Diagonal Transport and Spectral Fibers in Nonnegative Inverse Eigenvalue Problems

A pair of standalone working papers This deposit contains two standalone papers by Kevin Fathi: Diagonal Transport and Spectral Fibers in Nonnegative Inverse Eigenvalue Problems, and Moment Faces and Realization Dimension in Nonnegative Inverse Eigenvalue Problems. Each supplies its required arguments and constructions, with classical dependencies cited. Diagonal and Perron geometry The main paper proves a sharp finite diagonal-transport estimate and identifies sparse-support obstructions. For spectra with two exceptional eigenvalues above a repeated minimum, finite partition criteria determine complete prescribed-diagonal fibers and labeled principal-deletion decks. Exact counterexamples show that separately positive realizations may be incompatible and delimit nonnegative Gram reconstruction. Positive bordering and complementary commuting witnesses give constructive branches for trees, specified resonant systems, odd cycles, weighted eight-cycles, and alternating even cycles. Generic positive weights on connected bipartite witnesses with at least one edge force repeated Perron roots in complementary realizers. Finite stationary-value reductions address Perron thresholds; irreducible nonglobal local minima demonstrate the need for global comparison. Moment faces and realization dimension The companion establishes a sharp third-moment defect and its complete real-spectrum equality boundary, exact ordinary and irreducible cyclic zero-padding laws, and prescribed-order compression on maximal odd-moment faces through order nine. Hamilton necklaces and overlapping principal deletions expose remaining compatibility constraints. Conic compression bounds order at fixed zero-Jordan excess while preserving nontrivial zero chains. Three nonzero eigenvalues can force unbounded realization order and arbitrarily long zero chains. One specified spectrum has minimum order eight at its first feasible excess and seven at the next; repeated copies yield exact hierarchies with arbitrarily long decreasing branches. The seven-state exclusion is computer-assisted, and these hierarchies inherit that dependency. For fixed irreducible Frobenius blocks with positive spectral radii, nonnegative couplings attain every excess in an explicit interval. Scope and supporting material General NIEP and SNIEP classification remain open. Exceptional witness branches, higher-order odd-face compression, and general first-excess formulas remain incomplete. Schur–Horn, Boyle–Handelman, and Laffey results and earlier low-order classifications receive attribution; unrestricted claims in motivating working papers are not premises. The deposit includes manuscripts, editable sources, exact certificates, verification programs, and research records. Numerical search failures are not proofs of nonrealizability.