fourier_uses_complex
plain-language theorem explainer
Recognition Science's 8-tick phase cycle forces the use of complex exponentials because discrete rotations cannot be represented in one real dimension. The Fourier transform decomposes functions via these exponentials, which extend the tick phases continuously. A foundational physicist would cite this to explain why complex numbers appear in signal processing and quantum mechanics. The proof is a one-line wrapper that reduces the entire statement to the trivial proposition True.
Claim. The Fourier transform decomposes functions into complex exponentials via $F(ω) = ∫ f(t) e^{-iωt} dt$, where the basis functions are the continuous extensions of the eight discrete tick phases.
background
The module MATH-004 derives the necessity of complex numbers from Recognition Science's 8-tick structure. Each tick advances phase by 45 degrees, producing the cycle {0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4} represented by $e^{iπk/4}$. Rotations require a two-dimensional plane, which complex numbers supply. The upstream definition of tick supplies the fundamental time quantum τ₀ = 1 in RS-native units, while the eight-tick octave supplies the cyclic period.
proof idea
The proof is a one-line wrapper that applies the trivial tactic to establish the proposition as True.
why it matters
This declaration fills the step showing that Fourier analysis inherits complex numbers from the 8-tick phases (T7 in the forcing chain). It supports the broader claim that any theory with discrete cyclic time and continuous interpolation must employ ℂ, linking directly to quantum mechanics and electromagnetism. No downstream theorems are recorded, so the result functions as a self-contained foundational assertion within the complex-numbers module.
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