tick_phases_roots_of_unity
plain-language theorem explainer
Physicists tracing the origin of complex structure in quantum mechanics and electromagnetism cite this result to confirm that the eight recognition tick phases close under eighth powers on the unit circle. For each k in Fin 8 the phase defined by exp(i π k /4) satisfies z^8 = 1. The proof proceeds by unfolding the phase definition, rewriting the exponent via the exponential multiplication rule, simplifying the argument with ring arithmetic to 2π i k, and invoking the standard identity that exp(2π i m) equals 1 for integer m.
Claim. For each integer $k$ with $0 ≤ k < 8$, let $ω_k = exp(i π k / 4)$. Then $ω_k^8 = 1$.
background
The module MATH-004 derives the necessity of complex numbers from Recognition Science's eight-tick ledger cycle. Each tick corresponds to a 45° rotation, so the phases are the set {0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4}, represented by exp(i π k /4). The upstream definition tickPhase supplies exactly this map: noncomputable def tickPhase (k : Fin 8) : ℂ := Complex.exp (I * π * k / 4). The local setting states that rotations cannot be performed in one real dimension, therefore the ledger forces the plane and hence ℂ.
proof idea
The tactic proof unfolds tickPhase to expose the exponential, rewrites via Complex.exp_nat_mul to pull the integer power inside, introduces an auxiliary equality that ring reduces the coefficient 8 * (I * π * k /4) to k * (2 * π * I), and finishes with the exact application of Complex.exp_nat_mul_two_pi_mul_I.
why it matters
This theorem supplies the algebraic closure property required by the eight-tick octave (T7) in the forcing chain. It directly supports the module claim that the ledger phases are rotations and therefore demand complex numbers for any physical description that tracks them. No downstream uses appear in the current graph, yet the result anchors the later arguments that quantum mechanics and phasor representations inherit their complex character from the same discrete cycle.
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