spin_statistics_certificate
plain-language theorem explainer
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.
Claim. Let $P(k) = e^{i k π / 4}$ 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
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.
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