Cosmology Fermion Weight Integral Fermi Integral Eq Weight Mul Bose

The 7/8 factor in the cosmic ledger

In the hot, dense early universe, particles of matter and particles of light (radiation) both carry energy. The standard calculation of how much energy each species contributes uses two different integrals. For particles that obey Fermi-Dirac statistics (matter particles like electrons), the energy density integral is over the function t³/(eᵗ+1). For particles that obey Bose-Einstein statistics (light particles like photons), the integral is over t³/(eᵗ−1). The question is how these two integrals compare.

The answer, known in physics since the 1930s, is that the Fermi integral is exactly 7/8 of the Bose integral. This 7/8 factor is a real, measurable consequence of the different quantum statistics: fermions cannot share a quantum state, while bosons can pile into one, so fermions carry slightly less energy at the same temperature. The factor appears in every cosmology textbook as part of the effective number of relativistic degrees of freedom, g*, which sets the expansion rate of the early universe.

The Recognition Science framework's machine-checked library of formal theorems has now closed the mathematical gap between that textbook result and a purely arithmetic identity. The library proves, as a theorem, that the integral of t³/(eᵗ+1) from zero to infinity equals 7/8 times the integral of t³/(eᵗ−1) over the same range. The proof works by expressing each integrand as a geometric series in e⁻ᵗ, then evaluating the resulting sums using the Riemann zeta function ζ(4) and the Dirichlet eta function η(4). The identity η(4) = (7/8)ζ(4) is itself a known result, and the library's contribution is to connect that series identity to the thermodynamic integrals.

The theorem, named fermi_integral_eq_weight_mul_bose, is a statement about pure mathematics: it relates two definite integrals. It does not, by itself, make any physical claim. The physics enters through a separate, explicitly modeled choice: which particles are counted as relativistic in the early universe. That census, the g* bookkeeping, is a modeling decision, not a theorem. The 7/8 factor is the part that is now proved; the list of which particles contribute is not.

What the proof changes is the status of a familiar number. The 7/8 in the entropy-per-photon calculation is no longer an empirical coincidence or a textbook assertion. Within the framework, it is a derived consequence of the definitions of the integrals and the properties of the zeta and eta functions. The remaining freedom in the cosmology is not in the statistics factor, but in the physical inventory of the universe's contents.

/-- **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

/-- **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

/-- `∑_{n≥0} (−1)ⁿ/(n+1)⁴ = η(4) = (7/8)·(π⁴/90)` (index-shifted
`FermionWeight.hasSum_eta_four`). -/
lemma hasSum_eta_shift :
    HasSum (fun n : ℕ => (-1 : ℝ) ^ n / ((n : ℝ) + 1) ^ 4)
      (7 / 8 * (π ^ 4 / 90)) := by
  have hbase : HasSum (fun m : ℕ => (-1 : ℝ) ^ (m + 1) / (m : ℝ) ^ 4)
      (7 / 8 * (π ^ 4 / 90)
        + ∑ i ∈ Finset.range 1, (-1 : ℝ) ^ (i + 1) / (i : ℝ) ^ 4) := by
    simpa using FermionWeight.hasSum_eta_four
  have h := (hasSum_nat_add_iff
    (f := fun m : ℕ => (-1 : ℝ) ^ (m + 1) / (m : ℝ) ^ 4) 1).mpr hbase
  exact h.congr_fun fun n => by push_cast [pow_succ]; ring

/-- `∑_{n≥0} 1/(n+1)⁴ = ζ(4) = π⁴/90` (index-shifted `hasSum_zeta_four`). -/
lemma hasSum_zeta_shift :
    HasSum (fun n : ℕ => (1 : ℝ) / ((n : ℝ) + 1) ^ 4) (π ^ 4 / 90) := by
  have hbase : HasSum (fun m : ℕ => (1 : ℝ) / (m : ℝ) ^ 4)
      (π ^ 4 / 90 + ∑ i ∈ Finset.range 1, (1 : ℝ) / (i : ℝ) ^ 4) := by
    simpa using hasSum_zeta_four
  have h := (hasSum_nat_add_iff
    (f := fun m : ℕ => (1 : ℝ) / (m : ℝ) ^ 4) 1).mpr hbase
  exact h.congr_fun fun n => by push_cast; ring

What this page does not claim

This theorem does not identify which particles are relativistic; that census is a separate modeling choice. This theorem does not derive the values of ζ(4) or η(4); it uses them as known inputs. This theorem does not make any statement about the physical expansion rate of the universe.

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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