cyclePhase
plain-language theorem explainer
The cyclePhase definition maps each spin quantum number s to the complex factor exp(2π i s) that accumulates after one full 8-tick cycle. Researchers deriving the spin-statistics theorem from discrete time in Recognition Science cite this as the central phase map. The definition is a direct one-line application of the complex exponential to the scaled spin value.
Claim. For a spin quantum number $s$, the phase accumulated over one complete 8-tick cycle is $e^{2π i s}$.
background
The QFT module derives the spin-statistics connection from Recognition Science's 8-tick structure. The Spin structure encodes half-integer values via an integer field twice the spin, kept non-negative. Upstream, the tick constant supplies the fundamental time quantum τ₀ = 1, while EightTick.phase gives the increments kπ/4 for k = 0 to 7 and the Wedge phase supplies the map w ↦ e^{i w}.
proof idea
one-line wrapper that applies the complex exponential to twice π i times the spin value field.
why it matters
This definition supplies the phase factor used by the boson_symmetric and fermion_antisymmetric theorems, which establish +1 for integer spins and -1 for half-integer spins under the full cycle. It realizes the QFT-001 target of obtaining the spin-statistics link from 8-tick phase accumulation. In the framework it implements the eight-tick octave (T7) as the source of 2π periodicity, with the sign distinguishing bosons from fermions.
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