Astrophysics Planetary Formation From Jcost R Orbit Adjacent Ratio Band

The orbital ratio band

The Titius-Bode law is a 1766 observation that planetary distances from the Sun often follow a simple geometric progression. It was never derived from physics, and it skips a rung at the asteroid belt. The Recognition Science framework offers a structural reading: it models stable orbital radii as sitting on a ladder where each step multiplies the previous radius by the golden ratio φ, about 1.618.

The declaration r_orbit_adjacent_ratio_band is a theorem in the framework's machine-checked library of formal theorems. It states that for any positive inner scale r₀ and any integer rung k, the ratio of the next orbital radius to the current one, r_orbit(k+1) / r_orbit(k), lies strictly between 1.61 and 1.62. This is a pure structural fact about the ladder: it follows directly from the definition r_orbit(k) = r₀ · φ^k and the known numerical bounds on φ. The theorem does not depend on any particular planet or on the choice of r₀.

In plain terms, the theorem pins down the spacing between consecutive stable orbits. If one orbit sits at radius r, the next stable orbit must be at roughly r · 1.618, and no other ratio is allowed by the ladder. The band (1.61, 1.62) is the canonical Titius-Bode-compatible window: the classical pattern used a factor near 1.7, and the golden ratio sits comfortably inside it. The framework's own prediction table places Mercury at 0.40 AU, Venus at 0.65 AU (actual 0.72), Earth at 1.05 AU (actual 1.00), and Mars at 1.69 AU (actual 1.52), with the asteroid belt occupying the skipped rung at about 2.74 AU.

What the theorem does not claim is just as important. It does not prove that real planets must follow this ladder; that is a physical hypothesis with a named falsifier. The falsifier is any Solar-System planet whose semi-major axis is not within a half-rung tolerance, meaning within a factor of √φ (about 1.272) of some predicted rung. The theorem itself only establishes the internal consistency of the ladder, not its application to the sky. The empirical check against JPL data lives in a companion Python script, not in the Lean proof.

The gap-skip identity is a second structural result: skipping one rung gives a ratio of φ², which lies between 2.5 and 2.7. This is exactly the jump from Mars to Jupiter, with the asteroid belt as the missing rung. The framework presents this as a recognition-cost reading of the old empirical pattern, with no free parameters per planet beyond the single overall scale r₀. The one-statement certificate packages all of these facts together: the ladder is its own half-rung witness, meaning every prediction falls inside its own falsifier band by construction.

/-- Adjacent-rung ratio is in the band `(1.61, 1.62)`. -/
theorem r_orbit_adjacent_ratio_band (r0 : ℝ) (h : 0 < r0) (k : ℕ) :
    1.61 < r_orbit r0 (k + 1) / r_orbit r0 k ∧
    r_orbit r0 (k + 1) / r_orbit r0 k < 1.62 := by
  rw [r_orbit_adjacent_ratio r0 h k]
  exact ⟨phi_gt_onePointSixOne, phi_lt_onePointSixTwo⟩

/-- Adjacent-rung ratio is in the band `(1.61, 1.62)`. -/
theorem r_orbit_adjacent_ratio_band (r0 : ℝ) (h : 0 < r0) (k : ℕ) :
    1.61 < r_orbit r0 (k + 1) / r_orbit r0 k ∧
    r_orbit r0 (k + 1) / r_orbit r0 k < 1.62 := by
  rw [r_orbit_adjacent_ratio r0 h k]
  exact ⟨phi_gt_onePointSixOne, phi_lt_onePointSixTwo⟩

/-- 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

/-- Half-rung tolerance: a measured semi-major axis `r_meas`
agrees with the φ-ladder at scale `r0` iff there exists `k` with
`r_meas ∈ [r_orbit r0 k / √φ, r_orbit r0 k · √φ]`. The
half-rung width `√φ ≈ 1.272` is exactly half the adjacent ratio
in log-space. -/
def AgreesAtHalfRung (r0 r_meas : ℝ) : Prop :=
  ∃ k : ℕ, r_orbit r0 k / Real.sqrt phi ≤ r_meas ∧
           r_meas ≤ r_orbit r0 k * Real.sqrt phi

What this page does not claim

This theorem does not prove that real planets must follow the golden-ratio ladder. It does not derive the Titius-Bode law from first principles of physics. It does not claim that every star system's planets will obey this 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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: