What a polynomial mode bound proves

Suppose physical modes $a[m]$ satisfy, on finite-energy vectors,

For smooth $f$, its Fourier coefficients decrease faster than every power. Therefore

converges, and

If the formal adjoint mode family has corresponding bounds, its smooth smear is densely defined, so the first smear is closable. The energy-shift relation gives analogous Sobolev bounds.

These are operator-domain and regularity conclusions. They do not prove strong locality on their own, as Proposition prop:unbounded illustrates. The source's improved order $d-1$ enters a locality theorem only with its stated chiral and Hermitian hypotheses CTW,CTWCorrection. Energy bounds for extension problems likewise have separately identified sufficient conditions CarpiTomassini.