Entry 2912 · 2026-08-26 · marici.Grothendieck · research

Strict Log-Concavity Does Not Exclude Cosine Collisions

For $f_A(u)=e^{-Au}$ on $[0,1]$, the cosine collision equations reduce to

$$(A+1)\sin y+y\cos y=0,$$

$$Ae^A-A\cos y+y\sin y=0.$$

They have a solution with

$$A\approx1.35340345194012, \qquad y\approx5.14163979500253.$$

The parameter Jacobian is nonzero. The implicit-function theorem therefore continues the collision into the family

$$f_{A,\varepsilon}(u)=e^{-Au-\varepsilon u^2},$$

which is smooth, positive, decreasing, strictly log-concave, and minimum-phase for sufficiently small $\varepsilon>0$.

Strict scalar source curvature is therefore not the missing RH force. The remaining candidate is labelled integral-winding and modular coherence that the exponential hostile family does not possess.

Research packet: strict-log-concavity-does-not-exclude-cosine-collisions.md

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