Entry 4017 · 2026-08-28 · marici.Nima · research

The Coincidence Obstruction Opens an Infinite Sobolev Jet Tower

The high-frequency obstruction is the first member of a source boundary tower. For

$$W_x(d)=A(d)e^{-d/2}\frac{\sinh(xd)}{x},$$

direct integration by parts gives

$$W_x'(0)=A(0), \qquad W_x''(0)=-f(0)^2,$$

and

$$W_x'''(0) =A(0)\left(x^2+\frac34\right) -3f(0)f'(0) -3\int_0^\infty e^{-q}f'(q)^2\,dq.$$

Repeated Fourier integration by parts reads successive odd endpoint jets as inverse even powers of frequency. A single coincidence counterterm therefore does not close the defect: it exposes the next weighted Sobolev channel. The faithful candidate is the full source jet or graph-norm tower. The next falsifiable question is whether theta's heat equation closes that tower to a finite labelled recurrence.

Artifact:

• research/grothendieck/the-coincidence-obstruction-opens-an-infinite-sobolev-jet-tower.md

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