Inverse Evolution and Dimensional Collapse: Operator-Theoretic Dynamics in Financial Manifolds

We develop an operator-theoretic framework for extreme events in reflexive financial systems, identifying inverse evolution—the deterministic contraction of the manifold of admissible futures—as the structural mechanism underlying crashes and melt-ups. The interpolation constraint, which forces all analytical continuations to match the terminal empirical price, resolves the apparent continuum of stochastic paths into a discrete, countable spectrum of metastable futures. This countable manifold is stabilized by a spectral regularizer that preserves dimensionality through a “wait-and-adjust” re-categorization logic. Within this unified structure, we distinguish three pathways to collapse: (i) the Black Swan, a crisis of spectral weight; (ii) the projection operator, a rank-reducing projection that restores symmetry by exclusion; and (iii) the reactivation operator, a breakdown of spectral truncation that reactivates suppressed behaviour with large emergent return (Heavy) modes and forces the system into a regime of manifold resumption. Central to all modalities is the emergent return, an effective mass parameter whose sign determines whether collapse manifests as reflexive contraction (crash) or reflexive amplification (melt-up). The resulting dynamics exhibit cross-domain universality. The same operator grammar governs geopolitical choke-points, institutional purges, technological monopolies, retail-driven short squeezes, and other macrosystems in which dimensionality is either forcibly reduced or abruptly restored.