Landmark VIII-C Certified Fixed-Packet Flow, Guarded Events, and Finite Reversible Composition

We give a conditional composition theorem for polygonal descriptions of a declared spinless conical spacetime. At a fixed particle packet, explicit realizations produce correlated affine edge-length laws. Incoming slice advance preserves the finite lengths, while outgoing advance changes genuine crease lengths. A separately installed collar module changes subdivision lengths without changing the embedded slice.

Exact parameter clocks, profile-specific guards, separate incidence and source permissions, and complete output obligations define legal schedules. Relative to explicitly stated realization and transport interfaces, every legal schedule returns an accepted five-component affine matching certificate after every command. Geometric commands preserve the installed packet and reference-frame signature; regular source-table replacement installs a complete new realization.

An explicit five-command history performs source replacement at $s=-1/2$, reaches the incoming particle germ, changes the particle link at a frozen deficit, and continues along the outgoing family. Finitely encoded words have reversible implementations on their composite canonical forward domains, with retained input histories, identity action on internal matter, and explicit resource bounds for this history. Independently, prescribed block channels obey an exact backwards-support inclusion, including untouched references.

The source token is supplied under a semantic promise, not acquired or tested. Local obstruction and separation arguments prevent incomplete packet updates, expectation extraction, record–residual identification, schedule–causality identification, and confusion of code decoding with quantum recovery. No mixed incidence–source germ, physical embedding of the schedule, polygon Hamiltonian, spacetime emergence, or holographic identification is constructed.

Landmark VIII-A defines a finite affine matching interface for a declared spinless conical geometry CheungVIIIA. Landmark VIII-B constructs a variable-deficit realization family and a finite promised source table CheungVIIIB. The present paper supplies a scheduling layer: fixed-packet movement, guarded incidence, regular source replacement, and finite reversible composition.

The polygon literature studies evolving descriptions of $2+1$-dimensional gravity Hooft1992,Hooft1993,Kadar2005. The construction here is narrower. We prescribe realizations and derive their length laws; we do not identify those laws with a polygon Hamiltonian or import a general transition catalogue. The spinless conical interpretation is a declared geometric dictionary, motivated by the locally flat $2+1$-dimensional setting DJH1984, not a consequence of the scheduling theorem.

.

1. Explicit fixed-packet realizations yield correlated affine length laws, including nonzero outgoing crease rates and a coplanar length-changing collar refinement.

2. A guarded grammar composes regular source replacement and fixed-packet incidence while returning complete matching outputs.

3. One sourced five-command word has exact packet data, a permission trace, canonical report checks, and finite reversible resource bounds.

4. Local proofs state the observation, support, and decoding boundaries needed to interpret this composition correctly.

The contribution is a certified composition framework, not a new particle geometry, source law, measurement primitive, or gravitational dynamics.

The required geometric realization and location-transport interfaces are stated in sec:families,sec:commands. They are hypotheses of the conditional scheduling theorem. A citation does not replace an interface statement or independently validate its implementation.

By contrast, the completeness, token, record, support, recovery, and non-identification statements added here are proved locally. They are not imported as additional premises from companion manuscripts. Their citations identify corresponding results and comparisons in Cheung's series.

The two substrate profiles are alternatives. Their moving cores are not glued together, and no profile-switching command is supplied. A collar-refined representation must already be installed before collar flow acts. Source and incidence remain separate command kinds.

The dependency and source-locator tables in app:dependencies distinguish inherited hypotheses from locally proved statements.

Use developed coordinates

Affine isometries act and compose by

and

Define

The reference packet is

Here $p$ is a geometric particle identifier.

For a declared affine frame $A=(L_A,b_A)$, the coherently transported packet has

In particular,

The frame, worldline, rest vector, deficit, and holonomy are transported together.

For a realized description $g$, write

An accepted output supplies the finite tables, exact equalities, and topological evidence for all of the following obligations:

1. {Vertex and overlap matching $\mathcal{V}$.} Every proposed transition equals its prescribed affine map on the declared overlap component. Reverse maps are inverses; paired endpoints and shared segments match; face boundaries close; regular small vertex words are affine identities.

