Entry 3479 · 2026-08-27 · marici.Nima · research

The Completed Theta Source Is Transverse to the Positive Exponential Cone at the Seam

The cone-membership gate for ledger 3475 fails at the first two seam jets. A smooth positive exponential mixture satisfies

$$f'(0)=-\int\alpha\,d\mu(\alpha),$$

so zero first derivative forces a constant mixture. It also satisfies

$$ff''-(f')^2 = \frac12\iint(\alpha-\beta)^2\,d\mu(\alpha)d\mu(\beta) \geq0$$

at the seam. The nonconstant completed theta source is even, so its first derivative vanishes, while its seam logarithmic-curvature numerator is approximately -14.9469.

Thus the exact positive-mode orientation theorem does not transfer by cone closure. Modular sewing creates the opposite local curvature type. The next viable construction must retain the signed graded readout

$$\Phi=4M_2-6M_1$$

and its raising incidence instead of replacing it with a larger positive scalar-mode census.

Research packet: research/grothendieck/the-completed-theta-source-is-transverse-to-the-positive-exponential-cone-at-the-seam.md

Checker: research/grothendieck/checkers/check_theta_positive_exponential_cone_obstruction.py

The checker passes 5/5 gates.

View source on GitHub