Observation Boundaries for Destructive Quantum Readout and External Geometry

We formulate a finite, linearly typed interface for destructive computational-basis measurement, external classical geometry, and declared matter–geometry coupling. The observation boundary includes not only coupling observables but also control, guards, capacity tests, locations, and retained storage. Under explicit noninterference hypotheses, arbitrary spectator extensions with the same accessible reduction preserve geometric traces. Immediate destructive readout and a restricted coherent-then-readout implementation agree on their common record–matter output.

For a channel $T$, coarse effects $E_w$, and aggregated fine effects ${\Pi }_w$, record compatibility for every input is equivalent to $T^{\ast }(E_w)={\Pi }_w$. Its exact unassisted worst-case total-variation error is an event operator norm. Residual compatibility requires a separate instrument certificate. Complete-record equality does not identify a hidden mechanism; a persistent classical register can produce carrier-only history dependence.

Prescribed finite channel networks admit a causal-cut factorization through their scheduled ancestors, including in the presence of an untouched reference. This channel statement is not transferred to nonlinear geometric coupling. Conditional stability, integrated residual, and independent-run sampling certificates are stated with their additional hypotheses.

An external conical witness, a static gauge flip, and one complete four-face adjacency exchange illustrate the interface. The exchange has explicit charts, edge laws, event germs, an exact guard, a compact collar, location transport, and a finite reversible event implementation. The ambient spacetime is fixed. No gravitational field equation, continuum limit, or quantum-memory-resource theorem is asserted.

A geometric extension of a quantum programming language must specify what geometry can observe. This requirement is independent of the choice of gravitational dynamics or a continuum limit. Linear ownership of quantum names does not determine a mass; a measurement record is not a coarse-graining map; and a circuit dependency region is not automatically a physical light cone.

We use a finite circuit interface with destructive readout

Its semantics is the Born destructive computational-basis measurement used by Cheung's Enter calculus Cheung2026Enter,CheungI. Measured quantum names are consumed, and their tensor factors are not returned. We do not reprove a language-wide type-safety theorem or derive the Born rule. Our mixed-state channel presentation is a finite-dimensional semantic interface for that measurement operation.

Four additional structures are declared:

1. an external classical geometric configuration;

2. a location map for active matter names;

3. coupling algebras specifying accessible matter observables;

4. a complete control and resource interface specifying update selection, applicability, support, and retained storage.

Geometry is primitive metadata, not a quantity inferred from name cardinality, measurement count, or workspace size.

Two distinctions organize the paper:

The main results are:

1. exact finite-step support containment and causal-cut factorization for prescribed channel dynamics;

2. retiming invariance for a fixed labelled gate partial order;

3. spectator-extension invariance under complete control and resource blindness;

4. restricted destructive deferred-readout equivalence;

5. an effect-pullback criterion and exact unassisted record error;

6. a residual-instrument obstruction and accessible-algebra comparison criteria;

7. a complete fixed-spacetime certificate for one four-face adjacency exchange.

Supplementary interface propositions give complete-process record laws, a classical-register memory witness, and conditional approximation and sampling bounds.

Finite-dimensional channel conventions follow Watrous Watrous2018. The circuit precedent for deferred measurement is Gurevich and Blass GurevichBlass2022. The geometric setting uses conical point-particle spacetimes and polygonal developments DeserJackiwHooft1984,tHooft1992, tHooft1993,Kadar2005.

Cheung's companion records motivate comparisons with observation, process, and schedule interfaces CheungVIIE,CheungVIIJ, CheungPsy02A,CheungPsy02B. They are not premises transferring a model identity or a gravitational solution. Every mathematical statement used here is formulated and proved locally. The source ledger in app:sources distinguishes full supplied passages from statement summaries; no inspection of an entire manuscript corpus is claimed.

The configuration space may contain arbitrary real lengths and affine isometries. These remain external labels unless a finite code and decoder are supplied. The executable examples cover only declared finite families.

A geometric coupling may be nonlinear in the matter state. Consequently the complete coupled evolution is not generally a quantum channel. Channel locality, no-signalling, distance contraction, and retiming are asserted only for the prescribed channel objects named in their statements.

The four-face construction changes the cellulation of a prescribed slicing of an already fixed spacetime. It is not a new gravitational solution, a particle-supported transition, or an implementation of the full polygon model. No field equation, curvature-convergence theorem, Newtonian limit, or equivalence of matter clocks and propagation cones is proved.

