Astrophysics Tidal Locking From Phi Resonance Mercury Deviation Eq J Phi

The Mercury deviation

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 measured property of the planet, established by radar observations in the 1960s. The ratio 3/2, or 1.5, is a simple rational number. The golden ratio φ, approximately 1.618, is an irrational number that solves the equation r² = r + 1.

The machine-checked declaration mercury_deviation_eq_J_phi proves a precise arithmetic identity: φ minus 3/2 equals φ minus 3/2. This is a tautology, but it is a useful one because it defines the deviation between Mercury's resonance ratio and the golden ratio. A companion theorem proves this deviation lies strictly between 0.11 and 0.13. So the framework's library of formal theorems establishes that the difference between Mercury's ratio and φ is a positive number in a narrow band around 0.118, a value the framework calls the J-cost ceiling, a measure of the cost of recognition on a discrete ledger of events.

In the Recognition Science framework, this band is not a coincidence. The framework models spin-orbit resonance ratios as the minima of a cost function on a phase manifold. The Moon's 1:1 resonance sits at exactly zero cost. Mercury's 3:2 sits near φ, within the golden-section cost band. Venus's 4:1 retrograde ratio sits near φ³, within a related band. The framework's structural prediction is that every observed spin-orbit resonance in the Solar System has its ratio within about 0.118 of an integer or half-integer power of φ.

What the declaration does not claim is equally important. It does not prove that Mercury's resonance is caused by the golden ratio, nor does it explain the tidal evolution that produced the 3:2 lock. The theorem is a statement about real numbers, not about planetary dynamics. The physical bridge from recognition cost to actual tidal locking remains an open target in the framework. The measured 3:2 ratio is a fact; its placement near φ is a proved arithmetic observation that the framework interprets as structural.

/-- Mercury's `p/q = 3/2` is near `φ ≈ 1.618`; the deviation is
`|φ - 3/2| = J(φ) ≈ 0.118`. -/
theorem mercury_deviation_eq_J_phi :
    phi - mercury_resonance_pq = phi - 3 / 2 := by
  unfold mercury_resonance_pq
  ring

/-- 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 prove that the golden ratio causes Mercury's 3:2 resonance. The declaration does not explain the tidal evolution mechanism that produced the lock. The physical bridge from recognition cost to tidal locking is not established.

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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