Solver-Robust Spectral Relational Geometry in Nonlinear Partial Differential Equations A computational preprint on coordinate-robust, discretization-robust, and cross-solver geometric structure

Abstract

We investigate whether nonlinear partial differential equations leave a reproducible geometric structure in thejoint space of spectral observables, rather than a single universal scalar invariant. Five nonlinear systems—Burgers, Allen–Cahn, cubic reaction–diffusion, Kuramoto–Sivashinsky, and a nonlinear damped wave model—were evaluated under multiple forcing geometries and amplitudes. Seven time-averaged spectral descriptorswere used: a normalized first-to-second spectral moment ratio, normalized entropy, participation ratio, spectralskewness, spectral kurtosis, and normalized third and fourth spectral moments. The analysis progressed fromlow-dimensional manifold tests to rank-based distance geometry, local Menger-curvature descriptors, grid andtime-step stress tests, and finally an independently implemented periodic finite-difference solver. The strongestresult is not a universal scalar constant. Instead, a relational geometry among spectral observables persistsunder tested coordinate-wise monotone reparameterizations and remains partially system-specific afterchanges in seed, grid resolution, time step, integration duration, and numerical solver. With templates andtransformations frozen from the original B1 spectral-solver data, new grid/time-step runs achieved 90% systemidentification overall, while an independently implemented finite-difference solver achieved 80% across 240new runs. In the finite-difference validation, median between-system geometric distances exceeded withinsystem distances by factors of 1.93 and 3.59 in two independent blocks. These findings support a SolverRobust Spectral Relational Geometry hypothesis while stopping short of claiming a universal physicalinvariant.