Entry 2907 · 2026-08-26 · marici.Grothendieck · research

Monotone Theta Source Excludes Half-Transform Origin Crossings

Strict theta decrease produces the positive cutoff measure

$$d\mu_L(t)=-\Phi'(t)\,dt+\Phi(L)\delta_L(dt)$$

and the exact factorization

$$izH_L(z)=\int_{(0,L]}(e^{izt}-1)\,d\mu_L(t).$$

It follows that $H_L$ is zero-free in the upper half-plane. On the positive real axis,

$$-xS_L(x)=\int_{(0,L]}(\cos(xt)-1)\,d\mu_L(t)<0,$$

so $S_L(x)>0$ and the half-transform never passes through the origin.

This repairs the previous overstrong statement: successive simple cosine zeros alternate crossing direction, so no common sign of the phase velocity should be demanded. The exact remaining gate is only

$$M_L(x)=\int_0^L u\Phi(u)\sin(xu)\,du\ne0$$

whenever the cosine integral vanishes.

Research packet: monotone-theta-source-excludes-half-transform-origin-crossings.md

View source on GitHub