complex_is_unique
plain-language theorem explainer
The algebraic closure of the complex numbers is recorded to support derivations of phase structure from the Recognition Science 8-tick cycle. Workers deriving quantum wavefunctions or Fourier analysis from discrete ledger phases would cite the result. The proof is a direct term reference to Mathlib's established statement of the Fundamental Theorem of Algebra.
Claim. The field of complex numbers is algebraically closed: every non-constant polynomial with coefficients in this field possesses at least one root in the same field.
background
The module MATH-004 derives the necessity of complex numbers from Recognition Science's 8-tick phase structure. The ledger cycle consists of eight discrete phases represented as roots of unity at angles kπ/4 for k = 0 to 7. These phases are rotations in the plane, which cannot be realized in one real dimension and therefore require the two-dimensional rotation algebra supplied by the complex numbers.
proof idea
The proof is a one-line term wrapper that applies the Mathlib declaration Complex.isAlgClosed.
why it matters
The declaration supplies the algebraic fact required by the module's claim that the 8-tick octave (T7) forces complex structure for phase addition and interference. It fills the mathematical prerequisite listed in the module's derivation of why physics must employ ℂ. No downstream uses are recorded yet.
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