impulse_per_octave_nonneg
plain-language theorem explainer
The theorem establishes that the per-octave J-cost impulse from any positive VEI eruption is non-negative. Climate modelers using the Recognition Science eight-tick cascade cite it to bound the forcing term in the volcanic impulse ledger. The proof is a short tactic sequence that unfolds the product definition and combines Jcost non-negativity with octave period non-negativity via multiplication.
Claim. For any $vei > 0$, the per-octave impulse satisfies $0 ≤ J(veiRatio(vei)) · octavePeriod$, where $J$ is the J-cost function, $veiRatio$ is the eruption magnitude normalized to the saturation threshold, and $octavePeriod$ is the eight-tick period.
background
This declaration belongs to the Geology module that recasts volcanic forcing as a J-cost impulse on the eight-tick climate attractor. The module replaces an earlier placeholder exponent with an expression drawn from the Recognition Composition Law and the T7 octave forced at D = 3. The octave period is the constant 8 ticks, the minimal complete cover for three spatial dimensions. The J-cost function satisfies J(x) ≥ 0 for x > 0 by the AM-GM inequality, as recorded in the Cost module lemmas Jcost_nonneg and JcostCore.Jcost_nonneg. The impulse_per_octave definition is the product of J applied to the VEI ratio and this period.
proof idea
The proof unfolds impulse_per_octave to expose the product of Jcost(veiRatio vei) and octavePeriod. It applies Jcost_nonneg to the positive argument supplied by veiRatio_pos h. A separate have block casts octavePeriod_pos to a real-number inequality. The final step invokes mul_nonneg on the two non-negative factors.
why it matters
The result supplies the impulse_nonneg field required by the VolcanicForcingAsJCostImpulseCert structure that certifies the full replacement of the v4 skeleton. It confirms the impulse vanishes at saturation (VEI ratio = 1) and stays non-negative elsewhere, consistent with the T7 eight-tick octave and the non-negativity of J-cost throughout the Recognition Science chain. The parent certificate also records the minimal-cover property of the period and the ordering between Pinatubo-class and Tambora-class events.
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