Astrophysics Planetary Formation From Jcost Two Rung Gap Eq Phi Squared

The two-rung gap

In classical astronomy, the Titius-Bode law is a rough empirical pattern for the spacing of planetary orbits: each planet's distance from the Sun is roughly double the previous one after the first few. It was noticed in the 18th century, fits the known planets reasonably well, and famously predicted a missing planet at about 2.8 astronomical units, a spot later found to be occupied by the asteroid belt. Physics has never derived the pattern from first principles; it remains a numerical curiosity with no accepted theoretical basis.

The Recognition Science framework offers a derivation. Its starting point is a ledger, a discrete record of recognition events, where the cost of each recognition is forced by a proved theorem rather than chosen freely. From that cost function, the framework proves that the golden ratio φ, about 1.618, is the unique self-similar scaling for certain structures. Applied to a protoplanetary disk, the framework models stable orbital radii as sitting on a ladder: r_orbit(k) = r₀ · φ^k, where r₀ is a single overall scale and k is a nonnegative integer. The disk minimises the cost when neighbouring stable orbits are separated by the ratio φ, because any other ratio incurs a positive cost mismatch on the radial standing-wave pattern.

The declaration two_rung_gap_eq_phi_squared is a proved theorem inside this framework. It states that skipping one rung of the ladder, going from rung k to rung k+2, multiplies the orbital radius by φ², which is about 2.618. In the Solar System, this gap-skip is exactly the structure seen between Mars and Jupiter, with the asteroid belt sitting at the skipped rung. The theorem itself is a pure structural fact about the ladder: it does not depend on any particular value of r₀ or k, only on the definition r_orbit(r₀, k) = r₀ · φ^k. It is a consequence of the algebra of powers, not a new physical assumption.

What the theorem does not claim is equally important. It does not claim that any real planet actually sits on the ladder; that is an empirical check, not a proved result. The framework's companion Python script compares the ladder against JPL Horizons data for the Solar System, and the docstring reports that with r₀ = 0.4 AU, no planet falls outside the half-rung tolerance band. But that comparison is a measurement check, not part of the formal result. The theorem also does not claim that the Titius-Bode law is universally true for all planetary systems; it only states the algebraic identity for the ladder itself. The physical bridge from recognition cost to actual orbital formation is a separate, open question.

The consequence of this theorem is that the framework gives a precise, testable structure for planetary spacing: adjacent orbits differ by φ, and skipped rungs differ by φ². It turns a centuries-old empirical pattern into a specific prediction with a named falsifier. Any Solar-System planet whose semi-major axis is not within a factor of √φ, about 1.272, of r₀ · φ^k for some integer k would refute the ladder. No such planet is known, but the claim remains a structural prediction awaiting broader confirmation.

/-- Two-rung gap (e.g. Mars → Jupiter, skipping the asteroid-belt
rung) has cumulative ratio `φ²`. -/
theorem two_rung_gap_eq_phi_squared (r0 : ℝ) (k : ℕ) :
    r_orbit r0 (k + 2) = r_orbit r0 k * phi ^ 2 := by
  unfold r_orbit
  rw [pow_succ, pow_succ]
  ring

theorem r_orbit_succ (r0 : ℝ) (k : ℕ) :
    r_orbit r0 (k + 1) = r_orbit r0 k * phi := by
  unfold r_orbit
  rw [pow_succ]
  ring

/-- 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

/-- 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⟩

What this page does not claim

The theorem does not prove that any real planet lies on the ladder; that is an empirical check. The theorem does not derive the Titius-Bode law from first principles for all planetary systems. The theorem does not establish the physical bridge from recognition cost to orbital formation.

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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