Cosmology Radiation Entropy Relation Entropy Coeff From Functional

The entropy coefficient

In the standard physics of the early universe, the entropy density of a gas of massless particles is written as s = (2π²/45)·g·T³, where T is the temperature and g counts the particle types. The prefactor 2π²/45 has long been a textbook result, but in the Recognition Science framework it was previously an assumed thermodynamic input. The declaration entropy_coeff_from_functional proves that this coefficient follows from the microscopic definition of entropy in quantum statistics, with no separate assumption of the 4/3 factor that relates entropy density to energy density.

The proof works with the entropy integrand for a gas of bosons, particles that can share a quantum state. The integrand, σ_B(x) = x²[(1+f)ln(1+f) − f ln f] with f = 1/(eˣ−1), is split into an energy kernel plus a logarithmic kernel. The logarithmic kernel expands as a Mercator series, and its Mellin transform at s = 3 evaluates to π⁴/45. Adding the energy integral π⁴/15 gives the total 4π⁴/45, which is exactly 4/3 of the energy integral. Dividing by 2π² yields the coefficient 2π²/45, the prefactor in the entropy formula.

The same method applies to fermions, particles that cannot share a quantum state. Their entropy integrand evaluates to 7π⁴/90, again exactly 4/3 of their energy integral, and the ratio of fermion to boson entropy is 7/8. This 7/8 weight, familiar from the energy density of neutrinos, now holds at the entropy level as well. Both results are proved as theorems in the framework's machine-checked library of formal theorems, meaning the calculations are verified step by step with no gaps.

In Recognition Science, this result closes a gap in a longer chain: earlier steps used the 4/3 factor as an input, and now it is derived. What remains a definitional choice, not a theorem, is the phase-space normalization g/(2π²), which includes the degeneracy count and the choice of units where ℏ = c = k_B = 1. The identification of the physical entropy density with the ideal-gas entropy functional is also a definitional bridge. The numerical content of s = (4/3)ρ/T and the 7/8 weight, however, is proved.

/-- **THEOREM (the 2π²/45 entropy coefficient from the functional).**
`s_γ = (g/2π²)·T³·∫σ_B = (2π²/45)·g·T³`: dividing the derived entropy
integral by the phase-space normalization `2π²` yields exactly the `2π²/45`
prefactor of the photon entropy density, with the `4/3` factor never
assumed. -/
theorem entropy_coeff_from_functional :
    (∫ t in Ioi (0 : ℝ), boseEntropyIntegrand t) / (2 * π ^ 2) = 2 * π ^ 2 / 45 := by
  rw [bose_entropy_integral_value]
  rw [div_eq_iff (by positivity)]
  ring

/-- **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 (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 (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 a definitional choice, not a derived theorem. The identification of the physical entropy density with the ideal-gas entropy functional is a definitional bridge, not a theorem. The result does not derive the fine-structure constant or any other coupling constant.

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)

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: