Astrophysics Tidal Locking From Phi Resonance Tidal Locking One Statement

The one-statement theorem

Tidal locking is the reason the same face of the Moon always points at Earth. The Moon rotates once per orbit, a 1:1 spin-orbit resonance. Mercury spins three times for every two orbits around the Sun, a 3:2 resonance. Venus rotates slowly backward, about four rotations for every one orbit, a 4:1 retrograde resonance. Planetary scientists have long treated these integer ratios as accidents of tidal evolution, the outcome of eons of gravitational friction.

In Recognition Science, the framework proposes that these ratios are not accidents. It models the spin-orbit ratio as a point on a phase manifold, and evaluates it with a cost function, a measure of how expensive a given ratio is to recognize. The golden ratio φ, about 1.618, is the unique self-similar scaling forced by the framework's axioms. The framework's library, a machine-checked collection of formal theorems, proves that the three canonical ratios sit near powers of φ: the Moon's 1:1 ratio is exactly at cost zero, Mercury's 3:2 sits within 0.11 to 0.13 of φ, and Venus's 4:1 sits within 0.22 to 0.24 of φ³, which is about 4.236.

The declaration tidal_locking_one_statement bundles these three numerical facts into a single theorem. It states that the Moon-Earth ratio equals 1, that its cost is zero, that the Mercury deviation from φ is in the band (0.11, 0.13), and that the Venus deviation from φ³ is in the band (0.22, 0.24). These bands are not arbitrary; they are the framework's canonical golden-section cost band. The theorem is a structural statement about where the ratios sit, not a derivation of tidal locking itself.

What the theorem does not claim is just as important. It does not prove that tidal locking happens, nor does it explain the mechanism by which a planet reaches a resonant ratio. It offers no account of the tidal forces, orbital evolution, or timescales involved. The physical bridge from recognition cost to actual spin-orbit dynamics remains open. The theorem is a pattern-matching result: it shows the observed ratios fit the φ-ladder within the stated bands, and it names a falsifier. A confirmed resonance deviating from a φ power by more than about 0.118 would break the prediction. The Pluto-Charon 1:1 and the Galilean satellites' 1:2:4 resonance are cited as within the band, but the theorem itself only covers the three named bodies.

/-- Moon-Earth ratio is exactly 1. -/
theorem moon_resonance_eq : moon_resonance_pq = 1 := rfl

/-- 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

/-- 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

/-- 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

/-- **TIDAL LOCKING FROM φ-RESONANCE: ONE-STATEMENT THEOREM
(Track AS6).**

The three canonical inner-Solar-System spin-orbit resonance ratios
sit at φ-rational positions:

- Moon-Earth 1:1 ratio at J-cost zero (trivial 1:1 resonance).
- Mercury-Sun 3:2 within `J(φ) ∈ (0.11, 0.13)` of `φ`.
- Venus-Sun 4:1 (retrograde) within `1/φ² ∈ (0.22, 0.24)` of `φ³`.

All deviations sit at the canonical golden-section J-cost band,
forced by `Constants.phi` arithmetic. -/
theorem tidal_locking_one_statement :
    -- (1) Moon-Earth 1:1.
    moon_resonance_pq = 1 ∧
    -- (2) Moon at J-cost zero.
    Cost.Jcost moon_resonance_pq = 0 ∧
    -- (3) Mercury deviation in J(φ) band.
    (0.11 < phi - mercury_resonance_pq ∧
      phi - mercury_resonance_pq < 0.13) ∧
    -- (4) Venus deviation in 1/φ² band.
    (0.22 < phi_cubed - venus_resonance_pq ∧
      phi_cubed - venus_resonance_pq < 0.24) :=
  ⟨rfl,
   moon_J_cost_zero,
   mercury_deviation_in_J_phi_band,
   venus_deviation_in_inverse_phi_sq_band⟩

What this page does not claim

The theorem does not prove that tidal locking occurs or explain its mechanism. The theorem does not derive the spin-orbit ratios from the framework's axioms; it only verifies their positions against φ powers. The physical bridge from recognition cost to spin-orbit dynamics is not established by this declaration.

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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