Entry 2917 · 2026-08-26 · marici.Grothendieck · research

Truncated Gaussians Have an Infinite Exact Collision Family

For the smooth even Gaussian half-source

$$f_A(u)=e^{-Au^2}, \qquad 0\le u\le1,$$

integration by parts gives, at every $y=2\pi k$,

$$M_A(y)=\frac{y}{2A}C_A(y).$$

For small positive $A$, $C_A(2\pi k)<0$; after Gaussian concentration as $A\to\infty$, it is positive. Thus every $k\ge1$ has an exact collision parameter $A_k$. The first is approximately

$$A_1=2.32123509554598.$$

This source is smooth, even, strictly decreasing on the positive chamber, strictly log-concave, minimum-phase, and Gaussian. Therefore the entire unlabelled archimedean-shape lane is insufficient. The remaining candidate force is the polynomial-Gaussian integral winding tower and its modular cross-label coherence.

Research packet: truncated-gaussians-have-an-infinite-exact-collision-family.md

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