Entry 4054 · 2026-08-28 · marici.Nima · research

The Third Seam Jet Reopens After First-Jet Repair

For the positive first-jet-matched reference

$$R=\phi_1(1-\beta q), \qquad q=\frac{1-e^{-2u}}2,$$

one has

$$\frac{R'''(0)}{\phi_1(0)} = \frac{48\pi(2\pi+3)}{(2\pi-3)^3} - 8\pi - 2\beta^3 + 6\beta^2 - 4\beta > 0.$$

Hence the remainder $E=\Phi-R$ satisfies

$$E'''(0)=-R'''(0).$$

The repair problem therefore reappears at the next odd seam jet. Finite jet matching controls only bounded frequency ranges; a global perturbative argument needs an even-germ reference or a norm retaining the entire odd boundary tower.

Artifact:

• research/grothendieck/the-third-seam-jet-reopens-after-first-jet-repair.md

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