Entropy-Weighted Projection Algebras and Diagrammatic Hilbert Spaces

We present a mathematically explicit operator-theoretic framework in which entropic weighting on a diagrammatic Hilbert space induces a rich projection algebra with geometric and gauge-theoretic structure emerging from commutator relations. A separable Hilbert space is constructed whose orthonormal basis consists of equivalence classes of topological diagrams; on this space we define a family of entropy-weighted projection operators whose algebraic properties support scale-dependent coarse-graining. We derive general theorems for positivity, spectral structure, invariance under topological moves, and tensorial behavior of curvature-like operators defined from commutators with coarse-derivative maps. A renormalization-group (RG) flow on the projection algebra is introduced and shown to reduce to familiar beta-function behavior under appropriate truncations. Several examples illustrate how curvature operators, gauge-like phases, and constrained topological sectors arise from the algebraic structure without additional geometric assumptions. We focus on mathematical consistency, operator properties, spectral behavior, and structural results, providing a foundation for future physical interpretation. The manuscript is positioned as a contribution to operator algebra approaches to emergent geometric structures and information-theoretic formulations of effective field behavior.