A sourced spinless conical packet with a persistent three-to-six polyhedral slicing event Landmark VIII-B

We construct an explicit spinless conical family on a compact deficit interval about $\pi /2$ and prove persistence of one declared three-to-six polyhedral slicing event. Three sectors of angle $(2\pi -\mu )/3$ are used, with coefficients retuned in the physical oblique metric. The two nonconstant gradient norms are exactly $2\sqrt{2}/3$ and $3/5$. The slices are embedded, uniformly spacelike on regular faces, globally acausal, and strictly ordered. At every admitted fixed deficit, the declared cellulation has exactly one incidence-event parameter. Vertices, lengths, one-sided laws, areas, adapted affine charts, winding-marked words, and primitive core and collar holonomies are explicit.

A fixed Hermitian observable on a named surviving matter interface selects a member of this preconstructed family through its expectation. The resulting deterministic semantic constructor leaves matter unchanged and is invariant under spectator extensions that preserve the named source reduction. Fixed-presentation matching uniqueness, stale-rotation rejection, and time-shift rejection are proved for the whole deficit interval.

A separate finite restriction uses an algebraic packet table and a supplied classical source token. On canonical promised inputs, it has an exact NOT–CNOT–Toffoli realization with retained history, clean work returned to zero, and identity action on internal matter. The token's agreement with the expectation is a representation promise, not an implemented measurement. Affine packet selection, expectation acquisition, classical records, and quantum recovery are distinguished by local proofs.

No mixed incidence–deficit germ, hybrid evolution theorem, field-theoretic source admission, mass law, or derivation of Einstein dynamics is asserted.

Changing a conical deficit requires more than changing a label. The sector opening, seam identification, realized slices, adapted charts, and matching witnesses must change coherently. This paper supplies such a constructor on an explicit compact interval and proves that a specified particle-link event persists throughout it.

The argument has three independent stages:

1. construct and certify the geometry for every scalar deficit in the interval;

2. substitute the expectation of a fixed named Hermitian matrix into that scalar construction;

3. implement a finite algebraic restriction on classical codes carrying a compatible, externally supplied source token.

The geometric theorem does not depend on the source law. The finite implementation does not acquire the expectation and does not turn the continuum semantic rule into a quantum channel.

The affine matching interface of Landmark VIII-A CheungVIIIA is restated here. Its fixed-presentation uniqueness and two rejected updates are proved locally for variable deficit. The accessible-current and residual distinctions reported in VII-E CheungVIIE, and the affinity distinction reported in VII-G CheungVIIG, are also proved in the present notation. None of these companion results is a formal premise.

The physical setting is motivated by classical $2+1$-dimensional gravity with zero cosmological constant: source-free regions are locally flat DJH1984, and polygon descriptions can be reconstructed from first-order gravity Kadar2005. These references provide context. We import neither a mass normalization nor a Hamiltonian or constraint-algebra theorem.

Faces in the present construction may be nonconvex, and regular vertices need not be trivalent. The results concern the explicit matching contract below, not every more restrictive polygon gauge.

and conventions. Matter states are $\varrho$; the sector radial coordinate is $\rho$. Developed spatial rays are $e_i\in {\mathbb{R}}^2$, whereas $\mathbf{t}=(1,0,0{)}^{\mathsf{T}}$ is timelike. The second oblique coordinate is $\zeta$, distinct from the Minkowski metric $\eta$. All matrix operator norms are Euclidean spectral norms, including norms of homogeneous affine matrices; $\Vert \cdot {\Vert }_1$ is the trace norm. Report precision, clock precision, and permission width are $p_{\mathrm{rep}}$, $p_{\mathrm{clk}}$, and $w_{\mathrm{src}}$. Lengths and the slicing parameter are expressed in fixed normalized units.

Let $\Gamma$ be a finite named matter interface and $\varrho \in \mathcal{D}({\mathcal{H}}_{\Gamma })$. Registers declared classical are jointly block diagonal in their distinguished bases, including their correlations with other factors.

For destructive basis readout of a factor $S$, with complement $B$, the output channel is

The name $S$ is absent from the output interface. Forgetting $R$ gives ${\mathrm{Tr}}_S\varrho$, by summing the diagonal blocks.

Choose a unital $\ast$-subalgebra

on declared surviving names, with a fixed embedding into the complete current interface. Fix

The embedding is suppressed in formulas. Equivalently, the source input is the restricted state functional

The notation $j[f]$ permits a declared real linear combination of smearings, but only the resulting fixed finite matrix is used. Calling it a current is a declared interpretation, not a hypothesis needed to evaluate the trace.

A consumed factor, an inaccessible coherent carrier, spectator cardinality, and implementation history are not source variables. A produced record is eligible as a source only when its surviving register is explicitly named.

For a channel $\Phi$, its adjoint is defined by

The following finite-dimensional observation criterion makes the meaning of source agreement explicit.

Let $\Phi ,\Psi$ be channels with the same input and output interfaces, and let $j=j^{†}$ be a fixed output observable. On the full input state space,

Moreover,

Agreement on every observable in a unital $\ast$-algebra ${\mathcal{A}}_{\mathrm{src}}$ is equivalent to equality of the adjoints on that algebra.

If inputs are restricted to the complete classical block-diagonal state space, replace adjoint equality by

and replace the norm in (eq:current-error) by the norm of this pinched operator. Here ${\mathcal{P}}_{\mathrm{cl}}$ is the input block pinching.

Proof. Put $\Delta ={\Phi }^{\ast }(j)-{\Psi }^{\ast }(j)$, which is Hermitian. The expectation difference is $\mathrm{Tr}(\varrho \Delta )$. Pure states in the eigenspaces of $\Delta$ show that all these expectations vanish precisely when $\Delta =0$, and that the largest absolute expectation is $\Vert \Delta {\Vert }_{\mathrm{op}}$. Every algebra element is a complex linear combination of Hermitian elements, proving the algebra statement.

For block-diagonal $\varrho$,

All density operators within every classical block are admitted. Spectral testing within those blocks proves both restricted statements.

This is the local current version of the observation-boundary criterion documented in VII-E CheungVIIE. Equality on one current does not imply equality on the whole source algebra, equality of channels, or equality of residual instruments.

Two channels can have identical classical records for every input and different surviving residual states.

Proof. For a qubit, let

and define channels to a one-label record and a surviving qubit by

Both records are always $\ast$. For $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$, the residual states are $|+⟩⟨+|$ and $I/2$. Their difference is the Pauli matrix $X_{\mathrm{P}}/2$, with eigenvalues $\pm 1/2$. Thus their trace distance is $1/2$. The residual observable $X_{\mathrm{P}}$ distinguishes them.

The example restates the residual distinction in VII-E CheungVIIE. In particular, agreement of classical token values does not certify agreement of an unobserved residual algebra or verify a token's agreement with an expectation.

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

Affine maps and their composition are

Define

The reference worldline and rest vector are

The declared frame is restricted to

It preserves the metric, orientation, and time orientation. For a fixed particle identifier $\mathsf{p}$, put

Explicitly,

For $0<\mu <2\pi$, the fixed-point set of $J(\mu )$ is ${\mathcal{W}}_0$, its Lorentz part fixes $\mathbf{t}$, and its oriented rest-frame spatial rotation is $\mu$. Consequently $H_A(\mu )$ fixes $A{\mathcal{W}}_0$ pointwise and its Lorentz part fixes $C\mathbf{t}$.

For $\delta \ne 0$, the map $(R(\mu ),\delta \mathbf{t})$ has no fixed point and is not an admitted spinless peripheral map.

Proof. The spatial rotation has no nonzero fixed spatial vector for $0<\mu <2\pi$, while its time component is unchanged. Affine conjugation transports the fixed set. A fixed point of $(R(\mu ),\delta \mathbf{t})$ would require $T=T+\delta$, which is impossible.

A developed presentation fixes:

1. the developed realization and all face, edge, and vertex labels;

2. every oriented overlap component;

3. the affine extension ${\mathcal{A}}_f$ of each face-coordinate map;

4. its prescribed developed ambient identification $S_{gf}$, including the seam identification;

5. a frame and topological winding convention.

An accepted transition on that component must satisfy the ambient affine-map equality

Its restriction must also match the realized shared segment.

The ambient equality is important. Agreement on the endpoints of one segment does not generally determine a Lorentz-affine map: a nontrivial boost can fix a spacelike line pointwise.

Write the certificate as

Its components contain the following five checks:

1. Realized local matching: endpoints, shared segments, polygon closure, overlap components, canonical ambient transitions (eq:ambient-matching), and regular local vertex words.

