Positive Density for the First Laguerre Inequality of Hardy's Function

This preprint proves an unconditional positive-density result for the first Laguerre inequality associated with Hardy's function. It establishes that the inequality holds on more than 27.87622 percent of every sufficiently large dyadic interval in the lower-density sense. The proof combines mollified second and fourth moment estimates with an explicit degree-36 rational mollifier. The resulting density certificate is verified by exact rational arithmetic. The accompanying computation package independently reconstructs the mollifier, the relevant moment coefficients, the density ratio, several low-degree consistency checks, and the published special-case formula used in the fourth-moment calculation. The paper also develops the associated variational problem in its natural energy space. It proves attainment of the optimal value, equality of the polynomial, Sobolev, and completed energy infima, absence of a Lavrentiev gap, and strong precompactness of normalized minimizing sequences. A separate construction explains why direct compactness in the stronger Sobolev topology is unavailable. The result does not prove pointwise positivity on the whole real axis and does not resolve Csordas Open Problem 4.7. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.