We present a unifying operator-based framework for matter and gravity constructed from a Hilbert space of diagrams. In this setting, Feynman graphs, Wilson loops, and knot invariants span the basis, while projective operators act to generate effective Hamiltonians for observable dynamics. This construction is mathematically original, embedding the core structures of quantum field theory into a topological–entropic representation. Several demonstrated achievements establish its viability: the reproduction of the QCD one-loop β-function, including both gluonic and fermionic contributions, corrected to the exact beta coefficient, a projection-based Yukawa sector yielding mass hierarchies; a derivation of the exact Bekenstein–Hawking entropy prefactor from diagram counting; a derivation of the Einstein field equations via entropic flux operators; and the identification of Newton’s constant, the electromagnetic fine-structure constant, and the QCD coupling as emergent topological invariants. Moreover, the framework naturally reproduces the Wilson loop area law and string tension scaling. Together, these results provide a coherent, predictive, and testable operator formalism that unifies the thermodynamic and quantum views of gravity while remaining anchored in perturbative QCD and semiclassical gravitational benchmarks.