Cosmology Fermion Weight Has Sum Eta Four

The 7/8 fermion weight

The Dirichlet eta function η(s) is the alternating cousin of the zeta function ζ(s): where ζ(4) sums 1 + 1/16 + 1/81 + ..., η(4) sums 1 − 1/16 + 1/81 − ... . Both converge to finite values. In 1735 Leonhard Euler showed ζ(4) = π⁴/90. The alternating series η(4) is not independent: it equals exactly 7/8 of ζ(4).

This ratio matters in cosmology. In the hot early universe, particles and antiparticles filled space in thermal equilibrium. Photons obey Bose-Einstein statistics, while electrons, neutrinos, and other fermions obey Fermi-Dirac statistics. The entropy density of a fermion species is lower than that of a boson species at the same temperature. Standard statistical mechanics gives the difference as a factor of 7/8, which appears in the effective number of relativistic degrees of freedom.

The recognition framework, a system that derives physics from a discrete record of events, formalizes this in a machine-checked library of formal theorems. Its declaration hasSum_eta_four proves that the alternating series converges unconditionally to (7/8)·(π⁴/90). A companion theorem, eta4_div_zeta4, states the ratio η(4)/ζ(4) equals 7/8 as real numbers. A third, fermionWeight_eq_eta_zeta_ratio, ties the framework's model constant for fermion entropy to this series identity.

The framework's contribution is mathematical provenance. The 7/8 factor was previously a model input: an assumption imported into the entropy calculation. The theorem removes that assumption from the mathematics. What remains a model choice is the physics identification: that a fermion species contributes the Fermi-Dirac integral to entropy. The series identity itself is now derived, not assumed.

This does not claim that the framework derives the 7/8 factor from first principles of recognition alone. The statistical mechanics input, the Fermi-Dirac integral, is still a model choice. It also does not claim that the framework computes the observed entropy of the universe; that would require additional cosmological assumptions. What the theorem secures is narrower: the classical series identity is now part of the framework's proved foundation, not an imported constant.

/-- **THEOREM (η(4) as a `HasSum`).** The alternating series
`∑ (−1)^(n+1)/n⁴` converges unconditionally to `(7/8)·(π⁴/90)`,
i.e. `η(4) = (7/8)·ζ(4)`. -/
theorem hasSum_eta_four :
    HasSum (fun n : ℕ => (-1 : ℝ) ^ (n + 1) / (n : ℝ) ^ 4)
      (7 / 8 * (π ^ 4 / 90)) := by
  have he : HasSum
      (fun k : ℕ => (-1 : ℝ) ^ (2 * k + 1) / ((2 * k : ℕ) : ℝ) ^ 4)
      (-(π ^ 4 / 90 / 16)) := by
    have hfun : (fun k : ℕ => (-1 : ℝ) ^ (2 * k + 1) / ((2 * k : ℕ) : ℝ) ^ 4)
        = fun k : ℕ => -((1 : ℝ) / ((2 * k : ℕ) : ℝ) ^ 4) := by
      funext k; exact eta_term_even k
    rw [hfun]
    exact hasSum_even.neg
  have ho : HasSum
      (fun k : ℕ => (-1 : ℝ) ^ (2 * k + 1 + 1) / ((2 * k + 1 : ℕ) : ℝ) ^ 4)
      (π ^ 4 / 90 * (15 / 16)) := by
    have hfun : (fun k : ℕ => (-1 : ℝ) ^ (2 * k + 1 + 1) / ((2 * k + 1 : ℕ) : ℝ) ^ 4)
        = fun k : ℕ => (1 : ℝ) / ((2 * k + 1 : ℕ) : ℝ) ^ 4 := by
      funext k; exact eta_term_odd k
    rw [hfun]
    exact hasSum_odd
  have h := HasSum.even_add_odd
    (f := fun n : ℕ => (-1 : ℝ) ^ (n + 1) / (n : ℝ) ^ 4) he ho
  convert h using 1
  ring

/-- **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 framework derives the 7/8 factor from recognition principles alone without physics input. The framework computes the observed entropy of the universe from this identity alone. The Fermi-Dirac integral itself is derived rather than chosen as a model.

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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