Boundary 0/π logical subspace and bulk dynamical probes in flux-controlled anomalous Floquet quantum walks

We formulate a one-dimensional flux-controlled anomalous Floquet quantum walk and show that it admits a direct microscopic realization in a driven bipartite lattice. The walk consists of a coin-dependent drift step and a momentum-dependent coin mixing step, so the same evolution operator governs quasienergy bands, boundary modes, and bulk dynamics in real space. Because the walk is chiral, the quasienergy gaps at 0 and π/T carry independent topological information, which organizes trivial, 0-only, π-only, and coexistence sectors in the (M,φ) plane. In the coexistence sector, a 0 mode and a π mode reside on the same edge and span a natural boundary logical subspace. One Floquet period acts there as a relative phase operation and produces a clear 2T response in local boundary observables. In the bulk, the same anomalous Floquet structure is probed dynamically in two complementary ways. Frame-resolved mean chiral displacements approach the two winding numbers in the clean pre-reflection window of the symmetric time frames, while selected benchmark cuts at a representative 0 gap closing and a representative π gap closing exhibit distinct local stroboscopic responses, with the π gap benchmark showing a much stronger odd-even alternation. The boundary logical subspace and the bulk dynamical probes are therefore organized within one flux-controlled anomalous Floquet quantum walk, suggesting a symmetry-protected route to quantum-walk information primitives in driven microstructured lattices.