En08-B Resource-controlled operational approximation of null boundary transport

We specify a Euclidean-calibrated family of finite quantum circuits and prove one operational boundary-approximation theorem. At refinement $j$, the spatial mesh is $\ell /j^2$, the permitted interaction radius is $\ell /j$, and one circuit layer takes $\ell /(cj)$, where $c>0$ is supplied. Every prescribed future-null displacement is approximated by a payload-independent sequence of local SWAP operations and waits. The spatial error is $O(j^{-2})$, the nonnegative arrival delay is $O(j^{-1})$, and the ideal logical channel is exactly the identity on a fixed finite-dimensional payload, including every untouched reference. All bounds are uniform on bounded-distance target families. The construction uses $O(j)$ ticks and $O(j)$ initialized registers; a matching leading-order register lower bound holds in the declared static-site model. Complete implementation errors are charged by channel telescoping. In particular, per-gate half-diamond error $o(j^{-1})$ is sufficient when preparations, spectators, waits, and output identifications are exact. The result concerns operational approximation of a prescribed causal boundary. It does not derive the metric, the speed, Lorentz covariance, a continuum field theory, or gravitational dynamics.

Keywords: quantum transport; circuit locality; causal boundary; diamond norm; lattice refinement; resource accounting.

A geometrically specified causal cone and an operational transport certificate are different objects. A boundary vector in a cone, or in the closure of a set of displacements, does not by itself supply a finite protocol that transmits an unknown quantum state. The supplied description of Cheung's En08-A emphasizes this distinction, including the possibility that a generated null boundary contains no attained null event pair CheungEn08A.

The present paper supplies a connecting theorem in one explicitly declared model: \[

prescribed future-null displacement

finite local transport protocols with complete-channel
correctness, uniform geometric control, and a resource ledger.

The implication is constructed directly. No result concerning cone selection is used as a premise.

to earlier transport constructions. The supplied En05-B description already includes direction-uniform relays, reference-preserving quantum transport, and matched finite experiments. The supplied En06-B description already includes prescribed-speed finite-network transport CheungEn05B,CheungEn06B. Accordingly, En08-B is an independently proved quantitative specialization and synthesis, not a claim that either reference-preserving transport or increasing-stencil isotropy is new in general. Its particular content is a single-payload, two-site-gate model with explicit endpoint bounds, a future-directed arrival convention, a static-site memory lower bound, and complete noise accounting.

The distinction between scheduled ancestry and actual influence is also retained CheungEn05A,CheungVIIJ,CheungVIIIC. The support argument below gives an upper bound for every admissible circuit. A separately constructed SWAP circuit gives the positive transport witness.

context. Finite periodic transport structures generally have polyhedral large-scale velocity sets; Fritz gives a systematic treatment Fritz2013. Our directional permission set therefore grows with refinement. This does not contradict a fixed-stencil obstruction.

There are also constructive representation results relating unitary quantum causality to local circuit implementation ArrighiNesmeWerner. We do not invoke such a representation theorem: the local circuits used here are specified explicitly.

, rather than conclusions. Throughout, the following are externally supplied: the Euclidean spatial metric, spatial and temporal calibration, the length unit $\ell$, the speed $c$, finite-dimensional quantum channel semantics, and the permission rule. The target cone is

We approximate its prescribed null boundary. We do not infer this cone from scheduler consistency or from quantum computation.

Fix

The restriction $D\ge 2$ makes unknown-state transport and signalling nontrivial. The geometric estimates also hold for $D=1$, but a one-dimensional payload does not witness nontrivial influence.

For each integer $j\ge 1$, declare

The spatial permission lattice is

and the displacement stencil is

In particular, $0\in S_j$.

The stencil is finite at each $j$. Its physical radius $a_j$ tends to zero, whereas its radius in lattice units grows. No fixed-stencil or fixed-degree substrate is asserted.

At refinement $j$, a finite realization consists of:

1. a finite set $X\subset {\Lambda }_j$ containing a source $0$ and the declared destination;

2. one register $A_v\cong {\mathbb{C}}^D$ at each $v\in X$;

3. a prescribed finite sequence of partitions ${\mathcal{B}}_1,\dots ,{\mathcal{B}}_n$ of $X$;

4. for each tick $k$, a channel

   $$C_k=\underset{B\in {\mathcal{B}}_k}{⨂}{C}_{k,B}$$

   on $\mathcal{L}({\mathcal{H}}_X)$, where

   $${\mathcal{H}}_X=\underset{v\in X}{⨂}{A}_v;$$

5. blocks $B$ of cardinality at most two, with $\mathrm{diam}(B)\le a_j$, and completely positive trace-preserving maps $C_{k,B}$ on the fixed block registers.

Each tick is exactly one such partition layer and takes time ${\delta }_j$. The partitions and channels are fixed before the payload is supplied and do not depend on its state.

