A Complex Ginzburg-Landau Theory of Algorithmic Attention: Microfoundations, Topological Defects, and a Falsification Program, with a Synthetic Demonstration

This working paper proposes that collective algorithmic attention is well described as a complex, driven, spatially extended field on a latent stance manifold, with online controversies behaving as topological defects: phase vortices whose coherent amplitude collapses at the core through destructive interference of competing framings. The argument is one of minimality rather than inevitability, and once a complex order parameter and an oscillatory coherence onset are granted, the complex Ginzburg-Landau equation follows as the generic amplitude equation. The signature topological prediction is demonstrated synthetically by direct numerical integration of the governing equation, which produces integer-charged defects with collapsed cores and the expected count-versus-size transition from pure noise. The paper then executes a pre-registered, two-sector falsification program on real data. The temporal relaxation sector is a validated but underpowered instrument. The topological sector, using a detector validated to recover planted defects and to reject the principal artifacts, returns a robust null across seven scoped Bluesky controversies and three framing subspaces, twenty-one multiplicity-corrected tests in total, with no controversy showing the predicted phase winding co-located with an engagement collapse, consistent with the spectral one-dimensionality of controversy retweet graphs. By the decision rule fixed in advance, the strong reading of the controversy-as-vortex claim retreats to the weaker, still-novel claim that stance space carries measurable circular structure. The deposit provides the manuscript, the full analysis and collection code, the processed datasets, the figures, and the results writeups, so that the synthetic demonstration and both empirical sectors can be reproduced and re-run under alternative methodological choices.