Astrophysics Exoplanet Habitability Habitability Score At Zero Ecc

The zero-eccentricity baseline

In orbital mechanics, eccentricity measures how much an orbit deviates from a perfect circle, ranging from 0 for a circle up to values approaching 1 for highly elongated paths. The Recognition Science framework, a system that derives physical structure from a forced cost function, builds a habitability score for exoplanets from three ingredients: resonance with an Earth-referenced period, an eccentricity penalty, and a companion-moon mass ratio bonus. Its declaration habitability_score_at_zero_ecc establishes one clean fact: when eccentricity e equals 0, the score equals exactly 1, the maximum possible value.

The proof is short and mechanical. The framework defines the eccentricity penalty as J(1 + e), where J is its central cost function. A separate theorem shows this penalty vanishes at e = 0, because J(1) = 0 by construction. The habitability score is then defined as 1 divided by (1 + penalty), so at zero eccentricity the score becomes 1 divided by 1, which is 1. The declaration simply assembles these pieces into a single theorem, checked by the machine.

What the declaration does not claim is more interesting than what it proves. It does not say that a circular orbit makes a planet habitable in any astronomical sense. It does not assert that Earth's Moon, with a mass ratio around 0.0123, sits inside the framework's predicted bonus band between approximately 0.0344 and 0.0557; the framework's own documentation explicitly says Earth lies below that band. The score is a formal construction, not a measured physical quantity. The declaration proves a property of a definition, not a fact about the universe.

The framework's master certificate bundles this result with three others: the resonant period is positive, the eccentricity penalty is non-negative for physically meaningful orbits, and the year unit is set to 1. Together these form a consistent package, but the package's physical predictions, such as the claim that maximally habitable companion-mass ratios fall on specific golden-ratio rungs, remain hypotheses for empirical testing, not theorems. The zero-eccentricity result is a mathematical anchor point, a baseline from which the framework can compare penalized orbits, but it carries no observational weight on its own.

/-- At `e = 0`, habitability score is exactly 1 (maximum). -/
theorem habitability_score_at_zero_ecc :
    habitability_score 0 = 1 := by
  unfold habitability_score
  rw [eccentricity_penalty_zero]
  norm_num

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

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

/-- Habitability score (additive form).  Higher is better.  Built
    from `1 / (1 + eccentricity_penalty)` and `T_RS resonance`. -/
def habitability_score (e : ℝ) : ℝ :=
  1 / (1 + eccentricity_penalty e)

What this page does not claim

The declaration does not claim that zero eccentricity makes a planet habitable in any observational sense. The declaration does not claim Earth's Moon lies inside the framework's predicted bonus mass-ratio band. The declaration does not claim the habitability score is a measured physical quantity rather than a formal construction.

Verify this page

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:

$ lake env lean IndisputableMonolith/Astrophysics/ExoplanetHabitability.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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