cumulative_ratio
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.
claimThe 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 in Recognition Science
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.
scope and limits
- Does not compute numerical values for specific k.
- Does not incorporate empirical data from the GVP catalog.
- Does not prove the band bounds on the step ratio.
- Does not address non-adjacent VEI classes.
formal statement (Lean)
77def cumulative_ratio (k : ℕ) : ℝ := phi ^ (2 * k)
proof body
Definition body.
78