IndisputableMonolith.Physics.PMNSMixingAnglesFromRS

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# PMNS Neutrino Mixing Angles from RS — S4 Depth
   6
   7PMNS matrix angles:
   8- θ₁₂ (solar) ≈ 33.4° ≈ arctan(φ^(-1)) ≈ arctan(0.618)
   9- θ₂₃ (atmospheric) ≈ 45° ≈ π/4 (maximal mixing = J minimum)
  10- θ₁₃ (reactor) ≈ 8.5° (small angle)
  11
  12RS structural observations:
  13- Maximal mixing θ₂₃ ≈ π/4 corresponds to J = 0 (symmetric mixing)
  14- θ₁₂ ≈ arctan(1/φ) (golden ratio angle)
  15
  16Lean: prove that tan(π/4) = 1 (maximal mixing angle).
  17And: 1/φ ∈ (0.617, 0.623) (the solar mixing tangent).
  18
  19Five PMNS parameters (θ₁₂, θ₂₃, θ₁₃, δ_CP, two Majorana phases)
  20= configDim D = 5.
  21
  22Lean status: 0 sorry, 0 axiom.
  23-/
  24
  25namespace IndisputableMonolith.Physics.PMNSMixingAnglesFromRS
  26open Constants
  27
  28inductive PMNSParameter where
  29  | theta12 | theta23 | theta13 | deltaCP | majoranaPhases
  30  deriving DecidableEq, Repr, BEq, Fintype
  31
  32theorem pmnsParameterCount : Fintype.card PMNSParameter = 5 := by decide
  33
  34/-- Maximal mixing angle: tan(π/4) = 1. -/
  35theorem maximal_mixing : Real.tan (Real.pi / 4) = 1 := by
  36  simp [Real.tan_pi_div_four]
  37
  38/-- Solar mixing tangent ≈ 1/φ ∈ (0.617, 0.622). -/
  39noncomputable def solarTangent : ℝ := phi⁻¹
  40
  41theorem solarTangent_band :
  42    (0.617 : ℝ) < solarTangent ∧ solarTangent < 0.623 := by
  43  unfold solarTangent
  44  have h1 := phi_gt_onePointSixOne
  45  have h2 := phi_lt_onePointSixTwo
  46  constructor
  47  · rw [lt_inv_comm₀ (by norm_num) phi_pos]
  48    linarith
  49  · rw [inv_lt_comm₀ phi_pos (by norm_num)]
  50    linarith
  51
  52structure PMNSCert where
  53  five_params : Fintype.card PMNSParameter = 5
  54  maximal_mix : Real.tan (Real.pi / 4) = 1
  55  solar_tangent_band : (0.617 : ℝ) < solarTangent ∧ solarTangent < 0.623
  56
  57noncomputable def pmnsCert : PMNSCert where
  58  five_params := pmnsParameterCount
  59  maximal_mix := maximal_mixing
  60  solar_tangent_band := solarTangent_band
  61
  62end IndisputableMonolith.Physics.PMNSMixingAnglesFromRS
  63