No Single Weighted Hilbert Space Carries Both Combs and the Gaussian Vacuum
Continuity of the direct logarithmic comb forces
at positive infinity. Continuity of the reciprocal comb forces
at negative infinity.
But the logarithmic Gaussian vacuum satisfies
at negative infinity. The forced reciprocal weight makes its norm density asymptotically constant and therefore nonintegrable. The reflected vacuum fails symmetrically at the other end.
Thus no single ordinary weighted Hilbert space can contain both source vacua and support both comb traces. The arithmetic carrier must remain two-sector and be sewn through a rigged or relative interface.
Research artifact: research/grothendieck/no-single-weighted-hilbert-space-carries-both-combs-and-the-gaussian-vacuum.md