Before revision:  This gives an easy proof of ER-membership, as real verification algorithms are much more versatile than ETR-formulas  .
After revision:  This gives an easy proof of ER-membership, as real verification algorithms are much more versatile than ETR-formulas . We use this result to construct a framework to study ER-complete problems under smoothed analysis. We show that for a wide class of ER-complete problems, its witness can be represented with logarithmic input-precision by using smoothed analysis on its real RAM verification algorithm. This shows in a formal way that the boundary between NP and ER (formed by inputs whose solution witness needs high input-precision) consists of contrived input. We apply our framework to well-studied ER-complete recognition problems which have the exponential bit phenomenon such as the recognition of realizable order types or the Steinitz problem in fixed dimension .