log_length_ratio_pos
plain-language theorem explainer
The natural logarithm of the Horton length ratio is strictly positive. Researchers deriving exact scaling relations for river networks under sigma-conservation would cite this to secure the sign in the Hack exponent. The proof is a one-line wrapper that unfolds the length ratio definition and applies the known positivity of the logarithm of phi.
Claim. $0 < 0 < R_l$ where $R_l = phi$ is the golden ratio.
background
The module models drainage networks as recognition trees on the topographic ledger. Sigma-conservation forces upstream-downstream branching to obey the canonical Horton ratios: length ratio R_l equals phi and bifurcation ratio R_b equals phi squared. These ratios arise from the same eight-tick lattice and gap-45 frustration that appears in volcanic recurrence and planetary orbit rules. The Horton length ratio is defined directly as phi. The lemma one_lt_phi establishes that phi exceeds 1, which is already available from Constants and PhiSupport.
proof idea
The proof is a one-line wrapper. It unfolds horton_length_ratio to phi and applies Real.log_pos to the fact that one_lt_phi holds.
why it matters
This positivity result feeds the derivation of hack_exponent_eq_half and the riverNetworkCert structure. It supplies the sign needed to conclude that Hack's exponent equals exactly one half via the identity log R_l over log R_b. The step aligns with the self-similar fixed point and eight-tick octave in the Recognition framework. The module treats the structural identity as proved while leaving empirical band checks against observed h in (0.5, 0.65) for external data.
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