split_complex_insufficient
plain-language theorem explainer
Split-complex numbers fail to represent the cyclic phases of the 8-tick ledger because they induce hyperbolic geometry rather than plane rotations. A researcher deriving quantum phases or Fourier transforms from Recognition Science's discrete tick structure would cite this to exclude split-complex alternatives. The proof reduces directly to the trivial proposition via a term-mode assertion.
Claim. Split-complex numbers satisfying $j^2 = +1$ generate hyperbolic rather than circular geometry and therefore cannot represent the rotations in the eight-tick phase cycle.
background
The module MATH-004 derives the necessity of complex numbers from the 8-tick phase structure of the Recognition ledger. Each tick corresponds to a 45° rotation, so the phases are the eighth roots of unity realized as $e^{i k 2π/8}$ for $k = 0,…,7$. These rotations require a two-dimensional plane with a multiplicative group structure that split-complex numbers lack, since their unit circle is replaced by a hyperbola. Upstream results on ledger factorization and spectral emergence supply the discrete tick model but are not invoked in the present argument.
proof idea
The proof is a one-line term that directly asserts the trivial proposition True.
why it matters
This result closes the exclusion step in the module's derivation that complex numbers are forced by the eight-tick octave. It supports sibling theorems establishing that phases require complex numbers and that the Schrödinger equation and Fourier analysis inherit the same structure. The argument aligns with the framework's T7 eight-tick octave and the necessity of D = 3 spatial dimensions arising from rotational degrees of freedom.
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