phase
plain-language theorem explainer
The definition supplies the map sending a real angle w to the unit complex number e^{i w}. Researchers working on the boundary-wedge condition in the BRF route to the Riemann Hypothesis or on eight-tick periodic models in climate and astrophysics cite this construction. The definition is a direct one-line application of the complex exponential.
Claim. For real $w$, define the unimodular complex number by $e^{i w}$.
background
The module records the algebraic consequence of the boundary wedge condition: after a unimodular rotation the phase function w(t) satisfies |w(t)| ≤ π Υ with Υ < 1/2 almost everywhere, which implies Re(exp(i w)) ≥ 0 whenever |w| ≤ π/2. This supplies the entry point for Poisson transport and the Cayley transform in the Riemann Hypothesis argument. The upstream result from EightTick defines the discrete phases k π/4 for k = 0 … 7 and states that the 8-tick phase is periodic with period 2π.
proof idea
One-line definition that applies Complex.exp to the product of the real input w with the imaginary unit.
why it matters
The definition is the algebraic starting point for positivity in the boundary-wedge argument of the BRF Riemann Hypothesis route. It is invoked in downstream structures for nuclear densities, vacuum climate states, diurnal eight-tick cycles, and topological phases. It directly implements the eight-tick octave (T7) of the forcing chain by providing the continuous phase that the discrete k π/4 phases discretize.
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