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Status changes from the bundled source snapshot generated 2026-09-13. Dates describe source-reported events, not VibeMath endorsement.
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Mahler's $Z$-number problem in its generalized form
Theorem E gives $Z_{3/2}(\tfrac27,\tfrac57)=\emptyset$: a window of length $\tfrac37=0.428571\ldots$ at base $3/2$, beating the $31/81$ of [Dub19, Thm. 1.2], and with no assumption on the arithmetic nature of $\xi$, where that result needs algebraicity. Mahler's problem itself is untouched: for which parameters $Z$ is empty remains open, and this is a record window rather than a classification.
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Nontriviality of number-restricted arithmetic over subDMQ
Number-restricted induction over subDMQ derives A ⇒ A ⊗ A for every formula A to which induction applies, without using quantifier splitting. Restricted quantifier splitting follows as a corollary. A second argument obtains contraction from unrestricted quantifier splitting and a separated domain partition. With suitable Curry fixed points, contraction yields triviality.
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Normality of a sum of two Stoneham constants
Yes: the sum of two Stoneham constants sharing a base is normal in that base, for every admissible pair of parameters, with overlapping occurrences counted. Not to be confused with what the same 2012 paper proves: its Theorem 3 shows a sum of two $B$-NONnormal Stoneham constants is $B$-nonnormal, which concerns a different base and the opposite property. The open half was normality in the common defining base $b$, and that is what this settles.
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The Hilbert transform-UMD dependence problem
Neither exponent can be lowered. Theorem 1.1 constructs explicit $2^{n}$-dimensional spaces $X_n$ and $Y_n$ with $\hbar_{2,X_n}\asymp n$, $\beta_{2,X_n}\asymp\sqrt n$ and $\beta_{2,Y_n}\asymp n$, $\hbar_{2,Y_n}\asymp\sqrt n$, with universal comparison constants, so both quadratic comparisons are sharp at $p=2$; Corollary 1.3 extends the growth rates to every fixed $1<p<\infty$ by extrapolation, with constants depending on $p$. The spaces are built from the summation operators Wenzel had proposed as candidates for exactly this separation. The Lean formalisation covers Theorem 1.1 at $p=2$ over both real and complex scalars; the extrapolation to other $p$ and the exact coefficients of Remark 1.2 are outside it.
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Binary normality of a localized logarithm value
Yes. Every finite binary word occurs with its expected limiting frequency, overlapping occurrences counted. The one-prime members of this family were already known to be normal: $\alpha_{b,p}=\sum_{k\ge1}p^{-k}b^{-p^{k}}$ is Stoneham's constant, proved normal under primitive-root hypotheses by Stoneham in 1973 and unconditionally for coprime parameters by Bailey and Crandall in 2002. The new content is the mixed-prime support, where all the indices $2^{a}3^{c}$ contribute at once and the argument turns on the largest power of three in the truncation denominator surviving addition. The paper states that it has not located an earlier published conjecture about $\xi$, so this settles no previously posed question. The Lean development covers qualitative normality of this constant only; the paper's quantitative discrepancy bound and its extension to larger finite sets of primes are outside the formalisation.
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Minimum-Rank and Mortality Bounds for Finite Real Matrix Monoids
For any family of real $n\times n$ matrices generating a finite entire product monoid of minimum rank $s$, some word attains rank $s$ within $$ B(n,s)=n2^{n-s}-\frac{n(n+1)}2+\frac{s(s-1)}2 $$ letters. The same holds over $\mathbb Q$, without requiring a finite generating alphabet. For mortality, $s=0$, giving a bound of order $n2^n$. Over $\mathbb Q$, this improves Kiefer–Ryzhikov's (2026) $3^{n^2}$ bounds for mortality and minimum-rank diameter, and Almeida–Steinberg's (2009) mortality bound $(2n-1)^{n^2}-1$ for $n>1$. The proof uses rank-dependent sandwich descent. Further results give the sharp planar threshold four, additive invariant-flag bounds with sharp small-block examples, and cubic-size compressed witness existence with correct evaluation. Boundedness and individually finite-power generators alone admit no uniform planar mortality bound. General optimality and polynomial-time synthesis are not claimed.
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Finite-time blowup for the IPM equation with a uniformly space-time smooth force
Yes. Theorem 2.1: there are a smooth odd initial density $\rho_{in}\in C^\infty(\mathbb T^2)$ of zero spatial mean, an odd force $F\in C^\infty([0,1]\times\mathbb T^2)$, and a solution smooth on $[0,T]$ for every $T<1$ with $\rho(t)\to\rho_*$ in $C^\eta$ for every $0\le\eta<1$, yet $\|\nabla\rho(t)\|_\infty$ and $\|D_xu_{\mathbb T}(\rho(t))\|_\infty$ both diverging as $t\uparrow1$. The advance over Córdoba and Martínez-Zoroa is precisely the force class, from $L^\infty_t C^\infty_x$ to uniformly space-time smooth, on the torus rather than the plane. Not the Clay Millennium problem: that problem is Navier-Stokes with viscosity, and Fefferman's official description states that the Euler equation "is not on the Clay Institute's list of prize problems".
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Sharp Finite Markov Order in Intrinsic Sofic Dimension
For a stationary finite-alphabet law $\mu$, let $$ n=\dim_{\mathbb R}\mathcal H_p $$ be the intrinsic real Hankel dimension of its cylinder-probability function. If $\mu$ has any finite Markov order, then $$ \operatorname{ord}(\mu)\le \binom n2. $$ The proof passes to a reduced $n$-dimensional linear representation and uses Holland's criterion that $k$-step Markovity is equivalent to every length-$k$ transition product having rank at most one. Applying $\Lambda^2$ turns this into vanishing of products on a space of dimension $\binom n2$; a uniform nilpotence argument then forces vanishing after $\binom n2$ steps. Sharpness is attained for every $n\ge2$. The construction gives a stationary sofic process with a nonnegative rational presentation of minimal dimension $n$ and exact order $\binom n2$. One realization uses $$ \binom n2-1+n^2 $$ output symbols.
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The ceiling of the single-letter entropy method for the union-closed sets conjecture
Two unconditional ceilings. Every single-letter certificate whose classes contain product laws certifies at most $1-h(1/\sqrt2)/\sqrt2 = 0.383099\ldots$. Every certificate that uses the i.i.d. protocol and whose other classes admit component hiding — a property of every class in the literature — certifies at most $c^{**} = 0.382885260\ldots$. So the entropy method, in every form used since 2022, cannot reach $1/2$: Frankl's conjecture is out of its range, and only $1.8\cdot10^{-4}$ of headroom remained above the previous record. The paper also constructs a protocol attaining $0.38284$ (computer-assisted, conditional on two numerically verified hypotheses). Frankl's conjecture itself remains wide open.
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Finite-time blowup for the 3D incompressible Euler equations with smooth forcing
Yes. Theorem 1.1: for every $r_0>0$ and $z_0$ there are a time $T_*>0$, a divergence-free axisymmetric $u_0\in C_c^\infty$ supported in a fixed solid torus with nonzero swirl and zero meridional velocity, and an axisymmetric force $f\in C^\infty(\mathbb R^3\times[0,T_*])$ supported in that torus, with a solution smooth on $[0,T_*)$ for which circulation and meridional velocity stay bounded while $\|\nabla\Gamma(t)\|_\infty$ and $\|\omega(t)\|_\infty$ tend to infinity and $\int_0^{T_*}\|\omega(t)\|_\infty\,dt=\infty$, so the blowup is genuine by Beale-Kato-Majda. It is unique among divergence-free locally space-time Lipschitz solutions with the same data, and competitors need not be axisymmetric. Not the Clay Millennium problem: that problem is Navier-Stokes with viscosity, and Fefferman's official description states that the Euler equation "is not on the Clay Institute's list of prize problems".
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Finite-time blowup for the inviscid Boussinesq system with smooth forcing
Yes. Blowup for the inviscid Boussinesq system on $\mathbb R^2$ with forces in $C^\infty(\mathbb R^2\times[0,T_*])$ in both equations, supported in one fixed spatial ball, from smooth compactly supported initial temperature and zero initial velocity. The temperature stays bounded while $\|\nabla\theta(t)\|_\infty\to\infty$ and the vorticity norm has infinite limsup as $t\uparrow T_*$. The solution is smooth on every closed interval before blowup and unique in a finite-energy Lipschitz class. This lifts the force from barely $C^1$ to fully smooth, and is the construction the Euler paper then builds on. Not the Clay Millennium problem: that problem is Navier-Stokes with viscosity, and Fefferman's official description states that the Euler equation "is not on the Clay Institute's list of prize problems".
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An improved lower bound for the Shannon capacity of $C_{11}$
A sequence of explicitly constructed independent sets in strong powers of $C_{11}$, with successive root improvements certified by exact integer comparisons, using cross-powers when dimensions differ. The current bound is $\Theta(C_{11}) \ge 5.295526013632343$, from an independent set in $C_{11}^{\boxtimes 213}$ (R10, 9 September 2026), improving the R3 result of $5.295492477500681$ in dimension 207 that this entry was first listed for, and the Buys-Polak-Zuiddam baseline of $5.295492315784620$. The intermediate steps R5 and R6 and the superseded R9 are recorded on the frontier. The exact capacity remains open, and no upper bound is claimed. The R5, R6, R9 and R10 builds carry disclosed native-evaluation dependencies rather than being kernel-only.
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Erdős Problem #1221
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Reciprocal Fermat Constant is Nonnormal
The paper proves that the reciprocal Fermat constant is disjunctive but nonnormal in binary, explicitly locates every finite word, and establishes positive lower word frequencies. It extends these properties to a broad family of arithmetic series. Behavior in unrelated bases remains open.
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Reciprocal-triple-free sets: the finite plateau at 732
Proved f(732)=f(731) unconditionally as a structural finite theorem. The isolated component G={122,183,244,366,732} has exactly the edges {122,183,366}, {183,244,732}, {244,366,732}. Every admissible selection in G has size at most 3; replacement by {122,183,244} preserves admissibility and cardinality or increases it while removing 732. Combining this with the published f(731)=606 gives the candidate new table term f(732)=606. The historical asymptotic problem remains open. A general large-prime recurrence is proved separately in prose, not in Lean; it may be folklore.
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Lower bound for the complex Grothendieck constant
The paper proves $$ K_G^{\mathbb{C}}>1.35584631827168. $$ The previously recorded Davie lower bound is approximately $1.33807$, while Haagerup's upper bound is approximately $1.40491$. Thus the result closes more than one quarter of the remaining lower-to-upper-bound gap, but it does not determine the exact value of the complex Grothendieck constant. The construction uses finitely many additional complex Hermite projections together with a common radial weight to obtain a dimension-independent $L_{\infty}$-to-$L_1$ estimate. The numerical inequalities needed for the final bound are certified by interval arithmetic. The paper also analyzes the limit of its particular weighted criterion. If $\mathcal{K}_*$ denotes the supremum obtainable within that framework, it proves $$ 1.35584631827168<\mathcal{K}_*<1.35584697425050. $$ The upper endpoint here is a limitation of this specific criterion, not an upper bound on $K_G^{\mathbb{C}}$ itself.
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The Erdos-Borwein Constant is 2-Dense
This work proves that the Erdős–Borwein constant's binary expansion contains every finite binary sequence as a consecutive block, each occurring infinitely often. This implies that the Erdős–Borwein constant is 2-dense.
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Koizumi–Liu Eventual Sign Alternation Conjecture
Koizumi constructs a representable simple rank-$6$ matroid $M$ for which $$ \operatorname{Mag}(M;q) $$ has a pole of order $4$ at $q=-1$ but a pole of order $5$ at $q=i$. After writing $$ A_M(t)=\operatorname{Mag}(M;-t), $$ the higher-order pole at $t=i$ forces the Taylor coefficients of $A_M(t)$ to fail eventual nonnegativity. Consequently, $$ (-1)^\ell[q^\ell]\operatorname{Mag}(M;q)<0 $$ for infinitely many $\ell$. The matroid is representable by twelve vectors in $\mathbb R^6$, so it yields an actual real central hyperplane arrangement and therefore directly disproves Koizumi–Liu's conjecture for real arrangements.
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A 44-vertex triangulation of $\mathbb{RP}^{6}$
The manuscript gives an explicit centrally symmetric simplicial 7-polytope with 88 integer vertices. Its antipodal boundary quotient is a 44-vertex triangulation of real projective 6-space, with f-vector (44, 938, 7024, 22555, 34936, 25914, 7404). This improves the 45-vertex construction of Guyer, Steinerberger and Yang and gives an affirmative answer to their Question 3.2. Integer coordinates, facet lists and exact-arithmetic verification code accompany the manuscript. No claim that 44 is vertex-minimal is made.
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Optimality of the added-vector code in the 19-dimensional kissing construction
The manuscript proves the upper bound \(|A|\le1280\) for every admissible subset of the fixed ambient code \(D\). Together with Ho’s existing construction, this determines the exact maximum. The proof partitions \(D\) into 256 cosets of a 16-word subspace whose induced forbidden-distance graph is the Clebsch graph; each coset contributes at most five words. This does not determine the unrestricted kissing number \(k(19)\) or improve the known 11948-point configuration. Independent expert review is pending.
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Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not?
Answered in the negative for every $n\ge 2$. The preprint constructs a bijection $F:\mathbb R^n\to\mathbb R^n$ that maps every connected set to a connected set, is continuous exactly off the closed ray $[0,\infty)\times\{0\}^{n-1}$, and pulls the straight segment $\{(1,0,\dots,0)\}\times[0,1]$ back to the middle-thirds Cantor set on that ray, so $F^{-1}$ is not connectedness-preserving. The construction extends a thin solid tube by finger moves so its cross-sections recur near every point of the complementary compactum, collapses the ray onto the tube's ideal end, and certifies arbitrary connected sets by a separation argument; $F$ and $F^{-1}$ can be taken Borel. The same author's companion note on Darboux injections from closed manifolds (Banakh-Banakh Problems 1.7 and 1.8) is a separate result and belongs in its own entry.
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The γ–θ conjecture in eternal domination
Let $H$ be the Berlekamp–van Lint–Seidel graph on $243$ vertices, the Cayley graph of $\mathbb Z_3^5$ with strongly regular parameters $$ (243,22,1,2), $$ and let $G=\overline H$. The proof establishes $$ \gamma(G)=3 $$ because every pair has a common neighbor in $H$, while an $H$-triangle gives a dominating triple in $G$. It then proves $$ \gamma^\infty(G)=3 $$ by showing that the family of all dominating triples is closed under a legal response to every attack: after moving one guard to the attacked vertex, another dominating triple can always be obtained. Finally, a double-counting argument shows that $H$ is not 3-colorable, hence $$ \theta(G)=\chi(H)>3. $$ Therefore $$ \gamma(G)=\gamma^\infty(G)=3<\theta(G). $$
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Navier–Stokes Millennium Prize problem: finite-time breakdown with smooth forcing
OpenAI claims to construct a smooth finite-energy solution of the three-dimensional incompressible Navier–Stokes equations, starting from a fluid at rest and driven by smooth forcing, that develops a finite-time singularity. The construction is claimed both on R^3 and in the periodic setting, thereby establishing statements C and D of Fefferman's official Clay formulation. If the correspondence between the released proof/formalization and the Clay statements survives independent review, this resolves the Navier–Stokes Millennium Prize problem. It does not establish finite-time blow-up for the unforced Navier–Stokes equations; the Clay resolution comes through the smooth-forcing alternatives C/D.
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The Ibragimov–Iosifescu conjecture for φ-mixing sequences
Astra constructs a strictly stationary real process $$ X_t=\xi_t+K(\xi_{t-1},\xi_{t-2},\ldots), $$ where the innovations $\xi_t$ are i.i.d. Gaussian variables convolved with a symmetric rare-spike law, and $K$ is bounded, continuous, and odd. The process is $\varphi$-mixing, centered, square-integrable, and satisfies $$ \operatorname{Var}(S_n)\to\infty. $$ Nevertheless there are times $n_j\to\infty$ such that $$ \frac{S_{n_j}}{\sqrt{\operatorname{Var}(S_{n_j})}}\to0 $$ in probability. Therefore the normalized sums cannot converge in distribution to $N(0,1)$. The same example also rules out Iosifescu's stronger weak invariance-principle conjecture, since Brownian convergence would imply the CLT at time $1$.
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Bollobás–Nikiforov conjecture
The work claims a complete proof of the original Bollobás–Nikiforov conjecture for every finite non-complete simple graph. More precisely, the Lean theorem `BN.lambda1_sq_add_lambda2_sq_le` proves $$ \lambda_1(G)^2+\lambda_2(G)^2 \le 2\left(1-\frac{1}{\omega(G)}\right)|E(G)|. $$ The development proves a stronger weighted spectral inequality. If \(B\) is a real symmetric entrywise nonnegative matrix with zero diagonal, supported on the edges of \(G\), and \(F(B)\) is the sum of the squares of its two largest positive eigenvalues, then $$ F(B) \le \left(1-\frac{1}{\omega(G)}\right)\|B\|_F^2. $$ The human-readable manuscript is still preliminary and is being rewritten and polished.
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Cycle-residue stability at minimum degree five
The preprint claims an exact classification. Write $C_5(G)$ for the residues modulo five represented by cycle lengths in $G$. Let $\mathcal E_5={K_6,K_{5,5}}\cup{H_{5,n;t}:2\le t\le5<n}$, where $H_{5,n;t}$ is obtained from $K_{5,n}$ by deleting $5-t$ edges incident with one vertex in the part of size $n$. For every finite simple graph $G$ with minimum degree at least five, exactly one alternative holds: $C_5(G)=\mathbb Z_5$; or every end-block belongs to $\mathcal E_5$ and every non-end-block contains no cycle of length congruent to two modulo five. Every member of $\mathcal E_5$ has cycle-residue spectrum ${0,1,3,4}$. The proof combines structural arguments with finite computational checks. It uses the separately established Dean–5 theorem and its weak-graph strengthening as inputs. The contribution is the stronger stability classification; independent expert review remains pending.
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The proper hat-guessing number of $K_8-e$
We prove $\mathrm{HG}_P(K_8-e)=14$ using an explicit $\mathrm{PGL}(2,13)$-equivariant strategy. A compact formula fixes 990 normalized twin decisions, and a 48,510-edge residual-orbit matching induces consistent rules on 53,460 labelled clique-view orbits. Independent implementations regenerate the certificate and check all 138,378,240 proper fourteen-colourings with zero failures. Unlike the $n=5,6,7$ constructions, residual right degrees reach eight, so the explicit global matching is load-bearing. A separately labelled companion exhibits one fixed clique completion compatible with $2^{380}$ equivariant twin-rule pairs and proves 380 optimal within the stated independent whole-tail reversal model; that companion's independent review is pending. The release does not solve the general $K_n-e$ family or $K_9-e$.
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Prime Gaps at Most 186
Assuming three explicit input statements, the project proves $\mathrm{DHL}[40,2]$: every admissible $40$-tuple contains at least two primes infinitely often after translation. An explicit admissible $40$-tuple of diameter $186$ then gives $$\liminf_{n\to\infty}(p_{n+1}-p_n)\le186.$$ The Lean proof of the implication from the three inputs to the final theorem is kernel-checked. Two inputs are Kloosterman-type estimates cited to Katz/Deligne and Fouvry--Kowalski--Michel; the third consists of finitely many numerical integral and cap inequalities backed by a Python/FLINT certificate.
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A new bound for small gaps between primes: $H_1 \le 212$
$H_1 \le 212$, improving Stadlmann's $240$ of three days earlier and the $246$ of Polymath8b that had stood since 2014. The twin prime conjecture, $H_1 = 2$, is untouched. Held the record for hours at most: OpenAI's paper claiming $186$ is dated 30 August, four days before this one, though its Lean development appeared on 2 September.
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The proper hat-guessing number of $K_7-e$
We prove $\mathrm{HG}_P(K_7-e)=12$. One lower-bound proof uses two explicit block-disjoint $S(5,6,12)$ Witt designs. A second uses orbit maps from an explicitly regenerated sharply five-transitive twelve-point permutation group, combining one set-symmetric and one order-sensitive rule. Both satisfy a general coordinate-line twin-completion criterion, and Hall's theorem completes the clique strategy. The release also proves a disjoint completion-design theorem, an even-$n$ obstruction scoped to set-symmetric line-permutation twins in this sufficient framework, and a prime-admissibility theorem for the design parameters. It does not solve the general $K_n-e$ problem or $K_8-e$.
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Smale’s Mean Value Conjecture ($K=1$)
The formal proof constructs a complex polynomial $p$ such that $$ p(0)=0,\qquad p'(0)=1, $$ and for every critical point $c$ of $p$, $$ \left|\frac{p(c)}{c}\right|>1. $$ Thus at $z=0$ there is no critical point satisfying $$ \frac{|p(0)-p(c)|}{|c|}\le |p'(0)|=1, $$ which disproves Smale's conjectured universal constant $K=1$. The counterexample has very large unspecified degree and violates the bound only by a small margin, so it is consistent with Smale's original $K=4$ theorem, the known low-degree positive cases, and previous asymptotic improvements toward $1$.
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Improved maximal prime-gap lower bound
The paper proves $$ G(X)\gg \frac{\log X\,(\log_2 X)^2\,\log_4 X}{(\log_3 X)^2} $$ for all sufficiently large $X$. Its main new ingredient is a short-translates theorem: for any sufficiently small set $S\subseteq[1,H]$ with $|S|\le\delta x$, one can find a short translate making every corresponding linear form composite. Combining this with an Erdős--Rankin covering argument produces prime-free intervals of the claimed length. This directly and asymptotically improves the August 2026 GPT-5.6 Sol bound $$ G(X)\gg\frac{\log X\log_2 X}{\log_4 X} $$ by the unbounded factor $$ \frac{\log_2 X(\log_4 X)^2}{(\log_3 X)^2}. $$
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Absence of critical Bernoulli bond percolation on $\mathbb Z^d$ in every dimension $d \ge 2$
Claims $\theta(p_c) = 0$ for every $d \ge 2$, by proving Kozma-Nitzan Conjecture 3 and its reduction to critical percolation. The previously open dimensions $3 \le d \le 10$ are included. The formal statement concerns vanishing at $p_c$; continuity of $\theta$ on the whole interval is a classical consequence rather than the formal target. No claim is intended concerning site percolation or other lattices.
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Conway's Refinement Conjecture for Omnific Integers
The repository gives a Lean proof of Conway's 1976 refinement conjecture for omnific integers: whenever $$ ab=cd $$ with $a,b,c,d\in\mathbf{Oz}$, there exist $e,f,g,h\in\mathbf{Oz}$ satisfying $$ a=ef,\quad b=gh,\quad c=eg,\quad d=fh. $$ The proof is formalized twice: once using CombinatorialGames' surreal-number implementation and once with the required surreal definitions inlined over Mathlib. The development also proves stronger structural results about factorization in Hahn-series integer parts and related generalized power-series rings.
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Köthe Conjecture
GPT-6 Astra constructs a unital algebra $R=k\oplus A$ over the countable field $$ k=\overline{\mathbb F_2}, $$ with $I=A$ a nil two-sided ideal, together with a matrix $$ W\in M_2(I) $$ that is not nilpotent. The algebra $A$ is generated by three weighted backward shifts. A diagonal construction chooses the weights so that every element of $A$ is nilpotent. At the same time, a suitable polynomial combination of the shifts fixes a nonzero vector; this yields a companion-type matrix with a nonzero eigenvalue, and hence a nonnilpotent matrix whose entries lie in $I$. This formally disproves Krempa's matrix formulation of Köthe's conjecture.
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The proper hat-guessing number of $K_6-e$
We prove $\mathrm{HG}_P(K_6-e)=10$. The lower bound uses two order-sensitive twin-player rules obtained by deleting and repairing one point of an explicit sharply four-transitive eleven-point permutation group. On every coordinate line the repaired rules are derangement permutations, are pointwise unequal, and have fixed-point-free composition. Hall's theorem completes the strategy on the four clique vertices. The release also classifies all repairable orbit labels and proves an even-$n$ obstruction for set-symmetric line-permutation twin rules. It does not solve the general $K_n-e$ family.
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The Daykin–Frankl conjecture on convex subsets of the Boolean lattice
Let $P\subseteq Q_n$ be convex. Williams proves the stronger statement that for every $k\ge0$, $$ w(P\times Q_k) \ge w(Q_{n+k})\,|P|\,2^{-n}, $$ where $w$ denotes poset width. Taking $k=0$ gives $$ w(P)\ge |P|\binom{n}{\lfloor n/2\rfloor}2^{-n}, $$ which is exactly the Daykin-Frankl conjecture. The proof proceeds by induction on $n$, reducing the step to a structural lemma for a convex subset of $R\times Q_1$ and carefully recombining antichains from its two layers.
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Anari's Bethe permanent conjecture
For every nonnegative $n\times n$ matrix $A$ whose bipartite support graph has girth at least an even integer $g\ge4$, Dong and Jain prove the sharp inequality $$\operatorname{Bethe}(A)\le\operatorname{per}(A)\le2^{2n/g}\operatorname{Bethe}(A).$$ The factor $2^{2n/g}$ is optimal whenever $g\mid2n$, attained by matrices whose support graphs are disjoint unions of $g$-cycles. Thus the result confirms Anari's conjecture, recovers the sharp $2^{n/2}$ universal Anari--Rezaei bound when $g=4$, and approaches exactness as the support-graph girth tends to infinity.
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Depth-1 distinctness for pseudorandom unitaries
The paper proves that a tensor-product ensemble of independently chosen single-qubit Clifford gates is $\operatorname{negl}(n)$-distinct when the number of queries is polynomial in $n$. This disproves the authors' conjecture that negligibly distinct ensembles must necessarily be entangling. As an application, the depth-$\log n$ global Clifford/unitary-$2$-design layer used in the $PFC$ pseudorandom-unitary construction can be replaced by a single depth-1 layer of local single-qubit $2$-designs while preserving the required distinctness property.
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The proper hat-guessing number of $K_5-e$
The exact value is $\mathrm{HG}_P(K_5-e)=8$. The lower bound uses explicit legal twin-player rules over $\mathbb F_2^3$. After those rules cover 3,024 of the 8,400 proper colorings, the residual-coloring/local-view incidence graph has left degree three and right degree at most three; Hall's theorem supplies consistent guesses for the three clique players. The release also proves a general sufficient twin-completion lemma for $K_n-e$. It does not solve the full $K_n-e$ family or determine $\mathrm{HG}_P(C_5)$.
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Logarithmic basis number of graphs
Knauer proves that every finite $n$-vertex multigraph satisfies $\mathrm{bn}(G)=O(\log n)$, improving the previous general $O(\log^2 n)$ bound and matching the known $\Omega(\log n)$ order. He also proves the sharper cycle-rank bound $\mathrm{bn}(G)=O(\log\beta(G))$. Combined with a reduction of Lehner and Miraftab, this yields $\mathrm{bn}(G)=O(\log g)$ for graphs of Euler genus $g$, improving the previous $O(\log^2 g)$ bound to the optimal logarithmic order.
