Programmable Attractor Itinerancy via Symbolic-Energy Feedback in Curvature--Basin Dynamics

We present a curvature‑basin dynamical framework in which a slowly evolving symbolic‑energy variable reshapes the depths of competing attractor basins, producing **deterministic multi‑state itinerancy** without noise, external forcing, or parameter modulation. The key result is a **necessary and sufficient phase‑flow condition**: itinerancy occurs *iff* the sign pattern of symbolic‑energy drift coefficients generates a non‑vanishing velocity on the circle \(S^1\), preventing stable fixed points in the averaged slow flow. A complete sweep of all eight sign patterns for a three‑basin system confirms this prediction exactly. In the adiabatic limit, the cycling period obeys the closed‑form law \[T = 2π/ε |v|,\] verified numerically with \(R^2 = 1.0000\) and 0.5% mean residual. This establishes symbolic‑energy feedback as a **programmable itinerary selector** with precise control over both cycle existence and period. As a consistency check, we show via center‑manifold reduction and Hopf normal‑form analysis that the curvature‑basin equations share the same **local Hopf universality class** as unified breather models of mode‑locked fiber lasers. This provides local physical grounding without implying global equivalence. Symbolic‑energy feedback thus offers a compact, predictive mechanism for autonomous attractor sequencing, with potential applications in neuromorphic dynamics, reservoir computing, nonlinear photonics, and adaptive control.