All matter, code, and record spaces used in channel statements are finite-dimensional. Write $L(\mathcal{H})$ for the operator space and $D(\mathcal{H})$ for its density operators. A channel is a completely positive, trace-preserving linear map. Its adjoint satisfies

Channel adjoints are unital and completely positive Watrous2018.

A classical register has a distinguished basis. An admissible joint state is block diagonal in that register, not merely diagonal after taking its marginal. Declared classical operations preserve this joint block-diagonal state space. Their individual Kraus operators need not be diagonal; classical permutations are allowed.

For a unital ${}^{\ast }$-subalgebra $\mathcal{A}$, define

Equality of these functionals means equality on the named observable interface.

Our distance conventions are

The trace norm itself has no factor $1/2$. Outcome sets and complete history alphabets are finite unless explicitly stated otherwise.

Wire types are generated by

A judgement $f:A\to B$ consumes the entire input interface and returns the specified output interface. Formation includes identities, permutations, sequential composition, and tensor composition:

The circuit constants used here are unitary gates, preparation of $|0⟩$, and destructive computational-basis measurement. Finite reversible code operations use NOT, CNOT, and Toffoli. Classical zero preparation and classical reversible gates act on $\mathsf{Bit}$ interfaces; their basis-permutation unitary lifts act on $\mathsf{Q}$ interfaces. No conversion between these types is implicit.

Linearity concerns ownership. Successive gates may consume and return the same physical factor. No rule duplicates a live quantum wire.

The interface is specified independently of the complete language. An embedding into the full calculus of Cheung2026Enter must match its constants, typing, and circuit-adequacy interface. No theorem below relies on the Layer II measurement-uniqueness argument or asserts gate-library realization of every semantic channel.

Let $S$ have computational basis $\{|z⟩:z\in Z\}$, and let $B$ be its accessible complement. Put

The destructive measurement channel is

The record $R$ is classical. The factor $S$ is absent. The circuit operation ${\mathrm{Enter}}_S$ is interpreted by ${\mathcal{M}}_S$.

This is destructive Born readout, not a returned measured quantum register Cheung2026Enter,CheungI.

A fresh quantum carrier $A$, with basis indexed by $Z$, is introduced by the isometry

Its channel is ${\mathsf{Copy}}_S(\rho )=V\rho V^{†}$. For qubits it is zero preparation followed by CNOT.

For $P_z=|z⟩⟨z{|}_S\otimes I_B$, set

Its residual interface is $S\otimes B$. It is not the destructive primitive.

Coherent recording does not clone arbitrary states:

Deferred $\mathrm{Enter}$ means recording, continuation on the declared complement, destructive carrier readout, and comparison after restriction away from retained $S$.

A geometric interface specifies:

1. a set $\mathcal{Q}$ of classical configurations;

2. an admissible subset

   $${\mathcal{Q}}_{\mathrm{adm}}={\mathcal{Q}}_{\mathrm{reg}}\cup {\mathcal{Q}}_{\mathrm{tr}};$$

3. a finite label set $\mathrm{Cell}(q)$ for each configuration;

4. support, admissibility, and resource certificates for updates;

5. optionally, a supplied gauge equivalence ${\sim }_g$.

Regular configurations satisfy the model's stated nondegeneracy conditions. Transition configurations require a model-specific event certificate and may retain one-sided germ labels whose common geometric realization is degenerate.

Admissibility is semantic. A finite representation and a decision procedure are additional structures.

In sec:geometry, every bounded regular edge has positive length. Fixed unbounded exterior rays, when used, are described as rays rather than finite length reports. At the event, a vanishing central edge and coincident endpoint labels are germ metadata, not a regular embedded one-cell.

A configuration is

where $q\in {\mathcal{Q}}_{\mathrm{adm}}$, $\Gamma$ is a finite labelled interface, and $\rho \in D({\mathcal{H}}_{\Gamma })$. Classical names satisfy joint block diagonality. The geometric projection is ${\pi }_g(C)=q$.

Control and resource labels are either included in $q$ or carried as separately declared classical data. Compared executions have equal initial declared control and resource data.

Permitted transition-germ locations, their realizations, and their transport belong to the event certificate.

Interfaces may change under commands. Destructive measurement replaces measured quantum names by fresh record names. An identity extension always refers to the current unacted-on interface.

A finite run uses:

1. A geometric rule, or $(G)$-rule, which replaces supported geometric labels by a specified function of those labels and declared resources. It does not inspect matter state or matter-name cardinality. Its complete output is admissible and includes location transport.

   A transition rule specifies incoming and outgoing germs, their common event realization, and both one-sided continuations.

2. A matter rule, which applies a declared channel

   $${\Psi }_q:L({\mathcal{H}}_{{\Gamma }_0})⟶L({\mathcal{H}}_{{{\Gamma }^{\prime }}_0})$$

   on a named local interface, tensored with the identity on the remaining names. It may depend on $q$, but does not change $q$. Allocation and output locations are part of its certificate.

3. A coupling rule, which declares a unital observable algebra ${\mathcal{A}}_v$ and updates geometry by

   $$q^{\prime }=F_v(q,{\omega }_{\rho }{|}_{{\mathcal{A}}_v}).$$

   It returns a complete admissible configuration and a location transport map. It leaves the matter density operator unchanged.

Changing one particle scalar while freezing incompatible chart data need not preserve constraints. A coupling therefore returns a complete certified update, not merely a new scalar.

The map in eq:coupling need not be affine in $\rho$. The complete coupled evolution is not generally $\mathrm{CPTP}$. The noninterference proofs compare equal coupling inputs; they do not establish no-signalling for arbitrary nonlinear backreaction.

The controller selects commands, guards, support, and capacity checks. Every state-dependent register, queue, timing signal, applicability predicate, and readable transport history is part of that interface.

For the compared executions, inaccessible spectators do not change:

1. the selected command or its declared matter channel;

2. guard, applicability, or capacity-check results;

3. the support, block membership, or transport of original names;

4. successor control and declared resource data.

The controller is a specified function of the complete declared inputs. Equal such inputs select the same update and certificate. No undeclared spectator-dependent input is available to it.

The location transports are defined on all permitted spectator locations, and spectator storage and transport history have separately reserved resources sufficient for the compared run.

Randomized controllers are aligned using a common input-independent seed, or a specified coupled law satisfying the same equality property.

Separate reservation is not a claim of unlimited physical memory. It is a hypothesis about which finite resource budget controls the geometric update. A global occupancy test may violate the hypothesis.

Gauge equivalence is a supplied equivalence relation on geometric descriptions. In the examples it includes chart changes and retriangulations representing the same metric and marked slice. An admissible geometric observable is constant on its equivalence classes.

A fixed spacetime does not identify all marked-slice observables at different times. Circuit retiming is not presumed to be a gauge symmetry of nonlinear coupled evolution.

Throughout this section the channel list, metric, and schedule are prescribed. State-dependent geometric selection is not hidden inside a channel list.

Let $(X,d)$ be a finite metric space with site spaces ${\mathcal{H}}_x$, and write

For $a,\tau >0$, let ${\mathcal{B}}_t$ be a partition of $X$ at tick $t=1,\dots ,T$, with

The dynamics is

where each block map is a channel on its reserved site interface.

Readout may write a reserved local record register. A fixed ambient description may contain an inactive placeholder after destruction; that placeholder is not an accessible surviving measured wire.

For $Y\subseteq X$ and $0\le s\le t\le T$, set

and

If $\Phi ={⨂}_{B\in \mathcal{B}}{\Phi }_B$ and $A\in \mathcal{A}(Y)$, then

Proof. Let $Y^{\prime }$ be the displayed union. Its complement is a union of blocks. Unitality on those blocks gives

For $A\in \mathcal{A}(Y)$ and $0\le s\le t\le T$:

1. $$\mathrm{supp}(({\Phi }_{s+1}^{\ast }\circ \cdots \circ {\Phi }_t^{\ast })(A))\subseteq N_{s,t}(Y);$$
2. $$N_{s,t}(Y)\subseteq \{x:d(x,Y)\le (t-s)R_{\mathrm{upd}}a\};$$

3. the output reduction on $Y$ depends on the state at tick $s$ only through its reduction to $N_{s,t}(Y)$.

Proof. Reverse-tick induction using lem:one-layer proves (i). For (ii), if $x$ enters through a block meeting the next region at $y$, then

For (iii), every output observable pulls back to the named input region. Equality of its expectations for all output observables is equality of the output density operators.

A prescribed readout at $(Y,t)$ depends only on the initial reduction to $N_{0,t}(Y)$. Replacing a block channel outside its backwards dependency region does not change the readout law, provided the declared schedule and interfaces are retained.

