impulse_per_octave
plain-language theorem explainer
Geologists modeling volcanic forcing on climate cascades cite this definition to compute the impulse per eight-tick octave as the J-cost of the normalized VEI scaled by the fixed octave period. It replaces the earlier ad-hoc VEI^8 placeholder with a construction that vanishes exactly at saturation and remains non-negative for positive VEI. The definition is a direct multiplication of Cost.Jcost applied to veiRatio by the constant octavePeriod of eight ticks.
Claim. For volcanic explosivity index $v$, the per-octave impulse equals $J(r) · 8$, where $r = v / v_{sat}$ is the ratio to the saturation eruption (VEI 7), $J$ is the J-cost function, and the factor 8 is the octave period.
background
The Volcanic Forcing as J-Cost Impulse module frames eruptions as instantaneous σ-sources on the eight-tick climate attractor. The J-cost function measures deviation from the fixed point via the Recognition Composition Law, while veiRatio normalizes any VEI to the saturation value of 7. The octavePeriod is the constant 8 ticks, the minimal complete-cover period for D=3 forced by T7 and drawn from Patterns.eight_tick_min.
proof idea
One-line definition that applies Cost.Jcost to veiRatio vei and multiplies the result by the real number octavePeriod.
why it matters
This supplies the base impulse used by impulse_after_octaves for geometric decay over n octaves at rate 1/φ and by impulse_at_saturation to show the impulse vanishes at the J-cost minimum. It closes the v4 skeleton gap by deriving the exponent 8 from the T7 eight-tick octave rather than asserting it. The construction supports the empirical ordering of cooling between Pinatubo-class and Tambora-class events.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.