Eigenvalue Conditioning as Universal Optimizer: Cross-Domain Transfer Between Finance, Robustness, and Machine Learning

We prove that eigenvalue conditioning — decompose a structure matrix into eigenmodes, condition on the \(K\) dominant modes, solve \(K\) independent one-dimensional problems, and combine — is a universal optimization principle that transfers across five domains: portfolio Value-at-Risk, basket option pricing, adversarial robustness certification, SGD convergence, and transformer attention dynamics. The cross-domain transfer yields a provable improvement factor of \(I = \lambda_{\max} / L_{\text{eff}}\), where \(L_{\text{eff}} = \sqrt{\sum \lambda_k^2 / n}\) is the root-mean-square eigenvalue and \(\lambda_{\max}\) is the largest eigenvalue. This improvement factor satisfies \(I \geq 1\) with equality only for flat spectra, and approaches \(\sqrt{n}\) for rank-1 spectra. When the spectrum is sufficiently concentrated, \(I\) is well-approximated by \(\sqrt{n / K_{\text{eff}}}\) where \(K_{\text{eff}} = (\sum \lambda_k)^2 / \sum \lambda_k^2\) is the effective rank, but the two expressions are not identical in general. We establish a unified spectral gap theorem from which all five convergence results follow as one-line corollaries, and prove that the improvement factor depends only on the spectrum, not the domain — so a technique originating in adversarial robustness (Frobenius certification) provably tightens basket option pricing bounds, and vice versa. We further extend the Bellman equation with spectral decomposition (per-mode convergence rates), constraints (shadow price of risk limits), and robustness (model-free option pricing bounds). All results are machine-verified in Lean 4 with zero sorry. To our knowledge, this is the first formal proof that cross-domain eigenvalue transfer is valid.