An exact L2 criterion for periodic neutral equations with distributed delay

We establish a complete characterization of strong periodic solvability for linear neutral differential equations with distributed delay in Hilbert spaces. Our main result shows that strong well-posedness in the natural square-integrable setting is entirely determined by the invertibility and uniform control of the associated frequency-domain operators. In particular, this yields an exact criterion that removes the additional discrete difference condition required in the available neutral theory on UMD spaces. We also obtain the corresponding characterization for the fractional problem and show through explicit examples that the result is sharp in both the classical and fractional settings. The paper further clarifies the role of the discrete difference term in the Banach-space multiplier approach and identifies concrete kernel assumptions ensuring its boundedness.