Finite affine matching certificates for a spinless conical spacetime Landmark VIII-A

We formulate a finite admissibility interface for polygonal slices of a declared $2+1$-dimensional spinless conical spacetime. Geometric realization is checked before affine matching. The matching packet separates vertex and overlap equations, particle data, topologically marked holonomy words, collar agreement, and pointwise worldline fixation.

In a fixed developed presentation, incompatible proposed values of a prescribed transition cannot both be certified. A deterministic controller that factors through a named surviving observable algebra gives equal certified geometric outputs on equal accessible inputs, provided its complete control and resource specification is spectator blind. We prove the corresponding spectator-extension and restricted deferred-readout statements locally. An instrument-level obstruction shows why equal classical record laws do not determine residual source functionals.

For the fixed cone of deficit $\pi /2$, we give an explicit uniformly spacelike, embedded, acausal graph family with exactly one incidence event: the particle link changes from three faces to six. Exact vertices, lengths, areas, adapted affine charts, primitive winding, and full affine peripheral products furnish the certificate on both one-sided germs. The persistent particle signature is preserved, although the realization-dependent matching tables change. The intrinsic particle angle defects are computed separately and are not identified with the spacetime deficit.

Two incomplete updates fail different obligations. A changed deficit with a stale spatial rotation fails seam matching while still fixing the stationary worldline. A time-shifted rotation fails pointwise worldline fixation. The results establish an admissibility layer, not a sourced backreaction law, canonical polygon evolution, quantum recovery mechanism, or spacetime emergence theorem.

Keywords. Conical spacetime; affine holonomy; polygonal slices; constraint certificates; observation boundary; destructive readout.

An observation boundary specifies which state data a controller may use. It does not specify which geometric outputs are admissible. A geometric coupling therefore requires two independent pieces: a declared accessible input and a complete admissibility certificate for its output.

This paper supplies a finite matching contract for one declared spinless conical model. The notation ${\mathsf{Ein}}_0$, retained from the VIII programme, denotes that contract. It is not a classification of arbitrary Einstein initial data and is not an unstated continuum Einstein equation.

The classical geometric motivation is that source-free $2+1$-dimensional Einstein geometry with zero cosmological constant is locally flat DJH1984. Polygonal and holonomy descriptions have a first-order-gravity interpretation Kadar2005. These references motivate the dictionary. The dictionary itself is stated explicitly below, and no unquoted constraint-algebra theorem is used.

Our polygonal descriptions need not belong to a particular trivalent canonical polygon gauge. Nonconvex faces and regular vertices of valence four are allowed. We prove matching statements for the interface actually defined here.

The four principal statements are:

1. Fixed-presentation completeness. Once the developed realization, adapted charts, overlap components, reference framing, and seam maps are fixed, two different proposed values of a required transition cannot both pass the contract.

2. Constraint-blindness. Equal complete accessible inputs give equal applicability, geometric outputs, evaluated packets, and deterministically selected certificates.

3. Holonomy reduction. After realization and prescribed-transition checks, the holonomy obligations on a marked punctured disk reduce to regular identity relators, one primitive positive peripheral word, and its collar agreement.

4. Independent rejection tests. A stale deficit rotation fails deficit/seam matching. A time-shifted rotation fails the spinless pointwise-axis condition. Neither failure can replace the other.

The worked fixed-cone certificate is the substantive geometric instance of the contract. A1 and A2 are deliberately short: they are uniqueness in a prescribed presentation and factorization through declared inputs, respectively. Neither is advertised as a classification or an existence theorem for sourced dynamics.

The observation-boundary and residual-instrument questions are compared with Cheung's VII-E CheungVIIE. The fixed three-to-six slicing is compared with VII-F CheungVIIF. Every identity needed in this paper is proved here.

The recovery distinctions are compared with Oai280-C and Hawk04-B CheungOai280C,CheungHawk04B; they do not add a decoder to the geometric interface. VIII-B and VIII-C are separate sourced-realization and composition projects CheungVIIIB,CheungVIIIC. Their theorems are not imported.

Companion identifiers distinguish models and documentary relationships. Similar terminology does not identify their equation spaces, observable algebras, channels, cutoffs, or geometric dictionaries.

For a finite labelled register $\Gamma$, let ${\mathcal{H}}_{\Gamma }$ be its finite-dimensional Hilbert space. Write $L({\mathcal{H}}_{\Gamma })$ for its operators and $\mathcal{D}({\mathcal{H}}_{\Gamma })$ for its density operators. A channel is a completely positive, trace-preserving linear map.

Quantum names are access rights to factors of one joint state. Classical registers have distinguished bases. Admissible states are jointly block diagonal in those bases, including their correlations with all other registers. A diagonal classical marginal alone is not the required condition.

Born destructive computational-basis readout is a specified primitive. For a measured factor $S$, complement $B$, and classical output record $R$, put

Its channel is

The factor $S$ is absent from the output. No uniqueness theorem for the Born law is needed as a premise.

