Cosmology Baryogenesis Trajectory From Phi Ladder Bviolation Channel

The five channels

In particle physics, baryogenesis is the proposed process by which the early universe generated more matter than antimatter, leaving the surplus that became stars, planets, and us. The standard picture requires some interaction that violates baryon number, the conserved quantity counting particles like protons and neutrons. Without such a violation, matter and antimatter would have annihilated symmetrically, leaving a nearly empty cosmos.

The Recognition Science framework's machine-checked library of formal theorems contains a small definition called BViolationChannel. It is a list of five named channels through which baryon number might be violated: sphaleron processes, electroweak interactions, QCD (the strong force), leptogenesis, and neutrino mass. The definition is an inductive type, which in the framework's formal language means it is a closed enumeration: these five, and no others, are the channels the framework recognizes. A companion theorem proves the count is exactly five, and a certificate structure bundles this count together with three proved facts about a related quantity called etaB, the baryon-to-photon ratio.

What the definition establishes is purely structural. It fixes a vocabulary for discussing baryogenesis within the framework, and it ties that vocabulary to a specific mathematical trajectory. In the framework's account, the baryon-to-photon ratio etaB grows by exactly the golden ratio phi per temperature rung as the universe cools, reaching the value 1 at a specific rung indexed 44. The five channels are presented as the canonical set of rungs where baryon-number violation could act. The formal theorems prove the ratio's growth law, its positivity, and its completion value; they do not prove that any of the five channels actually operated in the early universe.

The distinction matters. The list of five channels is a definitional choice, a modeling decision about which physical processes to include. The growth law for etaB is a proved theorem within the framework's axioms. But neither the definition nor the theorem constitutes empirical evidence that baryogenesis occurred, that the golden ratio governs the real universe's matter-antimatter asymmetry, or that any specific channel is the one nature used. Those remain open questions in physics, and the framework's formal results do not close them.

/-- Canonical B-violation rungs (5 = configDim D): sphaleron, electroweak,
QCD, leptogenesis, neutrino-mass. -/
inductive BViolationChannel where
  | sphaleron
  | electroweak
  | qcd
  | leptogenesis
  | neutrinoMass
  deriving DecidableEq, Repr, BEq, Fintype

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

/-- 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'

What this page does not claim

The five channels listed are not proved to be the only baryon-number-violating processes in nature. The framework's theorems do not establish that baryogenesis actually occurred in the early universe. The golden-ratio growth law for etaB is not an empirical measurement of the baryon-to-photon ratio.

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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