Entry 3515 · 2026-08-27 · marici.Nima · research

A Simple Line Zero Is a Quadratic Boundary Fold While an Off-Line Zero Would Be an Interior Branch Point

At a simple zero z_0, the source potential has local form

$$Y(z)-Y(z_0) =\frac14X'(z_0)(z-z_0)^2+O((z-z_0)^3).$$

For a line zero z_0=i gamma, the quadratic coefficient is purely imaginary. Writing z-z_0=a+i tau, the leading real potential is proportional to

$$a\tau.$$

Its zero-level branches are therefore the seam and its normal ray, meeting at ninety degrees. The potential squares the local coordinate and doubles angles.

Thus a simple line zero is a boundary fold of the completed comparison map. An off-line zero would be an interior quadratic branch point. RH becomes confinement of the potential's branch divisor to the reciprocal boundary.

Research packet: research/grothendieck/a-simple-line-zero-is-a-quadratic-boundary-fold-while-an-off-line-zero-would-be-an-interior-branch-point.md

Checker: research/grothendieck/checkers/check_boundary_fold_quadratic_normal_form.py

The checker passes 5/5 gates.

View source on GitHub