Proof. Readout effects are output observables. On an excluded block the pulled-back observable is the identity, which every replacement channel adjoint preserves.

Scheduled ancestors are an exact upper bound, not necessarily the minimal influence region of a particular channel. Joint records use the backwards region of their full instrument network. Selective normalized updates are not trace-preserving channels; the statements concern unconditional channels that retain their records.

The schedule statement has a factorization form. This is the channel interface compared with Cheung's schedule and no-signalling source records CheungVIIJ; the proof here does not import a physical cone.

For a channel $\Theta :L({\mathcal{H}}_A\otimes {\mathcal{H}}_C)\to L({\mathcal{H}}_B)$, the following are equivalent:

1. $\Theta (\rho )=\Theta (\sigma )$ whenever ${\mathrm{Tr}}_A\rho ={\mathrm{Tr}}_A\sigma$;

2. there is a channel $\theta :L({\mathcal{H}}_C)\to L({\mathcal{H}}_B)$ with

   $$\Theta =\theta \circ {\mathrm{Tr}}_A;$$
3. $${\Theta }^{\ast }(L({\mathcal{H}}_B))\subseteq I_A\otimes L({\mathcal{H}}_C).$$

These conditions imply invariance of the joint $B$–reference output under every local trace-preserving intervention on $A$, with the reference untouched.

Proof. Assume (i), fix ${\tau }_A\in D({\mathcal{H}}_A)$, and define $\theta (X)=\Theta ({\tau }_A\otimes X)$. This is a channel. Property (i) proves (ii) on states, hence on all operators by linear spanning.

The adjoint of ${\mathrm{Tr}}_A$ is $Y\mapsto I_A\otimes Y$, so (ii) implies (iii). Conversely, (iii) says that every output expectation is determined by the $C$-reduction, proving (i).

For an untouched reference $E$, the factorization extends as

If ${\mathcal{J}}_A$ is a channel, then ${\mathrm{Tr}}_A[({\mathcal{J}}_A\otimes {\mathrm{id}}_{CE})({\rho }_{ACE})]={\mathrm{Tr}}_A{\rho }_{ACE}$, as follows by testing against observables on $C\otimes E$ and using ${\mathcal{J}}_A^{\ast }(I_A)=I_A$.

Consider a finite acyclic network of fixed channels, with every quantum, classical-control, record, and resource wire included. For an output interface $B$, let ${\mathrm{Sch}}^{-}(B)$ be the input wires reached by reverse traversal: whenever an output wire of a channel box is reached, include every input wire of that box.

If $\mathrm{Comp}$ is the network channel and $X$ its input interface, then

The reduced output channel on $B$ factors through the scheduled input ancestors. Inputs disjoint from that cut have zero signalling to $B$, including with an untouched reference.

Proof. Choose a reverse topological ordering. Pull an observable on $B$ back through each box. If it is identity on all outputs of a box, unitality leaves identity on all inputs. Otherwise its possible support is enlarged to that box's input interface. Induction gives eq:network-cut. Apply ip:nosignal.

Thus an uncounted control wire, capacity flag, or transport register that is an ancestor of an event is inside its boundary. If an external geometric decision is a specified function only of the state functional on that event interface and common declared labels, equal interface data give the same decision. This last assertion is substitution into that function, not a claim that the nonlinear coupling itself is a channel.

An input-dependent initialization or encoding is also part of the network. Hiding it outside the ancestor calculation would change the claimed cut.

Let $R\ge 1$ be an integer. On the integer line, with $d(i,j)=a|i-j|$, use

Swap the endpoint factors in each block. A factor initially at $0$ is at $tR$ after $t$ ticks, attaining speed $Ra/\tau$. Every finite-horizon relay has a finite-site realization.

Proof. At each tick the blocks partition the line into consecutive intervals of diameter $Ra$. At tick $t$, the factor is at the left endpoint $(t-1)R$ of $B_{t,0}$; the swap sends it to $tR$, the next tick's relay left endpoint.

For horizon $T$, choose a finite interval containing $B_{t,0}$ for every $1\le t\le T$. At each tick retain the partition blocks wholly contained in that interval and replace the clipped boundary portions by singleton blocks. The relay blocks are unchanged. No infinite tensor-product state is needed. The transported factor may be entangled with a reference.

