I'm sure someone has thought about this before or immediately dismissed it, but why does Schaefer's dichotomy theory along with Mahaney's theorem on sparse sets not imply P = NP ?
Here's my reasoning: Create a language $L$ which is equal to SAT intersected by an infinite decidable sparse set. Then $L$ must also be sparse. Since $L$ it is not trivial, affine, 2-sat, or Horn-sat, by Shaefer's theorem it must be NP-complete. But then we have a sparse NP-complete set so by Mahaney's theorem, P=NP.
Where am I going wrong here? I suspect that I am misunderstanding/misapplying Shaefer's theorem but I don't see why.