Cosmology Partition Kernels Bose Weighted Has Sum

The Bose weighted sum

In statistical mechanics, the average number of particles occupying a quantum state is a fundamental quantity. For bosons, particles that can share a state, this average is the Bose-Einstein distribution, ledger (a discrete record of events) of possible occupation numbers. The declaration bose_weighted_hasSum in the Recognition Science library proves that a specific infinite sum converges to a closed form.

The sum in question weights each possible occupation number n by the Boltzmann factor x^n, where x = e^(-t) and t is positive (inverse temperature units). The theorem states that Σ n·x^n = x/(1-x)². This is a standard result in statistical mechanics, and the proof in the library is a direct application of the geometric series. It requires only that 0 < t, ensuring the series converges.

In Recognition Science, this result is a stepping stone. The framework models the universe as a ledger of recognition events, and this theorem provides the mathematical floor for the thermal history of the cosmos. It connects to the Bose-Einstein occupation number, the average occupancy of a mode, which is ⟨n⟩ = 1/(e^t - 1). This is derived by dividing the weighted sum by the unweighted partition function.

The theorem does not claim that the universe is a bosonic gas, nor does it derive the Bose-Einstein distribution from first principles of recognition. It takes the occupancy range (all natural numbers) as a given, matching the standard bosonic assumption. The result is a piece of mathematics, not a physical postulate.

What this establishes is a clean, machine-checked foundation for a key formula. It shows that within the framework's library, the standard statistical mechanics result holds without hidden assumptions. The next step, connecting this to the full thermal history of the universe, remains a target for the framework's ongoing work.

/-- The occupancy-weighted Bose sum: `Σ n·xⁿ = x/(1-x)²` for `x = e^{-t}`. -/
theorem bose_weighted_hasSum (t : ℝ) (ht : 0 < t) :
    HasSum (fun n : ℕ => (n : ℝ) * Real.exp (-t) ^ n)
      (Real.exp (-t) / (1 - Real.exp (-t)) ^ 2) := by
  have hξpos : (0 : ℝ) < Real.exp (-t) := Real.exp_pos _
  have hξlt : Real.exp (-t) < 1 := Real.exp_lt_one_iff.mpr (by linarith)
  apply hasSum_coe_mul_geometric_of_norm_lt_one
  rw [Real.norm_eq_abs, abs_of_pos hξpos]
  exact hξlt

/-- **Bose-Einstein occupation number**: the mean occupancy of a bosonic mode
is `⟨n⟩ = (Σ n·xⁿ)/(Σ xⁿ) = 1/(e^t - 1)`. -/
theorem bose_occupation (t : ℝ) (ht : 0 < t) :
    (∑' n : ℕ, (n : ℝ) * Real.exp (-t) ^ n) / (∑' n : ℕ, Real.exp (-t) ^ n)
      = 1 / (Real.exp t - 1) := by
  have hξpos : (0 : ℝ) < Real.exp (-t) := Real.exp_pos _
  have hξlt : Real.exp (-t) < 1 := Real.exp_lt_one_iff.mpr (by linarith)
  have h1 : (1 : ℝ) - Real.exp (-t) ≠ 0 := by linarith
  have hE1 : Real.exp t - 1 ≠ 0 := by
    have h0 : Real.exp 0 < Real.exp t := Real.exp_lt_exp.mpr ht
    rw [Real.exp_zero] at h0
    linarith
  have hprod : Real.exp (-t) * Real.exp t = 1 := by
    rw [← Real.exp_add]
    simp
  rw [(bose_weighted_hasSum t ht).tsum_eq, bose_partition_tsum t ht]
  field_simp
  linear_combination hprod

What this page does not claim

The theorem does not derive the Bose-Einstein distribution from recognition principles. The theorem does not state that the universe is a bosonic gas. The result does not connect to the fine-structure constant or other derived constants.

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/PartitionKernels.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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