Cosmology Partition Kernels Fermi Exchange Sign

The exchange sign

In quantum mechanics, particles come in two kinds. Bosons, like photons, can occupy the same quantum state in unlimited numbers. Fermions, like electrons, obey the Pauli exclusion principle: at most one can occupy any given state. The standard explanation is antisymmetry: swapping two identical fermions multiplies their joint wavefunction by minus one, and a doubly occupied antisymmetric state is its own negative, hence zero.

Recognition Science (RS) models this minus one as a derived fact, not an assumption. The framework starts from a discrete record of recognition events, called the ledger, and its eight-tick cycle of operations. Within that cycle, the declaration fermi_exchange_sign establishes that the phase acquired by a fermion under exchange at the half-cycle is exactly -1. This is a proved theorem in the machine-checked library of formal theorems, derived from the framework's spin-statistics key, not an axiom added for convenience.

The physical consequence flows from this sign. Because the exchange phase is -1, the occupancy of a fermionic mode is restricted to {0, 1}, giving the Fermi-Dirac partition function Z_F = 1 + e^(-t) and the mean occupation number ⟨n⟩ = 1/(e^t + 1). The same derivation for bosons, whose occupancy ranges over all natural numbers, yields the Bose-Einstein distribution ⟨n⟩ = 1/(e^t - 1). The only physical fork between the two computations is the occupancy range, and that fork is the exchange sign.

What the declaration does not claim is equally precise. It does not prove the Pauli exclusion principle from scratch; it takes the standard antisymmetrization argument as the justification for choosing the two-state index type. It does not derive the fine-structure constant or any coupling strength. It does not claim that the minus one explains why fermions exist, only that within the framework's structure, the sign is forced, not chosen.

/-- **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

/-- 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]

/-- **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

What this page does not claim

The declaration does not prove the Pauli exclusion principle; it uses the standard antisymmetrization argument as the choice of statement. No claim is made about the origin of fermions themselves, only about the sign under exchange. The fine-structure constant and other coupling strengths are not derived from this declaration.

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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