Cosmology Entropy Per Photon Entropy Per Photon Gt

The entropy per photon

In standard cosmology, the entropy per photon is a measure of how much disorder, or entropy, is packed into each photon in the universe's radiation bath. It is a useful bookkeeping number because, after the early universe's heat death of particle creation, this ratio stays nearly constant as the universe expands. The conventional value quoted in textbooks is about 7.04, a number that appears in calculations of how the matter-antimatter asymmetry might have been generated.

The Recognition Science library, a machine-checked collection of formal theorems, does not take 7.04 as a given. Its declaration entropyPerPhoton_gt proves a lower bound: the ratio is greater than 7.0393. A companion theorem proves the upper bound, so the whole derivation pins the value to the narrow window (7.0393, 7.0396). The exact formula is s/nγ = π⁴ g*s / (45 ζ(3)), where ζ(3) is Apéry's constant, about 1.2020569, and g*s is the effective number of entropy degrees of freedom.

The derivation has three ingredients. First, the library proves a tight window for ζ(3) by splitting the infinite sum at 40 terms and trapping the tail between two telescoping series. Second, it uses a standard bound on π⁴. Third, it computes g*s = 43/11 from the Standard Model particle content: photons with 2 polarizations, electrons and positrons with 4 fermionic degrees of freedom, and 3 neutrino generations with 6. The 7/8 weight for fermions is itself proved from a series identity, and the neutrino dilution factor (4/11)³ follows from entropy conservation when electrons and positrons annihilate. Only the particle content and the conservation hypotheses remain as model inputs.

What the declaration does not claim is just as important. It does not prove that the entropy per photon is exactly 7.04; it proves a window around that value. It does not derive the particle content of the Standard Model; that is assumed. And it does not by itself explain why the universe has more matter than antimatter, though the value feeds into a larger framework that addresses that question.

/-- **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

/-- The ratio in terms of the derived `gStarS` (the `43/11` in the definition
is not bare: it is `gStarS`). -/
theorem entropyPerPhoton_eq_formula :
    entropyPerPhoton = Real.pi ^ 4 * (gStarS : ℝ) / (45 * zeta3) := by
  rw [gStarS_eq]
  norm_num [entropyPerPhoton]

/-- **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 declaration does not prove the entropy per photon equals exactly 7.04. The declaration does not derive the Standard Model particle content. The declaration alone does not explain the matter-antimatter asymmetry.

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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