Cosmology Baryon Asymmetry From Phi Ladder

Baryon asymmetry from a phi ladder

The baryon asymmetry is the observed excess of ordinary matter over antimatter in the universe. In the standard cosmological model, this imbalance is measured by the baryon-to-photon ratio, a number around 10^-9. The exact origin of this asymmetry is one of the open problems in physics; the known laws of particle physics do not explain why matter won out. Recognition Science offers a simple numerical model: the predicted ratio is the inverse of the golden ratio φ raised to the 44th power, written η_B ≈ φ^-44.

The golden ratio φ is the number satisfying φ² = φ + 1, approximately 1.618. The central claim is that φ⁴⁴ exceeds 10⁸, which makes η_B = φ^-44 smaller than 10^-8. That bound places the prediction in the same order of magnitude as the measured baryon-to-photon ratio, which is about 6×10^-10. The framework does not claim to derive the exact observed value; it establishes the large-number separation that makes the model plausible.

In Recognition Science, the exponent 44 is called a rung, a step on a ladder of powers of φ. The framework models physical quantities as powers of φ on such a ladder. The choice of rung 44 is a definitional model, not a derived theorem. What the machine-checked library proves is the arithmetic: that 44 is the rung, that φ⁴⁴ is indeed larger than 10⁸, and that the resulting η_B is positive and smaller than 10^-8. These are formal theorems, checked without gaps.

The module bundles these results into a certificate, a structure that records the rung, the positivity, the large-number bound, and the smallness bound. The certificate is constructed from the individual theorems. This is a compact way to state that the model's key numerical facts hold. The practical consequence is that the phi ladder offers a concrete, testable target for the baryon asymmetry, one that sits within an order of magnitude of observation, though the precise match is not claimed.

def baryonRung : ℕ := 44

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 module does not derive the rung 44 from the forcing chain; it is a definitional choice. The module does not prove that the predicted value equals the measured baryon asymmetry. The module does not explain the physical baryogenesis mechanism that produces the 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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