Astrophysics Tidal Locking From Phi Resonance Mercury Deviation In J Phi Band

The Mercury deviation band

Mercury rotates on its axis three times for every two orbits it makes around the Sun. This 3:2 spin-orbit resonance is a well-known fact of Solar System astronomy, and the ratio 3/2 equals 1.5. The golden ratio φ, approximately 1.618, is the number that satisfies φ² = φ + 1. The difference between φ and 3/2 is about 0.118, a value that falls inside the open interval (0.11, 0.13).

The Recognition Science framework, which models physical structure through a forced cost function J(x) = (x + 1/x)/2 - 1, treats this numerical proximity as a structural theorem. Its machine-checked library of formal theorems contains a declaration, mercury_deviation_in_J_phi_band, that proves the difference φ - 3/2 lies strictly between 0.11 and 0.13. The framework interprets this band as the canonical golden-section J-cost ceiling, the maximum deviation below which a resonance remains stable on its recognition lattice. The same library proves the Moon's 1:1 ratio sits at J-cost zero and Venus's 4:1 ratio lies within a related band of the cube of φ.

What the declaration does not claim is that this proximity explains why Mercury is locked in the 3:2 resonance. It establishes a numerical fact about a ratio, derived from the definition of φ and basic real arithmetic. It does not derive the resonance from tidal dynamics, nor does it predict the existence of any new resonance. The falsifier for the broader φ-resonance prediction is stated plainly: a confirmed Solar System spin-orbit resonance deviating from a power of φ by more than about 0.118 would falsify the prediction. The Pluto-Charon 1:1 resonance and the Galilean satellites' 1:2:4 resonance are cited as lying within the predicted band, but these are checks, not consequences of the theorem.

/-- The deviation `|φ - 3/2|` is positive and below `J(φ)`-band ceiling. -/
theorem mercury_deviation_in_J_phi_band :
    0.11 < phi - mercury_resonance_pq ∧
    phi - mercury_resonance_pq < 0.13 := by
  unfold mercury_resonance_pq
  have h1 := phi_gt_onePointSixOne
  have h2 := phi_lt_onePointSixTwo
  refine ⟨?_, ?_⟩ <;> linarith

/-- Moon sits at J-cost zero (trivial resonance). -/
theorem moon_J_cost_zero : Cost.Jcost moon_resonance_pq = 0 := by
  unfold moon_resonance_pq
  exact Cost.Jcost_unit0

/-- Venus deviation `|venus - phi^3|` is positive (slow retrograde
sits below the φ³ ratio). -/
theorem venus_deviation_in_inverse_phi_sq_band :
    0.22 < phi_cubed - venus_resonance_pq ∧
    phi_cubed - venus_resonance_pq < 0.24 := by
  unfold venus_resonance_pq
  have h := phi_cubed_band
  refine ⟨?_, ?_⟩ <;> linarith

What this page does not claim

The declaration does not explain the dynamical origin of Mercury's 3:2 resonance. It does not predict any new spin-orbit resonance. It does not claim that all Solar System resonances are exactly at powers of φ, only within a stated 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/TidalLockingFromPhiResonance.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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