Astrophysics Tidal Locking From Phi Resonance J Phi Ceiling Pos

The resonance ceiling

Spin-orbit resonance is the situation where a moon or planet rotates in a fixed ratio with its orbit around another body. The Moon always shows Earth the same face, one rotation per orbit, a 1:1 resonance. Mercury spins three times for every two trips around the Sun, a 3:2 resonance. Venus rotates slowly backward, about four rotations for every one orbit, a 4:1 retrograde resonance. These ratios are observed facts, and they have long been treated as accidents of tidal history.

In Recognition Science, the framework models these ratios as the cost, a measure of recognition effort, of a spin-orbit state. Its cost function J(x) = (x + 1/x)/2 - 1 assigns zero cost to the 1:1 ratio, and a positive cost to any deviation from it. The framework's library of machine-checked formal theorems proves that Mercury's 3:2 ratio sits within a specific band of that cost, and Venus's 4:1 ratio sits within a wider band. The declaration J_phi_ceiling_pos proves the narrowest fact that makes those bands meaningful: the ceiling value, defined as the golden ratio φ ≈ 1.618 minus 1.5, is positive.

That positivity is the hinge. The ceiling is φ - 3/2 ≈ 0.118. The theorem proves it is greater than zero, and a companion theorem proves it lies strictly between 0.11 and 0.13. Mercury's deviation from φ is exactly that ceiling, so Mercury sits at the edge of the stable band. Venus's deviation from φ³ is about 0.236, which is 1/φ², the next rung down the same ladder. The framework's structural prediction is that every observed spin-orbit resonance in the Solar System has its ratio within that 0.118 band of some integer or half-integer power of φ. The theorem J_phi_ceiling_pos alone does not assert that prediction; it only proves the ceiling is a positive number, which the larger certificate then uses.

What the declaration does not claim is just as important. It does not prove that any observed resonance is stable, nor that tidal locking in general follows from the cost function. It does not derive the resonance ratios from first principles; it takes the observed 1:1, 3:2, and 4:1 ratios as given and proves they sit in the predicted bands. The physical bridge from recognition cost to actual tidal dynamics remains an open target. The theorem is a piece of real arithmetic about a defined constant, not a physical law.

theorem J_phi_ceiling_pos : 0 < J_phi_ceiling := by
  unfold J_phi_ceiling
  linarith [phi_gt_onePointSixOne]

theorem J_phi_ceiling_band :
    0.11 < J_phi_ceiling ∧ J_phi_ceiling < 0.13 := by
  unfold J_phi_ceiling
  have h1 := phi_gt_onePointSixOne
  have h2 := phi_lt_onePointSixTwo
  refine ⟨?_, ?_⟩ <;> linarith

/-- 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

/-- 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

This does not prove that any observed resonance is physically stable. This does not derive the resonance ratios from first principles; it takes them as observed inputs. This does not establish the physical bridge from recognition cost to tidal dynamics.

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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