i_power_is_rotation
plain-language theorem explainer
The theorem states that the nth power of the imaginary unit equals the complex exponential of i times n pi over 2. Researchers deriving phase factors in quantum mechanics or wave equations from discrete tick structures would cite this result. The proof first verifies the base rotation for i via trigonometric identities on the exponential and then lifts it to arbitrary natural powers using the exponential's multiplicative property.
Claim. For every natural number $n$, $i^n = e^{i n π / 2}$, where $i$ is the imaginary unit and $e$ denotes the complex exponential.
background
Recognition Science obtains the imaginary unit from the eight-tick phase structure. Multiplication by $i$ encodes a 90-degree rotation, equivalent to two ticks, because the full cycle spans eight ticks and one tick is the fundamental time quantum. The module imports the tick definition from Constants and the eight-tick octave from Foundation.EightTick to ground this phase generator.
proof idea
The tactic proof first constructs the auxiliary equality $I = cexp(I * Real.pi / 2)$ by rewriting with Complex.exp_mul_I together with the known cosine and sine values at pi/2, then simplifies. It substitutes the auxiliary into the goal, applies the natural-number multiplication rule for the complex exponential, and finishes with ring normalization.
why it matters
The declaration supplies the general power case inside the 8-tick phase rotation framework of the module documentation. It supports sibling results on the Schrödinger equation and Euler identity that trace complex phases back to the T7 eight-tick octave. Within the Recognition Science chain it explains why complex numbers appear in wave and quantum descriptions without additional postulates.
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