Astrophysics Exoplanet Habitability T Rs Period Pos

The Earth-resonant period

In the Recognition Science framework, an exoplanet's habitability is scored from three orbital properties: its period, its eccentricity, and the mass of any companion moon. The framework defines a reference period called the Earth-resonant period, written T_RS, as one year multiplied by the golden ratio cubed and divided by 45. With the golden ratio near 1.618, that works out to roughly 0.094 years, or about five weeks. The declaration T_RS_period_pos is a machine-checked proof that this number is greater than zero.

The proof itself is simple. The golden ratio is positive, so its cube is positive, and dividing by 45 keeps it positive. The declaration confirms that the definition produces a meaningful, non-zero period rather than something degenerate. It is a small sanity check, not a physical discovery. The larger habitability score combines this period with an eccentricity penalty and a moon-mass bonus, and the framework's library proves several properties of those pieces, such as the penalty being zero for a perfectly circular orbit.

What the declaration does not claim is more interesting. It does not say that a five-week orbital period is habitable, nor that any real exoplanet has such a period. It does not assert that the Earth-Moon system falls inside the framework's favored moon-mass band; in fact, the framework's own notes state that Earth's Moon ratio sits below that band. The declaration merely guarantees that the defined period is a positive real number, a necessary but modest foundation for the framework's more ambitious habitability scoring.

/-- Earth-Moon mass ratio is approximately `1/81.3 ≈ 0.0123`.
    `φ^{-7} ≈ 0.0344` and `φ^{-6} ≈ 0.0557`.  So Earth-Moon ratio
    is *below* the predicted habitability band — the prediction is
    that the maximally habitable companion-mass ratio is between
    these two φ-rungs.  Earth's Moon is a borderline-favorable case.
    We expose the predicate without claiming Earth lies inside. -/
theorem T_RS_period_pos : 0 < T_RS_period := by
  unfold T_RS_period year_dimensionless
  have hphi_pos : (0 : ℝ) < phi := phi_pos
  have h3pos : (0 : ℝ) < phi ^ (3 : ℕ) := pow_pos hphi_pos 3
  positivity

/-- The Earth-resonant period: `year · φ^3 / 45`.  At `φ ≈ 1.618`,
    `φ^3 ≈ 4.236`, so `T_RS ≈ 0.094` (about 1/10 year, i.e., ~5 weeks).
    This is the "tick" of orbital coherence. -/
def T_RS_period : ℝ := year_dimensionless * phi ^ (3 : ℕ) / 45

/-- 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 that any real exoplanet has a five-week orbital period. The declaration does not assert that a five-week period is habitable. The declaration does not place the Earth-Moon system inside the framework's favored moon-mass band.

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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