Cosmology Fermion Weight Integral Fermi Integral Value

The closed form

Fermi-Dirac statistics describe how particles that cannot occupy the same quantum state, like electrons, distribute their energies. In the mathematics of the early universe, one integral appears constantly: the energy density of such particles. The integral is t cubed divided by e to the t plus one, integrated from zero to infinity. The declaration fermi_integral_value establishes that this integral evaluates exactly to 7 * π^4 / 120.

This is not a numerical approximation. It is an exact identity, proved in a machine-checked library of formal theorems. The proof works by recognizing the integral as a Mellin transform, a standard tool that converts this kind of integral into an infinite sum. At the value s = 4, the sum becomes the well-known Dirichlet series for the Riemann zeta function, with an alternating sign for the fermionic case. The result is a closed form: 7π^4/120.

The same technique gives the companion result for bosons, particles that can share quantum states, like photons. Their integral, t cubed divided by e to the t minus one, equals π^4/15. Taking the ratio of the fermion integral to the boson integral yields exactly 7/8. This is the famous 7/8 factor that appears in cosmological entropy calculations, connecting the statistical behavior of matter and radiation.

In Recognition Science, this theorem closes a mathematical gap. The framework's earlier work had derived the series identity η(4) = (7/8)·ζ(4). This result proves that the same 7/8 ratio holds for the actual thermodynamic integrals, not just the series. The framework models the early universe's entropy bookkeeping using this factor. The theorem guarantees that the 7/8 statistics factor is not a model choice but a proved mathematical fact. The only remaining model content in that chain is deciding which particle species are relativistic, a physical census, not a mathematical one.

The declaration does not claim to derive the fine-structure constant, nor does it prove the Riemann Hypothesis. It does not claim that the 7/8 factor applies to all particles in all conditions. It establishes one exact integral identity and its ratio to a related integral. That is a precise, bounded result, and it is complete.

/-- **THEOREM (Fermi–Dirac integral).** `∫_{0}^{∞} t³/(eᵗ+1) dt = 7π⁴/120`. -/
theorem fermi_integral_value :
    (∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t + 1)) = 7 * π ^ 4 / 120 := by
  have h := mellin_fermi_value
  rw [mellin_fermi_eq_integral] at h
  exact Complex.ofReal_inj.mp h

/-- **THEOREM (7/8 at the thermodynamic layer).** The Fermi–Dirac energy
integral is exactly 7/8 of the Bose–Einstein one. -/
theorem fermi_div_bose_integral :
    (∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t + 1))
      / (∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t - 1)) = 7 / 8 := by
  rw [bose_integral_value, fermi_integral_value]
  rw [div_eq_iff (by positivity)]
  ring

What this page does not claim

The declaration does not prove the Riemann Hypothesis. The declaration does not derive the fine-structure constant. The declaration does not claim the 7/8 factor applies to non-relativistic particles or in non-thermodynamic contexts.

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

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