Cosmology Radiation Entropy Relation Bose Log Integral Value

The Bose integral

The Bose–Einstein distribution describes how identical particles with integer spin, such as photons, occupy energy states at a given temperature. In the mathematics of blackbody radiation, integrals over this distribution appear constantly. One such integral, the subject here, is ∫₀^∞ t² · (−log(1 − e^(−t))) dt, where t is energy divided by temperature. The declaration boseLog_integral_value establishes that this integral equals π⁴/45, a clean closed form involving the fourth power of π.

This value is not an isolated curiosity. In the Recognition Science framework, it is a stepping stone in a chain that derives the radiation entropy relation s = (4/3)ρ/T from the microscopic entropy functional of quantum statistics. The entropy integrand for a Bose gas splits into an energy kernel plus this logarithmic kernel. The theorem proves the logarithmic part integrates to π⁴/45, and the energy part integrates to π⁴/15. Adding them gives 4π⁴/45, which is exactly 4/3 times the energy integral. That 4/3 factor, often assumed as a thermodynamic input, here emerges from the integral itself.

The proof method is classical. The logarithmic kernel expands by the Mercator series for log(1 − x), and the resulting Mellin transform at s = 3 evaluates via the gamma function and the Riemann zeta function at 4. The same technique handles the Fermi–Dirac case, where the analogous integral equals 7π⁴/360, and the ratio of the two entropy integrals comes out to 7/8, the familiar fermionic weight.

What the declaration does not claim is as important as what it proves. It does not assert anything about the physical entropy density of the universe; that identification requires additional definitional bridges, such as the phase-space normalization g/(2π²) and the choice of units. It does not claim to derive the 4/3 factor from nothing, only from the entropy functional once that functional is accepted. And it says nothing about the fine-structure constant or any other coupling; this is a pure mathematical identity about a specific integral.

/-- **THEOREM.** `∫₀^∞ t²·(−ln(1−e^{−t})) dt = π⁴/45`. -/
theorem boseLog_integral_value :
    (∫ t in Ioi (0 : ℝ), t ^ 2 * (-Real.log (1 - Real.exp (-t)))) = π ^ 4 / 45 := by
  have h := mellin_boseLog_value
  rw [mellin_boseLog_eq_integral] at h
  exact Complex.ofReal_inj.mp h

/-- **THEOREM (s = (4/3)ρ/T, Bose).** The Bose entropy integral is exactly
`4/3` of the Bose energy integral.  This is the dimensionless content of the
thermodynamic relation `s = (4/3)·ρ/T` for a massless boson gas, derived from
the microscopic entropy functional (never assumed). -/
theorem bose_entropy_eq_four_thirds_energy :
    (∫ t in Ioi (0 : ℝ), boseEntropyIntegrand t)
      = 4 / 3 * ∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t - 1) := by
  rw [bose_entropy_integral_value, FermionWeightIntegral.bose_integral_value]
  ring

/-- **THEOREM (7/8 at the entropy layer).** The Fermi entropy integral is
exactly `7/8` of the Bose one: the fermionic statistics weight of
`EntropyPerPhoton.fermionWeight` holds directly for entropy, not only for
energy. -/
theorem fermi_div_bose_entropy :
    (∫ t in Ioi (0 : ℝ), fermiEntropyIntegrand t)
      / (∫ t in Ioi (0 : ℝ), boseEntropyIntegrand t) = 7 / 8 := by
  rw [bose_entropy_integral_value, fermi_entropy_integral_value]
  rw [div_eq_iff (by positivity)]
  ring

What this page does not claim

This does not claim to derive the physical entropy density of the universe without additional model assumptions. This does not claim to derive the fine-structure constant or any other coupling constant. This does not claim the 4/3 factor is assumed rather than derived from the entropy functional.

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

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