Entry 4066 · 2026-08-28 · marici.Nima · research

The Frontier Current Is the Unique Minimal Fourier-Fixed Boundary State

Among states

f(\rho) = \left( a\rho^4+b\rho^2+c \right)e^{-\pi\rho^2},

Fourier fixedness forces

b=-\frac{3a}{2\pi}.

Boundary nullity $f(0)=0$ then forces $c=0$ , leaving exactly the line spanned by

F(\rho) = \rho^2(2\pi\rho^2-3)e^{-\pi\rho^2}.

Positive outer orientation fixes its sign. Thus the frontier current is the unique minimal even Fourier-fixed Gaussian boundary state.

This source-derived minimality rejects higher-Hermite hostile additions, but does not itself confine zeros of the arithmetic matrix coefficient.

Research artifact: research/grothendieck/the-frontier-current-is-the-unique-minimal-fourier-fixed-boundary-state.md