volcanicForcingAsJCostImpulseCert
plain-language theorem explainer
Volcanic forcing receives a certificate as a nonnegative J-cost impulse on the climate cascade whose period is fixed at 8 for D=3. Recognition Science geologists cite the certificate to replace ad-hoc VEI exponents with derivations from the eight-tick minimal cover and the J-cost functional. The definition is a structure instance that directly wires six sibling lemmas on period equality, minimality, impulse nonnegativity, saturation zero, Pinatubo positivity, and monotonic decay.
Claim. The volcanic forcing certificate asserts that the climate octave period equals 8, that any surjective cover of the 3-pattern by a finite set of size $T$ requires $T$ at least 8, that the per-octave impulse is nonnegative for every positive VEI, vanishes exactly at the saturation VEI, is strictly positive for VEI equal to 6, and decreases monotonically over successive octaves.
background
In Recognition Science the climate response to a volcanic eruption is modeled as an instantaneous source applied to the eight-tick attractor whose period is forced by T7 at spatial dimension 3. The impulse magnitude after one octave of relaxation is the product of the J-cost of the normalized VEI ratio and the period 8, with the J-cost taken from the multiplicative recognizer and shown nonnegative by Cost.Jcost_nonneg. The module replaces an earlier placeholder exponent with this derivation, pulling the period from Patterns.eight_tick_min and the nonnegativity from the ledger factorization upstream.
proof idea
The definition is a one-line structure instance that populates the six fields of VolcanicForcingAsJCostImpulseCert by direct reference to the sibling lemmas octavePeriod_eq_eight, octavePeriod_is_minimal_cover, impulse_per_octave_nonneg, impulse_at_saturation, impulse_pinatubo_pos, and impulse_after_octaves_mono_decay.
why it matters
This supplies the master certificate for volcanic forcing in the geology track, closing the path from the unified forcing chain (T7 eight-tick octave at D=3) to empirical VEI ordering. It feeds any downstream climate-cascade calculation that needs a J-cost impulse grounded in Cost.Jcost and Patterns.eight_tick_min. No open scaffolding remains in the module; the certificate is fully discharged by the referenced siblings.
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