The Metric Decay of Financial Manifolds: Topological Phase Transitions in Algorithmic Markets

Abstract

Clayton Wangsanata

As autonomous AI trading systems proliferate, their shared training corpora and correlated signal generation induce a structural convergence we term Algorithmic Consensus Homogenisation (ACH) — a collapse of effective strategy diversity that standard risk models are blind to. We propose a geometric framework for detecting and anticipating this fragility by embedding the Level 2 limit order book (LOB) as a point cloud in R and subjecting it to Vietoris-Rips persistent homology. We demonstrate that ACH acts as a curvature-concentrating operator on the market’s Fisher information metric tensor gij, driving the system toward a Ricci singularity — the mathematical signature of total liquidity extinction. Across three calibrated synthetic regimes (Normal, ACH Convergence, Flash Cascade), our pipeline yields four robust findings: average topological persistence lifetime contracts by 56% (0.109 to 0.048), total persistence collapses by 66% (32.95 to 11.29), the Wasserstein distance W = 1.448 provides 220 ticks of warning ahead of price dislocation, and mean. Ollivier-Ricci curvature undergoes a 129× amplification from +0.016 to −2.045. Taken together, these results reframe market crashes not as stochastic surprises but as deterministic geometric events — ones that leave measurable topological traces well before prices begin to move. We introduce the Decision Black Box (DBB), a curvature-triggered regulatory mechanism, as a concrete operational response.

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