Quantum Anomalous Hall Transport in Physical–Observation Dual-Axis Geometry: Mixed Chern Curvature, Fixed-Source Recovery, and a Quantized Observation Cycle

The quantum anomalous Hall effect is ordinarily written on the Brillouin torus of a fixed Hamiltonian. Here the occupied Chern band is held fixed while the state that forms the observation record is allowed to vary independently. Two projectors carry the two roles: an occupied-band projector PX(k) and an observation-response projector R(k, s). On every fixed observation section the construction reduces exactly to the standard theory, σxy = CXe 2 /h. When the observation state contains a periodic coordinate φ, the response bundle can carry a mixed first Chern number ν_iφ on \( S^1_{k_i}\times S^1_{\varphi} \) without changing the physical Chern number. Once the response connection enters the phase-transport law, one closed observation cycle transports Q_i = eν_iφ per equivalent transverse response channel. The integer collapses to ν_iφ = ±1 whenever the mixed response line bundle is topologically primitive; orientation fixes the sign. An elementary nontrivial response cycle therefore carries exactly one charge quantum per channel, while higher integers describe composite topological sectors. A two-level projector realizes the primitive class. The parameter-free relations Q/e = 1, I/(ef) = 1, and eV_H/(hf) = 1 follow, distinct from ordinary Thouless and Brouwer pumping. The experimental question is whether such a quantized, orientation-odd residue survives while the physical QAH Hamiltonian remains fixed.