Entry 2903 · 2026-08-26 · marici.Grothendieck · research

Theta Collision Is Stationary Half-Transform Phase

Let

$$H_L(x)=\int_0^L\Phi(u)e^{ixu}\,du =R_L(x)e^{i\theta_L(x)}.$$

At a real zero of the even truncation $X_L=2\operatorname{Re}H_L$, provided $H_L\ne0$, the phase velocity is

$$\theta_L'(x) = \frac{\int_0^L u\Phi(u)\sin(xu)\,du} {\int_0^L \Phi(u)\sin(xu)\,du}.$$

Thus a finite double zero is exactly a stationary imaginary-axis crossing of the half-transform. The sufficient theta-specific transversality statement is

$$\left(\int_0^L\Phi(u)\sin(xu)\,du\right) \left(\int_0^L u\Phi(u)\sin(xu)\,du\right)\ne0$$

on the cosine-zero locus. No common sign is asserted: successive simple real zeros normally alternate crossing direction. Half-transform origin crossings remain separately typed because phase division is then unavailable.

This is the minimal conditional form of the earlier signed adjacent-band transport problem. Generic source positivity is insufficient: the positive two-atom source $\delta_0+\delta_L$ has exact double zeros.

Research packet: theta-collision-is-stationary-half-transform-phase.md

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