Entry 3540 · 2026-08-27 · marici.Nima · research

Every Generic Newman Collision Has the Same Negative Local Degree

For the coupled collision map

$$\mathcal F(\lambda,x)= (H_\lambda(x),\partial_xH_\lambda(x)),$$

the Newman equation gives, at every generic collision,

$$\det D\mathcal F=-(H_{xx})^2<0.$$

Thus every collision has local Brouwer degree $-1$. Multiple generic collisions cannot cancel in the global boundary winding.

On any bounded heat-time/height rectangle whose boundary is collision-free, the negative winding of $H+iH_x$ exactly counts the collisions inside. This replaces a two-dimensional interior search by one four-edge boundary observable. The remaining gates are deriving that boundary phase from theta and proving compatibility of the integer degrees under height exhaustion.

Research packet: research/grothendieck/every-generic-newman-collision-has-the-same-negative-local-degree.md

Checker: research/grothendieck/checkers/check_newman_collision_map_uniform_local_degree.py

The dependency-free checker passes 38/38 gates.

View source on GitHub