this finite wired network. Input and output tensor products collect named ports mathematically; they do not assert that differently located ports are physically reunited at one instant.

We use the following explicit capacity convention.

Each finite experiment may reserve finitely many finite-dimensional register factors at each of its finitely many used sites. There is no bound, uniform over all experiments, on this finite number. A singleton block may implement an arbitrary complete instrument on its declared factors. A two-site block may implement typed swaps and act as the identity on all other factors. Unused factors have fixed product blank states.

No gauge constraint or shared central sector is included in this tensor-product model. Extending the construction to constrained regional algebras would require additional interfaces; the tensor-free comparisons in En05-C do not automatically provide such a routing construction.

A source experiment of resolution $\epsilon >0$ has tick duration $\tau =\epsilon /c$. At each tick, its active blocks are singletons or pairs $\{x,y\}$ with ${\Vert x-y\Vert }_2\le \epsilon$, completed by identity blocks.

The closed footprint of a block firing during $[k\tau ,(k+1)\tau ]$ is

For a singleton, $\mathrm{conv}(B)$ is that site.

Logical registers use fixed typed output ports. Between consecutive uses they remain at their designated sites; a register transfer inside a pair block is contained in that block's footprint. All such retained segments are included in the physical support $F$. Initial and terminal port events are also included.

A report acquired by a block is acquired at that block's declared output tick and output site. Its actual acquisition event is part of the source data. The declared support is any nonempty compact $K\supseteq F$.

The source need not use the target substrate introduced below. Its calibration and fixed-wire geometry, however, are required.

For a pair block, use its spatial midpoint; for a singleton, use its site. The event time is the midpoint of its firing interval. Thus a block $B=\{x,y\}$ during tick $k$ has midpoint

Connect consecutive uses of each logical register by a straight segment between the corresponding midpoint events. Add the initial and terminal port events and their half-tick connecting segments. Let $F^{\mathrm{mid}}$ be the resulting finite geometric graph.

For every calibrated source,

Every directed edge of the replacement graph is future causal. For a report acquired at an event $r$ by a block with midpoint $p$,

A block footprint is at Hausdorff distance at most $\epsilon$ from its midpoint singleton.

The constants are independent of the number of vertices, registers, or ticks.

Proof. Suppose a logical register leaves one block at site $q$ and next enters another block at the same site $q$. If their spatial midpoints are $u$ and $v$, then

Consecutive distinct block uses are separated by at least one tick. Consequently

so their midpoint segment is future causal. The same argument for an initial or terminal half-tick gives displacement at most $\epsilon /2$ in time $\tau /2$.

During the storage interval between the two block uses, the interpolated midpoint segment is a convex combination of $u$ and $v$, both within $\epsilon /2$ of $q$. It is therefore within $\epsilon /2$ spatial distance of the retained vertical segment at the same time. During the adjacent half-block intervals, comparison with an endpoint or with the block footprint costs at most $2\epsilon$ in $d_c$.

Conversely, each retained storage point has the corresponding midpoint-segment point at the same time, except within those half-block intervals, where the same bound applies. Every point of a block footprint lies within

of its midpoint. This also covers footprint points not selected by a particular logical wire. Taking unions uses a maximum, not a sum, proving (eq:midpointHausdorff).

A report output event has time displacement $\tau /2$ and spatial displacement at most $\epsilon /2$ from the block midpoint, proving (eq:acquisitionmidpoint). The block-footprint statement follows from the same estimate.

The construction puts a vertex at every active register use. It does not replace a many-tick zigzag by one endpoint chord. The uniform coarse bound would not follow from that different operation.

After applying $g\in {\mathsf{P}}_{+}^{\uparrow }$, Lemma lem:midpoint gives

The transformed graph remains future causal.

Extra points of $K\setminus F$ are support metadata. They need not be physically populated and will not be confused with the minimal physical support.

Fix a length ${\ell }_0>0$. At target scale $n\ge 1$, put

The sites are $h_n{\mathbb{Z}}^D$. The allowed generators are

A two-site block is legal whenever its displacement is $h_{n}s$ with $s\in S_n$. Singleton blocks are legal.

At each integer tick, the active pairs must be a matching. Every remaining used site is a singleton. The partition may depend on the deterministic experiment skeleton, but not on the payload values, input state, outcomes, or external reference.

The distinction between a skeleton-dependent schedule and a fixed autonomous periodic schedule is part of the model.

