Foundation Qrft Higgs Potential From Recognition Vacuum Higgs Potential Cert

The certificate

The standard model of particle physics describes the Higgs boson with a potential energy function V(H) = −μ²|H|² + λ|H|⁴, a quartic polynomial whose minimum sits at |H| = v/√2, about 174 GeV. This is the field configuration that breaks electroweak symmetry and gives mass to other particles. In the Recognition Science framework, the proposed replacement for this potential is the J-cost function applied to the ratio r := |H| / (v/√2), namely V_RS(r) = J(r) = ½(r + r⁻¹) − 1. The minimum of this function lies at r = 1, which is exactly the electroweak vacuum expectation value.

The declaration HiggsPotentialCert is a machine-checked certificate, a formal object in the framework's library of theorems, that bundles four structural facts about this proposed potential. First, the vacuum has zero potential: J(1) = 0. Second, the potential is symmetric about its minimum: J(r) = J(r⁻¹). Third, it is non-negative for all positive field ratios. Fourth, and most importantly, the vacuum is the unique minimum: J(r) = 0 if and only if r = 1. Each of these four facts is proved from the defining properties of the J-cost function, and the certificate bundles them into a single reusable object.

What the certificate does not do is derive the value of the Higgs mass or the vacuum expectation value. The mass squared of the Higgs boson is proportional to the second derivative of the potential at the minimum, V''(1) = 1 in RS units, but this is a calibration condition, a choice of units, not a prediction. The certificate records the shape and symmetry of the potential, not the physical scale at which it operates. The 174 GeV value is an input from measurement, not an output of the framework.

In plain terms, the certificate establishes that the J-cost function, originally derived from a set of axioms about recognition cost, has exactly the right qualitative shape to serve as a Higgs potential: it has a unique minimum at the right place, it is symmetric, and it never dips below zero. The framework's contribution is structural, a demonstration that the recognition cost function has the correct geometry for electroweak symmetry breaking, not a numerical prediction of the Higgs mass.

/-- Higgs potential = J-cost on the field ratio. -/
def higgsPotential (r : ℝ) : ℝ := Jcost r

/-- The vacuum has zero potential. -/
theorem vacuum_zero_potential : higgsPotential 1 = 0 := Jcost_unit0

/-- The potential is symmetric about the vacuum. -/
theorem higgs_symmetric {r : ℝ} (hr : 0 < r) :
    higgsPotential r = higgsPotential r⁻¹ := Jcost_symm hr

/-- The vacuum is the unique minimum. -/
theorem higgs_unique_minimum {r : ℝ} (hr : 0 < r) :
    higgsPotential r = 0 ↔ r = 1 := by
  unfold higgsPotential
  constructor
  · intro h
    by_contra hne
    exact absurd h (ne_of_gt (Jcost_pos_of_ne_one r hr hne))
  · rintro rfl; exact Jcost_unit0

What this page does not claim

The certificate does not predict the numerical value of the Higgs mass or the vacuum expectation value. The certificate does not claim that the standard model quartic potential is derived from recognition axioms; it only proposes a structurally similar alternative. The certificate does not establish that electroweak symmetry breaking actually occurs in the framework; it only records properties of a candidate potential.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/QRFT/HiggsPotentialFromRecognitionVacuum.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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