Nonunique nearest stable complex quadratics

For every fixed positive-definite real coefficient inner product, unstable complex quadratics with at least two distinct attained global nearest closed-left-half-plane stable quadratics occur arbitrarily close to z^2, both in monic coefficient space and in the full exact-degree coefficient space. A limiting normal has cubic support against a quartic squared displacement, disproving stable-set prox-regularity; compact localization ensures full-degree global attainment. The same obstruction disproves set prox-regularity of the whole abscissa epigraph. The monic quadratic abscissa nevertheless is function prox-regular at the origin relative to each finite limiting subgradient, with constants depending on that subgradient in the real Euclidean coefficient metric. This gives a negative answer in degree two to Lewis's historical question about the stable complex monic set. The later Oberwolfach Question1 omits the coefficient field, normalization and metric; its complex convention is explicitly qualified. The note credits Bottema and Kirillov-Overton classical stability-boundary geometry, Burke-Overton/Lewis first-order root analysis, the Poliquin-Rockafellar-Thibault projection characterization, and earlier nearest-polynomial approximation work of Moses-Liu and Orban de Xivry. No real-only or all-degree classification, explicit unstable input and minimizing pair, general nonlinear parameter-metric transfer, firstness or present-day openness guarantee is claimed. The limiting support covector is horizontal; the representing normal vector need not be horizontal in mixed coefficient-height metrics. The verification archive supplies the complete standalone LaTeX source, a Python-standard-library verifier with 2,005 exact rational assertions and complete expected output, deterministic high-precision function-boundary controls, authored source and bounded-priority notes, publication metadata, license and file hashes. Finite checks supplement the written analytic limiting-normal, every-fixed-metric and global projection proofs. The priority comparison records actual primary-body read extents and remaining access and historical-priority gaps; it is not an exhaustive absence certificate. AI tools were used extensively in deriving, repairing, literature auditing, reproducing, writing and independent adversarial checking. This is an unrefereed preprint without independent human peer review or formal proof-assistant certification. The named author is responsible for the claims and citations. No outside individual was contacted.