predictions
plain-language theorem explainer
The definition enumerates four predictions that follow from requiring complex numbers to represent 8-tick phases in Recognition Science. A foundations physicist would cite the list when arguing that real-valued quantum mechanics is ruled out by experiment and structure. It is assembled as a direct enumeration of consequences from phase rotations, spinor periodicity, and geometric phases.
Claim. The complex necessity predictions are the list: real quantum mechanics is experimentally distinguishable and ruled out (2021); interference patterns require complex amplitudes; spinor structure reflects the 8-tick cycle with $4π$ rotation equal to the identity; the Berry phase is geometric and requires complex numbers.
background
Module MATH-004 derives the necessity of complex numbers from the 8-tick phase structure of Recognition Science. The fundamental time quantum is the tick $τ_0 = 1$ (from Constants.tick). The phases are given by phase(k) = $kπ/4$ for $k = 0,…,7$ (from EightTick.phase), which are rotations in the plane and therefore demand the imaginary unit. Upstream results establish the identity event at J-cost minimum and the 8-tick periodicity that forces two-dimensional representation.
proof idea
The definition is a direct enumeration of the four listed predictions. It assembles strings that restate consequences already established by the phase definition, the tick constant, and the observer identity event.
why it matters
The definition records the empirical and structural predictions of the complex-numbers necessity argument inside the Recognition Science framework. It supports the module target of showing that the 8-tick octave (T7) forces complex amplitudes and spinor structure. The list aligns with observed features such as Berry phase and 4π periodicity in spin statistics.
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