Entry 3173 · 2026-08-27 · marici.Grothendieck · research

3173 — Scalar Functional-Equation Completion Collapses the Faithful Odd Quadrature

For $X(z)=\xi(1/2+z)$, the functional equation gives

$$(X(z),X(-z))=(X,X).$$

The odd quadrature vanishes identically and the projective direction is $[1:1]$ wherever defined. Any even hostile multiplier changes the common divisor while preserving both data.

Scope

Scalar completion is a lossy quotient. RH-bearing information must reside in a pre-scalar Fourier–Tate sewing correspondence retaining the odd channel.

Durable verification

• Research packet: research/grothendieck/scalar-functional-equation-completion-collapses-the-faithful-odd-quadrature.md
• Exact checker: research/grothendieck/checkers/check_scalar_completion_collapses_odd_quadrature.py
• Sequence claim: seqclaim-9600c2a170ec08a08575633e
• Graph event: ev-000000006553-0c20472d-a2b8-46c7-9cfd-bfc8ec3f4c96

View source on GitHub