impulse_after_octaves_mono_decay
plain-language theorem explainer
Monotonic decay of the post-eruption J-cost impulse holds over the phi-ladder: larger octave count yields strictly smaller remaining impulse when the per-octave term is positive. Geologists extending the eight-tick climate cascade to volcanic forcing cite this ordering to rank cooling magnitudes. The term proof reduces the claim to the monotonicity of phi-powers for phi >= 1 together with division by a positive quantity.
Claim. Let $I(k)$ be the impulse remaining after $k$ octaves for eruption magnitude VEI. If $0 <$ VEI, $n,m$ are natural numbers with $nleq m$, and the per-octave impulse is positive, then $I(m)leq I(n)$.
background
The module replaces an earlier placeholder exponent 8 with a derivation from Patterns.eight_tick_min (T7) and Cost.Jcost. An eruption supplies an instantaneous sigma-source to the eight-tick attractor; the impulse after one octave is J(VEI_ratio) times period, where period equals 2^D at D=3. The function impulse_after_octaves vei k is obtained by dividing the per-octave term by phi^k, placing the decay on the phi-ladder forced by T5 J-uniqueness. Constants.phi_ge_one and Constants.phi_pos supply the required inequalities on phi.
proof idea
Term-mode proof. Unfold impulse_after_octaves, obtain 0 < phi and 1 <= phi from Constants, establish phi^n <= phi^m by pow_le_pow_right₀, then apply div_le_div_of_nonneg_left to the positive per-octave impulse.
why it matters
Supplies clause 6 (decay_mono) of the Volcanic Forcing Master Certificate, which in turn populates volcanicForcingAsJCostImpulseCert. The certificate assembles the six structural facts needed for the geology depth extension in §XXIII.C. It rests on the eight-tick octave (T7) and the phi-ladder; the monotonicity guarantees that sub-saturation eruptions produce strictly smaller impulses than saturation events, reproducing the observed Tambora-Pinatubo ordering.
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