Boundary Conditions as Structural Selectors: A Formal Analysis of N₀, Operational Constraints, and N₁ in Self-Optimizing Systems

This paper formalizes a critical distinction within the Theory of Axiomatic Necessity (TNA): the difference between a system's internal dynamics ($N_0$), the operational constraints that condition its solution space (boundary conditions), and the selecting instance that imposes those constraints ($N_1$). Through analogies drawn from physics (Laplace equation, Euler-Bernoulli equation, Manning equation), hydrodynamics (channel flow), wave mechanics, and artificial intelligence (LLM system prompts), we demonstrate that systems with identical internal dynamics can produce radically different trajectories when their boundary conditions vary. We identify and respond to materialist objections, showing that the complete reduction of boundary condition selection to $N_0$ leads to either infinite regress or ontological collapse. The central thesis is that the identity of complex systems—biological, cognitive, or social—does not reside in their constituent matter (which is replaceable) but in the realizability structure that persists through material turnover.