A torus-collar proof of the real-spectrum compact-support classification for Nijenhuis operators
Read the paper (PDF, 5 pages) · Source and verification (ZIP) · Zenodo record
Plain-English explanation · Read on this page · Collaboration
Abstract
We give a complete smooth compact-support proof of the classification asked for in the historical integrable-systems problem source: a real Jordan type occurs on a nonempty open subset of ℝⁿ for a compactly supported smooth Nijenhuis operator exactly when its spectrum is real. The construction realizes arbitrary specified numerical real Jordan data in a commuting local frame. On a same-dimensional torus collar, the torsion equation reduces to (A-λ I)A'=0; an ordered finite path removes every Jordan block and reaches an exact zero band. The nonreal obstruction and earlier semisimple and two-dimensional examples are prior results. We include the precise zero-point rigidity deduction needed for necessity. The historical priority of the full result and of this construction remains unestablished. Extensive AI tools assisted solving, verification and writing. This preprint is unrefereed and has no external human peer review or formal proof-assistant certification.
The idea in plain English
A Nijenhuis operator assigns a linear transformation to each point of a space while satisfying a differential compatibility condition. This paper asks which matrix patterns can hold throughout an open region when the entire field must become exactly zero outside a bounded region.
Every pattern with real eigenvalues can be realized. The construction allows repeated eigenvalues and Jordan blocks of any size, including matrices that cannot be diagonalized. It fixes both the numerical eigenvalues and the block pattern in a suitable local frame.
The difficult step is turning the field off without breaking its compatibility condition. The paper makes that transition in a thin neighborhood of a torus, removing the connections within each Jordan block in a controlled order.
Outside the chosen region, the matrix pattern is allowed to change. The converse is an established obstruction: a smooth compactly supported real Nijenhuis operator cannot have a nonreal eigenvalue anywhere.
The precise result
For every integer n ≥ 1 and every prescribed numerical real Jordan matrix J whose eigenvalues are all real, there is a globally C∞, compactly supported real Nijenhuis operator on ℝⁿ whose matrix is J in a smooth commuting local frame on a nonempty open domain.
The construction includes repeated eigenvalues, multiple blocks belonging to the same eigenvalue, blocks of arbitrary size, negative or zero eigenvalues and the zero matrix.
Conversely, every eigenvalue at every point of any smooth compactly supported real Nijenhuis operator on ℝⁿ is real. Together, these statements give the classification asked for in the printed historical problem.
The prescribed numerical matrix is realized on an open domain in the same dimension. The theorem does not prescribe the ambient Cartesian-frame matrix or require one Jordan type throughout the whole space.
Why the result holds
In dimensions two and higher, a thin torus neighborhood lets all coefficients depend on one transverse variable. For the block matrix L = [[A(t), b(t)], [0, λ(t)]], the Nijenhuis condition reduces to the ordered equation (A − λI)A′ = 0.
A finite sequence first removes the transverse coupling, then turns off each Jordan block’s arrows in order and moves its scalar part to zero. Smooth transitions with constant end intervals join these stages and leave exact zero bands at both ends.
In these dimensions, an embedded torus collar lies in ℝⁿ in the required dimension. The tensor is pushed into that collar and extended by zero; the zero bands make the extension smooth and compactly supported. In dimension one, a smooth compactly supported scalar function suffices.
For necessity, the paper uses the established local splitting and holomorphic structure of nonreal spectral blocks in a zero-point rigidity argument. Compact support allows extension to a sphere with a neighborhood on which the tensor is zero.
Scope and limitations
The result is C∞ smooth. It does not give a nonzero real-analytic compactly supported operator.
The matrix J appears in a smooth commuting local frame on a domain chosen by the construction. The ambient Cartesian-frame matrix and an arbitrarily preassigned realization domain are not prescribed.
Rank, eigenvalues and Jordan type may change outside the realization domain. Global algebraic regularity and a single global Jordan type are not guaranteed; algebraic regularity is distinct from smoothness.
The nonreal obstruction concerns globally compactly supported operators. It does not forbid local models with nonreal eigenvalues or operators without compact support.
Earlier work supplies the nonreal obstruction, real semisimple and two-dimensional examples, local point realizations, and the splitting and holomorphic tools. Historical priority of the full result or this construction is not established.
Status and AI use
AI tools were used extensively in solving, source comparison, verification and writing. Internal AI reviews and finite computational controls do not constitute external human peer review or formal proof-assistant certification.
The manuscript and Zenodo metadata are dated 15 October 2026; the Zenodo record was created on 17 October 2026. The earlier date records manuscript preparation, not verified public availability. Neither date establishes first priority.
Read the paper
Open PDF in a full window · Download the original from Zenodo
Collaborate on this result
I welcome substantive collaboration from researchers in Nijenhuis geometry, differential geometry or integrable systems on independent verification, comparison with earlier constructions and preparation for journal review.
Authorship and responsibilities would be discussed in relation to the substantive contributions made.
Cite this paper
Alec Kriebel. (2026). A torus-collar proof of the real-spectrum compact-support classification for Nijenhuis operators (Version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23431313
Background and prior work
- Bolsinov, Matveev, Miranda and Tabachnikov — Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems: Problem 5.11 in the 2020 author version asks for compact support and a fixed algebraic type on an open domain. It already records the nonreal obstruction and real semisimple and two-dimensional examples.
- Bolsinov, Konyaev and Matveev — Nijenhuis Geometry: Provides local spectral splitting, the canonical complex and holomorphic structure for nonreal blocks, and earlier realizations of arbitrary algebraic types at individual points.
- Bolsinov, Konyaev and Matveev — Nijenhuis geometry III: gl-regular Nijenhuis operators: Established local results for regular Jordan layers provide related context. They are distinct from the compact-support transition with arbitrary repeated blocks considered here.
- Bolsinov, Konyaev and Matveev — Nijenhuis geometry IV: Develops symmetries and functional constructions subject to differential compatibility conditions. These earlier methods are credited; the page makes no exhaustive claim about their possible consequences.