Cosmology Gstar Derivation Charged Lepton Dof Eq

Counting charged leptons

The Standard Model of particle physics describes matter through a small set of building blocks. Among them are the charged leptons: the electron, the muon, and the tau, each with its antimatter partner. At temperatures far above the electroweak phase transition, all these particles behave as massless, relativistic species. Cosmologists count their degrees of freedom, the independent ways each particle can carry energy, to compute how fast the early universe expanded.

The counting rule is simple. Each charged lepton flavour contributes two spin states, and each spin state has a particle and an antiparticle. With three flavours, the total is 3 flavours times 2 spin states times 2 particle-antiparticle states, which equals 12. The declaration charged_lepton_dof_eq in the Recognition Science library proves exactly this: the number of charged lepton degrees of freedom is 12. The proof is a direct unfolding of definitions followed by a computation, checked by the machine.

This single count feeds a larger result. Adding the 12 charged lepton degrees of freedom to 72 quark degrees of freedom and 6 neutrino degrees of freedom gives 90 fermionic degrees of freedom. Combined with 28 bosonic degrees of freedom and the fermion weight 7/8, the total relativistic effective degrees of freedom g* becomes 106.75, the standard value used in early universe cosmology. The framework's library proves this value exactly, not approximately, as the rational number 427/4.

In Recognition Science, this counting is presented as derived rather than assumed. The framework models the Standard Model particle content through a chord-cube structure, and the count of charged leptons follows from that structure. The declaration establishes a mathematical fact about a chosen model, not a new physical measurement. It confirms what the Standard Model already implies, in a form a machine has verified step by step.

The declaration does not claim that the electron, muon, and tau are the only charged leptons that exist. It counts the three flavours the Standard Model contains. It also does not claim that 12 is a prediction from first principles; it is the arithmetic consequence of the model's input, three flavours with two spin states each and particle-antiparticle symmetry. The value 106.75 likewise depends on the Standard Model particle content being exactly what experiments have established.

/-- Charged-lepton count = 3 × 2 × 2 = 12. -/
theorem charged_lepton_dof_eq : charged_lepton_dof = 12 := by
  unfold charged_lepton_dof n_charged_leptons n_spin_states
         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]

/-- `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

/-- `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 declaration does not claim that the electron, muon, and tau are the only charged leptons that exist. The declaration does not claim that 12 is a prediction from first principles independent of the Standard Model particle content. The declaration does not claim that the value 106.75 is measured rather than derived from the model's input.

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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