A two-site block $\{u,v\}$ is permitted precisely when

Singleton channels include the identity and local state replacement. A circuit may use arbitrary finite $X$, but no physical register outside $X$ is available during that realization.

The phrase “one tick” is important. A sequence of overlapping two-site gates is not silently included inside one tick. If such gates require successive layers, each layer is charged its own ${\delta }_j$.

A target-dependent route may be selected before injection. The target and route are classical task data, not payload data. Preinstalled command lists do not receive the payload or measurement results during execution.

Every channel is unconditional and trace preserving. No successful branch, normalized postselection, instantaneous external feedback, or uncounted measurement-record wire is admitted. A more general adaptive model would have to place every quantum and classical control wire inside a separately specified local interface.

Only tick-boundary input and output access is defined. No continuous-time Hamiltonian implementation or within-tick access theorem is claimed.

Let $M\cong {\mathbb{C}}^D$ be the logical payload space. At every site use the same declared computational basis and the same blank state

For a realization $X$, the encoding channel is

All auxiliary registers are prepared before time zero, independently of the payload and of any reference. Transport time is measured from availability of this primed finite network and the source payload. Preparation time and physical apparatus construction are not included in the transport-time estimate.

For a destination $b\in X$, define

followed by the fixed basis identification $A_b\cong M$. This is an output-reduction convention, not a physical command that transports discarded systems to the destination.

The logical channel of a realization is

Thus the physical interface may grow with $j$, while the comparison interface $M$ remains fixed.

For finite-dimensional compatible channels $\Phi ,\Psi$, use

One definition of the diamond norm is

In finite input dimension the reference size can be bounded by the input dimension. Channels have diamond norm one, and compatible pre- and post-composition by channels are contractions Watrous2018 (Chapter 3).

The use of a complete channel, rather than selected input tests, is compatible with the finite-channel conventions in the supplied Cheung manuscripts CheungEn05C,CheungEn02C. The corresponding finite telescoping estimate is proved below.

For $Y\subseteq X$, write

For a circuit of $n$ ticks and an output set $Y$, define its scheduled ancestor sets backwards:

For every admissible realization,

and

Equal input reductions on $N_0(Y)$ give equal output reductions on $Y$. With an untouched reference $R$, equality of the joint $N_0(Y)$–$R$ input reductions gives equality of the joint $Y$–$R$ output reductions.

Proof. At one backwards tick, a block disjoint from the current support acts on an identity factor. Its adjoint is unital and therefore preserves that factor. All other blocks can enlarge the support only to their union. Induction proves (eq:adjoint-support).

Every point added at a backwards step is within distance $a_j$ of a point in the preceding backwards set. Following these steps and applying the triangle inequality proves (eq:metric-support).

The adjoint inclusion makes every output expectation depend only on the indicated input reduction. Output expectations determine the reduced output state. Tensoring the inclusion with the identity on $R$ proves the referenced assertion.

This proof is included in full. The similar prescribed-block statement in VIII-C is a comparison, not an imported hypothesis CheungVIIIC (Theorem thm:C5).

For a fixed circuit and fixed auxiliary preparation, the payload influences the destination at tick $n$ if its logical channel is not a replacement channel. Equivalently, some two payload states produce different unconditioned destination states. The circuit, auxiliary preparation, and output interface are held fixed under this intervention.

If ${\Vert b\Vert }_2>n{a}_j$, the source-to-$b$ logical channel is a replacement channel. In particular, no admissible protocol can transmit payload influence, or implement ${\mathrm{id}}_D$, in time strictly less than

Proof. For $Y=\{b\}$, the source does not belong to $N_0(Y)$. Every encoded payload state has the same reduction on that ancestor set, namely the fixed auxiliary reduction. Lemma lem:support therefore makes the destination state independent of the payload.

Since $a_j=c{\delta }_j$, the condition $n{\delta }_j<{\Vert b\Vert }_2/c$ is exactly $n{a}_j<{\Vert b\Vert }_2$. A replacement channel cannot equal ${\mathrm{id}}_D$ when $D\ge 2$.

For an encoded state ${\rho }_{MR}$, the auxiliary systems are independent of $MR$. If the source lies outside the ancestor set, the joint destination–reference output is ${\omega }_b\otimes {\rho }_R$. For local trace-preserving payload interventions, ${\rho }_R$ is unchanged. The reference is not an extra communication wire.

Suppose an admissible realization uses $q=\left|X\right|$ sites and transmits payload influence from $0$ to $b$. Then

Also, its number $n$ of ticks satisfies

Proof. Influence requires $0\in N_0(\{b\})$, since otherwise the argument of Corollary cor:speed gives a replacement channel. Unwinding the ancestor recursion supplies a scheduled walk connecting $0$ to $b$, with one displacement of length at most $a_j$ per tick.

Form the undirected graph on $X$ containing every two-site block used by the circuit. The walk places $0$ and $b$ in the same connected component. A shortest graph path between them is simple, so it uses at most $q-1$ edges. Each edge has Euclidean length at most $a_j$. Consequently

This gives (eq:memory-lower). The scheduled walk similarly gives ${\Vert b\Vert }_2\le n{a}_j$.

The bound is specific to the static-site realization. Moving carriers, undeclared storage, or allocation at new locations would change the model and its resource question.

Define coordinatewise nearest-lattice rounding by

Ties are rounded towards the larger coordinate. This map is coordinatewise monotone, fixes ${\Lambda }_j$, and satisfies

Let $j>\kappa$ and $b\in {\Lambda }_j$. Set $r={\Vert b\Vert }_2$. For $b=0$, set $N_j(b)=0$. For $b\ne 0$, set

There is an explicit route

whose increments belong to $h_j{S}_j$. Its time

satisfies

If $r\le L$ and $j\ge 2\kappa$, then

Proof. The zero case uses the single point $0$, no gates, and zero time.

For $b\ne 0$, let $N$ be given by (eq:route-count) and define

Because rounding fixes lattice points, $p_0=0$ and $p_N=b$. Write

Then

Both points are lattice points, so every increment is in $h_j{S}_j$. An increment may be zero; it is a permitted wait.

The triangle inequality along the route gives $r\le N{a}_j=cN{\delta }_j$, proving the lower bound. Also,

so

Subtracting $r/c$ proves (eq:lattice-delay). Finally, $j-\kappa \ge j/2$ gives

For the route (eq:route-points), every coordinate sequence $k\mapsto (p_k{)}_i$ is monotone. If $p_k=p_l$ for $k<l$, then every intermediate vertex equals that same point. Consequently, after deleting consecutive repetitions, the route has distinct vertices. If $m$ is its number of nonzero increments and $q$ its number of distinct sites, then

Proof. The affine coordinate $k{b}_i/N$ is monotone, and the scalar rounding rule is monotone. A monotone scalar sequence with equal values at two indices has the same value at every intermediate index. Applying this to every coordinate proves the assertion about repeated vertices. Each nonzero increment then introduces one new site.

Let $X$ be the set of distinct route sites. Prepare (eq:encoder). At tick $k$:

• if $p_k=p_{k-1}$, apply the identity to all sites;

• otherwise, use the block $\{p_{k-1},p_k\}$, apply its SWAP channel, and apply identity singleton channels elsewhere.

This is one partition layer. The block diameter is at most $a_j$, and the schedule is determined by the target, not by the payload.

The route protocol implements

More strongly, for every untouched reference $R$ and every state ${\rho }_{MR}$, its physical state after tick $k$ is

under the fixed register identifications. Every non-output route register is therefore in a payload-independent state, jointly independent of the payload and reference, at the terminal tick.

Proof. At tick zero, (eq:state-invariant) is precisely the declared product preparation.

A wait preserves the expression. For a nonzero increment, the target register is blank and independent of every other factor. SWAP exchanges the entire current payload factor, including its correlations with $R$, with that target factor. The former payload site becomes blank. This proves the invariant by induction.

At $k=N$, the payload-reference state is ${\rho }_{A_{b}R}$, and all other factors are blank. Taking the destination reduction and identifying $A_b$ with $M$ gives

The argument is an identity of tensor-factor permutations, so it holds for arbitrary reference Hilbert spaces carrying density operators, not only for references of a particular finite size.

The payload is not measured, and the protocol produces no destructive classical payload record. Its non-output quantum registers remain part of the resource ledger until terminal reduction. The distinction between a record and a residual system, emphasized in the supplied VIII-A material, is respected directly CheungVIIIA. Classical influence–reachability equality is not used to infer Lemma lem:identity.

Let $(T,x)\in K_c$ be null:

Define

First execute the route to $x_j$. If necessary, wait there until tick $n_j(T,x)$. Thus $T_j\ge T$.

This final waiting convention repairs a possible sign issue: rounding $x$ inward can make the unpadded route arrive slightly before the original continuum target time. Spatial rounding alone does not prove a nonnegative delay relative to $T$.

For $L>0$, define

For $j>\kappa$ and every null target with ${\Vert x\Vert }_2\le L$, the construction (eq:null-construction) satisfies

At most one extra terminal waiting tick is needed. The total tick count and the number $q_j$ of initialized sites obey

For $j\ge 2\kappa$,

Proof. The spatial estimate is (eq:rounding-error). Put $r={\Vert x\Vert }_2=cT$ and $r_j={\Vert x_j\Vert }_2$. Then

By Lemma lem:routing,

and

The upward-rounded target time has delay less than ${\delta }_j$. Taking the maximum of these two tick-boundary times proves (eq:time-bound).

If ${\tau }_j(x_j)<T$, then

Since ${\tau }_j(x_j)$ is already a tick-boundary time, one additional tick suffices. Otherwise no additional tick is needed.

The route uses at most

ticks and one more site than its number of nonzero increments. Including the possible terminal wait gives (eq:resource-H). The zero target uses one source register and zero ticks and also satisfies these estimates.

For $j\ge 2\kappa$, substitute $j-\kappa \ge j/2$ and ${\epsilon }_j=\kappa \ell /(2j^2)$ into (eq:B). This gives (eq:B-expanded). Similarly,

Let $E,D,G_1,\dots ,G_n$ be compatible channels, and let their implemented counterparts be compatible channels on the same comparison interfaces. Then

The left-hand side is also at most one.

Proof. Subtract the two compositions and replace one factor at a time. Each term contains exactly one difference of corresponding factors, with channels before and after it. Diamond-norm contraction under those compositions bounds that term by the diamond norm of the factor difference. The triangle inequality and division by two give (eq:telescoping). The distance between two channels in half-diamond normalization is at most one.

The corresponding Heisenberg complete-bounded-norm ledger is recorded in En05-C CheungEn05C (Proposition prop:telescoping). Equation (eq:telescoping) is used on the actual compatible interfaces, including any spectator registers.

In the model of Section sec:model, the following hold.

1. Exact locality ceiling. For every admissible $n$-tick realization, payload influence from $0$ to $b$ requires

   $${\Vert b\Vert }_2\le cn{\delta }_j.$$

2. Uniform lattice transport. For $j>\kappa$ and every $b\in {\Lambda }_j$, the explicit route implements ${\mathrm{id}}_D$, including every untouched reference, at time ${\tau }_j(b)$, with

   $$0\le {\tau }_j(b)-\frac{{\Vert b\Vert }_2}{c}\le \frac{\kappa {\Vert b\Vert }_2}{c(j-\kappa )}+{\delta }_j.$$

   It uses $N_j(b)$ ticks, $m_j(b)\le N_j(b)$ SWAPs, and $q_j(b)=m_j(b)+1\le N_j(b)+1$ initialized sites.

3. Uniform null-target approximation. For every $L>0$, every null target ${\Vert x\Vert }_2=cT\le L$, and every $j>\kappa$, there is a finite payload-independent admissible protocol with destination $x_j$, arrival time $T_j$, and logical channel ${\Phi }_j(T,x)$, such that

   $${\Vert x_j-x\Vert }_2\le {\epsilon }_j,0\le T_j-T\le B_j(L),{\Phi }_j(T,x)={\mathrm{id}}_D.$$

   It uses at most $H_j(L)$ ticks and $H_j(L)$ initialized sites.

4. Conditional implementation convergence. Suppose the encoding and decoding errors of these protocols are at most ${\alpha }_j$ and ${\beta }_j$, uniformly over the target family, and every complete tick map has error at most $e_j$. Then

   $$\underset{{\Vert x\Vert }_2=cT\le L}{\sup }{d}_{\diamond }({\tilde{\Phi }}_j(T,x),{\mathrm{id}}_D)\le \min \{1,{\alpha }_j+H_j(L)e_j+{\beta }_j\}.$$

   In particular, ${\alpha }_j\to 0$, ${\beta }_j\to 0$, and $j{e}_j\to 0$ are sufficient for uniform complete-channel convergence.

For a fixed nonzero null target $r={\Vert x\Vert }_2>0$, the constructed tick and initialized-site counts additionally satisfy

The site count has the optimal leading coefficient in the declared static-site model.

Proof. Part (i) is Corollary cor:speed. Part (ii) combines Lemmas lem:routing, lem:repetitions, and lem:identity. Terminal identity waits preserve the complete transport identity, so Lemma lem:endpoint gives part (iii). Lemma lem:telescoping and the bound on the total number of ticks give part (iv).

For the last assertion, set $r_j={\Vert x_j\Vert }_2$. The route has $m_j$ nonzero increments, so

For nonzero $x_j$,

Since $r_j\to r>0$, these inequalities imply

There is at most one additional wait, and $q_j=m_j+1$. This proves (eq:leading-resources).

For any other admissible static-site protocol that influences the same rounded destination, Proposition prop:memory-lower gives

Therefore its asymptotic site count cannot have a smaller leading coefficient.

For a target $(T,x,{\mathrm{id}}_D)$, define the dimensionless comparison error of an implemented witness by

Under the assumptions of Theorem thm:main(iv),

The right-hand side tends to zero whenever ${\alpha }_j,{\beta }_j\to 0$ and $j{e}_j\to 0$. In the ideal construction, the channel-error term is zero at every finite refinement.

Proof. Substitute the three estimates from Theorem thm:main into (eq:operational-error). Equations (eq:B-expanded) and (eq:H-expanded) give the stated convergence.

This is an endpoint transport limit on a common payload interface. It is not a construction of a spatial continuum observable algebra, and it does not supply a common family of global evolution channels at all intermediate times. In particular, it is not silently identified with the operational evolution-limit definition in XIX-C CheungXIXC (Definition def:limit).

For the constructed route, let $N$ be its unpadded tick count, $m$ the number of nonzero increments, $q=m+1$ the number of distinct route sites, and $w\in \{0,1\}$ its terminal padding count. The total number of ticks is $n=N+w$.

All $q$ registers exist at time zero and are retained until the terminal output reduction. Thus the global accounted peak is $q$, even though only two registers participate in a nontrivial gate at a given tick.

The total quantum Hilbert dimension is $D^q$. A fixed logical dimension is therefore not a constant global-memory bound. If a finite reference of dimension $r_R$ is included in a global Hilbert-space count, that dimension becomes $r_R{D}^q$. The reference is not initialized or operated on by the transport protocol.

The ledger concerns quantum register ownership and circuit resources. It assigns no heat, work, hardware fabrication cost, or Hamiltonian energy bound to a SWAP or to blank-state preparation.

The permission stencil obeys

Indeed, the integer cube with coordinate bound $\lfloor j/\kappa \rfloor$ lies inside the Euclidean radius-$j$ ball, and that ball lies inside the coordinate radius-$j$ cube. Consequently

Permission is tested by the integer inequality

The model may use this fixed membership rule instead of storing a fully enumerated stencil. If the stencil is explicitly materialized, its entry count is $\left|S_j\right|$, and its displacement-bit storage is $O_d(j^d\log (j+1))$. These documentary permission choices are not additional quantum registers.

Only the $q$ route sites are initialized in a given realization. The theorem does not require a quantum register at every site in a bounded continuum region, nor does it count a full infinite lattice as an initialized finite apparatus.

For a lattice destination $b$ with ${\Vert b\Vert }_2\le L$, every route coordinate in lattice units has magnitude at most $\lceil L/h_j\rceil +1$. A signed coordinate can therefore be stored with

bits. An explicit route list uses at most

coordinate bits, together with $O(\log (N+2))$ bits for its length and an execution index. For fixed $d,L,\ell$, this is $O(j\log (j+1))$ classical storage.

Given an exact lattice destination, the route is finitely computable: $N$ is determined by comparisons of algebraic numbers, and (eq:route-points) uses rational-coordinate rounding. No optimization oracle is needed.

For an arbitrary real continuum target, ${\mathsf{Q}}_{j}x$ specifies a mathematical family. Effective compilation additionally requires a target representation sufficient to certify the relevant rounding choices. No algorithmic access to arbitrary noncomputable real coordinates is asserted.

Assume exact encoding, exact output reduction, exact waits, and exact spectator storage. Suppose each implemented SWAP is a channel on the same two-site interface and differs from the ideal SWAP channel by at most ${\eta }_j$ in half-diamond distance. Then

Thus $j{\eta }_j\to 0$ is sufficient for uniform convergence on bounded-distance null targets.

Proof. Tensoring a local gate difference with identity channels on the other registers does not enlarge its diamond norm. All wait ticks have zero error under the stated assumptions. Apply Lemma lem:telescoping to the $m_j$ nontrivial gates.

A state-specific gate fidelity is not the hypothesis in this corollary. The allowance must hold for the complete local channel, including arbitrary extensions by the other route registers and the untouched reference.

Compatible per-step CPTP channels also constitute a model assumption. Arbitrary time-correlated environmental memory is not automatically covered by local channel benchmarks. Such memory must be included in a declared complete interface on which the comparison maps and error estimates actually hold.

The full estimate (eq:main-noise) charges complete tick maps, not merely active SWAP gates. This distinction is necessary when idle storage or the terminal wait is imperfect.

For example, suppose the $q-1$ auxiliary preparations are independent states ${\tilde{\sigma }}_v$ with

Product-state telescoping gives

The corresponding encoding-channel half-diamond error has the same bound: its difference tensors the input with this fixed state difference.

Consequently, if each complete tick map has error at most $e_j$ and the final output map has error at most ${\beta }_j$, then

Uniform conditions

are sufficient.

No independence between payload and reference is imposed. Independence is imposed on auxiliary preparation. Other preparation models may instead use a separately certified complete encoding allowance.

For every $0<ϵ\le 1$, there is a channel within half-diamond distance at most $ϵ$ of the identity that signals across an arbitrarily large spatial separation.

Proof. Place two $D$-dimensional registers $A,B$ at the chosen separation and define

It is a channel, and

because the half-diamond distance between any two channels is at most one.