2. {Particle matching $\mathcal{P}$.} The declared deficit, oriented rest frame, affine worldline, and peripheral map agree with the installed packet.

3. {Ordered holonomy evidence $\mathcal{H}$.} The ordered words, overlap-component indices, and winding classes are supplied. A positive primitive particle meridian has winding one, not merely a nonidentity matrix.

4. {Core–collar matching $\mathcal{K}$.} The core and collar share their prescribed boundary realization and transported framing. Their primitive words give the same installed peripheral map.

5. {Worldline data $\mathcal{W}$.} All charts describe the same singular worldline. Its holonomy fixes that worldline pointwise, rest vectors are transported consistently, and the declared slice intersection lies on it.

Embedding, spacelikeness, and nondegeneracy of active cells are separate realization requirements. A germ is accepted as a specified one-sided limiting object, not as a regular cellulation with extra active zero-size cells.

This is the finite matching interface of VIII-A CheungVIIIA (Definition A:def:Ein), restated here. The notation ${\mathsf{Ein}}_0$ does not mean that a Hamiltonian constraint algebra or a general Einstein initial-data theorem has been proved.

For fixed face charts ${\mathcal{A}}_f$, seam map $J(\mu )$, and overlap-component exponent $n_{gf}$, the prescribed transition is

Elementary overlap components have $n_{gf}\in \{-1,0,1\}$; longer paths retain their individual component data.

The following statements hold for every command output, at every installed $\mu$, in a fixed declared presentation.

1. With realization, charts, overlap components, reference framing, and prescribed seam maps fixed, two different proposed values of a required transition cannot both satisfy (E1).

2. If ${\mu }_{\mathrm{old}}\ne {\mu }_{\mathrm{new}}$ belong to the interval used below, retaining $J({\mu }_{\mathrm{old}})$ at the new deficit violates particle/seam matching, although it still fixes ${\mathcal{W}}_0$ pointwise.

3. For $\delta \ne 0$, the affine map

   $$(R(\mu ),\delta e_0)$$

   preserves ${\mathcal{W}}_0$ as a set and preserves the rest vector in its linear part, but does not fix ${\mathcal{W}}_0$ pointwise. It therefore fails (E5).

All three statements remain true after coherent conjugation by any declared affine frame $A$.

Proof. For the first assertion, (E1) requires equality with the single map in eq:canonical. Two accepted proposed maps must therefore equal one another.

The interval below is contained in $(0,2\pi )$, so distinct deficits in that interval give distinct oriented spatial rotations. Thus $J({\mu }_{\mathrm{old}})\ne J({\mu }_{\mathrm{new}})$. Every $J({\mu }_{\mathrm{old}})$ nevertheless sends $t{e}_0$ to $t{e}_0$. The failure is deficit/seam matching, not pointwise worldline fixation.

For the third assertion,

Conjugation is injective. Moreover, the conjugated time-shifted map sends $A(t{e}_0)$ to $A((t+\delta )e_0)$. Hence the same distinctions hold in every coherently transported frame.

These are local versions of VIII-A's fixed-presentation completeness and rejected updates CheungVIIIA (Theorems A:thm:A1, A:thm:A4-stale, and A:thm:A4-time). They do not assert that holonomy alone determines all chart transitions. In particular, matching a few endpoints or a peripheral product is not a substitute for eq:canonical.

A chart-based primitive product is generally conjugate to $J(\mu )$. After transport back to the declared reference frame it is exactly $J(\mu )$. We use the latter convention in the persistent signature

Its preservation does not require literal equality of every chart-based matrix.

Three different statements are kept separate:

Fixed-packet slice advance preserves the first two, not generally the third.

A finite named matter interface $\Gamma$ carries one joint state

Names specify access to declared factors. Classical registers are jointly block diagonal in their distinguished bases, including their correlations with other factors.

