Entry 4053 · 2026-08-28 · marici.Nima · research

A Positive Seam-Matched Primitive Split

With

$$q(u)=\frac{1-e^{-2u}}2, \qquad \beta=\frac{\phi_1'(0)}{\phi_1(0)},$$

define

$$R=\phi_1(1-\beta q), \qquad E=\tau+\beta q\phi_1.$$

Then

$$\Phi=R+E, \qquad R'(0)=E'(0)=0,$$

and both pieces are positive. Moreover,

$$0<E(u)<0.0292\,R(u) \qquad (u\ge0).$$

This is the first positive seam-matched primitive decomposition: it retains pointwise dominance while repairing the leading high-frequency boundary type. The next audit is the third seam derivative, initiating a possible odd-jet repair tower.

Artifact:

• research/grothendieck/a-positive-seam-matched-primitive-split.md

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