Proposition: Inversion is an input, not a conclusion of this record

Replace ${V_{\varphi }}^{†}$ on $A$ in (eq:inverse) by $V_{\varphi }$, keeping the same source family, diagonal $\mathrm{CZ}$, relaxation channel, and computational readout. The complete record law on every ${\rho }_{\theta }$ is unchanged.

Accordingly, the records used by the main theorem do not certify the sign of the antiqubit response.

Proof. The two rotations have identical squared matrix elements. The source family is diagonal. The computational populations after the local rotations therefore agree. The diagonal $\mathrm{CZ}$ leaves those populations unchanged, and amplitude damping transforms them by the same population rule. Computational readout consequently gives the same (eq:probabilities).