rung_slope_pos
plain-language theorem explainer
The rung_slope_pos theorem asserts that the natural logarithm of the golden ratio is strictly positive. Recognition Science geologists cite it to establish the positive slope in the rung-form Gutenberg-Richter relation derived from the phi-ladder. The proof is a one-line term that unfolds the definition of rung_slope and invokes the standard lemma that the logarithm is positive for arguments exceeding one.
Claim. $0 < ln φ$ where $φ = (1 + √5)/2$ is the golden ratio.
background
In the GutenbergRichterFromLedger module the rung-form of the Gutenberg-Richter law states that frequency N(r) satisfies ln N(r+1) − ln N(r) = −ln φ, so rung_slope is defined as Real.log phi. The module derives this from recognition-cost releases on the geophysical ledger where each magnitude unit corresponds to phi-ladder steps. The upstream lemma one_lt_phi establishes 1 < phi via the quadratic inequality satisfied by the golden ratio, already proven in Constants.lean.
proof idea
The proof unfolds the definition rung_slope := Real.log phi and then applies Real.log_pos directly to the fact 1 < phi supplied by one_lt_phi. It is a direct term-mode application with no additional tactics.
why it matters
This result supplies the rung_slope_pos field in the GutenbergRichterCert structure that certifies the base-10 slope b equals 1 in the empirical Gutenberg-Richter law. It fills the positivity requirement needed to force b = 1 from the change of base between the phi-ladder and Richter magnitude. In the Recognition framework it supports the structural prediction that the observed b-value is not free but follows from the phi fixed point and the eight-tick octave.
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