Proposition: Output completeness and two independent failures

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.