Cosmology Sakharov From Ledger Sphaleron Changes B By 3

The three-unit jump

Baryon number is a conserved quantity in most particle interactions, but certain non-perturbative processes, called sphalerons, can change it. In the standard model, a sphaleron transition changes baryon number by the number of fermion generations, which is three. The Recognition Science (RS) framework derives this same result from its own structure: the declaration sphaleron_changes_B_by_3 proves that a single sphaleron event changes baryon number by exactly three.

The proof is a direct consequence of the framework's ledger model. In RS, baryon number is a winding charge along one axis of a three-dimensional integer lattice. A sphaleron is a collective rotation of all three axes simultaneously, and the change in baryon number equals the number of face pairs in three dimensions, which is three. This is a formal theorem in the machine-checked library, not a postulate.

The declaration does not claim that this change is the full story of baryogenesis. It establishes only the magnitude of the change, not the rate at which sphalerons occur, nor the resulting asymmetry. The rate is given separately as proportional to α_W⁵ T⁴, and the framework also derives the other two Sakharov conditions, CP violation and departure from thermal equilibrium, in separate declarations.

What the theorem does not do is prove that baryon minus lepton number (B − L) is conserved. The module explicitly notes that stating B − L conservation requires separate baryon and lepton charge changes under a sphaleron transition, and the current framework only defines the baryon change. This remains an open target, not a derived result.

theorem sphaleron_changes_B_by_3 : deltaB_per_sphaleron = 3 := rfl

/-- The three independent conservation laws in D = 3. -/
theorem three_conservation_laws : face_pairs 3 = 3 := rfl

/-- 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⟩

What this page does not claim

The theorem does not prove baryon minus lepton number conservation, which requires separate lepton charge changes. The theorem does not establish the sphaleron rate or the resulting matter-antimatter asymmetry. The theorem does not state that sphalerons are the only source of baryon number violation.

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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