The Adaptive Kuramoto Model: A Computational Framework for Wound, Healing, and Integration in Coupled Dynamical Systems

This deposit presents the complete research record of the Adaptive Kuramoto Model (AKM), a computational framework extending standard Kuramoto synchronisation theory to model structural change, healing, and integration in coupled dynamical systems. The AKM introduces a five-layer hierarchy of adaptive variables — coupling (K), plasticity (P), frequency (ω), openness (η_ω), and annealing rate (α_g) — each operating on a distinct timescale and with distinct rheological properties. A companion framework, Adaptive Mismatch Feedback (AMF), provides bifurcation analysis and strange-attractor characterisation of the health regime. The deposit covers 26 numerical experiments across classical oscillator physics, quantum information theory (QuTIP), spectral analysis, fractional calculus, causal inference (do-calculus), fractal geometry, and stationarity analysis. Key results include: (1) formal proof of dipolarity — two acoustically opposite states (white noise α≈0.18, brown noise α≈1.89) with identical boundary metrics; (2) a five-state acoustic fingerprint distinguishing wound states that standard synchrony metrics cannot separate; (3) a Layer 5 P_max discovery — hostile episodes permanently scar the rate of future annealing rather than the accumulated record; (4) a causal priority result — CV_wound causally precedes future bandwidth W (do-calculus r=0.837 vs observed r=0.333); (5) a selectivity function separating earned receptivity from pathological permeability; (6) the Adaptive Mandelbrot — a new mathematical object in which coupling history deforms the fractal geometry of the integrable set; and (7) a complete anisotropic rheological classification of the five-layer hierarchy. Applications are proposed and formalised across power grid stability, financial systemic risk, cybersecurity intrusion detection, AI architecture, organisational health diagnostics, and clinical trauma science. The work engages the Jung-Pauli programme — specifically Atmanspacher's generalised quantum theory and Roderick Main's synchronicity research — as a theoretical context for the formal results. All findings are numerical observations under specific model assumptions; none constitute empirical validation of real-world phenomena.