Cosmology Occupation Energy Occupation Energy Cert

The occupation-energy certificate

A ledger, in this framework, is a discrete record of events. The declaration occupationEnergyCert is a machine-checked certificate that bundles two proven identities about thermal energy. The first identity rewrites the standard Bose and Fermi energy integrands, t³/(eᵗ − 1) and t³/(eᵗ + 1), as t³ times an occupation number. That occupation number is not assumed; it is derived from the partition function, the sum over all possible occupation states weighted by their probabilities. The second identity states that the ratio of the Fermi to Bose total energy integrals is exactly 7/8.

The 7/8 ratio is the famous low-temperature limit of the ratio of fermion to boson energy densities in a gas of massless particles. In the framework it appears as a theorem, not a coincidence: the Fermi integral evaluates to 7π⁴/120, the Bose integral to π⁴/15, and their quotient is 7/8. The certificate packages both the integrand identities and the ratio into a single declaration, so a reader can audit both claims at once.

What the certificate does not claim is just as important. It does not claim that the ratio 7/8 is derived from the framework's foundational forcing chain; it is a statement about thermal integrals, not about the origin of the partition function. It does not claim any physical measurement or cosmological application; the integrals are mathematical objects, and their identification with observed energy densities is a separate step. It does not claim that the Fermi integrand is the only possible one; it is one of two standard forms, selected by the two-state Pauli restriction.

The practical upshot is that the framework's thermal energy calculations rest on a single, auditable certificate. A reader who wants to check the chain from partition function to occupation number to the 7/8 ratio can do so in one place, without tracing separate declarations. That is the point of the certificate: it turns a chain of reasoning into a single object that can be inspected and trusted.

/-- The Bose energy integrand equals (dimensionless energy per mode `t³`) ×
(the occupation number derived from the partition function). -/
theorem bose_energy_kernel_eq (t : ℝ) (ht : 0 < t) :
    t ^ 3 / (Real.exp t - 1)
      = t ^ 3 * ((∑' n : ℕ, (n : ℝ) * Real.exp (-t) ^ n)
          / (∑' n : ℕ, Real.exp (-t) ^ n)) := by
  rw [PartitionKernels.bose_occupation t ht]
  ring

/-- The Fermi energy integrand equals `t³` × (the Pauli-restricted occupation
number derived from the two-state partition function). -/
theorem fermi_energy_kernel_eq (t : ℝ) (_ht : 0 < t) :
    t ^ 3 / (Real.exp t + 1)
      = t ^ 3 * ((∑ n : Fin 2, ((n : ℕ) : ℝ) * Real.exp (-t) ^ (n : ℕ))
          / (∑ n : Fin 2, Real.exp (-t) ^ (n : ℕ))) := by
  rw [PartitionKernels.fermi_occupation t]
  ring

/-- **The 7/8 ratio, stated at the partition-function level**: the ratio of
the thermal energy integrals, with each integrand written as
`t³ × ⟨n⟩` (occupation numbers from the derived partition functions), is
exactly `7/8`. -/
theorem energy_ratio_seven_eighths :
    (∫ t in Ioi (0 : ℝ),
        t ^ 3 * ((∑ n : Fin 2, ((n : ℕ) : ℝ) * Real.exp (-t) ^ (n : ℕ))
          / (∑ n : Fin 2, Real.exp (-t) ^ (n : ℕ))))
      / (∫ t in Ioi (0 : ℝ),
        t ^ 3 * ((∑' n : ℕ, (n : ℝ) * Real.exp (-t) ^ n)
          / (∑' n : ℕ, Real.exp (-t) ^ n)))
      = 7 / 8 := by
  have hf : (∫ t in Ioi (0 : ℝ),
      t ^ 3 * ((∑ n : Fin 2, ((n : ℕ) : ℝ) * Real.exp (-t) ^ (n : ℕ))
        / (∑ n : Fin 2, Real.exp (-t) ^ (n : ℕ))))
      = ∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t + 1) := by
    refine setIntegral_congr_fun measurableSet_Ioi fun t ht => ?_
    exact (fermi_energy_kernel_eq t ht).symm
  have hb : (∫ t in Ioi (0 : ℝ),
      t ^ 3 * ((∑' n : ℕ, (n : ℝ) * Real.exp (-t) ^ n)
        / (∑' n : ℕ, Real.exp (-t) ^ n)))
      = ∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t - 1) := by
    refine setIntegral_congr_fun measurableSet_Ioi fun t ht => ?_
    exact (bose_energy_kernel_eq t ht).symm
  rw [hf, hb]
  exact FermionWeightIntegral.fermi_div_bose_integral

/-- **Certificate** for the axiom audit. -/
theorem occupationEnergyCert :
    (∀ t : ℝ, 0 < t →
        t ^ 3 / (Real.exp t - 1)
          = t ^ 3 * ((∑' n : ℕ, (n : ℝ) * Real.exp (-t) ^ n)
              / (∑' n : ℕ, Real.exp (-t) ^ n)))
      ∧ ((∫ t in Ioi (0 : ℝ),
            t ^ 3 * ((∑ n : Fin 2, ((n : ℕ) : ℝ) * Real.exp (-t) ^ (n : ℕ))
              / (∑ n : Fin 2, Real.exp (-t) ^ (n : ℕ))))
          / (∫ t in Ioi (0 : ℝ),
            t ^ 3 * ((∑' n : ℕ, (n : ℝ) * Real.exp (-t) ^ n)
              / (∑' n : ℕ, Real.exp (-t) ^ n)))
          = 7 / 8) :=
  ⟨bose_energy_kernel_eq, energy_ratio_seven_eighths⟩

What this page does not claim

The certificate does not derive the partition function from the framework's foundational axioms. The certificate does not claim any direct measurement or cosmological prediction. The certificate does not claim that the Fermi integrand is the only possible form.

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/OccupationEnergy.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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