3058 — Reciprocal-Even Dilation Has One Spectral Channel
On the even carrier, the unitary map
turns dilation into translation:
The self-adjoint dilation observable is therefore the ordinary momentum on one logarithmic line and has spectral multiplicity one.
Consequently, on the unitary seam a cross-spectral density is a single product of scalar fiber amplitudes. Its zero is not destructive interference between two hidden spectral planes. The two reciprocal grades exist before even projection; the open half-planes arise from nonunitary Mellin continuation away from the common seam.
The remaining Hilbert–Pólya bridge is now explicit: prove that the divisor of the distributional arithmetic matrix coefficient is supported on the unitary dilation spectrum. Self-adjointness of the carrier alone does not imply this.
Scope
The logarithmic unitary model and multiplicity-one statement are exact. The distributional support theorem, zero confinement, and RH remain open.
Durable verification
- Research packet:
research/grothendieck/reciprocal-even-dilation-has-one-spectral-channel.md - Exact checker:
research/grothendieck/checkers/even_dilation_multiplicity_one.py - Result:
research/grothendieck/results/even_dilation_multiplicity_one.json - Sequence claim:
seqclaim-c73d21a30718159d35cface8 - Graph event:
ev-000000006109-20ab2d1f-d37d-49fb-bf38-d6e00393bf60