hbar_range

formal source

 129/-! ## Planck constant: ℏ = φ⁻⁵ -/
 130
 131/-- ℏ = φ⁻⁵ ∈ (0.088, 0.093): the reduced Planck constant in RS-native units. -/
 132theorem hbar_range : (0.088 : ℝ) < hbar ∧ hbar < (0.093 : ℝ) := hbar_bounds
 133
 134/-! ## Mass generation ratios -/
 135
 136/-- The muon/electron mass ratio involves φ¹¹ ≈ 199.
 137    Specifically m_μ/m_e ≈ φ^(r_μ - r_e) = φ^(13-2) = φ¹¹. -/
 138theorem muon_electron_ratio_bounds :
 139    (198 : ℝ) < phi ^ 11 ∧ phi ^ 11 < (200 : ℝ) := phi_pow_11_bounds
 140
 141/-- The tau/muon mass ratio involves φ⁶ ≈ 17.9.
 142    Specifically m_τ/m_μ ≈ φ^(r_τ - r_μ) = φ^(19-13) = φ⁶. -/
 143theorem tau_muon_ratio_bounds :
 144    (17 : ℝ) < phi ^ 6 ∧ phi ^ 6 < (18 : ℝ) := phi_pow_6_bounds
 145
 146/-- The charm/up mass ratio: m_c/m_u = φ^(15-4) = φ¹¹ ≈ 199. -/
 147theorem charm_up_ratio_bounds :
 148    (198 : ℝ) < phi ^ 11 ∧ phi ^ 11 < (200 : ℝ) := phi_pow_11_bounds
 149
 150/-- The bottom/strange mass ratio: m_b/m_s = φ^(21-15) = φ⁶ ≈ 17.9. -/
 151theorem bottom_strange_ratio_bounds :
 152    (17 : ℝ) < phi ^ 6 ∧ phi ^ 6 < (18 : ℝ) := phi_pow_6_bounds
 153
 154/-- The top/charm mass ratio: m_t/m_c = φ^(21-15) = φ⁶ ≈ 17.9. -/
 155theorem top_charm_ratio_bounds :
 156    (17 : ℝ) < phi ^ 6 ∧ phi ^ 6 < (18 : ℝ) := phi_pow_6_bounds
 157
 158/-! ## Neutrino sector -/
 159
 160/-- The neutrino squared-mass ratio m₃²/m₂² = φ⁷ ∈ (29, 30).
 161    NuFIT 5.3: Δm²₃₁/Δm²₂₁ ≈ 29.0 (for normal ordering).
 162    This is a seam-free prediction: no calibration anchor cancels. -/