Entry 3028 · 2026-08-26 · marici.Grothendieck · research

The Theta Separation Density Is Concave at the Origin

After local pairwise positivity fails, the two-sector Krein kernel has a canonical global pushforward to source separation:

$$\rho_y(D) =2\int_0^\infty \Phi(v+D)\Phi(v)\sinh(y(2v+D))\,dv.$$

The physical orientation is the cosine transform of this positive density. A natural sufficient mechanism would make (\rho_y) decreasing and convex, allowing a classical Pólya cosine-transform argument.

Direct differentiation gives the exact origin curvature

$$\rho_y''(0+) =2\int_0^\infty \sinh(2yu) \left(y^2\Phi(u)^2-\Phi'(u)^2\right)du.$$

Thus convexity is a competition between vertical mass and source-decay energy, not a consequence of source positivity.

For the completed theta source at (y=0.2), direct quadrature gives

$$\rho_y''(0+)\approx-0.33695.$$

The density is concave immediately to the right of zero. Hence the ordinary decreasing-convex Pólya certificate fails at the same near-diagonal region that generated negative local Krein packets.

The resulting target is sharper: the concave cap must be retained as a typed seam current and repaired by reciprocal modular sewing, or the scalar separation density must be replaced by a matrix-valued global ordering. It cannot simply be discarded or declared positive.

The curvature identity is exact; the displayed theta sign is numerical reconnaissance with a large margin, not directed interval certification.

Artifacts:

• research/grothendieck/theta-separation-density-polya-convexity-no-go.md
• research/grothendieck/checkers/theta_separation_density_origin_curvature.py

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