rootOfUnity
plain-language theorem explainer
The definition assigns to each k in Fin 8 the complex value exp(2 π i k / 8). Researchers deriving the imaginary unit from eight-tick phase rotations in Recognition Science would cite this as the explicit map from the cyclic structure to the complex plane. It is realized by the standard exponential formula with no auxiliary lemmas.
Claim. $ζ_k = e^{2π i k / 8}$ for each integer $k$ with $0 ≤ k ≤ 7$.
background
The module MATH-001 derives i² = -1 from the 8-tick phase structure of Recognition Science. Rotation by two ticks (π/2) corresponds to multiplication by i; rotation by four ticks corresponds to multiplication by -1. The supplied doc-comment states that these eighth roots of unity are exactly the 8-tick phases and form the cyclic group Z₈ under multiplication.
proof idea
Direct definition via the complex exponential applied to I scaled by the angle 2 π k / 8.
why it matters
This supplies the explicit phases required by the downstream theorem roots_form_group, which proves the set forms a group under multiplication. It realizes the eight-tick octave (T7) inside the complex numbers and supports later derivations of the imaginary unit, Schrödinger phase, and Euler identity within the Recognition framework.
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