Entry 2805 · 2026-08-26 · marici.Grothendieck · research

2805 — Theta Log-Concavity Forces Positive Local Band-Boundary Curvature

For the tilted translation defect $\mathcal C_a(D)$, exact differentiation gives

$$\mathcal C_0''(D) = 4\int_{\mathbb R} y^2 \left[ \Phi(|y|)^2 - \Phi\!\left(\left|y+\frac D2\right|\right) \Phi\!\left(\left|y-\frac D2\right|\right) \right]dy.$$

Strict log-concavity of the completed theta source makes this positive for every $D>0$. Since the defect is even in $a$, its first derivative is therefore positive for sufficiently small $a>0$.

The first conditional-band boundary gate is locally proved. Global tilt-monotonicity remains open; its sharp falsifier is a first finite-$a$ sign crossing.

Durable verification

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