VII-J-ST: Scheduled dependencies and certified causality on a prescribed conical spacetime A local Grade S Enter adaptation with complete clock instruments, directional certificates, and buffered regular-part transport

We give a self-contained local Grade S adaptation of VII-J's finite-network causality programme to I-B's prescribed $3+1$-dimensional conical product. The metric, deficit $\pi /2$, causal relation, slicing family, preparations, and communication permissions are inputs. Complete timed dependency graphs, including classical controllers, measurements, records, encoders, and decoders, bound the Heisenberg support of every jointly observed output. A causal embedding then bounds reference-assisted intervention influence by the prescribed spacetime causal relation.

The operational addition is an explicit regular-part realization through the parameter of VII-F's transverse three-to-six incidence. Unlike the source apex-register benchmark, no quantum register, gate, or communication path meets the singular worldsheet. A destructive two-qubit clock controls finite event metadata; a buffered gather–operate–distribute protocol implements a nontrivial two-qubit exchange, preserves arbitrary retained reference correlations, and has a unit signalling witness on an allowed route. The complete recorded channel, positive durations, blank registers, controller storage, copied records, and finite instruction reserves are specified.

Directional certificates retain both cycle-velocity polytopes and finite-time offsets. A finite potential inequality exposes transient directional costs that an asymptotic velocity inclusion can miss. Approximate simulation is treated separately: signalling changes by at most twice the half-diamond error, while a finite-volume Hamiltonian tail is reference-stable but not an exact Lorentzian cutoff. These assembled certificates may provide a useful reproducible benchmark for distinguishing logical simulation accuracy, operational influence, and elementary causal realizability. The novelty claimed is the supported assembly and regular-part correction, not the general principles of channel locality, periodic-graph velocities, or destructive measurement. No continuum-field simulation, spacetime emergence, Einstein dynamics, or inherited quantum response coefficient is constructed.

boundary{ This manuscript proves a finite operational certificate on an installed classical geometry. The geometric event is a prescribed incidence change. It is not caused, detected at the singularity, or reconstructed by the finite clock.

The source VII-J benchmark permits a register on the apex. That permission is incompatible with the regular-part laboratory used here. We therefore construct a different, explicitly regular benchmark rather than silently transferring the apex construction.}

VII-J separates scheduled dependencies, intervention influence, and an independently supplied causal structure CheungVIIJ. I-B equips a prescribed conical product with finite names, accessible algebras, clock calibration, and communication permissions CheungIB. We retain those distinctions and prove the interfaces needed for their local combination.

Cheung's Enter framework treats destructive readout as an input–output and access discipline: measured quantum names are consumed, classical outcomes are returned, and specified unmeasured factors survive CheungEnter,CheungI. This does not identify a classical record with the return of an unknown quantum input. Nor does a channel equality identify a thermal implementation, a geometric interpretation, or a gravitational dynamical law CheungPhilosophy.

We adopt an explicit finite Born instrument. Its channel identity is proved below. We do not rederive the structural measurement characterization of Manuscript I from an abbreviated hypothesis list, and do not extend that characterization to fields.

The operational motivation can be summarized as four questions: what is supplied, what may be accessed, what is returned, and what additional construction supports the proposed interpretation? The present answer supplies a classical background and a finite network; it returns classical records and surviving finite quantum payloads.

A local Grade S certificate consists of:

1. prescribed classical geometry and a supported regular operational presentation;

2. explicit finite channels and complete control interfaces;

3. a finite destructive-clock instrument with its calibration;

4. proved support, influence, and causal-embedding statements;

5. a nonempty finite transport realization with a resource ledger.

Derivation of the geometry from operational records, called Grade E here, is not included.

Let $\Psi$ be a declared finite target channel and

an implementation on the same comparison interfaces.

Target A requires a support or signalling statement for $\Phi$, together with a separate accuracy comparison to $\Psi$.

Target B additionally realizes every preparation, gate, communication, clock operation, controller read, record, reset, and decoder by the admitted local and causal primitives on the supplied spacetime.

Endpoint channel correctness does not by itself construct Target B. A physical embedding of dependency endpoints is a sufficient support certificate; an actual elementary realization also needs the declared implementation primitives.

. VII-J is the source causality manuscript; VII-F supplies the prescribed slicing family; I-B supplies the spacetime-interface pattern CheungVIIJ,CheungVIIF,CheungIB. Other sheets are documentary comparisons. Their citation does not import unprinted proof hypotheses. Every positive mathematical claim used in this release has a proof below.

Fix a length unit $\ell >0$, a speed $c>0$, and a time origin $T_{\ast }$. Write

The regular background is

where $r>0$, $\varphi$ has period $2\pi$, and the stipulated completion has singular locus

The defect is a worldsheet. Only its fixed-$z$ sections are worldlines. No smooth chart, pointwise Einstein tensor, or field boundary condition is asserted on $\mathcal{W}$.

On a lifted regular angular interval,

is a local Minkowski development. Static observers have $\mathrm{d}{\tau }_{\mathrm{proper}}=\mathrm{d}T$. The metric lapse in $T$ is one. The spinless seam gluing fixes $T,z$ and rotates the transverse developed plane by $\pi /2$ in the chosen gluing convention. These are the prescribed I-B data CheungIB.

