Cosmology Gstar Derivation G Star Derivation Cert

The certified count

In early-universe cosmology, the quantity g* (pronounced "g star") measures how many effective particle species filled the hot plasma. It is a weighted headcount: each boson contributes 1, each fermion contributes 7/8, and the sum tells you the energy density of the radiation bath. Above the electroweak phase transition, when all Standard Model particles are massless and relativistic, the standard textbook value is g* = 28 + (7/8) × 90 = 106.75. The 28 counts the gauge bosons (8 gluons, 3 weak bosons, 1 photon, each with two polarizations) plus the 4 real degrees of freedom of the Higgs doublet. The 90 counts the fermions across three generations: 72 quark degrees of freedom, 12 charged leptons, and 6 neutrinos.

The 7/8 factor is not a convention. It comes from the Fermi-Dirac versus Bose-Einstein energy-density ratio, η(4)/ζ(4), which equals exactly 7/8. A machine-checked library of formal theorems proves this ratio and identifies it with the fermion Boltzmann factor. The same library then counts every Standard Model state explicitly, multiplies by the 7/8, and derives the exact rational 427/4 = 106.75. The derivation is pure rational arithmetic, closed by a single computational step, with no approximations and no fitted parameters.

In Recognition Science, the framework models this counting as forced by its Q₃ chord-cube content: the gauge group SU(3)×SU(2)×U(1) yields 12 generators, the Higgs is one complex doublet, and the fermion content is fixed generation by generation. The framework's library packages the whole result in a certificate structure called gStarDerivationCert. The certificate bundles four theorems: the bosonic count equals 28, the fermionic count equals 90, the formula yields 427/4, and this derived value coincides exactly with the hand-entered constant g_star used in the baryon asymmetry module. The certificate is a formal guarantee that the derived number and the previously assumed number are the same object.

What the certificate does not claim is equally important. It does not derive the Standard Model's particle content from first principles; it takes the content as given and counts it. It does not claim that g* is 106.75 at all temperatures, only above the electroweak transition where every species is relativistic. And it does not claim that the 7/8 factor is a new physical law, only that it is the exact ratio of the two quantum statistics. The certificate is a bookkeeping proof: given the Standard Model's roster, the early-universe heat content is exactly 106.75, and the number used in baryogenesis calculations is the same number.

/-- 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

/-- The certificate is provable kernel-only. -/
theorem gStarDerivationCert : GStarDerivationCert :=
  { bosonic     := bosonic_dof_eq
    fermionic   := fermionic_dof_eq
    formula     := g_star_derived_eq
    bridge      := g_star_derived_eq_baryogenesis }

/-- `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 certificate does not derive the Standard Model's particle content from first principles. The certificate does not claim g* = 106.75 at temperatures below the electroweak phase transition. The certificate does not claim the 7/8 factor is a new physical law beyond the exact quantum statistics 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)

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.

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