vei_step_ratio
The definition sets the recurrence ratio between successive VEI classes equal to phi squared. Geologists modeling intervals from the Smithsonian GVP catalog cite it when deriving cumulative factors across k VEI steps. It enters as a direct abbreviation of the golden ratio power.
claimThe recurrence ratio $r$ between adjacent VEI classes satisfies $r = phi^2$, where $phi$ is the golden ratio.
background
The module treats volcanic eruptions as clustering on a phi-rational recurrence ladder. Each VEI step corresponds to one octave on the recognition lattice J-cost impulse spectrum. The phi squared ratio is the canonical two-phi-steps-per-octave structure that follows from the eight-tick lattice plus gap-45 frustration on long-period geophysical events.
proof idea
The declaration is introduced as a direct definition equating the ratio to phi raised to the power two. No lemmas are invoked; the abbreviation is unfolded in downstream statements such as cumulative_ratio_one_step and cumulative_ratio_factors.
why it matters in Recognition Science
This supplies the step ratio inside EruptionRecurrenceCert and feeds eruption_recurrence_one_statement, which asserts the band membership together with the cumulative identity. It realizes the module prediction that adjacent-VEI recurrence ratios equal phi squared, consistent with the eight-tick octave (T7) in the unified forcing chain. The cumulative ratio across k steps then becomes phi to the power 2k.
scope and limits
- Does not verify the numerical band 2.59 to 2.63; that is proved separately in vei_step_ratio_band.
- Does not fit or process raw Smithsonian GVP recurrence data.
- Does not extend the ratio to non-adjacent VEI classes without the cumulative_ratio definition.
- Does not claim applicability outside the volcanic recurrence ladder model.
Lean usage
unfold cumulative_ratio vei_step_ratio; rw [← pow_mul]formal statement (Lean)
52def vei_step_ratio : ℝ := phi ^ 2
proof body
Definition body.
53