| # Prior art and claim boundary | |
| ## What the release does not claim to originate | |
| Berkolaiko, Kennedy, Kurasov, and Mugnolo established the eigenvalue bounds used here in *Edge connectivity and the spectral gap of combinatorial and quantum graphs*, J. Phys. A 50 (2017), 365201, DOI 10.1088/1751-8121/aa8125; arXiv:1702.05264. | |
| Serio's *On extremal eigenvalues of the graph Laplacian*, J. Phys. A 54 (2021), 015202, DOI 10.1088/1751-8121/abc59c; arXiv:2008.01542, distinguishes “lower sharp,” “upper sharp,” and “sharp,” with the last meaning simultaneous equality at the ends of a degenerate block. The paper relates two-sided sharpness to maximal degeneracy. Its published/author-repository topology statement includes lasso trees, figure-eight graphs, and three-watermelon (theta) graphs. The initial preprint's abstract and final statement are narrower; relying only on that abstract would miss known exceptions. The proof of the new classification does not rely on that older topology argument. | |
| Kurasov and Serio's *On the sharpness of spectral estimates for graph Laplacians*, Rep. Math. Phys. 82 (2018), 63–80, DOI 10.1016/S0034-4877(18)30071-5, supplies prior sharp families. Infinite sequences of extremal modes and terminal loop attachments are not new concepts. | |
| Tree oscillation/nodal results, variational interlacing under surgery, coarea rearrangement, and the equilateral discrete-to-metric spectral correspondence are established tools. The release supplies the particular proofs needed but does not claim to invent those subjects. Relevant references include Schapotschnikow (2006), Nicaise (1987), Berkolaiko–Kuchment (2013), and the BKKM surgery paper (2019). | |
| ## Candidate contributions requiring priority review | |
| The principal claim to compare is: | |
| > In the stated high-index regime, lower sharpness alone forces maximal multiplicity, simultaneous upper sharpness, the complete known extremal topology class, and the explicit edge-length phase rules. | |
| The candidate is stronger than merely reconstructing the known topology class for a graph already assumed to be two-sided sharp. Other candidate contributions are the direct actual-metric nodal-inertia proof used to obtain mixed-tree equality, terminal evaluation-functional rigidity as a classification mechanism, the graph-specific collapse from R=0 and K>=0 to R=0 with K=0, and the explicit energy-normalized gap estimate with path-Gram normalization. | |
| The direct nodal identity or abstract gap estimate may have equivalent formulations in existing oscillation, constrained-form, or subspace perturbation literature. The release does not certify that no such formulation exists. | |
| ## 2026 context | |
| Harrell II, Kennedy, and Ramos, *Bounds on eigenvalue ratios of quantum graphs*, arXiv:2603.26172, explicitly ask for the equality classification in the Dirichlet-tree Pólya bound in Open Problem 1.12(3) of the inspected HTML version. The tree argument in this release gives that classification as a special case. The same paper separately asks for an optimal eigenvalue-ratio constant between 9/4 and 4. This release does not solve that ratio problem. | |
| ## Search record and its limits | |
| Current primary-source searches and inspections were performed on 25 September 2026, including the original Serio preprint, the author-repository indexed text of the published topology statement, BKKM bibliographic/bound sources, and the 2026 ratio preprint. Search phrases included “lower sharp” with “quantum graphs,” “lower sharp” with “maximally degenerate,” and quantum-graph equality/rigidity variants. The main one-sided theorem was not established as a duplicate by these checks. | |
| The author-repository PDF fetch did not complete through the web reader; its indexed primary-source text was available and was enough to confirm the existence of the known exceptional families. The original preprint PDF was inspected directly. There was no exhaustive MathSciNet/zbMATH or citation-network search. Absence from targeted search results is not proof of novelty. | |
| ## Public wording | |
| Accurate: “A proof-complete candidate extending one-sided lower equality to complete metric/topological rigidity, with an explicit quantitative closure obstruction.” | |
| Inaccurate: “First discovery of lasso trees, theta graphs, maximal spectral degeneracy, or sharp eigenvalue bounds”; “the first ever proof of all quantum-graph equality problems”; “independently verified world-first theorem”; “a demonstrated hardware breakthrough.” | |