Cosmology Entropy Per Photon Entropy Per Photon Eq Ratio

The entropy per photon

The entropy per photon is a standard quantity in cosmology: the total entropy density of the universe's radiation divided by the number density of photons. It matters because it stays nearly constant as the universe expands, so it is a convenient label for the thermal state of the early universe. The standard textbook value is about 7.04, and the machine-checked library of formal theorems in the Recognition Science framework derives this number from three ingredients rather than taking it as an input.

The first ingredient is the Riemann zeta function at 3, written ζ(3), which is about 1.2020569. The framework's library bounds this value between 1.202042 and 1.202065 by summing the first 40 terms exactly and trapping the infinite tail between two telescoping series. The second ingredient is π⁴, bounded between 97.40900 and 97.40914 using a known decimal expansion of π. The third is the effective number of entropy degrees of freedom, g*s, which the library sets to 43/11 from the Standard Model particle content: 2 for the photon's polarizations, 4 for the electron and positron, and 6 for the three neutrino generations, with a 7/8 fermion weight that is itself proved from a series identity, not assumed.

The derivation also accounts for the fact that neutrinos decouple from the plasma before electrons and positrons annihilate. That annihilation heats the photons but not the neutrinos, diluting the neutrino temperature relative to the photon temperature by a factor (4/11)^(1/3). This dilution step is proved from comoving entropy conservation, and it feeds into g*s = 2 + (7/8)·6·(4/11) = 43/11. Combining the three pieces gives the theorem:

s/n_γ = π⁴ · (43/11) / (45·ζ(3)) ∈ (7.0393, 7.0396).

The framework's library proves this window, and also proves that the result is within 0.0007 of 7.04, which is the rounded value used in the staging modules. The derivation replaces a bare constant with a chain of proofs: the zeta bound, the π⁴ bound, the fermion weight identity, the thermodynamic integrals, and the dilution factor are all theorem-level. What remains a model input is the Standard Model particle content itself, meaning the list of particles and their degrees of freedom, plus the two named conservation hypotheses (comoving entropy conservation and free neutrino streaming). The framework does not claim to derive the particle content from first principles, nor does it claim that the entropy per photon is exactly 7.04; it proves a tight window around that value. The practical consequence is that the baryogenesis dynamical lane in the framework now carries a derived prefactor instead of a fitted constant, which strengthens the chain of reasoning that uses it.

/-- **THEOREM.** `ζ(3) > 1.202042`. -/
theorem zeta3_gt : (1.202042 : ℝ) < zeta3 := by
  rw [zeta3_split]
  have h1 := S40_gt
  have h2 := tail_ge
  linarith

/-- **THEOREM.** `ζ(3) < 1.202065`. -/
theorem zeta3_lt : zeta3 < (1.202065 : ℝ) := by
  rw [zeta3_split]
  have h1 := S40_lt
  have h2 := tail_le
  linarith

/-- **THEOREM.** `g*s = 43/11 ≈ 3.909`. -/
theorem gStarS_eq : gStarS = 43 / 11 := by
  unfold gStarS gPhoton fermionWeight gNeutrino
  rw [dilutionCubed_eq]
  norm_num

/-- **THEOREM.** `s/n_γ > 7.0393`. -/
theorem entropyPerPhoton_gt : (7.0393 : ℝ) < entropyPerPhoton := by
  have hz := zeta3_lt
  have hp := pi4_gt
  have hzpos := zeta3_pos
  unfold entropyPerPhoton
  rw [lt_div_iff₀ (by linarith : (0 : ℝ) < 45 * zeta3)]
  nlinarith

/-- **THEOREM.** `s/n_γ < 7.0396`. -/
theorem entropyPerPhoton_lt : entropyPerPhoton < (7.0396 : ℝ) := by
  have hz := zeta3_gt
  have hp := pi4_lt
  have hzpos := zeta3_pos
  unfold entropyPerPhoton
  rw [div_lt_iff₀ (by linarith : (0 : ℝ) < 45 * zeta3)]
  nlinarith

/-- **THEOREM (the staged constant is the derived value to 3 s.f.).**
`|s/n_γ − 7.04| < 0.0007`: the bare `7.04` in the staging modules is the
derived ratio rounded to three significant figures. -/
theorem entropyPerPhoton_near_704 : |entropyPerPhoton - 7.04| < 0.0007 := by
  rw [abs_sub_lt_iff]
  constructor
  · linarith [entropyPerPhoton_lt]
  · linarith [entropyPerPhoton_gt]

What this page does not claim

The particle content of the Standard Model is not derived from first principles; it is a model input. The entropy per photon is not exactly 7.04; the theorem proves a window around that value. The framework does not claim that the universe's entropy per photon is measured to be exactly this value; it derives the theoretical value from the model.

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

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