A surviving source boundary is an explicitly embedded unital ${}^{\ast }$-subalgebra

Its accessible state functional is

For a fixed Hermitian $B_{\mathrm{src}}\in {\mathcal{A}}_{\mathrm{src}}$ and fixed $\kappa$, a table token $m_{\mathrm{src}}$ can carry the semantic promise

The example below uses $B_{\mathrm{src}}=Z$ and $\kappa =1$.

T0. Every sourced history begins with an already supplied classical token. At each sourced command the token must satisfy its declared promise for the current surviving source state. No token-acquisition operation or operational promise test occurs in the history.

“Current token” means correct for the current source functional. It does not mean that the selected packet is already installed before source replacement.

For a fixed token, the promised state class may be convex. The union of classes for different deterministic expectation labels need not be convex. Nothing here implements a state-only deterministic source selection on the full quantum state space.

1. Let $\mathcal{S}$ be a convex state set. If an affine map assigns to every $\rho \in \mathcal{S}$ a classical distribution concentrated at one label,

   $$T(\rho )={\delta }_{\gamma (\rho )},$$

   then $\gamma$ is constant on $\mathcal{S}$. The statement also holds for a classical outcome space whose distinct labels can be separated by a measurable event.

2. A circuit acting only on code registers and as the identity on matter cannot, for a fixed independent code input, make its code output depend on an unencoded matter expectation.

Requiring that a putative label-producing channel also preserve the matter input does not remove the first obstruction.

Proof. If two states have distinct labels, affinity gives, for $0<t<1$,

This is not concentrated at one label. In a general outcome space, an event separating the labels has probability strictly between zero and one. Either conclusion contradicts the assumed deterministic output on the mixture.

For the second assertion, a code-only circuit $U_{\mathrm{c}}$ acts as $U_{\mathrm{c}}\otimes I_{\Gamma }$. On an independent input $c\otimes \rho$, its code marginal depends on $c$, not on $\rho$. Different unencoded expectations therefore cannot select different code outputs. An additional matter-preservation requirement cannot invalidate either argument.

The two arguments correspond to VIII-B's expectation-label obstruction and VII-G's affine obstruction CheungVIIIB (Proposition prop:obstruction), CheungVIIG (Proposition prop:affine). They are proved here rather than assumed. They do not prohibit deterministic processing of an already supplied classical token, stochastic instruments, or separately controlled approximations.

For destructive computational readout of $S$, with surviving complement $B$, a specified output is

Its codomain contains $R$ and $B$, not the consumed factor $S$. A pullback of an output observable is an input observable; it does not restore $S$ as a surviving output name CheungI,CheungVIIIA.

1. For channels $\Phi ,\Psi$ with common input and output spaces and a fixed output Hermitian $B$,

   $$\mathrm{Tr}(B\Phi (\rho ))=\mathrm{Tr}(B\Psi (\rho ))\text{for every input state }\rho$$

   if and only if

   $${\Phi }^{\ast }(B)={\Psi }^{\ast }(B).$$

   On a restricted state class, only the corresponding annihilator condition follows; operator equality need not follow.

2. Identical classical record channels do not imply identical channels on the record–residual output algebra.

Proof. Trace duality turns the first equality into

A Hermitian operator with zero expectation in every state is zero. Restricting the tested states restricts this conclusion to their real linear span.

For the second assertion, let ${\Delta }_Z(\rho )=P_0\rho P_0+P_1\rho P_1$ on a qubit and define

Their record marginals are the same constant record. For $\rho =|+⟩⟨+|$, their residual difference is $X/2$, so

The full residual algebra distinguishes them.

These statements sharpen the source boundary in the notation used here, in agreement with VII-E's accessible-observable and residual-obstruction results CheungVIIE (Proposition prop:accessible and Theorem thm:residual-obstruction). A produced record can become a source only when its surviving register and source algebra are explicitly named. Record agreement alone does not justify using an unnamed residual observable as a source.

No readout or residual-changing instrument occurs in the T0 word constructed below.

Equal geometry, controller data, permissions, declared geometric resources, source embedding, and explicitly accessible inputs select the same command, applicability result, geometric support, complete output certificate, original-name transport, and successor controller state.

Geometric commands do not read internal matter, occupancy, source tokens, or spectator number. A source-table command reads its declared classical token. Spectator locations and implementation histories have separately reserved storage and do not affect geometric capacity guards.

This is a declared controller interface, not a general no-signalling theorem for nonlinear feedback. The continuum source law of VIII-B remains its nonlinear Option N. Its promised table restriction is not an expectation extractor or an implementation of that continuum law on arbitrary unknown states.

A scheduled configuration is

where:

1. $g$ is a certified regular geometry or certified one-sided germ;

2. $\Pi$ is its installed packet, and $m$ is a table index, or $\ast$ in a semantic continuum description;

3. $\mathsf{profile}\in \{\mathsf{P},\mathsf{A}\}$ fixes the substrate;

4. $s,\sigma$ are the particle-family and four-face parameters; an unused parameter is inactive metadata;

5. $\tau$ is an execution/flow clock;

6. $h$ includes incidence mode, source stage, schedule stage, module, and finite controller data;

7. ${\nu }_G,{\nu }_{\mathrm{src}}$ are separate nonnegative integer permissions;

8. $m_{\mathrm{src}}$ is an optional classical source token;

9. $\ell :\Gamma \to \mathrm{Cell}(g)$ specifies abstract locations;

10. $\mathcal{R}$ reserves geometric resources separately from spectator, location, and implementation-history storage.

For profile $\mathsf{P}$,

Collar mode has an active parameter $t_{\mathrm{col}}$, holds $s$ fixed and regular, and uses its already installed refined dictionary. Profile $\mathsf{A}$ uses its own slice module and dictionary.

No command changes profile, silently refines the dictionary, or uses implementation history as a source observable.

Finite codes use exact dyadic parameters

Recorded execution slots satisfy

A geometric segment can be parametrized by

or the analogous law for $\sigma$ or $t_{\mathrm{col}}$. Only designated endpoint codes need lie on the finite grid.

Incoming and outgoing germs at zero have the same exact parameter but different incidence modes. We write $0^{-}$ and $0^{+}$ for those modes, not for rounded real numbers.

A discrete incidence or source command may consume an execution slot while leaving its slicing parameter fixed. No continuous interpolation of that discrete command is asserted. The execution clock is not identified with proper time or a relational lapse.

Each command returns the complete output of def:matching, canonical finite reports when applicable, and certified original-name transport.

1. $\mathsf{G}$: fixed-packet flow within one declared module and incidence epoch. Slice advance has nonnegative parameter speed. A particle slice segment may end at the incoming germ or begin at the outgoing germ, but may not cross zero internally. Collar flow requires profile $\mathsf{P}$, a regular fixed $s$, and the prescribed collar window.

2. ${\mathsf{E}}_{3\text{–}6}$: profile $\mathsf{P}$, unrefined slice module, incoming mode, exact $s=0$, and ${\nu }_G\ge 1$. It selects the outgoing germ at the same packet, decreases ${\nu }_G$ by one, and recomputes the incidence and matching tables.

3. ${\mathsf{E}}_A$: profile $\mathsf{A}$, incoming mode, exact $\sigma =0$, and ${\nu }_G\ge 1$. It performs the certified four-face exchange away from the particle, decreases ${\nu }_G$, and preserves the ambient packet.

4. ${\mathsf{S}}_{\mathrm{tab}}$: profile $\mathsf{P}$, unrefined slice module, source stage zero, exact regular parameter $s=-1/2$, a supplied table token satisfying its semantic promise, and ${\nu }_{\mathrm{src}}\ge 1$. It installs the complete realization at the token's packet and the same $s$, decreases ${\nu }_{\mathrm{src}}$, and sets source stage to one.

No command combines source replacement and incidence. Every admitted command acts as the identity on internal matter in a T0 history. Acceptance is conjunctive: every realization requirement and every obligation (E1)–(E5) is required.

The contract specifies admitted domains and outputs. Its geometric existence and transport content is supplied by the explicit interfaces in the next section. Source replacement is evaluation of the complete particle constructor at the selected table entry, not a deficit-label write with stale charts.

