The Holding Equation: A Unified Framework for Viability Across Substrates

We present a unified mathematical framework for system viability across thermodynamic, biological, cognitive, and artificial intelligence domains. The core insight is that existence requires simultaneously maintaining two conditions: (1) a positive energy/resource buffer Ξ(t) > 0, and (2) structural integrity of a boundary operator M(t). We derive the fundamental dynamics from Onsager-class dissipation (Π ∝ σ‖∇Ψ‖²) and demonstrate that the coupled buffer-boundary system generically exhibits cusp catastrophe bifurcations, explaining abrupt collapse phenomena observed empirically. The framework bridges Prigogine's dissipative structures, Friston's Free Energy Principle, and catastrophe theory into a single formalism. We validate the framework against AI sycophancy data from transformer models, where predicted critical thresholds (H* ≈ 0.45, λ* ≈ 2–3) match empirical observations with high precision (ICC = 0.962, N > 33,000). The transformer-specific instantiation reveals that the Markov blanket corresponds to attention mechanisms, whose collapse under social pressure explains the phase-transition nature of sycophancy. We propose that this "Holding Equation" represents a substrate-independent law of viability—the mathematical form of persistence in being.