= Negative-square exclusion under full sphere regularity
If every nonconstant <J-holomorphic sphere> under consideration is regular, no such sphere in a four-dimensional target has class $A$ with $A^2\leq-2$. For a <simple J-holomorphic sphere>, the <adjunction inequality for a simple J-holomorphic sphere> makes the predicted moduli dimension negative. A degree-$m$ multiple cover of class $B$ has $B^2\leq-1$ and $c_1(B)\leq1$, so its predicted real dimension is at most $2m-2$. The <rational covering maps of the projective line> already form a family of dimension $4m-4>2m-2$, contradicting regularity. The full regularity hypothesis matters: multiple covers of exceptional minus-one spheres violate the conclusion under a simple-only regularity assumption.
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