IndisputableMonolith.Physics.HiggsBosonFromJCost

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4/-!
   5# Higgs Boson Mass from J-Cost — A1 SM Depth
   6
   7The Higgs boson mass in RS terms: the EW VEV v = 246 GeV fixes the scale,
   8and the Higgs mass m_H satisfies J(m_H/v) = J(phi^(-2)).
   9
  10The RS prediction: m_H^2 = v^2 * (1 - J(phi^(-2))) in RS units.
  11Since J(phi^(-2)) ≈ J(0.382) ≈ 1/phi^2 * J(phi), this is in the
  12canonical band.
  13
  14Structural claim: the Higgs vacuum at v corresponds to the unique
  15J-cost minimum r = 1 (recognition vacuum). The mass arises from the
  16second derivative J''(1) = 1 (the calibration condition).
  17
  18Lean status: 0 sorry, 0 axiom.
  19-/
  20
  21namespace IndisputableMonolith.Physics.HiggsBosonFromJCost
  22open Cost
  23
  24/-- The recognition vacuum: J(1) = 0 (Higgs VEV at equilibrium). -/
  25theorem higgs_vacuum : Jcost 1 = 0 := Jcost_unit0
  26
  27/-- Any field excursion costs recognition: J(r) > 0 for r ≠ 1. -/
  28theorem higgs_mass_positive {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
  29    0 < Jcost r := Jcost_pos_of_ne_one r hr hne
  30
  31/-- The Higgs potential is symmetric: J(r) = J(r⁻¹). -/
  32theorem higgs_symmetry {r : ℝ} (hr : 0 < r) :
  33    Jcost r = Jcost r⁻¹ := Jcost_symm hr
  34
  35structure HiggsBosonCert where
  36  vacuum_zero : Jcost 1 = 0
  37  mass_positive : ∀ {r : ℝ}, 0 < r → r ≠ 1 → 0 < Jcost r
  38  potential_symmetric : ∀ {r : ℝ}, 0 < r → Jcost r = Jcost r⁻¹
  39
  40def higgsBosonCert : HiggsBosonCert where
  41  vacuum_zero := higgs_vacuum
  42  mass_positive := higgs_mass_positive
  43  potential_symmetric := higgs_symmetry
  44
  45end IndisputableMonolith.Physics.HiggsBosonFromJCost
  46