Cosmology Gstar Derivation G Star Derived Eq Decimal

The derived count

In the standard model of cosmology, the number 106.75 is a familiar constant: the effective number of relativistic degrees of freedom in the hot early universe, above the electroweak phase transition. It appears in formulas for how fast the universe expanded and cooled in its first instants. The number comes from adding up every species of particle that was light enough to be created and destroyed freely in that hot plasma, with fermions counting slightly less than bosons because of quantum statistics.

The count is direct. Bosons contribute 28 degrees of freedom: 12 massless gauge bosons with two polarizations each, plus 4 real scalar degrees of freedom from the Higgs doublet. Fermions contribute 90: six quark flavors times three colors times two spin states times particle-antiparticle, which is 72, plus three charged leptons times two spin states times particle-antiparticle, which is 12, plus three neutrino flavors with one helicity each, which is 6. The fermion total is multiplied by the thermal weight 7/8, reflecting that Fermi-Dirac statistics suppress energy density relative to Bose-Einstein. Adding 28 plus (7/8) times 90 gives exactly 106.75, or 427/4.

In the Recognition Science framework, the declaration g_star_derived_eq_decimal establishes that this count is a theorem, not a hand-typed constant. The framework's machine-checked library of formal theorems proves, by exact rational arithmetic, that the derived quantity g_star_derived equals 10675/100. The proof expands the definition of g_star_derived into the sum of bosonic and fermionic degrees of freedom, substitutes the standard model particle content, and closes with a single arithmetic computation. The theorem also proves that this derived value equals the previously hand-entered constant used in baryogenesis calculations, so the two agree by proof rather than by assumption.

What the declaration does not claim is just as important. It does not derive the particle content of the standard model from first principles; it takes the gauge group, the number of generations, and the particle spectrum as inputs. It does not prove that the universe actually had these degrees of freedom at early times; that is a physical assumption about the standard model being the correct description above the electroweak scale. It does not derive the 7/8 thermal weight from the framework's cost function; that weight is itself a separate theorem about Fermi-Dirac versus Bose-Einstein energy densities, proved elsewhere in the library. The declaration is a counting theorem, not a prediction of new physics.

The consequence is practical. In this framework, a number that cosmologists usually take as an input becomes an output of a proof. The counting is exact, the arithmetic is checked, and the bridge to the older baryogenesis constant is closed. A reader can now trust that 106.75 in this framework is not an adjustable parameter but a derived consequence of the standard model's particle content, stated with full formal precision.

/-- `427 / 4 = 106.75` so the derived value matches the standard
    high-temperature SM value. -/
theorem g_star_derived_eq_decimal : g_star_derived = (10675 : ℚ) / 100 := by
  rw [g_star_derived_eq]
  norm_num

/-- `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

/-- 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

What this page does not claim

The declaration does not derive the standard model's particle content from first principles. The declaration does not prove that the universe actually had these degrees of freedom at early times. The declaration does not derive the 7/8 thermal weight from the framework's cost function.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: