Landmark VII-J Scheduled dependencies and certified physical causality in localized quantum simulation Exact support, directional reachability, approximation, and continuation through a fixed-cone slicing event

We distinguish scheduled reachability, intervention-based quantum influence, and an independently supplied physical causal relation. For finite quantum networks, including their classical controllers, we prove that the complete timed dependency graph bounds the Heisenberg support of every jointly observed output. Absence of a scheduled path implies absence of signalling, including with an untouched entangled reference. A causal embedding of the dependency edges then gives an exact physical containment theorem for the encoded–computed–decoded channel.

The directional route is developed separately. A finite, phase-expanded periodic graph has a cycle-velocity polytope and an explicit finite-time remainder. Support-function comparison certifies convex velocity inclusion, but an asymptotic inclusion alone does not certify short-time communication. A spatially varying example gives an exact earliest-arrival law. For approximate simulation we prove the signalling bound with consistent diamond-norm constants and separate simulation accuracy from causal safety. A finite-volume Hamiltonian argument gives a reference-stable influence-tail certificate; a product-formula estimate supplies a corresponding simulation error.

For VII-F's three-to-six particle-link event, the ambient spinless cone remains fixed. Its uniformly spacelike, ordered slices give a conservative causal gate speed $1/(1+2\sqrt{2}/3)$ in the slicing parameter. We certify physical inclusion across the event using persistent spatially marked registers, rather than abstract cell labels. An explicitly buffered, nontrivial two-qubit transport target is implemented exactly by causal routes and local gates, together with the separately localized event-code permutation. The target preserves arbitrary reference entanglement and has a unit signalling witness along an allowed causal route.

No spacetime metric, field equation, Lorentz invariance, or gravitational backreaction is derived. The finite causal carrier model is not a quantum-field discretization. The separate characteristic quantum-product programme is not advanced.

and scope{ Constructed: exact network-support and physical-containment theorems; operational signalling criteria; localization and clock contracts; periodic directional bounds; robust finite-time margins; approximate signalling and simulation estimates; and a buffered causal realization through VII-F's event.

Two targets remain distinct. Target A concerns the complete decoded simulation channel. Target B additionally embeds the actual elementary communication and gates in the supplied causal geometry.

Explicit model boundary. The cone application is a finite carrier-and-controller model on a prescribed metric cone. It does not derive matter propagation from Einstein dynamics or provide a continuum-field simulation.

No compilation, numerical execution, experimental implementation, or machine verification is asserted.}

The central distinction is

The aim is to prove relations among these objects after defining each one, not to identify them by terminology.

The finite-channel conventions are standard Watrous. Operational causal notions require a stated intervention class; broader notions of influence need not agree in every probabilistic theory Perinotti. Circuit paths can overestimate the influence of a unitary transformation LorenzBarrett. We use an intervention-based notion throughout.

The supplied texts permit exact, manuscript-specific attribution.

These statements are cited as manuscript results CheungVIIE,CheungVIIF. The network and causal arguments used below are proved here.

VII-G's history-stability theorem is a compatible finite-channel comparison, not the physical event theorem CheungVIIG. VII-H's response distribution is neither a schedule nor an instrument CheungVIIH. The classical, domain, and numerical results of VII-A–D likewise do not construct the causal embedding studied here CheungVIIA,CheungVIIB,CheungVIIC,CheungVIID.

Let ${\Psi }_{t,s}$ be a declared target channel. Let

be the implemented channel on the same model input and output spaces. Target A consists of:

1. a causal-support or signalling estimate for ${\Phi }_{t,s}$;

2. a separately proved accuracy estimate comparing ${\Phi }_{t,s}$ with ${\Psi }_{t,s}$.

Intermediate compiler gates need not represent model events.

Target B additionally assigns all implementation ports, communications, gates, preparations, measurements, and control records to events or regions in the supplied physical model. Each possible dependency has a certified causal route.

Target B is stronger. Endpoint causal support alone does not construct a local realization of an arbitrary channel. Our positive Target B constructions use a specified routing architecture and local gate primitives.

Let the input factors be indexed by a finite set $X$:

For $S\subseteq X$, define

Output factors may have a different index set $Y$.

A channel is completely positive and trace preserving. Its adjoint is defined by

Adjoints are unital. We use

The diamond norm, without the factor $1/2$, is denoted explicitly.

For input $A\subseteq X$, an intervention is any CPTP map

It is deterministic and nonselective. A locally measured outcome may be retained as an additional declared output, but conditional normalization of a selected branch is not an intervention in this definition.

For output $B\subseteq Y$, put

Let $R$ be an arbitrary finite untouched reference. Define

The observer receives $B$ and the untouched reference $R$. Thus $0\le {\mathrm{Sig}}_{\Phi }\le 1$.

The reference may initially be entangled with the input. It is not acted upon by either intervention. This definition measures deterministic signalling, not postselected steering.

The following are equivalent:

1. ${\mathrm{Sig}}_{\Phi }(A\to B)=0$;

2. there is a channel $\Xi :L({\mathcal{H}}_{A^c})\to L({\mathcal{H}}_B)$ with

   $${\Phi }_B=\Xi \circ {\mathrm{Tr}}_A;$$
3. $${\Phi }_B^{\ast }(L({\mathcal{H}}_B))\subseteq I_A\otimes L({\mathcal{H}}_{A^c}).$$

The factorization implies equality with every untouched reference.

Proof. Factorization implies (i), since a local trace-preserving intervention leaves the $A^{c}R$ reduction unchanged.

Suppose (i) holds. Compare the identity intervention with the replacement channel

With no reference, equality for every input state gives equality of the corresponding linear maps:

Define

This is CPTP and gives (ii).

Taking adjoints proves (ii) implies (iii). Conversely, (iii) makes the output expectation depend only on the $A^c$ reduction. The same replacement construction gives (ii). Tensoring the factorization with ${\mathrm{id}}_R$ proves the final assertion.

For a joint output $B$, its actual influence set consists of input factors $a$ for which ${\mathrm{Sig}}_{\Phi }(\{a\}\to B)>0$. Equivalently, it is the union of tensor factors occurring in the minimal support of ${\Phi }_B^{\ast }(L({\mathcal{H}}_B))$. The equivalence follows from Theorem thm:nosignal and

Separate no-signalling tests for $B_1$ and $B_2$ do not automatically certify the joint output $B_1{B}_2$. A classical bit can be encoded as the parity of two uniformly random shares: each share alone is independent of the bit, while their joint law reveals it. Our support certificates below apply to every declared jointly observed output.

A finite network is a directed acyclic circuit of channels with input and output ports, fixed initial ancillas, and declared discarded outputs. Its vertices include:

• quantum and classical register states;

• local gates and routed wire segments;

• ancilla preparations;

• destructive measurements and their outcome registers;

• resets, discards, guards, and classical feed-forward;

• every classical variable read to select a later operation.

For a conservative dependency graph, connect every input port of a gate to every output port of that gate. A tighter graph is allowed only with a proved joint-output support certificate for that gate. Identity wires have their ordinary directed edges.

Classical registers use a distinguished basis. Controller maps first read the declared classical blocks and implement fixed branch maps. They are channels independent of an unknown input density operator. A canonical pinching extension allows the finite matrix-channel arguments to apply on a common ambient matrix space.

A controller may inspect only declared classical inputs. Every variable affecting the gate choice, routing, guard, availability time, or reset is included among the control ports.

A finite adaptive execution is padded to a fixed finite instruction envelope. Its dependency graph contains every possible branch and the control reads selecting that branch. No unrecorded state-functional scheduling is permitted. Ancillas are supplied independently of the unknown input; correlated ancilla preparation and transport must be declared.

A local measurement instrument is represented by its complete record-retaining channel. A fixed controlled continuation is then a channel on the record and the registers it reads. Padding unused branches with identities and retaining the controller gives the required finite envelope.

For a set of output ports $B$, let

This graph depends on the prescribed implementation, not merely on the final channel.

For the channel $\mathcal{C}$ of a complete finite network,

Consequently,

and every intervention on inputs disjoint from ${\mathsf{Sch}}^{-}(B)$ has zero signalling to $B$, including with an untouched reference.

Proof. Pull an output observable backwards through a topological order of the network. If it is the identity on all outputs of a gate, unitality of the adjoint makes it the identity on that gate's inputs. Otherwise, its possible support expands only to the declared predecessors of the observed gate outputs.

For the conservative all-to-all gate graph, these are exactly all gate inputs. Tensoring with untouched factors does not add other support. Fixed preparations introduce no unknown input. A discard pulls a scalar identity backwards; a reset is treated as its stated channel. Measurement records and controlled continuations are included as ordinary channel ports.

Repeating this process leaves support only on the scheduled ancestors. This proves (eq:schedule-support). Theorem thm:nosignal gives the remaining assertions.

This proof extends the prescribed-block mechanism of VII-E CheungVIIE (thm:locality) by explicitly including the controller and all communication interfaces.

Declare a two-register SWAP, followed by the same SWAP. Its dependency diagram contains cross-register paths, but the total channel is the identity. An intervention on the first input cannot change the second output. Thus scheduled reachability can strictly exceed actual influence.

Measure a computational bit at $A$, and prepare a distant output bit $B$ equal to the result. The complete channel has $\mathrm{Sig}(A\to B)=1$. Omitting the classical outcome wire from a quantum-only diagram gives an invalid support certificate.

A fixed undirected register graph with positive edge weights has a shortest-path metric

Directed weights instead give an asymmetric extended distance. Weights representing routing cost are not automatically communication times or geometric lengths.

In a timed network, waiting and availability are represented explicitly. The earliest-arrival function is

The value is $+\infty$ if there is no path. On a finite time-expanded DAG, dynamic programming in topological order computes the arrivals exactly.

The three clocks are

A certificate assigns model times to ports and, separately, hardware start and finish times. Reordering independent gates can change depth or hardware time without changing the channel. It need not preserve the assigned model-time embedding.

A physical comparison structure consists of a set of events $\mathcal{M}$ and a reflexive transitive relation $⪯$. It is specified independently of the computational schedule.

For point-localized ports, assign an event $p_v\in \mathcal{M}$. A scheduled edge $u\to v$ is physically certified when

For uncertain or extended localization regions ${\mathcal{L}}_u$, a sufficient robust condition is

This all-point requirement is conservative. A smaller allowance must specify the correlated endpoint choices and their routes.

Encoding and decoding are additional network components:

Their spreads are included in the full network. There is no free nonlocal encoding before the support theorem begins.

Suppose:

1. the entire encoded–computed–decoded experiment is a complete finite network;

2. every dependency edge has a physical endpoint certificate;

3. input and output localizations use the same independent causal relation.

For a joint output $B$, let ${\mathsf{Phys}}^{-}(B)$ be the input factors whose assigned events can causally reach at least one output event in $B$. Then

Inputs outside ${\mathsf{Phys}}^{-}(B)$ cannot signal to $B$, even with an untouched entangled reference.

Proof. Theorem thm:schedule gives the first inclusion. Every scheduled path is a chain of certified causal endpoint relations. Transitivity gives a causal relation between its first and last events. Hence every scheduled ancestor belongs to the physical past. The signalling conclusion follows from Theorem thm:nosignal.

The theorem is sufficient, not necessary. Extra scheduled dependencies can cancel in the complete channel.

The implementation model permits:

1. exact carriage of a finite register along a prescribed future causal curve;

2. local unitary gates and complete instruments on registers meeting at the same station;

3. local preparation of a specified input-independent ancilla;

4. local storage, resets, and discards;

5. classical records carried along the same kind of causal communication routes.

Gate durations are positive and explicitly assigned. These are ideal model primitives, not claims of laboratory realization or a quantum-field description of a point register.

Consider a gate with input events $p_1,\dots ,p_m$ and output events $q_1,\dots ,q_n$. Suppose there are stations $r_{-}⪯r_{+}$ such that

with specified routes, and the gate has a finite local implementation between $r_{-}$ and $r_{+}$. Then Contract contract:routing gives a Target B realization of that gate.

Proof. Carry the inputs to $r_{-}$, retain them through the local implementation, and carry the outputs from $r_{+}$ to their destinations. Include every ancilla and outcome register in the local implementation. Each wire and elementary gate has causal endpoints and a declared support. The endpoint channel is the intended gate, including with arbitrary references.

A causal relation between every input and output does not by itself imply the common-rendezvous hypothesis. We do not use the proposition as a universal localization theorem. QCA localizability and quantum causal graph dynamics provide stronger structural theorems under their own hypotheses SchumacherWerner,ArrighiNesmeWerner,ArrighiMartiel. They are comparison results, not unproved substitutes here.

Let

where the three channels may act on differently indexed tensor products. Suppose their support predecessors are $P_{-}$, $P_{\mathcal{R}}$, and $P_{+}$, respectively: each output algebra pulls back into the algebra of its declared predecessor set. Then

If all three predecessor relations have physical causal certificates in one common event space, the composite does also.

Proof. Compose the three adjoint inclusions. Compose the corresponding causal chains and use transitivity.

A change in register number is allowed if allocation and discard are channels with stated local supports. A newly prepared independent factor has no model-input predecessors. A state-dependent remapping must include its controller.

Let $K_{\mathrm{phys}}(x,t)\subset {\mathbb{R}}^d$ be a declared compact convex velocity set. Assume that the paths used in a certificate are absolutely continuous and satisfy

Define $p{⪯}_{K}q$ when an admissible path joins the events with nondecreasing time. Concatenation makes this relation transitive.

For constant $K$, the reachable displacement set in time $T>0$ is exactly $TK$: averages of allowed velocities lie in $K$, and a constant velocity realizes every point of $TK$.

Suppose each scheduled communication has a specified continuous route with velocity in $K_{\mathrm{sched}}(x,t)$, and

at every point visited by those routes. Include gate, encoding, decoding, and storage dependencies in the same path contract. Then the full schedule is physically contained, and Theorem thm:physical applies.

Proof. Every scheduled route is an admissible physical path. Concatenated routes remain admissible piecewise almost everywhere. The physical relation of each dependency edge is therefore certified.

The hypothesis is local along actual routes. A comparison of two average speeds is not a replacement for it.

For a compact convex set $K$, write

For compact convex $K,L\subset {\mathbb{R}}^d$,

If $L=\{v:n_j\cdot v\le b_j,1\le j\le q\}$ is a polytope, it suffices to check $h_K(n_j)\le b_j$.

Proof. Inclusion gives the support inequalities. If $v\in K\setminus L$, let $z$ be the Euclidean nearest point in $L$. The minimizing property gives $(v-z)\cdot (w-z)\le 0$ for every $w\in L$. Thus the direction $n=v-z$ strictly separates $v$ from $L$, contradicting the support inequalities. The half-space claim is immediate.

Suppose for all unit $n$,

If

then true velocity inclusion holds. Strict inequality gives a positive certified margin.

Proof. Combine the inequalities and apply Proposition prop:supportfunction.

A periodic dependency model has a finite quotient graph $Q$. Its vertices include register type and schedule phase. Each edge $e$ has displacement ${\Delta }_e\in {\mathbb{R}}^d$ and duration $T_e\ge T_{\min }>0$. Its periodic lift permits translated copies of each transition. Waiting at a port is allowed.

Assume $Q$ is strongly connected. For a simple directed cycle $c$, set

There are finitely many simple cycles. Define

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

The set of limiting velocities along paths with elapsed times tending to infinity is exactly $K_{\mathsf{Sch}}$. These are statements about potential scheduled reachability, not about the channel carried by every path.

Proof. Remove directed cycles from a finite walk until its remaining walk has no repeated vertex. Decompose each removed cycle into simple cycles. The remainder has at most $q-1$ edges.

The cycle contribution is

The sum of those cycle times is at most the elapsed time $T$. Including $0$, the contribution belongs to $T{K}_{\mathsf{Sch}}$. The remaining displacement has norm at most $(q-1)D_{\max }$, proving (eq:finiteoffset).

The offset divided by $T$ tends to zero, so every limiting velocity lies in $K_{\mathsf{Sch}}$. Conversely, choose any finite convex combination of cycle velocities and zero. Repeat each chosen cycle sufficiently many times in the prescribed time proportions, connecting their quotient base vertices by fixed paths. Strong connectivity supplies the connectors. Their total duration and displacement are negligible relative to the repeated cycles. Additional waiting realizes the zero contribution. Integer repetition counts approximate arbitrary real proportions. This realizes every point of $K_{\mathsf{Sch}}$.

