Bounded Persistent Linear Dynamics Induce Inner-Product Geometry

This work investigates whether inner-product geometry can arise as a structural consequence of minimal dynamical requirements rather than as an independent assumption. For finite-dimensional linear systems, we prove that uniformly bounded persistent evolution admits a positive-definite quadratic invariant. This invariant induces an inner product under which the generator is skew-adjoint, forcing orthogonal flow and naturally admitting a complex unitary representation. The results identify inner-product structure as a compatibility condition for stable, non-dissipative dynamics and suggest that mathematical frameworks commonly treated as foundational may instead reflect deeper constraints on admissible evolution.