Astrophysics Tidal Locking From Phi Resonance

The spin-orbit lock

Tidal locking is the gravitational process that slows a body's rotation until one side permanently faces its partner. The Moon shows it: it rotates once per orbit, so we always see the same face. Mercury is locked to the Sun in a 3:2 spin-orbit resonance, meaning it rotates three times for every two orbits. Venus rotates slowly backward, nearly four rotations per orbit in the retrograde direction. These ratios, written as p/q, are usually treated as accidents of each world's tidal history.

In Recognition Science, the framework models these ratios as the minima of a forced cost function on a spin-orbit phase manifold. The cost, called recognition cost, is a discrete record of how far a ratio sits from a preferred scale. The framework's central theorem forces the golden ratio phi, about 1.618, as that preferred scale. The module then checks the three canonical ratios against powers of phi.

The Moon's 1:1 ratio sits at cost zero, the trivial case. Mercury's 3/2 = 1.5 lies within 0.118 of phi, and that 0.118 is exactly the golden-section cost ceiling. Venus's 4 lies within 0.236 of phi cubed, which is about 4.236, and 0.236 is the next step down the phi ladder. The library proves these bands hold: Mercury's deviation is between 0.11 and 0.13, and Venus's is between 0.22 and 0.24. The structural prediction is that every Solar System spin-orbit resonance has its p/q ratio within 0.118 of an integer or half-integer power of phi.

What this establishes in plain language is a pattern: the resonance integers are not arbitrary. They sit at phi-rational positions, and the deviations from those positions are themselves the canonical cost values. The module is a structural theorem, not a dynamical model; it does not explain how the locking happens. It certifies, with machine-checked proofs, that the observed ratios occupy the predicted bands. A confirmed resonance deviating from a phi power by more than 0.118 would falsify the prediction.

/-- 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

This module does not prove a mechanism for tidal locking, only the placement of observed ratios. The universal band for all Solar System resonances is a prediction, not a proved theorem for every body. The framework does not derive the fine-structure constant, and this module makes no claim about it.

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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