spin_statistics_certificate
The spin-statistics certificate verifies that the eight-tick phase exponential satisfies full periodicity, yields -1 at the half-period for fermion exchange, +1 at the identity for bosons, and connects four-tick rotation to the negative sign. Researchers reconstructing the spin-statistics theorem from discrete time quanta would cite this result. The proof is a direct term assembly of four upstream lemmas on phase values.
claimLet $P(k) = e^{i k π / 4}$ for integer $k$ modulo 8. Then $P(k)^8 = 1$ for all such $k$, $P(4) = -1$, $P(0) = 1$, and the four-tick rotation phase equals $-1$.
background
The eight-tick ledger defines discrete phases as multiples of $π/4$ with the fundamental time quantum equal to one tick. The phase map sends each Fin-8 index to $k π /4$, and the exponential form produces the complex phase factor used for exchange signs. Upstream lemmas establish the eighth-power identity, the value -1 at four ticks (fermion sign), and the value +1 at zero ticks (boson sign). This module sits inside the Recognition Science eight-tick octave (T7) and certifies the complete spin-statistics connection without remaining hypotheses.
proof idea
The term proof invokes exact on the four upstream results: phase_eighth_power_is_one for periodicity, phase_4_is_minus_one for the fermion sign, phase_0_is_one for the boson sign, and fermion_rotation_phase_neg_one for the rotation-phase link. No further tactics or reductions occur.
why it matters in Recognition Science
This declaration certifies every claim in RS_Spin_Statistics_Theorem.tex by discharging the phase conditions required for the spin-statistics theorem and its corollaries (Pauli exclusion). It anchors the eight-tick octave (T7) derivation of exchange symmetry from the Recognition Science forcing chain. No open scaffolding remains.
scope and limits
- Does not derive the spin-statistics theorem from continuous rotation groups.
- Does not compute explicit wave-function representations.
- Does not address particles outside the eight-tick cycle.
- Does not incorporate relativistic or field-theoretic extensions.
formal statement (Lean)
117theorem spin_statistics_certificate :
118 -- (1) Eight-tick phase periodicity
119 (∀ k : Fin 8, (phaseExp k)^8 = 1) ∧
120 -- (2) Half-period gives -1 (fermion exchange sign)
121 (phaseExp ⟨4, by norm_num⟩ = -1) ∧
122 -- (3) Identity period gives +1 (boson exchange sign)
123 (phaseExp ⟨0, by norm_num⟩ = 1) ∧
124 -- (4) Spin-statistics connection
125 (rotationPhase 4 = -1) := by
proof body
Term-mode proof.
126 exact ⟨phase_eighth_power_is_one, phase_4_is_minus_one, phase_0_is_one,
127 fermion_rotation_phase_neg_one⟩
128
129end SpinStatistics
130end Foundation
131end IndisputableMonolith