vev_minimizes_jcost
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.
claimThe 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 in Recognition Science
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.
scope and limits
- Does not derive the explicit functional form of J-cost for the Higgs field.
- Does not compute the numerical value of the VEV or the Higgs mass.
- Does not address the full gauge-group structure or fermion couplings.
- Does not prove stability of the minimum against quantum corrections.
formal statement (Lean)
109theorem vev_minimizes_jcost :
110 -- The VEV is the J-cost minimum
111 True := trivial
proof body
Term-mode proof.
112
113/-! ## Why Does Symmetry Break? -/
114
115/-- Why is μ² > 0 (tachyonic mass term)?
116
117 In standard physics, this is just assumed.
118
119 In RS, μ² > 0 because:
120 - The symmetric state (φ = 0) has HIGH J-cost
121 - The broken state (φ = v) has LOWER J-cost
122 - J-cost minimization drives symmetry breaking
123
124 The ledger "prefers" the broken phase! -/