Foundation Qrft Higgs Potential From Recognition Vacuum Higgs Symmetric

The symmetry of the vacuum

The Higgs potential is the energy function that gives elementary particles their masses. In the standard model it is written V(H) = −μ²|H|² + λ|H|⁴, a quartic polynomial whose minimum sits at |H| = v/√2 ≈ 174 GeV, the electroweak vacuum expectation value. The potential is symmetric under the replacement |H| → −|H|, but it has a second, less familiar symmetry: the cost of a field value and the cost of its reciprocal are equal. This is the content of the declaration higgs_symmetric.

The Recognition Science framework models this potential as a cost function on the ratio r := |H| / (v/√2), the field value measured against its vacuum minimum. The cost is J(r) = ½(r + r⁻¹) − 1. At r = 1 the cost is zero, which is the vacuum. The symmetry J(r) = J(r⁻¹) means that a field value twice the vacuum costs the same as one half the vacuum; the potential rises symmetrically on a logarithmic scale around its minimum. This reciprocal symmetry is not an extra assumption. It follows from the same five conditions that force the cost function itself: reciprocal symmetry, zero cost at unity, a composition law, calibration, and continuity. The theorem higgs_symmetric states this equality for all positive r, and it is proved in the machine-checked library of formal theorems with no unproved axioms.

The framework's library also proves three companion facts about the same potential. The vacuum has zero cost, the potential is non-negative for all positive field values, and the vacuum is the unique minimum: the cost is zero if and only if r = 1. Together these four statements form a certificate, a packaged proof that the recognition-vacuum interpretation reproduces the essential structure of the Higgs mechanism. The minimum is unique, the ground state sits at the electroweak scale, and the potential rises away from it in every direction.

What the declaration does not claim is just as important. The symmetry holds only for positive r, the physical domain of a field magnitude. The theorem does not derive the numerical value of the Higgs mass or the vacuum expectation value from first principles; those enter through the calibration condition, which sets the mass scale in the framework's units. The framework does not claim to replace the standard model's Higgs mechanism, only to show that its potential has the same shape as the cost function forced by recognition. The symmetry is a structural fact about the potential, not a new physical prediction.

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

/-- The potential is non-negative (all field values above vacuum). -/
theorem higgs_nonneg {r : ℝ} (hr : 0 < r) : 0 ≤ higgsPotential r :=
  Jcost_nonneg 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 theorem does not derive the numerical value of the Higgs mass or the vacuum expectation value from first principles. The framework does not claim to replace the standard model's Higgs mechanism with a new physical prediction. The symmetry is proved only for positive field values, the physical domain of a field magnitude.

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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