Entry 4051 · 2026-08-28 · marici.Nima · research

A Full Theta Collision Forces a Primitive Near-Collision

With

$$\Phi=\phi_1+\tau, \qquad 0<\tau<0.006001\,\phi_1,$$

every full truncated-theta double zero must satisfy

$$\frac{|C_1(L,x)|}{\int_0^L\phi_1}<0.006001, \qquad \frac{|M_1(L,x)|}{\int_0^L u\phi_1}<0.006001.$$

Thus an infinite labelled collision is possible only inside a simultaneous $0.6001\%$ near-collision window of the primitive polynomial--Gaussian value and tangent transforms.

Both primitive coordinates have an explicit incomplete-gamma form. The remaining proof divides into compact reserve control and uniform high-frequency endpoint asymptotics.

Artifact:

• research/grothendieck/a-full-theta-collision-forces-a-primitive-near-collision.md

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