Cosmology Fermion Weight

The 7/8 fermion weight

In the standard model of cosmology, the entropy of the early universe is carried by relativistic particles. Bosons, particles like photons that can share a quantum state, contribute one unit of entropy per degree of freedom. Fermions, particles like electrons that obey the Pauli exclusion principle, contribute less. The ratio is a familiar factor: 7/8. This number appears in textbooks as the relative weight of a fermion species in the total entropy density, and it comes from the Fermi-Dirac integral that governs how fermions distribute in energy.

The factor 7/8 is not a coincidence. It follows from a classical identity linking two infinite series. The Riemann zeta function at 4, ζ(4), sums 1/n⁴ over all positive integers n, and equals π⁴/90. The Dirichlet eta function at 4, η(4), sums the same terms but with alternating signs, 1 - 1/16 + 1/81 - ..., and it also converges. The identity states that η(4) = (7/8) · ζ(4). The derivation splits the zeta sum into even and odd parts: the even terms contribute 1/16 of the total, the odd terms the remaining 15/16, and the alternating series is the odd part minus the even part, giving 15/16 - 1/16 = 14/16 = 7/8.

In Recognition Science, this identity is no longer taken as an imported fact. The framework's machine-checked library of formal theorems proves it from the definitions. The theorem fermionWeight_eq_eta_zeta_ratio states that the rational 7/8 used in the entropy-per-photon model is exactly η(4)/ζ(4). The proof splits the series, re-indexes the even terms, and recombines using the uniqueness of sums, with no axioms beyond the base three of the ambient type theory.

What this changes is the status of a model input. Previously, the 7/8 was a MODEL choice, a number put in by hand because the physics said so. Now the mathematics behind it is derived. The remaining model content is only the physical identification: that a fermion species contributes the Fermi-Dirac integral, which is the statistical mechanics input. The series identity itself, the part that could be checked by pure mathematics, is now a theorem.

The practical consequence is that one less assumption sits at the base of the framework's cosmology. When the framework computes the entropy of the early universe, the 7/8 weight is not a free parameter or an imported classical result; it is a proved consequence of the series definitions. This is a small step, but it is the kind of step that builds trust: a textbook number, derived from first principles in a machine-checked way, with the remaining physics input clearly labeled.

/-- **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

/-- **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

/-- The odd-index part of `ζ(4)`: `∑_k 1/(2k+1)⁴ = (π⁴/90)·(15/16)`.
Derived by subtraction: full sum minus even part, using uniqueness of
unconditional sums in ℝ. -/
lemma hasSum_odd :
    HasSum (fun k : ℕ => (1 : ℝ) / ((2 * k + 1 : ℕ) : ℝ) ^ 4)
      (π ^ 4 / 90 * (15 / 16)) := by
  obtain ⟨B, hB⟩ := summable_odd
  have hfull : HasSum (fun n : ℕ => (1 : ℝ) / (n : ℝ) ^ 4) (π ^ 4 / 90 / 16 + B) :=
    HasSum.even_add_odd hasSum_even hB
  have hval : π ^ 4 / 90 / 16 + B = π ^ 4 / 90 := hfull.unique hasSum_zeta_four
  have hBval : B = π ^ 4 / 90 * (15 / 16) := by linarith
  exact hBval ▸ hB

What this page does not claim

The module does not derive the statistical mechanics statement that a fermion species contributes the Fermi-Dirac integral; that remains a model input. The module does not prove the value of ζ(4) itself; it uses Mathlib's existing theorem for that. The module does not compute any numerical entropy value for the early 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/FermionWeight.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: