Jacobian Conjecture
Prior state unknown→disproved
n ≥ 3; plane case open
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Mathematical findings
12 source-grounded records. Verification labels remain separate from source authentication and publication status.
Imported from VibeMathed under CC BY 4.0. Each record links to its registry entry and named primary source. Registry verification is preserved verbatim.
Erdős's Planar Unit Distance Conjecture
Prior state unknown→disproved
Conjectured upper bound on how many pairs among $n$ points in the plane can be exactly one unit apart.
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Crouzeix's Conjecture
Prior state unknown→proved
Two independent proofs within eight days, both with AI in the loop. Jin's (posted 27 July, preprints.org, submitted to Annals) is the first: its decisive theorem came out of an autonomous GPT-5.6 Sol run, and it is the proof Townsend, Greenbaum and Crouzeix have checked. Lorist and Schwenninger's five-page argument (arXiv, 4 August) is a genuinely different route - double-layer potentials plus a perturbation lemma…
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FullRSB Jamming Identity a + b = 1
Prior state unknown→proved
Can the critical-exponent relation $a + b = 1$ at the jamming transition, observed numerically to high precision in the full replica-symmetry-breaking solution of hard spheres, be derived analytically from the scaling equations?
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Huneke-Wiegand Conjecture
Prior state unknown→disproved
Verified by author of the conjecture
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Simplicity of the Hodge Bundle
Prior state unknown→proved
Does the Hodge bundle $\Omega_g$ over the moduli stack of genus $g \ge 2$ curves contain any nontrivial sub-bundles? Posed by Dawei Chen around 2015; the answer is no.
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The Middle Stair of Parallel Chip-Firing
Prior state unknown→proved
Ji, Li and Wang conjectured in 2024 that every parallel chip-firing game on a finite connected graph whose chip count lies strictly between $2|E|-|V|$ and $2|E|$ has period exactly 2, generalizing the middle rung of Levine's devil's staircase from complete graphs to all graphs. Known before only for trees, cycles, complete and complete bipartite graphs.
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Return Probability for the Lamplighter Walk on a Tree
Prior state unknown→proved
For the switch-walk-switch lamplighter walk on $\mathbb{Z}_2 \wr T_d$, prove the sharp asymptotic $p_{2n}(e,e) = \rho_d^{2n} \exp[-(\pi^2 (\log(d-1))^2 + o(1)) \frac{n}{\log^2 n}]$ with $\rho_d = \frac{2\sqrt{d-1}}{d}$.
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Total Variation for the Lamplighter Walk on Z
Prior state unknown→proved
For the switch-walk-switch walk on $\mathbb{Z}_2 \wr \mathbb{Z}$ started at $(0,0)$ and $(0,2)$, prove $\|P_t^x - P_t^y\|_{TV} \asymp t^{-1/2}$.
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Erdős Problem #1040
Prior state unknown→disproved
capacity alone does not determine the invariant; the zero-measure clause is a separate open question
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Erdős Problem #654
Prior state unknown→disproved
the strongest form is disproved via configurations where every point sees at most about 3n/4 distinct distances; the weaker improvement remains open
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Integral Local Invariant Cycles in Degree One
Prior state unknown→proved
For a semistable one-parameter family of complex projective varieties with smooth nearby fiber $X_t$ and monodromy $T$, is the map $H^1(X, \mathbb{Z}) \to H^1(X_t, \mathbb{Z})^T$ surjective? True in degree one, although the integral statement fails in higher degree.
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