Prepare $B$ in $|0⟩$. Inputs $|0⟩$ and $|1⟩$ at $A$ produce respective $B$-outputs

Their trace distance is $ϵ>0$. The separation can exceed $a_j$, making this channel inadmissible as one tick of the frozen model.

Thus implementation closeness alone is not a support theorem. Exact locality for noisy gates follows when their actual complete maps remain on the permitted blocks. Without that structural assumption, only an approximate observational statement follows. This is consistent with the approximate certificate distinctions in En05-A and En05-C CheungEn05A (Proposition prop:approx) CheungEn05C (Proposition prop:approx).

Fix a nonzero null target of distance $r>0$ and a number $\lambda >0$. There are local gate implementations with per-gate half-diamond error at most $\lambda /j$, for all sufficiently large $j$, whose logical channels do not converge to ${\mathrm{id}}_D$.

Proof. Let

At each nonzero route step, apply the ideal SWAP and then ${\mathcal{P}}_{p_j}$ to the arriving payload register, where $p_j=\lambda /j$ for sufficiently large $j$. This is a channel on the same permitted two-site block. Its distance from the ideal SWAP channel is at most $p_j$.

All other registers remain blank. If the route has $m_j$ SWAPs, the logical channel is

Theorem thm:main gives $m_j/j\to r/\ell$, so

To see a positive complete-channel discrepancy, apply the channel to one half of

The replacement component has overlap $D^{-2}$ with $|{\Omega }_D⟩$. Testing the projector $|{\Omega }_D⟩⟨{\Omega }_D|$ therefore gives

whose limit is positive.

The condition $j{\eta }_j\to 0$ is sufficient, not necessary for every possible implementation: errors can cancel or have additional structure. Proposition prop:accumulation shows why merely assuming ${\eta }_j\to 0$ is insufficient.

Suppose the available one-tick velocities at every refinement lie in one fixed finite set $V\subset c{\overline{B}}_2(0,1)$. If all positions are obtained by concatenating those steps, then every positive-time endpoint velocity lies in

For $d\ge 2$, these permissions cannot approximate every point of the Euclidean speed-$c$ sphere at its null travel time.

Proof. The endpoint velocity of an $n$-step route is the arithmetic mean of its $n$ step velocities, hence belongs to $P$.

The finite convex hull $P$ is a polytope contained in the Euclidean ball. It cannot contain the entire sphere when $d\ge 2$, since that would force it to contain the ball and hence equal the ball. Choose a unit vector $u$ with $cu\notin P$. If endpoints and positive arrival times approached $(ru,r/c)$, their endpoint velocities would approach $cu$, contradicting closedness of $P$.

This is the elementary version of the velocity-polytope issue treated more generally by Fritz Fritz2013. The growing $S_j$ in (eq:stencil) is an explicit resource choice, not a hidden exception to a fixed-stencil hypothesis.

A fixed tick cap $n\le n_{\max }$ gives

A fixed initialized-site cap $q\le q_{\max }$, even with unrestricted tick count, gives

for every destination influenced by the source. The second statement follows from Proposition prop:memory-lower. Thus the declared construction cannot be reinterpreted as a fixed-resource approximation of nonzero null distances.

For completeness, the relevant quantitative distinction can be proved on the same payload interface.

If a payload is converted into a finite classical record and a decoder prepares a $D$-dimensional quantum output from that record, then the resulting channel $\Lambda$ satisfies

Proof. The composite channel has measure-and-prepare form

Its output on half of $|{\Omega }_D⟩⟨{\Omega }_D|$ is separable between reference and output.

For unit product vectors,

by Cauchy–Schwarz. Convexity gives the same overlap bound for every separable state. Testing the Bell projector yields trace distance at least $1-1/D$ from $|{\Omega }_D⟩⟨{\Omega }_D|$. This is an admissible referenced channel test and proves (eq:record-bound).

Measure-and-prepare channels are the standard entanglement-breaking class HorodeckiShorRuskai2003. The quantitative distinction is also explicitly present in the supplied VIII-A and VIII-C material CheungVIIIA (Theorem A:thm:classical) CheungVIIIC (Proposition prop:recovery). The SWAP construction avoids the premise of Proposition prop:record: the payload remains a quantum register.

The main theorem is an approximation theorem. However, this particular permission family does have some exactly attained finite-scale null pairs. For example, the displacement $a_j{e}_1$ is permitted in one tick.

Fix $j\ge 1$, $n\ge 1$, and $b\in {\Lambda }_j$ with

There exists an admissible $n$-tick protocol transmitting payload influence from $0$ to $b$ if and only if

Whenever (eq:exact-null) holds, exact referenced identity transport is attainable.

