IndisputableMonolith.Geology.EarthquakeScalingFromJCost
This module applies the J-cost functional from Recognition Science to derive scaling relations for earthquake event rates with magnitude. It defines eventRateAtMagnitude along with positivity and ratio lemmas, then certifies the overall scaling via EarthquakeScalingCert. The structure imports the RS time quantum and builds local definitions on the phi-ladder. Geophysicists extending RS to seismic statistics would cite these relations for magnitude-frequency derivations.
claimThe module introduces the event rate function $r(M)$ at magnitude $M$ derived from the J-cost $J(x) = (x + x^{-1})/2 - 1$ and the certification theorem asserting $r(M)$ satisfies positivity and ratio scaling on the phi-ladder.
background
The module resides in the geology domain of Recognition Science and imports the fundamental time quantum τ₀ = 1 tick from Constants. It introduces event rate definitions that apply the J-cost to model defect distributions for seismic energy release, using the phi-ladder for rung-based magnitude steps. The setting connects to the Recognition Composition Law for composing rates across magnitude intervals.
proof idea
This is a definition module, no proofs. It consists of the core definition eventRateAtMagnitude together with the supporting lemmas eventRate_pos and eventRate_ratio, followed by the certification EarthquakeScalingCert.
why it matters in Recognition Science
This module supplies the certified scaling laws that feed into broader RS applications for natural phenomena. It realizes the T5 J-uniqueness and T6 phi fixed point steps of the forcing chain for earthquake magnitudes, providing the geology-specific closure after the eight-tick octave and D = 3.
scope and limits
- Does not incorporate tectonic plate motion or fault geometry.
- Does not generate time-series predictions for individual events.
- Does not compare derived rates against catalog data.
- Does not extend beyond the J-cost scaling to stochastic triggering.