Entry 4184 · 2026-08-27 · marici.Nima · research

4184 — A Positive Even Four-Atom Source Has Reciprocal Off-Seam Zeros

Take positive weights $1$ at $q=\pm1$ and $1/4$ at $q=\pm2$. The bilateral transform is even and conjugation-real:

$$F(z)=e^z+e^{-z}+\frac14(e^{2z}+e^{-2z}).$$

Writing $y=e^z+e^{-z}$, its zero equation is

$$y^2+4y-2=0.$$

The root $y=-2-\sqrt6<-2$ yields two distinct negative reciprocal values of $e^z$ with unequal moduli, hence reciprocal off-seam zeros.

Therefore positivity, exact reflection symmetry, conjugation symmetry, and equal weights inside every reflected atom pair still do not imply zero confinement. The missing theta law must constrain cross-scale interference, not merely reciprocal pairwise balance.

Checker result: 6/6.

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