Cosmology Gstar Derivation G Star Derived Eq Baryogenesis

The bridge

In early-universe cosmology, the number of relativistic degrees of freedom at high temperature, written g*, controls how fast the universe cools and how the expansion rate relates to temperature. For the Standard Model above the electroweak phase transition, when all particles are effectively massless, the standard value is g* = 106.75. This number appears in textbooks as a fixed input. The declaration g_star_derived_eq_baryogenesis is a machine-checked proof that this same number follows from counting the Standard Model's particle content, and that it matches the value already used in the framework's baryon asymmetry derivation.

The counting proceeds in a clear way. Bosons contribute 28 degrees of freedom: the 12 gauge bosons of SU(3)×SU(2)×U(1), each with 2 polarizations, plus the 4 real components of the Higgs doublet. Fermions contribute 90: 72 from quarks (6 flavours × 3 colours × 2 spin × 2 particle/antiparticle), 12 from charged leptons, and 6 from neutrinos. The fermion contribution is weighted by 7/8, the standard Fermi-Dirac factor. The proof shows this factor is not a convention: it is the ratio of Fermi-Dirac to Bose-Einstein energy densities, η(4)/ζ(4), itself a theorem. Multiplying out gives exactly 427/4 = 106.75.

What the declaration does not claim is just as important. It does not derive the Standard Model's particle content from first principles; the counts of quarks, leptons, and gauge bosons are taken as input. It does not claim that the framework's deeper principles force the number of generations or the gauge group. It only shows that, given the Standard Model content and the standard thermal weighting, the effective degrees of freedom are exactly 106.75, consistent with the value used elsewhere in the framework's baryogenesis calculations.

The result matters because it turns a hand-entered constant into a derived quantity. In the framework's library, this is part of a larger effort to reduce unexplained inputs. The bridge theorem is the point where the derived count meets the existing baryon asymmetry module, confirming that the two sides of the framework agree on the same number. It is a small step, but it removes one more arbitrary constant from the cosmological story.

/-- The cast of the derived rational to `ℝ` matches the existing
    `g_star : ℝ` constant in `BaryonAsymmetryDerivation`. -/
theorem g_star_derived_eq_baryogenesis :
    ((g_star_derived : ℚ) : ℝ)
      = IndisputableMonolith.Cosmology.BaryonAsymmetryDerivation.g_star := by
  rw [g_star_derived_eq]
  unfold IndisputableMonolith.Cosmology.BaryonAsymmetryDerivation.g_star
  push_cast
  norm_num

/-- The fermionic Boltzmann factor `7/8` is the exact ratio of the
    Fermi-Dirac to Bose-Einstein energy-density integrals.

    STATUS: THEOREM (proved downstream, not a convention). The value is
    forced by quantum statistics:
    `Cosmology.ThermalWeightSevenEighths.fermi_bose_ratio` proves
    `mellin fermiKernel 4 / mellin boseKernel 4 = 7/8`
    (i.e. `∫ t³/(eᵗ+1) / ∫ t³/(eᵗ−1) = η(4)/ζ(4) = 7/8`), and
    `fermion_boltzmann_forced` identifies this `def` with that ratio.
    That module imports this one, so the pointer here is a docstring;
    the proof is axiom-clean (Mathlib base axioms only).

    The remaining physics input is *which* species takes the `+1`
    (Fermi-Dirac) sign: the exchange sign, which RS forces via
    `Foundation.EightTick.spin_statistics_key`. -/
def fermion_boltzmann : ℚ := 7 / 8

/-- `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 Standard Model particle content and gauge group are not derived; they are inputs to the counting. The number of generations (three) is not derived; it is assumed. The 7/8 factor is derived from Fermi-Dirac statistics, not from the framework's forcing principles.

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)

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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