Cosmology Baryogenesis Trajectory From Phi Ladder Eta B At Gap45

The completion threshold

In the framework's account, the baryon asymmetry, the excess of matter over antimatter in the early universe, is tracked by a quantity called etaB. This quantity is defined along a ladder of temperatures: as the universe cools by a factor of phi (about 1.618) at each step, etaB grows by exactly that same factor. The declaration etaB_at_gap45 establishes the endpoint of this climb: at step 44, which the framework labels gap-45, etaB equals exactly 1. The proof is a direct consequence of the definition, which sets etaB(k) = phi^k / phi^44, so that at k = 44 the numerator and denominator cancel.

The number 1 carries a specific meaning in this framework: it is the recognition-complete threshold, the point where the asymmetry has reached its full, late-time value. The trajectory starts at the GUT scale with a negligible asymmetry, roughly phi^(-44), and grows monotonically toward this threshold. The framework's library, a machine-checked collection of formal theorems, also proves that etaB is strictly positive at every rung and that the ratio between consecutive rungs is exactly phi. These facts are packaged together in a certificate structure that bundles the ratio law, the completion value, positivity, and a count of five baryon-violation channels.

What the declaration does not claim is just as important. It does not assert that this mathematical trajectory describes the actual history of the universe; that would require identifying the framework's temperature ladder with physical temperatures and its rungs with real physical processes. The five channels named in the certificate, sphaleron, electroweak, QCD, leptogenesis, and neutrino mass, are listed as canonical examples, not as a proved derivation of baryogenesis. The theorem proves an algebraic identity about a defined function; the physical interpretation of that function as the baryon asymmetry of the cosmos remains a modeling choice.

/-- 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 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 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)

What this page does not claim

This theorem does not claim that the framework's trajectory describes the actual baryon asymmetry history of the universe. The five named channels are listed as canonical examples, not as a proved derivation of baryogenesis. No claim is made that the value 1 at gap-45 corresponds to any measured physical quantity.

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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