(gqr54) Floquet-Organized Singular-Value Ridges and the Geometry of Schr¨odinger–Heisenberg Divisibility Fracture in Qubit Dynamics

Recent work has shown that divisibility of quantum dynamical maps can differ betweenSchr¨odinger and Heisenberg pictures. In this work we investigate the geometric structure ofthis fracture in a minimal qubit model combining anisotropic contraction with time-dependentrotation. By analysing the singular values of the corresponding Bloch-space propagators weintroduce a singular-value gap field ∆(s, t) = σmax(MS(t, s)) − σmax(MH(t, s)) that quantifiesthe ordering asymmetry between the two intermediate maps. Numerical exploration revealsthat the fracture region in the (s, t) plane is organized by a system of ridge curves forminga comb-like structure. When expressed in terms of the phase difference ∆θ = θ(t) − θ(s),these ridges collapse onto an approximately periodic ladder condition reminiscent of Floquetresonances. The resulting geometry shows that the Schr¨odinger–Heisenberg divisibility fracturecan be interpreted as a phase-organized instability governed by a Floquet-like ladder of singular-value ridges modulated by anisotropic damping. Full 70 article programme https://github.com/jamessutton600613-png/GQR/blob/main/Programme_Index.md The programme contains both standalone and self referencing articles. These can all be read as notes in proof of GQR1.