Cosmology Fermion Weight Integral Summable Shift Rpow

The shifted sum

The declaration summable_shift_rpow is a lemma in the machine-checked library of formal theorems. It states that the infinite series whose terms are 1 divided by (n+1) to the fourth power, summed over all whole numbers n starting at 0, converges to a finite value. This is not a new physical principle; it is a technical fact about the behavior of a particular infinite sum, proved using the standard result that the series of 1 over n to the fourth power converges.

The lemma matters because it is the last mathematical gap between a series identity and a statement used in physics. Earlier work in the framework derived the identity η(4) = (7/8)·ζ(4), a relationship between two famous infinite sums. This new lemma, by showing that the shifted series converges, allows the framework to close the integrals that appear in the thermodynamic bookkeeping of the early universe. Specifically, it enables the proof that the integral of t³/(eᵗ+1) from 0 to infinity equals 7/8 times the integral of t³/(eᵗ−1). These integrals are the Fermi–Dirac and Bose–Einstein energy integrals, and their 7/8 ratio is the statistical weight of fermions relative to bosons.

In Recognition Science, this result is part of a chain that connects the framework's foundational mathematics to the entropy-per-photon calculation in cosmology. The framework models the early universe's particle census, and the 7/8 factor is the statistics factor that distinguishes fermions from bosons. With this lemma, the framework's library proves the entire mathematical content of that factor: the only remaining modeling choice is the physics bookkeeping of which species are relativistic, not the 7/8 statistics itself.

What the lemma does not claim is broader. It does not assert that the universe contains any particular set of particles, nor does it derive the value of any fundamental constant. It is a statement about the convergence of a specific infinite series, used as a stepping stone. The physical interpretation, the census of which particles are relativistic, remains a modeling choice, not a theorem.

/-- Summability of the shifted `p`-series with real (rpow) exponent, as
required by `hasSum_mellin`. -/
lemma summable_shift_rpow :
    Summable (fun n : ℕ => (1 : ℝ) / ((n : ℝ) + 1) ^ (4 : ℝ)) := by
  have h : Summable (fun n : ℕ => (1 : ℝ) / (n : ℝ) ^ (4 : ℝ)) :=
    Real.summable_one_div_nat_rpow.mpr (by norm_num)
  have h2 := (summable_nat_add_iff
    (f := fun m : ℕ => (1 : ℝ) / (m : ℝ) ^ (4 : ℝ)) 1).mpr h
  exact h2.congr fun n => by push_cast; ring_nf

/-- Summability of the shifted `p`-series with real (rpow) exponent, as
required by `hasSum_mellin`. -/
lemma summable_shift_rpow :
    Summable (fun n : ℕ => (1 : ℝ) / ((n : ℝ) + 1) ^ (4 : ℝ)) := by
  have h : Summable (fun n : ℕ => (1 : ℝ) / (n : ℝ) ^ (4 : ℝ)) :=
    Real.summable_one_div_nat_rpow.mpr (by norm_num)
  have h2 := (summable_nat_add_iff
    (f := fun m : ℕ => (1 : ℝ) / (m : ℝ) ^ (4 : ℝ)) 1).mpr h
  exact h2.congr fun n => by push_cast; ring_nf

/-- Summability of the shifted `p`-series with real (rpow) exponent, as
required by `hasSum_mellin`. -/
lemma summable_shift_rpow :
    Summable (fun n : ℕ => (1 : ℝ) / ((n : ℝ) + 1) ^ (4 : ℝ)) := by
  have h : Summable (fun n : ℕ => (1 : ℝ) / (n : ℝ) ^ (4 : ℝ)) :=
    Real.summable_one_div_nat_rpow.mpr (by norm_num)
  have h2 := (summable_nat_add_iff
    (f := fun m : ℕ => (1 : ℝ) / (m : ℝ) ^ (4 : ℝ)) 1).mpr h
  exact h2.congr fun n => by push_cast; ring_nf

/-- **THEOREM (7/8 at the thermodynamic layer).** The Fermi–Dirac energy
integral is exactly 7/8 of the Bose–Einstein one. -/
theorem fermi_div_bose_integral :
    (∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t + 1))
      / (∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t - 1)) = 7 / 8 := by
  rw [bose_integral_value, fermi_integral_value]
  rw [div_eq_iff (by positivity)]
  ring

What this page does not claim

The lemma does not identify which specific particles in the universe are relativistic. The lemma does not derive the value of any fundamental constant from the framework's axioms. The lemma does not prove the 7/8 ratio is physically realized; it proves the mathematical identity between the two integrals.

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/FermionWeightIntegral.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

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