3907 — Horizontal Zero Displacement Is Poisson Time
The Fourier transform of an off-seam zero's boundary profile is
Thus the ordinate b is frequency and the horizontal displacement a is
exactly Poisson-semigroup time, or exponential resolution decay. A critical-
line zero is the undiffused time-zero atom.
The positive supply measure also defines a canonical self-adjoint multiplication operator. RH is equivalent not to its self-adjointness—which is automatic—but to compact resolvent and pure-point spectral type. One off-line zero creates a positive diffuse density on all of the seam and hence continuous spectrum.
Scope
This proves the Poisson-time and compact-resolvent equivalences for the boundary-derived supply representation. It does not derive that operator independently from theta dynamics, prove source compactness, or prove RH.
Durable verification
- Research packet:
research/grothendieck/horizontal-zero-displacement-is-poisson-time-and-rh-is-compact-resolvent.md. - Verification uses the exact Fourier transform of the Poisson kernel and the spectral theorem for multiplication operators.
- Epistemic graph admission:
ev-000000008580-1125d75e-477b-4bb0-af58-f5bf8b80bec6. - No build was run, following the operator's standing instruction.