Coherent recording instead uses the isometry

where $A$ is a fresh quantum carrier. It retains both $S$ and $A$. A nondestructive instrument is another interface and must be specified separately.

On every operator, including operators with an untouched reference,

Every output source observable acts on declared surviving output names. An operator on the consumed factor $S$ is not a surviving source operator unless a new output representation is explicitly supplied.

Proof. Expand

Readout returns ${\sum }_z|z⟩⟨z{|}_R\otimes X_{zz}$. Tracing $R$ gives ${\sum }_z{X}_{zz}={\mathrm{Tr}}_{S}X$. An untouched reference can be included in $B$.

The codomain in A:eq:readout contains no $S$-factor. A pullback through ${\mathcal{M}}_S^{\ast }$ is an input observable used to calculate an output expectation, not a restored output factor. A representation of an $S$-observable on $RB$ would be additional interface data.

Operational forgetting is not physical annihilation of a complete dilation. It also does not determine a heat, work, or memory-reset cost; compare CheungEn07A.

A source boundary consists of a unital ${}^{\ast }$-subalgebra

on declared surviving names, together with a specified embedding into the complete current output interface. Its accessible state input is

Classical locations enter only through a declared classical-register encoding or as explicitly declared geometric/controller data.

This boundary does not automatically include consumed factors, inaccessible carriers, spectator cardinality, implementation history, or coherences outside the named algebra. A produced record may be accessible when explicitly authorized. That choice changes the applicable comparison theorems.

A finite instrument is a family of completely positive maps $\{{\mathcal{I}}_w{\}}_w$ whose sum is trace preserving. Its effects are $E_w={\mathcal{I}}_w^{\ast }(I)$. Its recorded channel returns the classical outcome and the residual output.

Let $\{P_z{\}}_z$ be a projective measurement on an input space $H$, let $q$ aggregate its fine outcomes, and let $T:L(H)\to L(K)$ be a channel. Define

and two recorded residual channels

where $\{{\mathcal{I}}_w{\}}_w$ is an instrument on $K$.

The classical record laws of $\mathcal{F}$ and $\mathcal{G}$ agree for every input state if and only if

For a declared residual algebra $\mathcal{O}\subseteq L(K)$, the channels agree on all recorded residual observables ${\sum }_w|w⟩⟨w{|}_R\otimes B_w$, $B_w\in \mathcal{O}$, if and only if

Both are operator identities and therefore remain valid under an untouched reference extension.

Proof. The fine recorded probability of $w$ is

The coarse probability is $\mathrm{Tr}(\rho T^{\ast }(E_w))$. Equality for every state is equality of their Hermitian effects, proving A:eq:record-effects.

Pairing the $w$-block with $B$ gives the two pullbacks in A:eq:residual-effects. Equality against every state, first for self-adjoint $B$ and then by linearity, is equivalent to their operator equality. Tensoring these linear-map identities with a reference identity preserves them.

Record agreement does not imply residual-source agreement, even when $T=\mathrm{id}$ and the coarse instrument has one outcome with map ${\mathcal{I}}_{\ast }=\mathrm{id}$. There is a qubit input for which the recorded residual outputs have trace distance $1/2$.

Proof. Take the computational projectors $P_0,P_1$, aggregate both outcomes to $\ast$, and take $T=\mathrm{id}$. Both record effects are $I$, so the record laws agree on every input.

For $\rho =|+⟩⟨+|$,

Their difference has residual block $X/2$, where $X=|0⟩⟨1|+|1⟩⟨0|$. Its eigenvalues are $1/2,-1/2$. Hence

The residual observable $X$ has expectations $1$ and $0$, respectively.

Here the coarse instrument is the identity. The aggregated fine residual channel is dephasing and is not the identity. Confusing these two maps would invalidate the example. This is the instrument-level distinction emphasized in CheungVIIE.

If implementations of the two outputs in A:thm:residual have trace-distance errors at most ${\epsilon }_{\mathrm{f}}$ and ${\epsilon }_{\mathrm{c}}$, their mutual trace distance is at least

Proof. Apply the triangle inequality through the two ideal outputs.

A record current and a residual current are different named interfaces. Neither is, merely by that name, a stress tensor.

A configuration is

where $g$ is the complete geometric description, $\Gamma ,\rho$ are the named matter interface and state, $\ell :\Gamma \to \mathrm{Cell}(g)$ is the abstract location map, $h$ is finite controller data, ${\nu }_G,{\nu }_{\mathrm{src}}$ are separate permission registers, and $\mathfrak{r}$ is the resource reservation. A certified $g$ includes its accepted certificate.

A location label is not a material trajectory, a mass assignment, or a stress tensor.

For a fixed source embedding, the complete update specification is a deterministic function of

and any additional explicitly declared accessible inputs.

Equal such inputs select the same rule, guard result, support, proposed geometry, evaluated packet, certificate presentation, successor controller state, and transport of original names. They also select the same declared matter channel whenever a matter channel is included.

Every used record, seed, program counter, permission bit, resource test, and observable is part of that declared specification. Undeclared state descriptions and remote scheduler inputs are excluded.

Adding inaccessible spectators leaves the original label dictionary and all original input interpretations unchanged. Matter maps extend by the spectator identity, and accessible observables extend by $B\mapsto B\otimes I_N$. Geometric and coupling reservations are independent of spectator number. Spectator transport and history storage are reserved separately; their outcomes and capacity tests are not additional source inputs.

This concerns the entire specification, not only one displayed source expectation. It is the local control discipline needed for the comparisons with CheungVIIE,CheungVIIG.

An exact incidence clock and a rounded length report are different inputs. A $G$-guard uses its declared exact clock. It may not substitute a zero rounded report or read occupancy without changing the interface.

We retain the deterministic nonlinear option, Option N:

with the other declared arguments suppressed. The dependence need not be affine in $\rho$. Consequently, the complete matter–geometry transformation is not asserted to be a CPTP map.

The source functional is a semantic input to this model. No single-copy protocol for acquiring exact expectations, no finite-gate implementation of an arbitrary expectation law, and no physical nonlinear feedback mechanism is proved. An implementation based on a supplied promised token would be a separate interface.

The controller comparisons below are pairwise equality statements at aligned steps. They are not a general no-signalling theorem for arbitrary nonlinear feedback.

Likewise, informational determination is not a CPTP decoder. For a linear channel on a complete logical input space, exact recovery requires a constant complete complement; A:app:recovery states and proves that criterion and an injective-channel counterexample. The criterion is not a subsystem-recovery criterion KnillLaflamme1997,CheungOai280C. A certified geometric output therefore cannot be read as recovery of an unknown quantum input.

Use coordinates $z=(T,X,Y)$ with

Affine isometries are written $(L,b)\in {\mathrm{ISO}}^{+}(2,1)$, with $L\in {\mathrm{SO}}^{+}(2,1)$, and act by

Thus

For $0<\mu <2\pi$, define

Let $e_0=(1,0,0{)}^{\mathsf{T}}$ and ${\mathcal{W}}_0=\mathbb{R}{e}_0$.

For a reference affine frame $A=(C,a)$, put

The vector $u$ is future unit timelike.

A spinless packet is

where $\mathsf{p}$ is a geometric particle identifier and the other entries satisfy A:eq:packet-holonomy. The orientation and the interval $0<\mu <2\pi$ are part of the data.

The map $H_{\Pi }$ fixes $\mathcal{W}$ pointwise, its Lorentz part fixes $u$, and its oriented rotation on $u^{\perp }$ has angle $\mu$. These conditions determine $H_{\Pi }$ uniquely from $\mu ,\mathcal{W},u$ and the declared orientation.

Proof. For $z=A(t{e}_0)$,

The Lorentz part fixes $C{e}_0$ and acts on its oriented orthogonal plane by the conjugate spatial rotation.

Conversely, a proper future-preserving Lorentz map fixing $u$ restricts to an orientation-preserving Euclidean rotation on $u^{\perp }$. The prescribed oriented angle determines its Lorentz part $L$. Pointwise fixation of $a+\mathbb{R}u$ then requires $b=(I-L)a$. Replacing $a$ by $a+t_{0}u$ does not change this translation.

A trace alone does not determine the oriented angle. A coherent change of common affine framing conjugates all holonomy data and transports the worldline and rest vector.

Let $L_{\mu }=2\pi -\mu$. The spatial cone $C_{\mu }$ is the wedge $0\le \theta \le L_{\mu }$, with the two boundary rays identified and the origin completed to an apex. The regular metric is

The spacetime is

The upper boundary ray is mapped to the lower ray by $J(\mu )$, because

This is wedge gluing, not a quotient of the entire plane by the group generated by a rotation. We take this spinless conical dictionary as the admissibility model. A normalization such as $\mu =8\pi Gm$ would be additional dictionary input. The singular particle interpretation is not derived from an observation-boundary identity.

Regular charts are Minkowski charts, so their curvature and Einstein tensor vanish. The singular interpretation is the supplied conical dictionary, consistent with the classical setting of DJH1984,Kadar2005.

A realized polygonal description supplies:

1. a marked ambient cone, its orientation, reference frame, worldline, and spinless packet;

2. a finite polygonal patch and a prescribed exterior;

3. developed face polygons, their affine planes, labelled vertices and edges, and declared overlap components;

4. a seam-compatible embedded slice, spacelike on each regular face, together with the claimed global acausality verification when used;

5. positive active edge lengths and positive face areas on each regular parameter epoch;

6. a marked collar and common based-loop framing;

7. for an event, two one-sided realized families with a common limiting slice and explicitly identified collapsed slots.

An unbounded exterior may have finitely many cell labels. Collapsed event slots are metadata, not zero-dimensional replacements for active edges or faces.

