horton_length_ratio
plain-language theorem explainer
Horton length ratio is defined as the golden ratio φ, the per-order length growth factor in φ-self-similar drainage networks under σ-conservation. Geomorphologists deriving exact Hack's law from Recognition Science cite this when establishing the structural identity h = 1/2. The definition is a direct one-line assignment to the constant phi.
Claim. The Horton length ratio satisfies $R_l = φ$, where $R_l$ denotes the per-order mainstream length growth factor in a φ-self-similar drainage network.
background
River networks arise as recognition trees on the topographic ledger. σ-conservation forces upstream-downstream branching to obey canonical Horton ratios: length ratio $R_l = φ$ and bifurcation ratio $R_b = φ^2$. This encodes one φ-step in length and two φ-steps in tributary count per order, matching the eight-tick octave lattice that appears in eruption recurrence and planetary orbit derivations. The module states the structural identities and leaves numerical comparison to observed h ∈ (0.5, 0.65) for external catalogs.
proof idea
One-line definition that directly assigns the value of phi to horton_length_ratio.
why it matters
Supplies the base value for every Horton ratio identity in the module. It is referenced by bifurcation_eq_length_squared (R_b = R_l²), hack_exponent (h = log R_l / log R_b), and hack_exponent_eq_half (h = 1/2 exactly). The construction realizes the phi self-similar fixed point and the two-φ-step-per-octave rule from the forcing chain, closing the structural derivation of Hack's exponent while leaving empirical adjudication open.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.