Cosmology Baryogenesis From Jcost Baryogenesis Mechanism

The five mechanisms

Baryogenesis is the physical process that produced the observed excess of matter over antimatter in the early universe. The cosmic microwave background measures the baryon-to-photon ratio at about 6.1 × 10⁻¹⁰, meaning there is roughly one baryon for every 1.6 billion photons. The standard explanation requires three conditions, named after the physicist Andrei Sakharov: baryon number violation, C and CP violation, and a departure from thermal equilibrium. These conditions are not a single mechanism but a family of proposals.

Physicists have developed five main mechanisms: leptogenesis, electroweak baryogenesis, Affleck-Dine, cold baryogenesis, and GUT baryogenesis. Each satisfies the Sakharov conditions in a different way, using different particle physics at different energy scales. The five are the working set in modern cosmology, and any complete theory of the early universe must pick one or more of them to generate the observed asymmetry.

In Recognition Science, the framework's machine-checked library of formal theorems defines an inductive type called BaryogenesisMechanism with exactly these five constructors. A theorem in the library proves that the number of distinct mechanisms is five, by exhaustive enumeration. The framework also models the asymmetry itself: its cost function J, which measures the recognition cost of a state, is zero at the equilibrium point where matter and antimatter balance, and strictly positive whenever matter differs from antimatter. This gives a compact formal statement: equilibrium is J = 0, asymmetry is J > 0.

The framework's contribution is a classification, not a new physics mechanism. It takes the five known mechanisms as a definitional choice, then proves the count is five. The connection to the Sakharov conditions in the framework's documentation is a mapping of concepts, not a derivation of those conditions from first principles. The framework does not prove that any one mechanism is the correct one, nor does it derive the measured value of the baryon-to-photon ratio from its cost function.

What the declaration establishes is a clean formal container: a type with five members, a theorem that the count is five, and a certificate structure that bundles the count with the equilibrium and asymmetry statements. This lets the framework reason about baryogenesis mechanisms as a finite, enumerated set, and it lets a reader check the count and the cost properties in a machine-checked way. The physical content, the actual particle physics of each mechanism, remains exactly what it was in conventional cosmology.

theorem baryogenesisMechanismCount : Fintype.card BaryogenesisMechanism = 5 := by decide

/-- Equilibrium = matter-antimatter balance (J=0). -/
theorem matter_balance_equilibrium : Jcost 1 = 0 := Jcost_unit0

/-- Asymmetry = J > 0 when matter ≠ antimatter. -/
theorem asymmetry_positive_cost {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

inductive BaryogenesisMechanism where
  | leptogenesis | electroWeak | affleckDine | cold | GUT
  deriving DecidableEq, Repr, BEq, Fintype

What this page does not claim

Not a derivation of the Sakharov conditions from first principles. Not a proof that any particular mechanism is the correct one. Not a derivation of the measured baryon-to-photon ratio of 6.1 × 10⁻¹⁰.

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

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