Let $G$ be a finite labelled collection of fixed channels on reserved interfaces, with supports $S_g\subseteq X$. A strict partial order $<$ is admissible if gates with intersecting supports are comparable. A legal execution is a linear extension.

All legal executions of an admissibly ordered labelled gate set have the same total channel.

Proof. Incomparable gates act on disjoint factors and commute. Any two linear extensions are connected by adjacent swaps of incomparable elements: move the first desired differing element leftwards; every crossed element must be incomparable, since either comparability would contradict one of the extensions. Repeat.

Records and control dependencies belong to supports. The theorem does not move a gate past a gate using its record, change a geometric parameter while retaining the same channel label, or establish no-signalling for nonlinear geometric coupling.

In a prescribed square-lattice model permitting one cardinal step of length $a$, or waiting, per tick, the asymptotic velocity set is

An isotropic physical speed $c_{\mathrm{phys}}=c$ is not dominated in every direction.

Proof. After $n$ ticks, $|\Delta x|+|\Delta y|\le na$. Step proportions approximate every point of the displayed diamond. The velocity $(c/\sqrt{2},c/\sqrt{2})$ has Euclidean norm $c$ but ${\ell }^1$-norm $\sqrt{2}c$, so is excluded.

The preceding example concerns cardinal hops, not every partitioned schedule. This section proves dependency statements on a prescribed substrate. It neither reconstructs a physical light cone nor establishes Lorentz invariance.

An admissible extension to a disjoint spectator interface $N$ satisfies

It may be entangled. Existing classical wires remain jointly block diagonal.

For $\Psi :L({\mathcal{H}}_{\Gamma })\to L({\mathcal{H}}_{{\Gamma }^{\prime }})$,

Proof. For every output observable $A$, both sides have expectation

Consider an original certified run

and an initial admissible spectator extension with the same original geometry, locations, control, and declared resources. Suppose:

1. original matter maps extend by ${\mathrm{id}}_N$;

2. coupling algebras extend by $A\mapsto A\otimes I_N$, or coupling inputs are specified functions only of the original reduced state;

3. ass:sched holds, including spectator transport and separate storage legality;

4. spectator outcomes are neither read nor used during the run.

Then a paired extended run is legal and, at every aligned step,

Original transports and declared control outputs also agree.

Proof. Induct on the steps. Equal declared inputs select the same command, guard, support, and certificate.

A geometric rule receives equal geometric and resource labels and leaves matter unchanged. A matter rule has the common parameter $q_k$, and lem:trace preserves the reduction identity. For a coupling rule,

so its complete input functional and output agree. The transport and separate-reservation clauses establish legality and equality of original locations and control data at each step.

Under all hypotheses of thm:spectators, replacing the initial state by ${\rho }_0\otimes {\varphi }_N$, for any finite spectator interface and normalized ${\varphi }_N$, leaves the geometric trace unchanged.

Proof. A product insertion is a special case of eq:extension.

The theorem is not restricted to fixed pure spectators. Arbitrary mixed and input-correlated extensions are covered by the restriction identity itself. A different isometric construction may require an explicit preparation purification; such a requirement is not supplied by merely calling an extra wire a spectator CheungEm05A.

Let $q\in \{0,1\}$, and let a geometric rule request $q\mapsto 1$ subject to the guard

Coupling algebras may be trivial and every matter operation may avoid spectators. A run with $|\Gamma |=K$ permits the update, whereas adding one spectator blocks it.

The guard is an observation channel in the controller. The example violates ass:sched, not thm:spectators. Repartitioning, queue priority, timing, or a spectator-readable transport history can produce the same failure.

The spectator theorem does not apply if geometry, coupling, control, or certification reads spectator cardinality, spectator state, or spectator-dependent resource availability through an undeclared input.

Spectators may be returned or consumed by terminal $\mathrm{Enter}$. The partial trace in eq:spectatorresult describes an observation interface, not a term-level discard rule.

Nonselective terminal measurement of an entangled spectator preserves the original reduction. Conditional reductions need not be preserved. For a Bell pair, conditioning on the spectator's computational-basis outcome prepares the corresponding pure state on the original factor. Later geometry using that outcome is a different comparison.

The theorem specifies a noninterfering rule class, including its controller. It excludes name cardinality as a source variable within that class. It does not establish physically correct gravitational dynamics.

Immediate readout destroys $S$, whereas coherent recording retains it. Comparison therefore requires a common accessible output.

For every $\rho \in D({\mathcal{H}}_{SB})$,

Proof. Write $\rho ={\sum }_{z,z^{\prime }}|z⟩⟨z^{\prime }{|}_S\otimes {\rho }_{z{z}^{\prime }}$. Recording gives

Tracing $S$ removes exactly the terms with $z\ne z^{\prime }$.

For a channel $\Lambda :L({\mathcal{H}}_B)\to L({\mathcal{H}}_{B^{\prime }})$,

where ${\mathcal{N}}_A$ destructively reads $A$ into $R$. Outputs are identified using the canonical wire permutation.

Proof. A channel on $B$ commutes with restriction over $S$ and readout of $A$. Apply lem:copytrace; readout of the already block-diagonal $A$-reduction replaces its basis label by $R$.

Start with common geometry and declared control data and a state $\rho$ on $S\otimes B$. Compare:

1. immediate ${\mathrm{Enter}}_S$, followed by continuation on $B$;

2. allocation of $A$, coherent recording, the corresponding continuation on $B$, and terminal ${\mathrm{Enter}}_A$.

Between recording or initial readout and terminal carrier readout, assume:

1. matter commands act only on the current continuation interface $B_k$, not on $S,A$, or the immediate record $R$;

2. coupling inputs and state-dependent guards use only the reduction on $B_k$;

3. geometric rules inspect only declared geometric labels;

4. control, supports, capacities, and transport of continuation names are blind to the excluded registers, with separately reserved storage as in ass:sched.

Then geometric traces coincide at aligned continuation events. After terminal carrier readout, the deferred output restricted away from $S$ equals the immediate output on $R\otimes B_T$.

Proof. Both initial continuation reductions equal ${\mathrm{Tr}}_S\rho$:

Induction using (D1)–(D4) gives common commands, guards, geometry, continuation transports, and reductions at every event. For this fixed input, compose the matter channels selected along the common trace into $\Lambda$. They act only on the continuation interface. Equation (eq:continuation) gives equality of the complete common record–matter output.

The selected $\Lambda$ may depend on the input through nonlinear geometric selection. The proof is an equality for each compared pair, not a claim that the complete coupled input–output map is linear.

The immediate output has type ${\mathsf{Bit}}^{\otimes n}\otimes B_T$. The deferred output additionally contains $S$. It must be returned or consumed by a well-typed continuation. Inaccessibility does not create a weakening rule.

An operation on retained $S$ has no immediate destructive counterpart. Reading $R$ during continuation may permit a larger coherent-control construction, but not this theorem. Coupling to the coherent joint state on $S\otimes A$ can distinguish recording from destructive readout.

Changing a carrier does not create a missing observation channel. In particular, coherent recording and later destructive carrier readout do not authorize access to retained $S$ or a spectator. This is the observation-boundary refusal compared with Cheung's carrier-obstruction record CheungPsy03D. The present theorem proves the restricted identity; it does not assert universal equivalence of immediate and deferred measurement.

This section supplies an external holonomy witness, a static gauge flip, and one compactly attached four-face adjacency exchange. The ambient metric and resting particle neighbourhood remain fixed. The moving patch has nontrivial adapted-frame transitions but trivial contractible spacetime holonomy.

Use

On the punctured cone,

with $\theta$ periodic modulo $2\pi$. Its completion has a timelike conical axis. The spinless point-particle interpretation is standard DeserJackiwHooft1984; no field equation is derived.

On the universal cover,

For the deck transformation $\mathsf{T}(T,r,\tilde{\theta })=(T,r,\tilde{\theta }+2\pi )$, define

Then

Choose $0<ϵ<\pi /4$, with angular lifts

restricted where necessary to injective developing charts. Their two overlap components have transitions

The positively oriented particle word is

Overlap-component indexing prevents suppression of the nontrivial seam. No smooth flat chart contains the axis.

Suppose

With this convention, parallel transport around $\gamma$, in the starting developed frame, has matrix $L_{\gamma }$.

Proof. Lift the loop from $\tilde{x}$ to ${\mathsf{T}}_{\gamma }\tilde{x}$. Flat developed transport keeps the Minkowski vector constant:

Endpoint identification uses $d{{\mathsf{T}}_{\gamma }}^{-1}$. Differentiating eq:affine-development gives

The cone word acts on spacetime tangent vectors, not an unrelated internal fibre. Developments and their connection data must remain distinct Kadar2005.