The regular spacetime $M$ is not globally hyperbolic.

Proof. Fix $r_0>0$, an angle, and $z=0$. Take two events at that spatial point with times $T_{\ast }\pm 2r_0/c$. Events at time $T_{\ast }$ and radii $r_n\downarrow 0$, at the same angle and axial coordinate, lie in their causal diamond by radial causal paths. They have no convergent subsequence in $M$. The diamond is therefore noncompact.

No global Green or Cauchy theorem is inferred. All operational paths below lie in a relatively compact regular developing tube.

Use a cut wedge of angle $3\pi /2$, divided into three right-angle sectors. In each sector let the dimensionless coordinates be $\xi ,\eta \ge 0$, and put

The functions agree on identified sector rays. The prescribed product markings are

with no axial dependence in $F_s$. The formula is VII-F's transverse family, explicitly lifted rather than identified with a three-dimensional cell count CheungVIIF.

Set

On every regular developed region, $F_s$ is spatially $\lambda$-Lipschitz, including the axial variable. For $s^{\prime }>s$,

The graphs are embedded, and their regular affine pieces are spacelike with induced dimensionless metric bounded below by $I/9$.

The prescribed transverse cellulation has one incidence-event parameter, $s=0$, where its singular transverse link changes from three incident faces to six.

Proof. The active gradients are

up to sector rotations. Minima and maxima preserve their common Lipschitz bound. The graph metric is $I-a{a}^{\mathsf{T}}$; its least eigenvalue is at least $1-{\lambda }^2=1/9$. Graph projection gives embeddedness. For fixed $x$, the term after the initial $s$ in (eq:family) is nondecreasing and 1-Lipschitz in $s$, proving (eq:time-order).

For the incidence, use

When $s\le 0$, the $h$-branch dominates and one face in each sector meets the apex. When $s>0$, in the half-sector $\xi \ge \eta$, the new boundary is $10\xi +\eta =15s$; its symmetric counterpart is $\xi +10\eta =15s$. Their vertices are

They define two distinct new apex triangles in each sector. Their outermost radial coordinate is $30s/11<3$, so they do not meet the persistent crest $\rho =3$. Within each open epoch these boundaries have fixed combinatorics, and all new coordinates vanish only at zero. Thus the normal transverse link changes exactly once from three to six.

The source's full lengths, areas, affine chart words, primitive winding, and peripheral matching remain documentary geometric data CheungVIIF. They are not needed to infer the finite-channel support theorem. In the product, the event locus is the line $\{\tau =0,x=o\}\subset \mathcal{W}$; it is not an isolated point event of the $3+1$-dimensional background.

A geometric atlas may describe a finite transverse patch and a finite axial slab in the conical completion, including its stipulated singular stratum.

The operational support is different: it is the closure of a finite time interval times a bounded convex developed spatial tube $U$, with $\overline{U}\cap \mathcal{W}=\varnothing$. All initial carriers, blanks, programme words, controller registers, and ancillary inputs are declared there. There is no undeclared incoming boundary data.

This is not an initial-boundary-value problem for a field. A meridian surrounding the defect needs a developing atlas; a single regular laboratory chart need not and does not contain the apex.

For finite input factors indexed by $X$, write

Classical registers have a distinguished basis and diagonal observable algebra. Their canonical matrix extension first pinches that basis. Every controlled branch map is fixed independently of the unknown input density operator.

A channel is completely positive and trace preserving (CPTP). We use

Standard finite-channel conventions may be found in Watrous. Instruments retain their complete records DaviesLewis.

For input region $A$, output region $B$, and ${\Phi }_B={\mathrm{Tr}}_{B^c}\Phi$, define

where $E$ is any finite untouched reference, $\rho$ is any input-reference state, and

The interventions ${\mathcal{I}}_A,{\mathcal{J}}_A$ are deterministic CPTP maps on the same interface. They do not act on $E$.

Thus $0\le {\mathrm{Sig}}_{\Phi }\le 1$. Postselected normalization is not an intervention in this definition.

For a finite channel $\Phi$, the following are equivalent:

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

2. a channel $\Xi :\mathcal{B}({\mathcal{H}}_{A^c})\to \mathcal{B}({\mathcal{H}}_B)$ satisfies ${\Phi }_B=\Xi {\mathrm{Tr}}_A$;

3. $${\Phi }_B^{\ast }(\mathcal{B}({\mathcal{H}}_B))\subseteq I_A\otimes \mathcal{B}({\mathcal{H}}_{A^c}).$$

The factorization holds with every untouched reference.

Proof. For (ii), local trace preservation leaves the $A^{c}E$ reduction unchanged, proving (i). Conversely, compare the identity intervention with the local replacement

Condition (i), even without a reference, implies ${\Phi }_B={\Phi }_B({\mathcal{R}}_A\otimes \mathrm{id})$ on all states and therefore on all operators. Set

It is CPTP and proves (ii). Taking adjoints gives (iii). Conversely, (iii) makes every output expectation depend only on the $A^c$ reduction; the same replacement argument gives (ii). Tensor the factorization with ${\mathrm{id}}_E$.

This is the precise finite criterion used in VII-J CheungVIIJ (:nosignal). It does not identify correlation with signalling.

