Accelerated Nonlinear Root-Finding via Arithmetic Modular Rigidities and Ramanujan Spectral Preconditioning

This work unites arithmetic algebraic geometry and numerical analysis to resolve initial-basin sensitivity and Jacobian ill-conditioning in nonlinear root-finding. By parameterizing nonlinear systems through the complex multiplication of class-number-one imaginary quadratic fields, truncated expansions of singular moduli generate an analytical starting guess that strictly satisfies Smale’s point-estimate criterion, guaranteeing quadratic convergence from the initial step without empirical guessing. For high-dimensional networked systems, Lubotzky–Phillips–Sarnak Ramanujan graph Laplacians provide explicit operator preconditioners whose spectral condition numbers remain strictly bounded independently of network dimension, ensuring rapid, dimension-free Krylov solver convergence. Furthermore, the framework formalizes the categorical distinction between arithmetic Arakelov heights and canonical phase-space invariants, proving long-term secular energy conservation on multi-scale symplectic manifolds through backward error analysis.