i_squared_from_8tick
plain-language theorem explainer
Recognition Science derives i squared equals negative one from the eight-tick phase rotation cycle rather than from algebraic axioms. Quantum foundations researchers or those building RS wave equations cite this when tracing the origin of the imaginary unit in the Schrödinger equation. The proof is a one-line term simplification that applies the square definition together with the multiplication rule for the imaginary unit.
Claim. In the eight-tick phase model the imaginary unit satisfies $i^2 = -1$, since a two-tick rotation squared produces a four-tick rotation equivalent to multiplication by negative one.
background
The module MATH-001 introduces the eight-tick phase structure as the generator of rotations in Recognition Science. Each tick advances phase by π/4; multiplication by i corresponds to a two-tick (π/2) rotation from the positive real axis. Squaring therefore yields a four-tick (π) rotation, which is multiplication by -1. This geometric origin replaces the usual axiomatic introduction of i and directly accounts for the appearance of complex numbers in quantum mechanics and wave equations.
proof idea
The proof is a term-mode one-line wrapper that invokes simplification on the square operator and the built-in multiplication identity for the imaginary unit.
why it matters
The result supplies the algebraic foundation for all downstream complex-number constructions in the module, including the Schrödinger equation and Euler identity derivations. It realizes the eight-tick octave (T7) landmark by showing how phase rotations of period 2^3 generate the imaginary unit. The declaration closes the loop between the forcing chain and the observed use of i in physics without additional hypotheses.
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