A Geometric Vector Framework for High-Dimensional Interaction Modeling Applications to Systemic Risk Using Dot and Cross Product Invariants

The quantification of structural resilience and sub-percentile tail risk represents a major challenge across both corporate financial engineering and modern industrial logistics. Traditional aggregation architectures—principally linear risk matrices and parametric extreme-value copulas—harbor critical, systemic blind spots. This paper highlights a numerical limitation of the Gumbel extreme-value copula in deep-tail regions (F≥0.999). Analytical results indicate that the logarithmic structure of the tail generator produces progressively higher sensitivity near the distribution boundary, yielding an empirical condition number greater than 1,220 at the regulatory 99.9% Value-at-Risk (VaR) threshold. This numerical conditioning issue increases sensitivity to sample noise and data scarcity, resulting in a 36.5% underestimation of systemic tail damage. The proposed model formalizes risk scenarios by mapping multi-node threats as normalized directional unit vectors within a compact 3D vector space. Interactions are then calculated algebraically using geometric invariants—the Dot Product for root-cause convergence and the Cross Product Norm for dynamic, second-order risk resonance—effectively contracting high-dimensional combinations into a stable framework. Rather than treating risks as frame-dependent scalar probabilities, this generalized High-Dimensional Geometric Invariant Operational Risk Framework replaces legacy structures with domain-agnostic invariants capturing dynamic risk resonance and multi-trigger cascades. Empirical results across rugged operational environments, acute data scarcity (Ntrain = 100), and high-dimensional scaling (50 risk factors) demonstrate that the proposed model outperforms standard alternatives by a factor of approximately 13 in out-of-sample predictive accuracy (MSE = 0.08193) while maintaining absolute parametric stability. Furthermore, a Taylor-series tensor contraction successfully collapses 1,275 second-order interactions into just 2 free parameters. This framework bypasses iterative Maximum Likelihood Estimation (MLE) bottlenecks, unlocking real-time, low-latency Monte Carlo stress testing for systemic banking compliance and global supply chain risk governance.