Proposition: Two rejected updates on the whole interval

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.