Eigenvalue Conditioning: A Universal Computational Primitive for Correlated Systems

We present eigenvalue conditioning as a universal computational primitive: given an \(n\)-dimensional problem governed by a positive semidefinite structure matrix \(\Sigma\), eigendecompose \(\Sigma\), project onto its \(K\) dominant eigenmodes, solve \(K\) independent one-dimensional problems, and combine. The method applies wherever \(\Sigma\) exists — covariance matrices (finance), Jacobian Gram matrices (machine learning), Hessians (optimization), generator matrices (dynamical systems), interaction kernels (physics), and connectivity matrices (biology, neuroscience, climate). We prove three results. First, the improvement factor \(I = \lambda_{\max}/L_{\text{eff}} \geq 1\) depends only on the eigenvalue spectrum and transfers unchanged between domains: an improvement discovered in adversarial robustness gives the same factor in option pricing, and vice versa (14 Lean files, zero sorry). Second, the number of modes \(K\) that suffice is bounded by \(N^* = \Theta(\log(1/\varepsilon)/\log\rho)\), where \(\rho > 1\) is the system's analyticity parameter and \(\varepsilon\) is the target accuracy — this \(K\) is independent of the ambient dimension \(n\) (the Latent Theorem). Third, the convergence rate of any iterative algorithm that admits eigenvalue conditioning depends on the effective rank \(K_{\text{eff}} = \text{tr}(\Sigma)^2/\|\Sigma\|_F^2\), not \(n\), yielding dimension-free convergence guarantees. We demonstrate the method across ten domains: portfolio risk, derivatives pricing, credit risk, adversarial robustness, neural network training, transformer dynamics, fluid dynamics, plasma confinement, molecular dynamics, and climate uncertainty propagation. In each case, eigenvalue conditioning reduces computational cost by \(O(n/K_{\text{eff}})\) while providing exact or provably bounded results, replacing stochastic approximations with deterministic computation.