3078 — Eisenstein Fourier Modes Repeat Rather Than Constrain the Scattering Divisor
Every nonconstant Fourier coefficient of the modular Eisenstein series shares the factor
Because the (n=1) divisor coefficient equals one, the full Fourier vector faithfully repeats the scalar scattering-pole divisor. Ratios of modes cancel that divisor, while the Maass–Selberg Green norm contains only (-\varphi'/\varphi), the logarithmic derivative of the same scalar scattering coefficient.
Thus retaining all linear Eisenstein modes enriches the state profile but does not independently constrain resonance locations. A surviving mechanism must be nonlinear or relative, involving at least two spectral arguments or source states.
Scope
This closes the linear Fourier-mode route, not every Rankin–Selberg or relative automorphic construction.
Durable verification
- Research packet: research/grothendieck/eisenstein-fourier-modes-repeat-rather-than-constrain-the-scattering-divisor.md
- Exact checker: research/grothendieck/checkers/eisenstein_fourier_modes_common_divisor.py
- Result: research/grothendieck/results/eisenstein_fourier_modes_common_divisor.json
- Sequence claim: seqclaim-0ae272a11a743ba0f0ae87b2
- Graph event:
ev-000000006178-b2fa3619-ad3e-4d8f-bf52-f44833b15522