releaseRate_succ
Engineers modeling Mars CO2 release schedules rely on the fact that the release rate at each successive phi-rung multiplies exactly by the golden ratio phi. This recurrence is cited inside the schedule certification and the one-statement terraforming summary. The proof is a one-line algebraic reduction that unfolds the explicit power definition and applies the successor exponent rule.
claimLet $r(n) = r_0 phi^n$ denote the release rate at rung $n$. Then for every natural number $n$, $r(n+1) = r(n) cdot phi$.
background
In the Mars Terraforming J-Cost Schedule the release rate is defined by the closed form $r(n) = r_0 phi^n$, where $r_0$ is the base rate and phi is the golden ratio. This implements the geometric progression required by the J-cost optimal schedule with a 45-year gap per rung. The upstream releaseRate definition supplies the explicit formula that the present theorem manipulates.
proof idea
The term proof unfolds the releaseRate definition to expose the power, rewrites the exponent via the successor rule, and closes the equality with ring arithmetic.
why it matters in Recognition Science
The result supplies the adjacent-rung ratio to the MarsAtmosphereJCostScheduleCert and to the mars_terraform_one_statement theorem. It realizes the self-similar scaling of phi inside the engineering track. The module doc-comment states that the full schedule is theorem-status once this recurrence and the positivity and monotonicity properties are in place.
scope and limits
- Does not fix the numerical value or units of the base rate r_0.
- Does not treat continuous-time limits or diffusion processes.
- Does not incorporate atmospheric loss or feedback terms.
formal statement (Lean)
56theorem releaseRate_succ (n : ℕ) :
57 releaseRate (n + 1) = releaseRate n * phi := by
proof body
Term-mode proof.
58 unfold releaseRate
59 rw [pow_succ]; ring
60
61/-! ## §2. Cumulative release -/
62
63/-- Cumulative CO₂ released over `n` rungs (geometric partial sum). -/