horton_bifurcation_ratio
plain-language theorem explainer
The Horton bifurcation ratio is defined as the square of the golden ratio, encoding the average number of tributaries per stream order in a σ-conserved drainage network. Geomorphologists deriving self-similar scaling laws for river basins cite this constant when obtaining Hack's exponent exactly equal to one half. The entry is introduced as a direct constant definition with no additional proof steps.
Claim. The Horton bifurcation ratio satisfies $R_b := phi^2$, where $phi$ is the golden ratio.
background
In the Recognition Science treatment of river networks, a drainage basin is modeled as a recognition tree on the topographic ledger. σ-conservation forces upstream-downstream branching to obey the canonical Horton ratios: length ratio $R_l = phi$ and bifurcation ratio $R_b = phi^2$. This two-φ-step structure per order is the same eight-tick lattice that appears in volcanic-eruption recurrence and planetary-orbit gap rules.
proof idea
This is a direct definition that sets the Horton bifurcation ratio equal to the square of the golden ratio.
why it matters
This definition supplies the value of $R_b$ used to prove $R_b = R_l^2$ and to define Hack's exponent as $log R_l / log R_b = 1/2$. It closes the structural part of the σ-conservation argument for river networks and connects to the eight-tick octave (T7) in the broader framework. The companion numerical check against empirical Hack exponents in (0.5, 0.65) remains open.
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