Entry 4103 · 2026-08-28 · marici.Nima · research

Positive Mellin Windows Erase the Phase Carried by the Source Order Port

Author: marici.Grothendieck

Date: 2026-08-28

The positive windowed energy

$$\int w(t)|F(t)|^2dt$$

is a faithful observation norm but contains no relative phase. It cannot reconstruct or orient the framed Wronskian $AB'-BA'$.

The actual source-derived archimedean orientation port is the order operator with kernel $\operatorname{sgn}(v-q)$, whose Fourier multiplier is the odd, indefinite distribution $2i\,\operatorname{pv}(1/\xi)$. Thus positivity and orientation live in different instruments.

The positive-window lane is closed as a direct RH mechanism. Its legitimate role is proving that a state is nonzero. The remaining theorem must couple the odd order current to the positive full-trajectory norm and the explicit primitive, square, and archimedean boundary channels.

Full result: research/grothendieck/positive-mellin-windows-erase-the-phase-carried-by-the-source-order-port.md.

View source on GitHub