vei_step_ratio_pos
plain-language theorem explainer
The theorem establishes that the recurrence-interval ratio between adjacent Volcanic Explosivity Index classes is strictly positive. Geophysicists fitting eruption catalogs to the phi-ladder model cite it to ensure the scaling is well-defined before band checks. The proof is a one-line term that unfolds the ratio definition and invokes positivity of powers of the positive golden ratio.
Claim. $0 < phi^2$, where $phi$ is the golden ratio fixed point of the recognition composition law.
background
The EruptionRecurrenceLadder module models volcanic eruptions as clustering on a phi-rational recurrence ladder derived from the eight-tick recognition lattice. Each VEI step represents one octave, so the ratio between successive classes is defined as phi squared. This rests on the local definition of the step ratio together with the band operation from PreLogicalCost, which performs arithmetic conjunction on stable states via 0/1 multiplication. The module documentation states that the ratio approximates 2.618 and lies in the empirical band (2.5, 2.7) for n greater than or equal to 4.
proof idea
The term proof unfolds the ratio definition to phi squared and applies the lemma establishing that powers of positive reals are positive.
why it matters
This positivity result supplies the step_ratio_pos field inside the EruptionRecurrenceCert structure that certifies the full recurrence model. It fills the T7 eight-tick octave step in the forcing chain by guaranteeing the ratio is positive before the numerical band is checked. The framework landmark is the forcing of phi as the self-similar fixed point, with the open question being direct comparison against the complete Smithsonian GVP database.
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