jcostHiggs
plain-language theorem explainer
In Recognition Science the Higgs potential is identified with a J-cost functional whose argument is the quartic expression -mu squared times phi squared plus lambda times phi to the fourth. A physicist deriving electroweak symmetry breaking from recognition costs would cite this definition to embed the Mexican-hat shape inside the J-cost framework. The definition is a direct one-line wrapper that applies the Jcost operator to the standard potential term.
Claim. The J-cost of the Higgs potential for field value $phi$ and parameters $mu^2$, $lambda$ is $J(-mu^2 phi^2 + lambda phi^4)$, where $J$ is the recognition cost function.
background
The module derives electroweak symmetry breaking by equating the Higgs potential to a J-cost functional. The potential is written $V(phi) = -mu^2 |phi|^2 + lambda |phi|^4$, and its minimum determines the vacuum expectation value. Upstream results define the cost of a recognition event as its J-cost and the cost induced by a multiplicative recognizer as the derived cost of its comparator.
proof idea
The definition is a one-line wrapper that applies the Jcost operator directly to the expression -mu_sq * phi^2 + lambda * phi^4. It draws on the imported Jcost from the Cost module together with the cost definitions in MultiplicativeRecognizerL4 and ObserverForcing.
why it matters
This definition supplies the explicit J-cost form for the Higgs sector and supports the later minimization step vev_minimizes_jcost. It carries out the SM-009 program of obtaining spontaneous symmetry breaking from J-cost, locating the Mexican-hat potential inside the Recognition Science chain that begins from T5 J-uniqueness and the Recognition Composition Law. It leaves open how the numerical values of mu squared and lambda are fixed by the phi-ladder.
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