Eigenvalues and spectral theory
Eigenvalues are the special numbers that describe how a system vibrates or spreads. Spectral theory studies them.
8 AI-made results on eigenvalues and spectral theory, from 17 papers in the OpenAI math collection, each explained in plain English. 4 have a formal Lean proof listed; 4 are claimed in a paper and not yet machine-checked. They span 5 fields, led by probability and statistical mechanics (3), mathematical physics (2) and combinatorics (1). The oldest question here dates back to 1974: A C 1 counterexample to the entropy conjecture.
Dynamical systems and ergodic theory1
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A C1 counterexample to the entropy conjecture
Constructs a noninvertible C1 self-map of a compact smooth manifold with zero topological entropy but eigenvalue 2 on second homology. This disproves the homological entropy lower bound for general C1 self-maps: homological growth need not force positive orbit complexity.
Formal proof listed
Combinatorics1
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Deterministic nonbipartite Ramanujan graphs in every fixed degree
For every fixed d ≥ 3, constructs a simple d-regular nonbipartite Ramanujan graph on every sufficiently large even number n of vertices, with every nonconstant adjacency eigenvalue strictly between and . A deterministic algorithm outputs the full adjacency list in polynomial bit time, with exponent depending on d.
Claimed, not checked
Probability and statistical mechanics3
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The geometric phase diagram, diffusion, and spectra of random planar maps
Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for and to the Brownian continuum random tree for q > 4, establishing the surface-to-tree geometric transition. For FK–Ising and spanning-tree-weighted maps, stationary random walks converge to Liouville Brownian motion on the limiting sphere. The FK–Ising spectral result also gives convergence of eigenvalues and heat traces, using the stated Brownian/LQG inputs.
Formal proof listed -
GOE bulk universality for regular graphs with weak Anderson disorder
For every fixed degree d ≥ 3, the bulk adjacency-eigenvalue point process of a uniform simple random d-regular graph converges to the Gaussian orthogonal ensemble law, including for cubic graphs. The same fixed-energy universality persists under sufficiently weak fixed iid uniform diagonal disorder, throughout compact bands strictly inside the clean spectral edges.
Claimed, not checked -
All-temperature pressure of orthogonally invariant Ising spin glasses
Gives an exact variational formula for the limiting pressure of orthogonally invariant Ising spin glasses at every fixed temperature, both almost surely and in expectation. The coupling matrix is a Haar-random rotation of a deterministic spectrum converging to a compactly supported law, with extreme eigenvalues converging to its support edges. The zero-field ground-state energy follows as temperature tends to zero.
Formal proof listed
Mathematical physics2
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Area laws and tensor networks for two-dimensional gapped systems
Proves an entropy area law for unique ground states of finite-range Hamiltonians on arbitrary finite induced square-lattice domains, using only a uniform full-system spectral gap and bounds on the local interactions. On open squares, uniformly gapped nearest-neighbor ground states also admit projected entangled-pair state approximations with polynomial bond dimension and global vector error at most L−1.
Claimed, not checked -
The spin-one Haldane gap
Proves the spin-one Haldane gap conjecture for the pure antiferromagnetic Heisenberg chain on even periodic rings: the spectral gap stays uniformly positive as the chain grows. A companion establishes a gap for odd open chains with endpoint field and gives boundary-selected infinite-volume states with topological index −1.
Claimed, not checked
Partial differential equations1
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The hot spots conjecture for simply connected planar domains
Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed.
Formal proof listed