veiRatio_at_saturation
plain-language theorem explainer
The volcanic explosivity index ratio equals one exactly at the saturation threshold. Climate modelers applying Recognition Science to stratospheric aerosol forcing cite this when reducing the per-octave J-cost impulse to its minimum. The proof is a one-line algebraic reduction that unfolds the ratio definition and invokes the division identity on a positive saturation value.
Claim. Let $r$ denote the volcanic explosivity index ratio defined by division by the saturation magnitude. Then $r$ evaluated at the saturation magnitude equals 1.
background
In the volcanic forcing module the climate impulse per octave is the J-cost of the VEI ratio multiplied by the eight-tick period. The VEI ratio normalizes eruption magnitude against the saturation reference (Tambora-class VEI = 7), so the ratio is unity precisely when the eruption sits at that reference. This normalization is required for the J-cost to vanish at saturation, consistent with the reciprocal symmetry of the cost function.
proof idea
The proof is a direct term-mode reduction. It unfolds the definition of veiRatio, then applies the div_self lemma using the positivity fact vei_saturation_pos supplied by the sibling declaration vei_saturation_pos.
why it matters
This identity is invoked by the downstream theorem impulse_at_saturation to conclude that the per-octave J-cost impulse is exactly zero at saturation. It thereby closes the J-cost minimum property for the volcanic model and links the forcing calculation to the eight-tick octave forced by T7 at D = 3. The result supports the empirical ordering that Pinatubo-class (VEI 6) impulses are strictly smaller than the Tambora-class reference.
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