Cosmology Fermion Weight Eta Term Odd

The odd terms

The Dirichlet eta function is the alternating cousin of the zeta function: where zeta sums 1/n⁴ over all positive integers, eta sums the same terms but with alternating signs, (−1)^(n+1)/n⁴. The fourth-power case is special because it appears in the thermodynamics of the early universe, where the energy density of a fermion species, such as an electron or neutrino, differs from that of a boson species by a factor of 7/8.

The declaration eta_term_odd is a lemma about the odd-indexed terms of that alternating sum. It states that for any nonnegative integer k, the term with index 2k+1, which carries a positive sign, is exactly the corresponding odd-indexed term of the zeta sum: (−1)^(2k+2)/(2k+1)⁴ equals 1/(2k+1)⁴. The proof is a one-line observation about parity: the exponent 2k+2 is always even, so the alternating sign is always +1. This is not a deep result on its own, but it is the precise statement needed to split the eta series into its positive odd part and negative even part.

In the Recognition Science framework, this lemma is part of a chain that upgrades a model input into a theorem. The framework's cosmology library previously used the number 7/8 as an unproved input when computing the entropy per photon in the early universe. The library now proves, in its machine-checked collection of formal theorems, that the eta series at 4 converges to exactly (7/8) times the zeta series at 4. The odd-term lemma is what makes that proof go through, by showing which zeta terms survive with a positive sign when the alternating series is recombined.

The declaration does not claim anything about the physical origin of the 7/8 factor. It is a statement about real numbers and infinite series, not about particle physics. The statistical-mechanics identification, that a fermion species contributes the Fermi–Dirac integral rather than the Bose–Einstein integral, remains a model input in the framework, not a derived theorem. The lemma also says nothing about other values of the eta function; it concerns only the fourth power, where the even-odd split yields the clean rational ratio.

/-- Odd-index terms of the alternating series are the odd-`ζ` terms:
`(−1)^(2k+2)/(2k+1)⁴ = 1/(2k+1)⁴`. -/
lemma eta_term_odd (k : ℕ) :
    ((-1 : ℝ)) ^ (2 * k + 1 + 1) / ((2 * k + 1 : ℕ) : ℝ) ^ 4
      = (1 : ℝ) / ((2 * k + 1 : ℕ) : ℝ) ^ 4 := by
  have heven : Even (2 * k + 1 + 1) := ⟨k + 1, by ring⟩
  rw [heven.neg_one_pow]

/-- **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]

What this page does not claim

The declaration does not claim that the 7/8 factor is derived from physics; the statistical-mechanics identification remains a model input. The lemma concerns only the fourth-power series, not other values of the eta function. No claim is made about the physical existence of fermions or the early universe; the theorem is about real-number series.

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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