Every register hop has displacement at most $a_n$ and duration ${\tau }_n$, hence speed at most $c$. Block footprints have $d_c$-diameter at most $\sqrt{2}{a}_n$.

The convex hulls

converge in Hausdorff distance to the Euclidean ball ${\overline{B}}_c(0)$.

Proof. The speed and footprint statements follow from (eq:scales)–(eq:stencil).

The inclusion $V_n\subseteq {\overline{B}}_c(0)$ is immediate. For a unit vector $u$, round $(n^2-\sqrt{D})u$ coordinatewise to an integer vector $s$. For sufficiently large $n$,

and

Thus every point on the sphere of radius $c$ is uniformly close to an allowed velocity. Convexity and the origin give the claim for the full ball.

This changing family does not contradict the fixed-periodic polytope obstruction Fritz,CheungEn05A. The number of available generator directions is unbounded.

Fix a finite causal graph. First merge vertices joined by zero-duration edges into local macrovertices. Such vertices occur at the same spacetime point. Retain their original payload mapsgical order inside the macrovertex. Distinct unrelated vertices at the same spacetime point may also be placed in one macrovertex.

After this operation, every remaining wire edge has positive duration. Write $E$ for the number of remaining edges and $V$ for the number of macrovertices. Choose an integer

Assign a different residue $r\in \{0,\dots ,L-1\}$ to every edge and every macrovertex.

For each assigned residue, define

These are pairwise disjoint subsets of $h_n{\mathbb{Z}}^D$. Let $Q_{r,n}$ be a fixed nearest-point rounding map onto ${\Lambda }_{r,n}$, with deterministic tie breaking. Then

Different assigned residues never share a physical lattice site, including when $D=1$.

For a vertex $v$, let its hub be $Q_{r_v,n}(x_v)$. For an incident edge $e$, its endpoint port is $Q_{r_e,n}(x_v)$. Their distance is at most $2b_n$. For sufficiently large $n$, one legal pair-block swap transfers a register between that port and the hub.

Proof. The first coordinate of a site in ${\Lambda }_{r,n}$, divided by $h_n$, is congruent to $r$ modulo $L$. Distinct residues are therefore disjoint. This argument does not use a transverse spatial dimension.

The access distance is at most $2b_n$ by (eq:roundingbound). Since

the access displacement lies in the allowed stencil for all sufficiently large $n$.

A private lane is a reserved subset of ordinary sites, not a separate coincident substrate. Geometric segments joining its sites may cross segments of another lane. Such a crossing does not identify site labels and does not introduce an interaction. The only interactions are the declared block maps.

Let $x,y\in {\mathbb{R}}^D$, let $N\ge 1$, and reserve one residue $r$. Define

If

then successive distinct positions are connected by legal generator-pair blocks. Equal successive positions are waits.

For every prefix $k$,

The piecewise affine interpolation stays within $b_n$ of the affine target path under the common affine parametrization.

Proof. Write

Then

which proves legality. Subtracting the expression for $y_0$ gives (eq:prefixestimate). Between successive samples, the error is a convex combination of $e_k$ and $e_{k+1}$, so its norm is at most $b_n$.

In the notation of digital generator steps, if the nonzero steps are $h_n{s}_i$, and zero steps represent waits, then

This is an every-prefix estimate, not an endpoint-only estimate.

Fix the finite causal graph after the zero-duration contraction. Let $p_v$ be the number of original payload maps at macrovertex $v$, and let $d_v^{-}$, $d_v^{+}$ be its incoming and outgoing wire counts. Choose

All constants $L$ and $B$ depend only on this fixed graph.

Let

Retard the geometry slightly by

The hub frame at vertex $v$ begins at

Its duration is $B{\tau }_n$.

During a hub frame, execute:

1. one incoming port-to-hub swap per incoming wire;

2. the original payload maps in their fixed topological order, one per singleton tick, with local allocations and consumptions;

3. one hub-to-outgoing-port swap per outgoing wire;

4. identity waits to fill the frame to $B$ ticks.

All maps act as the identity on undeclared factors. Acquisitions occur at the actual output tick of their corresponding local payload map.

The local frame is legal for every sufficiently large $n$. It gathers, processes, and separates any fixed finite set of typed registers without measuring them for routing.

The routing parts are tensor-factor permutations and preserve the transported state jointly with every untouched reference. Their time overhead is at most $B{\tau }_n$, and their geometric support shrinks to the vertex.

Proof. Each incoming or outgoing transfer is a legal access swap by Lemma lem:residues. Transfers involving the same hub are serialized. The singleton payload acts only after all its input factors are present and before its required outputs are dispersed. The chosen topological order implements the same internal zero-duration subnetwork.

A swap is unitary and merely relocates a typed factor. Its action on a logical state and an arbitrary reference is exactly the corresponding tensor-factor identification. No routing operation reads a quantum state or a measurement outcome.

The frame has $B$ ticks by construction. Every involved site is within $b_n$ of $x_v$, and all frame times are within $B{\tau }_n+{\tau }_n+c^{-1}c{\eta }_n{T}_{\Gamma }$ of the intended vertex time. These bounds tend to zero.

For an edge $e:v\to w$, put

Route from its outgoing port at time $(k_v+B){\tau }_n$ to its incoming port at time $k_w{\tau }_n$, using (eq:roundedpath) with $N=N_e$.

For every fixed finite graph, all sufficiently large $n$ satisfy $N_e\ge 1$ and the step criterion (eq:stepcriterion) for every edge.

Proof. The rounding in (eq:vertexclock) gives

Since the graph is fixed and has finitely many positive-duration edges, for sufficiently large $n$,

simultaneously for all edges. Hence

Causality implies

The remaining step margin is

On the other hand,

Thus $2b_n\le a_n{\eta }_n/(2+{\eta }_n)$ for all sufficiently large $n$, proving (eq:stepcriterion). Equation (eq:durationmargin) also proves $N_e\ge 1$.

For every fixed finite future-causal network, all sufficiently large target scales admit one deterministic legal partition schedule implementing all its routes and hub frames.

Long wire routes execute in parallel. Only the finite local access operations at each hub are serialized. The schedule is independent of the input state, payload values, outcomes, and external reference.

Proof. At each tick, a moving edge requests at most one pair, whose two sites lie in its assigned residue class. Different edge routes have disjoint residue classes, so their requested pairs are disjoint.

A hub-frame access pair joins its private hub residue to an incident private edge residue. An incoming route has ended before its gathering swap; an outgoing route begins after its dispersal swap. Thus an access pair never competes with movement on that same edge.

Different hub frames use different hub residues. An edge cannot be undergoing access at both endpoints at the same tick: Lemma lem:slack leaves at least one route tick between the two frames. Therefore all active pairs at a tick form a matching. Payload singleton ticks cannot compete with access at their own hub because the local frame serializes them.

Complete the matching by singleton identities at all remaining used sites. The experiment uses finitely many sites and factors. The unused lattice may equivalently be completed by singleton identities without constructing an infinite active tensor product.

Every decision was determined by the graph, assigned residues, rounding rules, and clock. No quantum or classical outcome was consulted.

Coincident geometric wires can be transported as distinct private residue lanes, or as a bundled typed register when their skeletons allow it. Crossing geometric segments need no special interaction because their site sets are disjoint.

At an actual payload junction, gathering and dispersal are exactly Lemma lem:junction. In particular, the proof does not rely on a false positive equal-time separation between crossing curves. It applies unchanged in $D=1$.

Let $F_{\Gamma }$ be the support of the finite point-vertex causal graph, including its straight wire and storage edges. Let $F_{\Gamma ,n}$ be the actual physical support of the constructed realization, including active block prisms, waits, hub frames, and retained paths.

Define

For the fixed graph, ${\delta }_{\Gamma ,n}\to 0$.

For every sufficiently large $n$,

The same upper bound applies to each individually labelled wire support, including its local endpoint waits.

Every actual payload acquisition is within ${\delta }_{\Gamma ,n}$ of its assigned graph vertex. The footprint of each original payload map, including a map inside a contracted macrovertex, is at Hausdorff distance at most ${\delta }_{\Gamma ,n}$ from its graph-vertex singleton.

Proof. For an edge $v\to w$, compare the original and implemented paths under the common affine parameter $\lambda \in [0,1]$. The original time is

The implemented route time is

Its time error satisfies

Lemma lem:pacing bounds the spatial error of the interpolated route by $b_n$.

An active block prism is within $2a_n$ in $d_c$ of the corresponding interpolated route. Endpoint waits, gathering, and dispersal occur within $b_n$ of the vertex position and within the frame's time bound. Payload singleton intervals have the same bound.

Conversely, every original edge point has the corresponding implemented route point, and every graph vertex is close to its hub frame. These comparisons establish both Hausdorff directions. The deliberately enlarged constant in (eq:fineerror) covers all the displayed bounds.

The argument applies edge by edge and vertex by vertex. Taking the union takes the maximum of their errors. The named wire estimates include the access and wait pieces attached to that wire. Acquisition times are actual frame output ticks, not replacement metadata, and satisfy the same estimate.

A transformed idle wire is a causal affine segment with some velocity $v$, generally nonzero after a boost. The construction samples the whole affine segment and obeys (eq:generatorprefix). It does not travel to its final position early and then remain there for a macroscopic wait. All waits and accesses are included in Lemma lem:finegeometry.

Let ${\hat{\Phi }}_{\Gamma }$ be the source network channel on its named logical ports. Let ${\hat{\Phi }}_{\Gamma ,n}^{\mathrm{phys}}$ be the constructed physical channel.

Let ${\mathcal{E}}_n$ insert the logical inputs into their reserved input factors and insert all required product blanks. Let ${\mathcal{D}}_n$ select and identify the named output factors, tracing unused factors. These are mathematical port comparison channels.

For every payload assignment on the fixed skeleton,

The identity holds after tensoring both sides with the identity on every finite-dimensional untouched reference.

Matched payload modifications use exactly the same routes, residues, schedule, encodings, and decodings.

Proof. Follow a topological ordering of the logical network. Every access or route operation moves a typed factor by swaps with reserved blank factors. On the logical factors this is the specified identity wire under tensor-factor relabelling.

When the inputs of a payload vertex have arrived, its original complete map is applied to precisely those factors. In a macrovertex, the original finite zero-duration subnetwork is applied in a topological order. This has the same contraction as the source subnetwork because operations on independent factors commute.

Induction over the vertices therefore gives the same complete map on all logical wires and records. Discarding unused blanks and identifying external ports gives (eq:logicalidentity). The argument is an equality of linear maps, so its tensor extension by ${\mathrm{id}}_R$ is immediate.

The schedule was constructed without inspecting payload maps. Replacing any specified complete payload map leaves every routing and comparison interface unchanged.

The decoding in (eq:logicalidentity) is not destructive readout. Readout followed by forgetting is a partial-trace operation, not an inverse of arbitrary quantum transport; this distinction is also recorded in En04-A CheungEn04A.

Suppose the exact payload channel at vertex $v$ is replaced by a channel with half-diamond error at most $e_{v,n}$, including all its classical controls and output records. Then

Proof. Replace payloads one at a time. The maps before and after a replacement are channels, including identity extensions by the other live factors. Composition with channels cannot increase the diamond norm. The triangle inequality gives (eq:telescoping).

Fix a calibrated source experiment of scale $j$, resolution ${\epsilon }_j$, skeleton ${\Gamma }_j$, physical support $F_j$, declared support $K_j\supseteq F_j$, and labelled acquisition tuple $R_j$. Fix $g\in {\mathsf{P}}_{+}^{\uparrow }$.

For every sufficiently large target scale $n$, the substrate of Section sec:substrate admits a payload-independent realization with the following properties.

1. Its logical complete instrument equals the source complete instrument exactly, as in (eq:logicalidentity), with every untouched reference.

2. Its physical support satisfies

   $$d_{\mathrm{H}}(F_{j\to n},g{F}_j)\le C_g{\epsilon }_j+r_{{\Gamma }_j,j,g}(n),$$

   where one may take

   $$C_g=2L_g,r_{{\Gamma }_j,j,g}(n)={\delta }_{g{{\Gamma }_j}^{\mathrm{mid}},n}⟶0.$$

3. A legal declared target support is

   $$K_{j\to n}=g{K}_j\cup F_{j\to n},$$

   and it obeys

   $$d_{\mathrm{H}}(K_{j\to n},g{K}_j)\le C_g{\epsilon }_j+r_{{\Gamma }_j,j,g}(n).$$

4. For the actual acquisition tuples,

   $$\underset{\alpha }{\max }{d}_c(R_{j\to n,\alpha },g{R}_{j,\alpha })\le C_g{\epsilon }_j+r_{{\Gamma }_j,j,g}(n).$$

5. The corresponding named retained-wire supports obey the same form of estimate. For every matched set of changed payload vertices, the source and target changed-payload footprint unions also obey that form of two-sided Hausdorff estimate.

The coefficient $C_g$ is independent of the source graph's complexity. The threshold and the fine error may depend on the entire fixed finite source.

Proof. Apply Lemma lem:midpoint to the source. Apply $g$ to the resulting causal graph. Since $g$ preserves future causality, Sections sec:routing–sec:schedule construct a legal target realization at every sufficiently large $n$.

The triangle inequality, (eq:transformedmidpoint), and Lemma lem:finegeometry give

This proves (eq:Tphysical).

The set in (eq:declaredtarget) is compact and contains the actual target support. Every point of $g{K}_j$ belongs to it. Every additional point lies in $F_{j\to n}$, hence is within the displayed bound of $g{F}_j\subseteq g{K}_j$. This proves (eq:Tdeclared). It does not assert that the extra declared points are active physical sites.

Each source acquisition lies within ${\epsilon }_j$ of its source payload midpoint. The target acquisition is the actual output event of the corresponding payload map in its hub frame. Lemma lem:finegeometry gives

which is bounded by (eq:Tacquisition).

The same argument applies to individual retained supports. A source payload footprint is within ${\epsilon }_j$ of its midpoint. Its target singleton payload footprint is within the fine error of the transformed midpoint. Taking unions over changed vertices again uses a maximum. Empty changed sets are handled as the unchanged case, without assigning a Hausdorff distance to an empty set.

Theorem thm:identity proves the channel and reference statements. It also proves matched-skeleton naturality.

The quantities $B$, $L$, $T_{\Gamma }$, and the refinement threshold may grow with a fixed source's complexity. They appear only in $r_{{\Gamma }_j,j,g}(n)$, which vanishes after that source is fixed. The coarse coefficient is $2L_g$, not a graph-dependent sum.

Theorem thm:T discharges the stated transport contract for the target family (eq:scales)–(eq:stencil). Applying it to a different prescribed target family requires checking that family's generator availability, clock, capacity, and partition rules. A documentary relationship between manuscripts does not perform that check.

A full-tail family consists of experiments at every sufficiently large scale $j$, on fixed finite external logical interfaces, such that

The finite skeleton may vary with $j$.

Let (eq:fulltail) hold for calibrated sources and fix $g\in {\mathsf{P}}_{+}^{\uparrow }$. There are experiments on the target substrate at every sufficiently large scale $n$, with

If two source families have matched skeletons and differ only on payload footprints contained in $(K^{\prime }{)}_{{\rho }_j}$, where ${\rho }_j\to 0$, their transformed target families can use matched skeletons and differ only in a neighborhood of $g{K}^{\prime }$ whose radius tends to zero.

Proof. For each fixed source $j$, Theorem thm:T supplies a threshold beyond which its fine error is at most $1/j$. Choose strictly increasing integers $M_j$, with $M_j\ge j$, so that this statement holds for every $n\ge M_j$.

For $n\ge M_1$, define

The maximum is finite, and $j(n)\to \infty$. Transport source $j(n)$ to target scale $n$. Then

which tends to zero. The acquisition estimate is identical in form. Exact instrument preservation gives

For matched modifications, choose each $M_j$ to work for the common skeleton and its changed-footprint union. The target changed footprints lie within

of $g{K}^{\prime }$. This radius tends to zero. Payload-independent transport supplies the common target skeleton.

On full-tail objects with common logical interfaces, define

Within the explicit source–target class of this paper, Theorem thm:diagonal supplies full-tail realizations of these transformed objects and preserves witnessed matched modifications. The transformations satisfy the group law on these limit data.

Proof. Realizability and modifications follow from Theorem thm:diagonal. On geometric data, $g(hK)=(gh)K$ and similarly for acquisitions. Logical channels are unchanged under the fixed port identifications, giving the group law.

This action is not an exact action on a single microscopic lattice or schedule. Nor does it reinstate unrestricted same-scale device boost closure.

We now construct one fixed continuum network, then realize it at every sufficiently fine target scale using the preceding theorem.

Fix $L_0>0$, $T>L_0/c$, and $H>0$. Let

The source-to-wing paths are

They are strictly timelike. Each wing has a vertical retained storage path from time $T$ to $T+H$.

At $A$, there are qubits $Q_L,Q_R$ and a memory qubit $W$. The memory accompanies the left-moving register and is never measured or acted on by the payload. Its purpose is to make the retained quantum-information content completely explicit.

At $B_L$ and $B_R$, respectively, local input registers $S_L,S_R$ select between two measurement settings. These are local input ports, not remotely supplied control wires. Default preparations are product preparations.

The named quantum output is $Q_L{Q}_{R}W$, and the named classical outputs include the retained settings and the two locally acquired outcomes.

Let

Write $X,Z$ for Pauli matrices and define

Since ${A_{\theta }}^2=I$, the $P_{a,\theta }$ are orthogonal projectors with sum $I$.

Choose local setting tables

The input space is

For a matrix $\rho$ on this space, let

Define

Using the label $a=\pm 1$ for a classical basis vector, set

This specifies the canonical full-matrix extension which dephases the setting inputs. On the declared classical input algebra, it is the controlled local instrument.

Equation (eq:Bellchannel) is CPTP. Its selective quantum maps are the declared local Lueders maps after the source gate. It is not a constant classical sampler.

Moreover, for product inputs between $W$ and the other input factors, its action on $W$ is exactly ${\mathrm{id}}_W$. With an external reference entangled with $W$, that entanglement is retained. Consequently the complete channel is not entanglement breaking.

Proof. The Kraus operators of the full channel are the operators which read the setting basis $⟨st|$, apply $K_{ab\mid st}$, and write the output setting and outcome basis vectors. They satisfy

for every $s,t$. Summing over the orthogonal input setting projectors gives the full identity. Thus the channel is CPTP.

The operators in (eq:BellKraus) contain $I_W$, so the channel factors as the displayed Bell instrument tensored with ${\mathrm{id}}_W$, with the conventional tensor ordering understood. This preserves every state of $W$ and its untouched reference when the other factors are fixed independently.

For an explicit nonconstant record example, take $s=1$, $t$ replaced, if desired, by an admitted right $X$-setting payload modification. The source inputs $|00⟩$ and $|10⟩$ produce ${\Phi }^{+}$ and ${\Phi }^{-}$, whose $X\otimes X$ correlations have opposite signs. More generally the projectors in (eq:BellKraus) directly distinguish input coherences in their admitted measurement bases. Thus the selective channel contains more than one fixed output law.

Finally, fix all other inputs and put $W$ in a Bell state with an untouched reference. The $W$–reference marginal remains that Bell state after the channel. An entanglement-breaking channel could not have this property.

The memory $W$ is not needed for the Bell correlations. It is included to prevent any ambiguity about the survival of unknown quantum information in the complete output.

For source input ${|00⟩}_{Q_L{Q}_R}$, any fixed memory state, and fixed local setting inputs $s,t$,

Both one-wing outcome marginals are uniform. The correlators

satisfy

Proof. The state ${\Phi }^{+}$ has

and vanishing one-qubit Pauli expectations. Expanding the two projectors in (eq:projectors) gives (eq:Belllaw). Summing over either outcome gives $1/2$.

The setting table gives

proving (eq:CHSH). The CHSH comparison is the standard one introduced in CHSH; the value here has been computed directly.

For a fixed state reaching the two wing interfaces, replacement of the left complete local instrument by any other trace-preserving left local instrument does not change the unconditioned right output, including any untouched reference. The analogous right-to-left statement holds.

Proof. Let ${\Lambda }_L$ be the complete left channel, including its record. Its adjoint is unital. For any observable $B$ on the right output and untouched reference,

Hence every such expectation is independent of ${\Lambda }_L$. Equality of expectations for all output observables proves equality of the right reduced state.

This is an unconditional statement. Conditional remote states may depend on a supplied local outcome. It also holds with local setting changes, provided the source state and remote input marginal are kept fixed.

At each wing add three blank qubits: the record carrier $R$, an outcome environment $E$, and a setting environment $F$. The setting input is $S$.

For $a=\pm 1$, let $r(a)=(1-a)/2\in \{0,1\}$. For each setting $\theta$, define a unitary on $QRE$ by

It is unitary because the projectors are orthogonal and the conditional bit flips are unitary.

Let $D_{SF}$ be ${\mathrm{CNOT}}_{S\to F}$, and put

with tensor order adjusted to the named factors. Define $U_R$ analogously using ${\phi }_t$. The two wing unitaries act on disjoint factor sets.

Their action on initialized apparatus factors is

On the common finite Hilbert space

let

Insert the six apparatus blank qubits, apply $U$, and trace $E_L{F}_L{E}_R{F}_R$. The resulting operational channel is exactly (eq:Bellchannel).

Proof. Equation (eq:dilationaction) copies the outcome into two orthogonal carriers $R,E$ and the setting into $S,F$. Tracing $E$ removes cross-outcome terms; tracing $F$ removes off-diagonal setting terms. The remaining terms have precisely the Kraus expression in (eq:Bellchannel). The identity on $W$ is retained throughout.

This is a concrete finite-dimensional instance of dilation Stinespring,Watrous, not an appeal to a localization theorem. Its locations have already been fixed: $U_{\mathrm{B}}$ acts at $A$, and $U_L,U_R$ act at the respective wing events.

The environments are inaccessible to the operational output but are retained in the complete dilation ledger. For its spacetime support, they have declared local storage segments to the terminal horizon. Operational tracing is not physical annihilation.

All observables are represented on the common pre-source Hilbert space ${\mathcal{H}}_0$. Dormant apparatus factors are not assigned a local observable algebra before their allocation events.

At $A$, assign the algebra

Before the wing measurements, assign along the left path

and along the right path

Let

At and after $B_L$, the full local dilation algebra is

and analogously

The source unitary is understood as acting trivially on all other factors.

These expressions include the full input and output algebra at a local measurement event. Conjugating a full wing algebra by $U_L$ or $U_R$ leaves that full wing algebra unchanged. Specific postmeasurement observables nevertheless have their specific Heisenberg representatives. For example, an output pointer projection is represented by

It belongs to ${{\mathfrak{A}}_{\mathfrak{L}}}^{\mathrm{post}}$.

For an open spacetime region $O$, let

where $F_{\mathrm{dil}}$ is the declared source, branch, measurement, and retained-storage geometry, and ${\mathfrak{A}}_p$ is the applicable point-labelled algebra above. If there are no such points, set $\mathfrak{A}(O)=\mathbb{C}I$.

This net is finite dimensional. Apparatus observables in the complete dilation need not all be accessible observables of the reduced operational instrument.

The assignment $O\mapsto \mathfrak{A}(O)$ is isotone. If $O_1$ and $O_2$ are spacelike separated, then

The net is noncommutative and contains nontrivial local quantum algebras.

Proof. Isotony follows immediately from the inclusion of generating point sets.

Every point of either active branch is causally related to $A$. Every two points on the same branch, including its vertical storage continuation, are causally related because that branch is future timelike. Thus two spacelike-separated regions cannot both contain active points of the same branch. Nor can one contain $A$ and the other an active branch point.

It remains to compare left and right generators. At the common cut immediately after the source gate, the entire left factor set ${\mathcal{H}}_L$ and right factor set ${\mathcal{H}}_R$ are disjoint tensor factors. Their full algebras commute. Conjugating both by the same source unitary preserves commutators:

The premeasurement algebras are subalgebras of these full wing algebras. Every left generator therefore commutes with every right generator. Products, sums, adjoints, and norm closure preserve this commutation, proving (eq:netmicrocausality).

A small region meeting the interior of the left branch contains a represented copy of $\mathcal{B}(Q_{L}W)$, and similarly for the right branch. These are noncommutative matrix algebras.

The left and right branches share the entangling source. Their backward ancestor sets are not disjoint. The proof uses disjoint factors at one common circuit cut and a common preceding unitary, not disjoint ancestor supports.

We prove isotony and spacelike commutation for the stated finite net. We do not assert a vacuum, positive-energy translation representation, time-slice axiom, field reconstruction, or arbitrarily many local degrees of freedom.

Apply Theorem thm:schedule directly to the fixed continuum Bell dilation graph. This gives physical realizations at every sufficiently large $n$. Let $K_{\mathrm{B}}$ be its compact declared support and $K_{\mathrm{B},n}$ the target declared support. Actual acquisition events are denoted ${B_{L,n}}^{\mathrm{acq}},{B_{R,n}}^{\mathrm{acq}}$.

The encodings and decodings of Theorem thm:identity give

for every sufficiently large $n$.

The Bell network has a full-tail continuum realization with

The equality includes every untouched reference. It also holds for matched local setting or measurement-payload modifications.

Proof. Lemma lem:finegeometry gives the geometric and actual acquisition convergence. Theorem thm:identity gives (eq:exactBelllimit) for each payload assignment on the fixed skeleton, together with its reference extension and matched-modification statement.