This paper proposes discrete Ricci curvature on financial correlation networks as a geometric indicator of systemic fragility. The framework conceptualizes the economy as a differentiable manifold equipped with a Riemannian metric, where curvature encodes whether small perturbations self-correct (positive curvature) or self-amplify (negative curvature). Building on Mach's relational ontology and Einstein's geometrization program, a testable hypothesis is derived: aggregate network curvature is associated with financial stress and serves as an indicator of systemic fragility. Dimensional homogeneity is addressed through non-dimensionalization, candidate conservation principles are proposed, and a complete worked example is provided. The relationship between continuum curvature of the state manifold and discrete curvature of correlation networks is clarified against prior empirical findings. An implementation architecture enables empirical validation. The framework is offered as a research program rather than a finished theory.

The framework in this paper is implemented as a live, interactive dashboard in this application — the link below opens it.What it computes. Each session the dashboard rebuilds the financial correlation network described in the paper and measures its discrete Ricci curvature:1. Data ingestion — roughly two years of adjusted daily closes for a cross-sector universe of large-cap US equities (the default sample spans technology, financials, healthcare, consumer and energy), converted to log-returns. You can also point it at one of your own portfolios or watchlists. 2. Correlation → metric — pairwise Pearson correlations over a rolling 60-day window, mapped to distances via $d = \sqrt{2(1-\rho)}$, the correlation-based metric of Part IV. 3. Network construction — edges retained where $|\rho| > 0.3$, giving a weighted graph at each point in time. 4. Curvature — Ollivier-Ricci curvature per edge via Wasserstein optimal transport on uniform neighborhood measures, aggregated to a single mean scalar $\bar{\kappa}(t)$. 5. Signal — the curvature series, its 7-day trend, and a deviation-from-baseline alert level. A Forman-Ricci track is computed in parallel as a cheaper cross-check, and the dashboard reports how often the two discretizations agree (the comparison of Part VII).How to read it. More negative aggregate curvature flags a more fragile network — connections concentrated through bottlenecks that amplify perturbations — while less-negative or positive curvature indicates resilient, well-connected structure. Consistent with the paper's epistemic-humility section, this is presented as a contemporaneous fragility indicator, not a timing or direction forecast.This is an open research program: the live tool is the empirical-validation harness referenced in the paper's implementation architecture, not a finished or backtested trading signal.