river_network_one_statement
plain-language theorem explainer
Sigma-conservation on a phi-self-similar drainage network forces the Horton bifurcation ratio to equal the square of the length ratio. Hack's exponent is then exactly one half under self-similar scaling. This value sits at the lower end of the empirical Hack band (0.45, 0.65). The proof is a direct term that assembles the ratio identity, the exponent equality, and the numerical band check.
Claim. Under sigma-conservation on a phi-self-similar drainage network, the Horton bifurcation ratio satisfies $R_b = R_l^2$, Hack's exponent is $h = 1/2$, and $0.45 < h < 0.65$, where $R_l = phi$ and $R_b = phi^2$.
background
In the Recognition Science treatment of river networks, a drainage basin is modeled as a recognition tree on the topographic ledger. Sigma-conservation enforces self-similar branching with Horton length ratio $R_l = phi$ and bifurcation ratio $R_b = phi^2$. Hack's law states that mainstream length scales with basin area as $L propto A^h$, and under Hortonian scaling the exponent is $h = log R_l / log R_b$ (from the hack_exponent definition). The module proves the structural identity $h = 1/2$ exactly, which lies in the empirical range (0.45, 0.65) reported by Hack (1957) and subsequent catalogs. Upstream results establish the ratio equality $R_b = R_l^2$ from sigma-conservation and the logarithm identity that reduces the exponent to one half.
proof idea
The proof is a term-mode wrapper that directly pairs the bifurcation-equals-length-squared identity, the theorem establishing the exponent equals one half, and the two sides of the empirical band membership.
why it matters
This statement consolidates the core results of the river-network module, linking sigma-conservation to observed Horton ratios and placing the derived Hack exponent in the empirical band. It parallels the phi-squared ratios arising in the eight-tick octave structure seen in volcanic recurrence and planetary formation. The partial closure notes that fractal-basin corrections, which would raise the exponent toward the upper band, remain outside the present formalization.
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