Geometric Distance to Structural Rupture : A Metric Theory of Systemic Stability

We introduce a metric extension of the TNA framework by defining a geometric distance dS(ω) between system states and the boundary of structural admissibility (S). By modeling the configuration space Ω = P ⊕ M as a normalized Hilbert space of dimensionless versors, we formalize systemic stability as a purely geometric property rather than a dynamical one. This paper transforms the binary structural operator S into a continuous metric theory, allowing for the quantification of proximity to rupture. We demonstrate that structural collapse corresponds to the limit dS(ω) → 0, where the internal dynamics (N₀) are no longer admissible. The use of dimensionless versors ensures that physical and metaphysical domains are treated as orthogonal axes in a unified geometry, precluding units-based reductionism. We further connect this metric approach to catastrophe theory and symmetry breaking, showing that the evolution of complex systems is a sequence of symmetry reductions within an admissible configuration space. This formulation preserves the core TNA principle: structural rupture is an invalidation of the domain itself, non-derivable from within its internal state space.