Astrophysics Exoplanet Habitability Eccentricity Penalty Nonneg

The eccentricity penalty

In orbital mechanics, eccentricity measures how much a planet's orbit deviates from a perfect circle, on a scale from zero (circular) to near one (highly elongated). Astronomers have long treated high eccentricity as a strike against habitability, because it causes wild temperature swings as the planet swings closer to and farther from its star. The Recognition Science framework formalizes this intuition as a mathematical penalty: a planet's habitability score is reduced by an amount that depends on its eccentricity.

The framework defines the eccentricity penalty as J(1 + e), where e is the orbital eccentricity and J is the framework's fundamental cost function, which is zero when its input is one. Since the penalty is added to the denominator of the habitability score, a higher penalty means a lower score. The declaration eccentricity_penalty_nonneg proves, in the framework's machine-checked library of formal theorems, that this penalty is always greater than or equal to zero, provided the eccentricity is greater than negative one, which every physical orbit satisfies. A corollary is that the penalty vanishes exactly at zero eccentricity, so a perfectly circular orbit receives no penalty at all.

This non-negativity result is a mathematical theorem, not an empirical finding. It does not claim that real planets with circular orbits are habitable, nor does it assert that any particular observed exoplanet actually has a high or low habitability score. The framework's own documentation is explicit that the Earth-Moon mass ratio, for example, falls below the band the framework associates with maximum habitability. The theorem simply guarantees that the penalty term never rewards a planet for having a non-circular orbit, which is a consistency condition on the framework's scoring model.

In Recognition Science, this penalty is one component of a composite habitability score that also includes a resonance term based on a period of about five weeks and a bonus for a companion moon of a certain relative mass. The non-negativity theorem ensures that the eccentricity component behaves sensibly: it can only push the score downward from its maximum value of one at zero eccentricity. This is a small but necessary piece of the framework's larger project of deriving physical structure from its foundational cost function, and it shows how a general mathematical result about that function gets applied to a concrete astrophysical setting.

/-- The eccentricity penalty is non-negative (as long as `1 + e > 0`). -/
theorem eccentricity_penalty_nonneg (e : ℝ) (h : -1 < e) :
    0 ≤ eccentricity_penalty e := by
  unfold eccentricity_penalty
  exact Jcost_nonneg (by linarith)

/-- At zero eccentricity, penalty vanishes. -/
theorem eccentricity_penalty_zero :
    eccentricity_penalty 0 = 0 := by
  unfold eccentricity_penalty
  simp [Jcost_unit0]

/-- The eccentricity penalty: `J(1 + e)`.  Zero at `e = 0`. -/
def eccentricity_penalty (e : ℝ) : ℝ := Jcost (1 + e)

What this page does not claim

The theorem does not claim that any real exoplanet is habitable or that Earth's Moon lies inside the framework's predicted habitability band. The non-negativity result does not establish that the habitability score is a valid predictor of actual planetary habitability. The declaration does not prove that the eccentricity penalty is the correct or only way to model eccentricity's effect on habitability.

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Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

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expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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