The Double Pendulum as a Homeostatic System: Geometric Stiffness, Information Topology, and the Onset of Hyperchaos

The double pendulum stands as a canonical pillar in the study of classical mechanics, histor-ically serving as the definitive pedagogical gateway into deterministic chaos. Within the frame-work of Axiomatic Physical Homeostasis (APH), we reinterpret this system not as a mechanicaloscillator, but as a homeostatic control system attempting to minimize a geometric hazard func-tion under energy constraints. We demonstrate that the transition from quasi-periodic motionto chaos corresponds to a phase transition from the Strong Buffer Regime (κ > 1/8, linearizedstability) to the Weak Buffer Regime (κ < 1/8, topological mixing). We define the “GeometricStiffness” β(θ) of the pendulum and show that the onset of chaos coincides with the collapse ofthis stiffness, akin to the vacuum flutter epoch in APH cosmology. Finally, we establish thatthe chaotic regime represents a “Computational Singularity” where the information density ofthe trajectory exceeds the holographic bound of the causal graph.