2. Packet matching: deficit, oriented rotation, and spinless peripheral affine map.

3. Marked ordered words: affine products with their overlap data and integer winding marks.

4. Peripheral agreement: core and collar primitive maps in the declared transported frame.

5. Particle data: worldline, rest vectors, slice intersection, and pointwise fixation of the worldline.

Embedding, spacelikeness, and active-cell nondegeneracy are separate geometric preconditions. Matrix matching alone proves none of them.

In a fixed presentation, two distinct proposed values for the same required transition cannot both be accepted. This is not uniqueness of transitions from peripheral holonomy alone.

Proof. Since ${\mathcal{A}}_f$ is invertible, (eq:ambient-matching) requires

Both accepted proposals must equal this map.

For the qualification, changing face coordinates by invertible maps $B_f$ changes transitions to $B_g{t}_{g\leftarrow f}{B_f}^{-1}$ and changes a based peripheral word by conjugation at its base. If that base map is the identity, the peripheral word is unchanged despite changes in individual transitions. Such a change is excluded only after the presentation has been fixed.

This is the precise local completeness statement used here, in the sense of VIII-A CheungVIIIA; it is not a classification of Einstein initial data.

At an incidence change, the raw cell tables change. For a fixed deficit, the preserved data are the particle and peripheral signature, not an unchanged list of vertex equations.

For a fixed named source embedding, equal incoming geometry, controller state, source permission, geometric resource reservation, declared frame, interpolation flag, and accessible source functional select equal applicability results, sourced rules, realizations, certificates, abstract original-name transports, and successor controller states.

Spectator locations and history storage are reserved separately. Their number and state do not change the geometric or sourced update budget.

The sourced step is performed at

Its formal input is

On the admitted domain, it returns:

1. a new packet and ambient cone, with seam, rest data, complete slicing family, charts, collar, and certificate;

2. the new slice at $s=s_c$;

3. permission ${\nu }_{\mathrm{src}}-1$ and the selected successor controller;

4. unchanged internal matter and identity transport of valid incoming abstract cell labels.

Identity label transport is not a pointwise identification of old and new cones.

This step consumes no incidence permission and fires no incidence rule. Future incidence germs are returned as family data; their installed deficit stays fixed when traversed.

Set

Then

Write

and retune the coefficients by

The developed rays are

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

Physical cone labels are cyclic modulo three. The fourth developed ray is an upper seam copy, not a fourth sector.

For every $\mu \in I$, the three sectors have total opening $2\pi -\mu$. Identifying the upper boundary with the lower boundary using $J(\mu )$ produces a spinless spatial cone $C_{\mu }$. The ray data, metric coefficients, seam, and packet vary continuously with $\mu$. The particle axis is fixed pointwise and the oriented rest-frame rotation is $\mu$.

Proof. Since $0<\alpha <\pi$, each sector is a convex wedge with independent boundary rays. Its opening is $\alpha$, so the total is $3\alpha =2\pi -\mu$. The upper ray has angle $3\alpha$; rotation by $\mu$ sends it to angle $2\pi$, the lower ray. All formulas are continuous, and $I\subset (0,2\pi )$, so Lemma lem:spinless applies.

Continuity across different cones can be understood on one common topological template: glue three copies of the nonnegative $(\xi ,\zeta )$ quadrant by their corresponding boundary parameters. Only their physical Gram matrices and developments vary.

The ambient spacetime is

on its regular part. The singular time axis is the particle.

Let $\mu ,\nu \in I$ with $\mu \ne \nu$. Using the stale rotation $J(\nu )$ for the $\mu$-cone fails seam and packet matching, although it still fixes ${\mathcal{W}}_0$ pointwise.

For every $\mu \in I$ and $\delta \ne 0$, replacing $J(\mu )$ by $(R(\mu ),\delta \mathbf{t})$ fails pointwise worldline fixation. Both statements persist under a common declared frame $A$.

Proof. The stale rotation sends the upper ray $e_3(\mu )$ to angle

For this to be the lower ray, $\nu -\mu$ must be a multiple of $2\pi$. But $|\nu -\mu |\le \pi /10$, so it must be zero, contrary to the hypothesis. Also $R(\nu )\ne R(\mu )$. Hence (E1) and (E2) fail. The stale rotation nevertheless fixes the time axis.

The time-shifted map sends $T\mathbf{t}$ to $(T+\delta )\mathbf{t}$. It fixes the rest vector in its linear part but not a single point of the worldline. Thus (E5) fails; unchanged seam heights also cannot match the shifted map. Conjugation by $A$ preserves these failures.

