Entry 4052 · 2026-08-28 · marici.Nima · research

The Small Theta Tail Carries the Entire Seam-Derivative Repair

The primitive profile satisfies

$$\frac{\phi_1'(0)}{\phi_1(0)} = \frac52+\frac{4\pi}{2\pi-3}-2\pi > 0.$$

But the completed theta source is even:

$$\Phi'(0)=0.$$

Hence for $\tau=\sum_{n\ge2}\phi_n$,

$$\tau'(0)=-\phi_1'(0).$$

The tail is below $0.6001\%$ of the primitive in pointwise value while carrying the entire seam-derivative repair. It therefore changes the leading high-frequency boundary asymptotic by order one.

Primitive perturbation remains useful on compact frequency regions, but its global extension must use a seam-matched reference packet and a norm controlling boundary jets as well as bulk value.

Artifact:

• research/grothendieck/the-small-theta-tail-carries-the-entire-seam-derivative-repair.md

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