Emergent Spectra from Non-Commutativity in Totally Constrained Matrix Systems

This work analyzes the emergence of discrete spectral structure in totally constrained matrix systems governed by non-commutative dynamics. Focusing on background-independent formulations such as BFSS-type matrix models, we show that linearized excitations around constrained background configurations are governed by a static, gauge-invariant operator: the constraint Hessian (adjoint Laplacian). The eigenvalues of this operator define a discrete stiffness spectrum that exists independently of time evolution, clock choice, or observer partitioning. Using explicit block-diagonal and fuzzy-sphere backgrounds, we demonstrate how non-commutativity and gauge constraints jointly generate quantized spectral scales at finite matrix size. In the large-NNN limit, these spectra approach a continuum, while at finite NNN they remain discretized, providing a natural algebraic origin for spectral separation without invoking external time or particle postulates. The analysis is strictly structural and does not address vacuum selection, dynamical stability, or observational interpretation. Instead, it isolates the algebraic mechanism by which constrained non-commutative systems admit well-defined spectral scales. This result serves as a foundational component for subsequent interpretive and phenomenological work connecting constraint-defined spectra to observable resonant structures in background-independent quantum theories.