Foundation Qrft Higgs Potential From Recognition Vacuum

The Higgs Potential as a Cost

The Standard Model of particle physics describes the Higgs field with a potential energy function V(H) = −μ²|H|² + λ|H|⁴. This function has a minimum at a nonzero field value, the vacuum expectation value, which is measured to be about 246 GeV. When the field settles into this minimum, it breaks the electroweak symmetry and gives mass to the W and Z bosons. The potential's shape, a sombrero with a central bump and a circular trough, is what makes this work.

Recognition Science offers a different starting point for the same curve. Its framework begins with a ledger, a discrete record of recognition events, and a forced cost function J(x) = ½(x + x⁻¹) − 1 for any positive ratio x. This cost is zero when x equals 1 and grows as x moves away from 1 in either direction. The framework models the Higgs potential as this cost applied to the ratio r = |H| / (v/√2), where v/√2 is the measured vacuum value of 174 GeV. In symbols, V_RS(r) = J(r).

This identification is a definitional choice, not a derivation from first principles. What the framework's machine-checked library of formal theorems proves are the properties of this modeled potential. The vacuum has zero potential: J(1) = 0. The potential is symmetric about the minimum, J(r) = J(r⁻¹), so moving the field above or below the vacuum by the same factor costs the same energy. The potential is non-negative for all positive field values, and the vacuum is the unique minimum, meaning J(r) = 0 if and only if r = 1. These four facts are assembled into a single certificate structure in the library, with zero unproven axioms.

In this account, electroweak symmetry breaking is the selection of r = 1 as the ground state. The Higgs mass squared is proportional to the second derivative of the cost at the minimum, J''(1) = 1 in the framework's units, which is the calibration condition that ties the cost to the measured mass scale. The framework's contribution is a compact restatement of a known physical structure: the Higgs potential is the simplest cost function that has a unique minimum and is symmetric about it. It does not predict the value of the Higgs mass; it shows what form the potential must take if it is to be a cost on a ratio.

/-- 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

This module does not derive the Higgs potential from the recognition axioms; it models the potential as the cost function and proves properties of that model. The framework does not predict the numerical value of the Higgs mass or the vacuum expectation value. The identification of the vacuum expectation value as the cost minimum is a definitional choice, not a proved physical law.

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)

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