Cosmology Baryogenesis Trajectory From Phi Ladder Eta B Ratio

The ratio theorem

The baryon asymmetry of the universe is the observed fact that matter vastly outnumbers antimatter. In the standard cosmological picture, this asymmetry is quantified by a number, η_B, the ratio of baryons to photons. Observations place its present value near 6 × 10^(-10), a small but important surplus that allowed atoms and galaxies to form. The Recognition Science framework models this asymmetry as a quantity that changes in discrete steps as the universe cools, rather than smoothly.

In this model, the cooling history is divided into rungs on a temperature ladder. At each rung, the temperature drops by a factor of the golden ratio, φ ≈ 1.618, and the asymmetry η_B increases by exactly the same factor. This is not an approximation: the theorem etaB_ratio proves, in the framework's machine-checked library of formal theorems, that for every integer step k, the ratio of the asymmetry at the next step to the asymmetry at the current step is precisely φ. The proof is a direct algebraic consequence of the definition of η_B as φ^k divided by φ^44.

The trajectory starts at the grand unified theory (GUT) scale, where the asymmetry is negligible, and climbs by a factor of φ at each of 44 rungs. At the 44th rung, the asymmetry reaches exactly 1, a threshold the framework calls recognition-complete. The late-time value, φ^(-44) ≈ 6 × 10^(-10), matches the observed order of magnitude. The framework also lists five named channels for baryon violation: sphaleron, electroweak, QCD, leptogenesis, and neutrino mass, and proves there are exactly five.

What etaB_ratio does not claim is that this trajectory is the actual history of the universe. It establishes a clean mathematical relationship within a specific model: if the asymmetry follows this ladder, then it grows by φ per rung. It does not prove that the ladder is realized in nature, nor does it derive the initial condition at the GUT scale. The match with the observed value is an empirical check, not a theorem. The framework's own documents mark the physical bridge from recognition events to this cosmological model as an open target, not a proved result.

/-- η_B grows by exactly φ per temperature rung. -/
theorem etaB_ratio (k : ℕ) :
    etaB (k + 1) / etaB k = phi := by
  unfold etaB
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  have hpos44 : (0 : ℝ) < phi ^ 44 := pow_pos phi_pos 44
  have hphi_ne : (phi : ℝ) ≠ 0 := phi_pos.ne'
  field_simp
  rw [pow_succ]
  ring

/-- η_B(T_44) = 1, the recognition-complete threshold reached at gap-45. -/
theorem etaB_at_gap45 : etaB 44 = 1 := by
  unfold etaB
  have h : (0 : ℝ) < phi ^ 44 := pow_pos phi_pos 44
  exact div_self h.ne'

noncomputable def etaB (k : ℕ) : ℝ := phi ^ k / phi ^ 44

theorem bViolationChannel_count : Fintype.card BViolationChannel = 5 := by decide

What this page does not claim

The trajectory is the actual history of the universe's baryon asymmetry. The observed value 6 × 10^(-10) is derived from first principles; it is only matched in order of magnitude. The physical bridge from recognition events to this cosmological ladder is proved; it remains open.

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

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