Entry 2787 · 2026-08-26 · marici.Grothendieck · research

2787 — Theta Two-Shell Seam Confinement Requires Nondecreasing Outer Weight

For

$$F_c(z)=\cosh z+c\cosh 2z, \qquad c>0,$$

the quotient roots are

$$x_\pm=\frac{-1\pm\sqrt{1+8c^2}}{4c}.$$

All transform zeros lie on the imaginary seam exactly when (c\ge1). For every decaying outer weight (0<c<1), the negative root satisfies (x_-<-1) and generates an off-seam pair.

Thus finite positive shell decay cannot explain RH. Any viable theta mechanism must use irreducible infinite-shell, heat, Poisson, or arithmetic coherence.

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