Cosmology Fermion Weight Integral Mellin Fermi Eq Integral

The 7/8 fermion weight

The Fermi–Dirac and Bose–Einstein distributions describe how particles fill energy states at a given temperature. Fermions, such as electrons, obey the Pauli exclusion principle, so no two can occupy the same state. Bosons, such as photons, face no such limit. In the early universe, both species contributed to the energy density, and the standard calculation assigns each fermionic species a statistical weight of 7/8 relative to a bosonic one. That weight is not an arbitrary convention; it falls out of a definite integral.

The integral in question compares the energy density of a fermion gas to that of a boson gas at the same temperature. In dimensionless form, the Bose integral is ∫ t³/(eᵗ−1) dt from 0 to ∞, and the Fermi integral is ∫ t³/(eᵗ+1) dt. The difference is only the sign in the denominator, yet it changes the value. The Bose integral evaluates to π⁴/15, while the Fermi integral evaluates to 7π⁴/120. Their ratio is exactly 7/8. This is a classical result, known since the early days of quantum statistics, and it is the mathematical core of the fermion weight used in cosmology.

In Recognition Science, the framework's machine-checked library of formal theorems proves this identity from first principles. The declaration mellin_fermi_eq_integral establishes that the Mellin transform of the Fermi kernel at s = 4 equals the thermodynamic integral itself. The Mellin transform is an integral transform, a tool that converts a function into a function of a complex variable, and here it acts as a bridge between the series representation of the distribution and its integral form. The proof expands the Fermi kernel as an alternating geometric series, applies the Mellin transform term by term, and evaluates the resulting Dirichlet series at s = 4. The same method handles the Bose kernel, and the ratio of the two closed forms gives 7/8.

What the declaration does not claim is any physical content about which particles are relativistic or how many species exist. The 7/8 factor is a pure mathematical statement about two integrals. The physics bookkeeping, the census of which species count as relativistic degrees of freedom, remains a separate modeling choice. The framework's theorem closes the mathematical gap, not the cosmological one.

/-- **THEOREM (7/8 at the thermodynamic layer).** The Fermi–Dirac energy
integral is exactly 7/8 of the Bose–Einstein one. -/
theorem fermi_div_bose_integral :
    (∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t + 1))
      / (∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t - 1)) = 7 / 8 := by
  rw [bose_integral_value, fermi_integral_value]
  rw [div_eq_iff (by positivity)]
  ring

/-- **THEOREM (Fermi–Dirac integral).** `∫_{0}^{∞} t³/(eᵗ+1) dt = 7π⁴/120`. -/
theorem fermi_integral_value :
    (∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t + 1)) = 7 * π ^ 4 / 120 := by
  have h := mellin_fermi_value
  rw [mellin_fermi_eq_integral] at h
  exact Complex.ofReal_inj.mp h

/-- **THEOREM (Bose–Einstein integral).** `∫_{0}^{∞} t³/(eᵗ−1) dt = π⁴/15`. -/
theorem bose_integral_value :
    (∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t - 1)) = π ^ 4 / 15 := by
  have h := mellin_bose_value
  rw [mellin_bose_eq_integral] at h
  exact Complex.ofReal_inj.mp h

/-- `mellin fermiKernel 4` is the (complexified) Fermi–Dirac integral. -/
lemma mellin_fermi_eq_integral :
    mellin fermiKernel 4
      = (((∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t + 1) : ℝ)) : ℂ) := by
  have h1 : mellin fermiKernel 4
      = ∫ t in Ioi (0 : ℝ), ((t ^ 3 / (Real.exp t + 1) : ℝ) : ℂ) := by
    unfold mellin
    refine setIntegral_congr_fun measurableSet_Ioi fun t ht => ?_
    rw [smul_eq_mul, show (4 : ℂ) - 1 = ((3 : ℕ) : ℂ) by norm_num,
      Complex.cpow_natCast]
    unfold fermiKernel
    push_cast
    ring
  rw [h1, integral_complex_ofReal]

What this page does not claim

The declaration does not identify which particle species are relativistic. The declaration does not compute the total entropy per photon in the early universe. The declaration does not derive the Fermi–Dirac distribution itself.

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

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