Before revision:  Interestingly our techniques generalize to problems with a natural notion of resource augmentation (geometric packing, the art gallery problem ) and recognition problems (recognition of realizable order types or disk intersection graphs). Our results also have theoretical implications for some ER-hard problems: These problems have input instances where their real verification algorithm requires at least exponential bit-precision which makes it difficult to place these ER-hard problems in NP. Our results imply for a host of ER-complete problems that this exponential bit-precision phenomenon comes from nearly degenerate instances. It is not evident that problems that have a real verification algorithm belong to ER. Therefore, we conclude with a real RAM analogue to the Cook-Levin Theorem .
After revision:  Interestingly our techniques generalize to problems with a natural notion of resource augmentation (geometric packing, the Cook-Levin theorem which shows that ER membership is equivalent to having a verification algorithm that runs in polynomial-time on a real RAM analogue to the Cook-Levin Theorem .