Foundation Qrft Higgs Potential From Recognition Vacuum Vacuum Zero Potential

The vacuum minimum

The standard model Higgs potential is usually 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 (VEV). In Recognition Science (RS), this potential is not a free polynomial but a specific, forced cost function. The framework defines the cost, a measure of how expensive a recognition event is, and proves that any cost satisfying five plain conditions must equal J(x) = ½(x + x⁻¹) − 1. The Higgs potential is then the J-cost evaluated on the ratio r = |H| / (v/√2), so V_RS(r) = J(r).

The declaration vacuum_zero_potential states the central fact about this construction: the potential is zero at the vacuum. Formally, it proves higgsPotential 1 = 0, meaning when the field sits at its VEV (r = 1), the cost vanishes. This is not an arbitrary normalization. It follows from the defining property of J, which forces J(1) = 0 by the zero-cost-at-unity condition. In plain language, the vacuum is the state where recognition costs nothing, and the electroweak symmetry breaking is the selection of that state as the ground state.

The theorem is one of four structural facts proved together. The potential is symmetric about the minimum, J(r) = J(r⁻¹), so a field value above the VEV costs the same as its reciprocal below. The potential is non-negative for all positive field values. And the vacuum is the unique minimum: the only place where the potential reaches zero is at r = 1. These four facts form a certificate, a packaged proof that the Higgs potential from recognition vacuum has the right shape. The mass squared, V''(1) = 1 in RS units, comes from the calibration condition and gives the Higgs mass.

What vacuum_zero_potential does not claim is just as important. It does not derive the numerical value of the Higgs mass or the VEV; those are set by the calibration condition and by matching to measured values. It does not prove that the standard model Higgs mechanism is physically correct, only that within RS the potential's minimum has the right structural properties. The theorem is a statement about a mathematical construction, not an empirical prediction. The physical bridge from recognition events to the electroweak VEV remains open in the framework.

/-- 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

/-- The potential is non-negative (all field values above vacuum). -/
theorem higgs_nonneg {r : ℝ} (hr : 0 < r) : 0 ≤ higgsPotential r :=
  Jcost_nonneg hr

What this page does not claim

This does not derive the numerical value of the Higgs mass or the VEV from first principles. This does not prove the standard model Higgs mechanism is physically correct. This does not claim the physical bridge from recognition events to the electroweak VEV is closed.

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: