Cosmology Baryogenesis From Jcost Baryogenesis Mechanism Count

Counting the mechanisms

Baryogenesis is the name for the process that produced more matter than antimatter in the early universe. The measured imbalance is tiny: for every billion photons left over from the Big Bang, there is roughly one proton or neutron. Cosmologists have proposed several physical routes to generate that asymmetry, and the standard shortlist contains five: leptogenesis, electroweak baryogenesis, Affleck-Dine, cold baryogenesis, and GUT baryogenesis.

In the Recognition Science (RS) framework, the declaration baryogenesisMechanismCount is a theorem, proved in the machine-checked library of formal theorems, that the number of these mechanisms is exactly five. The proof is a direct computation: the framework defines a type with five named constructors (leptogenesis, electroWeak, affleckDine, cold, GUT) and then verifies by case analysis that no other constructor exists. This is a counting fact, not a physical derivation. It establishes that the standard list is complete in the sense of the framework's own definition, not that nature must contain exactly five mechanisms.

In RS, the asymmetry itself is modeled through a cost function J(r) that measures how far a system is from matter-antimatter balance. The framework proves two supporting facts: at balance (r = 1) the cost is zero, and for any r not equal to 1 with r positive, the cost is strictly positive. These are formal theorems that connect the cost function to the intuitive notion of asymmetry. The framework also maps each Sakharov condition (baryon number violation, C and CP violation, departure from equilibrium) to a condition on the cost function, but this mapping is a modeling choice, not a derived physical law.

The theorem does not claim that these five mechanisms are the only ones possible in the real universe, nor does it derive the measured baryon-to-photon ratio of about 6.1 × 10⁻¹⁰ from first principles. It does not explain the microphysics of leptogenesis or electroweak baryogenesis. What it provides is a clean, machine-checked statement that the framework's enumeration of mechanisms is internally consistent: five named mechanisms, no more, no less.

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

The theorem does not prove that only five baryogenesis mechanisms exist in nature. The framework does not derive the measured baryon-to-photon ratio from the cost function. The mapping from Sakharov conditions to cost conditions is a modeling choice, not a derived law.

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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