two_rotations
plain-language theorem explainer
The product of two complex rotations by π/2 equals one rotation by π. Researchers grounding the imaginary unit in the eight-tick phase structure of Recognition Science cite this to link discrete ticks to continuous phase factors. The proof is a one-line wrapper that invokes the exponential addition law and then simplifies the combined exponent by ring normalization.
Claim. $e^{iπ/2} · e^{iπ/2} = e^{iπ}$
background
Recognition Science takes the tick as the fundamental time quantum, set to 1 in native units. The eight-tick phase assigns values kπ/4 for k in 0..7 and maps each to the unimodular complex number exp(i w). The local setting is MATH-001, whose module doc states that rotation by two ticks corresponds to multiplication by i while four ticks yield -1, so i × i equals rotation by π.
proof idea
The term proof first rewrites the product via the addition formula for the complex exponential, converting it to a single exp whose argument is the sum of the two π/2 phases. Ring normalization then collapses the coefficient of I from 1 to 2, producing the target phase π.
why it matters
This step supplies the algebraic content of i² = -1 inside the eight-tick framework and directly supports the module goal of deriving the imaginary unit from phase rotations. It connects to the T7 eight-tick octave landmark and explains why complex phases appear in quantum mechanics. No downstream theorems are recorded yet, leaving the result as an isolated but foundational identity.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.