Cosmology Baryogenesis Trajectory From Phi Ladder

The phi ladder

The golden ratio, φ ≈ 1.618, is famous for appearing in pentagons, sunflower spirals, and the Fibonacci sequence. In cosmology, it may also describe the growth of the matter-antimatter asymmetry, the reason the universe is made of matter rather than nothing. The asymmetry is measured by a quantity called η_B, the baryon-to-photon ratio, which today is about 6 × 10^(-10). In the Recognition Science framework, this value is not an accident but the endpoint of a precise trajectory.

The trajectory is a ladder on the temperature axis. As the universe cools from the grand unified theory (GUT) scale, the temperature drops by a factor of φ at each step: T_k = T_GUT · φ^(-k). At the same time, the asymmetry grows by exactly φ per rung: η_B(T_k) = φ^(k - 44). At k = 0, near the GUT scale, the asymmetry is negligible. It reaches the observed value φ^(-44) ≈ 6 × 10^(-10) at k = 44, which the framework calls gap-45. This is a monotone approach: each cooling step multiplies the asymmetry by φ, never decreasing.

The framework's machine-checked library of formal theorems proves this trajectory rigorously. It defines η_B(k) = φ^k / φ^44 and verifies three properties: the ratio between successive rungs is exactly φ, the value at k = 44 is exactly 1, and the value is always positive. These results are packaged in a certificate structure that bundles the proofs together. The framework also names five channels for baryon number violation, one for each of the five rungs: sphaleron, electroweak, QCD, leptogenesis, and neutrino mass. The number five matches the framework's internal configuration dimension.

In Recognition Science, this ladder is not an ad hoc fit. The framework derives the golden ratio as the unique self-similar scaling from a cost function that governs recognition events. The baryogenesis trajectory is a direct application: the same φ that appears in geometry and number theory also controls the growth of matter. The framework establishes that the observed asymmetry is the completion of a 44-step process, each step multiplying by φ. It is a structural statement about how the universe's asymmetry emerges from a simple, repeated scaling.

The practical consequence is a testable prediction: the baryon-to-photon ratio should be exactly φ^(-44) at the electroweak scale. This is not a free parameter but a forced value from the framework. The trajectory also predicts that the asymmetry grows monotonically with cooling, never oscillating or reversing. If future measurements find a deviation from this precise scaling, the framework would be falsified. The framework thus turns a cosmological observation into a sharp, checkable claim about the golden ratio.

/-- 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'

/-- η_B is strictly positive on the whole trajectory. -/
theorem etaB_pos (k : ℕ) : 0 < etaB k := by
  unfold etaB
  exact div_pos (pow_pos phi_pos k) (pow_pos phi_pos 44)

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

What this page does not claim

Not a claim that the observed baryon asymmetry has been measured to match φ^(-44) with high precision. Not a claim that the five named channels are derived from first principles. Not a claim that the framework explains why the universe started with a negligible 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/BaryogenesisTrajectoryFromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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