impulse_after_octaves_succ
plain-language theorem explainer
The recurrence shows that the impulse magnitude after n+1 octaves equals the value after n octaves scaled by the reciprocal of the golden ratio. Climate modelers tracking stratospheric aerosol decay on the eight-tick cascade would cite this to confirm consistent geometric reduction across relaxation steps. The argument follows directly from unfolding the geometric definition and applying the successor rule for exponentiation.
Claim. Let $I(v,n)$ be the impulse after $n$ octaves for VEI ratio $v$. Then $I(v,n+1)=I(v,n)/φ$, where $φ$ is the golden ratio.
background
In this module volcanic eruptions are treated as instantaneous σ-sources whose J-cost contribution is evaluated on the eight-tick diurnal-seasonal attractor forced by spatial dimension three. The impulse after n octaves is obtained by dividing the single-octave J-cost term by φ^n, replacing an earlier placeholder exponent with a derivation from the minimal complete-cover period. The upstream definition supplies the base geometric form impulse_after_octaves vei n = impulse_per_octave vei / (φ^n).
proof idea
The proof unfolds the definition of the impulse function, rewrites the power via the successor identity, and simplifies the resulting expression by field arithmetic. It is a direct algebraic reduction requiring no external lemmas beyond the definition itself.
why it matters
The recurrence anchors geometric decay of volcanic impulses across the phi-ladder, ensuring the eight-tick exponent derived from the forcing chain remains consistent under iteration. It supports the ordering that sub-saturation events produce strictly smaller cumulative forcing than saturation-class eruptions after any number of octaves. The result closes the replacement of the v4 skeleton with a derivation tied to the Recognition constants and the T7 octave period.
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