The Gaia mission is delivering exquisite astrometric data for 1.47 billion sources, which are revolutionizing many fields in astronomy. For a small fraction of these sources the astrometric solutions are poor, and the reported values and uncertainties may not apply. For many analyses it is important to recognize and excise these spurious results, commonly done by means of quality flags in the Gaia catalog. Here we devise and apply a path to separating good from bad astrometric solutions that is an order-of-magnitude cleaner than any single flag: we achieve a purity of 99.7% and a completeness of 97.6% as validated on our test data. We devise an extensive sample of manifestly bad astrometric solutions: sources whose inferred parallax is negative at >= 4.5 sigma; and a corresponding sample of presumably good solutions: the sources in HEALPix patches of the sky that do not contain extremely negative parallaxes. We then train a neural net that uses 14 pertinent Gaia catalog entries to discriminate these two samples, captured in a single astrometric fidelity parameter. An extensive and diverse set of verification tests show that our approach to assessing astrometric fidelity works very cleanly also in the regime where no negative parallaxes are involved; its main limitations are in the very low S/N regime. Our astrometric fidelities for all EDR3 can be queried via the Virtual Observatory. In the spirit of open science, we make our code and training/validation data public, so that our results can be easily reproduced.
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