Seconds Instead of Hours: A Multi-Pass Cognitive Preconditioning Framework for Large-Scale Eigenvalue Problems

Before a physicist runs a million-dollar simulation, she sketches the answer on a napkin. She uses dimensional analysis, symmetry arguments, and limiting cases to know roughly where the answer should land. If the simulation returns something wildly different, she knows to check her setup. That napkin sketch takes seconds and saves weeks. This paper builds the computational equivalent of that napkin sketch. We introduce a rapid multi-pass approximation framework for large-scale eigenvalue problems — the kind of problems that typically require hours or days of high-performance computing (HPC) on dedicated clusters. The method produces spectral estimates with relative errors on the order of 10⁻³ within seconds, not by replacing numerical solvers, but by providing a fast, structured "cognitive preconditioning" step that constrains the plausible outcome region before expensive computation begins. The approach works through successive refinement passes that exploit geometric constraints, convergence patterns, and extrapolation — analogous to how an experienced scientist narrows down the answer through structured reasoning before committing to detailed calculation. Benchmarks on challenging 2D and 3D domains (including the Fichera corner, a standard stress-test for eigenvalue solvers) demonstrate that reliable estimates are achievable at negligible cost. The key insight is a reframing of the role of computation in scientific workflows. Traditional pipelines assume that numerical simulation comes first and interpretation follows. We propose the opposite: a low-cost reasoning phase that forms expectations first, with full-scale HPC serving as the final confirmation step. This mirrors expert human practice — but externalized, systematized, and reproducible. The method does not replace high-fidelity solvers. It makes them faster to use, cheaper to run, and harder to misconfigure — by ensuring that the scientist already knows roughly what to expect before the first simulation starts.