Proof. Influence requires an ancestor walk from $0$ to $b$. Write its $n$ displacements, including possible zero steps, as $v_1,\dots ,v_n$, where

The null equality gives

Hence every step has length $a_j$, and equality holds in the Euclidean triangle inequality. All steps have the same direction and, because their lengths agree, the same vector. Each is a lattice displacement $h_{j}s$ with ${\Vert s\Vert }_2=j$. This proves necessity.

Conversely, initialize the $n+1$ sites $0,h_{j}s,\dots ,n{h}_{j}s$, and apply a SWAP along each successive edge in one tick. Lemma lem:identity proves complete identity transport.

At each fixed $j$, the attained null directions form a finite set. Across all $j$, they form a countable set with rational coordinate directions $s/j$. Targets in other directions are nevertheless approximated by Theorem thm:main. Accordingly, the correct conclusion is neither “all null pairs are attained” nor “no discrete null pair is attained.”

Let $t_j^{\ast }(b)$ be the smallest arrival time of any admissible finite protocol implementing ${\mathrm{id}}_D$ at $b$. This minimum exists: coordinate steps provide a finite route, and the possible times are nonnegative integer multiples of ${\delta }_j$.

For $j>\kappa$ and $b\in {\Lambda }_j$,

Consequently, for every $L>0$,

For rounded continuum endpoints $x_j={\mathsf{Q}}_{j}x$, the same conclusion holds uniformly for ${\Vert x\Vert }_2\le L$, after adding the endpoint allowance ${\epsilon }_j/c$.

Proof. The lower bound is Corollary cor:speed; the explicit route gives the upper bound. Lemma lem:routing supplies the uniform estimate. For rounded targets, $\left|{\Vert x_j\Vert }_2-{\Vert x\Vert }_2\right|\le {\epsilon }_j$.

The uniquely selected object in this statement is an arrival-time function inside the declared model. It is not an unrestricted uniqueness theorem for continuum evolutions.

The supplied XIX-C example exhibits different refinement choices with a common coarse channel but different limiting evolutions; its interpolation statement likewise prevents finitely many readings from selecting an unprescribed interpolation CheungXIXC (Propositions prop:two-limits and prop:interpolation). Here the refinement, calibration, rounding map, comparison interface, and route rule are fixed before the limit.

1. No predecessor proof transfer. The geometric, support, transport, and error arguments needed for Theorem thm:main are proved locally. En05-B and En06-B are documentary comparisons CheungEn05B,CheungEn06B. Their networks are not identified with the present route family.

2. No universal network realization. The positive construction transports one fixed-dimensional payload along one target-dependent route. It does not establish simultaneous routing, universal adaptive experiment matching, or a target-independent autonomous schedule.

3. No native incidence realization. No conical packet, three-to-six incidence structure, affine-conical certificate, source-dependent lapse, or residual-matter instrument is used. The finite support and recovery comparisons in VIII-A and VIII-C do not identify their geometric models with this Euclidean lattice CheungVIIIA,CheungVIIIC.

4. No derived relativity or gravity. No spatial metric, numerical speed, observer-invariant speed, Lorentz action on experiments, field regulator limit, Einstein equation, or quantum-gravitational coefficient is derived. The time parameter is external.

5. No catalogue premise. The supplied OpenAI catalogue descriptions, including Family 271, are not mathematical inputs to any argument here. No catalogue claim is verified or transferred.

6. No external obligation is discharged. The result makes no admission into an inherited characteristic map and does not discharge the separate obligations designated O.1, XV-B, XVI-N, or the series Route B. Those designations refer to external manuscript projects, not to additional conclusions of this theorem.

In one frozen Euclidean-calibrated model, every prescribed future-null target admits finite local complete-channel witnesses with

Ticks and initialized registers grow linearly with refinement; their leading coefficient is necessary in the declared static-site model. Compatible complete implementation errors are accumulated explicitly, rather than hidden in ideal identity transport.

The theorem supplies operational approximability of a prescribed causal boundary. It does not derive that boundary, identify it with another manuscript's generated cone, or establish spacetime emergence.

The following points are part of the mathematical specification, rather than informal implementation assumptions.

1. One layer per tick. The support bound requires a partition layer. It would be false if arbitrarily many overlapping gates were allowed inside an uncharged tick.

2. One preparation per distinct site. Repeated route vertices are waits under the chosen monotone rounding convention. A fresh auxiliary is prepared at each distinct non-source site, not at every repeated occurrence.

3. Future-directed endpoint timing. The padded time (eq:null-construction), rather than the raw route time alone, ensures $T_j\ge T$. At most one padding tick is required for null targets.

4. Nontrivial payload dimension. The operational lower-bound and memory statements use $D\ge 2$. A trivial one-dimensional register cannot signal.

5. Complete versus reduced residual accounting. The ideal whole-output factorization (eq:state-invariant) is proved explicitly. Ignoring non-output factors does not mean that they were never initialized or retained.

