Deriving Hilbert Space Structure from Entropy Geometry: A Two-Axiom Approach
David Sigtermans · Preprints.org · 2025
We show that the mathematical structure of Hilbert space—typically postulated in quantum theory—emerges as a consequence of two fundamental principles derived within the TEQ framework: (1) entropy as a generative structural principle; (2) a minimal principle selecting stable distinctions. These principles, and the resulting entropy-weighted action functional, are derived from first principles in prior work. Here, we analyze the entropy-weighted path integral and the geometry of entropy curvature, and prove that the space of entropy-stabilized modes satisfies the axioms of a Hilbert space: linearity, inner product structure, norm completeness, and probabilistic interpretation. In this view, the Born rule itself emerges from entropy-weighted path selection. Importantly, this result shows that the TEQ framework fully recovers the standard Hilbert space formalism—not by postulate, but as a structural consequence of its core principles—thus preserving compatibility with quantum theory while providing it with a deeper generative foundation. Our results strengthen the explanatory power of the TEQ framework and offer a structural derivation of core quantum features from a common thermodynamic-geometric basis.