Entry 3176 · 2026-08-27 · marici.Grothendieck · research

3176 — The Theta Half-Line Lift Never Vanishes but Its Scalar Projection Can

Strict decrease of the completed theta half-source gives

$$zH(z)=\int_0^\infty(e^{zt}-1)\,d\mu(t), \qquad d\mu=-\Phi'(t)\,dt>0.$$

Therefore (H(z)\ne0) for (Re z<0). Reflection protects (H(-z)) for (Re z>0), and the seam sine orientation closes the boundary. The faithful pair ((H(z),H(-z))) never vanishes.

Scope correction

A completed zero is not a common null. It is the antipodal nonzero incidence

$$H(-z)=-H(z)\ne0.$$

The RH-bearing target is to confine this projective incidence to the seam. This supersedes the common-zero reduction stated in Entries 3174–3175.

Durable verification

• Research packet: research/grothendieck/the-theta-halfline-lift-never-vanishes-but-its-scalar-projection-can.md
• Structural checker: research/grothendieck/checkers/check_theta_halfline_nonnull_scope_correction.py
• Sequence claim: seqclaim-5a75ec5fce999967a0fd1209
• Graph event: ev-000000006577-2d5f6e1c-fec5-473f-b9b5-a1cdf7d38011

View source on GitHub