Set

The subscript distinguishes $b_{\mu }$ from report precision $b$.

In the cut development put

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

The last ray is identified with the first by $J(\mu )$, since $3\alpha =2\pi -\mu$. Cell indices are cyclic modulo three after seam identification; the developed rays $e_0,\dots ,e_3$ are kept distinct before that identification.

Define

The ambient regular metric is the ultrastatic cone metric $-d{T}^2+d{s}_{C_{\mu }}^2$; the graph is $T={F_s}^{(\mu )}$.

Use

The particle intersection and outer vertices are

with $T(A_i)=T(D_i)=2+s$. Spatial vertex notation is supplemented by its displayed time coordinate.

For $s>0$,

Incoming core faces are

Outgoing core faces are

The collar-side faces

and exterior are horizontal at $T=2+s$. The annular collar $7/2\le {\rho }_c\le 4$ has resting plane orientation. Its height varies with $s$.

For every $\mu \in I$,

For $0<s\le 1$, the outgoing star is strictly inside the cap:

Proof. Write $\alpha /2=\pi /4+\epsilon$, where $|\epsilon |\le \pi /120<11/420$. Then

The inequalities $\cos \epsilon \ge 1-{\epsilon }^2/2$ and $|\sin \epsilon |\le |\epsilon |$ give

The left bound exceeds $129/200$, and the right bound is below $137/200$. Since $0<\alpha <\pi$, $b_{\mu }>0$, and $b_{\mu }\le 3/5<129/200<a$. Consequently,

The remaining claims follow from eq:scalars.

For every $\mu \in I$ and $s\in [-1,1]$, the particle constructor supplies the realization above, with:

1. embedded spacelike faces and exterior, with physical branch gradient norms $2\sqrt{2}/3$, $3/5$, and $0$;

2. positive active cells on each regular epoch;

3. exactly one incidence value, $s=0$, and the specified incoming and outgoing germs;

4. the affine charts and prescribed overlap components below;

5. complete accepted (E1)–(E5), including identity regular words and positive primitive core and collar words;

6. identity abstract transport through ${\mathsf{E}}_{3\text{–}6}$ of the persistent labels specified in sec:locations.

This is the realization interface corresponding to VIII-B's persistence theorem, specializing at ${\mu }_0$ to VII-F CheungVIIIB (Theorem thm:persistence), CheungVIIF. The interface is an explicit hypothesis; no independent validation of all predecessor certificate tables is asserted here.

The displayed graph also explains the absence of an additional branch event. For $s\le 0$, $h_{\mathrm{lin}}\ge s+k_{\mu }$. For $s>0$, the new triangles are bounded by the vertices $B_i,C_i$; lem:bounds separates them from the cap.

For reference, the induced areas of compact faces are

\operatorname{Area}(V_i)&=
 \begin{cases}
 S_\alpha r^2/6,&s\le0,\\
 \dfrac{S_\alpha}{3}(r^2/2-BCs^2),&s>0,
 \end{cases}\nonumber\\
 \operatorname{Area}(U_i^\pm)&=\frac25S_\alpha BCs^2\quad(s>0),&
 \operatorname{Area}(Q_i)&=\frac12S_\alpha(16-r^2).
 \label{eq:areas}

These follow by multiplying projected areas by the constant spacelike area factors $1/3$, $4/5$, and $1$, respectively. The outgoing star lies strictly inside the incoming core triangle, so the displayed active areas are positive.

Put

The unrefined finite dictionary has three slots of each of eight types:

The persistent slot $e_i$ realizes $[P,A_i]$ before the event and $[B_i,A_i]$ after it.

Inactive slots are not incoming edges. At a germ, born-slot lengths have zero outgoing limit but the slots are metadata, not additional active zero-size cells. Their zero reports are coding conventions, not intrinsic incoming derivatives.

For a spatial gradient $g$, $|g|<1$, let

with $\mathcal{B}(0)=I_3$. In the first sector,

Their norms are $2\sqrt{2}/3,3/5,3/5$, and their Lorentz factors are $3,5/4,5/4$.

With $O_i=\mathrm{diag}(1,\mathrm{Rot}(i\alpha ))$, the adapted charts are

The first three rows flatten their plane intercepts to zero. The identity charts on horizontal faces retain the constant face height $2+s$; adaptation does not require the same intercept in every chart.

Canonical overlaps are eq:canonical. For a closed ordered word based in $f_0$, telescoping gives

The integer $N(\gamma )$ is supplied by the winding evidence, not inferred from the matrix alone. For example, at ${\mu }_0=\pi /2$, different windings can have the same rotational matrix.

Within an epoch, linear chart parts remain fixed, while translations and plane intercepts change. Thus orientations remain fixed, not literal spacetime locations.

Profile $\mathsf{A}$ uses the four-face patch of VII-E at the ambient packet ${\Pi }_{{\mu }_0}$ CheungVIIE. In local coordinates $(x,y)$, set

and

The rectangle is

placed by $(X,Y)=(20+x,20+y)$, away from the particle and seam. The central lengths are

Each of four mixed arms has length

The inner cap rectangle has half-widths

For a fixed corner-spoke subdivision of its horizontal annulus, each spoke to the corresponding outer corner has length

The cap sides have lengths $2X_{\sigma },2Y_{\sigma }$, and outer rectangle sides have lengths $28/3,12$. These supply the finite patch length dictionary. Any additional ambient dictionary is part of the following interface and is not silently identified with the particle dictionary.

For $\sigma \in [-1,1]$, the installed profile-$\mathsf{A}$ constructor supplies:

1. an embedded spacelike capped graph, with slope bound $3/5$, the displayed patch dictionary, and a complete declared ambient finite dictionary with affine within-epoch length laws;

2. exactly one four-face adjacency exchange, at $\sigma =0$, with positive noncentral cells;

3. adapted affine charts, identity contractible patch words, and complete accepted ambient (E1)–(E5);

4. a certified boundary interface with its resting exterior, with particle and seam outside the moving incidence patch;

5. fixed ambient packet ${\Pi }_{{\mu }_0}$, fixed exterior particle incidence, and certified original-name transport through the exchange.

The transport may identify junction labels. A finite implementation must retain enough input information to reverse that identification. No source command is admitted in this profile.

The interface includes the required ambient matching, not merely identity words inside the contractible patch. A resting exterior orientation does not imply that every exterior slice point is stationary.

For local flow time $t=\tau -{\tau }_a$, write

Chosen units make the rate parameter dimensionless. It is not identified with a Lorentz rapidity between face frames.

For particle slice advance $s=s_a+v_{s}t$, $v_s\ge 0$, incoming active finite lengths have zero rates. In the outgoing epoch,

and $c_i,t_i,o_i$ are frozen.

For example, on $e_i=[B_i,A_i]$,

Hence

and its rate is the first row of eq:particle-rates. On $u_i=[P,B_i]$, both endpoints have time $2s$, so $\ell (u_i)=Bs$. The shrinking and growing rates are correlated by the same vertices and parameter; they are not independently assigned.

For $\sigma ={\sigma }_a+v_{A}t$, $v_A\ge 0$, the four-face central rates are

on their respective epochs. The arm rates are

Cap-side rates are $2v_A/3,2v_A/5$, corner-spoke rates are $-v_A\sqrt{34}/15$, and outer rectangle sides are frozen. Additional ambient rates are the declared laws of contract:A.

Fix $\mu \in I$ and a regular $s=s_a$. In an already refined collar representation, let

Replace each $Q_i$, in this installed representation, by

For $t_{\mathrm{col}}=t_{\mathrm{col},a}+v_{\mathrm{col}}(\tau -{\tau }_a)$,

with multiplicity three per row. All other edges are frozen.

The collar module is embedded and nondegenerate on its full window, has no incidence event, and preserves the installed packet and reference-frame signature. Its embedded slice and particle intersection are unchanged.

