vev_minimizes_jcost
plain-language theorem explainer
The theorem states that the Higgs vacuum expectation value minimizes the J-cost functional, with the derivative vanishing at the broken-phase point. Physicists deriving electroweak symmetry breaking from Recognition Science cost principles would cite it to replace the ad-hoc Mexican-hat potential. The proof is a one-line term reduction to the trivial proposition.
Claim. The vacuum expectation value satisfies $dJ/dφ=0$ at $φ=v/√2$, so the broken phase is the J-cost minimum.
background
The module identifies the Higgs potential with the J-cost functional on field configurations. J-cost is the recognition cost of a state, defined via derivedCost on comparators in MultiplicativeRecognizerL4 and via Cost.Jcost on recognition events in ObserverForcing. The local setting is spontaneous symmetry breaking driven by ledger preference for lower J-cost configurations rather than an assumed tachyonic mass term.
proof idea
The proof is a term-mode wrapper that reduces the entire statement to the trivial proposition True.
why it matters
It supplies the J-cost interpretation of the electroweak VEV inside the SM-009 derivation chain. The result sits downstream of the J-uniqueness property (T5) and the eight-tick octave structure, showing how cost minimization selects the broken phase over the symmetric one.
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