Cosmology Baryon Asymmetry From Phi Ladder Phi44 Gt 1e8

The large-number bound

The golden ratio, usually written φ, is the number satisfying φ² = φ + 1, approximately 1.618. Its powers grow quickly: φ² is about 2.618, φ⁸ is about 46.98, and φ⁴⁴ is already larger than 100 million. That last inequality, φ⁴⁴ > 10⁸, is the content of the theorem named phi44_gt_1e8 in the Recognition Science framework's machine-checked library of formal theorems. It is a simple statement about a real number, proved from the defining equation of φ and basic arithmetic, with no additional assumptions.

In the framework's cosmology, the baryon asymmetry η_B, the small excess of matter over antimatter in the universe, is defined as the reciprocal of that power: η_B = φ⁻⁴⁴. The theorem phi44_gt_1e8 directly implies that η_B is less than 10⁻⁸. The framework's library also bundles this into a certificate structure that records the rung value 44, the positivity of η_B, the large-number bound, and the smallness inequality. This certificate is a formal object, checked by the machine, but it does not by itself explain why the asymmetry should equal that particular reciprocal; it only states the numerical relationship and its immediate consequences.

The theorem does not claim that the golden ratio is the cause of the baryon asymmetry, nor that the framework has derived the physical mechanism of baryogenesis. The value 44 is chosen as a definitional rung in the framework's ladder, not derived from first principles in this file. The comparison to the measured baryon asymmetry, which is on the order of 10⁻¹⁰, is an empirical check that lies outside the formal theorem. The theorem phi44_gt_1e8 is a pure inequality about φ; its cosmological interpretation is a separate, interpretive step that the framework makes but does not prove in this declaration.

/-- 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]

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

/-- 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 theorem does not claim that the golden ratio causes the baryon asymmetry. The theorem does not claim that the value 44 is derived from first principles in this file. The theorem does not claim any agreement with the measured baryon asymmetry.

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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