rungs_per_magnitude
plain-language theorem explainer
The definition supplies the exact conversion factor from phi-rungs on the recognition lattice to standard Richter magnitude units as the ratio of natural logs of 10 and phi. Seismologists and Recognition Science researchers cite it when mapping the native frequency drop of 1/phi per rung onto the base-10 magnitude scale to recover the Gutenberg-Richter slope. It is a direct real-number definition with no reduction steps or lemmas.
Claim. The number of φ-rungs per Richter magnitude unit is given by the exact ratio $R = (ln 10) / (ln φ) ≈ 4.78$.
background
In the Recognition Science treatment of earthquake statistics the Gutenberg-Richter law is read as a direct consequence of stress-drop events on the phi-ladder of the J-cost lattice. Each rung is one phi-step in recognition cost and the frequency per rung falls by the fixed factor 1/phi. Standard Richter magnitude is measured in base-10 units of energy release, so the change of variable from rung count r to magnitude M requires the conversion factor R such that one magnitude unit spans exactly R rungs.
proof idea
The declaration is a one-line definition that directly sets the ratio of the natural logarithm of 10 to the natural logarithm of phi.
why it matters
This definition supplies the change-of-base constant that forces the empirical Gutenberg-Richter b-value to equal 1. It is used to define richter_b_value and is invoked in the identity rungs_per_magnitude_times_rung_slope and in the packaged statement gutenberg_richter_one_statement. The parent result shows that the RS-native rung slope ln phi together with R recovers the observed b = 1 inside the Aki window, closing the loop from the phi fixed point to the geology prediction.
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