IndisputableMonolith.StandardModel.WeakCoupling

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Constants.Alpha
   4import IndisputableMonolith.Masses.ElectroweakMasses
   5
   6/-!
   7# The Weak Coupling Constant α_W from RS First Principles
   8
   9This module defines the SU(2) weak coupling constant α_W by combining
  10two independently RS-derived quantities:
  11
  12- α (EM fine-structure constant) from `Constants/Alpha.lean`
  13- sin²θ_W = (3 − φ)/6 from `Masses/ElectroweakMasses.lean`
  14
  15via the tree-level electroweak identity: α = α_W · sin²θ_W.
  16
  17Both inputs are zero-parameter derivations from Q₃ cube geometry:
  18- α comes from the EMAlphaCert chain (44π · exp(−f_gap/44π))
  19- sin²θ_W comes from the gauge embedding: (D − φ)/(2D) at D = 3
  20
  21## Main Results
  22
  23- `alpha_W`: the weak coupling constant = α / sin²θ_W
  24- `alpha_W_pos`: α_W is positive
  25- `alpha_W_gt_alpha`: α_W > α (since sin²θ_W < 1)
  26- `alpha_W_structural`: α_W is fully RS-derived
  27
  28## Status: 0 sorry, 0 axiom
  29-/
  30
  31namespace IndisputableMonolith
  32namespace StandardModel
  33namespace WeakCoupling
  34
  35open Constants Masses.ElectroweakMasses
  36
  37noncomputable section
  38
  39/-! ## Part 1: Definition -/
  40
  41/-- The weak coupling constant α_W = α / sin²θ_W.
  42    From the tree-level electroweak identity: α_EM = α_W · sin²θ_W,
  43    so α_W = α_EM / sin²θ_W. -/
  44def alpha_W : ℝ := alpha / sin2_theta_W_rs
  45
  46/-- α_W expressed in terms of RS primitives:
  47    α_W = (1/alphaInv) / ((3 − φ)/6) = 6 / (alphaInv · (3 − φ)) -/
  48theorem alpha_W_expanded :
  49    alpha_W = alpha / ((3 - Constants.phi) / 6) := rfl
  50
  51/-! ## Part 2: Positivity and Bounds -/
  52
  53private lemma alpha_pos_aux : 0 < alpha := by
  54  unfold alpha alphaInv alpha_seed; positivity
  55
  56/-- α_W is positive (both α and sin²θ_W are positive). -/
  57theorem alpha_W_pos : 0 < alpha_W := by
  58  unfold alpha_W
  59  exact div_pos alpha_pos_aux sin2_theta_positive
  60
  61/-- α_W > α (since sin²θ_W < 1, dividing by it increases α). -/
  62theorem alpha_W_gt_alpha : alpha < alpha_W := by
  63  unfold alpha_W
  64  rw [lt_div_iff₀ sin2_theta_positive]
  65  calc alpha * sin2_theta_W_rs
  66      < alpha * 1 := by {
  67        apply mul_lt_mul_of_pos_left _ alpha_pos_aux
  68        linarith [sin2_theta_lt_half]
  69      }
  70    _ = alpha := mul_one _
  71
  72/-- sin²θ_W > 0 (needed for division). -/
  73theorem sin2_pos : 0 < sin2_theta_W_rs := sin2_theta_positive
  74
  75/-- sin²θ_W < 1/2 (the weak mixing is mild). -/
  76theorem sin2_lt_half : sin2_theta_W_rs < 1/2 := sin2_theta_lt_half
  77
  78/-- α_W > 2α (since sin²θ_W < 1/2). -/
  79theorem alpha_W_gt_two_alpha : 2 * alpha < alpha_W := by
  80  unfold alpha_W
  81  rw [lt_div_iff₀ sin2_theta_positive]
  82  calc 2 * alpha * sin2_theta_W_rs
  83      < 2 * alpha * (1/2) := by {
  84        apply mul_lt_mul_of_pos_left sin2_lt_half
  85        exact mul_pos (by norm_num) alpha_pos_aux
  86      }
  87    _ = alpha := by ring
  88
  89/-! ## Part 3: Structural Certificate -/
  90
  91/-- α_W is fully RS-derived: no free parameters enter its computation.
  92    - α comes from the EMAlphaCert (44π seed + f_gap from 8-tick)
  93    - sin²θ_W = (3 − φ)/6 from gauge embedding geometry
  94    Both trace to Q₃ cube structure + golden ratio φ. -/
  95structure WeakCouplingCert where
  96  alpha_from_cube : alphaInv = alpha_seed * Real.exp (-(f_gap / alpha_seed))
  97  sin2_from_phi : sin2_theta_W_rs = (3 - Constants.phi) / 6
  98  alpha_W_def : alpha_W = alpha / sin2_theta_W_rs
  99  alpha_W_positive : 0 < alpha_W
 100  alpha_W_exceeds_alpha : alpha < alpha_W
 101
 102theorem weak_coupling_cert : WeakCouplingCert where
 103  alpha_from_cube := rfl
 104  sin2_from_phi := rfl
 105  alpha_W_def := rfl
 106  alpha_W_positive := alpha_W_pos
 107  alpha_W_exceeds_alpha := alpha_W_gt_alpha
 108
 109end
 110
 111end WeakCoupling
 112end StandardModel
 113end IndisputableMonolith
 114