Cosmology Gstar Derivation Neutrino Dof Eq

The neutrino count

In the hot early universe, particle species were massless and moved freely, and the energy they carried shaped how fast the cosmos expanded. Cosmologists compress that energy into a single number, the effective degrees of freedom g*, which counts how many particle states contribute. The Standard Model's high-temperature value is 106.75, and the framework's machine-checked library of formal theorems derives that value by counting the particle content explicitly.

The count starts with the bosons: 12 gauge generators from SU(3)×SU(2)×U(1), each with 2 polarizations, gives 24, plus the 4 real scalar degrees of freedom of the Higgs doublet, for 28 bosonic states. The fermions are heavier to count. Quarks contribute 6 flavors × 3 colors × 2 spins × 2 particle/antiparticle = 72. Charged leptons give 3 × 2 × 2 = 12. Neutrinos give 3 flavors × 1 helicity × 2 particle/antiparticle = 6. The fermionic total is 90, and the Fermi-Dirac to Bose-Einstein energy ratio is exactly 7/8, a proved theorem. The formula g* = 28 + (7/8) × 90 = 427/4 = 106.75 follows by exact rational arithmetic.

In Recognition Science, the declaration neutrino_dof_eq is a proved theorem, not an input: it derives the count of 6 from the definitions. The proof unfolds the definitions of three flavors, one helicity state per neutrino, and particle/antiparticle doubling, then closes by computation. The same library proves the full fermionic count of 90 and the bosonic count of 28, and exhibits a bridge theorem showing the derived 106.75 coincides with the value used in the baryon asymmetry module. This turns a hand-entered constant into a derived quantity.

The theorem does not claim that neutrinos are massless, that they have no other helicity states, or that the framework derives the Standard Model's particle content from first principles. It counts the degrees of freedom given the Standard Model's three generations, one helicity per neutrino, and the doubling for antiparticles. The 7/8 factor is proved as a ratio of energy densities, not assumed. The result is a bookkeeping identity: given the particle content, the count is exact.

/-- Neutrino count = 3 × 1 × 2 = 6. -/
theorem neutrino_dof_eq : neutrino_dof = 6 := by
  unfold neutrino_dof n_neutrino_flavours n_particle_antiparticle
  decide

/-- Fermion count = 72 + 12 + 6 = 90. -/
theorem fermionic_dof_eq : fermionic_dof = 90 := by
  unfold fermionic_dof
  rw [quark_dof_eq, charged_lepton_dof_eq, neutrino_dof_eq]

/-- Bosonic count is the standard 28. -/
theorem bosonic_dof_eq : bosonic_dof = 28 := by
  unfold bosonic_dof gauge_dof gauge_generators gauge_polarisations higgs_dof
  decide

/-- `g_⋆ = 28 + (7/8) × 90 = 28 + 78.75 = 106.75 = 427/4`. -/
theorem g_star_derived_eq : g_star_derived = (427 : ℚ) / 4 := by
  unfold g_star_derived fermion_boltzmann bosonic_dof gauge_dof
         gauge_generators gauge_polarisations higgs_dof
         fermionic_dof quark_dof charged_lepton_dof neutrino_dof
         n_quark_flavours n_colours n_spin_states n_particle_antiparticle
         n_charged_leptons n_neutrino_flavours
  norm_num

What this page does not claim

The theorem does not claim neutrinos are massless or that they have no other helicity states. The theorem does not derive the Standard Model's particle content from first principles. The theorem does not claim the 7/8 factor is a convention; it is proved as an energy-density ratio.

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

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