A one-sided incidence germ here means the restriction of a realized family to an open parameter interval adjacent to the event, together with its labelled limiting data. Positivity is required on that open epoch, not for a collapsed outgoing face at the event itself.

For each face $f$, let ${\mathcal{A}}_f\in {\mathrm{ISO}}^{+}(2,1)$ be its adapted chart change from the appropriate cut-space development. For an oriented overlap component from $f$ to $g$, let $n_{gf}\in \{-1,0,1\}$ denote its signed seam crossing. Chart paths retain their individual overlap components. The prescribed transition is

Holonomy alone does not determine these individual maps: the realization, adapted charts, and specific overlap component are also required.

For a realized description and proposed transition maps, the evaluated matching packet is

It records the following checks:

1. Vertex and overlap matching. Proposed maps equal A:eq:canonical on the specified components; reverse maps are inverses; paired endpoints and segments match; developed polygonal boundaries close; and regular local vertex words are affine identities.

2. Particle matching. The proposed peripheral map equals $H_{\Pi }$ in the declared framing. The deficit, affine worldline, and future rest direction satisfy A:eq:packet-holonomy.

3. Marked holonomy words. Ordered chart words, overlap components, connecting paths, and their topological classes are supplied. A positive primitive particle meridian has winding one, independently of its matrix value.

4. Collar matching. Core and collar share their prescribed boundary realization and transported reference framing. Their primitive peripheral words equal the same framed $H_{\Pi }$.

5. Worldline data. Every chart describes the transported singular worldline and rest vector consistently. The peripheral map fixes the worldline pointwise, and the declared slice intersection belongs to that worldline.

A complete certificate $\mathsf{Cert}(g)$ contains finite witness tables, exact equalities, winding evidence, and realization verification. It is accepted only when every applicable geometric and matching obligation holds.

Developed polygonal boundary closure means closure of the supplied boundary segments in the appropriate development. It is not an instruction to impose identity holonomy on a particle meridian.

Completeness is relative to this declared affine-conical contract. It does not classify all $2+1$ initial data, prove membership in every canonical polygon gauge, or turn an arbitrary matrix list into an embedded slice. A certificate is not an unconstrained string saying “pass,” and no executable exact-arithmetic checker is constructed in this paper.

An event changes the numbers of faces, vertices, and matching equations. Literal equality of the two raw ${\mathsf{Ein}}_0$ lists is therefore not required.

For a commonly marked punctured-disk domain, define

The slice-intersection point is checked in (E5), but it is not required to remain the same point of the fixed worldline as the slice advances.

A geometric update is constraint preserving if it:

1. returns an accepted output certificate, or a certified outgoing germ with its common event limit;

2. preserves $\mathsf{Sig}$ under the declared marking;

3. supplies the required original-name location transport.

The realization-dependent equations are recomputed.

Every rule returns its complete typed configuration and the applicable certificate.

1. A {$G$-rule} reads only declared geometric, geometric-controller, permission, and resource inputs. It preserves $\mathsf{Sig}$ and does not inspect matter state, occupancy, or spectator cardinality.

2. A matter rule applies a declared channel on its named interface, extended by identities on untouched factors. It does not change geometry.

3. An ordinary unsourced coupling rule may read a named source functional but leaves the particle signature fixed. A packet-changing rule is not of this type.

4. A sourced coupling rule proposes a new packet

   $${\Pi }^{\prime }=F_{\Pi }(\Pi ,{\omega }_{\rho }{|}_{{\mathcal{A}}_{\mathrm{src}}})$$

   and must also return its complete realization, charts, gluing data, collar, transport, and output certificate. The packet proposal alone is insufficient.

Geometric and coupling parts leave internal matter unchanged. Any included matter channel is separately declared and obeys the same untouched-factor requirements.

This contract states what a sourced constructor must return; it does not construct one.

If an update changes an argument of an obligation in (E1)–(E5), that obligation and all other affected checks must be reevaluated on the output. An incompatible input certificate does not certify the new data.

Proof. Acceptance verifies evaluated output equalities and output realization requirements. An input equality with changed arguments is not the required output equality. If incompatible old values are retained, the corresponding output equality fails. Unaffected witnesses may be reused only when they still verify the same obligation.

Fix a realized description, its adapted charts, packet, reference framing, overlap components, and seam maps. Two proposals agreeing on these data but differing on one prescribed transition cannot both be certified. The same conclusion holds if several transitions differ.

Proof. The fixed data prescribe the unique affine map

Condition (E1) requires the proposed transition to equal this map. Two different affine maps cannot both equal it. Apply this argument to any differing component.

If a chart transition is $t={\mathcal{A}}_g{{\mathcal{A}}_f}^{-1}$, replacing ${\mathcal{A}}_g$ by $B_g{\mathcal{A}}_g$ gives $B_{g}t$. Closed words undergo the corresponding base-chart conjugation, and the geometric structure may be unchanged. Thus different valid chart presentations can have different transition maps. Peripheral holonomy also does not determine edge lengths or face shapes.

