Furey (Cl(6)) Conjugate-Ideal Extension
Date: 16 October 2026 Branch: B03 Furey interface Claim status: color/charge conjugate pair encoded; anti-linear and electroweak structure still open
1. Pairing the ideal with its conjugate
The first minimal ideal contains sectors
Its charge/color conjugate contains
Together they form a sixteen-complex-state color/charge module. The encoded sector table is the conjugate-pair JSON file.
2. Complex-linear gauge commutant
Six charged irreducible sectors occur once. The neutral singlet occurs twice. Therefore
with complex dimension
The neutral multiplicity permits mixing between the two neutral singlets. Color and charge alone do not fix that mixing.
3. Anti-linear conjugation changes the question
The physical conjugation relating the two ideals is anti-linear. Requiring an operator to commute with that real structure does not simply reproduce the complex-linear commutant above; it selects a real form and relates coefficients in conjugate sectors.
The next calculation must specify the exact involution—complex, octonionic and Hermitian conjugations must not be conflated—and then solve
This is also where a possible compatibility with the external Jordan involution can be tested.
4. What this contributes to the interface
The conjugate pair supplies the first nontrivial place where an external real selector could lock to an internal real structure. A successful locking must:
- preserve electric charge assignments;
- map color triplets to the correct conjugates;
- retain the neutral degrees needed for neutrino structure;
- remain compatible with weak chirality after the electroweak layer is added.
5. Pin and unlock conditions
The anti-linear commutant is pinned until the precise Furey involution matrices are encoded.
- Last established point: complex-linear commutant dimension is 10.
- Unlock condition: explicit conjugation matrix/action on the sixteen-state basis.
- Unsafe shortcut: treating the two neutral singlets as freely mixable after imposing a reality condition.
6. Certificate
0 verifies 16 states, conjugate sector pairing, neutral multiplicity two and the dimension-10 complex-linear commutant.