omega_lambda_bounds

formal source

 129  linarith
 130
 131/-- **BOUNDS**: 0 < Ω_Λ < 11/16 -/
 132theorem omega_lambda_bounds : 0 < 11/16 - (alpha / Real.pi) ∧ 11/16 - (alpha / Real.pi) < 11/16 :=
 133  ⟨omega_lambda_positive, omega_lambda_lt_11_16⟩
 134
 135/-! ## Section P-002: Fermion Mass Hierarchy -/
 136
 137/-- **CALCULATED**: φ^6 bounds: 17 < φ^6 < 18.
 138    Uses φ^6 = (φ^3)^2 = (2φ+1)^2. -/
 139theorem phi_6_hierarchy_bounds : (17 : ℝ) < (phi : ℝ)^6 ∧ (phi : ℝ)^6 < (18 : ℝ) := by
 140  have h3 := phi_cubed_eq  -- phi^3 = 2*phi + 1
 141  have hphi_lo : phi > 1.61 := phi_gt_onePointSixOne
 142  have hphi_hi : phi < 1.62 := phi_lt_onePointSixTwo
 143  have h6 : phi^6 = (2 * phi + 1)^2 := by nlinarith [phi_pos]
 144  constructor
 145  · rw [h6]; nlinarith
 146  · rw [h6]; nlinarith
 147
 148/-- **CALCULATED**: φ^11 > 180 (conservative lower bound for large hierarchies).
 149    Uses Fibonacci: phi^11 = 89*phi + 55 > 89*1.618 + 55 > 180. -/
 150theorem phi_11_hierarchy_lower : (180 : ℝ) < (phi : ℝ)^11 := by
 151  rw [phi_eleventh_eq]
 152  linarith [phi_gt_onePointSixOne]
 153
 154/-- **HIERARCHY STRUCTURE**: Mass ratios are φ-powers of integer differences. -/
 155theorem hierarchy_phi_power_structure (Δr : ℕ) (hΔr : Δr > 0) :
 156    ∃ (ratio : ℝ), ratio = (phi : ℝ)^Δr ∧ ratio > 1 := by
 157  use (phi : ℝ)^Δr
 158  refine ⟨rfl, ?_⟩
 159  have h1 : phi > 1 := one_lt_phi
 160  have h_ge : Δr ≥ 1 := hΔr
 161  calc 1 = 1^Δr := (one_pow Δr).symm
 162    _ < phi^Δr := by