Cosmology Partition Kernels Partition Kernels Cert

What the certificate proves

In statistical mechanics, a partition function is a sum over all the ways a system can be arranged, each arrangement weighted by a Boltzmann factor. For a single mode of a bosonic field, particles can occupy any nonnegative integer number of states, so the sum is the geometric series partition function Z_B = 1 + x + x² + ... = (1 - x)⁻¹, where x = e⁻ᵗ and t is positive. For a fermionic mode, the Pauli exclusion principle restricts occupancy to zero or one particle, giving Z_F = 1 + x. The mean occupation numbers follow from these sums: Bose-Einstein gives ⟨n⟩ = 1/(eᵗ - 1), and Fermi-Dirac gives ⟨n⟩ = 1/(eᵗ + 1). These are textbook results, known since the 1920s.

The Recognition Science framework's machine-checked library of formal theorems contains a declaration, partitionKernelsCert, that bundles four of these identities into one theorem. It states that the logarithm of the bosonic partition function equals a kernel used in the framework's phase-space reduction, the logarithm of the fermionic partition function equals its corresponding kernel, and the two occupation numbers are exactly the Bose-Einstein and Fermi-Dirac distributions. The theorem is proved for all positive t, so every series converges. The proof is a direct calculation from the definitions; no additional physical assumptions enter after the initial setup.

In Recognition Science, the only physical fork between the two cases is the occupancy range: all natural numbers for bosons versus the two-element set {0, 1} for fermions. The framework derives the fermionic restriction from an exchange sign: a fermion acquires a phase of -1 under exchange at the half-cycle of the framework's eight-tick structure. The step from that phase to occupancy at most one is the standard antisymmetrization argument, where a doubly occupied antisymmetric state equals its own negative and therefore vanishes. The framework treats that argument as the justification for choosing the two-element index type, not as a hidden axiom. Everything after that choice is mathematics with no free input.

The certificate matters because it closes a gap in the framework's cosmology chain. Earlier work used the log kernels as definitions; this theorem shows they are not arbitrary choices but the actual logarithms of the derived partition functions. The occupation numbers are likewise derived, not postulated. The declaration is also used in an axiom audit: checking the theorem's axioms should show only the framework's three base axioms, with no extra assumptions. That audit is a claim about postulates, not about the ambient type theory.

What the certificate does not claim is as important as what it proves. It does not derive the Pauli exclusion principle from scratch; it takes the exchange sign as the physical input and uses the standard argument to justify the occupancy restriction. It does not prove that the partition functions describe any particular particle in the real universe; it establishes the mathematics of a single mode. It does not claim that the framework's eight-tick structure is physically real; that bridge remains open. The theorem is a precise, limited result: given the occupancy ranges, the standard formulas follow.

/-- The Bose partition function as a `tsum`: `Z_B(t) = (1 - e^{-t})⁻¹`. -/
theorem bose_partition_tsum (t : ℝ) (ht : 0 < t) :
    (∑' n : ℕ, Real.exp (-t) ^ n) = (1 - Real.exp (-t))⁻¹ :=
  (bose_partition_hasSum t ht).tsum_eq

/-- The Fermi single-mode grand partition function: Pauli restricts occupancy
to `{0, 1}` (see `fermi_exchange_sign`), so the sum is two terms:
`Z_F(t) = 1 + e^{-t}`. -/
theorem fermi_partition_two_state (t : ℝ) :
    (∑ n : Fin 2, Real.exp (-t) ^ (n : ℕ)) = 1 + Real.exp (-t) := by
  simp [Fin.sum_univ_two]

/-- **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

/-- **Fermi-Dirac occupation number**: the mean occupancy of a fermionic mode
is `⟨n⟩ = (0·1 + 1·x)/(1 + x) = 1/(e^t + 1)`. -/
theorem fermi_occupation (t : ℝ) :
    (∑ n : Fin 2, ((n : ℕ) : ℝ) * Real.exp (-t) ^ (n : ℕ))
        / (∑ n : Fin 2, Real.exp (-t) ^ (n : ℕ))
      = 1 / (Real.exp t + 1) := by
  have hξpos : (0 : ℝ) < Real.exp (-t) := Real.exp_pos _
  have hEpos : (0 : ℝ) < Real.exp t := Real.exp_pos _
  have h1 : (1 : ℝ) + Real.exp (-t) ≠ 0 := by positivity
  have hE1 : Real.exp t + 1 ≠ 0 := by positivity
  have hprod : Real.exp (-t) * Real.exp t = 1 := by
    rw [← Real.exp_add]
    simp
  simp only [Fin.sum_univ_two]
  norm_num
  field_simp
  linear_combination hprod

/-- **Certificate**: the partition-kernel layer in one bundle. The log kernels
used upstream equal the logs of the derived partition functions, and the two
occupation numbers are the Bose-Einstein and Fermi-Dirac distributions. Used
by the loop's axiom audit (`#print axioms` must show only the base three). -/
theorem partitionKernelsCert :
    (∀ t : ℝ, 0 < t →
        Real.log (∑' n : ℕ, Real.exp (-t) ^ n)
          = PhaseSpaceReduction.boseLogKernel t)
      ∧ (∀ t : ℝ,
        Real.log (∑ n : Fin 2, Real.exp (-t) ^ (n : ℕ))
          = PhaseSpaceReduction.fermiLogKernel t)
      ∧ (∀ t : ℝ, 0 < t →
        (∑' n : ℕ, (n : ℝ) * Real.exp (-t) ^ n) / (∑' n : ℕ, Real.exp (-t) ^ n)
          = 1 / (Real.exp t - 1))
      ∧ (∀ t : ℝ,
        (∑ n : Fin 2, ((n : ℕ) : ℝ) * Real.exp (-t) ^ (n : ℕ))
            / (∑ n : Fin 2, Real.exp (-t) ^ (n : ℕ))
          = 1 / (Real.exp t + 1)) :=
  ⟨boseLogKernel_from_partition, fermiLogKernel_from_partition,
   bose_occupation, fermi_occupation⟩

/-- **RS input (re-export)**: the exchange phase at the half-cycle is `-1`
(fermionic sign), from the eight-tick structure. This is the physical fact
that forces the Pauli occupancy restriction used in the Fermi partition
function below. -/
theorem fermi_exchange_sign :
    Foundation.EightTick.phaseExp ⟨4, by norm_num⟩ = -1 :=
  Foundation.EightTick.spin_statistics_key.1

What this page does not claim

The certificate does not derive the Pauli exclusion principle from first principles; it uses the exchange sign as input. The certificate does not prove that any real particle obeys these distributions; it establishes the mathematics of a single mode. The certificate does not claim the eight-tick structure is physically real.

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)

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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