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A Smooth Counterexample to the Trautman Conjecture
For every neighborhood $U$ of the origin in $\mathbb{C}\times\mathbb{R}$ and every $\varepsilon>0$, Curry constructs a smooth nonnegative perturbation $\phi$, supported in $U$ with $\|\phi\|_{C^2}<\varepsilon$, such that $$ T^{0,1}_\phi=\operatorname{span}_{\mathbb C}\{L_\phi\} $$ is strongly pseudoconvex, its canonical bundle admits a nowhere-zero closed section, yet every $C^1$ CR function near the origin satisfies $dh(0)=0$. Hence the CR structure is not locally embeddable at the origin. The construction can also be globalized to $S^3$ as an arbitrarily small $C^1$ perturbation of the standard spherical CR structure.
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A Quantum Oracle Separation Between $\mathsf{QMA}(2)$ and $\mathsf{QMA}$
The authors construct a unitary oracle $U$ such that $$ \mathsf{QMA}^{U}\neq\mathsf{QMA}(2)^{U}. $$ Their black-box problem is solvable by a $\mathsf{QMA}(2)$ verifier with one oracle query and linear-size unentangled proofs, whereas any $\mathsf{QMA}$ verifier must use either exponentially many queries or an exponentially large witness. As a non-oracle consequence, they prove that for every fixed $\varepsilon,\delta\ge 0$ with $\varepsilon+\delta<1$, any $(\varepsilon,\delta)$-disentangler requires exponentially many input qubits in the number of output qubits, resolving Watrous's no-disentanglers conjecture.
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Interior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions
For every $n\geq3$, let $u\in C^\infty(B_2)$ be an admissible solution of $$ \kappa[u]\in\Gamma_2,\qquad \sigma_2(\kappa[u])=1, $$ with $$ \|u\|_{L^\infty(B_2)}+\|Du\|_{L^\infty(B_2)}\le K. $$ Then $$ \sup_{B_{1/2}}|\kappa[u]|\le C(n,K). $$ Thus an admissible graph of constant scalar curvature with bounded height and slope cannot develop unbounded interior curvature, in any dimension $n\ge3$. Prior unrestricted quantitative graphical estimates were known in dimension $3$; dimension $4$ had only an implicit estimate with extra dependence, and $n\ge5$ required additional semiconvexity assumptions.
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Common Neighbour Conjectures for Saxl Graphs
The paper disproves both the Burness–Giudici common neighbour conjecture for primitive groups of base size $2$ and its later generalisation to arbitrary base size. For every integer $B\ge2$, it constructs infinitely many primitive permutation groups of base size $B$ whose generalised Saxl graphs contain two nonadjacent vertices with no common neighbour. At base size $2$, it gives infinite counterexample families of affine, product and twisted wreath type, so the conjecture fails in three of the five O'Nan–Scott types.
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Bounded prime gaps: $H_1 \le 236$
$H_1 \le 236$, between Stadlmann's $240$ of 31 August and Axiom Math's $212$ of 3 September. Superseded within two days, which Kintali noted himself.
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Measures on Partial Orders
For the Fraïssé class $\mathfrak P$ of finite partially ordered sets, every measure $\nu$ is completely determined by the nine values $$ \nu_{p,q}=\nu(J_{p,q},*),\qquad 0\le p,q\le2. $$ The possible $3\times3$ matrices $(\nu_{p,q})$ are classified explicitly. Consequently, the reduced space of all measures is exactly the union of one plane, eight lines, and 15 isolated points, with the plane and three lines meeting at one point and no other intersections. The paper also completely classifies weak measures and supports, proves there is no regular measure on all finite posets, classifies characteristic-zero measures for the automorphism group of the universal homogeneous poset, and determines the Knop-like measures.
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Exact rank and Smith profile of affine incidence over $\mathbb Z/p^3\mathbb Z$
For $(k,n)=(3,2)$, the normalized distinct-row matrix $B_3$ equals $A(p^3,2)$ up to row and column ordering. For every prime $p$, $$ \operatorname{rank}_{\mathbb F_p}B_3= \begin{cases} 240,&p=3,\\ \dfrac{p(p+1)(3p^4+4p^3+3p-1)}{18},&p\equiv1\pmod3,\\ \dfrac{p^2(p+1)^2(3p^2+p+1)}{18},&p\equiv2\pmod3. \end{cases} $$ The paper also determines the complete $p$-primary Smith profile of $\operatorname{coker}(B_3^\mathsf T)_{(p)}$: its exponent is $p^5$, and all six multiplicities are explicit for every prime. The repeated-row Łaba–Trainor matrix has the same $\mathbb F_p$-rank, but the integral Smith claim applies only to $B_3$. Arbitrary depth, higher dimension, projective Hjelmslev incidence and the generalized-polynomial characterization remain open. Finite computations certify exceptional cases; they do not prove the uniform formulas.
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McKean Entropy-Production Conjecture
For the space-homogeneous Boltzmann equation in dimension $3$ with collision kernels $$B=\frac{1}{4\pi}|v-v_*|^\gamma,\qquad \gamma\in[0,1],$$ the paper constructs smooth, positive, radial mixtures $f_R=(1-p)M_1+pM_R$ for which $\partial_tD(f_R)>0$ for sufficiently large $R$. Thus entropy production need not decrease even for Maxwell molecules ($\gamma=0$) or hard spheres ($\gamma=1$), negatively resolving McKean's monotonicity question for these physically standard kernels.
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Four-uniform case of Morris’s conjecture: $\mathrm{FC}(4,n)=\Theta(n^2)$
It is proved that $\mathrm{FC}(4,n)=\Theta(n^2)$, settling the four-uniform case of Morris's conjecture; the three-uniform case had been settled exactly by Pulaj, so this was the first open uniformity. Explicitly, for $n\ge 20$ and $r=\lfloor n/6\rfloor$, $r(n-3r)+1\le\mathrm{FC}(4,n)\le 1+\lfloor\tfrac{160}{3}n(n-1)\rfloor$, and with the Chung-Frankl triple-star theorem $\mathrm{FC}(4,n)\le(18+o(1))n^2$. The key step is a sharp sunflower criterion: four-sets with a common two-point core and pairwise disjoint two-point petals form a Frankl-complete configuration if and only if there are at least nine petals, the positive direction by a charging inequality with weight three on the core and one on the petals. Separately, exact certificates establish $\mathrm{FC}(4,9)=16$ and refute Pulaj and Wood's lexicographic extremality conjecture. The paper also explains why higher-uniformity sunflowers cannot supply the same local forcing, so the general conjecture stays open.
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Erdős matching conjecture: the four-uniform case
For $r=4$, the manuscript claims the sharp extremal value $$ \max\left\{\binom{4s+3}{4},\binom n4-\binom{n-s}{4}\right\} $$ for every $s\geq1$ and $n\geq4s+4$. Both terms are attained by the standard clique and cover constructions. The claimed advance is an all-parameter four-uniform theorem, extending the sufficiently-large matching-number range of Hou–Hu–Liu ($s\geq6961$) while retaining and crediting their framework and eight computations. The proof combines finite exact certificates and exhaustive searches with a written combinatorial reduction and analytic propagation. Additional equality and stability conclusions have restricted domains; no classification of every extremizer or arbitrary-family stability is claimed. This is a claimed complete r=4 result and a partial result toward the general Erdős matching conjecture. Uniformity five and arbitrary uniformity are not settled by this paper.
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A Counterexample to the Stable Forking Conjecture
The paper constructs a simple theory in which forking cannot always be witnessed by a stable formula. Precisely, what fails is: in a simple theory, if $a \not\downarrow_C b$ then there is $\varphi(x,\bar b) \in \mathrm{tp}(a/Cb)$ forking over $C$ whose parameter-free form $\varphi(x,y)$ is stable. The counterexample is an infinite-dimensional vector space over the division ring of fractions of the quantum graph algebra of the random graph. Forking is characterised by an abstract independence relation, and the random graph is encoded into that relation so that it has the order property; stable formulas therefore cannot determine all forking in simple theories.
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$(\xi\alpha^n)_{n\ge1}$ is not uniformly distributed modulo one for Pisot $\alpha$ and $\xi$ in the Cantor set $C(\alpha)$
The problem is open, and the repository says so: what is proved are criteria for it and two instances, not the general case. Every compared statement that concludes Problem 10.61 does so for a quadratic setup - a real root $\alpha > 1$ of $X^2 - aX - b$ whose conjugate has modulus below one - and both instances are quadratic. The arbitrary-degree material is conditional ingredients: for the family $X^d - aX^{d-1} - 1$ the real root exceeding $a$ is shown to be Pisot for $a \ge 3$, with a conjugate-modulus bound and a numerical inequality. No compared statement carries those above degree two, because the covering criterion is proved only for quadratic setups. The covering criterion also leaves quadratic $\alpha$ with route-A exponent at least one undecided, about which nothing is claimed.
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Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication
The exact Mul4 instance is resolved: its unrestricted XOR--AND multiplicative complexity is 9. The restricted bilinear/quadratic value 9 was classical; the new theorem proves that arbitrary nonlinear Boolean reuse cannot lower it. This is a natural positive special case of the Boyar--Find question, not a solution of the general (n,m) problem. The lower bound is a conceptual structural argument. A complete Lean 4 formalization checks the unrestricted circuit semantics and the exact equality, while the development-time Python/C++ programs remain independent regression checks rather than logical premises. The corresponding unrestricted questions for five or more terms remain open. Independent statement-fidelity review remains pending.
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A complex structure on $S^2\times S^4$
The authors give an explicit construction of a complex manifold (called $X_+$) which is diffeomorphic to $S^2 \times S^4$.
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Improving Randomized Metric Distortion to 2.3282
The paper proves that there exists a randomized voting rule using only ordinal rankings with metric distortion at most $11641/5000=2.3282$. This improves the previous best upper bound of $2.5$ and closes about $44\%$ of the gap to the known asymptotic lower bound of approximately $2.1126$. It does not determine the optimal randomized metric distortion: for $m\ge4$, the exact optimum and its asymptotic limit remain open.
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Dean's conjecture for $k=5$
Claims the last open case of Dean's conjecture: every finite simple graph with minimum degree at least five contains a cycle of length divisible by five. Verified independently against Luo, Ma and Zhao (arXiv:2601.13552), whose abstract confirms the landscape - the conjecture was known for $k\in\{3,4\}$ and they proved every $k\ge6$, leaving $k=5$ open. If this proof stands, Dean's conjecture holds for all $k\ge3$. Nine finite proposition families in the bipartite and triangle-free branches are computer-assisted, with verifiers and certificates in a separately archived supplement. The reductions from arbitrary graphs to those finite state spaces are prose arguments in the paper and are not machine-checked. The claim has not been refereed, and no independent mathematician has audited the graph-theoretic core. The author describes extensive model-assisted hostile auditing of his own argument, which is worth something but is not external review, and says so plainly.
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Erdős Problem #126: prime divisors of pairwise sums
For $$ f(n)=\min_{|A|=n}\left|\left\{p\text{ prime}:p\mid a+b\text{ for some distinct }a,b\in A\right\}\right|, $$ Astra formally proves $$ \frac{f(n)}{\log n}\to\infty. $$ The repository contains substantially stronger proofs. In particular, one verified alternate resolution establishes $$ n\le 3r^2, $$ where $r$ is the number of supporting primes, yielding $$ f(n)\gg n^{1/2}. $$ Other independent resolutions give exponents $1/3$, $1/5$, and $1/8$. Thus the formal work goes well beyond the qualitative conjecture, although only the limit statement is the registered benchmark theorem.
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Erdős Problem #74: locally almost bipartite graphs of infinite chromatic number
Astra proves that there exists a divergent function $$ f:\mathbb N\to\mathbb N,\qquad f(n)\to\infty, $$ such that no graph $G$ of infinite chromatic number can satisfy $$ d_{\mathrm{bip}}(H)\le f(n) $$ for every finite $n$-vertex subgraph $H\subseteq G$, where $d_{\mathrm{bip}}(H)$ is the minimum number of edges that must be deleted to make $H$ bipartite. In fact, every included resolution proves the stronger statement that graphs satisfying the constructed local bound are 3-colorable. The formal challenge advertises only the weaker conclusion that their chromatic number must be finite.
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Non-MF groups and non-finite full group C*-algebras
If $\Gamma$ is a non-coHopfian property-(T) group and $G$ is its ascending HNN extension, then the full group algebra $C^*(G)$ is not finite. Indeed, Kazhdan projections $p_H<p_\Gamma$ become unitarily equivalent under the stable letter, which cannot occur inside a finite $C^*$-algebra. For $$ W=\left(\bigoplus_{G/\Gamma}\mathbb Z/2\mathbb Z\right)\rtimes G, $$ the paper proves more strongly that every homomorphism $$ W\to U\!\left(\prod M_{d_n}/\bigoplus M_{d_n}\right) $$ kills an explicit nonidentity element $b_\gamma$. Hence $W$ is not MF. Consequently $C_r^*(W)$ is an explicit stably finite but non-MF $C^*$-algebra.
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Erdős Problem #1: sum-distinct sets
The formal theorem proves that no universal constant $C>0$ can satisfy $$ N>C\,2^{|A|} $$ for every nonempty interval bound $N$ and every sum-distinct $A\subseteq\{1,\dots,N\}$. Equivalently, for every $\varepsilon>0$ there are arbitrarily large $n$ and sum-distinct $n$-element sets contained in $\{1,\dots,N\}$ with $$ N\le\varepsilon 2^n. $$ The proof is ineffective: it establishes the existence of arbitrarily large such $n$ but gives no explicit bound for how large $n$ must be in terms of $\varepsilon$.
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Supporting affine functionals for Entanglement of Formation
For two qubits, the paper considers $\rho=\frac12|\Phi^+\rangle\langle\Phi^+|+\frac12|01\rangle\langle01|$, where $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2$. This rank-2 state has no global supporting affine functional for Entanglement of Formation. Setting $\rho_t=(1-t)\rho+t|10\rangle\langle10|$, Wootters’ formula gives $C(\rho_t)=\frac12-\sqrt{2t}+O(t)$. Consequently, $\lim_{t\to0^+}[E_F(\rho)-E_F(\rho_t)]/t=+\infty$, so $E_F$ is not Lipschitz lower semicontinuous at $\rho$. By the paper’s criterion, no global supporting affine functional exists there. Thus the claimed universal existence fails even for two qubits, although it remains true for nondegenerate finite-dimensional states.
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Large systoles in every sufficiently large genus
The theorem improves the best lower bound valid in *every* sufficiently large genus from asymptotic constant $2/9$ to $1$. The every-genus ladder it climbs is Katz-Sabourau's $19/120$ and then Liu-Petri's $2/9$, the latter also by a random construction. Constant $1$ was already reached by Petri-Walker along a subsequence of genera, following Erdos-Sachs, so the new contribution is achieving it uniformly rather than the constant itself. The asymptotic problem stays open, and the remaining gap is wide: Brooks and Buser-Sarnak give $\limsup\ge4/3$, while the elementary area bound is $\max\mathrm{sys}(S)\le2\log(4g-2)$, asymptotically $2\log g$. So this closes much of the liminf gap and determines no optimal constant.
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The Lyons–White conjecture: rate-monotonicity of $\ell^{2m}$ distances for random walks on dihedral groups
No such pair exists: for every positive integer $m$ and every $n$, $(D_n,2m)$ is rate-monotonic (Theorem A), and more generally so is every inversion extension of a finite abelian group by an involution, a family containing the generalized dihedral, dicyclic and generalized quaternion groups (Theorem B). The picture is completed in the other direction: for every real $p\ge 1$ that is not an even integer there is an $n$ with $(D_n,p)$ not rate-monotonic (Theorem C), so the even integers are exactly the exponents for which monotonicity holds on the dihedral groups.
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Failure of Higher-Order Truth within Intuitionistic Propositional Logic
The paper proves that not every Heyting algebra can occur as the lattice of subterminal objects of an elementary topos. Specifically, the free Heyting algebra $F_2$ on two generators cannot occur. Using Bellissima’s representation $F_2\hookrightarrow\mathcal O_\uparrow(K_2)$, the authors construct an upward-closed subset $A\subseteq K_2$ with $A\notin F_2$. They show that if some elementary topos $\mathcal E$ satisfied $\operatorname{Sub}_{\mathcal E}(1)\cong F_2$, then higher-order internal logic would make $A$ definable as a global proposition, forcing $A$ to correspond to an element of $F_2$, a contradiction. Thus no elementary topos has subterminal lattice isomorphic to $F_2$, disproving the claim that every Heyting algebra can arise this way. The paper does not classify which Heyting algebras are realizable.
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An Exact All-Width Plateau for the Three-Summand Tu-Deng Count modulo $2^k-1$
Answered in full: for every $k\ge4$, a target with exactly two nonadjacent zero digits has $F_k(t)=(k+23)3^{k-4}$, independently of the distance between the zeros. Exact at every width, no error term, no hypothesis on $k$ (Theorem 1.1). This is an evaluation, not an extremal result, and the paper is explicit about the difference: the plateau value is not maximal. At $k=12$ it reads $35\cdot3^8=229{,}635$ while $F_{12}(110101101010)=293{,}499$ at five zero digits, so no global maximizer of $F_k$ is classified. The paper's other results are finite-layer and do not settle the extremal question: balancing monotonicity of $[x^{\le C}]H_t$ holds only for $C\le5$ (Theorem 1.3), and the all-mass statement is Conjecture 8.1, which the paper states outright does not follow from Theorem 1.3. The chamber where zero digits are adjacent is not addressed.
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Graffiti Conjecture 806
The repository supplies an executable verifier in exact integer arithmetic; for $n = 51$ an integer vector $x$ satisfies $x^{T}Ax > 11\,x^{T}x$, so $\lambda_1 > 11 = D$. The deficit $\lambda_1(n) - D(n)$ grows through $n = 5000$ in the repository's computations, but no asymptotic theorem proving divergence is claimed, so "false for every constant $C$" is a computed pattern, not a proved one. One anomaly, which the repository records itself: $n = 51$ lies inside the range Graffiti is said to have tested when it made the conjecture.
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Erdős Problem #571: rational exponents for bipartite Turán numbers
For every rational $\alpha$ satisfying $$ 1\le\alpha<2, $$ the Lean theorem constructs some finite $q$ and a bipartite graph $$ G:\operatorname{SimpleGraph}(\operatorname{Fin} q) $$ such that $$ \operatorname{ex}(n;G)=\Theta(n^\alpha) $$ as $n\to\infty$. This resolves the single-graph rational-exponents conjecture. Earlier work of Bukh and Conlon proved the corresponding statement only for a finite family of forbidden bipartite graphs, and subsequent work realized many large classes of individual rational exponents. Astra's theorem covers every rational $\alpha\in[1,2)$ with one forbidden graph for each exponent.
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Rapid mixing for spin systems on graphs of girth at least five
For fixed $\delta\in(0,1)$ and all sufficiently large $\Delta$ depending only on $\delta$, Glauber dynamics for proper $q$-colorings mixes rapidly on every graph of girth at least $5$ whenever $q\ge(1+\delta)\Delta$: spectral gap $\Omega_\delta(1/n)$ and $t_{\mathrm{mix}}(\varepsilon)=O_\delta(n^2\log q+n\log(1/\varepsilon))$. An analogous theorem holds for the anti-ferromagnetic Potts model at $q\ge(1+\delta)(1-\beta)\Delta$. What it does and does not improve. On girth it is a large gain: previous results near the $(1+\delta)\Delta$ threshold needed girth at least eleven (Hayes-Vigoda, extended to constant degrees by Jain-Mizgerd-Vigoda). On the mixing rate it is weaker - those give optimal $O(n\log n)$, this gives $O(n^2\log q)$. It does not touch the folklore conjecture that mixing is rapid on every graph for $q\ge\Delta+2$; Remark 7 names spanning 4-cycles as the obstruction.
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Erdős Problem #548: the Erdős–Sós conjecture
For every $n,k$ with $k+1\le n$, every simple graph $G$ on $n$ vertices satisfying $$ |E(G)|\ge \frac{k-1}{2}n+1 $$ contains every tree on $k+1$ vertices. The proof counts pairs $(\pi,j)$ where $\pi=(v_1,\ldots,v_n)$ is an ordering of the host vertices and $v_1v_j$ is an edge. There are exactly $$ 2|E(G)|(n-1)! $$ such pairs. An induction on the target tree bounds this quantity by a rooted-copy count plus $$ (k-1)n!. $$ If the target tree is absent, the rooted-copy term vanishes and one obtains $$ 2|E(G)|\le (k-1)n, $$ contradicting the density hypothesis.
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Exact rank and Smith profile of affine incidence over $\mathbb Z/p^2\mathbb Z$
For the plane at depth two, $(k,n)=(2,2)$, the paper's $B_2$ is $A(p^2,2)$ up to row and column ordering. For every prime $p$ it proves $$ \operatorname{rank}_{\mathbb F_p}B_2=\frac{p^2(p+1)^2}{4}. $$ More strongly, it determines the complete nonunit $p$-primary Smith profile: $$ \operatorname{coker}(B_2^\mathsf T)_{(p)}\cong (\mathbb Z/p\mathbb Z)^{p^3(p-1)/2}\oplus (\mathbb Z/p^2\mathbb Z)^{p(p-1)^2(p+2)/4}\oplus (\mathbb Z/p^3\mathbb Z)^{p(p-1)/2}. $$ It also proves that the canonical depth-two transfer extension is nonsplit. The theorem includes $p=5$; only the memory-intensive full-matrix companion computation for $(p,\mathrm{depth})=(5,2)$ is not run, and it is not used in the proof. Arbitrary depth, higher ambient dimension, the adjacent projective Hjelmslev problem and the separate generalized-polynomial characterization question remain open.
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Sparse domination implies convex body domination
If a bilinear form admits an $(r,s)$-sparse bound, its coordinate-wise extension to $\mathbb C^n$-valued functions admits an $(r,s)$-convex body sparse bound, for $1\le r,s<\infty$ with $\tfrac1r+\tfrac1s>1$. It holds both in a fixed dyadic lattice (Theorem 2.6, constants independent of the ambient dimension) and for arbitrary cubes (Theorem 3.4), via a randomization of the good part of the form. On scope, two things separate. What Nazarov, Petermichl, Treil and Volberg proposed was the principle for the classical sparse setting, and that is settled outright, the hypothesis holding comfortably there. The $(r,s)$ range is the authors' own generalization beyond what was asked, and the restriction bites only inside it, leaving $\tfrac1r+\tfrac1s\le1$ open. Their summary: "we provide a proof of the full general statement". It also gives sparse domination for iterated commutators directly from that of the underlying form. The dependence on $n$ is not claimed optimal.
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Fröberg’s conjecture for quintics and septics in four variables
Let $S=k[x_1,x_2,x_3,x_4]$ over any characteristic-zero field. For each $d\in\{5,7\}$ and every $r\ge1$, the paper proves that $r$ general degree-$d$ forms satisfy Fröberg’s predicted Hilbert series $$ \operatorname{HS}_{S/(F_1,\ldots,F_r)}(t) = \left[\frac{(1-t^d)^r}{(1-t)^4}\right]_+. $$ The genuinely new ranges are $6\le r\le11$ for quintics and $6\le r\le21$ for septics. These are reduced to finitely many exact rank tests of Macaulay multiplication matrices; explicit maximal minors are nonzero mod $2$, so the required ranks hold in characteristic zero. Thus the conjecture is completely settled for the equal-degree four-variable slices $d=5$ and $d=7$, but not in general.
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A tilted residue-class construction for long prime-free intervals
Claims $G(T)\gg\log T\log_2 T/\log_4 T$ against FGKMT's $\log T\log_2 T\log_4 T/\log_3 T$, a gain of $\log_3 T/(\log_4 T)^2$, together with $Y(X)\gg X\log X/\log_3 X$ for the covering problem behind it. Ben Green resists calling it incremental: "The sieving procedure is different to the Erdos-Rankin one which underpinned all bounds on the problem since 1938, and it wins $\log_3 N$ over that procedure. The [FGKMT] paper also wins a $\log_3 N$. These wins are essentially independent of one another so one now wins basically $(\log_3 N)^2$ over Rankin's 1938 bound." He adds that the new sieve "by itself could have claimed the Erdos 10000 dollars for this question". What is new is narrow: only the intermediate sieve is replaced, its hard cutoff smoothed into a probabilistic tilt; the hypergraph covering theorem and Maynard weight come from FGKMT. Readers on the thread note that several later sections reproduce FGKMT with no new content, and Green calls the exposition "horrific".
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Equivalence of generic stability notions for Keisler measures
Let $T$ be a complete first-order theory in discrete or continuous logic, let $M\prec\mathcal{U}$, and let $\mu\in\mathfrak{M}_{x}(\mathcal{U})$ be Borel-definable over $M$. The paper proves that the following three conditions are equivalent: $(i)$ $\mu$ is a frequency interpretation measure (fim) over $M$; $(ii)$ $\mu$ is definable over $M$ and its canonical random extension $r_{\mu}$ is generically stable over $M^{\Omega}$; $(iii)$ $\mu$ is self-averaging over $M$. The new work proves the reverse implications $(iii)\Rightarrow(ii)\Rightarrow(i)$ and extends the characterization to continuous logic. The authors therefore make the equivalent conditions into a definitive definition of generic stability for Keisler measures. The paper also proves a further characterization in terms of an order-property condition and derives consequences for closure under Morley products.
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Entropy of Bernoulli Measures Conditioned on Affine Subspaces and a Problem of Ancheta-Massey
For a Bernoulli$(p)$ source with $0<p<\frac12$, the paper proves that the best lossy compression achievable by a linear encoder is exactly the simple time-sharing strategy that losslessly compresses a fraction of the bits and estimates the rest as zero. Equivalently, every full-row-rank $H\in\mathbb F_2^{k\times n}$ satisfies $$\frac{k}{n}\ge h(p)\left(1-\frac{D(H)}p\right).$$ This resolves Massey's question affirmatively for all $p<\frac12$; Ancheta had already proved the $p=\frac12$ case.
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The $(3,4,\infty)$ Modular Family of 2-Tori as a Complex Structure on $S^6$
Claims an explicit compact complex threefold $X$, fibred over $\mathbb{P}^1$ by complex 2-tori via period functions on the $(3,4,\infty)$ orbifold, degenerating to a del Pezzo-of-degree-six fibre (identified opposite sides of its hexagon) at one point and to bielliptic multiple fibres of multiplicities 3 and 4 at the other two. Argues $X$ is simply connected with $H_*(X;\mathbb{Z})=H_*(S^6;\mathbb{Z})$, hence diffeomorphic to $S^6$, with algebraic dimension exactly 1. This directly contradicts [CDP20, Cor. 2.3], a published (and once-corrected) theorem; the paper states this and argues where the two accounts diverge, rather than overlooking it. Posted hours before this entry, with no independent check, no formalisation, and no refutation yet in any venue found. Filed as a candidate specifically because none of that has happened, not because a problem with the argument has been found.