For a quotient that is not strongly connected, one must respect the order of reachable strongly connected components. The convex hull of every cycle in the graph can overestimate attainability from a specified initial component.

The theorem is the timed, phase-explicit version of the periodic-graph mechanism discussed by Fritz Fritz. It does not infer finite-time physical containment from an asymptotic polytope.

Take two quotient states and edges with displacements $L$ and $-L$, each of duration one, together with waiting. Every nontrivial cycle has zero displacement, so $K_{\mathsf{Sch}}=\{0\}$. The first edge nevertheless moves a dependency by $L$ in one unit of time. For $L>c$ it violates a physical speed $c$. The finite offset in (eq:finiteoffset) cannot be omitted.

For cardinal hops of length $a$, one per model interval $\tau$, or a wait,

For an isotropic physical speed $c=a/\tau$,

Then

Indeed,

whereas the disk velocity $(c/\sqrt{2},c/\sqrt{2})$ lies outside the diamond.

This corrects an orientation ambiguity in the proposed plan: the diamond is safe relative to the equal-speed disk; it does not cover every physically possible direction. VII-E's directional example concerns that failed reverse coverage CheungVIIE (prop:direction).

Thus

is causal safety, while

is an information-coverage requirement. Neither proves simulation accuracy.

Choose $c_0>0$ and $0\le t\le T$. In one spatial dimension let

This is also the coordinate causal velocity set of the regular metric

Communication from $(s,x)$ to $(t,y)$, $t\ge s$, is possible exactly when

Its earliest arrival is

provided that value lies in the declared time window.

Proof. Integrating the velocity bound proves necessity. For sufficiency, put

Its speed is bounded by $c_0/(1+r)$, and it reaches $y$. Solving the equality in (eq:varyingreach) gives the arrival formula.

A register route with this printed profile is certifiable even when a constant-speed route with the same endpoints is not. A controller may select among finitely many such routes only after its selecting record reaches the route's departure port. Preloading a known function $c(t)$ is not an instantaneous global feed-forward operation.

Taking an independently declared finite sequence of transport and local gates along these routes gives a nontrivial exact discrete target. Applying Theorem thm:physical proves its causal safety; equality of the implemented gates with the declared target proves its accuracy. No wave-equation discretization is claimed.

Let $\Phi ,\Psi$ have the same input and output spaces. For a fixed output $B$, suppose

Then

In particular, a half-diamond error bound $d_{\diamond }({\Phi }_B,{\Psi }_B)\le ϵ$ contributes $2ϵ$, not $ϵ$, on the right.

Proof. For a fixed reference, input, and two interventions, let ${\rho }_{\mathcal{I}}$ and ${\rho }_{\mathcal{J}}$ be the resulting states. The two simulation-error terms obey

and the same holds for ${\rho }_{\mathcal{J}}$. The triangle inequality for the output difference therefore adds $2{ϵ}_B$ in trace norm and ${ϵ}_B$ after the prefactor $1/2$. Take the supremum and use ${\mathrm{Sig}}_{\Phi }\le 1$.

An expectation estimate for one state or one observable does not supply the diamond hypothesis. The local norm above still quantifies over every input and reference.

For channels ${\Phi }_j,{\Psi }_j$ between matching successive spaces, if

then

This includes separate encoding, discretization, transition, compilation, decoding, and hardware stages when corresponding channel comparisons have actually been supplied.

Proof. Insert one difference at a time between unchanged channel prefixes and suffixes. Channels have diamond norm one, and the norm is submultiplicative. Apply the triangle inequality Watrous.

If a target has leakage at most ${ϵ}_{\mathsf{Phys}}$, the implemented leakage is at most

in the full-diamond conventions above.

An uncertainty in a geometric arrival time is not automatically a channel-norm uncertainty. It can instead be handled by the robust geometric margins below, or by a separate dynamical continuity bound.

For bounded self-adjoint time-dependent generators $H(t),\tilde{H}(t)$, with the same initial unitary,

Consequently,

Proof. Differentiate $U(t,r)\tilde{U}(r,s)$ and integrate. All surrounding propagators are unitary. For channels, expand $UX{U}^{†}-\tilde{U}X{\tilde{U}}^{†}$ into two terms, also after tensoring with an identity.

This is one valid way to turn a specified finite-dimensional geometry-dependent generator estimate into a channel error.

Continuous-time local Hamiltonian evolution generally has tails, unlike an exact bounded-range circuit. Lieb–Robinson theory supplies sharper volume-uniform bounds under suitable interaction assumptions BravyiHastingsVerstraete. We give a deliberately conservative finite-volume proof sufficient for this paper.

Let $G=(X,E)$ be a finite graph and

where $h_x$ acts at one site, $h_e$ at the two endpoints of $e$, and all terms are bounded, self-adjoint, and piecewise continuous. Put

Let $d_G(A,B)$ be the minimum graph distance between the two nonempty disjoint regions.

For the unitary evolution channel ${\Psi }_{t,s}$, and $d=d_G(A,B)\ge 1$,

For every $\mu >0$, the tail is also bounded by

The bounds are independent of the dimension of an untouched reference. They are not claimed uniform in volume.

Proof. Pass to the interaction picture for the onsite terms. Their propagator is a product of onsite unitaries and preserves every support. It conjugates each two-site interaction without changing its norm or support.

