Theorem: Certified fixed-packet flow

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.