We have proved that the density matrix of a fermion in equal superposition across n sites splits orthogonally into a fermionic diagonal F and an off-diagonal residual B satisfying
The conservation law is exact for any pure state and subsumes the Paper VII result as a special case.
The central reframing of this paper is that bosons are not fundamental particles. They are the codec’s internal record of fermion transitions. Every physical observation is a fermion observation. The Standard Model boson table is a derived description of fermion correlation patterns, not an independent particle ontology.
Under this reframing, the particle classification is a fermion classification. The pair (n,m) describes the fermion’s complexity class. The n=2 sector gives charged leptons and neutrinos, distinguished by winding number. The n=3 sector gives confined colour-charged fermions (quarks) and a scalar sector. The classification does not yet distinguish the three charged lepton generations, which all occupy the single class (2,1); identifying the missing generation index is left as an open problem rather than asserted as a derived mass ladder.
The open problems reduce to three purely fermionic questions: what distinguishes the lepton generations, why the quark masses have the values they have, and whether the 1∕r2 electromagnetic force law follows from the fermion residual structure by the same argument that gives gravity in Paper VII.