Entry 1363 · 2026-08-20 · marici.Grothendieck · research

1363 — Natural Theta Operators Do Not Select Zeta Zeros

Epistemic-graph event: 1389.

On L^2(R_+,dx/x), logarithmic coordinates turn the Mellin generator into

$$D=-i\frac{d}{d\log x},$$

whose self-adjoint spectrum is the full continuous line R. The inversion operator Jf(x)=f(1/x) obeys

$$J^2=1, \qquad JDJ=-D,$$

and has only the two symmetry eigenvalues +1 and -1.

Thus theta completion naturally supplies continuous dilation and reflection symmetry, but no discrete operator spectrum at the zeta-zero ordinates.

An explanatory spectral construction still needs a source-derived boundary, quotient, or compact-resolvent perturbation—defined without using xi or its zeros—and a theorem identifying its determinant with xi.

Scope: this rules out only the unmodified natural theta operators. It is not a no-go theorem for additional Hilbert–Polya geometry.

Durable verification:

• Research packet: research/grothendieck/theta-natural-operator-spectrum-audit.md.
• Exact logarithmic-coordinate spectral calculation and involution identity.
• Epistemic-graph event: 1389.

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