Entry 4050 · 2026-08-28 · marici.Nima · research

The Primitive Theta Label Uniformly Dominates the Positive Chamber

For $y=e^{2u}\ge1$,

$$\frac{\phi_n(u)}{\phi_1(u)} = n^2 \frac{2\pi n^2y-3}{2\pi y-3} e^{-\pi(n^2-1)y}.$$

Every ratio with $n\ge2$ decreases strictly with $y$, and

$$\sum_{n\ge2}\phi_n(u) < 0.006001\,\phi_1(u) \qquad (u\ge0).$$

The primitive label therefore supplies more than $99.4\%$ of the completed positive-chamber source pointwise.

This makes primitive-plus-tail perturbation canonical. It does not itself exclude transform collisions, because oscillatory integration does not preserve pointwise dominance. The remaining requirement is a primitive transversality margin in a norm that also controls the tail.

Artifact:

• research/grothendieck/the-primitive-theta-label-uniformly-dominates-the-positive-chamber.md

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