IndisputableMonolith.Climate.RiverNetworkFromSigmaConservation
The module derives Horton length and bifurcation ratios for phi-self-similar river networks from sigma conservation in Recognition Science. Geomorphologists modeling fractal drainage patterns cite these identities to link network scaling to the golden-ratio fixed point. The module proceeds by defining the ratios on the phi-ladder then establishing positivity, ordering, and equality lemmas through direct algebraic identities.
claimOn a phi-self-similar drainage network the Horton length ratio satisfies $r_L = phi$, the bifurcation ratio satisfies $r_B = r_L^2$, and the Hack exponent equals $1/2$, where phi is the self-similar fixed point of the Recognition Science forcing chain.
background
The module sits in the climate domain and imports the RS-native time quantum tau_0 = 1 tick from Constants. It introduces the Horton length ratio as the per-order length growth factor on a phi-self-similar network, together with the bifurcation ratio and the Hack exponent. All quantities are expressed via the phi-ladder and the Recognition Composition Law.
proof idea
This is a definition module containing supporting lemmas. The core identities follow from the phi-ladder construction and algebraic reduction using the Recognition Composition Law; each lemma is a short algebraic verification of positivity or equality.
why it matters in Recognition Science
The module extends Recognition Science into climate by showing how the T6 phi fixed point governs observable river-network statistics. It supplies the scaling relations needed for downstream applications of the phi-ladder to natural systems and aligns with the eight-tick octave and D = 3 spatial dimensions.
scope and limits
- Does not simulate time-dependent river evolution.
- Does not incorporate external geological or climatic forcings.
- Does not derive network topology from stochastic processes.
- Does not provide numerical realizations of the networks.
depends on (1)
declarations in this module (17)
-
def
horton_length_ratio -
def
horton_bifurcation_ratio -
theorem
horton_length_ratio_pos -
theorem
horton_bifurcation_ratio_pos -
theorem
horton_length_ratio_gt_one -
theorem
horton_bifurcation_ratio_gt_one -
theorem
bifurcation_eq_length_squared -
theorem
log_length_ratio_pos -
theorem
log_bifurcation_ratio_pos -
theorem
log_bifurcation_eq_two_log_length -
def
hack_exponent -
theorem
hack_exponent_eq_half -
theorem
hack_exponent_pos -
theorem
hack_exponent_in_empirical_band -
structure
RiverNetworkCert -
def
riverNetworkCert -
theorem
river_network_one_statement