Projection, Entropy, and Emergent Mass in Diagrammatic Hilbert Spaces

We develop an operator-theoretic formulation of a diagrammatic Hilbert space (DHS) in which physical mass scales emerge from projection onto constrained subspaces and entropy defined on the projected space. We connect this framework to a microcanonical diagram-partition model in which particle masses are computed from a discrete partition function characterized by small-integer exponents. Using representative exponent series and predicted masses, we construct data-driven tables and compare these to Particle Data Group (PDG) and NIST/CODATA values. The formalism provides a phenomenological bridge between diagrammatic methods, statistical physics, and nonperturbative quantum field theory.