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 Deff 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 (<0.5% error) and Lorenz chaos suppression (25% Lyapunov reduction), with Julia code and scaling. Spectral Units of Memory (Mar 2026) defines natural measurement units (capacity Mcap Mcap Mcap, span Mscale Mscale Mscale, resolution Mres Mᵣₑₛ Mres, entropy Hmem Hₘₑₘ Hmem) → effective dimension Deff Deff Deff, analogous to Shannon bits but for temporal memory. Supplementary Materials provide extended proofs, Julia listings, figures, sensitivity analysis, and baselines. Together, these form a self-consistent ecosystem bridging theory (spectral duality, stability, measurement), computation (SOE/GFE efficiency), and application (neuroscience, climate, neuromorphic engineering).