higgsSectorCount
plain-language theorem explainer
The theorem shows that the Higgs field in the recognition vacuum has exactly five sectors. Researchers deriving the electroweak Lagrangian from the J-cost minimum would cite this result to set the sector count. The proof is a direct term evaluation that decides the cardinality from the inductive definition of the sector type.
Claim. The finite type enumerating the neutral, charged-positive, charged-negative, Goldstone-positive, and Goldstone-negative Higgs sectors has cardinality five.
background
In the module on the Higgs field from the recognition vacuum, the electroweak vacuum expectation value arises as the minimum of the J-cost function. The potential takes the form V(φ_H) = J(φ_H/v) = ½(φ_H/v + v/φ_H) - 1, with the minimum at φ_H = v corresponding to 246 GeV. Spontaneous symmetry breaking is induced by boundary conditions that select a nonzero value for v. The Higgs field sector type enumerates five components corresponding to the neutral Higgs, the two charged Higgs bosons, and the two Goldstone modes. This count is identified with the configuration dimension D = 5. This construction relies on the J-cost appearing in the inflaton potential, defined as V(φ_inf) = J(1 + φ_inf), and on the vertex count in spectral emergence given by 2^D for dimension D.
proof idea
The proof is a term proof that applies the decide tactic to establish the cardinality equality. It relies on the automatically generated Fintype instance for the inductive sector type.
why it matters
This result provides the sector count for the Higgs field certificate structure, which collects the vacuum, off-vacuum, and symmetry properties. It fulfills the module's assertion that the five sectors match the configuration dimension D = 5. In the Recognition Science framework it supports the derivation of the Standard Model from the recognition vacuum, aligning with the phi-ladder and the eight-tick octave for the internal structure. It leaves no open questions in the sector enumeration.
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