mars_terraform_one_statement
plain-language theorem explainer
The declaration asserts that the Mars CO2 release rate forms a geometric ladder with common ratio phi, is strictly monotonic, and induces a strictly monotonic cumulative release function. Terraforming engineers would reference it when optimizing J-cost schedules under the Recognition Science phi-ladder. The proof is a one-line term that directly assembles the successor relation for the rate together with the two strict-monotonicity lemmas on the rate and its partial sums.
Claim. Let $r(n) := r_0 phi^n$ be the release rate at rung $n$. Then $r(n+1) = r(n) phi$ for every natural number $n$, the function $r$ is strictly increasing, and the cumulative release $C(n) := sum_{k=0}^{n-1} r(k)$ is strictly increasing.
background
The module defines the release rate at rung $n$ by $r(n) = r_0 phi^n$, where $r_0$ is a positive base rate and $phi$ is the Recognition Science self-similar fixed point. Cumulative release is the partial geometric sum $C(n) = sum_{k < n} r(k)$. The local setting is an engineering derivation of a J-cost-optimal CO2 release schedule for Mars terraforming, with 45-year gaps per phi-rung and total release given by the closed-form geometric sum $r_0 * 45 * (phi^n - 1)/(phi - 1)$ (MODULE_DOC).
proof idea
The proof is a one-line term that packages three sibling results: the successor lemma establishing the exact ratio phi between adjacent rungs, the strict monotonicity of the release rate (proved by unfolding and applying pow_lt_pow_right), and the strict monotonicity of the cumulative sum (proved by induction on the difference m - n).
why it matters
This packages the core algebraic and ordering properties of the phi-ladder release schedule inside the Recognition Science framework (T6 self-similar fixed point, eight-tick octave). It supplies the monotonicity facts required for any quantitative model of J-cost-optimal terraforming. No downstream theorems yet depend on it; the module doc states the concrete falsifier as any deployed schedule diverging by a factor of two from the phi scaling at any rung.
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