phi_11_hierarchy_lower

formal source

 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
 163        apply pow_lt_pow_left₀ h1 (by norm_num)
 164        exact Nat.pos_iff_ne_zero.mp hΔr
 165
 166/-! ## Section: Combined Registry Certificate -/
 167
 168/-- **CERTIFICATE**: Registry predictions with calculated bounds. -/
 169structure RegistryPredictionsCert where
 170  omega_lambda_upper : 11/16 - (alpha / Real.pi) < 11/16
 171  omega_lambda_lower : 0 < 11/16 - (alpha / Real.pi)
 172  phi_6_lower : (17 : ℝ) < (phi : ℝ)^6
 173  phi_6_upper : (phi : ℝ)^6 < (18 : ℝ)
 174  phi_11_lower : (180 : ℝ) < (phi : ℝ)^11
 175  hierarchy_exists : ∀ (Δr : ℕ), Δr > 0 →
 176    ∃ (ratio : ℝ), ratio = (phi : ℝ)^Δr ∧ ratio > 1
 177
 178/-- **THEOREM**: Registry predictions certificate is inhabited. -/
 179theorem registry_predictions_cert_exists : ∃ _ : RegistryPredictionsCert, True :=
 180  ⟨⟨omega_lambda_lt_11_16,