The two omissions are independent, as emphasized in VIII-A CheungVIIIA. Worldline fixation alone does not detect a stale spatial angle, and a correct spatial angle alone does not exclude a time shift.

Fix $\mu \in I$ throughout this section and let $s\in [-1,1]$. Abbreviate $a=a(\mu )$, $b=b(\mu )$, and $\alpha =\alpha (\mu )$. Define sectorwise

The slice is the graph

The function $\rho$ is a continuous piecewise-linear radial coordinate; it is not the Euclidean radius $|x|$.

The functions in (eq:family) agree across sector rays and the seam. The oblique spatial metric is

For a covector $p$, its physical squared norm is $p^{\mathsf{T}}{G_{\alpha }}^{-1}p$. In particular,

Proof. On a sector ray, $\rho$ is its ray parameter, $h_{\mu }=a\rho$, and $k_{\mu }=0$. Adjacent sectors and the seam-identified boundaries therefore agree.

The Gram matrix of the unit boundary rays is $G_{\alpha }$. Substitute $(a,a{)}^{\mathsf{T}}$, $(b,0{)}^{\mathsf{T}}$, and $(0,b{)}^{\mathsf{T}}$ into its inverse. Equation (eq:coefficients) gives the constant values.

Keeping the coefficients fixed while varying $\alpha$ would not preserve these physical norms.

Put

Uniformly for $\mu \in I$,

Consequently,

For $s>0$, the region $s+k_{\mu }\ge h_{\mu }$ has largest $\rho$-coordinate $s/c<r$ and is not cut by the cap. For $\rho \ge r$, the graph is resting:

In particular, $7/2\le \rho \le 4$ is a fixed resting collar.

Proof. Write $\alpha /2=\pi /4+t$, with $|t|\le \pi /120<27/1000$. Then

The inequalities $\cos t\ge 1-t^2/2$, $|\sin t|\le |t|$, and $\cos t\le 1$ give the stated strict bounds on $a$. Since $0<\alpha <\pi$, $b=(3/5)\sin \alpha \in (0,3/5]$, whence $c>a-3/10>69/200$.

The comparisons in (eq:loop-clearance) follow from $r=2/a$ and $R_{\ast }=1/c$, together with direct rational comparisons. In particular, $200/69<29/10$ is equivalent to $2000<2001$.

Since $\min \{\xi ,\zeta \}\le \rho /2$,

Thus $s+k_{\mu }\ge h_{\mu }$ implies $\rho \le s/c$. Equality occurs on the bisector. The inequality $b<a$ gives $a<2a-b=2c$, hence $1/c<2/a=r$. Finally, $h_{\mu }\ge 2$ when $\rho \ge r$, so the outer minimum in (eq:family) equals $2$.

Causality at the singular axis is understood in the ultrastatic length-space sense: a future causal curve has nondecreasing time and spatial speed at most its time speed. This agrees with ordinary Lorentzian causality on the regular part.

For every $\mu \in I$ and $s\in [-1,1]$:

1. ${F_s}^{(\mu )}$ is globally ${\lambda }_{\ast }$-Lipschitz in the intrinsic cone distance;

2. graph projection ${{\Sigma }_s}^{(\mu )}\to C_{\mu }$ is a homeomorphism;

3. on each regular face, the induced metric is at least $1/9$ times the spatial metric;

4. the graph is acausal.

For $s_1<s_2$,

The particle intersection is

No regular Minkowski tangent plane is asserted at the particle.

Proof. The affine spatial branches have gradient norm at most ${\lambda }_{\ast }<1$, by Lemma lem:gradients; the constant branch has norm zero. Minimum and maximum preserve a common Lipschitz bound. Integrating along paths, using continuity at sector rays, the seam, and the apex, proves the intrinsic global bound.

The inverse projection is $x\mapsto ({F_s}^{(\mu )}(x),x)$, so projection is a homeomorphism. For a graph plane with Cartesian gradient $g$, its induced metric is

This is a coordinate-independent lower bound relative to the spatial metric.

A causal curve between distinct spatial points has time difference at least their intrinsic spatial distance. The graph's time difference is strictly smaller, since ${\lambda }_{\ast }<1$. Distinct graph points therefore cannot be causally related.

