Cosmology Baryon Asymmetry From Phi Ladder Phi32 Gt 4 M

The large-number bound

The golden ratio, φ = (1 + √5)/2 ≈ 1.618, is the number whose square equals itself plus one. Its powers grow quickly. The declaration phi32_gt_4M is a machine-checked theorem stating that φ raised to the 32nd power is greater than 4,000,000. The proof is short: it squares an earlier bound that φ^16 exceeds 2,000, and since squaring a positive number larger than 2,000 gives a result larger than 4,000,000, the conclusion follows. The theorem is part of a chain of such bounds in the framework's machine-checked library of formal theorems.

This bound is a stepping stone. The framework, called Recognition Science, derives physical structure from a single starting point: reality keeps a ledger, a discrete record of recognition events, and the cost of recognition is forced, not chosen. From that cost function, a chain of theorems forces the golden ratio as the unique self-similar scaling, an eight-tick recognition cycle, and three spatial dimensions. The framework then models particle masses on a phi-power ladder, where each rung is an integer power of φ. The baryon asymmetry, the observed excess of matter over antimatter in the universe, is predicted to sit on the 44th rung of this ladder.

In Recognition Science, the predicted value of the baryon asymmetry is η_B ≈ φ^(-44), the reciprocal of φ raised to the 44th power. The theorem phi32_gt_4M contributes to showing that φ^44 is large, specifically greater than 10^8. From that, the framework derives that η_B is smaller than 10^(-8), a bound consistent with the observed cosmic ratio of baryons to photons, which is about 6 × 10^(-10). The declaration itself does not compute the asymmetry; it only establishes a numerical lower bound on a power of φ, a necessary ingredient for the later bound on η_B.

The theorem does not claim that the baryon asymmetry is measured, nor that the framework's prediction matches observation. The comparison against measured values is an empirical check, not a theorem. The declaration also does not assert that φ^32 is the asymmetry; it is merely an intermediate bound. The framework's derivation of the phi-power ladder itself, and the identification of the 44th rung with the baryon asymmetry, are not proved in this declaration. The theorem only certifies a fact about the golden ratio's powers, a small but machine-checked piece of a larger, mostly unformalized physical account.

/-- φ^32 > 4000000. -/
theorem phi32_gt_4M : phi ^ 32 > 4000000 := by
  have h : phi ^ 32 = (phi ^ 16) ^ 2 := by ring
  rw [h]; nlinarith [phi16_gt_2000, sq_nonneg (phi^16 - 2000)]

def baryonRung : ℕ := 44

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 declaration does not prove that the baryon asymmetry is measured or that the framework's prediction matches observation. The declaration does not assert that φ^32 is the baryon asymmetry; it is only an intermediate numerical bound. The derivation of the phi-power ladder and the identification of the 44th rung are not proved in this declaration.

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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