Dual Tests and Spatial Obstructions for Positive-Real-Part Majorants

For a nonnegative measurable function f on the unit circle, we give an attained variational test for almost-everywhere domination by the boundary modulus of a holomorphic function with positive real part. With h = log max(1,f), the logarithm of the least center modulus for a majorant of max(1,f) equals the supremum of integral(hg) - (pi/2) norm(Hg,1) over smooth nonnegative tests g of integral one, where H is the periodic Hilbert transform. Finiteness is the exact existence criterion, the coefficient pi/2 is sharp, and the tests can be restricted to a countable family of rational squared-polynomial moduli. We construct explicit separated spikes in weak L1 with no such majorant, although their symmetric decreasing rearrangement does have one. For N spikes, the weak L1 quasinorm remains at most two while the least clamped center modulus lies between N/8 and N+1. This addresses the almost-everywhere majorant class in Korenblum’s Problem 5.74 in Hayman–Lingham (UnsolvedMath AMR-022-5074) through a dual variational characterization. It is not a finite decision algorithm or a claim of the first historical resolution. The five-page English preprint contains complete proofs; AI-assisted and self-audited, without independent peer review, formal verification, or exhaustive priority certification.