Cosmology Ewphase Transition G Star Ew Matches Threshold Fn

The g-star bridge

In the hot early universe, the expansion rate depends on how many particle species are present. Cosmologists package this count as g*, the effective number of relativistic degrees of freedom. At the electroweak phase transition, around 100 GeV, the Standard Model value is 106.75: photons, quarks, leptons, gluons, and the W and Z bosons all contribute. This number enters the Friedmann equation, which governs how fast the universe expands during radiation domination.

The machine-checked library of formal theorems contains a declaration, g_star_ew_matches_threshold_fn, that proves a consistency statement: the fixed 106.75 used at the transition temperature equals the high-temperature evaluation of a step function g_star(T). This step function models how the effective degrees of freedom change as the universe cools and particles drop out of thermal equilibrium. The theorem is a bridge: it confirms that the constant used in the electroweak calculation is not a free-floating number but a specific value of a defined function.

The proof is a simple computation. It unfolds the definition of g_star_ew as 106.75, applies the high-temperature branch of the step function, and evaluates the result. The declaration carries no axioms beyond the standard ones and has zero "sorry" placeholders. It is a formal certificate that the two descriptions agree.

In Recognition Science, this theorem is part of a larger scaffold for electroweak baryogenesis, the proposed mechanism for the universe's matter-antimatter asymmetry. The scaffold builds the sphaleron-to-Hubble ratio and a washout efficiency factor, both positive-definite quantities. But the scaffold is explicitly not connected to the framework's Planck-matched expression for the baryon asymmetry η_B = φ⁻⁴⁴·(1−φ⁻⁸)². That expression contains no g* and no sphaleron rate; the quantities built here are separate. A genuine connection requires a full Boltzmann transport treatment through the transition, which remains an open target.

The practical payoff is a clean separation of what is established from what is hoped. The theorem establishes a numerical identity within a defined model. It does not establish that the model's washout calculation is physically correct, nor that the framework's baryon asymmetry formula follows from it. The step function itself is a definitional choice, not a derived law.

/-- 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]

/-- 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]

/-- The washout efficiency factor: sphaleron_hubble_ratio / g★.

    In standard electroweak baryogenesis, the baryon asymmetry is
    η_B ∝ (ε_CP / g★) × min(1, R) where ε_CP is the CP asymmetry and R the
    sphaleron-Hubble ratio; this quantity is the combination R / g★.

    HONEST SCOPE (per the 2026-06-25 review): this is a positive-definite
    SCAFFOLD, not a thermal washout calculation. It is NOT connected to the
    Planck-matched η_B = φ⁻⁴⁴·(1−φ⁻⁸)² expression, which contains no g★ and
    no Γ_sph/H. A genuine connection requires Boltzmann transport through
    the transition, which is OPEN. -/
def effective_washout : ℝ := sphaleron_hubble_ratio / g_star_ew

What this page does not claim

This theorem does not prove that the washout calculation is physically accurate. This theorem does not derive the framework's baryon asymmetry formula. This theorem does not establish the step function g_star(T) as a derived law; it is a definitional choice.

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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Derived articles

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