For fixed $x$, the function $\min \{2,\max \{h_{\mu }(x),s+k_{\mu }(x)\}\}$ is nondecreasing and $1$-Lipschitz in $s$. Adding $s$ proves (eq:ordering). At the apex $h_{\mu }=k_{\mu }=0$, giving (eq:particle-time).

The source clock $s_c=-1/2$ is a parameter value, not a lapse. On a regular branch, the future unit normal is

so the normal component of the deformation $({\partial }_{s}F,0)$ is

It equals $3$ on a core branch, $5/2$ on an inner branch, and $1$ on a resting branch. These are local kinematical quantities of this prescribed family. No matter-defined relational clock or relational lapse is constructed. The relational expansion reported in Em02-E CheungEm02E is a different object.

The compact patch is $\rho \le 4$. Define

For $s>0$, define

Vertex symbols mean their lifted graph points when used in cells.

For $s\le 0$, the active core face in sector $i$ is

For $s>0$, the sector faces are

For all $s$, retain

and the three resting unbounded sector faces beyond $\rho =4$, with the exterior ray subdivisions.

The displayed active cells cover the graph, with disjoint face interiors. The incoming core faces are triangles; the outgoing $V_i$ are simple pentagons and may be nonconvex.

Proof. For $s\le 0$, $s+k_{\mu }\le h_{\mu }$, because $h_{\mu }-k_{\mu }\ge c\rho \ge 0$. Thus the uncapped core is $T=s+h_{\mu }$.

For $s>0$, in the half-sector $\xi \ge \zeta$, equality of the nonconstant branches is

Its endpoints are $(\beta s,0)$ and $(\chi s,\chi s)$. The other half-sector has boundary

The dominance region is the union of the two stated inner triangles. Its complement within $\rho \le r$ is the stated pentagon. The inequalities $a-b>0$ and $\rho (C_i)=s/c<r$ establish the placement and prevent a cap intersection. Outside $\rho =r$, the graph is resting.

Set

The persistent radial slot has epoch-dependent interpretation

{The 24 finite edge slots: three cyclic copies of each row. The first four types are active at $s=0$; the final four types are outgoing-only slots.}

The finite lengths are those of Table tab:lengths. Every active finite edge has positive length. The compact face areas are

Every active compact face has positive area.

Proof. For oblique spatial displacement $(\Delta \xi ,\Delta \zeta )$ and time displacement $\Delta T$,

Substitution gives the first four rows and $P{B}_i$. For $P{C}_i$, the spatial displacement is $(\chi s,\chi s)$ and the time displacement is $b\chi s$. For $B_i{C}_i$, these are $((\chi -\beta )s,\chi s)$ and $b\chi s$; the reflected edge has the same length.

Every active spatial edge is nonzero and tangent to a spacelike face, so its induced length is positive. In particular the displayed radicals are positive. Also

The spatial area element is $S_{\alpha }d\xi d\zeta$. Its graph multiplier is $\sqrt{1-|g{|}^2}$: $1/3$ on $V_i$, $4/5$ on ${U_i}^{\pm }$, and $1$ on $Q_i$. The incoming projected triangle has area $S_{\alpha }{r}^2/2$. Each projected inner triangle has area $S_{\alpha }\beta \chi s^2/2$. Subtracting both inner triangles gives the outgoing core area. The collar is the difference of projected triangles of radial sizes $4$ and $r$.

Since $b<a$,

Therefore, for $0<s\le 1$,

Finally $S_{\alpha }>0$ and $r<4$, proving all area assertions.

The persistent radial slot has one-sided laws

The crest, connector, and outer lengths have zero derivative. The born slots have right derivatives

Born slots have no incoming geometric edge derivative, since they are inactive rather than zero-length incoming edges.

The physical normal speed of the moving half-sector boundary is

the other half-sector has the reflected expression. These are slicing-parameter laws, not matter-velocity laws.

An incidence epoch is an open parameter interval on which the declared active-cell pattern is constant. The germs at $s=0$ consist of the one-sided labelled cell data, their limits, and their one-sided laws. Outgoing cells whose size tends to zero are germ metadata, not additional active zero-size cells of the limiting slice.

For the declared cellulation, the only incidence-event parameter in $[-1,1]$ is $s=0$. At that parameter, six outgoing vertex slots, six particle-star radial edge slots, six perimeter edge slots, and six inner face slots collapse at $P$. The particle link changes from three to six faces.

Proof. Lemma lem:decomposition gives a constant pattern for each sign of $s$. For $s>0$, the three $B_i$ lie on distinct cone rays and the three $C_i$ lie strictly in their sectors. Their coordinates are positive multiples of $s$, so they coalesce only at $s=0$.

All particle-star edges are genuine creases. Across $P{C}_i$, the gradients are those of $b\xi$ and $b\zeta$, which are distinct. Across a cone ray, unfolding the adjacent sectors gives inner transverse gradients of opposite nonzero signs. Incoming core gradients also have opposite nonzero transverse components after unfolding: their magnitude is $a\tan (\alpha /2)>0$.

No inner region meets the cap, by Lemma lem:bounds. Persistent radial lengths remain positive, all born edges and inner areas are positive for $s>0$, the outgoing core area is bounded below by a positive coefficient, and the collar never degenerates, by Lemma lem:lengths. The exterior is resting with fixed incidence. Thus no other collision, edge collapse, face collapse, or crease loss occurs in the window.

For $s<0$ the three $V_i$ meet $P$. For $s>0$ the six ${U_i}^{\pm }$ meet $P$, proving the link change.

The common limiting slice consists of the regular core triangles, collar, and resting exterior. The labels $P,A_i,D_i$, the persistent slots, and the core/collar/exterior face labels persist. The outgoing-only $B_i,C_i$, $P{B}_i,P{C}_i$, $B_i{C}_i,C_i{B}_{i+1}$, and ${U_i}^{\pm }$ belong to outgoing germ metadata. A zero report in an inactive slot is a coding convention, not an additional geometric cell.

In the first developed sector,

They are orthonormal only at $\alpha =\pi /2$. The Cartesian gradients are

For $0<|g|<1$, define

In an orthonormal spatial basis along and perpendicular to $g$, this is a standard two-dimensional boost plus an unchanged perpendicular coordinate. Consequently,

Thus $\mathcal{B}(g)\in {\mathrm{SO}}^{+}(2,1)$. Here ${\gamma }_h=3$ and ${\gamma }_{\pm }=5/4$.

Put $O_i=\mathrm{diag}(1,\mathrm{Rot}(i\alpha ))$. The affine extensions of the adapted face-coordinate maps are

If a face is $T=q+g\cdot x$, its first boost coordinate is $\gamma (T-g\cdot x)=\gamma q$. Therefore nonresting faces have adapted time $T^{\prime }=0$, whereas resting faces have $T^{\prime }=2+s$.

These are regular ambient affine coordinates away from the singular axis and algebraic extensions to face closures. They do not supply a regular Minkowski neighbourhood of the apex.

Positive traversal means increasing developed angle. For each oriented overlap component, let $n_{gf}=1$ for an upper-to-lower marked seam crossing, $n_{gf}=-1$ for its reverse, and $n_{gf}=0$ for an ordinary crossing. The fixed presentation prescribes $S_{gf}=J(\mu {)}^{n_{gf}}$. Hence

The last chronological transition acts on the left.

For a closed ordered chart word based in $f_0$,

The integer $N$ is retained as a topological mark, not reconstructed from the matrix. In the angular covering of the punctured cone, a loop of winding $N$ has angular lift increment $N(2\pi -\mu )$.

Transport from the base-face coordinates to the developed rest reference frame is conjugation by ${{\mathcal{A}}_{f_0}}^{-1}$. Transport onward to the declared frame is conjugation by $A$.

The transitions (eq:transitions) match every shared developed segment, satisfy (eq:ambient-matching), and have inverse reverse transitions. Face polygons close and regular local vertex words are affine identities.

The loops $\rho =29/10$ and $\rho =15/4$ are primitive positive meridians, lying respectively in the core and resting collar. Their products, transported to the reference frame, are both $J(\mu )$. For $s>0$, the six-face particle star has the same primitive peripheral product.

Proof. An ordinary overlap shares one developed segment. On a seam overlap, $J(\mu )$ identifies the upper segment with the lower segment, including its height by descent. Applying the two face maps gives (eq:transitions) and (eq:ambient-matching). Reversing the overlap gives the inverse map.

Every polygon is realized by the displayed endpoint coordinates. Its oriented endpoint displacements telescope to zero. Shared endpoints coincide, including seam copies. In an ordered word, consecutive chart factors cancel, leaving (eq:word). A sufficiently small loop about a regular vertex has winding zero, so its word is the identity.

The bounds

place the core loop outside all inner regions and inside the cap, and place the collar loop in the resting region. Write ${\mathcal{A}}_i={\mathcal{A}}_{V_i}$. The core's chronological crossings give

Transport gives $J$. The collar charts are identities, so its chronological crossings are $I,I,J$, with product $J$. Both loops traverse all three sectors once and have angular lift increment $3\alpha =2\pi -\mu$, proving primitivity independently of matrix values.

For the outgoing star use $\rho =s$, where $s>0$. Since $a<1$,

so this loop lies strictly in the inner region. In angular order set

Its six-factor word is

It has one positive seam crossing, winding one, and transported product $J$. At $s=0$ the nonvanishing core meridian certifies the common peripheral signature of both germs.

A power $J(\mu {)}^k$ can be the identity at a rational rotation angle. That equality does not erase winding: $k$ remains part of the marked certificate.

After the realized geometry, overlap components, charts, and seam have been fixed, every closed word is determined by its winding through (eq:word). Regular words have winding zero and one primitive meridian determines all peripheral powers. Core/collar agreement must still be checked in the same transported frame.

Proof. Equation (eq:word) gives the first two assertions. The punctured cone is topologically a punctured plane; angular lifting identifies its loop classes with integer winding. The core and collar checks are the two explicit winding-one words of Lemma lem:holonomy.

This reduction is a statement after realization. It does not recover embedding or individual transitions from one holonomy.

For every $\mu \in I$, formulas (eq:coefficients), (eq:rays), and (eq:family) construct a spinless conical slicing family with the following properties:

1. all affine spatial branch gradients have norm at most ${\lambda }_{\ast }<1$, uniformly in $\mu$;

2. graphs descend through rays and the seam, are embedded, and are globally acausal;

3. regular-face induced metrics are at least $1/9$ times the corresponding spatial metrics;

4. the family is strictly ordered by (eq:ordering);

5. exactly one event of the declared cellulation occurs, at $s=0$, changing the particle link from three to six;

6. every regular slice, the common limiting slice, and both fixed-deficit incidence germs have complete certificates (E1)–(E5).

The particle intersection is (eq:particle-time). Primitive core, collar, and outgoing-star maps are $J(\mu )$ in the reference framing and $AJ(\mu )A^{-1}$ in the declared framing.

At $\mu ={\mu }_0$, all displayed data specialize to the reference formulas in Appendix app:reference.

Proof. Lemmas lem:gradients, lem:bounds, and lem:embedding establish descent, uniform gradients, localization, embedding, the metric lower bound, acausality, ordering, and particle times. Lemmas lem:decomposition, lem:lengths, and lem:event establish the realized cellulation, nondegeneracy, and unique event.

Lemma lem:holonomy verifies endpoints, segments, closure, canonical transitions, regular local words, marked ordered words, and primitive core/collar agreement. Lemma lem:spinless verifies spinless packet matching and pointwise worldline fixation. The graph supplies the slice intersection, and the rest vector is the declared transported $\mathbf{t}$. These are all five checks.

A common $A\in {\mathrm{ISO}}^{+}(2,1)$ transports the spacetime, faces, rest splitting, and affine words coherently. Spacelikeness, embedding, and acausality are preserved. The metric comparison is with the transported spatial metric; it is not an assertion about an unrelated fixed Cartesian time splitting after a boost.

At the event, outgoing collapsed slots are germ metadata, while the limiting slice uses the active incoming triangles. The core meridian supplies its noncollapsing peripheral witness. Finally, substituting $\alpha =\pi /2$ proves the appendix identities.

On each incidence epoch and its one-sided germ, the finite coefficient, ray, vertex, gradient, chart, length, and area tables are Lipschitz in $\mu$, uniformly over the allowed $s$. The graphs are jointly continuous on the common topological template.

Proof. The finite table formulas are smooth in $\mu$ on a neighbourhood of $I$. Denominators stay away from zero by Lemma lem:bounds. Each active edge radical has a positive continuous coefficient on the compact interval, so its square root is smooth there. Born lengths are smooth coefficients times $s$ and born areas are smooth coefficients times $s^2$. Bounded derivatives prove the uniform Lipschitz assertions. Joint graph continuity follows directly from continuous minimum and maximum on the glued coordinate template.

This compares labelled data, not identical incoming and outgoing raw cell lists.

Fix $\kappa \in \mathbb{R}$ and define the admitted state domain

with the declared classical restrictions when present. For a fixed interpolation flag $\theta \in [0,1]$, put