vub_derived

formal source

  65    - Geometric Origin: The first and third generations are separated by the maximum
  66      diameter of the cube (√3 units). The recognition overlap is mediated by the
  67      vacuum polarization term α, with a 1/2 factor from the parity split. -/
  68theorem vub_derived :
  69    V_ub_pred = fine_structure_leakage := by
  70  unfold V_ub_pred fine_structure_leakage
  71  rfl
  72
  73/-- **Geometric Interpretation**:
  74    - 12 edges in a cube.
  75    - Each edge has 2 vertices.
  76    - Total coverage space = 24 (vertex_edge_slots).
  77    - V_cb represents the single-edge contribution. -/
  78theorem vcb_geometric_origin :
  79    V_cb_pred = 1 / vertex_edge_slots := by
  80  -- 1/24 = 1/(2 * 12) = 1/vertex_edge_slots
  81  -- Reduce both sides to 1/24 using the proven slot count.
  82  have hslots : (vertex_edge_slots : ℝ) = 24 := by
  83    -- vertex_edge_slots = 24 as a Nat; cast to ℝ.
  84    have h := vertex_edge_slots_eq_24
  85    norm_cast at h
  86  -- Now `V_cb_pred` is `1/24` (as a real), so both sides match.
  87  simp [CKMGeometry.V_cb_pred, CKMGeometry.V_cb_geom, edge_dual_ratio, hslots]
  88
  89/-! ## Neutrino Sector (PMNS) -/
  90
  91/-- The PMNS mixing weights follow the Born rule over the ladder steps.
  92    Weight W_ij = exp(-Δτ_ij * J_bit) = φ^-Δτ_ij. -/
  93noncomputable def pmns_weight (dτ : ℤ) : ℝ :=
  94  Real.exp (- (dτ : ℝ) * J_bit)
  95
  96theorem pmns_weight_eq_phi_pow (dτ : ℤ) :
  97    pmns_weight dτ = phi ^ (-dτ : ℤ) := by
  98  -- Algebraic identity: exp(-dτ * ln(φ)) = φ^(-dτ)