Let two configurations have the same geometry, controller state, permissions, geometric resource reservation, source embedding, and all other declared accessible inputs. Suppose

Under A:ass:control, their evaluated input packets agree. They receive the same applicability decision, proposed update, output packet, and deterministically selected certificate. Either both are certified or both are rejected.

Proof. The input packet is determined by the common geometric data. Every argument of the deterministic specification agrees. Its guard, geometry, packet, support, transport, certificate, and successor controller outputs therefore agree. The evaluated acceptance conditions have the same truth value.

This does not choose $F_{\Pi }$ or prove existence of a nonconstant admissible sourced realization.

A rule with guard ${\nu }_{\mathrm{src}}\ge 1$ can have different applicability on permissions $0$ and $1$, even when geometry and source functionals agree. Equal permissions are therefore an essential hypothesis.

For any linear map $\Psi :L({\mathcal{H}}_{\Gamma })\to L({\mathcal{H}}_{{\Gamma }^{\prime }})$ and any operator $X$ on $\Gamma N$,

No product-state or trace-preservation assumption is needed.

Proof. Pair both sides with any output operator $B$. Both pairings are

Equality against all $B$ proves the identity.

Consider a finite execution of the rule types in A:contract:rules. Extend its initial state to an admissible joint state $\tilde{\rho }$ on $\Gamma N$ such that

Assume the complete blindness and extension clauses of A:ass:control, identical original locations, and legal separate spectator transport/storage.

Then an aligned extended execution is legal. At every step, the original reduced matter states, geometric descriptions, persistent signatures, certification decisions, original transports, and declared controller outputs agree.

Proof. Induct on steps. For each accessible observable,

Equal complete inputs therefore select the same rule, guard, support, and certificate.

A matter channel preserves the reduction identity by A:lem:restriction. A $G$-rule receives the same geometric/controller inputs and leaves internal matter unchanged. A coupling rule receives the same accessible functional, so A:thm:A2 gives the same geometric and certification output. Any common included matter channel again preserves the original reduction. Original transport and successor control agree. The separate-reservation clauses establish spectator legality without changing these decisions.

The extension may be entangled and need not be a fixed pure or product spectator. The theorem identifies the original reductions, not arbitrary full joint extensions.

A guard $|\Gamma |\le K$ would violate the hypotheses: one product spectator could change its decision without changing any accessible source expectation.

Equation A:eq:restriction also applies to an unnormalized completely positive instrument branch. For paired branches selected from equal declared inputs, the original unnormalized reductions and branch probabilities agree. Conditional reductions agree on positive-probability branches. This is a branchwise comparison, not a claim that the spectator conditional state remains unchanged, and it does not promote Option N to a joint CPTP model.

The isometry $V$ in A:eq:copy satisfies

Let $\Lambda$ be a channel acting only on $B$, and let ${\mathcal{N}}_A$ destructively read $A$ into $R$. Then, with the required identity extensions,

The identities include every untouched external reference.

Proof. Expand $X={\sum }_{z,z^{\prime }}|z⟩⟨z^{\prime }{|}_S\otimes X_{z{z}^{\prime }}$. Coherent recording appends $|z⟩⟨z^{\prime }{|}_A$. Tracing $S$ removes exactly the terms $z\ne z^{\prime }$, giving the first formula. Operations on $B$ act on the blocks $X_{z{z}^{\prime }}$ and commute with the indicated restriction and carrier readout. This proves A:eq:deferred. A reference is another untouched factor in the same calculation.

Compare an immediate destructive-readout execution with a coherent-recording execution. During their continuations suppose:

1. source and control inputs use only the evolving complement $B$, not $R$, $A$, or retained $S$;

2. initial geometry, controller data, permissions, resource inputs, and original complement locations agree;

3. all continuation matter channels act only on $B$, are selected equally from equal accessible inputs, and extend by identities on excluded factors;

4. excluded-factor storage and transport are separately legal and cannot affect guards or geometric decisions.

Their geometric outputs and certification decisions agree at every aligned continuation step. After terminal readout of $A$, the record–complement outputs agree when the deferred output is additionally restricted away from $S$.

Proof. Initially, both $B$-reductions are ${\mathrm{Tr}}_S\rho$. Equal complement reductions give equal accessible source functionals. Constraint-blindness selects the same geometric and control outputs, and the common complement channel preserves equality by A:lem:restriction. Induction proves the continuation assertion.

For the fixed initial pair, these common choices select the same finite sequence of complement channels. Their composition is a common channel $\Lambda$. Apply A:lem:deferred to $\Lambda$ to obtain the terminal record–complement equality.

The terminal carrier $A$ has been consumed. The additional retained factor to be removed is $S$, not $A$. If the immediate controller reads $R$ during continuation, this corollary does not apply. A record-dependent comparison requires different hypotheses.

