Theorem: The five-command T0 history

The data above make eq:hybrid a legal five-command history. Every command returns complete accepted ${\mathsf{Ein}}_0$. Its only deficit change occurs at the regular source command $s=-1/2$. The particle-link event occurs at the frozen installed deficit ${\mu }_1$.

Internal matter is unchanged, persistent original locations remain valid under identity abstract transport, and permissions have exactly the displayed trace. The uniform particle branch-gradient bound is $2\sqrt{2}/3<1$.

Proof. The first flow remains in the incoming reference epoch and ends at the regular source boundary. The token satisfies the promise, source stage is zero, and source permission is one. Thus ${\mathsf{S}}_{\mathrm{tab}}$ installs the complete ${\mu }_1$ family at the same parameter, decreases only source permission, and changes source stage to one.

The next flow reaches the incoming germ of the installed family. The incidence guard is exact, incidence permission is one, and ${\mathsf{E}}_{3\text{–}6}$ holds ${\mu }_1$ fixed while selecting the outgoing germ. The final flow lies in the outgoing epoch. Theorem C3 supplies all matching and location conclusions.