Fixed-preparation short-time reconstruction of class-D BdG endpoint masses

Abstract In one-dimensional class-D superconductors, the relative signs of the Bogoliubov--de Gennes (BdG) masses at particle--hole-invariant momenta fix the bulk $\mathbb Z_2$ invariant. Band energies and transition-frequency peak locations alone cannot determine these signs. We reconstruct each signed endpoint mass from the equal-time responses of two fixed transverse preparations to a calibrated probe. Within this two-preparation class with equal independent Gaussian slope noise, orthogonal axes maximize the worst-case Fisher information over the probe orientation and attain the Cram'er--Rao bound. The endpoint product gives the class-D invariant but only the parity of a general BDI winding number. In the four-band Rashba model, a full-matrix identity gives positive finite-packet attenuation for sinusoidal spin--orbit coupling and a wrapped-Gaussian momentum distribution. A conditional reference-dot construction maps the two-level algebra to charge readout. Full finite-ring calculations quantify single-pulse charge deviations and verify that preparation-phase cycling is equivalent to analysis-phase cycling for a normal reference dot. The results establish a symmetry-constrained reconstruction and its scope, not a sensitivity advantage over phase-sensitive Ramsey readout or a direct electric-current measurement.