{'fraction_non_alphanumeric': 0.017582417582417582, 'fraction_numerical': 0.0, 'mean_word_length': 4.922077922077922, '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}}
Second-order phase theory in multiple-aperture systems

We establish the groundwork for a phase theory applicable to multiple-aperture systems. We examine the phase behavior of a reference system with rotational symmetry, and then examine the phase behavior of a system with only a single plane of symmetry assumed. A comparison of the two systems leads to the mathematical conditions for which one branch of a multiple-aperture system has ideal behavior.