Furey's Higgs-Like Doublets Are Nonunique and Dagger-Paired with Leptons
Date: 17 October 2026 Branches: B03 and B11 Claim status: exact block-map enumeration; source grading and statistics assignment remain unencoded
1. Higgs-like candidates
In the convention
the natural endomorphism inventory contains three color-singlet weak doublets with \(Y=+1/2\):
They have the gauge representation of a Higgs doublet, but gauge quantum numbers do not choose among them.
2. Exact dagger pairing
Their blockwise adjoints are
which are precisely the three lepton-like \((\mathbf1,\mathbf2)_{-1/2}\) candidates.
Thus
This is consistent with the earlier theorem that \(\dagger\) pairs mirror orientations rather than selecting chirality.
3. Statistics are not encoded by the gauge inventory
An endomorphism block is not automatically a bosonic field or a fermionic particle. Assigning
- one orientation as a Higgs scalar, and
- the reverse orientation as a lepton fermion
requires a grading, superselection rule, module distinction or dynamical interpretation beyond the ordinary matrix algebra and its \(\dagger\).
Without that extra structure, the inventory supplies representations but not spin-statistics roles.
4. Consequences for the scalar completion
The separate weak doublet required by ordinary Yukawas can be found representation-theoretically inside \(\operatorname{End}(\mathbb V)\), but not uniquely. There are three immediate possibilities:
- one source grading selects a unique Higgs map and one chiral lepton orientation;
- several Higgs doublets survive, producing a multi-doublet scalar sector;
- the physical Higgs lives outside the natural endomorphism inventory.
Each possibility changes the six doublet–lock portals and the flavor interpretation.
5. Branch decision
Stop-everything positive: the missing chiral/statistics projector selects one Higgs-like orientation, three complete lepton generations and compatible Yukawa maps in one construction.
Negative: choosing one Higgs candidate requires an arbitrary deletion among gauge-identical dagger pairs.
Current result: the required Higgs representation occurs three times, but neither uniqueness nor bosonic status follows from gauge symmetry and \(\dagger\).
Unsafe shortcut: pick one \((\mathbf1,\mathbf2)_{1/2}\) block as “the Higgs” while interpreting its reverse block as a fermion without an explicit grading.
6. Certificate
The Higgs-doublet candidate certificate exhausts the color-singlet weak-doublet block maps, identifies all three positive- and negative-hypercharge candidates and verifies their one-to-one dagger reversal.