riverNetworkCert

formal source

 162  hack_eq_half : hack_exponent = 1 / 2
 163  hack_in_band : (0.45 : ℝ) < hack_exponent ∧ hack_exponent < 0.65
 164
 165def riverNetworkCert : RiverNetworkCert where
 166  length_ratio_pos := horton_length_ratio_pos
 167  bifurcation_ratio_pos := horton_bifurcation_ratio_pos
 168  length_ratio_gt_one := horton_length_ratio_gt_one
 169  bifurcation_ratio_gt_one := horton_bifurcation_ratio_gt_one
 170  bifurcation_eq_length_squared := bifurcation_eq_length_squared
 171  log_length_pos := log_length_ratio_pos
 172  log_bifurcation_pos := log_bifurcation_ratio_pos
 173  log_bifurcation_eq_two_log_length := log_bifurcation_eq_two_log_length
 174  hack_eq_half := hack_exponent_eq_half
 175  hack_in_band := hack_exponent_in_empirical_band
 176
 177/-- **RIVER NETWORK ONE-STATEMENT.** σ-conservation on a φ-self-similar
 178drainage network forces Horton length ratio `R_l = φ` and bifurcation
 179ratio `R_b = φ² = R_l²`. Under self-similar Hortonian scaling, Hack's
 180exponent is `h = log R_l / log R_b = 1/2` exactly, sitting at the lower
 181end of the empirical Hack band `(0.45, 0.65)`. The upper end of the
 182empirical range is attributed to fractal-basin-area corrections not
 183formalized here (PARTIAL CLOSURE). -/
 184theorem river_network_one_statement :
 185    horton_bifurcation_ratio = horton_length_ratio ^ 2 ∧
 186    hack_exponent = 1 / 2 ∧
 187    (0.45 : ℝ) < hack_exponent ∧ hack_exponent < 0.65 :=
 188  ⟨bifurcation_eq_length_squared, hack_exponent_eq_half,
 189   hack_exponent_in_empirical_band.1, hack_exponent_in_empirical_band.2⟩
 190
 191end
 192
 193end RiverNetworkFromSigmaConservation
 194end Climate
 195end IndisputableMonolith