rung_slope
plain-language theorem explainer
rung_slope defines the natural logarithm of the golden ratio as the constant slope in the rung-form Gutenberg-Richter law. Seismologists and Recognition Science researchers modeling earthquake frequency-magnitude distributions from ledger topology would cite this value when converting phi-rung steps to standard Richter magnitudes. The definition is a direct one-line assignment from the imported phi constant.
Claim. The natural-log slope of the rung-form Gutenberg-Richter law is defined as $ln phi$, where $phi$ is the golden ratio.
background
The module derives the Gutenberg-Richter law from ledger topology in Recognition Science. Fault-surface stress drops are modeled as recognition-cost releases on the geophysical ledger, with each magnitude unit corresponding to a fixed number of phi-ladder rungs. The natural unit is one rung equals one phi-step in stress-drop energy, and frequency per rung drops by the recognition penalty 1/phi.
proof idea
The declaration is a direct definition assigning the natural logarithm of phi to the rung slope.
why it matters
This definition supplies the slope parameter used by the GutenbergRichterCert structure and the gutenberg_richter_one_statement theorem, which force the empirical b-value to equal 1 via the change of variable from phi-rungs to Richter magnitudes. It connects the RS phi-ladder (self-similar fixed point from the forcing chain) to geophysical observations, realizing the prediction that b equals 1 exactly.
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