Foundation Spin Statistics Spin Statistics Certificate

The spin-statistics certificate

The spin-statistics theorem is a central result of quantum physics. It states that particles with half-integer spin (fermions) must have antisymmetric wavefunctions, meaning that swapping two identical fermions multiplies the wavefunction by -1. Particles with integer spin (bosons) have symmetric wavefunctions, so swapping them leaves the wavefunction unchanged. This distinction explains why electrons cannot occupy the same quantum state (the Pauli exclusion principle), while photons, for example, can pile up in a single state to form a laser beam.

In Recognition Science (RS), the framework models physical states as discrete records called ledger states, where each state has a minimal recognition cycle measured in ticks. A state is fermionic if its cycle completes in 4 ticks, and bosonic if it completes in 8 ticks (or 1 or 2 for spin-0). The framework's library, a machine-checked collection of formal theorems, proves that a 4-tick state acquires a phase of -1 under a full 2π rotation, while an 8-tick state returns to itself with phase +1. These rotation phases are then identified with the exchange sign: swapping two identical fermions gives -1, and swapping two bosons gives +1.

The declaration spin_statistics_certificate bundles four results into a single theorem: the eighth-power periodicity of the phase, the half-period giving -1, the identity period giving +1, and the spin-statistics connection itself. It also includes the Pauli exclusion principle as a corollary: if a fermion amplitude ψ satisfies ψ = -1 * ψ, then ψ must be zero. The certificate is a formal statement that these claims follow from the framework's axioms with no additional hypotheses. It is a theorem about the framework's internal model, not a direct measurement of physical particles.

What the certificate does not claim is just as important. It does not claim that the framework's tick-based model is the only way to derive spin-statistics, nor that it replaces the standard relativistic quantum field theory proof. It does not assert that physical electrons are literally 4-tick cycles; the framework models them that way. The certificate is a statement about the consistency of the framework's own definitions, not an empirical test. It shows that if you accept the framework's starting points, the spin-statistics connection follows. The physical relevance of those starting points remains a separate question, one the framework addresses elsewhere but which this certificate itself does not settle.

/-- **SPIN-STATISTICS CERTIFICATE**:
    All claims in `RS_Spin_Statistics_Theorem.tex` are certified by this module
    and `Foundation.EightTick`. No hypotheses remain. -/
theorem spin_statistics_certificate :
    -- (1) Eight-tick phase periodicity
    (∀ k : Fin 8, (phaseExp k)^8 = 1) ∧
    -- (2) Half-period gives -1 (fermion exchange sign)
    (phaseExp ⟨4, by norm_num⟩ = -1) ∧
    -- (3) Identity period gives +1 (boson exchange sign)
    (phaseExp ⟨0, by norm_num⟩ = 1) ∧
    -- (4) Spin-statistics connection
    (rotationPhase 4 = -1) := by
  exact ⟨phase_eighth_power_is_one, phase_4_is_minus_one, phase_0_is_one,
         fermion_rotation_phase_neg_one⟩

/-- **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

/-- For bosons (8-tick period), the 2π rotation gives phase +1 (via two half-cycles). -/
theorem boson_rotation_phase_pos_one :
    phaseExp ⟨0, by norm_num⟩ = 1 := phase_0_is_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 certificate does not claim that physical electrons are literally 4-tick cycles. It does not claim to replace the standard relativistic quantum field theory proof of spin-statistics. It does not claim that the framework's starting points are empirically established.

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

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