Encode a classical input bit $b$ as $(R,R\oplus b)$, with $R$ uniform. Each individual share is independent of $b$, but their joint parity equals $b$. Separate marginal tests cannot replace a joint-output certificate.

For an entangled pair, selecting a local measurement outcome can change a conditional remote state. Summing the outcomes leaves the remote marginal unchanged. The latter is the deterministic comparison in Definition def:signal; an outcome used remotely requires a declared classical wire.

A complete network is a finite directed acyclic circuit of CPTP maps, including:

• quantum and classical carrier ports and identity wires;

• fixed independent ancillary preparations;

• gates, storage, allocations, resets, and discards;

• destructive measurements and all retained outcomes;

• every controller variable read to choose a gate, guard, route, reset, or availability time;

• the encoder and decoder, with their actual support.

A finite adaptive instruction tree is padded to a finite common envelope containing every branch and its selecting controls. There is no hidden state-functional scheduler.

For a conservative dependency graph, connect every input of a gate to every output. A reduced dependency graph is admissible only if it has a proved support statement for every joint subset of that gate's outputs, including references. Single-output no-signalling alone is insufficient.

For joint terminal output $B$, let ${\mathrm{Sch}}^{-}(B)$ be the model input factors with a directed dependency path to $B$.

For the channel $\Phi$ of a complete network,

Inputs outside ${\mathrm{Sch}}^{-}(B)$ cannot signal to $B$, including when the observer also retains an arbitrary untouched reference. Replacing a gate outside the backward region by any CPTP map on the same ports does not change that output reduction.

Proof. Pull an output observable backward in reverse topological order. A gate whose outputs all carry identity is invisible, because its adjoint is unital. Otherwise the conservative graph adds only that gate's input ports. Certified reduced graphs use their joint-output support hypothesis instead. A fixed preparation creates no unknown model-input predecessor. Measurements, classical feed-forward, and resets are ordinary complete channel nodes in this argument. Induction gives (eq:schedule). Apply Theorem thm:nosignal. The same reasoning with reference observables proves complete equality. On an excluded gate the pulled-back observable is identity, also preserved by any replacement channel.

This reproduces the support mechanism of VII-J and the partitioned-channel comparisons of En03-A and En05-A CheungVIIJ,CheungLocality.

For layers ${\Phi }_j={⨂}_{B\in {\mathcal{B}}_j}{\Phi }_{j,B}$ on a finite metric site set $(X,d_{\mathrm{comp}})$, assume $\mathrm{diam}B\le \Delta$. Define backward sets by

Then

Proof. The support assertion is Theorem thm:schedule. Each backward layer adds at most one block diameter. Apply the triangle inequality inductively.

The radius is $(t-s)\Delta$, not $t-s$ without the normalization $\Delta =1$. The computational metric is an input; it is not inferred from the conical metric.

Two successive SWAPs have a diagram with cross-register dependencies but total channel $\mathrm{id}$. The first input has no influence on the second output. Thus scheduled ancestry can strictly overestimate actual influence.

Swapping adjacent independent channel nodes acting on disjoint factors preserves the complete channel, including records and references. It need not preserve an assigned physical timing certificate.

Proof. The two tensor-factor maps commute, also after amplification. A finite sequence of such swaps preserves their composition. Their new assigned port times need not satisfy the original causal inequalities, which are separate hypotheses.

Logical depth, hardware execution time, slicing parameter $s$, geometric time $T$, and clock-code labels are not identified.

Give each port $v$ a prescribed event $p_v\in M$. Every dependency edge $u\to v$ must have a certified future-causal route in the operational tube. For uncertain regions $L_u,L_v$, a conservative sufficient condition is

A weaker allowance must specify the correlated endpoint choices and their actual routes.

Suppose the full encoded–computed–decoded network is complete, and every dependency edge has the stated causal certificate. For a joint output $B$, let ${\mathrm{Phys}}^{-}(B)$ contain input factors whose assigned events can reach at least one output event of $B$. Then

Every intervention supported outside ${\mathrm{Phys}}^{-}(B)$ has zero signalling to $B$, with arbitrary untouched references.

Proof. Theorem thm:schedule gives the first inclusion. Concatenate the supplied causal routes along a scheduled path. Transitivity gives the second inclusion. Apply Theorem thm:nosignal to the complement of the physical past.

The distinction in (eq:physical) is the central VII-J certificate CheungVIIJ (:physical). A physically possible path with no installed wire shows that the second inclusion need not be equality.

The model permits exact carriage of a finite register along a specified future-causal curve; positive-duration local gates and complete instruments on co-located registers; local independent preparation; local storage, resets, and discards; and classical records carried on explicitly scheduled causal wires.

These are ideal finite-model permissions, not a field-theoretic or laboratory construction of the communication mechanism.

Suppose all gate inputs have supplied causal routes to one station event $r_{-}$, all outputs have supplied routes from a later station event $r_{+}$, and a finite local implementation fits between $r_{-}$ and $r_{+}$. Then Assumption ass:routing realizes that gate, with its records, ancillary factors, and references, as Target B.

Proof. Carry the input factors to the station, execute the stipulated local channel, and distribute its outputs. Identity carriage and the local channel give the intended complete composition on all operators. Every elementary dependency has an explicit causal route.

Pairwise causal input–output possibilities alone do not establish this common-station hypothesis.

If $\Phi ={\Phi }_{+}\mathcal{R}{\Phi }_{-}$ and the three channels have predecessor maps $P_{+},P_{\mathcal{R}},P_{-}$, then

Causal certificates compose in the same order. Independent allocations have no model-input predecessors; discards and all remapping controls must be included.

Proof. Compose the adjoint inclusions and then the corresponding causal paths.

All spatial coordinates in the next statements are dimensionless. Let a regular spatial route in the supported presentation have length $D$.

A spatial route parametrized over $s\in [a,b]$, with metric speed at most

lifts under $s\mapsto (F_s(x(s)),x(s))$ to a future-causal curve. In particular, if an actual regular route of length $D$ joins $x$ to $y$, then

is sufficient for a causal route between its corresponding slice ports. The statement holds across $s=0$.

Proof. Lemma lem:slices gives, for an increment $\delta s>0$,

The lift is Lipschitz. At almost every parameter its time derivative is at least $1-\lambda |\dot{x}|$. When $|\dot{x}|\le v_{\ast }$, this is at least $|\dot{x}|$, so the lifted tangent is future causal. For the final assertion, traverse the supplied spatial route at constant arc-length speed $D/(b-a)$, allowing waits.

The source's conservative speed is retained CheungVIIJ (:conespeed). It is an inner sufficient bound in the slicing parameter, not a derived invariant speed or the full physical velocity set.

A spatial distance defined as an infimum need not be realized by a path in the regular part: the limiting minimizer could meet the removed apex. Consequently a statement at exact distance equality needs a realized regular path. Our benchmark supplies such paths in a convex regular chart.

In a convex regular developed chart, let a nominal slice-time gap be $h$, nominal spatial endpoint distance at most $D$, spatial localization radii $r_{-},r_{+}$, slicing-time errors at most ${\epsilon }_s$ at each endpoint, and additional dimensionless geometric-time errors at most ${\epsilon }_{\tau }$ at each endpoint. All endpoint choices remain causally admissible if

Strict inequality gives positive slack.

Proof. The actual spatial distance is at most $D+r_{-}+r_{+}$, and the actual slicing gap is at least $h-2{\epsilon }_s$. The two-family estimate gives actual geometric-time difference at least

Under (eq:robust) it exceeds the actual spatial distance. The straight developed segment is available within the chart.

This is an availability certificate, not permission to replace an exact event guard by a rounded geometric report.

These are kinematic graph statements. An arbitrarily long periodic lift is not asserted to be embedded in the one bounded conical laboratory. Only finite prefixes supplied with actual routes enter Theorem thm:physical.

Let $Q$ be a finite strongly connected directed multigraph. Its vertices include schedule phase and register type. Edge $e$ has displacement ${\Delta }_e\in {\mathbb{R}}^d$ and duration $t_e\ge t_{\min }>0$. The translation lift permits the declared copies of these transitions. Waiting is permitted in a manner compatible with the phase model.

For each simple directed cycle $C$, define

This is the timed version of the periodic-graph velocity construction Fritz,CheungVIIJ.

Let $q=|V(Q)|$, $D_{\max }={\max }_e|{\Delta }_e|$, and $R_Q=(q-1)D_{\max }$. Every path completed within elapsed time $t$ has displacement

From any specified initial quotient vertex, the set of subsequential limiting displacement/time ratios, for unbounded elapsed times, is exactly $K_Q$.

Proof. Remove closed subwalks until the remaining walk has no repeated vertex. Decompose removed closed walks into simple cycles. Their total displacement is a time-weighted sum of cycle velocities, with total cycle time at most $t$. The zero vector accounts for unused time. The remaining simple walk has at most $q-1$ edges and displacement norm at most $R_Q$, proving (eq:offset).

Divide by $t$ to obtain one inclusion for limiting ratios. Conversely, take any convex combination of finitely many cycle velocities and zero. Repeat the cycles in integer counts whose durations approximate the chosen proportions on a scale tending to infinity. Strong connectivity supplies fixed connecting walks from the chosen start and between cycles. Their total cost is bounded for this fixed combination. Waiting realizes its zero component. The normalized displacement tends to the desired combination.

For a graph with several strongly connected components, only cycles that can be visited in the allowed component order may be combined. The convex hull of every cycle can overestimate attainability from a specified start.

Fix $n\in {\mathbb{R}}^d$ and $\beta \ge 0$. The following are equivalent:

1. $n\cdot v(C)\le \beta$ for every simple cycle;

2. finite potentials ${\pi }_v$ exist such that

   $$n\cdot {\Delta }_e-\beta t_e\le {\pi }_{\mathrm{head}(e)}-{\pi }_{\mathrm{tail}(e)}(e\in E(Q)).$$

Every path from $u$ to $v$, of duration at most $t$, then satisfies

Proof. Summing (eq:potential) around a cycle proves (i). Conversely, set $w_e=n\cdot {\Delta }_e-\beta t_e$. Every closed walk has nonpositive weight by cycle decomposition. Add a root with zero-weight edges to all vertices. Let ${\pi }_v$ be the maximum weight of a root-to-$v$ walk. Removing nonpositive closed subwalks does not decrease its weight, so a maximum exists among finitely many simple walks. Appending an edge gives ${\pi }_{\mathrm{head}}\ge {\pi }_{\mathrm{tail}}+w_e$, which proves (ii). Summation gives (eq:potentialpath); use $\beta \ge 0$ for a path completed before $t$.

