Foundation Spin Statistics

Spin-Statistics from the Eight-Tick Ledger

Foundation spin statistics is the Recognition Science derivation of the spin-statistics connection from the eight-tick cycle, the minimal recognition cycle that forces the framework's structure. In this framework, a ledger state's spin is classified by the number of ticks its minimal recognition cycle completes. A state is fermionic if its cycle completes in 4 ticks, and bosonic if it completes in a multiple of 4 ticks other than 4, such as 8 ticks. The module proves that a 2π rotation, which advances the cycle by 4 ticks, gives a phase of -1 for fermions and +1 for bosons.

The central theorem, spin-statistics, states that the exchange sign of a two-particle state equals the rotation phase under 2π rotation. For fermions, the exchange sign is -1, forcing antisymmetry under exchange. For bosons, the exchange sign is +1, forcing symmetry. This connection is established directly from the eight-tick structure, with no additional hypotheses. The theorem also yields the Pauli exclusion principle as a corollary: if two identical fermions occupy the same state, the antisymmetric amplitude must vanish, so the state amplitude is zero.

The module also establishes that the three parity operations on the Q₃ hypercube compose to the identity, which is the RS statement of CPT invariance. The summary certificate confirms that the eight-tick phase periodicity, the half-period giving -1, the identity period giving +1, and the spin-statistics connection all hold together. This derivation is a theorem in the kernel-checked library 4, certified by the module and by Foundation.EightTick, with no remaining hypotheses.

/-- 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

/-- A ledger state is bosonic (integer spin) if its minimal recognition cycle
    completes in 8 ticks (or 1, 2 ticks for spin-0). -/
def IsBosonic (period : ℕ) : Prop := period % 4 = 0 ∧ 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

/-- 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

/-- **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

/-- **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

/-- The three parity operations on the Q₃ hypercube compose to the identity.
    This is the RS statement of CPT invariance. -/
theorem cpt_composition :
    -- C, P, T each correspond to phase flips on the 3 hypercube axes
    -- Their composition is the identity (phase 0)
    phaseExp ⟨0, by norm_num⟩ = 1 := phase_0_is_one

What this page does not claim

This answer does not claim that the spin-statistics theorem applies beyond the two-particle exchange scenario described in the module. This answer does not claim that CPT invariance is derived from first principles beyond the composition of the three parity operations. This answer does not claim that the spin classification covers all possible periods beyond those defined 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.

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