A fixed-area exceptional set for positive squared-pole sums

A full affirmative mathematical deduction for Baernstein's question, Hayman–Lingham Problem 7.37 (2307037 / AMR-022-7037). For every finite sum g(z)=sum lambda_j/(z-z_j)^2 with positive weights summing to one and arbitrary, possibly repeated, complex poles, there is one compact exceptional set S(g) of exact planar area pi, independent of R, such that the ordinary finite integral over D_R outside S(g) is at most 2 pi log R + pi(log 2 + 1/e) for every real R greater than one. The leading coefficient 2 pi is sharp; no optimality is claimed for the additive constant. The deduction combines the established Eremenko–Hamilton 1995 indicator-transform inequality and the Levine–Peres 2009 quadrature/smash-sum construction, with their earlier Astala, Sakai and partial-balayage antecedents credited. No new singular-integral theorem, quadrature construction or general balayage/operator mechanism is claimed. Related recent work by Yongxi Lin and André Guerra is credited. The ingredients were logically available by 2009; a new target-specific application is distinguished from an earlier recorded full solution. The extensive bounded priority audit identified no verified pre-PR full-target answer but does not certify historical novelty, worldwide first discovery/publication or global openness. Leung preliminary reports titled On a fixed area problem were located in AMS Abstracts 11 (1990), p.181, and 13 (1992), p.440; their mathematical text was inaccessible and relevance remains unresolved. Other unavailable full/final bodies and exact inspection limits are documented. Git version dates do not certify historical public visibility or causal independence. The source archive preserves twelve original PR artifacts, three distinct mathematical audits, three priority approaches, an independent check that the 2002 EH correction does not affect the imported indicator bound, and the focused Leung supplement. Three standard-library mathematical control programs run offline in ordinary and optimized modes with complete-output comparison and actual failure guards where supplied. A fourth independent target control is reproducible with an explicitly requested, hash-pinned primary-source download. Floating calculations are diagnostics, not validated numerical or formal proof certificates. The universal proof is contained in the note. AI tools were used extensively in proof development, computation, adversarial verification, literature research and writing. This note and supporting materials are unrefereed; independent AI reviews are not external human peer review, journal acceptance or formal proof-assistant certification.