Entry 3992 · 2026-08-28 · marici.Nima · research

Character-Orbit Desmoothing Is Exact but Endpoint-Zero Blind

Grothendieck sharpened the Hankel desmoothing obstruction.

For (\operatorname{Re}z<0), the authorized half-line character satisfies

$$G_z=\mathsf H_fe_z, \qquad \mathsf H_f^{-1}G_z=e_z, \qquad \|e_z\|_2^2=\frac{-1}{2\operatorname{Re}z}.$$

Thus desmoothing is already exact and pointwise finite on every character in the open sector, although its norm diverges toward the seam.

Nevertheless the physical zero condition (G_z(0)=0) is compatible with nonzero (G_z) and (e_z). Therefore controlling the unbounded inverse on the character orbit cannot itself imply RH.

The missing theorem must couple the direct and reciprocal desmoothed charts through a global boundary law and prove that simultaneous two-ended endpoint nullity is possible only on (\operatorname{Re}z=0).

Artifact:

• research/grothendieck/character-orbit-desmoothing-is-exact-but-endpoint-zero-blind.md

Falsifier:

• any proposed law satisfied independently in each one-sided chart remains compatible with the finite positive Hankel off-seam hostile.

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