For an output observable $O_{BR}={O_{BR}}^{†}$ with $\Vert O_{BR}\Vert \le 1$, expand the interaction-picture Heisenberg evolution in its norm-convergent time-ordered nested-commutator series. An order-$n$ term has norm at most

Summing the ordered integrals gives $(2\Lambda {)}^n/n!$.

A nonzero commutator with an edge term requires that edge to meet the current support. Each commutator can extend the graph distance from $B$ by at most one. Thus every term of order $n<d$ acts as the identity on $A$. The same statement holds when the coefficients are arbitrary operators on $R$.

Let $O_{<d}$ be the partial sum of those terms, including the onsite conjugation. Its support excludes $A$, and the remainder has norm at most the tail in (eq:tail). The two intervention adjoints agree on $O_{<d}$ and are contractions in operator norm. Their difference on the remainder has norm at most twice the tail. Duality with the trace norm and the factor $1/2$ in (eq:signal) give (eq:tail).

For $n\ge d$, insert $1\le e^{\mu (n-d)}$ and sum the exponential series.

One may choose an error-dependent influence radius by requiring the tail to be at most a prescribed ${ϵ}_{\mathsf{Phys}}$. This is an approximate dependency region of the named Hamiltonian model, not a strict Lorentzian cone.

For two sites separated by a positive assigned geometric distance, take $H=J{X}_A{X}_B$, $J>0$. Prepare $B$ in $|0⟩$, and use replacement interventions preparing $A$ in $|+⟩$ or $|-⟩$. The two $B$ outputs are

Their trace distance is $\left|\sin (2Jt)\right|$, nonzero for arbitrarily small positive $t$. A finite-dimensional interaction of this type therefore does not have an exact geometric propagation cutoff.

Let $H={{\sum }_{j=1}}^m{H}_j$ be time independent and define

For $t\ge 0$,

Proof. For two terms, a double Duhamel expansion compares their joint exponential with the ordered product; the error is bounded by ${\delta }^2\Vert [H_1,H_2]\Vert /2$ on an interval of length $\delta$. Iterating over the finite sum gives the one-step bound ${\delta }^2{C}_H/2$, since $\Vert [H_j,{\sum }_{k<j}{H}_k]\Vert \le {\sum }_{k<j}\Vert [H_j,H_k]\Vert$. Telescoping $n$ unitary steps yields the first estimate. The second follows from ${\Vert {\mathrm{Ad}}_U-{\mathrm{Ad}}_V\Vert }_{\diamond }\le 2\Vert U-V\Vert$.

Combining Theorem thm:tail, Proposition prop:trotter, and Theorem thm:signal-stability certifies the Hamiltonian route at Target A, including reference systems.

It does not automatically give Target B at the target Hamiltonian time: assigning every small nonlocal block gate an arbitrarily short geometric duration can violate the physical communication bound. A causal routing implementation needs a separate clock and latency certificate. More efficient local-Hamiltonian simulation algorithms use Lieb–Robinson decompositions HaahHastingsKothariLow; no efficiency claim from that literature is needed for our elementary estimate.

Let $C_{\pi /2}$ be the spatial Euclidean cone of total opening $3\pi /2$, defined by gluing the boundary rays of that wedge, not by quotienting the whole plane by a finite rotation group. Let $d_C$ be its intrinsic length distance. Define

Away from the axis the metric is

At the axis we use its metric-length completion.

A future causal curve is a locally absolutely continuous curve whose spatial metric speed is at most its nonnegative time derivative.

For $p=(T,x)$, $q=(T^{\prime },y)$,

This relation is reflexive and transitive, including at the axis.

Proof. Integration of the speed inequality proves necessity. The cone is a proper length space: bounded closed sets are compact. A minimizing sequence of spatial paths can be parametrized with uniformly bounded speed in a common compact ball; uniform convergence gives a length-minimizing path. Parametrize that path at speed at most one in the available time, inserting a wait if necessary. This proves sufficiency. The triangle inequality gives transitivity.

This is the physical comparison relation of the cone application. It is not inferred from a circuit graph. The spinless conical interpretation is classical DJH; no field dynamics is derived here.

In each of VII-F's three developed quadrants, write $x=\xi v_i+\eta v_{i+1}$, $\xi ,\eta \ge 0$, and set

The functions agree on identified rays. For $-1\le s\le 1$, define

VII-F proves the complete geometric certificate, including its affine holonomy, collar, and three-to-six particle link CheungVIIF (thm:certificate). For self-containment of the causal argument, the required estimates are verified here.

Let

For every $s,s^{\prime }\in [-1,1]$, $s^{\prime }\ge s$,

Every face is spacelike, with induced-metric lower bound $1-{{\lambda }_0}^2=1/9$. At $s=0$ the limiting graph remains embedded and uniformly spacelike in the stratified Lipschitz sense.

Proof. The nonconstant spatial gradients are

up to rotations. Their norms are at most ${\lambda }_0$. Maximum and minimum preserve the common Lipschitz bound. Integrate it along spatial paths and take their length infimum.

For fixed $x$, the minimum–maximum expression after the leading $s$ is nondecreasing and one-Lipschitz in $s$, proving the time estimates. The induced metric on an affine graph with gradient $a$ is $I-a{a}^{\mathsf{T}}$, bounded below by $(1-{{\lambda }_0}^2)I$. Projection of the graph to the cone is a homeomorphism. No vanished germ cell is counted as a positive-dimensional regular face at the event.

The spacetime axis and deficit remain fixed. The incidence changes at $s=0$; it is not a degeneracy of the supplied causal relation.

Choose finitely many persistent spatial marks

Register $v$ at slicing parameter $s$ is localized at

These marks are additional simulation data.

VII-F's abstract location transport is the identity on persistent labels, but some labels change their geometric realization. For example, an edge that formerly meets the particle need not meet it after the event. Identity of a cell name does not prove identity of a physical localization or a causal transport route. We therefore do not use that transport as a worldline certificate.

For an input mark $x$ at $s$ and output mark $y$ at $s^{\prime }\ge s$, it is sufficient that

to certify $p_x(s)⪯p_y(s^{\prime })$. Consequently every complete network satisfying this inequality for all dependency edges, including its encoding, decoding, controls, and transition, obeys (eq:main-containment) across $s=0$.

Proof. By Lemma lem:slice-estimates,

Condition (eq:conegate) makes the right side at least $d_C(x,y)$. Apply Lemma lem:conecausal and then Theorem thm:physical. The estimates hold on both sides and at the event.

A spatial route parametrized by $s$, with metric speed at most

lifts to a future causal curve $(F_s(x(s)),x(s))$. In each regular developed chart, the Euclidean ball $\{v:\Vert v\Vert \le v_{\ast }\}$ is an admissible conservative velocity set in the slicing parameter.

Proof. For a short increment $\delta s>0$,

Taking the almost-everywhere metric derivative gives

The speed restriction makes this at least $|\dot{x}|$. Thus the lifted curve is causal.

This is an inner causal bound, not the entire physical velocity set. No equality with a schedule cone is asserted.

A nominal dependency has slicing-time gap $h$, nominal endpoint distance at most $D$, spatial localization radii $r_{-},r_{+}$, slicing-time errors at most ${ϵ}_s$ at each endpoint, and additional absolute spacetime-time errors at most ${ϵ}_T$ at each endpoint.

It remains causally admissible if

Strict inequality is a positive robustness margin.

Proof. The actual spatial distance is at most $D+r_{-}+r_{+}$, and the actual slicing gap is at least $h-2{ϵ}_s$. The time-difference estimate of Theorem thm:conegates then loses at most another $2{ϵ}_T$. Rearrangement gives (eq:robustcone).

The event guard of VII-F is an exact stored-clock test. This margin concerns physical port placement and availability. It is not a replacement of that guard by a rounded edge report.

Fix any one finite forward-domain event circuit $U_{\mathrm{code}}$ from VII-F. Its input, history, work, and code registers are finite; work is returned to zero on the forward domain, and history is retained CheungVIIF (thm:implementation).

Two additional choices are essential:

1. all code and location-descriptor bits are placed at a declared controller station;

2. a model-time interval is assigned to their actual finite computation.

A location descriptor stored at the station is not a remote quantum register. No instantaneous acquisition of unknown remote location data is implied.

For Target A, one may interpret $U_{\mathrm{code}}$ as a macro event on local code data with identity matter channel. For Target B, positive-duration gates require a time buffer.

The code's germ timestamp $s=0$ is data describing the geometric event. It is not the time at which every computed code bit becomes physically available. A computation announced after the event is not an instantaneous global geometric update.

Let $o$ be the cone apex and choose the persistent ray mark

Set

Place one input qubit at each of $p_o(-h_{\ast })$ and $p_e(-h_{\ast })$. The independent target is:

1. update the local event code by its specified $U_{\mathrm{code}}$;

2. exchange the two quantum states at the output ports $p_o(h_{\ast })$, $p_e(h_{\ast })$;

3. retain code history and return clean work.

Thus its matter channel is ${\mathrm{Ad}}_{\mathrm{SWAP}}$, not the identity. This is a declared discrete carrier target, not an Einstein or scalar-field evolution.

The encoding is the identity on each input qubit, with local clean code/work initialization. The decoding reads the returned target factors and the declared event record; it has no nonlocal preprocessing.

The four matter-port times are

Both cross-port dependencies are causal. Their conservative slicing-time margin obeys

Proof. On a ray, $K=0$ and $H=2r_{\ast }/3$. At negative $s$, the $H$ branch is active; at positive $h_{\ast }>2r_{\ast }/3$, the $s$ branch is active. The apex time is $s$ before and $2s$ after the event. Since $d_C(o,e)=r_{\ast }$, the causal claims follow from Theorem thm:conegates. Finally, ${\lambda }_0<1$ gives

Use the apex as the controller and rendezvous station. The model admits local finite quantum registers on its timelike axis under Contract contract:routing; no quantum-field boundary condition at the singular axis is asserted.

The qubit starting at $e$ travels radially inward at unit spacetime speed. It can arrive at the apex at

The other qubit is already there. Both wait until $T=0$.

Let $L$ be the finite number of elementary gates in the event-code circuit plus three CNOT gates implementing SWAP. Execute them at the apex, each with assigned spacetime duration

All gates finish before $T=h_{\ast }/2$. Route the appropriate output qubit to $e$ and retain the other at the apex. Since

both output ports are reached in time, with waiting as needed. The code remains at the apex.

For this declared target and architecture:

1. every route and every elementary gate is causally admissible in the independently supplied cone;

2. the implemented endpoint channel equals

   $${\mathrm{Ad}}_{U_{\mathrm{code}}}\otimes {\mathrm{Ad}}_{\mathrm{SWAP}},$$

   with histories retained and work returned on the canonical forward domain;

3. accuracy is exact, including with every untouched entangled reference;

4. the two cross-port matter signalling strengths are one;

5. the model-time computation is compatible with the geometric event at $s=0$, without assigning a geometric meaning to intermediate compiler code states.

