Cosmology Fermion Weight Eta Term Even

The even terms

The Dirichlet eta function is the alternating cousin of the zeta function: where zeta sums 1/n⁴ over all positive integers n, eta sums the same terms but with alternating signs, (−1)^(n+1)/n⁴. The fourth power makes both sums converge, and classical analysis gives eta(4) = (7/8)·zeta(4), with zeta(4) = π⁴/90. This exact rational ratio is what makes the number 7/8 appear in the entropy of a fermion gas: fermions obey the Fermi–Dirac distribution, and the integral that computes their energy density produces eta(4), while the corresponding boson integral produces zeta(4).

The lemma eta_term_even states a pointwise fact about the terms of that alternating series: for every even index 2k, the term (−1)^(2k+1)/(2k)⁴ equals −1/(2k)⁴. This is the algebraic observation that the alternating sign is negative at every even position. It is a single, small step in a longer proof, but it is the step that lets the even-index part of the eta series be identified with the negative of the even-index part of the zeta series, which is what makes the subtraction in the final ratio work. The declaration is a lemma in the framework's machine-checked library of formal theorems, and its proof is verified by the kernel with no additional axioms.

What the lemma does not claim is any physics. It says nothing about fermions, entropy, or cosmology. The physical content enters only in a separate, explicitly modeled step: the statistical-mechanics statement that a fermion species contributes the Fermi–Dirac integral to the entropy. That step is a MODEL input, not a theorem. The lemma's role is narrower and purely mathematical: it certifies the sign of the even terms in the series, so that the derived identity eta(4) = (7/8)·zeta(4) can replace the 7/8 constant as a proved result rather than an assumed one.

/-- Even-index terms of the alternating series are negatives of the
even-`ζ` terms: `(−1)^(2k+1)/(2k)⁴ = −1/(2k)⁴`. -/
lemma eta_term_even (k : ℕ) :
    ((-1 : ℝ)) ^ (2 * k + 1) / ((2 * k : ℕ) : ℝ) ^ 4
      = -((1 : ℝ) / ((2 * k : ℕ) : ℝ) ^ 4) := by
  rw [(odd_two_mul_add_one k).neg_one_pow]
  push_cast
  ring

/-- **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 (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 lemma does not establish any physical fact about fermions or entropy. The statistical-mechanics identification of the fermion entropy integral remains a MODEL input, not a theorem.

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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