Cosmology Sphaleron Rate

The sphaleron rate

A sphaleron is a short-lived, unstable configuration of the electroweak field, a lump of energy that can violate baryon number and thereby change the amount of matter in the universe. In the standard cosmological picture, these configurations were abundant just after the big bang, when the universe was hotter than the electroweak phase transition. The rate at which they appeared, per unit volume, is 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 number of order one.

The prefactor κ_sph is not a free parameter in the standard theory; it must be computed from the gauge field dynamics, and lattice QCD estimates place it somewhere between 0.1 and 1.0. The Recognition Science framework derives a specific value from the topology of the configuration space. A sphaleron transition changes all three winding numbers of the SU(2) gauge field at once, which the framework models as a path through a graph with four vertices, the even sign-flip subgroup (ℤ/2ℤ)². The number of Hamiltonian cycles on the complete graph K₄ is 3, and each cycle traverses 4 edges. The prefactor is then (3 × 4) / 4² = 3/4.

In Recognition Science, the machine-checked library of formal theorems proves κ_sph = 3/4, that this prefactor is positive and less than one, and that the dimensionless rate Γ_sph / T⁴ = (3/4) · α_W⁵ is positive. The value 0.75 sits inside the lattice QCD range of 0.1 to 1.0. The framework also records an honest provenance note: the structure of the rate, the power of α_W and the combinatorial prefactor, is derived, but the input value of α_W itself is a boundary datum, not a derived constant.

What this means for a reader is that the framework offers a concrete, testable prediction for a quantity that standard theory leaves to expensive numerical simulation. The number 3/4 is not fitted; it falls out of a graph-theoretic count. Whether that count captures the real physics of the electroweak field is a separate question, one the framework's own certificate does not pretend to settle.

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)

/-- Sphaleron-rate provenance (honest, 2026-07-06):
    - κ_sph from Q₃ Hamiltonian cycles (combinatorial, structural)
    - α_W from α / sin²θ_W, where α is the RS CONSTRUCTION value whose
      exact value is a boundary datum in RS, not a derived constant
      (`Constants.AlphaGenesis.KappaGammaIrreducibility`, `MeasurementVerdict`).
    The structure is RS-derived; the α input carries one boundary datum. -/
structure SphaleronRateCert where
  kappa_from_Q3 : kappa_sph = 3 / 4
  kappa_positive : 0 < kappa_sph
  alpha_W_positive : 0 < alpha_W
  rate_positive : 0 < sphaleron_rate_dimensionless
  rate_formula : sphaleron_rate_dimensionless = kappa_sph * alpha_W ^ 5

What this page does not claim

The framework does not derive the value of the weak coupling α_W; that remains a boundary datum. The framework does not prove that sphalerons exist in the physical universe, only that the rate formula with its prefactor is positive and structurally correct. The graph-theoretic count is a model of the topology, not a derivation from the full Standard Model Lagrangian.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: