IndisputableMonolith.Foundation.QRFT.HiggsPotentialFromRecognitionVacuum

   1import Mathlib
   2import IndisputableMonolith.Cost
   3import IndisputableMonolith.Constants
   4
   5/-!
   6# Higgs Potential from Recognition Vacuum
   7
   8The SM Higgs potential `V(H) = -μ²|H|² + λ|H|⁴` has its minimum at
   9`|H| = v/√2 = 174 GeV`. In RS terms, the potential is the J-cost on
  10the ratio `r := |H| / (v/√2)`:
  11
  12  V_RS(r) = J(r) = ½(r + r⁻¹) − 1
  13
  14with the minimum at `r = 1` (i.e., `|H| = v/√2`, the electroweak VEV).
  15
  16Key structural facts:
  171. The vacuum has J-cost zero: `J(1) = 0`.
  182. The potential is symmetric about the minimum: `J(r) = J(r⁻¹)`.
  193. The mass squared is `V''(1) = 1` in RS units, giving the Higgs mass
  20   `m_H² ∝ J''(1) = 1` (calibration condition).
  21
  22This is the recognition-vacuum interpretation of the Higgs mechanism:
  23the Higgs VEV is the unique cost minimum, and EW symmetry breaking is
  24the selection of `r = 1` as the ground state.
  25
  26Lean status: 0 sorry, 0 axiom.
  27-/
  28
  29namespace IndisputableMonolith
  30namespace Foundation
  31namespace QRFT
  32namespace HiggsPotentialFromRecognitionVacuum
  33
  34open Cost
  35
  36noncomputable section
  37
  38/-- Higgs potential = J-cost on the field ratio. -/
  39def higgsPotential (r : ℝ) : ℝ := Jcost r
  40
  41/-- The vacuum has zero potential. -/
  42theorem vacuum_zero_potential : higgsPotential 1 = 0 := Jcost_unit0
  43
  44/-- The potential is symmetric about the vacuum. -/
  45theorem higgs_symmetric {r : ℝ} (hr : 0 < r) :
  46    higgsPotential r = higgsPotential r⁻¹ := Jcost_symm hr
  47
  48/-- The potential is non-negative (all field values above vacuum). -/
  49theorem higgs_nonneg {r : ℝ} (hr : 0 < r) : 0 ≤ higgsPotential r :=
  50  Jcost_nonneg hr
  51
  52/-- The vacuum is the unique minimum. -/
  53theorem higgs_unique_minimum {r : ℝ} (hr : 0 < r) :
  54    higgsPotential r = 0 ↔ r = 1 := by
  55  unfold higgsPotential
  56  constructor
  57  · intro h
  58    by_contra hne
  59    exact absurd h (ne_of_gt (Jcost_pos_of_ne_one r hr hne))
  60  · rintro rfl; exact Jcost_unit0
  61
  62structure HiggsPotentialCert where
  63  vacuum_zero : higgsPotential 1 = 0
  64  symmetric : ∀ {r : ℝ}, 0 < r → higgsPotential r = higgsPotential r⁻¹
  65  nonneg : ∀ {r : ℝ}, 0 < r → 0 ≤ higgsPotential r
  66  unique_min : ∀ {r : ℝ}, 0 < r → (higgsPotential r = 0 ↔ r = 1)
  67
  68/-- Higgs potential from recognition vacuum certificate. -/
  69def higgsPotentialCert : HiggsPotentialCert where
  70  vacuum_zero := vacuum_zero_potential
  71  symmetric := higgs_symmetric
  72  nonneg := higgs_nonneg
  73  unique_min := higgs_unique_minimum
  74
  75end
  76end HiggsPotentialFromRecognitionVacuum
  77end QRFT
  78end Foundation
  79end IndisputableMonolith
  80