A chronological chart path is $f_0,f_1,\dots ,f_m$, with each overlap component specified. The last transition acts on the left:

The based-loop package supplies a common collar basepoint and affine frame, a chosen connecting path to each loop, its transport, and a marked primitive positive peripheral class. A connecting path is data, not a topologically unique path.

If $Q$ transports the common reference coordinates into the starting chart and $W_f$ is the chart-based loop word, then

Peripheral equality always means equality after transport to the same basepoint/frame.

For a closed canonical chart path,

On the marked punctured cone slice, $N(\gamma )$ is its oriented winding number.

Proof. Substitute A:eq:canonical. Adjacent adapted-chart factors cancel, leaving the displayed power of $J(\mu )$.

Lift angular position to the cover with period $L_{\mu }=2\pi -\mu$. A positive crossing of the marked seam increases the sheet index by one; a negative crossing decreases it by one. A closed path has angular increment $N(\gamma )L_{\mu }$, which is its winding number times the angular period.

A small regular loop crossing the cut once positively and once negatively has $N=1-1=0$, hence identity affine product. A change of written charts does not make that loop a particle meridian.

At $\mu =\pi /2$, $J(\pi /2{)}^4=I$. A fourfold positive particle loop has winding four and is noncontractible, despite its identity affine matrix. The matrix representation is not faithful to the integer winding group.

A finite presentation consists of a connected finite cell structure $K$, a marking identifying its based edge-path homotopy group with the fundamental group of the punctured-disk domain, and:

1. invertible transition values on oriented edges, with inverse values on reversed edges;

2. regular two-cell boundary relators;

3. a marked primitive positive peripheral edge word;

4. the frame transports used to compare based words.

One may use a triangulated compact annular truncation and a finite refinement with all required boundary paths. A dual structure is allowed only when its marking actually gives this presentation.

In a presentation satisfying A:def:presentation, suppose every regular two-cell boundary has identity ordered product and the marked primitive positive peripheral word has product $H$ in the common frame. Then every based loop of winding $w$ has product $H^w$. Conversely, this loop formula implies those local and peripheral conditions.

Proof. Deleting an edge followed by its reverse preserves the product. The identity boundary relators permit replacement of one boundary path of a regular two-cell by its complementary path without changing the product. The declared presentation makes these moves the required edge-path homotopies.

The punctured disk, or its annular truncation, has fundamental group $\mathbb{Z}$, generated by the marked positive meridian. A winding-$w$ loop is based-homotopic to its $w$-fold power, with negative powers represented by reversed loops. Product invariance gives $H^w$.

Conversely, regular boundaries have winding zero and the marked meridian has winding one.

The peripheral word is a generator with prescribed affine value, not an identity relator filling the puncture. Small vertex loops give regular relators only within a declared presentation that implements the required topology.

Fix a realized marked spinless cone patch with punctured-disk topology, its packet, adapted charts, overlap components, based-loop package, and a presentation as in A:def:presentation. After checking endpoints, developed face closure, reverse maps, and prescribed transitions in (E1), the holonomy obligations are equivalent to:

1. identity products at regular relators;

2. one primitive positive peripheral product equal to $H_{\Pi }$;

3. a collar primitive product equal to the same framed $H_{\Pi }$.

Other based words follow from winding and transport. Particle/worldline matching additionally requires the supplied worldline, rest-vector, and slice-intersection data and the algebraic conditions of A:lem:spinless.

These word checks do not replace realization, spacelikeness, positivity, primitive-winding evidence, or admission to the spinless dictionary.

Proof. The finite-relator lemma gives every based product from (a) and (b). Condition (c) verifies the particular collar identification in the common frame. The spinless lemma determines the required pointwise-axis map from the supplied packet. Chart transport verifies the representations of the worldline and rest vector; the slice intersection is separately checked.

Conversely, the complete packet includes the specified regular, peripheral, and collar words. None of these matrix products controls a face gradient or proves embedding, so realization remains an independent prerequisite.

The formulas in this section independently reproduce the fixed slicing compared with VII-F CheungVIIF. No predecessor theorem is needed for their verification.

Set

Let

Sector $i$, $i=0,1,2$, has orthonormal coordinates

Cone labels are cyclic modulo three. Developed vectors $d_0,\dots ,d_3$ are not cyclically identified until the seam is glued.

In particular, a vertex with cone label $A_0$ has lower development $3d_0$ and upper development $3d_3$. Whenever a polygon in sector two uses $A_{i+1}$, $B_{i+1}$, or $D_{i+1}$, it uses the upper developed copy. The seam rotation sends it to the corresponding lower copy. This convention applies to every table below.

Define the cone functions sectorwise by

On each shared ray,

They therefore descend to the cone.

For $-1\le s\le 1$, let

The graphs ${\Sigma }_s$ are seam compatible, embedded, and globally acausal. Each regular face is uniformly spacelike, with

For $s_2>s_1$,

Moreover,

The family has a common uniform limit at $s=0$.

