Cosmology Entropy Per Photon Entropy Per Photon Near 704

The entropy per photon

In the standard hot Big Bang picture, the early universe was a hot, dense plasma of particles and radiation. As it expanded and cooled, species of particles annihilated or decoupled, leaving behind a relic sea of photons and neutrinos. A useful bookkeeping number describes this aftermath: the entropy per photon, usually written s/nγ. It counts, roughly, how much disorder or information capacity remains in the radiation bath for each photon that survives today. Standard cosmology textbooks give the value as about 7.04, and observations of the cosmic microwave background are consistent with that figure.

The number 7.04 is not pulled from a vacuum; it follows from the particle content of the Standard Model and from thermodynamics. The formula is s/nγ = π⁴ g*s / (45 ζ(3)), where ζ(3) is Apéry's constant, about 1.2020569, and g*s counts the effective number of entropy-carrying degrees of freedom at late times. For the present-day universe, g*s equals 43/11, or about 3.9. That value comes from counting the photon's two polarizations, the electron-positron pairs, and three generations of neutrinos, with a 7/8 factor for fermions, and then accounting for the fact that neutrino decoupling leaves them cooler than photons by a factor of (4/11)^(1/3).

In Recognition Science, the framework's machine-checked library of formal theorems contains a module that derives this ratio from first principles rather than taking 7.04 as an input. The declaration entropyPerPhoton_near_704 is a theorem stating that the absolute difference between the derived entropy per photon and 7.04 is less than 0.0007. The derivation bounds ζ(3) between 1.202042 and 1.202065 by summing the first 40 terms exactly and trapping the infinite tail with telescoping series. It bounds π⁴ between 97.40900 and 97.40914 using standard inequalities on π. It then proves that the effective degrees of freedom g*s equals 43/11, not as an assumption but as a consequence of the particle content and of entropy conservation through electron-positron annihilation, which heats photons but not the already-decoupled neutrinos.

What the theorem does not claim is just as important as what it proves. The value 7.04 is not a prediction of a new physical law; it is a derivation of a known cosmological quantity using the Standard Model particle content and Fermi-Dirac statistics as inputs. The theorem does not say that the universe must have exactly this entropy per photon, nor does it derive the particle content itself. Those are model choices, not conclusions of the framework. The theorem also does not claim that the true value is exactly 7.04; it proves only that the derived value lies within a narrow window, between 7.0393 and 7.0396, which rounds to 7.04 at three significant figures.

Within the framework, this result matters because it replaces a bare constant in a larger dynamical calculation with a derived quantity. The bridge module propagates the window into a prefactor P = (28/79) · s/nγ and re-proves the rung-selection theorems with the fully derived prefactor. For a reader outside the framework, the takeaway is simpler: a number that cosmologists have used for decades can be traced, step by step, to the particle inventory of the Standard Model and to the laws of thermodynamics, with every step checked by a machine.

/-- **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]

/-- **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

What this page does not claim

The theorem does not derive the Standard Model particle content itself. The theorem does not claim the entropy per photon is a fundamental constant of nature. The theorem does not assert that the observed universe has exactly this entropy per photon.

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.

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: