Entropy-Originated Topological Framework for Unified Matter–Geometry Dynamics and Complex Mass–Charge Interactions

We present a unified theoretical framework in which space-time geometry, gauge interactions, particle masses, and dark sectors emerge from a diagrammatic Hilbert space governed by a holomorphic operator Z=M+iQ. The real operator M  encodes mass and gravitational content, while the imaginary operator Q  encodes electric, weak, color, and possible GUT charges. Diagrammatic basis states are weighted graphs, and their entanglement structure defines emergent spatial distances. The projection from the diagrammatic Hilbert space to effective continuum fields induces gravitational dynamics as a consequence of entropy maximization. Gauge interactions arise from the imaginary part of  via projection-preserving topological invariants. Fermion masses result from overlap integrals between diagrammatic flavor motifs and a scalar projection sector, providing a dynamical alternative to arbitrary Yukawa matrices. Hidden imaginary-sector excitations generate dark-matter-like and dark-energy-like behavior. We develop the mathematical structure, derive limiting cases including Friedmann–Robertson–Walker cosmology and U(1) gauge emergence, construct toy models, formulate renormalization-group flow from diagram coarse-graining, and propose phenomenological predictions. The framework is compared to string theory, loop quantum gravity, and holographic tensor-network approaches.