The norms ∥F∥ and ∥B∥ depend only on n — the number of sites across which the fermion is delocalised.
The internal structure of the compression residual is determined by the winding number m. Together,
(n,m) defines a fermion complexity class.
The symmetry of the off-diagonal B under transposition determines whether the fermion can propagate
freely or is confined:
- B = −B⊤ (antisymmetric, m odd for n=2): the fermion’s residual has self-contained
antisymmetric structure and can propagate as a free asymptotic state.
- B = B⊤ (symmetric, m=0): the residual has self-contained symmetric structure and
propagates freely with scalar character.
- Mixed symmetry (n ≥ 3, m ≥ 1): the residual has no self-contained symmetry class. The
fermion requires reference to its source configuration to be described — it cannot propagate as
a free asymptotic state. This is the information-theoretic basis of confinement (see Section 8).