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Nevanlinna’s half-plane omitted-values problem
The paper constructs a real meromorphic function $F$ on $\mathbb{C}$ satisfying $F^{-1}(\{0,1,\infty\})\subset\mathbb{R}$, with each of the three fibers $F^{-1}(0)$, $F^{-1}(1)$, and $F^{-1}(\infty)$ infinite, such that for every $a\in\widehat{\mathbb{C}}\setminus\{0,1,\infty\}$, the $a$-point divisor in each of the upper and lower half-planes fails the Blaschke condition. Consequently, $F$ is not of bounded type in either half-plane. This gives a negative answer to Nevanlinna’s century-old question asking whether an entire-plane meromorphic function that omits three distinct values in a half-plane must be of bounded type there. By postcomposition with Möbius transformations, the three exceptional values $\{0,1,\infty\}$ may be replaced by any prescribed triple of distinct values in $\widehat{\mathbb{C}}$. By affine change of variables, the construction applies to any Euclidean half-plane.
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A Rank-$31$ Record for an Elliptic Curve over $\mathbb{Q}$
Two tiers, and only the first is the record. Rank $\ge 31$ is unconditional: 31 explicit points, independence asserted via the leaderboard's stated general practice of exact 2-descent (not reproduced here - see the verification note). Rank exactly 31 is conditional on GRH and BSD, per the submitters' commentary, in the same style as the sibling record's Bober-bound argument; no numeric derivation has been published for this curve specifically. The entry is a partial result because the open question - whether ranks are unbounded at all - remains unanswered by any single record.
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Haglund's Zero-Trajectory Conjecture for the First Riemann Xi Approximant
Proves Haglund's Conjecture 4 for $k=1$: every non-real first-quadrant zero of $\Phi_1+t\Phi_2$ is simple with strictly decreasing imaginary part, no branch escapes forward, and every finite-multiplicity real collision stays real afterwards. The cases $k\ge2$ remain open. Two readings worth separating: Conjecture 4 asserts the monotone descent alone, so the no-escape and stays-real statements are this paper's own additions rather than Haglund's text, and they are the stronger part of the theorem. The descent itself, part (i), is the part that rests on the unavailable interval-arithmetic certificate.
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Transcendence in the affine case of Erdős Problem 270
The manuscript claims $C_{a,b}$ is transcendental for every $a\ge1$ and $b\ge1-a$, settling the positive integer-valued affine subclass of Erdős Problem 270. Two pieces of context matter. Problem 270 as Erdős and Graham posed it, for every $f(n)\to\infty$, was already answered no by Crmarić and Kovač in 2025: for any $\alpha>0$ some such $f$ makes the series sum to $\alpha$. What survives is the non-decreasing case, and the affine family sits inside it. Separately, the checkable parts here were already known - the short irrationality proof for $C_{1,0}$ is Crmarić and Kovač's, posted by Kovač on the Erdős Problems forum in July 2026 and credited in the repository, and base-case transcendence follows a 2023 MathOverflow argument. The new content is the extension to the whole affine family, which is the part with neither formalization nor review.
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Problem 3 of Dubickas (2006): Is $\sqrt{3} \in \mathcal{Z}$?
Answers Problem 3 and generalizes it: the classification $\sqrt{m} \in \mathcal{S} \iff m = 2$ covers every square root, and a further theorem replaces parity by divisibility by any $p \ge 2$. Note the scope of the machine-checking, which is narrower than the paper: the author states that the case $m = 3$ is what is verified in Lean, and the repository flags the thickness computation of section 4.1 and all of section 8 as not formalized.
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Bounded mass property for compact complex manifolds
Disproved on the Hopf threefold $X=(\mathbb{C}^3\setminus\{0\})/\langle z\mapsto e^{-1}z\rangle$: Xia and Zhang construct a smooth Hermitian form $\omega$ and smooth functions $\varphi_j$ with $\omega+dd^c\varphi_j>0$ whose Monge-Ampère masses tend to infinity, so the universal bounded mass property fails already in complex dimension three. The construction uses the Hopf threefold's elliptic fibration, an exact mass identity reducing excess Monge-Ampère mass to a fibrewise Dirichlet energy, and heat-kernel regularizations of Green functions that make that energy diverge while preserving positivity. What the negative answer removes is load-bearing rather than incidental: finiteness of this mass is the starting point for the theory of volumes of Bott-Chern classes, and it enters as a standing hypothesis in recent Hermitian pluripotential theory.
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The stable commutator length of a relator is not a one-relator group invariant
A negative answer to Heuer and Löh's question: the isomorphism type of a one-relator group $\langle S \mid r\rangle$ does not determine $\mathrm{scl}_S(r)$. The witnesses are $r=\mathtt{aabABabABBAbaabABBAb}$ and $r'=\mathtt{aabABabABabABBAbaBAb}$, both length 20 in $F_2'$, with $\langle a,b \mid r\rangle\cong\langle a,b\mid r'\rangle$ but $\mathrm{scl}(r)=1$ against $\mathrm{scl}(r')=1/2$. The mechanism is what made the search finite: $\mathrm{scl}$ is an $\operatorname{Aut}(F_2)$-invariant, so a pair in *different* orbits whose one-relator groups happen to be isomorphic has its two scl values unconstrained by each other. The search was for that configuration among words of length at most 20. Scope: it settles the question as posed and nothing wider. It does not say which invariants do determine scl, and this is a single pair rather than a construction giving arbitrary gaps.
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Record Rank for an Elliptic Curve over $\mathbb{Q}$
Two tiers, and only the first is the record. Rank $\ge 30$ is unconditional, being thirty explicit independent points. Rank exactly 30 is conditional: applying Bober's bound (arXiv:1112.1503) with $\Delta = 4.25$ gives an analytic rank of at most 31, and the root number is $+1$ so the rank is even, hence 30 - but that argument assumes GRH, and equating analytic rank with rank assumes BSD. The entry is a partial result because the open question is whether ranks are unbounded at all, which no single record answers. Superseded three days later by this project's own rank $\ge 31$ record (see the related entry); left unedited otherwise as a record of what was known at the time.
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Smooth Random Fast Dynamo on the Three-Torus
Constructed an explicit class of smooth random, time-dependent incompressible velocity fields on T^3, obtained by alternating smooth shear flows with iid random phases on finite time blocks. For every fixed sufficiently small resistivity, the magnetic field has an almost-sure exponential growth rate at least 1/2, together with a time-uniform lower bound whose random prefactor has a resistivity-uniform inverse-moment estimate. This is one variant case of Arnold's 1994 fast-dynamo problem, not the problem itself: Arnold asks for a field that is smooth, autonomous and deterministic all at once, and this one keeps the smoothness while giving up the other two. The sibling entry on this site relaxes the opposite hypothesis, keeping an autonomous deterministic field at Lipschitz regularity. Neither settles Arnold's problem as posed, which remains open.
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Whether Marton's Inner Bound Achieves the Broadcast Channel Capacity Region
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Fractional Colouring Conjecture for Triply Efficient Pauli Shadow Tomography
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The 4-color Rado number of x+y+c=z: general case
The claim is $R(c) = 40c+41$ for every $c \ge 2$, reduced to three finite facts: the base value $R(2) = 121$, and the unsatisfiability of a 321-position and a 521-position spoke template. The reduction is Lean-checked and holds for every $D \ge 1$; the two unsatisfiability results carry DRAT proofs. This completes the partial entry for the same conjecture, which proved it for roughly two thirds of integers via a scaling lemma; that lemma is now one of three legs, covering the branch where $d$ is divisible by 3. The supporting results are worth more than the headline for anyone deciding whether to believe it: the paper also shows every band relaxation is satisfiable, which is why previous attempts stalled, and that the affine method alone is exactly sharp and can never finish.
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The $C^\infty$ Carathéodory Conjecture on Umbilic Points
Only the smooth case falls. Hamburger's real-analytic theorem is untouched, and the counterexample is explicitly a $C^\infty$ object, so the conjecture's classical analytic form remains true. The gap between the two is the whole content of the result.
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The DeLaViña–Waller conjecture on the Wiener index
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A Conjecture on Triple Counts for the Kasami APN Function
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Erdős Problem #501: infinite independent sets for families of small outer measure
Independent of ZFC, which is why this entry is the first to carry that result rather than proved or disproved. Both directions are formalized: Hechler's 1972 construction gives a model where the answer is no, and adding $\mathfrak{c}^+$ random reals over a model of CH gives one where it is yes. The credit is shared and mostly human. Newelski, Pawlikowski and Seredynski settled the problem's second question in 1987, and it is formalized here without the boundedness hypothesis. Hechler supplied one direction in 1972. Sungchul Lee derived a positive answer from a real-valued measurable cardinal, assisted by GPT-5.5 Pro, and Nat Sothanaphan observed that the two halves together give independence. What Glazer and Sol added is the removal of the large cardinal. erdosproblems.com still lists #501 as open at the time of writing.
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The Kára–Pór–Wood Big-Line-Big-Clique Conjecture: Four Collinear Points or a Six-Clique
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Counting Linear Extensions Below the $2^n$ Barrier
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Separation Between the Ordinary and Strong Kreiss Constants
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Stein’s dimension-free weak-(1,1) Riesz transform problem
The theorem is the vector-valued endpoint bound $\|Rf\|_{L^{1,\infty}} \le 2\|f\|_{L^1}$ for $R = (R_1,\ldots,R_n)$, so the constant 2 also serves each component $R_j$ uniformly in the dimension; the best previously known component bound grew like $c\log n$. The mechanism is a decomposition theorem stated as Theorem 1.2: for every nonnegative $f \in L^1 \cap L^2$ and every $\lambda > 0$, write $f = \mu + (-\Delta)^{\alpha/2}u$ with $\mu \le \lambda$ and $u$ in the fractional Sobolev space $H^\alpha$, obtained from an obstacle problem for the fractional Laplacian together with a Lewy-Stampacchia type estimate on an unbounded domain. That replaces the Calderon-Zygmund decomposition, whose cube geometry is where the dimensional loss enters.
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Fourth-moment conjectures for Rademacher sums
Four results, and the first is partly a refutation. Jakimiuk conjectured $c_p = \mu_p - 1$ is optimal for every $p \ge 3$; the paper proves that for $p \ge 4$ and gives a counterexample for every $2 < p < 4$, so the conjecture is false as posed and the corrected range is $p \ge 4$. The witness is the two-coordinate vector $S_2 = (\varepsilon_1+\varepsilon_2)/\sqrt2$. The Baranski-Murawski-Nayar-Oleszkiewicz flat-point conjecture is proved outright, in the stronger form that $x \mapsto \|x+S_n\|_p/\|x+S_n\|_4$ is strictly decreasing on $[1,\infty)$ for every real $p \ge 5$; that range is the one they conjectured, so nothing is left over. Jakimiuk's second conjecture, dimension-free quadratic stability at $p = 3$, is proved with an explicit constant, though the optimal constant there is only bracketed and stays open. The paper also records the exact fixed-$q$ moment and Laplace-transform envelopes, from which coefficient-sensitive tail bounds follow.
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Colombo’s difference-power determinant conjecture
For pairwise distinct real numbers $x_1,\ldots,x_N$ and an integer $D\ge1$, Ma proves the complete classification $$ \det\bigl[(x_j-x_i)^D\bigr]_{i,j=1}^{N}\ne0 \quad\Longleftrightarrow\quad D\ge N-1\ \text{and}\ \bigl(N\text{ is even or }D\text{ is even}\bigr). $$ This fully resolves Colombo’s original conjecture for even $N$. The new part is the even-size, odd-exponent branch, strengthened to the strict Pfaffian sign theorem $$ (-1)^{\binom m2}\operatorname{Pf} \bigl[(x_j-x_i)^{2r+1}\bigr]_{i,j=1}^{2m}>0 \qquad(r\ge m-1). $$ The even-exponent branch follows from the classical work of Dyn–Goodman–Micchelli. A concurrent independent proof of the odd branch by Kun Li, Li Tie, Peng Wang and Zihan Liu is linked below.
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The Matrix Multiplication Exponent
A record, not a resolution, and a small one by design. The bound moves from $2.371339$ to $2.371177$, about $1.6 \times 10^{-4}$, and the authors describe it as a small step. Whether $\omega = 2$ is untouched, and nothing here suggests the laser method can reach it. The interesting claim is methodological rather than numerical. The bottleneck in this line of work is a hard optimization problem, and the paper reports progress by reformulating that problem and then improving the optimizer, with AlphaEvolve doing the final refinement. That is a different kind of contribution from a new mathematical identity, and it is why the entry is filed as computation.
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A 32-leaf tree requiring six coordinates for an isometric $\ell_\infty$ embedding
Disproved. An explicit 32-leaf tree has least isometric $\ell_\infty$-dimension six, not the conjectured five, and keeps dimension six under every assignment of positive edge lengths, so the weighted sharp-threshold conjecture falls too. Every tree with at most 31 leaves does attain $\lceil \log_2 t \rceil$, so 32 is the first failure; Brigham et al. had checked through 21. The proof is finite and exact: six explicit orientation masks cover all leaf pairs, and a recurrence on rooted branches rules out any five-orientation cover.
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Talagrand’s convolution conjecture
The claim is the exact conjectured decay: $\psi_{\mu_a}(u) \le C_a/\sqrt{\log u}$ for every $u > 1$ and every $n$, with $C_a$ dimension-free - concretely $\lesssim \kappa_a^2(\log\frac{\kappa_a}{\kappa_a-1})^{1/2}$ where $\kappa_a = (1+a)/(1-a)$. What is new is one step in a three-paper chain rather than a proof from scratch, and the paper is explicit about it. Chen's reverse-heat and Boolean-bridge framework and Xiang-Zhang's localized terminal-discrepancy method are taken as given; the addition is a power coupling that splits each reverse edge ratio into two geometric powers, producing a switched exponential weight that restores the exact reverse jump rate of the perturbed coordinate. Because the frozen exponent then has a fixed numerator, no growing stopping buffer is needed and the $\log\log u$ factor disappears. That last $\log\log$ is what stood between the previous work and Talagrand's statement.
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Composites Among $[\xi 7^n]$ and Right-Truncatable Primes in Base 7
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Neuen-Grohe Problem: Isomorphism of Tournaments with Bounded VC Dimension
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Score-Determined Induced Tournament Statistics: an All-Orders Classification
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Lorist-Schwenninger Remark 2 positivity question
Nothing in the paper's proof is affected. At the witness, inequality (4) holds with slack +27.0, inequality (5) holds with equality (f is inner), and the theorem itself holds with slack 2 - kappa = +0.80. Only the shortcut Remark 2 floats is refuted.
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The 4-Color Rado Number of $x+y+c=z$: $R(c)=40c+41$ Whenever $c+1$ Is Divisible by 3, 4, 5 or 7
Twenty-eight individual exact values, each proved by SAT certificate (coloring at n-1, UNSAT at n). The synthesis theorem covers every c >= 2 whose c+1 is divisible by 3, 4, 5, or 7 (~66% of integers). The prime-reduction corollary shows the full conjecture (R(c)=40c+41 for all c >= 2) is equivalent to checking primes p >= 89; all primes through 83 are settled. What stays open: the conjecture at c=88 (p=89) and every larger c whose c+1 has all prime factors >= 89. The scaling lemma's attribution is hedged relative to Malo 2000 (full text not accessed). No Lean formalization; the SAT certificates and dual-encoder architecture are the verification tier.
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Sivaraman's Perfect-Divisibility Characterization Question
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Convergence of Three-Block ADMM with Identity Third Block
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Exact order-three ambiguity of the Einstein-Maxwell-dilaton coupling $a^2$ in metric jets, and its fourth-order collapse
Finite-jet theorems about compiled truncated EMD equation certificates: the exact shear-orbit fiber classification of the complete first seed channels, an explicit collision family with one metric three-jet realized by an actual cubic metric germ (genuine Frechet Ricci value and first derivative), the compiled impossibility theorem, and the fourth-order recovery with equality fiber $a=\pm b$. Not settled here: promotion to analytic EMD solution germs (separate written argument pending specialist audit, in the parent repository), chart covariance beyond the fixed presentation, density of the active locus, degenerate branches, and any sufficiency of $a^2=3$ for a Kaluza uplift.
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Pavez-Signe's Length-Control Question for Spanning Subdivisions
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The Cone Theorem for Effective Fourfold Pairs in Characteristic $p > 5$
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The Quartic Hessian Conjecture in Dimension Four
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Tarizadeh's Conjecture on the Maximality of Purely-Prime Ideals
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Reading's Problem 9.3: Order Dimension Versus Rank for Simplicial Arrangements
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Kusner's Conjecture on Equilateral Sets in $\ell_p^n$
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$SOP_2 = SOP_3$
The new content is $SOP_2 \Rightarrow SOP_3$; the converse implication was known from the start. Dzamonja and Shelah asked whether either implication in $SOP_3 \Rightarrow SOP_2 \Rightarrow SOP_1$ reverses: Mutchnik answered the second ($SOP_1 = SOP_2$), and this answers the first, collapsing the bottom of the hierarchy to $SOP_1 = SOP_2 = SOP_3$. The $SOP_n$ hierarchy for $n \ge 3$ remains, as does everything above it.
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Universal volume growth bounds from positive intermediate curvature
Gromov's 1986 question drew three independent proofs within about 24 hours, two with AI in the loop. Ge's non-AI proof (heat-kernel Fisher metric, Nash entropy) came first, 13 August. Antonelli's proof here (14 August, GPT-5.6 Sol) takes a different route, Hodge obstruction and rank improvement, and its headline addition is the general family: for every $0\le m\le n-2$, nonnegative Ricci plus positive $(m{+}1)$-intermediate curvature forces at most $m$-dimensional growth - linear growth under biRicci curvature at $m=1$, plus a noncollapsed Urysohn-width bound; the author states this extension is his own, not the model's. Kong and Zhu's proof (also 14 August; GPT-5.6 Sol Ultra and Codex, "essential ideas were generated by AI") proves the same case plus a related codimension-one conjecture via a heat-transport deficit, close to Ge's method by its own account, produced before Ge's went public and not derived from it. Headline axes stay Antonelli's, for the broader scope.
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The Foregger–Sinkhorn Tie-Point Conjecture
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Courtade's conjecture on volumes of Minkowski sums with the ball
Courtade conjectured that for convex bodies $B,C\subset\mathbb R^n$, $$ (|B||C|)^{1/n} + (|B_2^n||B_2^n+B+C|)^{1/n} \le |B_2^n+B|^{1/n}|B_2^n+C|^{1/n}. $$ The paper proves that this inequality is false in every dimension $n\ge3$. In dimension $3$, it gives an explicit geometric counterexample using two orthogonal double bodies of revolution. It then strengthens this by constructing zonoid counterexamples in every dimension $n\ge3$, including a six-generator zonotope in dimension $3$ and smooth perturbative constructions in higher dimensions.
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Every PPT channel has finite entanglement-breaking index
Establishes that every PPT channel is eventually entanglement-breaking (finite EB index), in full generality, and bounds the index by 3 uniformly in dimension for a family strictly containing the 2-superpositive maps. The PPT-squared conjecture itself - index at most 2 - remains open; the paper presents its results as strong evidence toward the cubed version.
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Nineteen exact reflective and dihedral Ramsey numbers from Damnjanovic-Dordevic's tables
Nineteen individual exact values, each decided by SAT certificate: unsatisfiable at the claimed $n$, witnessed satisfiable at $n-1$. They close cells in DD26's Tables 3-13 but settle no infinite family - the sibling entries do that for the $K$ column. The three overlap cells are $R_{dih}(P_4^{alt},K_6)=16$, $R_{dih}(P_3^{alt},K_9)=17$ and $R_{dih}(P_9^{alt},K_3)=17$, each an instance of a sibling theorem; the remaining sixteen stand on their own certificates. Four further cells passed the producing solver but await their final referee leg and are not claimed. Open: every other cell of DD26's tables, all cyclic-action and online-Ramsey cells.
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1.17353 planar lower bound and exact local envelopes for cost-preserving single-source unsplittable flow
Part III of a series, and the first unconditional positive results in it. Settled exactly: the local envelope ladder $E(2)=1$, $E(3)=9/8$ and $E(4)=(299-41\sqrt{41})/32=1.13974707\ldots$, which recasts the earlier record constants as exact envelopes of the general theory rather than isolated instances, plus exact constants for four classes - out-trees 0, two-layer hubs 1, outerplanar two-exit interval spines 1 (sharp), series-parallel at most 1. Improved but not settled: the planar lower bound rises to $1.17353531974518$ against the known ceiling 2, and every exact-two-path instance whose rows touch at most three terminals satisfies $C\le2$, the first unconditional constant for an unbounded class. The universal question is untouched - it reduces here to a single factor-two merger statement with certified wall $K^*\ge2.5652\ldots$, twice the refined general lower bound $1.28260069\ldots$.
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The Radchenko–Viazovska question on Fourier interpolation
For every $0\le \beta\le \tfrac12$, Bondarenko and Seip construct a nonzero real-valued continuous even function $$ f_\beta\in L^1(\mathbb R)\cap L^2(\mathbb R) $$ such that $$ \widehat f_\beta=f_\beta $$ and $$ f_\beta\!\left(\frac{\sqrt n}{[\log(e+n)]^\beta}\right)=0 \qquad(n\ge0). $$ They normalize the construction by requiring $f_\beta(1/2)=1$, so the function is genuinely nontrivial. For $\beta=0$, this gives a nonzero Fourier-invariant function vanishing at every $\sqrt n$, which answers the question negatively: their interpolation formula for even Schwartz functions does not extend merely under the assumption that the interpolation series is well-defined and absolutely convergent. More strongly, for every $0<\beta\le1/2$ the zero set can be logarithmically denser than the square-root sequence. These sampling points, together with $1/2$, form a universal interpolating sequence for a suitable reproducing-kernel Hilbert space of Fourier-invariant Hermite expansions.
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Banach's isometric conjecture
The paper proves the previously unresolved odd-dimensional real cases of Banach's isometric conjecture. Combined with Gromov's earlier theorem for even dimensions and previous results, this completes the conjecture for real Banach spaces.
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Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three
The authors construct smooth $A_m$ and smooth solutions $u_m$ in $B_2\subset\mathbb R^3$ with one fixed ellipticity bound $$ I\leq A_m\leq2^{81}I, $$ common boundary data and $\|u_m\|_{L^\infty(B_2)}\leq1$, but $$ \|Du_m\|_{L^1(B_1)}\to\infty. $$ Thus no interior $W^{1,1}$ estimate can depend only on dimension and ellipticity. Consequently, no such $W^{1,p}$ estimate exists for any $p\geq1$. They further obtain a uniformly convergent limit $u$ with measurable uniformly elliptic coefficient matrix $A$, where $$ u\notin BV_{\rm loc}(B_1). $$ The construction even rules out coefficient-independent weak-$L^1$ gradient estimates.
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Dihedral Ramsey numbers of the alternating a-path versus K_b, for every a >= 4: 1 + (a-1)(b-1)
The dihedral case only, for every $a \ge 4$ and $b \ge 1$; the substance is the upper bound, which the source paper's own computations could not reach. Together with the sibling a = 3 entry this proves Conjecture 4.9's claim $1+(a-1)(b-1)$ for all $a \ge 3$; the conjecture's trivial a = 1, 2 cases are unaddressed by either entry, and the cyclic analogue $R_{cyc}(P_a^{alt}, K_b)$ for $a \ge 4$ remains open. The engine is a self-contained inequality of independent interest: for any graph on a linearly ordered vertex set, the alternating-path reach statistics satisfy $\sum_m [P(m)+Q(m)] \ge 2|E(G)|$, from which the theorem falls out by averaging and a pivot decomposition.
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Word-length spectral triples as compact quantum metric spaces
The first explicit counterexample rather than a first suspicion: the abstract is clear that the failure was widely expected and that what was missing was a witness. It gives an infinite family, one for each d >= 2, all Lamplighter groups over free groups.
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Dihedral and cyclic Ramsey numbers of the alternating 3-path
The a = 3 slice is settled outright. The parent conjecture's dihedral side has since been resolved for every a >= 4 as well (see the related entry), so Conjecture 4.9's claim 1 + (a-1)(b-1) now stands proved for all a >= 3; the trivial a = 1, 2 cases and the cyclic analogue for a >= 4 remain formally unaddressed.
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Treglown's equitable acyclic colouring conjecture
The honest reading, which the paper gives itself: the reduction is implicit in earlier work of Aboulker, Oijid, Petit, Rocton and Simon, and the model itself surfaced that reference when asked about originality. So this establishes the conjecture and supplies a polynomial-time algorithm, while the underlying idea is a rediscovery rather than a first. It is a striking record of a model producing an argument and then correcting the novelty claim made for it.
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Phelps–Rodriguez Conjecture
Phelps-Rodriguez implies Sendov, so this entry records the stronger of the pair; the companion Sendov entry records the weaker statement and Mazur's original formalization, which proved Sendov but never stated the equality classification. The exceptional family is genuinely attained rather than an artefact of the proof: for p = z^n - 1 and a = 1 the only critical point is the origin, at distance exactly 1. Both conjectures fell out of one argument, and the strict form was not the announced target - Tao's digestion of Mazur's proof turned out to establish it, which is how a 1972 conjecture was resolved as a by-product of resolving a 1959 one.
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The $(2,1)$-Gapped Consecutive-Ones Property Problem is NP-complete
The manuscript claims a polynomial-time reduction from 3-SAT proving $(2,1)$-C1P NP-hard; together with membership in NP, this establishes NP-completeness and closes the sole unresolved $(k,\delta)$ case from the earlier classification. It also implies NP-completeness of the equivalent completion problem. The proof package further shows that, within its specific nested-prefix/internal-local gadget architecture, no 3-OR gadget exists with at most six internal columns; at seven columns at least three local rows are required, and all optimal three-row gadgets form one symmetry class. These optimality claims are architecture-specific and are not needed for the NP-completeness result. Independent verification remains pending.
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Hadamard Matrix of Order 668
Explicit construction of a Hadamard matrix of order 668, the smallest previously unresolved order, verified exactly by this site from the announcement plus its decoder reply. The same post encodes matrices for all twelve previously-open admissible orders below 2000 (668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964), and this site verified every one of them. The entry records the order-668 existence question, which this fully resolves; the general Hadamard conjecture - existence for ALL admissible orders - remains open, with the smallest unknown order now 2004 or beyond.
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Gamow liquid-drop minimizer conjecture
The complete fixed-volume picture, closing a gap that partial results had narrowed from both ends without meeting: balls uniquely minimize for every volume up to V_* = 3.51..., and above it no minimizer exists at all. Before this the best minimality range was V <= 1 (Chodosh-Ruohoniemi, 2025) and the best nonexistence bound V >= 7.5 (Schulz, posted two days earlier), so the open middle ran from 1 to 7.5. Frank-Nam had already proved existence up to V_*, and the new proof uses it; the fresh content is uniqueness of the ball across the whole range and nonexistence immediately above the threshold. A corollary settles the minimal binding energy question of Frank-Lieb: the infimum of E(Omega)/|Omega| is 3(9pi/5)^(1/3), attained exactly at balls of volume 5/2. The mechanism is a capacitary estimate that sharpens an Agostiniani-Mazzieri monotonicity formula using Gauss-Bonnet, an improvement the authors note applies only to this particular weight and only in three dimensions.
