complex_from_ledger
plain-language theorem explainer
The declaration asserts that Recognition Science's 8-tick ledger forces cyclic phases whose additive structure requires the complex numbers. Foundational physicists would cite it when tracing the origin of quantum amplitudes to discrete time. The proof is a one-line wrapper that reduces the entire claim to the tautology True.
Claim. The 8-tick ledger of Recognition Science induces cyclic phases whose differences are additive under composition; the unique algebraic structure satisfying these conditions is the field of complex numbers.
background
Module MATH-004 derives complex numbers from the 8-tick phase structure. The ledger cycle consists of eight equally spaced phases {0, π/4, …, 7π/4}, each a 45° rotation; rotations in the plane cannot be represented in one real dimension. The upstream definition tick : ℝ := 1 supplies the fundamental time quantum, while the eight-tick octave (T7) supplies the period 2^3 that closes the cycle.
proof idea
Term-mode proof consisting of the single expression trivial applied to the proposition True.
why it matters
The theorem occupies the MATH-004 slot that converts the eight-tick octave (T7) into the necessity of ℂ for phase and interference. It directly supports the listed predictions that real-only quantum theory is inconsistent and that phase appears ubiquitously. No downstream uses are recorded; the result therefore functions as a high-level marker rather than a lemma inside a larger derivation.
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