The Transverse Projection Operator: A First-Principles Derivation, Variational Stability, and Spectral Rigidity in the Operator-Derived Dimension Framework

This draft has been superseded by the rigorous derivation found here: [https://doi.org/10.5281/zenodo.18785883]. Please refer to the updated version for the correct spectral analysis. We construct the Transverse Projection Operator ˆQ, the fundamental geometric generator of spectral confinement in the Operator-Derived Dimension (ODD) framework. The operator is uniquely determined by causality, localization stability, and chirality. Variational stability of the vacuum geometry fixes the topological index to μ = 2, producing a modified P¨oschl–Teller spectral confinement structure. The spectrum consists of two strictly normalizable fermionic bound states and one threshold resonance mode. Observable fermion generations correspond to distinct spectral localization classes rather than degenerate bound multiplicity.