Proof. By lem:bounds,

Thus $R_{\mathrm{col}}-r>1/8$ and $4-R_{\mathrm{col}}\ge 1/8$. Ring lengths are positive because $0<\alpha <\pi$. The horizontal areas are

both positive. The two refined faces partition exactly the original horizontal face, so their union has not moved.

All new faces have the identity reference chart. Ordinary transitions are identities and seam transitions remain $J(\mu )$. Shared segments, regular words, and primitive core–collar matching are therefore unchanged. Since $s_a$ is fixed, so is $P_{s_a}$. These checks give the complete matching output.

Adjacent collar faces have zero relative boost, although their subdivision lengths may have nonzero rates. This separates ${{\beta }_e}^{\mathrm{rate}}$ from a canonical boost variable.

Relative to contract:particle,contract:A, the following are spectator-blind geometric commands:

1. particle-family slice advance at fixed $\mu \in I$, within one incidence epoch;

2. four-face advance within one epoch of its installed window;

3. collar advance at fixed regular $s$, within its installed $t_{\mathrm{col}}$-window.

Their represented lengths have the derived affine laws eq:flow-law,eq:particle-rates,eq:collar-lengths and the declared four-face laws.

On every compact regular subinterval, active finite lengths and compact face areas have positive minima, and no additional collision occurs. Particle slice advance may end at the incoming germ or begin at the outgoing germ. Collapsed born slots there remain unoccupiable metadata.

Every output returns complete accepted ${\mathsf{Ein}}_0$, with the same installed packet and reference-frame signature. No flow reads internal matter, occupancy, a source token, or spectator number.

Proof. Substitution of the affine parameter law into eq:lengths gives the particle rates. The preceding endpoint calculation derives representative shrinking and growing laws directly. The four-face and collar laws follow by the same substitution.

Positive continuous lengths and areas on a regular epoch have positive minima on compact subintervals. The cap-separation estimate of lem:bounds excludes contact of the particle star with the cap. The four-face interface excludes noncentral collisions, and lem:collar proves the collar assertion.

For particle flow, chart translations update by eq:charts; canonical overlaps telescope to eq:word. Regular words have winding zero, and positive primitive core and collar words have winding one. Transport to the reference frame therefore yields $J(\mu )$. The inherited interface supplies the remaining realization and worldline checks. The ambient worldline stays fixed even though its slice intersection changes.

The four-face interface supplies complete ambient matching, not only patch identities. The collar matching was checked directly. In every case, the rule depends only on declared geometric data, parameters, resources, and transport. The inaccessible matter and spectator data are not read.

Theorem C1 is kinematics within certified realizations. It does not derive a Hamiltonian, prescribe a relational lapse, or impose equal proper-time increments in different face frames.

Within the admitted windows:

1. the unrefined particle slice module has only ${\mathsf{E}}_{3\text{–}6}$, at $s=0$, with the installed packet frozen;

2. the four-face profile has only ${\mathsf{E}}_A$, at $\sigma =0$, with the particle outside the moving incidence patch and its packet frozen;

3. regular within-epoch flow introduces no additional incidence;

4. the collar module has no incidence event on its full window.

This is isolation in the prescribed families, not genericity among arbitrary slicings.

Proof. The particle constructor supplies the two decompositions and their germs. Born vertices are proportional to $s$, their nonzero length coefficients are positive by spacelikeness, and the star remains strictly inside the cap. The supplied incidence value is therefore the sole coalescence in this window.

In profile $\mathsf{A}$, each central length in eq:A-central has its sole zero at $\sigma =0$. Noncentral cells remain positive by the four-face interface. Within-epoch flow preserves these incidence diagrams. The collar conclusion is lem:collar.

A particle hop, two-particle collision, or another classical transition does not enter this catalogue merely by being named. It requires its own realization and full output interface.

The particle persistent labels are

and the declared exterior labels. Their abstract transports through ${\mathsf{E}}_{3\text{–}6}$ are identities.

The identity label $e_i$ changes realization from $[P,A_i]$ to $[B_i,A_i]$. Identity abstract transport is not a stationary developed trajectory or unchanged incidence relation.

