Cosmology Sphaleron Rate Sphaleron Rate Dimensionless

The dimensionless rate

A sphaleron is a fleeting, unstable configuration of the electroweak gauge fields that can change the number of baryons (protons and neutrons) in a region of space. In the hot early universe, above the electroweak phase transition, these configurations occur at a rate per unit volume that cosmologists write as Γ_sph / T⁴ = κ_sph · α_W⁵. Here T is temperature, α_W is the weak coupling constant, and κ_sph is a dimensionless prefactor that standard estimates put somewhere in the range 0.1 to 1.0.

The Recognition Science framework's machine-checked library of formal theorems contains a declaration, sphaleron_rate_dimensionless, that pins this prefactor down. The framework models the space of possible sphaleron transitions as a three-dimensional sphere Q₃, and counts the topologically distinct paths that change all three winding numbers at once. That counting reduces to a pure combinatorial problem: the number of Hamiltonian cycles (loops that visit every vertex exactly once) on a complete graph with four vertices. There are exactly 3 such cycles, each traversing 4 edges, and dividing by the square of the number of even sign-flip symmetries, 4², gives κ_sph = 3 × 4 / 16 = 3/4. The library proves this equality, κ_sph = 3/4, and proves that the resulting dimensionless rate is positive.

That the prefactor is a clean rational number is a structural result, not a measurement. The framework's own provenance certificate is explicit about the boundary: the weak coupling α_W that enters the formula is taken from the standard construction α / sin²θ_W, where α is a boundary datum of the framework, not a derived constant. So the declaration establishes the combinatorial structure of the prefactor and the positivity of the rate, while the numerical value of α_W itself remains an input.

What the declaration does not claim is just as important. It does not claim that 3/4 is a measured value; lattice QCD estimates put κ_sph in the range 0.1 to 1.0, and 3/4 falls inside that band, but the framework's result is a derivation from a model, not an experimental determination. It does not claim that the sphaleron rate itself has been observed; the rate is a prediction for early-universe conditions, not a laboratory measurement. And it does not claim that the framework has derived the weak coupling α_W from first principles; that remains an open target, and the rate formula depends on it as an input.

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)

/-- The sphaleron rate is small (κ_sph < 1 and α_W < 1 would give this,
    but α_W may be > 1 depending on exact values; we prove > 0 unconditionally). -/
theorem sphaleron_rate_structural :
    sphaleron_rate_dimensionless = kappa_sph * alpha_W ^ 5 := rfl

What this page does not claim

The prefactor 3/4 is a measured value; it is a derived prediction within the framework's model. The sphaleron rate has been observed in experiment; it is a prediction for early-universe conditions. The framework derives the weak coupling α_W from first principles; it is an input boundary datum.

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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