Mathematical Description of the Diagram–Hilbert–Space Framework Theory

We present a comprehensive mathematical formulation of the Diagram–Hilbert–Space (DHS) Framework Theory, a unifying approach in which spacetime geometry, gauge interactions, and fermionic matter arise from algebraic relations among projection operators acting on a composite Hilbert diagram. The formalism replaces the conventional separation between geometry and quantum fields by a single operator algebra defined through diagrammatic relations and information–theoretic consistency conditions. We construct the fermionic substructure of the theory, define a projective Dirac operator that generates the observed chiral hierarchy, and show that renormalization–group (RG) flows of coupling operators converge to a universal fixed point determined by topological invariants of the diagram algebra. The resulting low–energy limit reproduces the Standard Model gauge group and predicts small deviations in neutrino mixing and gravitational coupling at accessible scales. The framework is mathematically renormalizable, phenomenologically testable, and provides a direct algebraic bridge between quantum field theory and emergent geometry.