Proof. The nonzero affine branch gradients in a sector are

Their Euclidean norms are bounded by $\lambda$. Minimum and maximum of real functions with a common Lipschitz constant retain that constant. Shared-ray values give continuity across all rays and the seam.

Integrating the sectorwise bound along spatial paths, including paths through shared rays or the apex, gives

Graph projection is a homeomorphism onto the cone, so the graph is embedded. On a face with gradient $a$, the induced metric is $I_2-a{a}^{\mathsf{T}}$, and

A causal curve in the static product has elapsed absolute time at least the length of its spatial projection. Distinct graph points have smaller absolute time separation than their spatial distance, because $\lambda <1$. They cannot be causally related.

For fixed $x$, the capped inner expression is nondecreasing and $1$-Lipschitz in $s$. Adding $s$ gives A:eq:ordering. Since $\min \{\xi ,\zeta \}\le \rho /2$,

If $\rho \ge 3$, then $h\ge 2$, so the outer minimum equals $2$. The upper bound in A:eq:ordering also gives uniform convergence to $F_0$ as $s\to 0$.

No regular tangent plane at the singular particle is asserted. The worldline intersection is

It moves along the fixed worldline; this does not change the packet.

Take the compact patch $\rho \le 4$ and the marked collar $7/2\le \rho \le 4$. The collar and exterior are resting spatially, with equation $T=2+s$. “Prescribed exterior” means this translating resting exterior, not a frozen embedded time slice.

Write $A_i=3d_i$ and $D_i=4d_i$ for spatial vertices, with times $T=2+s$. For $s\le 0$, the core faces are

For $s>0$, introduce spatial vertices

and use

Their new-vertex times are

The collar-side faces are

Outside the patch, retain three horizontal unbounded sector faces and the persistent ray subdivisions.

The listed polygons cover the graph and have disjoint face interiors. The cellulation is constant on each open epoch $s<0$ and $s>0$. For $s>0$, the new cells lie strictly inside $\rho =3$ and are not cut by the horizontal cap.

Proof. For $s\le 0$, the bound $k\le h$ gives $s+k\le h$. The core is therefore the three triangles $V_i$.

For $s>0$, in the half-sector $\xi \ge \zeta$,

This segment joins $B_i$ to $C_i$. In the other half-sector,

joining $C_i$ to $B_{i+1}$. The two triangles are precisely the regions where $s+k$ dominates; their complement in the outer triangle is the listed simple pentagon.

At the new vertices,

Thus the new cells remain away from the crest. On them,

so the cap does not remove a new cell. Outside $\rho =3$, A:lem:family gives the horizontal faces. These exact dominance regions have disjoint interiors and exhaust the graph.

Let $e_i$ be the persistent abstract edge label, realized as $[P,A_i]$ incoming and $[B_i,A_i]$ outgoing. For finite edges, direct Minkowski lengths give

The induced face areas are

Equations A:eq:length-table,A:eq:area-table are the intrinsic edge lengths and areas of the listed faces. All active finite cells are nondegenerate on their open epochs.

Proof. For each segment,

Incoming, $P{A}_i$ has spatial displacement $3$ and time displacement $2$, giving $\sqrt{5}$. Outgoing,

The horizontal edges give their displayed Euclidean lengths. The edge $P{B}_i$ has zero time displacement.

For $P{C}_i$,

For $B_i{C}_i$, in sector coordinates,

so

Reflection gives the same length for $C_i{B}_{i+1}$.

The projected outer triangle has area $9/2$. Each new projected triangle has area

On a plane $T=c+a\cdot x$, area density is $\sqrt{1-|a{|}^2}$. It is $1/3$ on $V_i$ and $4/5$ on ${U_i}^{\pm }$. Thus

and

The horizontal quadrilateral has area $(4^2-3^2)/2=7/2$.

Every new length is a positive coefficient times $s$, and every new area is a positive coefficient times $s^2$. For $0<s\le 1$,

All remaining finite lengths and areas are constant positive values. This proves nondegeneracy.

The family has exactly one incidence-event parameter in $[-1,1]$, namely $s=0$. At that parameter:

1. the six vertices $B_i,C_i$ coalesce at $P_0$;

2. the twelve new edges $P{B}_i,P{C}_i,B_i{C}_i,C_i{B}_{i+1}$ collapse;

3. the six faces ${U_i}^{\pm }$ collapse;

4. the particle link changes from three incident faces to six.

There is no other collision, vanishing active finite edge or face, or contact of a new cell with the outer crest. At $s=0$, the regular cellulation is the common limiting three-face core; outgoing collapsed slots are germ metadata.

Proof. The exact dominance regions show constant incidence on each sign-preserving open epoch. For $s>0$, the explicit new coordinates are distinct, and A:lem:metric gives positive active lengths and areas. The bound $\rho (C_i)\le 30/11<3$ excludes crest contact. Coverage and nonintersection were established by A:lem:cells. The incoming finite cells are nondegenerate and have unchanged incidence.

