Astrophysics Exoplanet Habitability Moon Mass Ratio In Band

The moon-mass band

The habitability of an exoplanet depends on more than its star. A large companion moon can stabilize a planet's axial tilt and tides, and the mass ratio between the moon and its planet is one factor in that stability. In the Recognition Science framework, the declaration moonMassRatioInBand defines a specific interval for this ratio: a planet's companion-moon mass ratio is "in band" when it lies between φ^-7 and φ^-6, where φ is the golden ratio, approximately 1.618. Numerically, that band runs from about 0.0344 to 0.0557, meaning the moon's mass is roughly between 3.4 and 5.6 percent of the planet's mass.

This band is not arbitrary. It comes from the framework's broader claim that the golden ratio structures physical scales, and it sits inside a composite habitability score that also includes an orbital resonance period and an eccentricity penalty. The declaration itself is a definition, not a theorem: it establishes what the band is, and it is used as a predicate in the framework's habitability model. The framework's machine-checked library of formal theorems proves supporting facts about that model, such as the eccentricity penalty being zero at zero eccentricity and the habitability score being exactly 1 at that same point. These are the proved results. The band itself is a modeling choice.

A striking consequence follows directly from the numbers. Earth's Moon has a mass ratio to Earth of about 1/81.3, roughly 0.0123. That value falls below the band's lower edge of 0.0344. The framework's own documentation states this plainly: the Earth-Moon ratio is below the predicted habitability band. The declaration exposes the predicate without claiming Earth lies inside it. The framework's prediction is that the maximally habitable companion-mass ratio sits between those two φ-rungs, and Earth's Moon is described as a borderline-favorable case, not a member of the band.

What the declaration does not claim is therefore as important as what it defines. It does not assert that any real exoplanet system, including our own, falls inside the band. It does not claim that a moon in the band guarantees habitability, nor that a moon outside it precludes it. The declaration is a formal predicate used within a specific model; it carries no empirical evidence about actual exoplanets. The framework's library proves facts about the model's internal consistency, not about the distribution of moons in the galaxy.

/-- Moon-mass-ratio is in the habitability band `[φ^{-7}, φ^{-6}]`. -/
def moonMassRatioInBand (ratio : ℝ) : Prop :=
  phi ^ (-(7 : ℤ)) ≤ ratio ∧ ratio ≤ phi ^ (-(6 : ℤ))

/-- Moon-mass-ratio is in the habitability band `[φ^{-7}, φ^{-6}]`. -/
def moonMassRatioInBand (ratio : ℝ) : Prop :=
  phi ^ (-(7 : ℤ)) ≤ ratio ∧ ratio ≤ phi ^ (-(6 : ℤ))

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

/-- 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

/-- Moon-mass-ratio is in the habitability band `[φ^{-7}, φ^{-6}]`. -/
def moonMassRatioInBand (ratio : ℝ) : Prop :=
  phi ^ (-(7 : ℤ)) ≤ ratio ∧ ratio ≤ phi ^ (-(6 : ℤ))

What this page does not claim

The declaration does not claim Earth's Moon lies inside the band. The declaration does not claim any real exoplanet system falls inside the band. The declaration does not prove that a moon in the band guarantees habitability.

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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