This is a finite difference-inequality witness, not a new general graph-theoretic principle. It supplements the cycle polytope by explicit endpoint-dependent transient costs. For a physical directional bound $b(n)t$, the sufficient endpoint check is

A constant potential would instead certify every edge in that direction directly. Neither potential feasibility nor a cycle average supplies the missing physical routes.

Two quotient states have transitions $+L$ and $-L$, each of duration one, and waits. Every cycle velocity is zero. The first hop still travels $L$ in one unit. For a prescribed speed $c_0<L$, it is unsafe. The finite offset or potential difference detects the transient.

For compact convex $K\subset {\mathbb{R}}^d$, define $h_K(n)={\sup }_{v\in K}n\cdot v$.

For nonempty compact convex sets $K,L$,

If $L$ is a polytope, its defining facet normals suffice. If true support functions have certified errors ${\delta }_K(n),{\delta }_L(n)$, the nominal gap $h_L(n)-h_K(n)\ge {\delta }_K(n)+{\delta }_L(n)$ certifies true inclusion.

Proof. Inclusion implies the inequalities. If $v\in K\setminus L$, choose its nearest point $z\in L$. Then $(v-z)\cdot (w-z)\le 0$ for all $w\in L$, which strictly separates $v$ from $L$. The facet and error statements follow by substitution.

For cardinal hops of length $a$, one per interval $\delta$, or a wait, in two dimensions,

Their support functions are $(a/\delta ){\Vert n\Vert }_{\infty }$ and $(a/\delta ){\Vert n\Vert }_2$. The disk is not contained in the diamond: $(a/(\sqrt{2}\delta ),a/(\sqrt{2}\delta ))$ is a counterexample. Thus the diamond is safe but does not cover all physical directions.

If a constant physical velocity set $K_{\mathrm{phys}}$ contains $K_Q+\overline{B}(0,R_Q/t)$, then (eq:offset) certifies endpoint reachability within time $t$. It does not certify the intermediate implementation paths. For Target B, check the actual route velocities locally.

On a finite time-expanded DAG, earliest arrivals are obtained by the recurrence

including all waits and availability restrictions. Topological induction proves the recurrence exactly. The value is $+\infty$ when no path exists.

Two closed-form comparisons can be made without changing the prescribed local Minkowski geometry. In a one-dimensional regular rod coordinate $x$, the time reparametrization $\tau =a\log (1+u)$, $a>0$, gives

Therefore

This is time-varying, not spatially varying.

For a genuinely spatially varying coordinate speed, set $x=a\log (1+q)$, $q>-1$. Then

Both formulas follow by integrating the rod-distance bound; monotone null motion attains equality. They are coordinate descriptions of the installed regular geometry, not derived gravitational dynamics.

For channels on the same interfaces,

In particular, if $\Psi$ is nonsignalling and $d_{\diamond }({\Phi }_B,{\Psi }_B)\le \epsilon$, then ${\mathrm{Sig}}_{\Phi }\le \min \{1,2\epsilon \}$.

Proof. For fixed reference, state, and interventions, the two comparison errors each have trace norm at most ${\Vert {\Phi }_B-{\Psi }_B\Vert }_{\diamond }$. The triangle inequality and the $1/2$ trace-distance factor give one such full-diamond norm for the difference of the two distinguishabilities. Take suprema and exchange the channels.

This retains VII-J's norm convention CheungVIIJ (:signal-stability). An expectation bound for one state or one observable does not supply its diamond-norm hypothesis.

For compatible sequences of channels,

Every encoding, clock, controller, transition, communication, decoding, and failure output must be compared on matching interfaces.

Proof. Replace one factor at a time. The resulting telescoping sum has one channel difference in each term. Composition with channels contracts the diamond norm. Use the triangle inequality and divide by two.

A causally embedded implementation has exact zero signalling outside its certified physical past, irrespective of its accuracy. If its target has signalling $s_0$ on a forbidden pair, any half-diamond approximation to that target has error at least $s_0/2$. Conversely, an arbitrarily accurate approximation to a nonsignalling target need not be exactly nonsignalling.

Proof. Use Theorems thm:physical and thm:signal-stability. For the last assertion,

on a qubit differs from the constant channel ${\mathcal{R}}_0$ by half-diamond distance at most $ϵ$, but has signalling $ϵ$ between its input and output.

Let $G=(X,E_G)$ be finite and

with bounded piecewise-continuous self-adjoint onsite and two-site edge terms. Let ${\Psi }_{t,s}$ be its unitary channel.

For disjoint nonempty $A,B$, let $d=d_G(A,B)\ge 1$. Then

For every $\gamma >0$, the series is at most

The estimate is independent of the untouched reference dimension, but is not uniform in the number of edges.

Proof. Pass to the interaction picture for the onsite propagator, a product of onsite unitaries. It preserves support and all edge-term norms.

For a Hermitian output observable $O_{BE}$ of norm at most one, expand the Heisenberg evolution in its norm-convergent ordered nested-commutator series. At order $m$, summing edge choices and ordered integrals bounds the norm by $(2\Lambda {)}^m/m!$. Each nonzero commutator must meet the current site support, and can extend its graph distance from $B$ by at most one. Every term of order below $d$ is therefore identity on $A$, also when its coefficients are arbitrary operators on $E$.

The two intervention adjoints agree on that partial sum. They are operator-norm contractions, so their difference on the remainder is bounded by twice the series in (eq:tail). Trace-norm duality and the $1/2$ normalization in Definition def:signal give (eq:tail). Insert $1\le e^{\gamma (m-d)}$ for $m\ge d$ and sum the exponential series for the final bound.

This is the conservative finite-volume route of VII-J CheungVIIJ (:tail). Sharper quasi-local estimates require additional interaction hypotheses BHV,NachtergaeleSims. We do not claim a new Lieb–Robinson theorem.

For two assigned separated sites, take $H=J{X}_A{X}_B$, $J>0$. Prepare $B$ in $|0⟩$, and replace $A$ by $|+⟩$ or $|-⟩$. Their output states on $B$ are $e^{\mp iJtX/\hslash }|0⟩$, with trace distance

It is nonzero for arbitrarily small positive $t$. Such a Hamiltonian cannot be identified with an exact geometrically local communication primitive at positive separation.

For bounded self-adjoint generators on a fixed finite space,

For time-independent $H={{\sum }_{j=1}}^m{H}_j$, put

Then

Proof. Differentiate the mixed propagator $U(t,r)\tilde{U}(r,s)$, integrate, and use unitarity. The channel comparison follows by expanding $UX{U}^{†}-VX{V}^{†}$ into two terms, also after amplification.

For two summands, the double integral comparing the joint exponential and ordered product has norm at most ${\delta }^2\Vert [H_1,H_2]\Vert /(2{\hslash }^2)$. Iterating over the finite sum gives one-step error ${\delta }^2{C}_H/(2{\hslash }^2)$. Telescoping $n$ unitary steps with $\delta =t/n$ proves (eq:trotter).

The product-formula channel is a logical approximation. Its elementary implementation has its own positive travel and gate times. Approximating a Hamiltonian to accuracy (eq:trotter) does not embed its numerical clock into the physical clock.

Let $C\cong {\mathbb{C}}^{2^m}$ have an installed orthonormal basis $\{|{\tau }_a⟩\}$. The surviving space $S$ includes matter and every retained control or location/history register. Set

The basis change $|{\tau }_a⟩\mapsto |a⟩$, followed by destructive Enter on all clock names, has channel

It is CPTP on all operators, including every reference extension. Its output has no quantum $C$ factor, and

For branch probability $p_a>0$, the conditional surviving state is $D_a\rho {D_a}^{†}/p_a$. Zero-probability branches are not normalized.

Proof. Kraus operators are $|a{⟩}_R\otimes D_a$, with squared moduli summing to ${\sum }_a{P}_a\otimes I=I$. Expansion in the clock basis shows that both expressions in (eq:clock) return its $aa$ block. Summation gives the partial trace. The calculation is unchanged after adjoining an arbitrary untouched factor. The destructive output type removes the measured quantum names.

This is the I-B/En09-A finite interface CheungIB,CheungEn09; its conditional block is not a metric, stress tensor, or singular detector.

For fixed continuation channels ${\mathcal{F}}_a$, the full programme channel is

It is CPTP because each continuation preserves the trace of its unnormalized block. Routing choices and record delivery must be implemented before the receiver uses $a$. An arbitrary state-functional selection of ${\mathcal{F}}_a$ is not this channel.

Let distinct numbers ${\theta }_a$ label clock codes. At one actual reading event $e$, exact event-time calibration

forces a single deterministic outcome. If no label equals $T(e)$, it is impossible for a normalized input.

A history or programme dictionary may instead associate $a$ with a different labeled event, arrival, route, or stored incidence parameter. It need not equal the actual reading time.

Proof. Distinctness allows at most one matching index and the probabilities sum to one. The dictionary interpretation makes no claim that several different labels are coordinates of the same reading event.

A programme label for a future arrival is a planned target, not evidence that the arrival already occurred. A history claim requires its independently specified preparation and provenance.

The channel ${\mathcal{M}}_C$ is not invertible on all clock inputs. Finite matrices cannot satisfy $[T_C,H_C]=i\hslash I$ when $\hslash \ne 0$. Consuming $C$ does not remove $T$, $M$, or $g$.

Proof. The clock states $(|{\tau }_0⟩\pm |{\tau }_1⟩)/\sqrt{2}$, with the same payload, have identical outputs. A left inverse is impossible. Taking the trace proves the commutator assertion. The final statement follows because these geometric objects are prescribed parameters, not output factors of ${\mathcal{M}}_C$.

For completeness, the additional hypothesis needed to turn conditioning into relational evolution can be stated precisely PageWootters,CheungEn09.

Suppose finite Hermitian $H_C,H_S$ and ${\rho }_{CS}$ satisfy

Then

For a complete orthonormal reading basis all probabilities equal $2^{-m}$. For an extended $SE$ assertion, stationarity of the full extension is required.

