Foundation Spin Statistics Pauli Exclusion Simple

The exclusion principle

The Pauli exclusion principle states that two identical fermions cannot occupy the same quantum state. In the Recognition Science framework, this is derived from a basic property of complex numbers. The declaration pauli_exclusion_simple proves that if a complex number ψ satisfies ψ = -1 * ψ, then ψ must be zero. This is a purely algebraic statement, and it is the kernel of the exclusion principle.

The framework models particles as states on a discrete recognition cycle, called a ledger, a record of events. A fermion is defined as a state whose cycle completes in 4 ticks, half of the 8-tick period. The theorem fermion_rotation_phase_neg_one shows that a 2π rotation of such a state multiplies it by -1. When two identical fermions are exchanged, the two-particle amplitude must change sign, but exchanging identical particles must leave the state unchanged. The only way both can be true is if the amplitude is zero, which is exactly what pauli_exclusion_simple proves.

The declaration itself is a lemma about complex numbers, not a statement about physical space. It proves the algebraic identity that if ψ = -ψ, then ψ = 0. The physical interpretation, that this represents the exclusion of two fermions from the same state, comes from the surrounding framework. The theorem pauli_exclusion in the same module makes this connection explicit, using the fermion rotation phase. The simple version isolates the algebraic core.

Within the framework, this result is a consequence of the eight-tick structure, but the simple declaration does not itself reference ticks, spin, or particles. It is a standalone fact about multiplication by -1. The framework's library, a machine-checked collection of formal theorems, proves the full spin-statistics connection elsewhere, but this declaration is the minimal algebraic step.

What this declaration does not claim is that it proves the full Pauli exclusion principle for physical fermions. It proves the algebraic condition that makes exclusion possible. The step from this lemma to the physical principle requires the identification of fermions with 4-tick states and the exchange argument, which are established in the broader module. The simple declaration is a building block, not the whole edifice.

/-- Simplified Pauli exclusion: ψ = -1 * ψ implies ψ = 0. -/
theorem pauli_exclusion_simple (ψ : ℂ) (h : ψ = -1 * ψ) : ψ = 0 := by
  have h2 : (2 : ℂ) * ψ = 0 := by linear_combination ψ + h
  exact (mul_eq_zero.mp h2).resolve_left two_ne_zero

/-- A ledger state is fermionic (spin-1/2) if its minimal recognition cycle
    completes in 4 ticks (half the 8-tick period). -/
def IsFermionic (period : ℕ) : Prop := period = 4

/-- **KEY**: For fermions (4-tick period), the 2π rotation gives phase -1. -/
theorem fermion_rotation_phase_neg_one :
    rotationPhase 4 = -1 := by
  unfold rotationPhase
  exact phase_4_is_minus_one

What this page does not claim

This declaration does not prove the physical Pauli exclusion principle for actual fermions. It does not establish that any physical particle has a 4-tick recognition cycle. It does not derive the spin-statistics connection; that is a separate theorem in the module.

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/Foundation/SpinStatistics.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: