Cosmology Partition Kernels Bose Partition Has Sum

The Bose sum

In statistical mechanics, a partition function is a weighted count of all the states a system can occupy. For a single bosonic mode, a particle can hold any number of quanta: zero, one, two, and so on without limit. The Boltzmann weight for each occupation number n is x to the power n, where x = e^−t and t is a positive real number encoding the energy and temperature. The partition function is the sum of all those weights, and the theorem bose_partition_hasSum proves that this infinite sum converges to the closed form (1 − e^−t)⁻¹. This is the standard geometric series result, and the proof in the machine-checked library of formal theorems verifies it for every positive t.

The statement is precise about its domain. It requires t > 0, which guarantees that x lies strictly between 0 and 1. That condition is what makes the series converge; without it, the sum would not have a finite value. The theorem is a statement about the real numbers, and it uses the standard notion of an infinite sum as a limit of partial sums. It does not depend on any physical assumption beyond the occupancy range: the sum runs over all natural numbers, which is the defining feature of a bosonic mode.

In Recognition Science, this theorem is one step in a larger chain. The framework models the universe as maintaining a discrete record of recognition events, and it derives the statistical mechanics floor beneath its cosmology thermal-history chain. The theorem establishes that the Bose log kernel, a function used in pressure and entropy integrals upstream, is not a definitional choice but the logarithm of a derived partition function. A companion theorem, bose_partition_tsum, states the same result using the library's notation for infinite sums, and a further result shows that the mean occupation number of a bosonic mode is 1/(e^t − 1), the Bose-Einstein distribution.

What the theorem does not claim is just as important. It does not prove that bosons exist, nor does it derive the Pauli exclusion principle. The occupancy range over all of ℕ is assumed as the definition of a bosonic mode, not derived from anything deeper. The theorem also does not address fermions, which occupy only zero or one quantum states; that is a separate result with its own two-term sum. The theorem is a mathematical fact about a geometric series, and its physical significance comes from the framework's choice to use it as the partition function for a bosonic mode.

/-- The Bose single-mode grand partition function: occupancies range over all
of `ℕ`, and the Boltzmann-weighted sum is the geometric series with ratio
`x = exp (-t) < 1`. -/
theorem bose_partition_hasSum (t : ℝ) (ht : 0 < t) :
    HasSum (fun 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)
  apply hasSum_geometric_of_norm_lt_one
  rw [Real.norm_eq_abs, abs_of_pos hξpos]
  exact hξlt

/-- The Bose single-mode grand partition function: occupancies range over all
of `ℕ`, and the Boltzmann-weighted sum is the geometric series with ratio
`x = exp (-t) < 1`. -/
theorem bose_partition_hasSum (t : ℝ) (ht : 0 < t) :
    HasSum (fun 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)
  apply hasSum_geometric_of_norm_lt_one
  rw [Real.norm_eq_abs, abs_of_pos hξpos]
  exact hξlt

/-- `log Z_B` is exactly the Bose log kernel used in
`PhaseSpaceReduction.boseLogKernel` (hence in the pressure/entropy integrals
upstream): the kernel is no longer a definitional choice but the log of the
derived partition function. -/
theorem boseLogKernel_from_partition (t : ℝ) (ht : 0 < t) :
    Real.log (∑' n : ℕ, Real.exp (-t) ^ n)
      = PhaseSpaceReduction.boseLogKernel t := by
  rw [bose_partition_tsum t ht, Real.log_inv,
      PhaseSpaceReduction.boseLogKernel]

What this page does not claim

The theorem does not prove that bosons exist or that the occupancy range over all natural numbers is physically necessary. The theorem does not derive the Pauli exclusion principle or the Fermi-Dirac distribution. The theorem does not state that the partition function is the only possible way to model a bosonic mode.

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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