Cosmology Ewphase Transition Sphaleron Hubble Ratio Pos

The washout ratio

In the first instants after the Big Bang, the universe was a hot soup of particles and antiparticles. As it cooled through the electroweak phase transition, around 100 GeV, a process called sphaleron transitions briefly became important. Sphalerons are field configurations that can change baryon number, the count of quarks and leptons. If they act too fast, they wash out any matter-antimatter asymmetry that has formed. If they act too slowly, they cannot generate one. The competition is captured by a single number: the ratio of the sphaleron transition rate to the Hubble expansion rate, the rate at which the universe itself is growing.

The Recognition Science framework formalizes this ratio in its machine-checked library of formal theorems. The framework's library defines the sphaleron-to-Hubble ratio, sphaleron_hubble_ratio, as the dimensionless product of the sphaleron rate per unit temperature to the fourth power, the cube of the transition temperature, and the inverse square root of the squared Hubble rate at that temperature. The declaration sphaleron_hubble_ratio_pos proves that this ratio is strictly greater than zero. This is a theorem, meaning it is a logical consequence of the definitions, with no additional assumptions. The proof chains together the positivity of the sphaleron rate, the transition temperature, and the Hubble rate, each of which is itself shown to be positive.

The framework models the transition temperature on its phi-ladder, a scale where particle masses and temperatures are expressed as powers of the golden ratio. It sets the electroweak temperature T_EW equal to phi to the 51st power, and the Hubble rate is computed from the radiation-dominated Friedmann equation with the full T^4 factor included. The framework also fixes the effective degrees of freedom g* at the standard high-temperature value of 106.75, and proves this constant matches the high-temperature limit of its threshold step function. These are definitional choices, not derived physical laws.

What the declaration does not do is connect this ratio to the final observed matter-antimatter asymmetry. The framework explicitly states that this ratio is a positive-definite scaffold, not a thermal washout calculation. It is not the source of the framework's Planck-matched baryon asymmetry expression, which contains no g* and no sphaleron-to-Hubble ratio. A genuine connection would require a full Boltzmann transport calculation through the transition, which remains an open target. The theorem proves the ratio is positive, and nothing more.

/-- The ratio is positive (both numerator and denominator are positive). -/
theorem sphaleron_hubble_ratio_pos : 0 < sphaleron_hubble_ratio := by
  unfold sphaleron_hubble_ratio
  exact div_pos (mul_pos sphaleron_rate_pos (pow_pos T_ew_pos 3))
    (Real.sqrt_pos.mpr hubble_sq_at_ew_pos)

/-- The RS-native EW temperature: T_EW = φ⁵¹ on the ladder (the EW-sector
    unit prefactor is absorbed into the unit choice; only φ-power structure
    matters for the RS-native ratios below). -/
def T_ew : ℝ := phi ^ (51 : ℕ)

/-- The effective degrees of freedom at the EW scale: the standard
    high-temperature SM value 106.75. Honest provenance (see
    StandardModel.RelativisticDOF header): the gauge GROUP and generation
    COUNT are RS-derived; the matter representations, minimal-neutrino
    convention, and the 7/8 thermal integral are imported SM content. This
    fixed number is the high-T evaluation of the temperature-dependent step
    function Cosmology.GStarThresholds.g_star (bridge theorem below). -/
def g_star_ew : ℝ := 106.75

/-- The fixed g★ = 106.75 used at T_EW equals the high-temperature
    evaluation of the g_star(T) threshold step function: the constant is a
    function value, not a free-standing number. -/
theorem g_star_ew_matches_threshold_fn :
    g_star_ew = ((GStarThresholds.g_star 200 : ℚ) : ℝ) := by
  rw [GStarThresholds.g_star_high]
  norm_num [g_star_ew]

What this page does not claim

The theorem does not prove the sphaleron-to-Hubble ratio is large or small, only that it is positive. The declaration does not show that the framework's baryon asymmetry expression is correct. The ratio is a scaffold, not a prediction of the observed matter-antimatter asymmetry.

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/EWPhaseTransition.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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