vei_step_ratio_band
plain-language theorem explainer
The theorem proves that the recurrence-interval ratio between successive Volcanic Explosivity Index classes equals φ² and falls strictly inside (2.59, 2.63). Geophysicists testing Recognition Science predictions against Smithsonian GVP data would cite this bound when validating the φ-rational ladder. The proof unfolds the definition to φ², imports the tight bounds phi > 1.61 and phi < 1.62, then closes both sides via pow_lt_pow_left₀ and nlinarith.
Claim. $2.59 < φ² ∧ φ² < 2.63$, where φ denotes the golden ratio.
background
The module treats volcanic recurrence intervals as steps on the recognition lattice. The step ratio between adjacent VEI classes is defined by vei_step_ratio := φ², the two-φ-step structure per octave on the eight-tick lattice. Upstream lemmas supply the numerical bounds φ > 1.61 and φ < 1.62 together with positivity of φ. The module setting states that T_VEI(n+1)/T_VEI(n) = φ², approximately 2.618 and consistent with the empirical 2.5–2.7 ratio for Smithsonian GVP classes n ≥ 4.
proof idea
The term proof unfolds vei_step_ratio to φ². It obtains the lower bound from Constants.phi_gt_onePointSixOne and the upper bound from phi_lt_onePointSixTwo, plus phi_pos. The lower inequality applies pow_lt_pow_left₀ to (1.61)² < φ² and finishes with nlinarith. The upper inequality proceeds symmetrically with φ² < (1.62)².
why it matters
This supplies the numerical band required by eruptionRecurrenceCert and the parent theorem eruption_recurrence_one_statement, which packages the band with the cumulative ratio φ^(2k). It realizes the Geology track E6 prediction that adjacent-VEI ratios cluster at φ² ∈ (2.59, 2.63), drawn from the eight-tick octave (T7) and the self-similar fixed point φ (T6). The result closes one falsifiable statement for the Smithsonian catalog.
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