cumulative_ratio
plain-language theorem explainer
The definition sets the cumulative recurrence ratio across k VEI steps to phi raised to the power 2k. Volcanic recurrence modelers cite it to express the product of successive phi-squared ratios in closed form. The declaration is a direct abbreviation with no computational steps.
Claim. The cumulative recurrence ratio across $k$ VEI steps is $phi^{2k}$.
background
In the volcanic eruption recurrence ladder each VEI step corresponds to one octave on the recognition lattice J-cost impulse spectrum. The ratio between successive recurrence intervals is phi squared, the canonical two-phi-steps-per-octave structure derived from the eight-tick lattice plus gap-45 frustration on long-period geophysical events. The cumulative ratio for k steps is therefore the product of k such factors.
proof idea
One-line definition that directly assigns phi^(2 * k) to cumulative_ratio k.
why it matters
This definition supplies the closed-form expression for cumulative ratios that appears in the EruptionRecurrenceCert structure and the eruption_recurrence_one_statement theorem. It completes the structural prediction for VEI recurrence intervals clustering at phi^2, consistent with Smithsonian GVP data in the band (2.59, 2.63). It ties the geology track to the phi-ladder and eight-tick octave from the forcing chain.
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