The Source Hankel Correspondence Unifies Evans and Curvature, but Selfadjointness Does Not Confine Zeros
Define the source Hankel history operator
On the character e_z(a)=e^{za}, it produces the complete tail path G_z.
Endpoint evaluation gives the Evans transform, while source pairing gives the
ordered autocorrelation. Thus Evans and curvature are two covector faces of
one source-derived history correspondence.
For real theta forcing, the kernel is symmetric. Superexponential decay makes the operator Hilbert--Schmidt and selfadjoint. Nevertheless selfadjointness has no zero-confinement force for the endpoint matrix coefficient. The exact two-cell Hankel matrix
is real symmetric and comes from nonnegative source data, while its endpoint
coefficient on (1,r) vanishes at r=-1/2, corresponding to
z=-log 2+(2k+1)pi i off the seam.
The canonical selfadjoint carrier is therefore not a Hilbert--Polya operator whose spectrum is the Riemann zero set. Zeros remain cancellations of one rigged matrix coefficient. A further theta/modular law must constrain the distinguished endpoint covector and character orbit.
Research packet:
research/grothendieck/the-source-hankel-correspondence-unifies-evans-and-curvature-but-selfadjointness-does-not-confine-zeros.md
Exact checker:
research/grothendieck/checkers/check_selfadjoint_hankel_endpoint_hostile.py
The checker passes 6/6 exact tests.