Astrophysics Planetary Formation From Jcost R Orbit Closed

The closed form of the orbit ladder

In the Recognition Science framework, a ledger, a discrete record of events, is used to model the structure of a protoplanetary disk. The framework proposes that stable orbital radii, the distances at which planets form, are not arbitrary but follow a specific multiplicative pattern. This pattern is a ladder of rungs, where each rung's distance is the previous one multiplied by the golden ratio, φ, approximately 1.618. The declaration r_orbit_closed is the formal, machine-checked statement that this ladder has a simple closed form: the radius at rung k is equal to the inner reference radius r₀ times φ raised to the power k.

This is a purely structural theorem. It states that the sequence of radii is exactly geometric, with no additional terms or corrections. The proof is a direct equality, meaning the definition of the ladder and its closed form are the same thing. This is not a claim about the actual positions of planets in our Solar System. The theorem does not say that Mercury is at 0.4 AU or that Jupiter is at 5.2 AU. It only establishes the algebraic relationship between the rungs of the ladder, given a starting scale r₀. The choice of r₀ and the identification of which rung corresponds to which planet is a separate, empirical step.

The theorem also guarantees basic properties of this ladder. For any positive starting scale, every rung is a positive distance, and the rungs are strictly increasing: each outer rung is farther out than the one before. The ratio between adjacent rungs is exactly φ, and skipping one rung gives a ratio of φ², which falls in the band between 2.5 and 2.7. These are all consequences of the closed form, and they are what the framework calls the recognition-cost reading of the historical Titius-Bode pattern. The framework's library proves these structural facts; it does not prove that any particular planet will be found at a specific rung.

The framework's claim about the Solar System is a prediction with a named falsifier. A planet's measured semi-major axis must lie within a half-rung tolerance band of a predicted rung, otherwise the prediction fails. The half-rung width is the square root of φ, about 1.272. The framework's own ladder is a witness to this condition, meaning the predicted rungs are always within their own tolerance band. This is a consistency check, not a confirmation. The empirical comparison against actual planetary data is carried out by a separate Python script, not by the formal theorem.

/-- Closed form. -/
theorem r_orbit_closed (r0 : ℝ) (k : ℕ) :
    r_orbit r0 k = r0 * phi ^ k := rfl

/-- Adjacent ratio is exactly φ. -/
theorem r_orbit_adjacent_ratio (r0 : ℝ) (h : 0 < r0) (k : ℕ) :
    r_orbit r0 (k + 1) / r_orbit r0 k = phi := by
  have hk : r_orbit r0 k > 0 := r_orbit_pos r0 h k
  rw [r_orbit_succ]
  field_simp

/-- Strict monotonicity: outer rung is strictly farther than inner rung. -/
theorem r_orbit_strict_mono (r0 : ℝ) (h : 0 < r0) (k : ℕ) :
    r_orbit r0 k < r_orbit r0 (k + 1) := by
  rw [r_orbit_succ]
  have hk : 0 < r_orbit r0 k := r_orbit_pos r0 h k
  have h1 : 1 < phi := one_lt_phi
  nlinarith [r_orbit_pos r0 h k, one_lt_phi]

/-- Gap-skip ratio (skipping one stable rung) is `φ² ∈ (2.5, 2.7)`. -/
theorem r_orbit_gap_skip_band (r0 : ℝ) (h : 0 < r0) (k : ℕ) :
    (2.5 : ℝ) < r_orbit r0 (k + 2) / r_orbit r0 k ∧
    r_orbit r0 (k + 2) / r_orbit r0 k < 2.7 := by
  have hk : 0 < r_orbit r0 k := r_orbit_pos r0 h k
  have hr0 : r0 ≠ 0 := ne_of_gt h
  have hphi_k : phi ^ k ≠ 0 := pow_ne_zero _ phi_ne_zero
  have hratio :
      r_orbit r0 (k + 2) / r_orbit r0 k = phi ^ 2 := by
    unfold r_orbit
    rw [show (k + 2) = k + 2 from rfl, pow_add]
    field_simp
  rw [hratio]
  exact phi_squared_bounds

What this page does not claim

This theorem does not predict the actual orbital radii of any specific planet in the Solar System. This theorem does not provide a physical mechanism for why a disk would minimize J-cost. This theorem does not confirm the Titius-Bode law; it only provides a structural model that matches the pattern.

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/PlanetaryFormationFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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