6. Gate error versus full implementation error. Equation (eq:gate-noise) assumes exact preparations, spectators, waits, and output maps. Equation (eq:main-noise) is the complete-tick alternative. Small channel error alone does not establish causal containment.

7. Exact null attainment. Some finite-scale null pairs occur in this family. Proposition prop:exact-null characterizes them. The approximation theorem does not require their absence.

8. Restricted limit claim. The established limit compares endpoints and logical payload channels. It is not an unrestricted interpolation theorem, a global evolution semigroup, or a continuum observable-net limit.

9. Resource scope. The ledger includes initialized and retained quantum registers, the finite route description, and stencil permissions. It does not claim a physical energy, fabrication, or preparation-time budget.

10. Verification scope. The paper gives finite analytic proofs. No proof-assistant certificate, numerical execution, successful pdfLaTeX build, or laboratory realization is asserted.

The external references below identify standard published or author-posted sources. Cheung's manuscripts are cited by the identifiers and versions given in the supplied material. Their dates are recorded as undated where no bibliographic date was supplied. The Enter Calculus Companion is a documentary locator, not an independent verification of those manuscripts. No comparison citation substitutes for a missing proof in this paper.

Watrous2018

J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018. Author's manuscript and chapter files: https://cs.uwaterloo.ca/ watrous/TQI/.

Fritz2013

T. Fritz, Velocity polytopes of periodic graphs and a no-go theorem for digital physics, Discrete Mathematics 313 (2013), 1289–1301. doi: 10.1016/j.disc.2013.02.010. Author's revised preprint: https://arxiv.org/abs/1109.1963v3.

ArrighiNesmeWerner

P. Arrighi, V. Nesme, and R. Werner, Unitarity plus causality implies localizability, arXiv:0711.3975, 2007; revised version 3, 2009. https://arxiv.org/abs/0711.3975v3.

HorodeckiShorRuskai2003

M. Horodecki, P. W. Shor, and M. B. Ruskai, Entanglement breaking channels, Reviews in Mathematical Physics 15 (2003), 629–641. doi: 10.1142/S0129055X03001709. Preprint record, titled “General Entanglement Breaking Channels”: https://arxiv.org/abs/quant-ph/0302031.

CheungEn08A

N. Cheung, En08-A, version 1, undated manuscript. Documentary comparison: scheduler consistency, conditional Lorentz-cone representation, and the distinction between a generated null boundary and attained null pairs. Enter Calculus Companion: https://enter.grok.me.

CheungEn05A

N. Cheung, En05-A, version 1, undated manuscript. Relevant supplied labels: thm:cone and prop:approx. Enter Calculus Companion: https://enter.grok.me.

CheungEn05B

N. Cheung, En05-B, version 1, undated manuscript. Relevant supplied section: “Introduction, target, and scope”; documentary comparison with increasing-direction relays, reference-preserving transport, and matched finite experiments. Enter Calculus Companion: https://enter.grok.me.

CheungEn06B

N. Cheung, En06-B, version 1, undated manuscript. Documentary comparison with prescribed-speed finite-network transport; relevant supplied label: thm:T. Enter Calculus Companion: https://enter.grok.me.

CheungVIIJ

N. Cheung, VII-J, undated manuscript; version not specified in the supplied material. Relevant supplied label: thm:release. Documentary distinction between scheduled ancestry, operational influence, and a separately supplied causal relation. Enter Calculus Companion: https://enter.grok.me.

CheungXIXC

N. Cheung, XIX-C, version 1, undated manuscript. Relevant supplied section: “Why finite certification does not select a continuum evolution”; labels def:limit, prop:two-limits, and prop:interpolation. Enter Calculus Companion: https://enter.grok.me.

CheungEn05C

N. Cheung, En05-C, version 1, undated manuscript. Relevant supplied section: “Operational no-signalling and approximate certificates”; labels prop:approx and prop:telescoping. Enter Calculus Companion: https://enter.grok.me.

CheungVIIIA

N. Cheung, VIII-A, version 3, undated manuscript. Relevant supplied labels: A:thm:classical and A:thm:complement; documentary comparison concerning records, residual systems, and complete quantum recovery. Enter Calculus Companion: https://enter.grok.me.

CheungVIIIC

N. Cheung, VIII-C, version 3, undated manuscript. Relevant supplied labels: thm:C5 and prop:recovery; sections “Prescribed schedule support” and “Code decoding is not a quantum-return decoder”. Enter Calculus Companion: https://enter.grok.me.

CheungEn02C

N. Cheung, En02-C, version 1, undated manuscript. Relevant supplied label: thm:C1, reference-preserving sequential soundness and actual-schedule storage ledgers. Enter Calculus Companion: https://enter.grok.me.