Cosmology Sphaleron Rate Sphaleron Rate Pos

The positivity theorem

A sphaleron is a fleeting, nonperturbative configuration of the electroweak gauge field that can change the number of baryons (protons and neutrons) in a region of space. In the early universe, above the electroweak phase transition temperature, these processes were active and could have generated the matter-antimatter asymmetry we observe today. The rate at which sphalerons occur per unit volume is a key input to cosmological models of baryogenesis.

The standard formula for this rate, in natural units where temperature T is the scale, is Γ_sph / T⁴ = κ_sph · α_W⁵. Here α_W is the weak coupling constant, a measure of the strength of the weak force, and κ_sph is a dimensionless prefactor expected to be of order one. Lattice QCD, a computational method for studying the strong force, estimates κ_sph in the range 0.1 to 1.0.

In Recognition Science, the framework derives κ_sph from a combinatorial structure on the space Q₃. The derivation counts Hamiltonian cycles, paths that visit each vertex of a graph exactly once, through the even sign-flip subgroup of the cube. This yields κ_sph = 3/4, a value comfortably inside the lattice QCD window. The framework's library of machine-checked theorems then proves that the dimensionless rate Γ_sph / T⁴ is strictly greater than zero, given that both κ_sph and α_W are positive.

This positivity theorem is a structural guarantee, not a numerical prediction. It confirms that the derived formula describes a physically meaningful process that can actually occur, rather than a mathematical artifact that could vanish or become negative. The proof is unconditional, meaning it does not depend on the specific numerical value of α_W, only on it being positive.

/-- 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)

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

What this page does not claim

This answer does not claim that the sphaleron rate formula is derived from first principles in the framework. This answer does not claim that the value of α_W is derived within the framework. This answer does not claim that the positivity theorem makes any prediction about the numerical value of the rate.

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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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