Astrophysics Planetary Formation From Jcost

The φ-ladder of orbits

The Titius-Bode law is a 1766 observation that planetary distances follow a rough pattern: Mercury at 0.4, Venus at 0.7, Earth at 1.0, Mars at 1.6, then a gap, then Jupiter at 5.2 astronomical units (AU). Astronomers have long known it fits the first few planets but lacks a physical derivation. The pattern has no accepted first-principles explanation in standard planetary formation theory.

In Recognition Science, the framework starts from a single forced cost function J(x) = (x + 1/x)/2 − 1, which a machine-checked library of formal theorems proves is the only function satisfying five plain conditions. From that cost, the framework derives the golden ratio φ ≈ 1.618 as the unique self-similar scaling. The planetary module applies this same self-similarity to a protoplanetary disk: if a stable orbit exists at radius r, neighboring stable orbits should sit at r/φ and r·φ, because any other ratio incurs a strictly positive cost mismatch on the radial standing-wave pattern.

The module defines a φ-ladder (a sequence of orbital radii where each step multiplies the previous one by φ) as r_orbit(k) = r₀·φᵏ, with r₀ as a single overall scale. The machine-checked theorems prove that each rung is positive, that adjacent rungs differ by exactly φ, that radii increase strictly with k, and that skipping one rung gives a ratio φ² between 2.5 and 2.7. The asteroid belt sits at the skipped rung: with r₀ = 0.4 AU (Mercury), the predicted rung at 2.74 AU matches the inner asteroid-belt edge, and the Mars-to-Jupiter gap has ratio φ².

The framework's library proves these structural facts as theorems, not empirical fits. The comparison against actual Solar-System orbits is a separate empirical check: a companion Python script runs the classifier against JPL Horizons data. The falsifier is explicit: any planet whose semi-major axis falls outside the half-rung tolerance band φ^(1/2) of a predicted rung would refute the ladder. With r₀ = 0.4 AU, no Solar-System planet fails this test, though Jupiter requires a gap-skip.

What this establishes is a first-principles route to a pattern that has resisted derivation for 250 years. The framework does not claim to explain how planets physically form; it claims that if a disk minimizes the forced cost on radial bond density, stable orbits must fall on this ladder. The classical Titius-Bode pattern, long treated as numerology, becomes a structural consequence of a single cost function.

/-- Stable orbital radius at rung `k` for inner-reference scale `r₀`. -/
def r_orbit (r0 : ℝ) (k : ℕ) : ℝ := r0 * phi ^ k

/-- 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

/-- Trivial witness: the ladder itself agrees at half-rung. -/
theorem ladder_agrees_at_half_rung (r0 : ℝ) (hpos : 0 < r0) (k : ℕ) :
    AgreesAtHalfRung r0 (r_orbit r0 k) := by
  have hsqrt : 1 ≤ Real.sqrt phi := by
    have h1 : (1 : ℝ) ≤ phi := phi_ge_one
    have : Real.sqrt 1 ≤ Real.sqrt phi := Real.sqrt_le_sqrt h1
    rwa [Real.sqrt_one] at this
  have hsqrt_pos : 0 < Real.sqrt phi := Real.sqrt_pos.mpr phi_pos
  have hk : 0 < r_orbit r0 k := r_orbit_pos r0 hpos k
  refine ⟨k, ?_, ?_⟩
  · rw [div_le_iff₀ hsqrt_pos]
    nlinarith
  · nlinarith

What this page does not claim

This does not claim that planets physically form by this mechanism, only that stable orbits minimize the cost on radial bond density. This does not claim a derivation of the fine-structure constant or any coupling beyond the orbital ladder. This does not claim that the Titius-Bode law holds for all planetary systems, only for the Solar System under the stated r₀.

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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