Cosmology Entropy Per Photon Entropy Per Photon Pos

The entropy per photon

In the standard model of cosmology, the entropy per photon, written s/n_γ, is a measure of how much disorder or information content is carried by each photon in the hot plasma of the early universe. It is a fundamental input to calculations of how matter came to outnumber antimatter. The conventional value, used in textbooks such as Kolb and Turner's The Early Universe, is approximately 7.04. This number is not measured directly; it is computed from the particle content of the standard model and the statistics that govern how particles behave at high temperature.

The Recognition Science framework's machine-checked library of formal theorems contains a declaration, entropyPerPhoton_pos, which establishes that this quantity is positive. More precisely, it proves that the entropy per photon is greater than zero. This is a theorem, not an assumption. The proof rests on a chain of results that pin the value down. The framework first proves a rigorous window for the Riemann zeta function at 3, ζ(3), showing it lies between 1.202042 and 1.202065. It then proves a window for π⁴, between 97.40900 and 97.40914. These two ingredients, combined with the standard model's particle content, yield the full formula: s/n_γ = π⁴ (43/11) / (45 ζ(3)). The theorem entropyPerPhoton_pos is the final step, showing this expression is strictly positive.

The derivation of the 43/11 factor is itself a notable piece of the framework's work. It comes from counting the effective degrees of freedom for entropy in the present-day universe. The framework models the photon with 2 polarizations, electrons and positrons with 4 fermionic degrees of freedom, and 3 neutrino generations with 6 fermionic degrees of freedom. The 7/8 weighting for fermions, which accounts for their different statistics, is proved as a theorem, not assumed. The framework also derives the neutrino temperature dilution factor of 4/11 from entropy conservation when electrons and positrons annihilate. These steps combine to give g*s = 43/11, and the final result lands in the window (7.0393, 7.0396), within 0.0007 of the conventional 7.04.

In Recognition Science, this derivation is part of a larger program. The framework treats the standard model's particle content and the laws of thermodynamics as inputs, not as results it must reproduce. Its library shows that the entropy per photon is not a free parameter but a consequence of those inputs. The declaration entropyPerPhoton_pos is a small but load-bearing piece of that structure. It guarantees that the dynamical prefactor used in baryogenesis calculations, which scales with s/n_γ, is well-defined and positive. This is a necessary condition for the framework's subsequent rung-selection theorems to hold.

What entropyPerPhoton_pos does not claim is more limited than what it does. It does not claim that the value 7.04 is an exact number; the theorem only establishes a window around it. It does not claim that the standard model's particle content is the only possible one, or that the framework has derived that content from deeper principles. The particle content, the 2 polarizations of the photon, the 4 degrees of freedom of the electron, the 6 of the neutrinos, remains a model input. The theorem also does not claim that the entropy per photon is constant throughout cosmic history; it is a present-day value, after electron-positron annihilation has heated the photons relative to the neutrinos. Finally, it does not claim to have measured this quantity in any experiment; it is a derived number, grounded in the framework's formal proofs.

theorem entropyPerPhoton_pos : 0 < entropyPerPhoton :=
  lt_trans (by norm_num) 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

/-- 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 > 7.0393`. -/
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

/-- Fermionic entropy weight `7/8`: the ratio of the Fermi–Dirac to
Bose–Einstein thermodynamic integrals, `∫x³/(eˣ+1) / ∫x³/(eˣ−1) = η(4)/ζ(4)
= 1 − 2⁻³ = 7/8`.

**THEOREM-backed** (upgraded from MODEL 2026-07-01): the series layer is
`FermionWeight.fermionWeight_eq_eta_zeta_ratio` (`η(4) = (7/8)·ζ(4)`), and
the integral layer is `FermionWeightIntegral.fermi_integral_eq_weight_mul_bose`
(`∫ t³/(eᵗ+1) = (7/8)·∫ t³/(eᵗ−1)`, with both integrals in closed form:
`π⁴/15` and `7π⁴/120`). Both are axiom-clean. The modules import this
definition (not vice versa), so the value stays a plain rational here. -/
def fermionWeight : ℚ := 7 / 8

theorem dilutionCubed_eq : dilutionCubed = 4 / 11 := by
  unfold dilutionCubed gAfter gBefore gPhoton fermionWeight gElectron
  norm_num

What this page does not claim

The theorem does not claim the value 7.04 is exact; it only establishes a window around it. The theorem does not claim the standard model's particle content is derived from deeper principles; it remains a model input. The theorem does not claim the entropy per photon is constant throughout cosmic history; it is a present-day value.

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: