releaseRate_zero
Engineers modeling J-cost optimal CO2 release schedules for Mars terraforming rely on the fact that the release rate at the initial rung equals the base rate. This anchors the geometric progression used for cumulative release calculations. The proof is a one-line wrapper that unfolds the definition and simplifies.
claimLet $r_0$ be the initial release rate and let $r(n) := r_0 · ϕ^n$ for $n ∈ ℕ$. Then $r(0) = r_0$.
background
This theorem appears in the Mars Terraforming J-Cost Schedule module, which derives an engineering plan for releasing CO2 on Mars to follow a J-cost optimal path under the Recognition Science framework. The module sets the initial release rate to 1 and defines the release rate at φ-rung n as the initial rate multiplied by phi to the power n, corresponding to self-similar scaling with 45-year gaps per rung. The module documentation states that the total CO2 released over n rungs is the geometric sum given by the initial rate times 45 times (phi to n minus 1) divided by (phi minus 1).
proof idea
The proof is a one-line wrapper that unfolds the definition of the release rate function and applies simplification to verify the equality at rung zero.
why it matters in Recognition Science
This base case populates the rate_zero field of the MarsAtmosphereJCostScheduleCert structure, which bundles the zero, positivity, successor, and monotonicity properties for the release schedule. It enables the geometric sum identity for cumulative release in the J-cost framework. The construction relies on the phi self-similar fixed point from the forcing chain.
scope and limits
- Does not establish positivity of the release rate for positive rungs.
- Does not compute the cumulative CO2 release over multiple rungs.
- Does not incorporate the 45-year time gap between rungs.
- Does not prove strict monotonicity of the release rate.
Lean usage
def cert : MarsAtmosphereJCostScheduleCert where rate_zero := releaseRate_zero rate_pos := releaseRate_pos rate_succ := releaseRate_succ rate_strict_mono := releaseRate_strict_mono cumulative_zero := cumulativeRelease_zeroformal statement (Lean)
43theorem releaseRate_zero : releaseRate 0 = r_0 := by
proof body
One-line wrapper that applies unfold.
44 unfold releaseRate; simp
45