Cosmology Sakharov From Ledger Baryogenesis Possible

What the theorem proves

In 1967, the physicist Andrei Sakharov listed what any process must have to create more matter than antimatter from an initially symmetric universe. Three things are needed: a way to change the total number of baryons (particles like protons and neutrons), a violation of the combined symmetries C and CP, and a departure from thermal equilibrium. Without all three, any early-universe mechanism would either conserve baryon number, produce equal amounts of matter and antimatter, or be washed out by thermal processes. Sakharov's conditions remain the standard checklist for baryogenesis, the hypothetical generation of the observed matter-antimatter asymmetry.

The Recognition Science (RS) framework claims to derive all three conditions from a single structural starting point. In RS, the universe is modeled as a discrete ledger, a record of recognition events with a forced cost structure. The framework's machine-checked library of formal theorems contains a declaration, baryogenesis_possible, which packages the three Sakharov conditions as a single provable statement. The first condition, baryon number violation, is established by showing that baryon number is a topological charge on a three-dimensional lattice that can change by exactly 3 units per sphaleron event, a type of non-perturbative transition. The second condition, CP violation, is derived from a positive Jarlskog invariant, a measure of CP violation in the quark sector. The third condition, departure from equilibrium, is asserted by defining the electroweak phase transition as first-order, which allows bubble nucleation to create out-of-equilibrium conditions.

It is important to state what this theorem does not do. The declaration baryogenesis_possible proves that the three conditions are consistent with the RS framework, not that baryogenesis actually occurred. The theorem's third condition is a definition set to True, not a derivation from the ledger structure; the framework's own documentation marks the physical bridge from recognition to linking as open. The statement also does not prove that the baryon asymmetry has the observed magnitude, nor does it show that the sphaleron rate or CP violation strength match experimental values. It establishes possibility, not actuality.

Within the framework, this result is a structural consistency check. It shows that the ledger's axioms do not forbid the ingredients needed for baryogenesis. The practical consequence is that RS can engage with a central question in cosmology: why is there more matter than antimatter? The framework provides a formal language in which the necessary conditions are provable, but it leaves the quantitative question of the observed asymmetry open. The theorem is a foundation stone, not a complete cosmological model.

/-- The master theorem: baryogenesis is possible in RS because all
    Sakharov conditions are derived (not postulated). -/
theorem baryogenesis_possible :
    deltaB_per_sphaleron = 3 ∧ cp_asymmetry_parameter ≠ 0 ∧ ew_transition_is_first_order :=
  ⟨rfl, cp_asymmetry_nonzero, out_of_equilibrium⟩

theorem sphaleron_changes_B_by_3 : deltaB_per_sphaleron = 3 := rfl

/-- CP violation source: J_CP > 0 from JarlskogInvariant. -/
theorem cp_source_positive : jarlskog_structural > 0 := jarlskog_positive

/-- The Higgs VEV in RS-native units is on the φ-ladder.
    v_EW = 246 GeV, and φ^{rung} × (anchor) gives the scale.

    The critical temperature for the EW phase transition is:
    T_EW ∝ v_EW ∝ φ^{rung_EW}

    At T > T_EW: symmetric phase, sphalerons active
    At T < T_EW: broken phase, sphalerons exponentially suppressed -/
def ew_transition_is_first_order : Prop :=
  True

What this page does not claim

The theorem does not prove that baryogenesis actually occurred in the early universe. It does not derive the observed magnitude of the matter-antimatter asymmetry. It does not establish that the electroweak phase transition is first-order from first principles.

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

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