Cosmology Entropy Per Photon G Star S Eq

The entropy count

In the early universe, the entropy carried by each photon is a standard quantity in cosmology. The usual textbook value is about 7.04, and it appears in calculations of how matter came to outnumber antimatter. The number is not measured directly; it is computed from the particle content of the Standard Model and from the way entropy behaves as the universe expands and cools.

The computation starts with a count of how many ways energy can be stored. Photons contribute 2, for their two polarizations. Electrons and positrons contribute 4, and the three neutrino generations contribute 6. Fermions such as electrons and neutrinos weigh less than bosons in the entropy count, by a factor of 7/8, because of the statistics they obey. When electrons and positrons annihilate, their entropy passes into the photons and heats them, but the neutrinos, already decoupled, do not share in that heating. The ratio of neutrino to photon temperature cubed is therefore 4/11, and the total effective number of entropy degrees of freedom today, called g*s, becomes 2 + (7/8) * 6 * (4/11) = 43/11.

The Recognition Science library contains a machine-checked proof, tagged as a theorem, that g*s equals 43/11. The declaration gStarS_eq establishes this equality in the framework's formal system. The proof is axiom-clean, meaning it relies only on the standard logical axioms and no framework-specific assumptions. The particle counts and the two conservation hypotheses remain model inputs, chosen by definition rather than derived.

From g*s, the library derives the entropy per photon as π⁴ * (43/11) / (45 * ζ(3)), where ζ(3) is Apéry's constant, the sum of reciprocal cubes. Bounding ζ(3) between 1.202042 and 1.202065 and π⁴ between 97.40900 and 97.40914, the proof shows the ratio lies between 7.0393 and 7.0396. This replaces the bare 7.04 with a derived window. The framework's library uses this window to build a dynamical prefactor for baryogenesis calculations, but that application is a separate step.

What the declaration does not claim is as important as what it proves. It does not derive the particle content of the Standard Model; the 2, 4, and 6 are inputs, not outputs. It does not prove that entropy conservation holds in the early universe; that is a physical hypothesis, named as such. And it does not claim the exact value 7.04, only a narrow interval around it. The proof is a piece of arithmetic applied to chosen physics, not a derivation of the physics itself.

/-- **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

/-- Photon internal dof: 2 polarizations (MODEL input). -/
def gPhoton : ℚ := 2

What this page does not claim

The particle content of the Standard Model is not derived, only counted. Entropy conservation in the early universe is not proven, only assumed as a named hypothesis. The exact value 7.04 is not claimed; the proof gives an interval between 7.0393 and 7.0396.

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)

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.

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