Cosmology Baryon Asymmetry From Phi Ladder Phi8 Gt 46

A large-number bound

The golden ratio φ, approximately 1.618, is the positive solution to r² = r + 1. Its powers grow quickly: φ² is about 2.618, φ⁴ about 6.854, and φ⁸ is about 46.98. The theorem phi8_gt_46 in the framework's machine-checked library of formal theorems proves the exact inequality φ⁸ > 46, using only the defining quadratic equation and the fact that φ > 1.61. This is a pure number statement, independent of any physical interpretation.

The inequality matters because it is the first rung of a ladder. The framework defines a recognition event as a discrete record of a comparison, and its cost function forces a specific scaling constant. From that constant, the framework derives a particle mass ladder where each rung is a power of φ. The baryon asymmetry rung is set at 44, so the framework's predicted value for η_B, the ratio of matter to photons in the universe, is φ⁻⁴⁴. The theorem φ⁸ > 46, combined with squaring steps, yields φ⁴⁴ > 10⁸, which in turn proves η_B < 10⁻⁸. This is the framework's way of showing its predicted asymmetry is small, as observations require.

The proof of φ⁸ > 46 is elementary. The library first proves φ⁸ = 21φ + 13 by repeated squaring of the defining equation. Then, using the separately proved fact that φ > 1.61, a linear arithmetic step gives 21φ + 13 > 21 × 1.61 + 13 = 46.81 > 46. The entire chain, from the quadratic equation to the final inequality, is checked by the Lean kernel with no axioms beyond the standard three. The theorem itself is a tautology about real numbers; it does not assert anything about the physical universe.

What the theorem does not claim is just as important. It does not prove that the baryon asymmetry is actually φ⁻⁴⁴; that is a definitional choice, not a derived result. The rung number 44 is an identification, not a consequence of the forcing chain. The theorem also does not compare its value to any measured cosmological data. The measured η_B is about 6 × 10⁻¹⁰, and φ⁻⁴⁴ is about 1.3 × 10⁻⁹, a factor of about two off. The framework treats this as an empirical check, not a theorem. Finally, the inequality alone says nothing about why there is more matter than antimatter; it only bounds a number the framework defines.

theorem phi8_gt_46 : phi ^ 8 > 46 := by
  rw [phi8_val]; linarith [phi_gt_onePointSixOne]

noncomputable def etaB_RS : ℝ := (phi ^ baryonRung)⁻¹

/-- φ^44 > 10^8. -/
theorem phi44_gt_1e8 : phi ^ 44 > (10:ℝ)^8 := by
  have h12 : phi ^ 12 > 321 := by
    have h2 := phi_sq_eq
    have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
    have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
    have h5 : phi ^ 5 = 5 * phi + 3 := by nlinarith
    have h6 : phi ^ 6 = 8 * phi + 5 := by nlinarith
    have h8 := phi8_val
    have h12v : phi ^ 12 = phi ^ 6 * phi ^ 6 := by ring
    rw [h12v]; nlinarith [phi_gt_onePointSixOne]
  have h44 : phi ^ 44 = phi ^ 32 * phi ^ 12 := by ring
  rw [h44]
  norm_num
  nlinarith [mul_pos (by linarith [phi32_gt_4M] : (0:ℝ) < phi^32) (by linarith : (0:ℝ) < phi^12),
             phi32_gt_4M, h12]

/-- η_B < 10^(-8). -/
theorem etaB_small : etaB_RS * (10:ℝ)^8 < 1 := by
  unfold etaB_RS baryonRung
  rw [inv_mul_lt_iff₀ (pow_pos phi_pos 44)]
  simp only [mul_one]
  exact phi44_gt_1e8

What this page does not claim

The rung number 44 is derived, not chosen. The framework's η_B matches the measured cosmological asymmetry. The theorem explains why matter dominates antimatter.

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

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