Idempotent-Numeric: An Invariant-Manifold and Idempotent Projection Computing Architecture for Robust Scientific Analysis

Standard textbook numerical algorithms—spanning nonlinear root-finding, ill-conditioned linear solvers, polynomial interpolation, numerical calculus, and ordinary differential equation (ODE) integrators—suffer from pervasive numerical pathologies: division-by-zero explosions near critical points, conditioning decay in dense linear systems, Runge oscillations and unphysical overshoots in splines, truncation-roundoff tradeoffs in numerical derivatives, and secular energy drift in Hamiltonian dynamics. In this work, we present Idempotent-Numeric (IdemNumeric), a high-performance scientific computing architecture that unifies numerical analysis under the geometric framework of closed invariant-manifold projections satisfying the algebraic idempotency condition Π2 ≡ Π. Rather than proposing disconnected heuristic patches, IdemNumeric establishes an integrative framework across five core pillars: Metric-clamped Newton root enclosure preventing singularity divergence; Scale-invariant Ruiz-Birkhoff equilibration (ΠBirkhoff) and pre-factored linear constraint manifold projectors (Πconstraint), achieving up to 1.8× direct speedup over unprojected pseudo-inverses with exact machine-zero feasibility; Chebyshev-Gauss-Lobatto spectral fitting paired with Fritsch-Carlson shape-preserving monotone cone splines (Πmono) that strictly eliminate unphysical overshoots (0.00% overshoot); Pseudospectral differentiation matrices and positive Clenshaw-Curtis quadrature; and Energy-conserving invariant manifold integrators (|ΔE| ≤ 10-14), numerical Jacobian Hurwitz/Schur stability regularizers for stiff continuation, and C1-smooth permutation-ring CPG oscillators on invariant tori TK, coupled with Takens permutation entropy chaos diagnostics. Benchmarks against standard SciPy and NumPy implementations confirm up to 4× faster root convergence, exact energy preservation, and significant condition improvements across 64 automated verification tests.