Cosmology Baryogenesis From Jcost Matter Balance Equilibrium

Matter balance in the ledger

In cosmology, the baryon-to-photon ratio η ≈ 6.1 × 10⁻¹⁰ measures how many more particles of matter than antimatter existed in the early universe, relative to the number of photons. That small surplus is why any matter exists today. The Recognition Science framework models this asymmetry as a σ-imbalance in its central object, the ledger, a discrete record of recognition events. The framework's cost function J(x) assigns a price to any imbalance, where x is the ratio of matter to antimatter. The theorem matter_balance_equilibrium proves that J(1) = 0, meaning a perfectly balanced universe, equal amounts of matter and antimatter, has zero cost.

The classical mathematics here is simple. The cost function J(x) = (x + 1/x)/2 − 1, which the framework proves is the only function satisfying five plain conditions. At x = 1, the formula gives (1 + 1)/2 − 1 = 0. This is the equilibrium point. A companion theorem, asymmetry_positive_cost, proves that for any positive ratio r not equal to 1, the cost J(r) is strictly greater than 0. So the framework's ledger charges a positive price for any departure from perfect matter-antimatter balance, and zero only for exact balance. The two theorems together define what the framework means by equilibrium: it is not a physical process but a property of the cost function at the ratio 1.

In Recognition Science, the Sakharov conditions for baryogenesis, the physical processes that generated the asymmetry, are mapped onto properties of this cost function. Baryon number violation becomes σ-export above a threshold; CP violation becomes J(r) ≠ J(r*) for the CP-conjugate ratio r*; and departure from equilibrium becomes J(r) > 0. The framework also counts five known baryogenesis mechanisms, leptogenesis, electroweak, Affleck-Dine, cold, and GUT, and proves in a machine-checked library of formal theorems that this count equals exactly 5, matching the number of mechanisms physicists list. These are definitions and identifications, not derivations of the physical mechanisms themselves.

What the declaration does not claim is equally important. The theorem proves a property of the framework's cost function, not a physical law. It does not derive the measured value η ≈ 6.1 × 10⁻¹⁰ from first principles; that number remains an empirical input. It does not prove that any of the five baryogenesis mechanisms actually occurred in the early universe. The framework's contribution is a consistent formal language in which the asymmetry has a zero point and a positive cost, and a proof that this language is internally coherent. The physical bridge from the ledger to the early universe remains open.

/-- 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

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

structure BaryogenesisCert where
  five_mechanisms : Fintype.card BaryogenesisMechanism = 5
  equilibrium : Jcost 1 = 0
  asymmetry : ∀ {r : ℝ}, 0 < r → r ≠ 1 → 0 < Jcost r

What this page does not claim

The theorem does not derive the measured baryon-to-photon ratio η ≈ 6.1 × 10⁻¹⁰ from the framework's axioms. The framework does not prove that any of the five baryogenesis mechanisms actually occurred. The mapping of Sakharov conditions to cost-function properties is a definitional choice, not a derivation.

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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