bifurcation_eq_length_squared
plain-language theorem explainer
Recognition Science models river networks as recognition trees on the topographic ledger under σ-conservation. The identity asserts that the Horton bifurcation ratio equals the square of the Horton length ratio, both defined as powers of phi. Geomorphologists deriving Hack's exponent from self-similar scaling would cite this step. The proof is a one-line reflexivity that follows immediately from the ratio definitions.
Claim. $R_b = R_l^2$, where $R_l = phi$ is the Horton length ratio (per-order length growth) and $R_b = phi^2$ is the Horton bifurcation ratio (tributary count per order).
background
The module treats a drainage network as a recognition tree forced by σ-conservation to obey canonical Horton ratios on the φ-self-similar lattice. Upstream definitions set horton_length_ratio to phi (one φ-step per Horton order) and horton_bifurcation_ratio to phi squared (two φ-steps per order). These ratios trace to the self-similar fixed point in PhiForcingDerived.of and the ledger factorization in DAlembert.LedgerFactorization.of, which calibrate J-cost on the positive reals. The local setting is Track P2 of Plan v7, where σ-conservation produces the same two-φ-step structure seen in volcanic recurrence and planetary gap rules.
proof idea
The proof is a one-line reflexivity that applies the definitions horton_bifurcation_ratio := phi ^ 2 and horton_length_ratio := phi directly.
why it matters
This identity is invoked inside the RiverNetworkCert structure and the river_network_one_statement theorem, which packages the full claim that σ-conservation forces R_b = R_l^2 together with Hack's exponent exactly 1/2 inside the empirical band (0.45, 0.65). It supplies the algebraic link h = log R_l / log R_b = 1/2 that places the self-similar value at the lower edge of observed Hack exponents. The step closes the structural part of the RS reading of Hack's law while leaving fractal-basin corrections for later work.
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