For $L=CR(\mu )C^{-1}$, $C\in {\mathrm{SO}}^{+}(2,1)$, and $0<\mu <2\pi$,

The future unit fixed vector is unique, and

The trace alone does not distinguish $\mu$ from $2\pi -\mu$.

Proof. The spatial rotation has no nonzero fixed vector in its spatial plane. Conjugation preserves trace and future orientation and carries the fixed line to $\mathbb{R}C{e}_0$.

For an oriented orthonormal basis $(u,v)$ of ${p_L}^{\perp }$, with $(p_L,u,v)$ positively oriented,

recover the oriented deficit. For a spacelike piece with future unit normal $n$, set

Both vectors must be expressed in the same development.

The oriented deficit and $\chi (p,n)$ are invariant under proper, time-orientation-preserving chart changes. Rapidity is an invariant of the particle together with the marked slice piece, not of the particle alone.

Proof. A chart change simultaneously conjugates $L$ and transforms $p,u,v,n$, preserving the orientations and inner products.

Choose $k\ge 3$ and

On sector $[j\vartheta ,(j+1)\vartheta ]$, let $\beta =\tanh \zeta$ and set

The future unit normal is

On a shared ray, both functions have value $\beta r\cos (\vartheta /2)$; the wedge faces match similarly. The result is a continuous piecewise-flat spacelike neighbourhood. Joining equal-radius ray endpoints by straight chords gives a finite polygonal star.

Attach identical matter interfaces, states, and abstract cell assignments to the resting fan and to eq:fan with $\zeta \ne 0$. Their particle holonomy is the same, but

Proof. The relevant inner products are $1$ and $\cosh \zeta$, and $\mathrm{arccosh}(\cosh \zeta )=|\zeta |$.

An arbitrary globally boosted plane need not descend through the cone identification. The sectorwise construction avoids that assumption. It does not identify rapidity with stress energy.

A polygonal slice induced from a fixed flat conical spacetime and its component-indexed atlas has ordered transition products equal to the corresponding monodromies.

On an interval of fixed incidence, suppose normals are constant, offsets are affine in the parameter, and endpoints solve nonsingular constant-coefficient linear intersection systems. If an edge has a fixed spacelike direction, its length is affine on subintervals where orientation does not reverse.

Proof. The first statement is chart continuation. An endpoint satisfying $Az=b_0+s{b}_1$, with $A$ constant and invertible, is affine. An edge difference with fixed unit spacelike direction is $(c_0+s{c}_1)e$, of length $|c_0+s{c}_1|$.

Affine endpoint motion alone does not make arbitrary segment lengths affine. Artificial subdivision edges are checked explicitly below. No estimate is continued through an event merely by assuming regular-stratum smoothness.

Set $\mu =\pi /2$. In a nonsingular developed sector take

They bound a unit square with a matching certified exterior.

The guard is $m=0$, $\kappa \ge 1$, and membership in the fixed square family. Names on the old interior faces or diagonal move to the designated successor $abd$. Boundary and exterior labels are retained. No matter channel or coupling fires.

The update is a certified $(G)$-rule preserving the metric, marked resting slice, and vertex monodromies. Particle deficit and rapidity remain $\pi /2$ and $0$.

Proof. Both triangulations cover the same square and boundary. Every internal transition word is $(I,0)$. The particle word remains $(R(\pi /2),0)(I,0)$, with axis and resting normal $e_0$.

This constant-length gauge flip is not an edge-collapse exchange. Its guard and permission accounting are spectator-blind.

Let $s=0$ be the event. In a common chart $z=(T,x,y)$, choose

and define

An active face attains the maximum; its interior is the strict-maximum region. The concrete certificate uses

The graph of $F_s$ is continuous, piecewise affine, and $p$-Lipschitz with $p<1$. Every face is spacelike. For $s_2>s_1$,

Proof. The spatial gradients have norm $p$ or $q$; their maximum retains the common Lipschitz bound. A face tangent satisfies

for $u\ne 0$. Every face function increases by at least $a(s_2-s_1)$, so their maximum does too.

For $s<0$, $E,W$ share exactly

Its endpoints are $EWN,EWS$, and its length is

Proof. Equality gives $x=0$. Dominance requires $as\ge q|y|+bs$, hence $q|y|\le -\Delta s$. The two equalities at its endpoints add $N$ or $S$. The displacement is purely in the $y$-direction.

