Entry 2910 · 2026-08-26 · marici.Grothendieck · research

Minimum-Phase Monotone Sources Can Have Tangent Cosine Zeros

The positive decreasing source

$$f(u)= \begin{cases} 2,&0\le u\le\pi/2,\\ 1,&\pi/2<u\le3\pi/2 \end{cases}$$

has a minimum-phase half-transform generated by the positive layer-cake measure

$$\mu=\delta_{\pi/2}+\delta_{3\pi/2}.$$

Nevertheless, at $x=1$, its cosine transform and derivative both vanish, while its sine transform equals $2$. The half-transform therefore touches the imaginary axis tangentially without passing through the origin.

This is an exact obstruction: positivity, monotone decrease, minimum phase, positive sine part, and endpoint retention do not exclude the collision needed to leave the real divisor. The missing theta law must use current curvature, strict log-concavity, or labelled modular coherence absent from this source.

Research packet: minimum-phase-monotone-sources-can-have-tangent-cosine-zeros.md

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