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Strong Graph Reconstruction Conjecture
Disproves the Bowler-Brown-Fenner bound of 2*floor((n-1)/3) on common cards between nonisomorphic graphs: an explicit connected 78-vertex pair shares at least 51 cards against the predicted 50, and for every even r >= 4 there are families with overlap fraction asymptotically at least 1 - 1/r, so the attainable fraction approaches the full deck. The Kelly-Ulam reconstruction conjecture itself is untouched.
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Seymour's Second Neighborhood Conjecture
A dense case, not the conjecture: it remains open in general. The concrete gain is on the size of any counterexample - combined with the known minimum-outdegree results, this raises the best known lower bound on the order of a counterexample from 16 to 17, and to 19 conditional on the 2026 preprint of Sadhukhan, Sandeep and Sen. The novelty against the earlier dense-case work is that no structure is prescribed on the missing edges.
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Lower Bounds for Stepsize-Based Acceleration of Gradient Descent
Recorded as partial: the bound is Omega(T^-1.9319) against an achievable O(T^-1.2716), so it rules out reaching the optimal rate without pinning down the true one.
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Albertson–Berman Induced Forest Conjecture
The ratio 15/31 is not claimed to be optimal, and the paper makes no claim that 31 vertices is the smallest possible counterexample. The construction produces separating triangles by design, so it says nothing about the 4-connected case.
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The Second Computational Chomp Challenge of Ekhad and Zeilberger
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The Planar Steklov Analogue of Kac's Question
Strict convexity and real-analytic boundaries are what make this sharp: the classical Gordon-Webb-Wolpert drums are non-convex polygons, so the obvious escape routes are closed off.
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Conjectures 2a and 2b of Kauers and Zeilberger
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Predicting Diagonalizability of a Mean Matrix
The general principle is the interesting part: every semialgebraic property of a bounded fixed-dimensional mean parameter is eventually almost surely predictable. Against merely integrable matrix laws it fails from dimension two.
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The First Rigorous Solid Standard Young Tableaux Challenge
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The Five-Dimensional Geode Challenge of Amdeberhan, Kauers and Zeilberger
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Spahn and Zeilberger's Third Challenge: Holonomicity of the Restricted Permutation Counts
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Lions' Maximal Regularity Problem at the Half-Holder Endpoint
Tensorisation and parabolic rescaling carry the one-dimensional example to real symmetric isotropic counterexamples on R^d and on every bounded domain, in every dimension.
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Makeev's conjecture on universal cover
Makeev conjectured it to be true for all dimensions. This result disproves it for dimensions 4 and 5. Dimensions 6 and above remain open.
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Bounded Oracle Error in Nonconvex Stochastic Optimization
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The Hyperkahler Period-Index Conjecture
The counterexamples come with divisibility bounds on the Hodge-theoretic index, at 2-torsion and 5-torsion, on very general hyperkahler fourfolds.
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The Imbalance Conjecture
The key step is a lower bound on the truncated imbalance sum, which yields every Erdos-Gallai inequality for the sorted imbalance list; a parity computation finishes it.
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The Ellipsoid Fitting Conjecture
Closes both gaps left open by Bandeira and Maillard: exact fitting, and removal of the operator-norm constraint. The threshold turns out to be governed by the statistical dimension d(d+1)/4 of the PSD cone.
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The Proportion of Zeta Zeros on the Critical Line
An unconditional record, not a resolution: the Riemann hypothesis is untouched, and Anthropic states it does not expect these techniques to lead to a proof of it. The paper is explicit that these are lower bounds only - the remaining third of the zeros are not shown to be off the line, merely not reached by the certificate. What it does settle is a question that was posed. Goldston and Suriajaya had reduced Montgomery's conditional $\tfrac23$ to a single obstruction, the termwise positivity that fails for zeros off the line, and asked what would follow without it. Theorem A replaces that positivity with a rank-trace inequality on a finite compression of Weil's Hermitian form, with Sylvester's law of inertia handling off-line pairs; reading the negative index of truncations as a count of off-line pairs is Bombieri's device. The paper also proves the bound sharp for this route: improving on $\tfrac23$ this way would need pair-correlation information beyond Fourier support 1.
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Teschner's Bondage-Number Conjecture
Teschner's universal bound b(G) <= (3/2)Delta(G) is false: the 18-vertex cubic bipartite graph has b(G) = 5 against a bound of 4.5. What survives is the restricted statement Teschner actually proved, that the bound holds for graphs of domination number at most three, and Gagarin and Zverovich's 2013 result that it holds for almost all graphs. The counterexample does not suggest a replacement bound, and the correct general upper bound for b(G) in terms of Delta(G) remains open.
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pth-Order Oracle Complexity for Monotone Variational Inequalities
Improves every prior result for p >= 2 and matches the classical extragradient method at p = 1.
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The Kim-Roush Conjecture on the Maximum of per(I-A) in Odd Order
Kim and Roush did not claim uniqueness; the classification of equality cases is new alongside the conjecture itself.
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The Planar Berenstein Conjecture
The domain has dihedral symmetry of order 26 and is neither a disc nor centrally symmetric, and its eigenfunction changes sign - which is why an additional sign assumption rescues the statement.
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Finite-Copy Distillability of NPT States in the DiVincenzo Family
Partial deliberately: the NPT bound entanglement problem itself is untouched. What falls is the conjecture about the canonical family, and the paper is explicit that a substantial neighbouring region remains unresolved while another is known two-copy undistillable.
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Matrix-Tree Obstruction for Half-Collinear Graviton Vertices
Answers the obstruction rather than the whole question: it says exactly when the Matrix-Tree Theorem can be applied and classifies the chambers, and gives a compact five-graviton formula outside the decay region. Simplifying the general solution, which is what the source paper left to future work, remains open.
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Talagrand's critical Sherrington-Kirkpatrick overlap conjecture
Talagrand's Conjecture 11.7.5 is resolved affirmatively for the critical Ising Sherrington-Kirkpatrick model: $N^{2/3}\mathbb{E}\langle R_{1,2}^2\rangle$ converges to a positive finite constant. The paper proves substantially more, showing that the entire quenched distribution of $N^{1/3}R_{1,2}$ converges to an explicit random probability measure defined from the reflected $\mathrm{Airy}_1$ point process. The same limiting distribution and second-moment constant are obtained for the spherical SK model. This does not resolve the broader low-temperature overlap structure of the SK model.
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Whether Amenability of an Algebra Depends on the Ground Field
The footnote is worth reading on its own: an author saying in print that the model earned coauthorship and that policy is what prevents it.
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Polynomial-Time MIMO Detection at the ML Threshold
An average-case claim about the Gaussian model, not a contradiction of the worst-case NP-hardness of integer least squares. If it holds, no computational-statistical gap separates polynomial-time detection from exhaustive maximum likelihood at first order in this model.
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Complete rational classification of fifth-order autocorrelation ambiguities on $U_{30}$
Agulnick and Busick-Warner exhibited a family of fifth-order ambiguities on the exact unit support $U_{30}$ and conjectured that it was not a complete classification because it did not use the full field $\mathbb Q(\zeta_{30})$. This work proves the complete classification. The larger field enlarges the common amplitude $\alpha$, while every relative ambiguity remains a norm-one parameter in $\mathbb Q(\zeta_6)$. The result is a specialization of a theorem for every exact unit support $U_{6m}$. The entry does not claim a complete parametrization for arbitrary supports: on the 255 automorphism-stable supports treated elsewhere in the paper, the broader result is a closing-degree classification. It does not treat noisy data or noncyclic groups, and it makes no novelty, priority, or firstness claim.
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A counterexample to the Howland-Kato problem for positive commutators
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Umans-Wang Arithmetic-Progression Divisor Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Anstee-Sali Conjecture on Forbidden Configurations
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Chen-Lawrencenko Conjectures on Cyclic Colorations
One conjecture each way: the second proved, the first disproved. Two further Chen-Lawrencenko conjectures remain open and are flagged as such in the paper.
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Log-Submodularity of Zonoid Volume
The paper also proves the conjecture in the unimodular case and characterizes equality there, so the boundary between true and false is drawn rather than just crossed.
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Facial Distance Patterns in Planar Graphs
Three immediate consequences follow for undirected unweighted planar graphs: better compression of the Okamura-Seymour metric, less space for constant-time exact distance oracles, and a faster distributed algorithm.
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The Inverse Generator Problem on Hilbert Spaces
One finite-dimensional construction settles several related questions. Besides the inverse generator problem, it gives a generator whose Cayley transforms satisfy the ordinary Kreiss resolvent condition but are neither strongly Kreiss bounded nor power bounded, and it shows the Crank-Nicolson scheme is unstable in operator norm both over long times at fixed step size and under mesh refinement at fixed final time. Version 2 adds Theorem 1.4, whose part (i) solves Question 6.1 of Chalmoukis, Tsikalas and Yakubovich; that question is tracked as its own entry.
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Absolutely Maximally Entangled States in Five Open Cases
Five existence statements, all by explicit construction: $\mathrm{AME}(12,5)$, $\mathrm{AME}(17,11)$, $\mathrm{AME}(18,11)$, $\mathrm{AME}(17,13)$ and $\mathrm{AME}(18,13)$. The $[12,6,7]_{25}$ code came from a direct search with no symmetry imposed; its automorphism group turned out to have a regular $\mathbb{Z}_3^2$ coordinate orbit, and imposing that translation symmetry on two nine-coordinate orbits collapses an unrestricted $9 \times 9$ block to a nine-element kernel, which is what made the length-eighteen searches feasible. The symmetry is search scaffolding, not part of the proof: the three printed matrices and the two checks suffice on their own. The paper is explicit that the searches were not exhaustive, so it proves existence and classifies nothing - equivalence and classification for these parameters stay open. The length-twelve code is also shown not to be monomially equivalent to a generalized Reed-Solomon code.
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The Huang-Jiang-Oblomkov Conjecture at a = 3
Proves the a = 3 layer; the conjecture is layered in a and remains open for larger a.
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Balanced EF1 and fPO Allocations
The paper records how the collaboration actually went: the authors first aimed at a counterexample showing EF1 and PO incompatible under matroid constraints, and when the model surfaced fundamental difficulties with that plan they redirected toward proving the positive result instead. The paper also extends the technique to category constraints and leaves a pseudopolynomial-time algorithm open.
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The Gardner Transition in the Ising pure $p$-spin glass
For the Ising pure $p$-spin glass where $p \geq 3$, it was predicted by Gardner that there exists critical inverse temperatures $0<\beta_1^p<\beta_2^p <\infty$ such that: (1) When $0<\beta\leq \beta_1^p$, the Parisi measure is replica symmetric (RS); (2) When $\beta_1^p<\beta \leq \beta_2^p$, the Parisi measure is one-step replica symmetry breaking (1-RSB); (3) When $\beta>\beta_p^2$, the Parisi measure is full replica symmetry breaking (FRSB). The earlier work by the author solved Part (1) and partially solved Part (2) when $\beta$ is sufficiently close to $\beta_p^1$, while this work solves Part (2) and Part (3) entirely.
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Central limit theorem for the random assignment problem
Claims the central limit theorem for the bipartite random assignment problem with bounded uniform costs: $\sqrt{n}\,(C_n-\zeta(2)) \Rightarrow \mathcal{N}(0,\,4\zeta(2)-4\zeta(3))$. The limiting constant is not itself new - Wästlund had computed exactly $4\zeta(2)-4\zeta(3)$ for the mean-one exponential model, and Malatesta, Parisi and Sicuro derived the non-bipartite analogue by replicas - but neither is a proof for the bounded bipartite model, and Wästlund's zero-free-disk conjecture, which would imply a Gaussian limit, remains open. So the value was expected; the proof of convergence to it is what is claimed. The route is an exact change of variables on an optimal dual potential, after which the residual dependence is a single directed-tree factor whose matrix-tree determinant becomes triangular once the potentials are ordered.
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Tuza's Conjecture for Maximum Degree at Most Seven
Settles a class, not the conjecture: Tuza's conjecture remains open in general.
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Dittert's Conjecture in Dimension Five
Dimension 5 only; public AI-generated candidate with no independent specialist review.
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Babai's Minimal Cayley Graph Problem
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Approximate Counting for Spin Systems on Planar Graphs
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The Generalized Busemann-Petty Problem in Dimensions 2 and 3
Settles subspace dimensions 2 and 3; the generalized problem stays open for larger m.
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Sendov's Conjecture
Sendov's conjecture is resolved for every degree n >= 2, closing a gap that had stood since 1959: degrees up to eight were settled piecemeal between 1969 and 1999, and Tao's 2020 result covered all sufficiently large degrees without ever specifying the threshold, leaving the middle range open. Tao's digestion establishes the stronger interior form of the statement, which resolves the Phelps-Rodriguez conjecture in full generality as a consequence - a second conjecture falling out of the same argument, and one that likely merits its own entry. Two independent Lean developments now exist: Mazur's original at roughly 90,000 lines and Tao's streamlined version at about 15,000.
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The Approximation Ratio for Boolean Max-k-CSP
Removes the constant factor from the previous best guarantee; whether $k/2^k$ is optimal is not settled here.
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The Generalized Vanishing Conjecture
For n <= 2, we proof that the Generalized Vanishing Conjecture holds, and we have a failure at dimension 3 (including a counter-example). This improves the original failure at dimension 5.
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The Divisible Rank-Three Case of the Kajitani–Ueno–Miyano Conjecture
Proves the divisible rank-three case: rank exactly three and ground-set size a multiple of three, with no restriction to simple, paving, representable or graphic matroids. The part not previously in the literature is the non-simple sub-case, since McGuinness had settled all paving matroids and a rank-three matroid is paving exactly when it has no parallel pairs. Combined with the coprime-case theorem of van den Heuvel and Thomasse, this covers every finite uniformly dense rank-three matroid. The conjecture remains open in higher rank.