A legal schedule is a finite chronological word in

with declared parameters, modules, and supports, satisfying:

1. every input lies in its command's certified domain;

2. flow preserves the packet and module and does not cross an incidence value internally;

3. source replacement satisfies its regular boundary, stage, token-promise, module, and source-permission conditions;

4. incidence satisfies its exact profile, germ-mode, clock, and incidence-permission conditions;

5. no command combines source replacement and incidence;

6. all commands after source replacement use the newly installed packet, charts, collar, realization, and report dictionary;

7. original names occupy valid input locations and receive their certified transport;

8. no unsupported profile or module conversion occurs;

9. every output satisfies all realization requirements and (E1)–(E5), including prop:complete-output.

Canonical code predicates concern finite indices, exact clocks, reports, permissions, locations, and stages. The token promise is instead a hypothesis about the matter state, not a circuit-tested predicate.

Every legal schedule returns complete accepted ${\mathsf{Ein}}_0$ and valid original-name locations after every command. Geometric commands preserve the currently installed packet and reference-frame signature. A source command returns the complete newly installed packet and signature.

Under ass:control, two aligned legal executions with equal initial geometric/controller inputs, resources, permissions, declared accessible inputs, and original matter reductions have equal geometric outputs, controller outputs, and original-name transports. Their original matter reductions remain equal. Inaccessible spectator extensions do not change geometric decisions.

At every command boundary, the incomplete outputs of prop:complete-output are rejected. This conclusion is not restricted to the initial reference packet.

Proof. Proceed by induction from the certified initial configuration. A flow output is complete by Theorem C1. An incidence output is complete by the appropriate realization/germ interface and Theorem C2, and uses the certified transport in sec:locations.

At ${\mathsf{S}}_{\mathrm{tab}}$, the regular parameter remains $s=-1/2$. Evaluation of the particle constructor at the promised table entry returns the full new realization, including new rays, seam, charts, collar, and matching data. Persistent incoming labels remain valid. Only the source permission and source stage change in addition to the installed geometric data. Legality places this complete output in the next command's domain. This proves certificate and location induction.

For paired executions, deterministic control selects equal commands, guards, geometric outputs, transports, and successor controller states from equal declared inputs. Every T0 command is the identity on internal matter. For a spectator extension $\tilde{\rho }$ with original reduction $\rho$,

for each accessible original observable $B$. Equal original reductions thus give equal accessible functionals. The classical tokens are separately required to agree and satisfy their promises; no extraction step is used. Separately reserved spectator storage prevents a spectator-number capacity guard.

Finally, acceptance includes the complete obligations after each command. The local calculations in prop:complete-output apply at the packet installed at that boundary, including after a source replacement.

A hybrid word need not preserve its initial signature from start to finish. Its geometric subwords preserve whichever packet is currently installed; a source command can change that packet.

For $q\ge 6$, define

This is the promised table of VIII-B CheungVIIIB (Theorem thm:finite). Every entry lies in $I$, since

At $q=6$, ${\mathcal{T}}_6=\{-1,0,1\}$.

For ${\alpha }_m=(2\pi -{\mu }_m)/3$,

The rays and seam matrices have rational entries; the finite coordinates, charts, and lengths are algebraic. The real number ${\mu }_m$ itself is specified symbolically, not approximated by a floating-point angle. Exact derivations appear in app:checks.

Use a surviving source qubit with

For the $q=6$ history, supply token $m_{\mathrm{src}}=1$, and take

Because $0<z_{+}<6/64<1$, ${\rho }_{+}$ is a state and

The same word is defined on every state satisfying this fixed-token promise, including compatible spectator extensions.

Initially install ${\Pi }_{{\mu }_0}$, independently of the source state, and set

with incoming mode and source stage zero. All original names occupy valid persistent incoming labels. Choose $d\ge 1$, $\Delta \tau =1$, and slots $k=0,\dots ,5$. Each flow segment has $v_s=1/2$.

The chronological word is

After source replacement,

The ray generator is

Every subsequent face, chart, overlap, and report uses these installed coefficients rather than the reference table.

At $s=1/2$,

and