Proof. The incoming and outgoing radial trajectories obey $|\dot{x}|\le \dot{T}$. Storage is along the fixed timelike axis or the fixed ray station. Every elementary gate is local at the apex with positive duration. Thus Target B holds.

Transport carries the same tensor factors without measurement. At the rendezvous, the three CNOT gates give the SWAP exactly, while the independent code gates give $U_{\mathrm{code}}$. The two operations act on distinct factors, so the endpoint channel is their tensor product. Unitary identities remain identities with every reference.

For signalling from the apex input to the ray output, compare replacement interventions preparing $|0⟩$ and $|1⟩$. SWAP gives orthogonal ray outputs, with trace distance one. The reverse direction is identical. These signals follow causal routes, rather than exceeding physical possibility.

The code describes the event germ at timestamp zero and becomes available after its finite computation. The supplied slicing passes continuously through zero during the bracket. Its metric and holonomy do not depend on the compiler states. The decoder statement concerns the canonical completed code, exactly as in VII-F.

A hardware implementation may give each local gate a hardware duration ${\delta }_{\mathrm{hw}}>0$. The certificate records

during the local stage. Along the positive-time apex worldline, $s_j=T_j/2$. Logical gate depth, hardware duration, and the model's slicing parameter are therefore explicitly different quantities.

The buffer is not a change in the cone's event. It is an added implementation latency around that event. An exact event code at model timestamp zero does not justify exposing its arbitrary compiler intermediate states as geometric configurations.

Place any finite pre-event and post-event networks on the slices (eq:F), with complete control ledgers and the edge certificates (eq:conegate). Insert either:

1. the local identity-on-matter macro event with independently certified code availability; or

2. the buffered benchmark of Theorem thm:eventbenchmark.

The complete decoded network satisfies (eq:main-containment). If its three stage channels have full-diamond errors ${ϵ}_{-},{ϵ}_{\mathcal{R}},{ϵ}_{+}$ against a declared target, the total error is at most their sum.

Proof. Use Proposition prop:eventcomposition, Theorem thm:conegates, and Proposition prop:telescope.

This certificate does not require the graph of cell incidence to remain fixed. It requires the registers, routes, and causal port relation to remain specified. A new graph edge cannot be used as an instantaneous communication link merely because a geometric cell has been born.

Two distinct points on any one slice ${\Sigma }_s$ are spacelike separated. A same-slice remote SWAP has unit signalling between them and is not a causally faithful transition for the cone relation.

Proof. The slice estimate gives

for distinct points. Lemma lem:conecausal therefore excludes either causal direction. The replacement-state SWAP test gives unit signalling, as in Theorem thm:eventbenchmark.

Thus pre-event and post-event spacelikeness, event reversibility, and an unchanged deficit do not by themselves certify transition locality.

A cardinal-hop schedule has the velocity diamond of (eq:diamond-disk). Replacing its two-site operations by identities leaves the same conservative support envelopes but removes cross-site influence. Replacing a permitted operation by SWAP can realize unit influence on that edge.

Assigning every cardinal dependency duration $\tau \ge a/c$ gives causal endpoint inclusion in a flat speed-$c$ geometry. Changing only hardware parallelism does not change that requirement. This finite example separates all three structures.

For a nontrivial accuracy test, prescribe SWAP as the target. The identity implementation is safe, but

on two qubits: the input $|01⟩$ gives orthogonal outputs $|01⟩$ and $|10⟩$, while the half-diamond distance between channels is at most one.

Let two model sites be spacelike separated during a short comparison interval. An encoding that swaps their factors, followed by an identity computation and local decoding, gives a decoded SWAP. The computation alone has no communication, but the encoded–computed–decoded channel does. This violates the localization contract and has unit unwanted signalling.

A distant controlled gate can signal through its classical control even if its quantum data wires are local. It must receive the control record after a certified causal route. Declaring the controller “external” does not remove the dependency if its value depends on model inputs.

Measurement on one half of an entangled state can change the conditional remote state. Ignoring the outcome preserves the remote reduction. A causal claim about an outcome-conditioned output must include the success flag and its communication. The deterministic intervention theorem does not assert absence of steering after conditioning.

A long-range SWAP can have constant abstract gate depth. If its port separation is $D$, its geometric communication latency cannot be inferred from that depth. Conversely, many local compiler gates at one station can have large depth without expanding the spatial influence region.

VII-F's seam-compatible family is essential: its unequal-slope vacuum maximum fails to descend through the cone seam CheungVIIF (lem:transplant). A causal certificate cannot repair a multivalued geometric localization.

Nor may one replace the fixed full affine seam word by a time-shifted word with the same rotation. The marked fixed axis and the actual spacetime are part of the physical model. A different deficit or time translation is a different comparison problem, not a scheduling convention.

A certificate consists of the following finite data and proved bounds.

1. Physical structure: event space, causal relation, time window, and continuation through any singular or incidence event.

2. Target dynamics: a named channel $\Psi$, independent of the certificate's causal-safety conclusion.

3. Localization: model, encoding, ancilla, routing, controller, and decoding ports, with any uncertainty regions.

4. Complete dependency network: jointly valid support envelopes, including classical branches, resets, and discarded systems.

5. Clocks: logical depth, hardware availability, model port times, and their declared calibration.

6. Finite-time causal proof: edge routes or endpoint certificates, with margins. An asymptotic velocity polytope is recorded separately.

7. Event transition: its channel, locality, record-availability time, and allocation or remapping of every affected factor.

