Determination of the transient information contribution

Determine the saturation value $\mathcal T(0)$ of the replica-connected transient contribution to the record fidelity in the SU(2)-symmetric monitored circuit, particularly its behavior across the spin-fuzzy and spin-sharp phases.

Background

The record fidelity separates into a contribution from the leading transfer-matrix eigenvalue and a transient term associated with the diffusive background sector. The theory argues that replica pairing suppresses this transient contribution in the fuzzy phase, whereas it should be generically of order one in the sharp phase. However, the large-loop-fugacity expansion does not calculate the saturation amplitude $\mathcal T(0)$, leaving the quantitative distinction unresolved.