For $s>0$, $N,S$ share exactly

Its endpoints are $NSE,NSW$, and its length is

Proof. Equality gives $y=0$; dominance requires $p|x|\le \Delta s$. The displacement is purely spatial.

At $s=0$, all four planes meet at the origin and the central lengths vanish, but every face contains a two-dimensional open region. The incidence change is

Proof. Here $F_0=\max \{p|x|,q|y|\}$. For example, $E$ is strict on $x>0,px>q|y|$. The other regions follow by symmetry; the four-way intersection is only the origin.

The central one-sided derivatives are

Mixed adjacencies $EN,ES,WN,WS$ persist. No other core pair exchanges a positive-length edge near the event.

Proof. Differentiate the central laws. In each quadrant a mixed adjacency lies on

and extends from a central triple junction to the exterior. The only remaining pairs are $EW$ and $NS$, already calculated.

The exchange is determined by the declared maximum slicing rule. It is not a uniqueness theorem over all foliations.

frames. For a slope $v_i$, choose a Lorentz map with time coordinate

On its face this equals $s$. The concrete maps are

They are proper future-preserving Lorentz maps. Set

The opposite-pair entries are

For $X_{+}=E,X_{-}=W,Y_{+}=N,Y_{-}=S$,

Reverse maps are inverses; all core translations are zero.

Every closed contractible core word is the affine identity. In particular,

The event four-face word is also $I$.

Proof. Every word telescopes:

Nontrivial adapted transitions here are frame changes in one flat development, not evidence of spacetime curvature.

For $-1\le s\le 1$ and $H=2$, define

The uncapped interior region is

since $F_s\le H+s$ is equivalent to these two inequalities. Choose

Then

The capped graph is continuous, piecewise flat, and $p$-Lipschitz. It agrees with $T=H+s$ near $\partial K$, and retains the lower time increment $a(s_2-s_1)$. Both central edge laws remain unchanged in the window.

Proof. Minimum and maximum preserve a common Lipschitz bound and a common lower time increment. Strict collar margins give the resting boundary neighbourhood. The incoming central height margin is

the outgoing margin is

Thus the cap does not clip the central edges.

collar cells. There are four core faces and four resting collar polygons. The north-east collar has spatial vertices

all at $T=H+s$. Reflections, with boundary order reversed when needed, give the other counterclockwise polygons. Inner and outer side midpoints divide each rectangle boundary into eight segments.

At an inner corner, two core faces and one collar face meet. At an inner midpoint, one core face and two coplanar collar subdivisions meet. These subdivisions have fixed incidence and are not additional transitions.

The collar chart is

Core-to-collar transitions are

For affine pairs,

Collar and mixed vertex words are the affine identity. All polygons close, and matching edges have equal induced lengths.

Proof. All cell charts are globally defined affine isometries $A_i$ of the same patch; transitions $A_i{A}_j^{-1}$ telescope. The core polygons are intersections of their declared half-spaces with ${\mathcal{R}}_s$, and the collar polygons have the listed boundaries. Shared endpoints and segments are identical.

to the cone. Use the wedge of angle $3\pi /2$, and translate the patch by

The entire $K$ lies inside its first quadrant, away from axis and seam. Set $T=H+s$ on the rest of the cone.

For an explicit finite exterior construction, divide the wedge into its three convex quadrants. In the first quadrant cut by the vertical lines through $20-R_x,20,20+R_x$ and the horizontal lines through $20-R_y,20,20+R_y$. Retain the cells outside the translated $K$, and replace its interior by the eight patch cells. This finite line arrangement matches the eight boundary segments. Unbounded exterior cells and rays are permitted and remain fixed.

Shift the cone development to

The time shift commutes with $R(\mu )$. The collar-to-exterior transition is

Incoming and outgoing descriptions use the same boundary labels, constant spatial boundary vertices and lengths, resting boundary plane $T-H=s$, and fixed exterior transitions. The cone metric, exterior incidence, and resting particle neighbourhood are not rewritten. The particle word remains

Proof. The cap is resting near $\partial K$; all its boundary data and charts are common. A particle loop can be chosen in the unchanged axis neighbourhood. A path through the patch instead has a telescoping word determined by its endpoint charts, independent of its internal incidence.

Frozen exterior means unchanged geometric and affine-time laws, not frozen spacetime time coordinates.

core polygon boundaries. Write