Cosmology Sphaleron Rate Sphaleron Rate Structural

The structural rate formula

A sphaleron is a special, unstable configuration of the electroweak gauge fields. In the standard cosmological picture, these configurations mediate reactions that violate baryon number, the quantum number carried by protons and neutrons. At temperatures above the electroweak phase transition, roughly 160 GeV, these reactions proceed at a rate per unit volume given by Γ_sph / T⁴ = κ_sph · α_W⁵. Here T is temperature, α_W is the weak coupling constant, a number measuring the strength of the weak force, and κ_sph is a dimensionless prefactor, a pure number expected to be of order one. The formula itself is standard in thermal field theory; the prefactor is the part that requires calculation.

The declaration sphaleron_rate_structural in the framework's machine-checked library of formal theorems establishes that this rate formula holds in the Recognition Science account: the dimensionless rate is exactly the product of the prefactor κ_sph and the fifth power of the weak coupling α_W. The theorem is a definitional identity, a statement that the rate is defined to be this product, and it is proved by unfolding the definitions. It does not by itself fix the numerical value of the rate, because that depends on the values of κ_sph and α_W. The declaration also bundles positivity: it proves that the rate is strictly positive, since both the prefactor and the coupling are positive.

Within the framework, the prefactor is not left as a free parameter. A separate combinatorial argument, also machine-checked, derives κ_sph = 3/4 from the topology of the Q₃ configuration space. The counting runs over Hamiltonian cycles, paths that visit each vertex of a graph exactly once, on the complete graph with four vertices, which has three such cycles, each traversing four edges. The prefactor is the product of these two numbers divided by the square of the size of the even sign-flip subgroup, giving 12/16 = 3/4. This value sits inside the range 0.1 to 1.0 estimated by lattice QCD, the numerical simulation of the strong force on a discrete grid.

The structural theorem is the load-bearing part of the derivation. It guarantees that whatever the inputs κ_sph and α_W turn out to be, the rate formula has exactly this shape, with no extra terms and no other factors. The combinatorial result then supplies a concrete value for κ_sph. The weak coupling α_W, however, is not derived from first principles in this file. The provenance certificate attached to the code states plainly that α_W is a boundary datum, an input taken from measurement, not a derived constant. The structure of the rate is derived; the strength of the weak force is carried in from outside.

What the declaration does not claim is as important as what it proves. It does not claim that the sphaleron rate itself has a specific numerical value, because that would require a numerical value for α_W. It does not claim that the combinatorial prefactor 3/4 is the measured value of κ_sph; it claims only that this value is consistent with the lattice range. And it does not claim that sphalerons are the actual cause of the matter-antimatter asymmetry in the universe. That remains a question for cosmology, not for a theorem about a rate formula.

/-- 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

/-- 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

/-- 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 declaration does not claim a specific numerical value for the sphaleron rate. The declaration does not claim that the prefactor 3/4 is the measured value of κ_sph. The declaration does not claim that sphalerons caused the matter-antimatter asymmetry in the 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)

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.

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