Cosmology Radiation Entropy Relation Fermi Entropy Eq Four Thirds Energy

The 4/3 factor for fermions

In statistical mechanics, the entropy density s of a gas of massless particles at temperature T is related to its energy density ρ by the simple formula s = (4/3)·ρ/T. This holds for both bosons (particles like photons that can share a quantum state) and fermions (particles like electrons that obey the Pauli exclusion principle). The formula is a standard result, but in the Recognition Science framework it is now proved, not assumed, for the fermionic case.

The proof starts from the microscopic definition of entropy for a quantum gas. For fermions, the entropy integrand is written as σ_F(x) = x²[−f ln f − (1−f)ln(1−f)], where f = 1/(eˣ+1) is the Fermi-Dirac occupation number. The theorem fermi_entropy_eq_four_thirds_energy states that the integral of this entropy integrand over all energies equals exactly 4/3 times the integral of the energy density integrand x³/(eˣ+1). The value of the entropy integral is 7π⁴/90, while the energy integral is 7π⁴/120, and their ratio is precisely 4/3.

The proof splits the entropy integrand into two pieces: the energy kernel plus a logarithmic term. The logarithmic kernel, x²·ln(1+e⁻ˣ), is expanded using the Mercator series and evaluated via its Mellin transform at s = 3, yielding Γ(3)·η(4) = 7π⁴/360. Adding this to the energy integral gives the total. The same structure works for bosons, where the entropy integral is 4π⁴/45 and the energy integral is π⁴/15, again with ratio 4/3. The framework's machine-checked library of formal theorems verifies every step.

In Recognition Science, this closes a gap: earlier work used the 4/3 factor as an assumed thermodynamic input, but now it is derived from the entropy functional of quantum statistics. The theorem also proves the fermionic 7/8 weight at the entropy level, not just the energy level. The entropy coefficient 2π²/45 emerges from the functional as well. However, the framework does not claim to derive the phase-space normalization g/(2π²) or the identification of the physical entropy density with the ideal-gas entropy functional; those remain definitional choices, not theorems.

/-- **THEOREM (s = (4/3)ρ/T, Fermi).** The Fermi entropy integral is exactly
`4/3` of the Fermi energy integral: the same 4/3 law holds for a massless
fermion gas. -/
theorem fermi_entropy_eq_four_thirds_energy :
    (∫ t in Ioi (0 : ℝ), fermiEntropyIntegrand t)
      = 4 / 3 * ∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t + 1) := by
  rw [fermi_entropy_integral_value, FermionWeightIntegral.fermi_integral_value]
  ring

/-- **THEOREM (Fermi entropy integral).**
`∫₀^∞ t²[−f ln f − (1−f)ln(1−f)] dt = 7π⁴/90` with `f = 1/(eᵗ+1)`. -/
theorem fermi_entropy_integral_value :
    (∫ t in Ioi (0 : ℝ), fermiEntropyIntegrand t) = 7 * π ^ 4 / 90 := by
  have hsplit : (∫ t in Ioi (0 : ℝ), fermiEntropyIntegrand t)
      = ∫ t in Ioi (0 : ℝ),
          (t ^ 3 / (Real.exp t + 1) + t ^ 2 * Real.log (1 + Real.exp (-t))) := by
    refine setIntegral_congr_fun measurableSet_Ioi fun t ht => ?_
    exact fermi_entropy_pointwise ht
  rw [hsplit, integral_add integrableOn_fermi_energy integrableOn_fermiLog,
    FermionWeightIntegral.fermi_integral_value, fermiLog_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

The phase-space normalization g/(2π²) is not derived, only the functional and numerical content of the 4/3 factor. The identification of the physical entropy density with the ideal-gas entropy functional is not proved. The theorem does not apply to massive particles or interacting gases.

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