Cosmology Ewphase Transition G Star Ew Pos

The electroweak heat count

The electroweak phase transition, around 100 GeV in the early universe, is the epoch when the electromagnetic and weak forces separated. Cosmologists describe the radiation-dominated plasma at that moment with a single number, the effective number of relativistic degrees of freedom, written g⋆. It counts how many particle species are effectively massless and contributing to the energy density. In the Standard Model, at temperatures above about 160 GeV, that number is 106.75, a sum over quarks, gluons, gauge bosons, Higgs, and leptons, each weighted by its spin and statistics.

The Recognition Science framework, a formal system for deriving physical structure from a cost function, models this epoch in its own units. Its library of machine-checked theorems defines the electroweak temperature as a power of the golden ratio, φ⁵¹, and the effective degrees of freedom as the standard 106.75. The declaration g_star_ew_pos is a small but load-bearing result: it proves, from the definition alone, that this number is greater than zero. The proof is a single line of arithmetic, but it certifies that the framework's model of the early universe has a positive energy density, a necessary condition for any sensible cosmology.

The framework does not stop at the sign. It builds a chain: the positive degree count feeds into the Friedmann equation for the Hubble expansion rate, which combines with a sphaleron rate to form a dimensionless washout ratio, and finally a washout efficiency factor. Each step carries its own positivity theorem, and a certificate structure bundles them all together. The chain is internally consistent: every quantity is positive, every definition is explicit, and the whole module compiles with no unproved axioms.

What the declaration does not claim is just as important. The value 106.75 is not derived from first principles within the framework; it is imported from the Standard Model as a definitional choice. The framework's own contribution is the golden-ratio temperature scale and the formal scaffolding around it. Critically, the washout efficiency computed here is a scaffold, not a physical prediction. It is not connected to the framework's separate expression for the baryon asymmetry η_B, which contains no g⋆ and no sphaleron rate. A genuine calculation of how sphalerons wash out the asymmetry would require a full Boltzmann transport treatment through the transition, and that remains an open target.

theorem g_star_ew_pos : 0 < g_star_ew := by norm_num [g_star_ew]

/-- 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

/-- H² at T_EW in RS-native units, with the T⁴ factor INCLUDED:
    H² = friedmann_coeff · G_rs · g★ · T_EW⁴.
    (The 2026-06-25 review caught the earlier omission of T⁴; this
    definition now matches the header formula exactly.) -/
def hubble_sq_at_ew : ℝ := friedmann_coeff * G_rs * g_star_ew * T_ew ^ 4

/-- 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

The value 106.75 is not derived within the framework; it is imported from the Standard Model. The washout efficiency is not a physical prediction and does not feed into the framework's baryon asymmetry expression. The positivity proof does not establish that the electroweak phase transition actually occurred in the early 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/EWPhaseTransition.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

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