Diagram–Hilbert Space (DHS) with Entropy-Originated Topological Projection (EOTF) Unified kinematics, projected partition functions, and emergent geometry

We formulate a unified framework in which microscopic degrees of freedom are encoded as equivalence classes of labeled diagrams spanning a Hilbert space (Diagram–Hilbert Space, DHS), while physicality and effective dynamics arise from a scale-dependent projection implementing gauge constraints, topological consistency, and entropy stationarity under coarse-graining (Entropy-Originated Topological Framework, EOTF). The projection induces a projected partition function and projected correlators. An emergent geometric sector is defined on a coarse macrostate manifold with a metric given by an entropy Hessian, linking the construction to information geometry and maximum entropy. A particle spectrum can be organized by a discrete “diagram complexity” operator; masses correspond to eigenvalues times a universal scale.