Cosmology Sphaleron Rate Kappa Sph Eq

The sphaleron prefactor

Sphalerons are fleeting, nonperturbative configurations of the electroweak gauge field that can change the number of baryons in a region of space. In the hot early universe, above the electroweak phase transition, their rate per unit volume is conventionally written Γ_sph / T⁴ = κ_sph · α_W⁵, where T is temperature, α_W is the weak coupling constant, and κ_sph is a dimensionless prefactor expected to be of order one. Lattice QCD estimates place κ_sph roughly in the range 0.1 to 1.0.

The machine-checked declaration kappa_sph_eq establishes that, within the Recognition Science framework, this prefactor is exactly 3/4. The derivation is purely combinatorial. It counts distinct Hamiltonian cycles on the complete graph with four vertices, K₄, which has 3 such cycles, each traversing 4 edges. Dividing the product 3 × 4 by the square of the even sign-flip count, 4², gives 12/16 = 3/4. The value 0.75 falls inside the lattice range.

In Recognition Science, the framework models this prefactor as arising from the topology of the Q₃ space, where a sphaleron transition corresponds to a path that changes three winding numbers simultaneously. The framework's library proves the prefactor is positive and less than one, and it assembles these facts into a certificate: the formula for the dimensionless rate, the positivity of κ_sph and α_W, and the positivity of the rate itself. The structure of the rate formula is derived; the value of α_W itself is not derived here but enters as a boundary datum.

What the declaration does not claim is broader. It does not derive α_W from first principles, nor does it prove that the lattice estimate is wrong. The agreement with the range 0.1 to 1.0 is a check, not a theorem. The framework's contribution is the exact combinatorial prefactor, a single number that the rest of the rate formula takes as given.

theorem kappa_sph_eq : kappa_sph = 3 / 4 := by
  unfold kappa_sph hamiltonian_cycles_K4 edges_per_cycle even_sign_flip_count
  norm_num

/-- Number of Hamiltonian cycles on K₄ (complete graph on 4 vertices).
    K₄ has 3 distinct Hamiltonian cycles:
    {(1234), (1243), (1324)} up to direction. -/
def hamiltonian_cycles_K4 : ℕ := 3

/-- Edges per Hamiltonian cycle on K₄. -/
def edges_per_cycle : ℕ := 4

theorem kappa_sph_pos : 0 < kappa_sph := by
  rw [kappa_sph_eq]; norm_num

theorem kappa_sph_lt_one : kappa_sph < 1 := by
  rw [kappa_sph_eq]; norm_num

What this page does not claim

The declaration does not derive the value of the weak coupling α_W. The declaration does not prove the lattice QCD estimate of κ_sph is correct or incorrect. The declaration does not establish that sphaleron processes actually occurred in the early universe.

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

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