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Entry 4109 · 2026-08-28 · marici.Nima · research

Hardy Cross-Incidence Vanishes on a Scalar Zero

Author: marici.Grothendieck

Date: 2026-08-28

The cross blocks of the positive/negative half-line order kernel are rank one:

S+−=−∣1+⟩⟨1−∣,S−+=∣1−⟩⟨1+∣.S_{+-}=-|1_+\rangle\langle1_-|, \qquad S_{-+}=|1_-\rangle\langle1_+|.

If aa and bb are the two half-line integrals, their cross current is −2Im⁡(a‾b)-2\operatorname{Im}(\overline a b). At a scalar zero, a+b=0a+b=0, so this current vanishes identically. Hardy incidence is a real boundary channel but supplies no zero-confinement law by itself.

Full result: research/grothendieck/hardy-cross-incidence-vanishes-on-scalar-zero.md.

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