eruption_recurrence_one_statement
plain-language theorem explainer
The declaration packages the band (2.59, 2.63) for the VEI-adjacent recurrence ratio together with the power law for cumulative ratios over k steps and the base case at k=1. Geophysicists modeling volcanic recurrence on the phi-ladder would cite it to connect the recognition lattice to Smithsonian GVP statistics. The proof is a one-line term that directly assembles the three upstream results into a single conjunction.
Claim. Let $r = phi^2$. Then $2.59 < r < 2.63$, and for every natural number $k$ the cumulative recurrence ratio satisfies $C(k) = r^k$, with the base case $C(1) = r$.
background
The module models volcanic eruptions as a phi-rational ladder on the recognition lattice. Each VEI step corresponds to one octave in the J-cost impulse spectrum, so the recurrence-interval ratio between adjacent classes is defined as vei_step_ratio := phi^2. The cumulative ratio across k steps is defined as cumulative_ratio k := phi^(2*k), which is algebraically identical to (phi^2)^k. Upstream results supply the numerical band for the ratio, the factorization identity, and the one-step identity.
proof idea
The proof is a term-mode conjunction that applies vei_step_ratio_band for the interval, cumulative_ratio_factors (which unfolds both definitions and rewrites via pow_mul) for the general power law, and cumulative_ratio_one_step (which reduces to reflexivity) for the base case. No additional tactics are required.
why it matters
This theorem is the master certificate for Track E6 of the eruption-recurrence prediction. It supplies the concrete phi^2 ratio (approximately 2.618) required by the eight-tick octave plus gap-45 frustration, and it matches the empirical 2.5-2.7 band reported for n >= 4 in the Smithsonian GVP catalog. The result closes the structural link between the self-similar fixed point phi and observable geophysical recurrence intervals.
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