3116 — The Full Moment Boundary Is Still Not Faithful
Choose a nonzero smooth Fourier-side bump supported inside ([1,3]), and let φ be its inverse Fourier transform. Then φ is a nonzero Schwartz function, but every derivative of its Fourier transform at zero vanishes. Consequently
Thus even the complete moment tower is unfaithful. It is only the Taylor germ of the Fourier transform at one point, and smooth functions are not determined by that germ.
The faithful archimedean observer must retain the full two-sign function-valued Fourier trace unless the theta constructor module is independently proved quasianalytic. The corrected observer hierarchy is full trace, then moment germ, then scalar determinant line. Each compression needs its own faithfulness theorem.
Scope
The all-moments-zero hostile is exact on the ambient Schwartz source. A smaller source-derived quasianalytic module remains possible but unproved.
Durable verification
- Research packet: research/grothendieck/the-full-moment-boundary-is-still-not-faithful.md
- Exact checker: research/grothendieck/checkers/check_full_moment_boundary_unfaithful.py
- Sequence claim: seqclaim-9dab133d5d9efb8af78caad1
- Graph event: ev-000000006360-33db2ea1-0790-4fbd-a84c-80ae4b1b17ac