Python: two_fermion_hop.py
We test whether the universal π∕2 phase breaks for two fermions, and whether the breaking has physical meaning.
Every two-fermion boson matrix contains entries with phase −π∕2, inherited from the individual single-fermion hops. It also acquires new phases 0 and +π∕2, corresponding to inter-fermion correlations with no single-particle analogue. The two-fermion boson is a superposition of single-hop bosons plus an interaction residual.
Fermions crossing (passing through each other) produce an identical boson matrix to fermions moving in the same direction. The codec records only initial and final pixel configurations, not trajectories. Path independence emerges for free.
Two fermions assigned to the same site produce a wavefunction ψ ∝ej (localised on a single site), for which B = 0 exactly. The density matrix of a single-site state is ρik = |α|2 δijδkj, which vanishes off-diagonal for i≠k since both i = j and k = j cannot hold simultaneously. Double occupancy produces no boson: it has zero compression residual and is informationally invisible. Pauli exclusion is not a law imposed on the system; it is the statement that double occupancy has nothing for the codec to encode.