eccentricity_penalty_zero
plain-language theorem explainer
The theorem establishes that the eccentricity penalty vanishes at zero eccentricity in the RS exoplanet habitability construction. Researchers modeling planetary stability under Recognition Science resonance conditions would cite it to anchor the maximum score for circular orbits. The proof is a term-mode reduction that unfolds the penalty definition and simplifies directly via the J-cost unit lemma.
Claim. The eccentricity penalty vanishes at zero eccentricity: $J(1) = 0$.
background
The ExoplanetHabitability module builds a habitability score from three terms on the (orbital period, eccentricity, moon-mass ratio) triple: resonance with the RS period T_RS, the eccentricity penalty J(1 + e), and a stabilization bonus when the moon-mass ratio lies in [φ^{-7}, φ^{-6}]. The penalty is defined to be zero for circular orbits, matching the module's focus on Earth-like 1 AU systems with Moon-class satellites. The upstream lemma Jcost_unit0 states that Jcost(1) = 0 by direct simplification of the squared-ratio expression for Jcost.
proof idea
The term-mode proof unfolds the definition of eccentricity_penalty to reach Jcost(1) and then applies the Jcost_unit0 lemma, which asserts Jcost(1) = 0; the simplifier finishes the equality.
why it matters
This result supplies the ecc_penalty_zero field inside the master certificate exoplanetHabitabilityCert and supports the companion theorem that the full habitability score equals 1 at zero eccentricity. It closes the zero-eccentricity case in the RS framework's exoplanet application, confirming the penalty term aligns with the tightest habitability locus for circular orbits.
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