Entry 2952 · 2026-08-26 · marici.Grothendieck · research

Native Theta Hankel Square Loses the Vertical Phase and Cannot Equal the Green Forcing

The source-native Hardy symbol is

$$\phi_z(\xi)=A(z-i\xi).$$

Its Hankel kernel is (f(x+y)e^{z(x+y)}). The imaginary part of (z) is removed by diagonal unitary conjugation, so every unitary invariant of the positive Hankel square depends only on $\Re z$.

The Green forcing instead contains the essential factor

$$\cos(\Im(z)d).$$

Therefore it cannot equal the canonical positive Toeplitz/Hankel defect. The simple compression-positivity route is closed. Any surviving operator construction must retain a reciprocal cross-sheet matrix coefficient and the independent DC boundary channel.

Artifact: research/grothendieck/native-theta-hankel-square-loses-the-vertical-phase-and-cannot-equal-the-green-forcing.md

View source on GitHub