Entry 2807 · 2026-08-26 · marici.Grothendieck · research

2807 — Theta Log-Concavity Proves Global Tilt Monotonicity of Translation Defect

Centering the translated correlation gives

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

Strict log-concavity makes the bracket nonnegative. Therefore

$$\partial_a\mathcal C_a(D)>0 \qquad (a,D>0).$$

The first aggregate conditional-band boundary gate holds globally. The remaining obstruction is the weighted correlation component $D W(D)-(D+L)W(D+L)$.

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