# Eigenvalues and spectral theory > 8 AI-made results on eigenvalues and spectral theory, each explained in plain English. Eigenvalues are the special numbers that describe how a system… Canonical: https://closedform.nishachay.tech/topic/eigenvalues-and-spectral-theory/ Last updated: 2026-10-14 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](https://closedform.nishachay.tech/e/openai-math-2026-151/). ## Results (8) - [A C 1 counterexample to the entropy conjecture](https://closedform.nishachay.tech/e/openai-math-2026-151/) - Entry 151 · A smooth-enough map can double a shape's homology while its orbits stay orderly. (Formal proof listed) - [Deterministic nonbipartite Ramanujan graphs in every fixed degree](https://closedform.nishachay.tech/e/openai-math-2026-178/) - Entry 178 · A fixed recipe builds networks with best-possible mixing, in every degree and every large even size. (Claimed, not checked) - [The geometric phase diagram, diffusion, and spectra of random planar maps](https://closedform.nishachay.tech/e/openai-math-2026-211/) - Entry 211 · Random FK planar maps are claimed to look like random surfaces for q ≤ 4 and like a random tree above 4. (Formal proof listed) - [GOE bulk universality for regular graphs with weak Anderson disorder](https://closedform.nishachay.tech/e/openai-math-2026-219/) - Entry 219 · Eigenvalue spacings of random regular graphs, even with weak random potentials, match those of Gaussian random matrices. (Claimed, not checked) - [All-temperature pressure of orthogonally invariant Ising spin glasses](https://closedform.nishachay.tech/e/openai-math-2026-234/) - Entry 234 · A formula for the pressure of orthogonally invariant Ising spin glasses holds at every temperature. (Formal proof listed) - [Area laws and tensor networks for two-dimensional gapped systems](https://closedform.nishachay.tech/e/openai-math-2026-265/) - Entry 265 · Claims the 2D area law: gapped ground states on a square grid hold entanglement proportional to boundary, not volume. (Claimed, not checked) - [The spin-one Haldane gap](https://closedform.nishachay.tech/e/openai-math-2026-268/) - Entry 268 · Claims the spin-one Haldane gap: the Heisenberg chain on even rings has an energy gap that stays above log(20)/784. (Claimed, not checked) - [The hot spots conjecture for simply connected planar domains](https://closedform.nishachay.tech/e/openai-math-2026-369/) - Entry 369 · On smooth flat shapes without holes, the lasting hot and cold spots lie on the edge. (Formal proof listed) ## Related - [Lean-checked results](https://closedform.nishachay.tech/trust/lean-checked/) - Trust · 242 results - [Claimed, not yet checked](https://closedform.nishachay.tech/trust/claimed/) - Trust · 130 results - [Open since the 1970s](https://closedform.nishachay.tech/open-since/1970s/) - Era · 63 results - [Open since the 1980s](https://closedform.nishachay.tech/open-since/1980s/) - Era · 60 results - [Erdős problems](https://closedform.nishachay.tech/topic/erdos-problems/) - Topic · 13 results - [Named conjectures](https://closedform.nishachay.tech/topic/named-conjectures/) - Topic · 85 results - [Ramsey theory](https://closedform.nishachay.tech/topic/ramsey-theory/) - Topic · 5 results - [Prime numbers](https://closedform.nishachay.tech/topic/prime-numbers/) - Topic · 12 results - [Graph theory](https://closedform.nishachay.tech/topic/graph-theory/) - Topic · 40 results - [Distances and packing](https://closedform.nishachay.tech/topic/distances-and-packing/) - Topic · 5 results - [Probability and statistical mechanics](https://closedform.nishachay.tech/science/mathematics/probability-and-statistical-mechanics/) - Field · 3 here - [Mathematical physics](https://closedform.nishachay.tech/science/mathematics/mathematical-physics/) - Field · 2 here - [Dynamical systems and ergodic theory](https://closedform.nishachay.tech/science/mathematics/dynamical-systems-and-ergodic-theory/) - Field · 1 here - [All topics](https://closedform.nishachay.tech/topics/) - Every way to browse