Foundation Spin Statistics Fermion Rotation Phase Neg One

The half-turn sign

In quantum mechanics, a particle's wavefunction does not always return to itself after a 360-degree rotation. For a spin-1/2 particle such as an electron, the wavefunction picks up a factor of -1 after one full turn; only after 720 degrees does it come back exactly. This half-turn sign is not a mathematical oddity. It is the reason two identical electrons cannot occupy the same quantum state, the principle Wolfgang Pauli stated in 1925 and the foundation of the structure of atoms and the periodic table.

The standard derivation of this sign in physics textbooks uses the rotation operators of the Lorentz group. The Recognition Science framework reaches the same conclusion from a different starting point. The framework models a particle's internal state as a ledger, a discrete record of recognition events, and assigns each particle a period: the number of ticks in its minimal recognition cycle. A fermion is defined as a state whose cycle completes in 4 ticks, half of the 8-tick period that defines a full cycle. The declaration fermion_rotation_phase_neg_one proves, within the framework's machine-checked library of formal theorems, that the phase accumulated under a 2π rotation, computed as a function of this 4-tick period, equals -1.

The proof is short and direct. The theorem unfolds the definition of the rotation phase and applies a prior lemma, phase_4_is_minus_one, which states that the fourth tick of the eight-tick cycle carries the phase -1. The framework then connects this rotation phase to exchange statistics: swapping two identical fermions is equivalent to a 2π rotation of their relative coordinate, and since that rotation contributes -1, the two-particle wavefunction must be antisymmetric under exchange. From this antisymmetry, the framework derives Pauli exclusion as a corollary: if two identical fermions occupy the same state, the amplitude ψ must satisfy ψ = -ψ, and the only complex number equal to its own negative is zero.

What the declaration does not claim is just as important. It does not claim to have measured the electron's spin or to have observed the half-turn sign in an experiment. It is a theorem about the framework's own definitions: given the ledger model and the definition of a fermion as a 4-tick state, the rotation phase must be -1. The framework's broader claim, that this discrete ledger structure is the origin of the spin-statistics connection in physical reality, is a model choice, not a theorem. The declaration also does not claim that the framework's derivation supersedes the standard quantum field theory result; it offers an alternative route to the same conclusion, and the two agree.

/-- 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

/-- Two-particle exchange involves one 2π rotation of the relative coordinate,
    contributing the rotation phase. For fermions: -1. For bosons: +1.

    This is the fundamental RS derivation of the exchange sign. -/
theorem exchange_sign_fermion :
    rotationPhase 4 = -1 := fermion_rotation_phase_neg_one

/-- **PAULI EXCLUSION**:
    If two identical fermions occupy the same state, the antisymmetric
    two-particle amplitude must vanish.

    In RS: if ψ₁ = ψ₂ = ψ, then exchange gives ψ → -ψ (from exchange_sign_fermion),
    but exchange of identical particles gives ψ → ψ.
    So ψ = -ψ → ψ = 0. -/
theorem pauli_exclusion (ψ : ℂ) (h_fermion : ψ = rotationPhase 4 * ψ) :
    ψ = 0 := by
  rw [fermion_rotation_phase_neg_one] at h_fermion
  -- h_fermion : ψ = -1 * ψ, so 2ψ = 0, so ψ = 0
  have h2 : (2 : ℂ) * ψ = 0 := by linear_combination ψ + h_fermion
  exact (mul_eq_zero.mp h2).resolve_left two_ne_zero

What this page does not claim

The declaration does not claim to have measured the electron's spin or observed the half-turn sign experimentally. The declaration does not claim that the ledger model is the unique or true description of physical reality; it is a definitional choice within the framework. The declaration does not claim the framework's derivation supersedes the standard quantum field theory result; it offers an alternative route to the same conclusion.

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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