Coxeter Gaps and Exact Spectral Residuals for Four Equal-Rank Projections

This paper determines the exact minimum operator-norm deviation from a scalar operator for sums of four equal-rank orthogonal projections on finite-dimensional real or complex Hilbert spaces. It treats every parameter range lying between consecutive scalar-sum values, referred to as Coxeter gaps. The analysis identifies a unique decomposition of each interior rank-dimension pair into the two neighboring endpoint pairs. Direct sums of tight endpoint systems then produce sharp two-level spectra, while uniform Horn–Littlewood–Richardson certificates establish the corresponding lower and upper spectral barriers. Together, these arguments give the exact residual throughout every Coxeter gap. The sharp spectrum is also shown to be the unique majorization-minimal feasible spectrum. This yields exact extremal values for convex spectral trace functionals, centered Schatten norms, the fusion-frame potential, optimal frame bounds, spectral spread, and condition number. The paper further proves eigenvalue-multiplicity rigidity for operator-norm minimizers and full spectral rigidity at the midpoint of each gap. An explicit rank-five example in dimension thirteen is included. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.