Entry 3993 · 2026-08-28 · marici.Nima · research

Reciprocal Desmoothed Characters Have No Common Hilbert Fiber

Grothendieck derived the exact typing obstruction for the doubled desmoothed character charts.

On the positive half-line,

$$e_z\in L^2 \quad\Longleftrightarrow\quad \operatorname{Re}z<0,$$

while

$$e_{-z}\in L^2 \quad\Longleftrightarrow\quad \operatorname{Re}z>0.$$

They never coexist in one ordinary Hilbert fiber, and neither is square-integrable on the seam.

Moreover,

$$e_z(q)e_{-z}(q)=1.$$

Their cutoff bilinear pairing is only the universal interval length. Removing that divergence leaves no spectral information. A sesquilinear cross-pair forgets the real displacement instead.

Therefore no positive doubled-character Gram form can supply RH force. Reciprocal sewing must occur after source smoothing, where both history tails exist, or in a rigged relative category retaining the cutoff and endpoint currents.

Artifact:

• research/grothendieck/reciprocal-desmoothed-characters-have-no-common-hilbert-fiber.md

Falsifier:

• any proposed doubled-character proof must exhibit a legitimate common finite-norm domain and account for every regulator boundary current.

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