Cosmology Baryon Asymmetry From Phi Ladder Phi16 Gt 2000

A large-number bound

The golden ratio φ, the number satisfying φ² = φ + 1 and approximately 1.618, has a striking property: its powers grow quickly. The sixteenth power exceeds 2000. This is not a numerical coincidence but a proved statement, derived from the defining equation and the fact that φ is greater than 1.61. The proof is short: φ⁸ exceeds 46, and since φ¹⁶ is the square of φ⁸, it must exceed 46², which is 2116, comfortably above 2000.

This bound is a rung in a ladder of similar statements. The thirty-second power exceeds four million, and the forty-fourth power exceeds one hundred million. These are not isolated facts; they build on each other, each squaring the previous result. The chain culminates in a statement about the baryon asymmetry, the observed excess of matter over antimatter in the universe. The framework's prediction is that this asymmetry, denoted η_B, is approximately φ⁻⁴⁴, a very small number.

In Recognition Science, a framework that derives physical structure from a forced cost of recognition, this small number is not arbitrary. The framework models the baryon asymmetry as the inverse of a large power of the golden ratio. The theorem phi16_gt_2000 is a supporting step: it helps establish that φ⁴⁴ is large, and therefore that η_B is small, less than 10⁻⁸. This is a machine-checked result, part of a library of formal theorems with no unproved assumptions.

What this declaration does not claim is equally important. It does not claim that the baryon asymmetry is measured to be exactly φ⁻⁴⁴. The framework's prediction is a hypothesis, not a measured value. The theorem only establishes a mathematical bound: that the proposed value is less than 10⁻⁸. It does not prove that this is the actual value in our universe, nor does it derive the golden ratio from cosmology. The comparison to observation is a separate empirical check, not part of the formal proof.

The practical consequence is a clear, testable prediction. The framework says the matter-antimatter imbalance should be about one part in ten billion, a specific number derived from a mathematical constant. This is a concrete target for cosmologists. The theorem phi16_gt_2000 is a small but necessary piece of that prediction, ensuring the proposed value is indeed as small as claimed.

/-- φ^16 > 2000. -/
theorem phi16_gt_2000 : phi ^ 16 > 2000 := by
  have h : phi ^ 16 = (phi ^ 8) ^ 2 := by ring
  rw [h]; nlinarith [phi8_gt_46, sq_nonneg (phi^8 - 46)]

/-- 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 prove the measured baryon asymmetry equals φ⁻⁴⁴. The theorem does not derive the golden ratio from cosmological observations. The theorem does not establish that the baryon asymmetry is the only possible value.

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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