8. Accuracy: exact equality or a justified channel-norm bound against $\Psi$, including all stated references.

9. Scope: output regions, intervention class, parameter domain, and unsupported physical identifications.

Suppose the finite data satisfy Definition def:certificate, with exact edge containment and target leakage ${ϵ}_{\mathsf{Phys}}(A,B)$. Then:

1. the complete implementation has the scheduled support of Theorem thm:schedule;

2. exact causal edge containment gives (eq:main-containment);

3. if the reduced full-diamond simulation error is at most ${ϵ}_B$, then

   $${\mathrm{Sig}}_{\Phi }(A\to B)\le \min \{1,{ϵ}_{\mathsf{Phys}}(A,B)+{ϵ}_B\};$$

4. certified pre-event, transition, and post-event stages compose without a new instantaneous remapping assumption;

5. a routing-and-local-gate realization satisfying Contract contract:routing also meets Target B.

Proof. The assertions are Theorems thm:schedule, thm:physical, and thm:signal-stability, Proposition prop:eventcomposition, and the elementary routing construction. Each ingredient is proved for complete channels, so entangled input references are retained.

The certificate is an analytical mathematical object. A script checking a finite example can corroborate its arithmetic, but a pass flag is not a proof of its universal hypotheses.

Within the finite models and contracts stated here:

1. actual reference-assisted influence is bounded by the complete scheduled ancestor set;

2. an independently certified physical embedding yields exact directional causal containment;

3. approximate decoded simulation obeys the norm-consistent signalling bound (eq:signal-stability);

4. finite-time periodic propagation, varying-velocity communication, and Hamiltonian-tail routes have the stated independent certificates;

5. the fixed-cone slicing event preserves the supplied causal relation, and causally certified transition channels compose through it;

6. the printed buffered event benchmark is both causally realizable and exactly accurate for its declared nontrivial quantum transport target.

No identification of scheduled reachability with the whole geometric causal past is made.

Proof. Combine Theorems thm:schedule, thm:physical, thm:polytope, thm:signal-stability, thm:tail, thm:conegates, and thm:eventbenchmark, with Propositions prop:varying, prop:trotter, and prop:eventcomposition.

The result closes the requested causal-comparison problem for explicitly declared finite architectures. It does not close every possible simulation problem on the cone.

A continuum-field successor would additionally need a specified field algebra and state, local encoding of its admitted observables, a target propagator or evolution, a discretization error, and a physical detector model. The finite transport target here supplies none of those by analogy.

Likewise, this paper does not construct the separate characteristic quantum coefficient left open in the supplied XIV–XV material. Its causal network proofs do not define a missing distributional product, choose an extension, or supply a quantum Ward identity. No O.1 or quantum admission status is changed.

The distinction from gravitational dynamics is unchanged:

The cone metric, deficit, and worldline are inputs. No Lorentz invariance or spacetime emergence is derived from the schedule. No black-hole, recovery, Page-curve, or holographic claim is made.

The first line is causal safety. A separate comparison with $\Psi$ is simulation accuracy. The last line is the explicit bridge to VII-F's event, not an assertion that its cell adjacency graph is a physical light cone.

scope. The new results concern finite channels and complete control networks, timed causal embeddings, directional and finite-time bounds, norm-stable signalling, an elementary Hamiltonian-tail estimate, and a buffered fixed-cone carrier realization.

scope. The causal relation and the ideal transport architecture are declared model inputs. Their consistency is proved, but a laboratory realization or quantum-field derivation of those primitives is not asserted.

scope. VII-F supplies the geometric three-to-six certificate and finite event-code permutation. This paper adds physical localization, positive-duration implementation timing, causal routing, and transition containment. The stored event timestamp is not an instantaneous broadcast.

scope. References to Cheung's works identify the supplied manuscripts and their stated results. They are not claims of journal publication, independent verification, or transfer between inequivalent models.

. No successful pdflatex execution, proof-assistant check, solver run, independently executed numerical certificate, referee review, or physical implementation is asserted.

Fix one VII-F code instance with report precision $b$, clock precision $d$, permission width $r$, and $n$ original location descriptors. Its displayed bound is

including retained history and clean work. Let $G_{\mathrm{code}}$ be any finite elementary gate count furnished by that instance.

The additional transport benchmark uses:

• two quantum matter registers;

• three CNOT gates for SWAP;

• causal carriage and storage of those two tensor factors, without measurement;

• no auxiliary matter register.

Thus $L=G_{\mathrm{code}}+3$ is a finite sufficient count for the local apex stage.

All code wires are at the apex station. The two matter wires are gathered there before $T=0$. With

the local stage ends before $T=1/8$, since $h_{\ast }=1/4$. The output deadline is $T=1/2$. The radial route of length $1/64$ then leaves strict positive slack.

The event code is decoded only at the completed forward boundary, with history omitted from the geometric observation but retained physically. No name descriptor is used to change the matter routing; the two-ray transport script is fixed in advance. If a later script reads code-dependent routing, the corresponding record wires and their arrival times must be added to the network.

Suppose a macro channel satisfies a physical endpoint no-signalling condition. This says what its outputs can depend upon, not how to generate them. A realization may require resource states, communication, or a decomposition whose ordinary circuit paths overestimate its actual influence.

The gather–operate–distribute proof avoids this gap by imposing a common station and explicit routes. The QCA localizability theorem ArrighiNesmeWerner and quantum causal graph dynamics ArrighiMartiel address broader settings under additional hypotheses. Their graph notion of causality does not automatically supply the geometric port timings of this paper.

Consequently: