Cosmology Fermion Weight Integral Mellin Bose Eq Integral

The 7/8 fermion weight

In the early universe, particles of two kinds filled space: fermions (matter particles like electrons and quarks) and bosons (force carriers like photons). Both contribute energy, but not equally. The question is how much less energy fermions carry. The answer, a factor of 7/8, is a standard result in statistical mechanics, and it appears in cosmology when counting the energy density of the hot primordial soup.

The standard derivation starts with the two energy distribution functions. For bosons, the average number of particles at a given energy follows the Bose-Einstein distribution; for fermions, the Fermi-Dirac distribution applies. Integrating the energy times the number of particles over all energies gives the total energy density. The two integrals look similar, but the fermion integral is smaller because the Fermi-Dirac distribution excludes states already occupied by another fermion, an effect with no boson analogue.

The declaration mellin_bose_eq_integral is a formal lemma in the framework's machine-checked library of formal theorems. It proves that a certain Mellin transform, a complex-analytic tool, equals the ordinary thermodynamic integral for the Bose-Einstein kernel. This is the bridge that lets the framework compute both energy integrals exactly. The Bose integral evaluates to π⁴/15, and the Fermi integral to 7π⁴/120, so their ratio is exactly 7/8. The framework's library proves this ratio, and it also proves that this 7/8 factor is precisely the fermion weight used in the entropy-per-photon bookkeeping.

What the declaration does not claim is important. It does not claim that the 7/8 factor applies to any specific particle census; that is a separate modeling choice about which species are relativistic. It does not claim that the integrals converge for all complex values of the Mellin parameter; the proof is for s = 4. And it does not claim that the 7/8 ratio is a new physical discovery; it is a rigorous restatement of a known statistical result, now with every step checked by a computer.

/-- **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

/-- **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 (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 (fermion weight provenance, integral layer).** The `7/8` MODEL
constant of `EntropyPerPhoton.fermionWeight` is the ratio of the actual
thermodynamic integrals: `∫ t³/(eᵗ+1) = fermionWeight · ∫ t³/(eᵗ−1)`.
Together with `FermionWeight.fermionWeight_eq_eta_zeta_ratio` (series layer)
this makes the full mathematical content of the 7/8 factor THEOREM; the
remaining MODEL content of the entropy chain is the relativistic-species
census (`g*`), not the statistics factor. -/
theorem fermi_integral_eq_weight_mul_bose :
    (∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t + 1))
      = ((EntropyPerPhoton.fermionWeight : ℚ) : ℝ)
          * ∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t - 1) := by
  rw [bose_integral_value, fermi_integral_value]
  unfold EntropyPerPhoton.fermionWeight
  push_cast
  ring

What this page does not claim

The 7/8 factor applies to any specific list of particle species. The Mellin transform equality holds for all complex values of the parameter. The 7/8 ratio is a new physical discovery rather than a rigorous restatement of a known result.

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)

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: