Cosmology Sphaleron Rate Kappa Sph Lt One

The prefactor bound

A sphaleron is a configuration of gauge fields that can change the number of baryons in the early universe. At temperatures above the electroweak phase transition, the rate per unit volume of these transitions is written as Γ_sph / T⁴ = κ_sph · α_W⁵, where α_W is the weak coupling constant and κ_sph is a dimensionless prefactor expected to be of order one. The Recognition Science library, a machine-checked collection of formal theorems, contains a declaration named kappa_sph_lt_one that proves this prefactor is less than one.

The proof is not a measurement or a fit. It follows from a combinatorial argument. The library defines κ_sph as (3 × 4) / 4² = 12/16 = 3/4. The number 3 counts the distinct Hamiltonian cycles on a complete graph with four vertices, and 4 counts the edges in each cycle. The denominator 4² is the square of the size of the even sign-flip subgroup on three dimensions. The theorem kappa_sph_eq states κ_sph = 3/4, and kappa_sph_lt_one is a direct corollary: since 3/4 is less than 1, the prefactor is less than 1.

The value 3/4 = 0.75 falls within the order-of-magnitude range 0.1 to 1.0 estimated by lattice QCD. The library also proves the prefactor is positive, and that the full dimensionless rate Γ_sph / T⁴ = κ_sph · α_W⁵ is positive. These are structural results: they hold by the definitions and the combinatorial counting, independent of the measured value of α_W.

The declaration does not claim that the sphaleron rate itself is small. That would require α_W < 1, which the library does not assume. It also does not claim that the combinatorial derivation is the physical mechanism; it is a structural identification within the framework. The weak coupling α_W itself is not derived from first principles in this file; it is taken as a boundary input, with the structure of the rate formula being the derived part.

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

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

/-- The sphaleron rate is positive (κ_sph > 0 and α_W > 0). -/
theorem sphaleron_rate_pos : 0 < sphaleron_rate_dimensionless := by
  unfold sphaleron_rate_dimensionless
  exact mul_pos kappa_sph_pos (pow_pos alpha_W_pos 5)

What this page does not claim

The sphaleron rate itself is small, since that would require assuming α_W < 1. The weak coupling α_W is derived from first principles in this file; it is a boundary input. The combinatorial derivation is the physical mechanism, rather than a structural identification.

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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