We follow the conventions of Paper VII exactly. A fermion is a localised excitation encoded as a normalised complex wavefunction ψ ∈ℂn. The density matrix is ρ = |ψ⟩⟨ψ|, with components ρij = ψiψj∗. We split ρ into two orthogonal parts:
where F is the fermionic diagonal (per-site occupancy, real and non-negative) and B is the off-diagonal matrix encoding inter-site correlations.
An n-site equal superposition is
for arbitrary real phases ϕk. The natural phase patterns on an n-site ring are the discrete Fourier modes
where m is the winding number — the number of full phase rotations around the ring. These are the minimum-Cs states on the ring, ordered by ascending spectral complexity.