Priority, significance and scope assessment

Henry Zweiman. September 16, 2026.

This is a provisional originating-assistant assessment of one coherent preprint, not independent certification of a solved open problem or a prediction of acceptance by a particular journal.

Problem and current primary frontier

The problem is whether a single homothetically shrinking hole at a prescribed planar point can completely split every finite-multiplicity Robin eigenvalue, including eigenspaces with high-order common zeros. The parameter is fixed and nonzero on both boundaries, of either sign. The shape must be selected once, before the hole-size limit and independently of the spectral window.

Felli-Roychowdhury-Siclari's final June 2026 article explicitly leaves multiple Robin eigenvalues and splitting for future work in Remark 2.10. Its sharp general nodal result requires dimension at least three; the planar arbitrary-hole result loses a small power, while a disk has an explicit sharp coefficient. The present all-order planar estimates and complete-splitting construction address that research direction. This is a precise theorem within that direction, rather than a claim to settle every possible Robin perturbation question. Final source.

Felli-Liverani-Ognibene's 2026 Neumann paper establishes generic reduction of multiplicity as the center of a spherical hole varies, and complete splitting for double eigenvalues. The present result fixes the center, allows all finite multiplicities and controls each positive nodal order through a shape choice. Its nonzero Robin parameter is essential to the treatment of the nonnodal direction. Current preprint.

The official arXiv histories were rechecked on September 16, 2026: 2504.05994 has v1 only; 2311.05999 has v3 as its latest version; 2601.13129 and 2606.03996 have v1 only. The final Robin article, rather than only its older preprint, is the principal comparison.

Prior mechanisms and access limits

The corrected Dirichlet-hole analysis of Abatangelo-Léna-Musolino supplies essential prior projection, filtration and Schur-complement ideas. They are explicitly credited. The present finite-dimensional proof is included to handle both signs and disparate scales, not claimed as a new matrix theory. Corrigendum.

Shape differentiation of generalized polarization tensors is prior work of Ammari-Kang-Lim-Zribi. The disk calculation here is a self-contained application, checked by an exact conformal family. Generic simplicity and complete splitting under other singular perturbations also predate this work. Neither is claimed as a general new principle.

The full ETH report by Dabrowski, A variational principle for the perturbation of repeated eigenvalues and applications (2017-37), was retrieved and compared. Its grounded-hole application identifies one leading evaluation branch. Its conductivity application gives a gradient polarization matrix, with directions invisible to that matrix remaining in the remainder. Its stated applications do not give the all-order fixed-center Robin construction. The report is cited separately; the final 2022 article was not available and is not assumed identical. ETH report.

Older Robin-hole statements of Ozawa and Ozawa-Roppongi were compared in their accessible final PDFs. Full theorem-level comparisons with AKLZ's 2010 complete-expansion paper, the 2012 polarization paper, Lanza de Cristoforis's 2012 simple-Neumann paper, Dabrowski's final 2022 article and Gadyl'shin's 1993 boundary-patch paper remain incomplete. These limits matter and are retained in the manuscript and source audit. Failed access and empty searches are not evidence of novelty. The scalar logarithmic term is not the basis of admission and is not claimed as a separately established priority result.

Why this is a substantial single contribution

The main difficulty is preserving relative precision for every vanishing order, including when other branches move on much larger scales. The planar constant ambiguity is handled by a decaying exterior normalization and quantitative residual estimates. The weighted error bounds then allow the corrected Schur mechanism to identify every branch. Finally, one analytic Fourier deformation splits every possible two-dimensional homogeneous block; analytic discriminants make one selected shape work across all finite windows.

This combination gives more than a simple-eigenvalue specialization or a finite computation. It provides reusable coefficients, correctors and a prescribed-center spectral construction. Those features support pursuing a Transactions-of-the-AMS-or-comparable audience. The methods build substantially on existing perturbation theory, so editorial judgment could still regard the contribution as better suited to a specialist journal. No citation count or acceptance outcome is predicted.

Decision and remaining frontier

Provisionally admit the combined manuscript as one preprint, P52, contingent on successful artifact checks and verified GitHub publication. The originating-assistant proof review identified no fatal gap. This remains a revisable internal judgment; human expert verification has not occurred.

No uniform threshold for the infinite spectrum, quantitative lower bound on the admissible size for arbitrary spectral windows, mixed nonnodal Neumann theorem, higher-dimensional complete-splitting theorem, or parameter-uniform result is proved. These are future problems, not additional papers. The simple, multiple, second-order and shape results in this bundle count only once.