Proof. Stationarity gives

Insert the covariant projector into the clock partial trace and use unitary invariance of that trace. The resulting blocks are unitarily related, so their traces are equal and sum to one. The same calculation applies to an extension exactly under the extended stationarity hypothesis.

No such dynamical covariance is assumed for the programme table below. The external schedule remains an input.

Two record-retaining instruments may have the same outcome law and different residuals. For example, with one coarse outcome, the identity qubit channel and computational dephasing have identical records but, on $|+⟩⟨+|$, output $|+⟩⟨+|$ and $I/2$. Their trace distance is $1/2$. Implementation errors ${\epsilon }_1,{\epsilon }_2$ leave a gap at least $1/2-{\epsilon }_1-{\epsilon }_2$, by the triangle inequality. This is the observation-boundary discipline compared with VII-E, VII-G, and VII-E-ST CheungObservation.

More generally, a complete recorded instrument that recovers every input in the full quantum algebra can have only input-independent record effects. Indeed, for Kraus operators $A_i$ of the instrument and $B_j$ of a proposed recovery, all $B_j{A}_i$ are scalar multiples of $I$, because the identity channel has rank-one Choi matrix. Completeness of the recovery then gives ${A_i}^{†}{A}_k={\lambda }_{ik}I$. Summing the diagonal terms for one record makes its effect scalar. This full-input criterion is consistent with standard quantum error correction KnillLaflamme,Watrous. It does not prohibit transport of the surviving payload.

We now give a nonempty Target B instance. It deliberately replaces VII-J's singular-axis rendezvous CheungVIIJ (:eventbenchmark).

In sector zero choose the dimensionless transverse mark

At that mark,

Choose two regular persistent ports

and slice parameters

Both ports have identical time coordinates on each marking. Their input and output times are

Take a convex developed spatial tube containing

Its closure is entirely regular and lies inside the first sector. Choose a finite time interval containing $[{\tau }_{\mathrm{in}},{\tau }_{\mathrm{out}}]$. All operational paths and gates remain in this tube.

The marks $A,B$ are spatial locations, not evolving abstract cell labels. The local branch at $x_{\perp }$ changes at $s=h-k=7/640$, but the mark and its quantum register do not cease to exist.

The complete VII-F metadata circuit is not silently inferred from these few bits. We use its following explicit fixed-instance restriction.

At the controller $A$, the exact stored incidence parameter is $j=0$; the fixed geometric packet specifies the prescribed cone and zero germ. Reserve classical mode, permission, and history bits

and one scratch bit $E=0$. A two-qubit computational-basis clock returns $a=(a_1,a_0)\in \{00,01,10,11\}$. Only $a=01$ requests the event update. Set

On the promised initial bits the update is

Thus the canonical programme $01$ performs the exact incoming-to-outgoing zero-germ mode update, spends its one permission, and retains a history bit. Other programmes request no update. No length report is used as a clock guard.

The permutation is implemented by:

1. NOT on $a_1$, Toffoli $(a_1,a_0;E)$, NOT on $a_1$;

2. CNOT from $E$ to each of $M,N,H$;

3. the inverse of step 1.

There are nine elementary NOT/CNOT/Toffoli gates. The record is restored and $E$ returns to zero. Off the promised domain this remains a reversible bit permutation, but is not assigned a geometric interpretation.

Any separately specified finite source circuit $U_{\mathrm{code}}$ with $k$ local code/history/work carriers and $G$ gates can replace this restriction. Co-locate its carriers at $A$ and give each gate a positive duration $1/(16(G+4))$ in the buffer below. This localizes the supplied finite map; it does not prove an unsupplied decoder theorem or its forward promise. The explicitly proved numerical ledger below concerns (eq:event-code).

Fix an integer $n\ge 1$. Install a chain of qubit registers $Q_0,\dots ,Q_n$ at

Here $Q_0$ is a temporary register at $A$; $Q_n$ is the input register at $B$. A separate qubit $P$ is the input at $A$. Initialize $Q_0,\dots ,Q_{n-1}$ in $|0⟩$.

Let

During the inbound interval of length $D$, use $n$ successive content exchanges

each of duration $D/n$, realized by opposite causal carriage of the two contents along the regular axial segment. Their physical dimensionless speed is $v_{\ast }/4<1$.

The unknown qubit reaches $A$ by

Both matter qubits then remain stored at $A$. Destroy the two clock qubits in two local slots of duration $1/64$, from $\tau =-1/32$ to $0$. The complete record is available at $\tau =0$.

Wait until $\tau =1/16$. Run the nine event-code gates, three local CNOTs implementing ${\mathrm{SWAP}}_{P,Q_0}$, and the classical-record controlled gate ${Z_P}^{a_1}$. The thirteen local slots each have duration $1/208$, so they finish at $\tau =1/8$.

Finally move $Q_0$'s content outward by the $n$ successive exchanges

Allow duration $D/n$ per outbound stage. Its first half copies the five-bit classical word

into the next site's blank record buffer by five sequential classical CNOTs. Each copy slot has positive duration $D/(10n)$. The second half carries the qubit exchange and the copied record over the segment of length $L/n$. The communication speed is $v_{\ast }/2<1$.

All records and payloads arrive by

Wait at the destination ports until ${\tau }_{\mathrm{out}}$. Records at earlier sites are retained, not reset.

The specified execution is a finite Target B schedule. The matter is buffered across both the worldsheet incidence time $\tau =0$ and the local slice parameter $s=0$. No operation meets $\mathcal{W}$.

The moving routes also satisfy the sufficient slicing speed (eq:vstar).

Proof. The entire axial chain is in the displayed regular tube. Every local operation is on one station worldline with strictly positive duration. The inbound exchange routes have speed $v_{\ast }/4$, and the outbound communication portions have speed $v_{\ast }/2$; both are future timelike.

The incoming branch has ${\partial }_s{F}_s=1$. On the outgoing communication interval, ${\partial }_s{F}_s=2$. Thus the route speeds in $s$ are respectively $v_{\ast }/4$ and $v_{\ast }$. Stationary waits and gates have zero spatial speed. Theorem thm:conegates also certifies them.

The qubits are at $A$ before $\tau =0$, and remain there until $\tau =1/16$. At $A$, the $s=0$ slice has time $1/64$, which lies inside that storage interval. All control copies occur before their communication departures, and every remote use waits for the record to arrive.

The classical metadata describes the zero germ. Its positive-duration computation finishes later. It is neither an instantaneous global update of geometry nor acquisition of data from the singularity.

Write the input matter order as $P,Q_n$, and define

The programme $a=01$ has $V_{01}=\mathrm{SWAP}$. The other programmes provide declared phase-controlled variants, not alternative spacetime metrics.

For every input operator $X$ on $C\otimes P\otimes Q_n$, the complete output of the benchmark is

where $D_a=⟨a{|}_C\otimes I_{P{Q}_n}$, and ${\omega }_{\mathrm{blank}}$ contains the final blank $Q_0,\dots ,Q_{n-1}$ and scratch $E=0$. The two matter output factors are at $A,B$, respectively. Equation (eq:benchmark-channel) holds with every untouched reference.

On the canonical clock preparation $|01⟩⟨01|$, the matter channel is exactly ${\mathrm{Ad}}_{\mathrm{SWAP}}$. It has

No input outside the physically certified backward region signals to the declared joint output.

Proof. Successive exchanges with blank chain registers move the entire $Q_n$ factor, with all its correlations, into $Q_0$. The other chain registers become blank. The clock identity gives $D_{a}X{D}_a^{†}$ on branch $a$. The nine-gate calculation gives (eq:event-code). The three CNOTs exchange the local matter factors and the record-controlled phase gives $V_a$. Outbound exchanges move the second factor into $Q_n$, again returning all intermediate qubits to their initial blanks. The record is already classical; copying its specified basis word creates precisely the repeated projectors in (eq:benchmark-channel), including their correlations with matter. No unknown quantum state is copied. This proves the formula on all operators and references.

For the witness, fix the clock to $01$, hold the initial $B$ qubit fixed, and replace the $A$ qubit by $|0⟩$ or $|1⟩$. The $B$-outputs are orthogonal, hence their trace distance is one. Theorem thm:physical gives the final no-signalling assertion.

The pre-existing reference can be entangled with the clock and both inputs. A conditional reference state may change at clock readout, but the channel acts as identity on the reference. No gravity-generated entanglement is claimed.

A zero-duration remote SWAP between distinct ports of a single regular marking cannot satisfy this causal architecture.

Proof. Lemma lem:slices makes the marking strictly spacelike: its time difference is bounded by $\lambda$ times spatial path length. The ports are spacelike. A SWAP would have the unit intervention witness just used, contradicting physical containment.

The preceding buffer is deliberately timelike. Exact local null transport is also available under the I-B routing permission, but only for a supplied target-adapted chain.

Let a developed segment of length $L_0>0$ remain in a convex regular chart. Install $m+1$ equal finite payload registers along it and initialize all but the source in a fixed blank. Successive content exchanges over intervals $L_0/(mc)$ implement the logical identity at the null-related target, including every reference, using $m$ exchanges.

An already available classical record and finite location/history payload may be included in the transported packet. Nontrivial preparation, copying, or decoding time is additional.

Proof. At each hop the complete source factor replaces the next blank, and the previous site becomes blank. Induction proves the complete identity. Each communication path is the regular straight segment traversed at speed $c$. The same argument applies to a joint quantum–classical packet.

This is not an exact-attainment theorem for a previously fixed lattice or for records produced too late to enter the packet.

For the benchmark's two-qubit clock, the exact table is

Each signalling entry is evaluated for that fixed programme. The full programme channel retains the record and is (eq:benchmark-channel).

The event indicator is the finite matrix

A nominal label, route length, duration, cycle reward, or potential value can similarly be stored as a diagonal matrix over a finite programme set. Such matrices encode declared evaluations; they are not measurements of an unknown field.

A finite time-expanded dependency graph with $N$ vertices can be represented by its Boolean adjacency matrix $A$. Its reachability table is the Boolean matrix

Every reachable pair has a path of at most $N-1$ edges in the DAG, which proves this finite test without numerical execution.

For the explicit $n$-segment benchmark: