Potential Stability and Signed Nests in Negative-Edge Trees

This working paper investigates when a prescribed tree sign pattern admits a real matrix whose eigenvalues all have strictly negative real parts. The underlying graph is a tree, and each edge corresponds to two nonzero matrix entries of opposite signs. Diagonal entries may be prescribed positive, negative, or zero. The paper establishes complete classifications when every diagonal entry is nonzero, both for ordinary potential Hurwitz stability and for a prescribed rooted cofactor. It also constructs an eight-vertex spectrally arbitrary pattern with no properly signed nest, disproving the unrestricted signed-nest conjecture of Das. Further results address zero-diagonal configurations, sharp instability indices, passive branch attachment, prescribed spectra, and exact geometric certificates. PRINCIPAL RESULTS Complete classification with nonzero diagonals A negative-edge tree sign pattern with no zero diagonal entries is potentially Hurwitz stable if and only if it has at least one negative diagonal and a balanced matching. Here, balance means that the matching leaves equally many positive-diagonal vertices unmatched in the two bipartition classes. These conditions are also equivalent to the existence of a properly signed nest and a winning static-port growth order. The classification holds at arbitrary order, without restrictions on branching or the number of negative diagonal entries. Mixed-edge matching numbers determine a sharp minimum instability index. Its bound is attained by rational realizations with distinct real eigenvalues. Complete rooted classification For a prescribed root, the paper characterizes the existence of a signed-nest realization with a positive signed root-deleted principal determinant. In the zero-free setting, the criterion requires a negative diagonal and balanced matchings of both the tree and its root-deleted forest. The deleted forest retains the original bipartition. Balance is a global condition across that forest: individual components may have compensating imbalances. Rational realizations with distinct negative real eigenvalues are available. The result concerns one prescribed root at a time; simultaneous cofactor requirements impose additional constraints. Spectral arbitrariness without a signed nest An explicit eight-vertex negative-edge tree pattern realizes every monic real polynomial of degree eight as a characteristic polynomial while admitting no properly signed nest. This disproves the conjecture that potential stability and signed nests are equivalent for all negative-edge tree patterns. The construction includes an exact stable realization, a nilpotent realization, a nonsingular coefficient Jacobian, and determinant identities supporting the pattern-wide obstruction to nests. Related constructions show that one zero diagonal already suffices for potential stability without a nest. A separate sharp boundary occurs at order seven, where a negative diagonal and a balanced matching can first fail to imply stability. Structural classes with zero diagonals and essential extensions Complete criteria are obtained for spiders, double stars, trees whose zero-diagonal vertices all have degree two, and specified families of pendant-zero decorations. The paper also constructs essential spectral extensions for which deleting a negative leaf or a passive branch destroys potential stability. These examples identify limitations of recursive approaches that require stable principal cores and show why unrestricted zero-diagonal configurations demand additional structure. Weak damping, passive synthesis, and robustness Matching deficiencies determine sharp weak-damping indices and associated completion costs. The analysis distinguishes stability along a small-damping regime from stability obtainable at finite damping. For passive attachments, the paper develops exact spectral tests and finite synthesis procedures under explicit assumptions on the core, storage, branches, and attachment ports. It distinguishes the existence of stabilizing joining gains from stability for every positive joining gain. Further criteria address heterogeneous branches, marginal cores, reciprocal multiport systems, and fixed branches attached to an arbitrary real core. The respective hypotheses, including the stable-reference-extension condition where required, remain part of each result. Inverse reconstruction and prescribed Hurwitz spectra Rooted polynomial recursions and Hankel signatures provide inverse-spectral reconstruction criteria within their stated regular domains. A stronger realization theorem applies whenever a tree has a prescribed negative root and a matching covering every other vertex. Such a pattern realizes every monic real Hurwitz polynomial of the appropriate degree while retaining all prescribed diagonal signs. Repeated characteristic roots are allowed, and nonroot diagonal magnitudes can be made arbitrarily small. This strengthens previously known potential-stability results for the same matching-covered family by prescribing the entire characteristic polynomial. Adapted geometry and rational certificates An adapted J-orthogonal decomposition expresses stability through compatible matrix inequalities. The construction combines classical Lyapunov theory with properties of positive matrix geometric means. The paper also establishes limitations on uniformly local adapted supports and gives a fixed, explicitly bounded word of tree-edge transformations for constructing exact certificates. Rational certificates are available for rational data. The word-length bound does not imply a bound on parameter bit length, and these geometric formulations retain continuous choices of realization magnitudes and adapted coordinates. RELATION TO EXISTING WORK The proofs use established tools from diagonal scaling, signature-inertia theory, the nilpotent–Jacobian method, Hermite–Biehler theory, skew-symmetric inverse spectral reconstruction, dissipativity, Lyapunov inequalities, and semidefinite separation. The contributions concern the exact matching classifications, rooted conditions, sharp indices and boundaries, explicit counterexamples and extensions, prescribed-spectrum strengthening, structured synthesis and robustness criteria, and tree-edge certificate constructions. The manuscript explicitly separates these claims from the classical machinery on which their proofs depend. CONTENTS AND REPRODUCIBILITY The deposit contains the 221-page working paper and a complete project archive with LaTeX sources, certificate data, verification programs, documentation, and recorded computational outputs. A one-page theorem map connects the principal results to their hypotheses, proof locations, and computational checks. The principal verification suite is run from the project root with: python3 verify.py The documented dependency baseline is Python 3.10 or later with assertions enabled and SymPy 1.14.0. Dependencies can be installed with: python3 -m pip install -r requirements-verification.txt The suite runs 21 principal checks and records pass/fail results, logs, input hashes, and regenerated certificate reports. Computations run in an isolated project copy to preserve the distributed inputs and archived outputs. The supplied reports record all 21 checks passing, together with a separate successful order-nine zero-free phase audit. Exact finite certificates and bounded implementation checks are distinguished from arbitrary-order proofs. A successful computational run verifies the stated certificates and tested cases; it does not replace the general mathematical arguments. SCOPE AND REMAINING OPEN PROBLEM The zero-free and rooted zero-free classifications are established in the manuscript, together with complete results for the specified zero-diagonal classes. A necessary-and-sufficient structural classification for negative-edge trees with arbitrary zero diagonals remains open. The principal remaining challenge is to replace continuous spectral and geometric feasibility choices by a general structural criterion that accommodates essential branches and nonlocal interactions.