Cosmology Fermion Weight Eta4 Div Zeta4

The 7/8 fermion weight

In the standard model of cosmology, the number of relativistic particle species in the early universe is often summarized by an effective count. For fermions, the entropy contribution carries a factor of 7/8 relative to bosons. That factor comes from Fermi-Dirac statistics: the integral ∫ x³/(eˣ+1) dx evaluates to Γ(4)·η(4), while the boson integral ∫ x³/(eˣ−1) dx equals Γ(4)·ζ(4), where η is the Dirichlet eta function and ζ is the Riemann zeta function. The ratio of the two integrals is therefore η(4)/ζ(4).

The classical identity η(4) = (7/8)·ζ(4) has been known since the 18th century. With ζ(4) = π⁴/90, the alternating series η(4) = Σ (−1)^(n+1)/n⁴ converges to (7/8)·(π⁴/90). The proof splits the zeta series into even and odd terms: the even part is ζ(4)/16, the odd part is 15ζ(4)/16, and the alternating series is odd minus even, giving (15/16 − 1/16)ζ(4) = 7/8·ζ(4).

In Recognition Science, the framework's machine-checked library of formal theorems contains a declaration named eta4_div_zeta4 that states exactly this ratio: the sum of (−1)^(n+1)/n⁴ divided by the sum of 1/n⁴ equals 7/8 as real numbers. The proof is fully formal, with no axioms beyond the base three of the logic system. A companion theorem, fermionWeight_eq_eta_zeta_ratio, connects this identity to the cosmological model: the constant 7/8 used as the fermion weight in the entropy-per-photon calculation is exactly η(4)/ζ(4).

What this does not claim is important. The declaration does not derive the physics of Fermi-Dirac statistics; that remains a model input, the identification that a fermion species contributes the fermionic integral. The series identity itself is now derived rather than imported, but the statistical mechanics is still a modeling choice. The theorem also does not say anything about the actual early universe, particle content, or the numerical value of the entropy; it is a statement about two infinite series and their ratio.

The consequence is that one less number in the cosmological ledger is a free parameter. The 7/8 factor, which might have been fitted to observations, is now a proved consequence of the series definition. This matters because it shows how a classical identity can be embedded in a formal framework, turning a numerical input into a theorem.

/-- **THEOREM (the eta/zeta ratio).** `η(4) / ζ(4) = 7/8` as real numbers. -/
theorem eta4_div_zeta4 :
    (∑' n : ℕ, (-1 : ℝ) ^ (n + 1) / (n : ℝ) ^ 4)
      / (∑' n : ℕ, (1 : ℝ) / (n : ℝ) ^ 4) = 7 / 8 := by
  rw [hasSum_eta_four.tsum_eq, hasSum_zeta_four.tsum_eq]
  have hz : (π : ℝ) ^ 4 / 90 ≠ 0 := by positivity
  rw [mul_div_assoc, div_self hz, mul_one]

/-- **THEOREM (fermion weight provenance).** The `7/8` MODEL constant in
`EntropyPerPhoton.fermionWeight` is exactly the eta/zeta ratio:
`fermionWeight · ζ(4) = η(4)`. The series identity is now derived, not
imported; the remaining MODEL content of the weight is only the
statistical-mechanics identification of the fermionic entropy integral. -/
theorem fermionWeight_eq_eta_zeta_ratio :
    ((EntropyPerPhoton.fermionWeight : ℚ) : ℝ)
        * ∑' n : ℕ, (1 : ℝ) / (n : ℝ) ^ 4
      = ∑' n : ℕ, (-1 : ℝ) ^ (n + 1) / (n : ℝ) ^ 4 := by
  rw [hasSum_eta_four.tsum_eq, hasSum_zeta_four.tsum_eq]
  unfold EntropyPerPhoton.fermionWeight
  push_cast
  ring

What this page does not claim

The declaration does not derive the statistical mechanics of Fermi-Dirac statistics; that remains a model input. The theorem does not make any claim about the actual particle content of the early universe. The identity is classical and not unique to the framework.

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

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