Multiplicity Transfer and Regularity Defects for Semilocal Weil Operators

We study the complete semilocal Weil operator on a symmetric logarithmic interval, retaining the archimedean contribution, the effective prime powers and the pole terms. A bounded primitive operator has a commutator whose bounded extension has rank two, with an explicit odd defect vector. This identity implies that the multiplicities of any eigenvalue in the even and odd sectors differ by at most one. We also describe exactly the eigenfunctions whose zero extensions belong to $H^1(\mathbb R)$ and whose derivatives belong to the form domain: their codimensions in the two sectors sum to one at every eigenvalue. Consequently, simplicity of the bottom eigenvalue at a fixed parameter is equivalent to inequality of the two sector bottoms, and a simple even ground state is equivalent to strict even preference. For the genuine prime family, continuity and a purely analytic small-parameter anchor make global simplicity equivalent to global even simplicity. If the sector bottoms meet, their first contact is detected by precisely one of two explicit spectral projections. The structural results hold for arbitrary finite real atomic perturbations of the archimedean operator. No noncrossing assertion for all parameters is assumed or proved.