Finite-Time Sampling of Dynamic Spectral Information: Early Blindness, Local Identifiability, and Global Collisions

We investigate whether a finite set of temporal slices of a dynamic spectral feature canpreserve the distinguishing power of the full trajectory. The basis is a two-qubit family ofgeneratorsK(J, h) = JX1X2 + h(Z1 + Z2), J > 0, r = h/J > 0,for which a precise three-branch formula for the maximized commutator magnitude waspreviously obtained in [1]:D(x, r) = max{F0(x), F1(x, r), Fmix(x, r)}.For two subsystems, the spectrum of the weighted Laplacian is fully equivalent to D(x, r),which reduces the identification problem to the analysis of a single scalar function.The paper distinguishes three levels of temporal design properties: early blindness, localidentifiability, and global distinguishability. We prove analytically the existence of earlyblindness with respect to the field parameter and of universal blind lines. We numericallyfind and certify regular collisions for a six-slice design: two distinct models coincide on allslices but differ along the full trajectory.The main result is a rigorous example of a finite-time design in a fully controllable twoqubit quantum model that is locally informative but globally does not distinguish two distinctmodels. We find 20 distinct collisions for the six-slice design; one of them is certified viathe Krawczyk method (existence and uniqueness of the root). The exact minimal number ofdistinguishing temporal slices remains open.