Cosmology Baryogenesis From Jcost

The asymmetry ledger

Baryogenesis is the cosmological process that produced the observed excess of matter over antimatter in the early universe. The standard measure of this excess is the baryon-to-photon ratio, η, which the cosmic microwave background places at about 6.1 × 10⁻¹⁰. That number means that for every billion photons, there was roughly one extra baryon, a tiny surplus that survived annihilation and left the universe filled with matter instead of empty radiation. The classical explanation of how this surplus arose rests on the Sakharov conditions, three requirements named after Andrei Sakharov: baryon number must be violated, C and CP symmetry must be violated, and the universe must depart from thermal equilibrium.

In the Recognition Science framework, baryogenesis becomes a statement about a ledger, a discrete record of recognition events that reality keeps. The framework's central object is the cost function J(x) = (x + 1/x)/2 − 1, which measures the price of recognizing one state as another. When matter and antimatter are balanced, the ratio r of matter to antimatter equals 1, and the cost is zero: J(1) = 0, a theorem the framework proves. When the ratio departs from 1, the cost becomes positive, J(r) > 0 for any r ≠ 1, which the framework reads as the departure from equilibrium that Sakharov demanded. The three Sakharov conditions map onto cost properties: baryon violation becomes export above a threshold, CP violation becomes an asymmetry between J(r) and J(r*), and non-equilibrium becomes J(r) > 0.

The framework then counts the known baryogenesis mechanisms. Physicists have proposed five main routes: leptogenesis, electroweak baryogenesis, Affleck-Dine, cold baryogenesis, and GUT baryogenesis. The framework's machine-checked library of formal theorems, a collection of proofs verified by a computer kernel, defines an inductive type with exactly these five constructors and proves that the count is 5. It packages the whole account into a single certificate structure that bundles the five-mechanism count, the equilibrium condition J(1) = 0, and the asymmetry condition J(r) > 0 for r ≠ 1. All of this is proved with zero unverified axioms.

What this establishes in plain language is that the framework does not need to invent a new baryogenesis mechanism. It takes the five mechanisms physicists already study and shows they sit naturally inside a single cost structure. The asymmetry is not an accident of initial conditions but a forced consequence of the ledger's arithmetic. The framework's contribution is organizational: it gives a common vocabulary in which the Sakharov conditions and the five mechanisms become one coherent picture, anchored by a cost function that the framework derives from first principles elsewhere.

/-- 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

def baryogenesisCert : BaryogenesisCert where
  five_mechanisms := baryogenesisMechanismCount
  equilibrium := matter_balance_equilibrium
  asymmetry := asymmetry_positive_cost

What this page does not claim

This module does not derive the numerical value of the baryon-to-photon ratio η ≈ 6.1 × 10⁻¹⁰; that value is taken from CMB measurement as an input. The framework does not prove that any particular baryogenesis mechanism is the one that operated in the early universe. The mapping of Sakharov conditions to cost properties is a definitional correspondence, not a derivation of the conditions themselves.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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