eruptionRecurrenceCert
Eruption recurrence ratios between successive VEI classes are certified to equal φ² in the interval (2.59, 2.63). Recognition theorists working on the geology track would cite this certificate when invoking the φ-ladder prediction for volcanic intervals. The definition is a direct record constructor that assembles six sibling lemmas on positivity, band membership, and power factorization.
claimThe certificate asserts that the VEI step ratio $r = φ^2$ satisfies $0 < r$ and $2.59 < r < 2.63$, the cumulative ratio across $k$ steps equals $φ^{2k}$, this equals $r^k$ for all $k$, and the base case $k=1$ recovers $r$.
background
The module models volcanic recurrence on the recognition lattice, with each VEI step as one octave in the J-cost impulse spectrum. It defines cumulative_ratio k := φ^(2k) and the structure EruptionRecurrenceCert with fields for step-ratio positivity, the numerical band (2.59, 2.63), cumulative positivity, factorization into powers of the step ratio, and the one-step equality. Upstream lemmas establish vei_step_ratio = φ² together with its positivity and band membership, plus the power identities for the cumulative function.
proof idea
This is a definition that constructs the certificate by direct field assignment: step_ratio_pos is taken from vei_step_ratio_pos, the band from vei_step_ratio_band, cumulative_pos from cumulative_ratio_pos, factors from cumulative_ratio_factors, and the one-step equality from cumulative_ratio_one_step.
why it matters in Recognition Science
This supplies the master certificate for Track E6, realizing the structural prediction that adjacent-VEI recurrence ratios cluster at φ² and cumulatives at φ^(2k). The module doc ties the ratio to the eight-tick octave plus gap-45 frustration on long-period events, with the explicit falsifier that any median GVP ratio for n ≥ 4 falls outside (2.5, 2.7).
scope and limits
- Does not include empirical verification against Smithsonian GVP data.
- Does not derive the ratio from the J-cost forcing chain T0-T8.
- Does not extend to non-adjacent VEI steps.
- Does not address recurrence for VEI classes below 4.
formal statement (Lean)
103def eruptionRecurrenceCert : EruptionRecurrenceCert where
104 step_ratio_pos := vei_step_ratio_pos
proof body
Definition body.
105 step_ratio_band := vei_step_ratio_band
106 cumulative_pos := cumulative_ratio_pos
107 cumulative_factors := cumulative_ratio_factors
108 one_step_eq := cumulative_ratio_one_step
109
110/-- **ERUPTION RECURRENCE ONE-STATEMENT.** Adjacent-VEI eruption
111recurrence ratios cluster at `φ² ∈ (2.59, 2.63)`; cumulative
112recurrence across `k` VEI steps equals `φ^(2k) = (φ²)^k`. -/