GFE–Aberconics Series: Spectral Calculus, Stability, and Measurement of Non-Markovian Adaptive Systems

This open collection compiles the core publications and supplementary materials (2025–2026) developing the Geometric Fourier Extension (GFE) and Aberconics frameworks for nonlinear, non-Markovian, and adaptive dynamical systems. GFE v1 (Oct 2025) introduces the foundational spectral calculus: field-dependent Laplace–Beltrami geometric transform + sum-of-exponentials (SOE) embedding for converting nonlocal memory into local auxiliary ODEs, with well-posedness, energy identities, and applications (Burgers, NLS, fractional GR n-body). GFE II (Mar 2026) extends to non-Weyl geometries (fractals, quantum graphs, sub-Riemannian), proves completeness for all completely monotone kernels, establishes the nonlinearity–non-Markovianity duality via Mori–Zwanzig, and uses effective memory dimension Deff D_{\rm eff} Deff as a blow-up/regularity diagnostic.\ Aberconics (Mar 2026, v2.0) unifies explicit-memory adaptive systems: convolutional kernels + internal model m(t) m(t) m(t) + bidirectional environment coupling, with GFE tractability, existence theorems, emergent non-Markovian behavior (echo tasks), and neuromorphic/hardware readiness. Stabilizing Memory Kernels (Mar 2026) identifies/corrects fundamental instability in naive SOE feedback, derives three stable formulations, validates on OU colored noise (