Foundation Spin Statistics Spin Statistics Theorem

The exchange sign

The spin-statistics theorem is a central result of quantum physics. It connects two properties of elementary particles: spin, an intrinsic form of angular momentum, and statistics, the rule governing what happens when two identical particles are swapped. The theorem states that particles with half-integer spin (fermions, like electrons) must obey the Pauli exclusion principle: no two can occupy the same quantum state. Particles with integer spin (bosons, like photons) can pile into the same state without limit. This distinction shapes the structure of atoms, the behavior of lasers, and the stability of matter.

In Recognition Science, the framework models physical structure from a discrete record of recognition events, called a ledger. Within this account, the spin-statistics connection is not assumed but derived from a fundamental cycle. The framework's library, a machine-checked collection of formal theorems, defines a fermionic state as one whose minimal recognition cycle completes in 4 ticks, half of the fundamental 8-tick period. A bosonic state completes its cycle in 8 ticks (or 1 or 2 for spin-0). The key step is the phase a state acquires under a full 2π rotation. For the 4-tick fermion, this phase is -1; for the 8-tick boson, it is +1.

The theorem then states that the exchange sign of a two-particle state equals this rotation phase. Swapping two identical fermions multiplies the two-particle amplitude by -1, which forces the amplitude to zero if both are in the same state, yielding the Pauli exclusion principle. Swapping two bosons multiplies by +1, allowing them to coincide. The formal declaration spin_statistics_theorem in the library's SpinStatistics module proves exactly this: the fermion rotation phase is -1, and the boson phase is +1. A corollary, pauli_exclusion, shows that a complex amplitude ψ satisfying ψ = -1 * ψ must be zero.

What this does not claim is that Recognition Science has discovered a new physical effect. The spin-statistics theorem is a well-established result of relativistic quantum field theory, confirmed by decades of experiment. The framework's contribution is a derivation from its own axioms, showing that the eight-tick structure forces the correct exchange signs. It does not, for instance, explain why the fundamental period is 8 ticks, nor does it address the full proof of the spin-statistics theorem in the context of Lorentz invariance. The derivation is a formal result within the framework's model, not a new experimental finding.

/-- 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

/-- **SPIN-STATISTICS THEOREM** (RS version):
    The exchange sign of a two-particle state equals the rotation phase
    of each particle under 2π rotation.

    - Fermions (4-tick): exchange sign = rotationPhase(4) = -1 → antisymmetric
    - Bosons (8-tick): exchange sign = rotationPhase(0)² = 1 → symmetric

    This is certified by `spin_statistics_key` in `Foundation.EightTick`. -/
theorem spin_statistics_theorem :
    -- Fermions antisymmetrize under exchange
    (rotationPhase 4 = -1) ∧
    -- Bosons symmetrize under exchange
    (phaseExp ⟨0, by norm_num⟩ = 1) :=
  spin_statistics_key

/-- 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

What this page does not claim

No new experimental prediction is made; the spin-statistics theorem is already established in relativistic quantum field theory. The derivation does not explain why the fundamental period is 8 ticks, nor does it derive the value from a deeper principle. The framework's proof does not address the standard proof's reliance on Lorentz invariance, causality, or the spinor formalism.

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.

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