{'fraction_non_alphanumeric': 0.029782359679266894, 'fraction_numerical': 0.006872852233676976, 'mean_word_length': 3.88268156424581, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The nonlow computably enumerable degrees are not invariant in ℰ

We study the structure of the computably enumerable (c.e.) sets, which form a lattice E under set inclusion. The upward closed jump classes Ln and Hn have all been shown to be definable by a lattice-theoretic formula, except for L1, the nonlow degrees. We say a class of c.e. degrees is invariant if it is the set of degrees of a class of c.e. sets that is invariant under automorphisms of E. All definable classes of degrees are invariant. We show that L1 is not invariant, thus proving a 1996 conjecture of Harrington and Soare that the nonlow degrees are not definable, and completing the problem of determining the definability of each jump class. We prove this by constructing a nonlow c.e. set D such that for all c.e. A ≤T D, there is a low set B such that A can be taken by an automorphism of E to B.