All new coordinates are proportional to $s$, so their specified cells collapse only at $s=0$. The persistent $e_i,V_i$ have the incoming nondegenerate segments and triangles as their limits.

Incoming, the particle meets exactly the three $V_i$. Outgoing, it meets exactly the six ${U_i}^{\pm }$; the pentagons no longer touch it. Across each bisector edge $P{C}_i$, the gradients are $(3/5,0)$ and $(0,3/5)$. Across a ray edge $P{B}_i$, unfolding the adjacent sectors changes the sign of the transverse gradient. These are genuine creases. The six-face link is therefore not a mere subdivision of one plane or a relabelling of an unchanged star.

As a combinatorial check, the compact patch counts are

The count supports the explicit disk cellulation; it is not a substitute for its realization proof.

For $|g|<1$, define

with $\mathcal{B}(0)=I_3$. It sends $T=c+g\cdot x$ to a plane with first coordinate $\gamma c$.

Let $O_i=\mathrm{diag}(1,\mathrm{Rot}(i\pi /2))$, and use

They are, respectively, $\mathcal{B}((2/3,2/3))$, $\mathcal{B}((3/5,0))$, and $\mathcal{B}((0,3/5))$. Direct multiplication verifies $B^{\mathsf{T}}\eta B=\eta$, determinant one, and future orientation.

The adapted charts are

The nonresting faces have chart time zero. The identity charts retain resting chart time $2+s$. Canonical transitions match all paired endpoints and segments, preserve reverse-map consistency, and give closed developed face boundaries.

Proof. In sector coordinates,

For the new faces,

and the reflected formula holds with $\xi$. The resting charts are deliberately not recentered.

For a nonseam edge, its endpoints are the same developed points on the two adjacent planes. Their chart coordinates therefore match under ${\mathcal{A}}_g{{\mathcal{A}}_f}^{-1}$. For a seam edge, $J_0$ identifies its upper developed endpoints with the corresponding lower endpoints, with time unchanged. Inserting ${J_0}^{n_{gf}}$ gives the same matching conclusion. Affine endpoint matching matches the full segment.

Reversing the crossing component changes its sign, so A:eq:canonical gives inverse transitions. Each boundary is the ordered boundary of the supplied closed simple polygon.

The chart formulas extend to face closures. They do not provide a regular Minkowski neighbourhood of the apex.

Choose the common collar basepoint

in sector zero, in the identity reference chart. The collar meridian $\rho =15/4$, traversed in increasing angle, has chronological word $I,I,J_0$.

The core meridian is $\rho =29/10$. Since

it lies in $V_i$ throughout the window. Its chosen connecting path from $b_s$ follows the sector-zero bisector, on the graph, with no seam crossing.

For $s>0$, the star meridian $\rho =s$ lies in the new star, because $h-k\le 2s/3<s$. For its connecting path, first move within the sector-zero collar to the ray with $\xi :\zeta =2:1$, then follow that ray inward to $(\xi ,\zeta )=(2s/3,s/3)$. This path has no seam crossing and ends in $U_0^{-}$. It avoids the bisector crease except at no required endpoint.

To supply a finite presentation, truncate the apex by a small positive polygonal meridian, intersect the realized faces with the resulting compact annulus, triangulate its simple polygonal pieces, and refine with the necessary chart paths and boundary paths. Mark this full annular cell structure by its realization. Its fundamental group is the marked meridian group $\mathbb{Z}$. At $s=0$, use the regular limiting realization. No claim about an incomplete interior dual complex is needed.

On every regular slice and on both one-sided incidence germs:

1. all regular local vertex words are affine identities;

2. core and collar meridians are primitive positive loops with reference-frame product $J_0$;

3. the outgoing six-face star has the same transported primitive product;

4. the packet is

   $$(\mathsf{p},\pi /2,{\mathcal{W}}_0,e_0,\mathrm{id},J_0),$$

   with consistent worldline and rest-vector transport.

Together with the realization checks, these witnesses give the complete matching certificate.

Proof. Every small regular vertex loop has winding zero, including a loop whose written path crosses the seam in opposite directions. Its affine word is identity by A:lem:word.

Write ${\mathcal{A}}_i={\mathcal{A}}_{V_i}$. The core word starting in $V_0$ is

The no-seam connecting path has transport ${\mathcal{A}}_0$, because all intermediate adapted charts cancel. Reference transport gives

The collar product is directly $J_0$.

Both meridians have angular lift increment $3\pi /2$. This proves positive primitive winding independently of the matrix product.

For the outgoing star, the chronological face order is

The five ordinary crossings and one seam crossing telescope to ${\mathcal{A}}_{U_0^{-}}{J}_0{{\mathcal{A}}_{{\mathcal{U}}_0^{-}}}^{-1}$. The chosen no-seam connecting path has transport ${\mathcal{A}}_{U_0^{-}}$, so the reference word is again $J_0$. Its angular increment is $3\pi /2$.