HydraulicExponents
plain-language theorem explainer
The structure of hydraulic geometry exponents encodes the Leopold-Maddock triple for width, depth, and velocity scaling in single-thread channels. Fluvial geomorphologists cite it to enforce discharge conservation without empirical fitting. It is introduced as a record type carrying three positive reals that sum to unity.
Claim. A triple of positive real numbers $(b, f, m)$ satisfying $b + f + m = 1$, where $b$ is the width exponent, $f$ the depth exponent, and $m$ the velocity exponent in the relations $w = k_w Q^b$, $d = k_d Q^f$, $v = k_v Q^m$.
background
The module derives hydraulic geometry from σ-conservation on the discharge ledger. From $Q = w · d · v$, taking logarithms produces the exponent sum $b + f + m = 1$. This supplies the local setting in which the Leopold-Maddock exponents are treated as an algebraic identity rather than a statistical fit. The upstream positivity lemmas extract the strict inequalities directly from the structure fields.
proof idea
The declaration is a structure definition. It directly assembles the three exponent fields with their positivity and closure constraints; no tactics or lemmas are invoked beyond the field projections.
why it matters
This structure is the parent type for the closure identity and the equipartition witness in the same module. It fills the structural theorem slot in the hydrology extension of Recognition Science, showing that σ-conservation alone constrains the exponents. The construction aligns with the framework's emphasis on deriving scaling laws from conservation without additional parameters.
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