Intrinsic Negative Sobolev Norms at a Fixed First Eigenvalue

We construct smooth positive unit-area conformal metrics on the two-sphere with first nonzero Laplace eigenvalue exactly 6*pi, while the intrinsic H^-1 norm of the conformal factor diverges uniformly over all round conformal parametrizations. The norm uses the scalar-function pairing with the varying metric's area measure. An explicit family of antipodal polynomial caps is small in L^(3/2) but large in L2. A variance-continuity lemma allows exact eigenvalue tuning, and the Mobius Jacobian formula prevents any conformal reparametrization from removing the growth. The five-page manuscript gives a complete proof and exact rational diagnostics. It distinguishes this counterexample from known positive bounds for the area measure in a fixed round-background Sobolev dual. The relation to Nadirashvili's printed 2009 conjecture, represented by UnsolvedMath OWR-3389-011, is explicitly qualified by the choice of pairing; no unqualified original-source closure is claimed. This is an unrefereed preprint, without a claim of absolute priority or formal verification.