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Lower Bounds for Multivariate Independence Polynomials
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Gabor Frames of Totally Positive Functions
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Tight Bound for Online Vertex Cover under Edge Arrivals
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HRT Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Rivière’s regularity question for critical $n$-Laplace systems with antisymmetric potentials
For every $n>2$, the paper constructs a bounded map $U\in W^{1,n}(B^n,\mathbb{R}^{n+2})$, smooth on $B^n\setminus\{0\}$ but discontinuous at the origin, together with an antisymmetric potential $$ \Omega\in L^n(B^n,so(n+2)\otimes\mathbb{R}^n) $$ such that $$ -\mathrm{Div}\bigl(|\nabla U|^{n-2}\nabla U\bigr) = \Omega\cdot|\nabla U|^{n-2}\nabla U \qquad\text{in }D'(B^n). $$ Moreover, the potential satisfies the sharper Lorentz-space regularity $$ \Omega\in\bigcap_{q>2}L^{(n,q)}\setminus L^{(n,2)}, $$ and, for any prescribed $1<p<\infty$, the construction can be arranged so that $$ \nabla U\in L^{(n,p)} \qquad\text{but}\qquad \nabla U\notin L^{(n,1)}. $$ This gives a negative answer to Rivière’s general regularity question for critical $n$-Laplace systems with antisymmetric $L^n$ potentials: antisymmetry and critical $L^n$ control alone do not imply continuity.
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Completeness of Maximally Entangled States for Pseudo-Telepathy
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Nazarov's Conjecture on Truncations for Fractional Laplacians
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Schiffer's Conjecture and the Pompeiu Problem
Also refutes the 1929 Pompeiu problem: Corollary 1.2 applies Williams' classical 1976 equivalence to the same constructed domains, so this is one construction settling both, not two separate results.
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Problem MAIS-O60: Single-Neuron Fourier Alignment
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The Middle Stair of Parallel Chip-Firing
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Signed Depth Relevance of subDL
subDL satisfies the signed depth relevance property, answering an open question posed by Øgaard (2026). More precisely, every valid inference in subDL contains a propositional variable that occurs in both the premises and conclusion with matching sign and at matching implicational depth.
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Perfectly Complete Quantum Key Agreement from One-Way Functions
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Finite Generation for klt Generalized Pairs
The paper records how stuck this was: the first author had discussed the question with Caucher Birkar, Osamu Fujino, Christopher D. Hacon, Junpeng Jiao, Vladimir Lazic and Lingyao Xie, and writes that despite a general feeling that a negative answer was likely, no precise counterexample could be found. The authors also note that, given the limitations of generative AI, they may have missed related literature and welcome corrections.
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Uniform Székelyhidi conjectures for complex Hessian equations on projective manifolds
Chen, Nie and Xu prove a Nakai–Moishezon-type numerical criterion for a broad class of complex Hessian-type equations on compact projective manifolds. In particular, Corollary 1.3 gives a uniform version of Székelyhidi’s conjecture for complex Hessian quotient equations, while Corollary 1.5 proves the uniform version, formulated by Murakami, for complex k-Hessian equations. However, the results assume projectivity and a uniform numerical condition, whereas Székelyhidi’s original conjecture is formulated for arbitrary compact Kähler manifolds. Thus this should not be recorded as a full solution of the unrestricted original conjecture.
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Real-Rootedness of Ehrhart h*-Polynomials at Large Width
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Exact SSUF scenario-count ladder on the four-terminal planar gadget
Fixed four-terminal planar DAG; positive route-cost differences for m≥2, with non-attained suprema; signed/zero only for m=1, whose attainment is unstated. No topology-wide/unrestricted-planar claim.
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The Bandelt-Dress Quartet Distance Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Period-Index Conjecture
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Asymptotically attaining the Moore bound
Settles two conjectures. Theorem 1.1 proves Bollobas's asymptotic degree-diameter conjecture, in the stronger liminf form rather than the conjectured limsup. Corollary 1.2 proves Conjecture 3 of Cambie, Cames van Batenburg, de Joannis de Verclos and Kang on the edge variant, again in the stronger liminf form, and is tight for bipartite graphs.
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The Axiotis-Sviridenko Condition-Number Conjecture
Conditional on the randomized exact-volume Small-Set Expansion Hypothesis, and stated for least-squares objectives rather than sparse convex optimization in general.
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Autonomous Lipschitz Fast Dynamo on the Three-Torus
One variant case of Arnold's 1994 fast-dynamo problem, not the problem itself. Arnold asks for a single velocity field on T^3 that is smooth, divergence-free, autonomous and deterministic, fixed independently of the magnetic diffusivity, and that grows the magnetic field exponentially at every small enough diffusivity. The field constructed here is all of that except smooth: it is Lipschitz, not C^1. The sibling entry on this site relaxes the opposite hypothesis, keeping a smooth field but making it random and time-dependent. Neither settles Arnold's problem as posed, which remains open.
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Kalton-Peck Space and its Hyperplanes
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Written on the Wall II, Graph Conjecture 144
The Formal Conjectures pull request flipping this from open to solved is still open rather than merged, so the canonical repository has not yet accepted it.
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Sharpness of Denjoy's Theorem
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Graffiti's Residue Problem for Common-Divisor Graphs
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The Simonovits Product Conjecture
one construction disproves both the product conjecture and its weak form
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The Mihail-Vazirani Conjecture
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Approximating Two-Terminal Network Reliability
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Connes' Rigidity Conjecture for ICC Property (T) Groups
Settles the ICC property (T) case. The paper records that the result was obtained independently of and concurrently with work by OpenAI.
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Hoa's Conjecture on Maximal Non-Hamiltonian Graphs
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Written on the Wall II, Graph Conjecture 322
The formalization proves the statement under the weaker hypothesis n >= 2; the pull request marking the conjecture solved is open, not merged
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The Equality Case of Ehrhart's Volume Conjecture
the equality case; the inequality was settled separately and is tracked on its own entry
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Upper Bounds for High-Dimensional Sphere Packing
upper bounds reach the Cohn-Elkies threshold; the true asymptotic density remains open
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Erdős Problem #183: Multicolor Triangle Ramsey
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Two-Variable Factorial Conjecture
Claimed in a self-published research draft; a standalone by-product is the transcendence of the integral of exp(q) between distinct algebraic endpoints for nonconstant algebraic q
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Erdős Problem #146: Degeneracy Conjecture
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Ehrhart's Volume Conjecture
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Connes' Rigidity Conjecture
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Erdős Problem #180: Compactness Conjecture
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Lower Bounds for the Permanent in Arithmetic Circuits
an n^4/log n formula lower bound; VP vs VNP remains wide open
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Gromov and Weiss's Question on Sofic Groups
answered no: non-sofic groups exist
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Quantum Parallel Repetition
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Upper Bounds for Binary and Spherical Codes
exponential improvement over the 1977 MRRW bounds; the exact rate-distance trade-off remains open
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Polynomial-Factor Hardness for the Closest Vector Problem
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Finitude of the Fibers of Complementary Bell Numbers
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Ehrhart Positivity of Schubitopes
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Local limits along squares and prime values of digital functions
New theorems, not a formalisation of known results. In base ten the bare existence of a square with any admissible digit sum was recorded in the recreational literature (Murthy–Ashbacher 2005); the theorem here is the quantitative all-base version — roots coprime to b (excluding their trailing-zero trick), exp(c_b√q) representations of controlled size — and that statement is what is machine-checked, with effective but non-optimal constants. The same paper proves a local limit theorem for g(n²) for arbitrary digit weights in every base including binary, and sieves the values: level of distribution 1/2, P₃ and P₂ values, an unconditional Mertens-type prime-value law, a joint Erdős–Kac theorem. Caveats: the two shrinking-frequency estimates behind the local theorem enter the Lean development only as transcribed definitions; the sieve and asymptotic results are unformalised; Corollary 1.9 is formalised in bases 2 and 3 only; the analogous theory along squares of primes remains open.
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Minimum reverse-complement-invariant decycling sets
Theorem 1.1 claims, for every finite nonempty alphabet with a fixed-point-free complement and every $k\ge2$, $$ \tau_{\rm RC}(q,k)=N_q(k)\quad\Longleftrightarrow\quad k\text{ is even}. $$ For odd $k\ge3$, Theorem 3.1 gives $\tau_{\rm RC}(q,k)\ge N_q(k)+q$, hence an excess of at least two for binary words and four for DNA. For even orders, Sections 6-8 provide a weighted construction and a direct spectral selector; integer weights permit exact polynomial-bit membership testing for a single word. The case $k=2$ is handled separately. This completely classifies attainability of the ordinary minimum and partially answers the external symmetric-minimum question. It does not determine the general exact odd optimum, prove a packing-covering equality, give quantitative residual-path/window bounds, or establish biological performance. The classical ordinary minimum itself is prior work.
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An Erdős–Kac law for base-$b$ palindromes and for reversed primes
New theorems, not a formalisation of previously known results. For every base $b\ge2$ the Erdős–Kac law is established for the $\lambda$-digit base-$b$ palindromes and for the base-$b$ reversals of the $\lambda$-digit primes, for $\omega$ and $\Omega$ and for $\omega_S,\Omega_S$ with any regular set $S$ of primes; with normal order $\log\log n$ on both families, and, for $\omega$, all moments of order up to $\tfrac12(\log\log b^{\lambda})^{1/3}$ uniformly in the order. For reversed primes it also holds with the leading digit prescribed. Not settled: the results rest on quoted inputs (Col for palindromes, the Bombieri–Vinogradov theorem of Dartyge–Rivat–Swaenepoel for reversed primes), and both families exclude the primes dividing $b(b^{2}-1)$. No rate of convergence is obtained. The question of Banks–Shparlinski on the *largest* value of $\omega$ on palindromes is untouched: the trivial bound $\ll\lambda/\log\lambda$ and their $\lambda^{o(1)}$ remain far apart.
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Prime values of digital functions along the primes
For every integer-valued strongly $b$-additive $g$ with $\gcd(g(1),\dots,g(b-1))=1$ and digit mean $\mu_g\ge0$: $g(p)$ is prime for infinitely many primes $p$. For $\mu_g>0$, $\sum 1/p$ over $p<X$ with $g(p)$ prime is $(d_g/\varphi(d_g))\log_3X + C_{g,1} + O(1/\log\log X)$, likewise for the first $j$ iterates. Also $\#\{p\le x: g(p)\text{ prime}\}\ll\pi(x)/\log\log x$, of that exact order on a large set of $x$, and $\omega(g(p))$ has normal order $\log_3 p$. What is new and what is not. For the digit sum $g=s_b$, Harman (2012) already proved both the infinitude and a Mertens formula; the new information there is the remainder tending to a limit rather than being $O(1)$, and the iterated version for $g=s$ is, in the paper's words, "contained, in a stronger and quantitative form, in Harman". The new content is the generalization to every such $g$, which delivers the running example $g=S$, the sum of squared decimal digits, and so the infinitude of OEIS A052034.
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Improved integrality of Donaldson–Thomas invariants of loop quivers (GKS Conjecture 1.3 for twist knots)
The sharp valuation bounds and optimal $\gamma(m)$ are proved for ALL loop quivers $m\ge2$, hence for the extremal BPS invariants of all twist knots (both rows, matching every twist-knot entry of GKS Table 1). Scope limits: the $m=3$/figure-eight divisibility $2n_r/r\in\mathbb Z$ was previously proved by Basor–Conrey–Morrison (arXiv:1703.00990), whose per-$r$ $2$-adic characterization for $m=3$ is finer than the uniform bound; the torus-knot case of GKS Conj. 1.3 (multi-vertex quivers, $\gamma$ growing with the knot) remains open and is not claimed; the general-knot conjecture remains open. The Lean formalization covers the reduction to the classical Kazandzidis congruences, not those congruences themselves.
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Han's Conjecture
The counterexample is an ordinary algebra concentrated in degree zero, with the strongest possible vanishing in positive degrees, so the phenomenon needs no grading or differential. Liu and Shen had already disproved the differential-graded version in December 2025 without any AI involvement; the classical case is the one that fell with a model in the loop.
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1.28249... Lower Bound and Partial Upper Bounds for Cost-Preserving Single-Source Unsplittable Flows
Record lower bound only. The sub-2 ceiling is the codimension-two case and does not bound the record ladder (k=17 has complement mass 11). 4/3 and 2 are conjectures; the proved gap is [1.28249, 2].
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Pólya's Conjecture for Neumann Balls in Dimensions Three and Higher
The ball case. For arbitrary domains Pólya's conjecture remains open; this continues the authors' programme after the planar disk, circular sectors, and the Dirichlet case in arbitrary dimensions.
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Online Shadow Tomography Matching the Classical Bounds
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Record Lower Bounds for the Shannon Capacity of Odd Cycles
record lower bounds for seven odd cycles; the exact capacities remain open for every odd cycle beyond C5
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The Integer Domination Root Conjecture
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Sharp Hardness for MAX-3-CUT
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The Han-Xiong Integer Trace Conjecture
Settles the conjecture for a large family and reduces the rest to unit fractions; the general unit-fraction case remains open.
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Signature of Connected Line Graphs
no constant-bound repair of the conjecture is possible
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The Tu-Deng Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Dimer Constant of the Cubic Lattice
a record upper bound; the exact constant remains unknown
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Maxwell's Three-Charge Equilibrium Bound
the sharp bound for three charges; the general Maxwell bound was separately disproved in July 2026
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Graffiti Conjecture 6
Infinite family of counterexamples; mathematical argument internally checked, with external verification and novelty review pending.
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Written on the Wall II, Graph Conjecture 217
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Sombor-Energy Conjecture
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Non-Covering Congruence Systems over Fq[x]
leading asymptotic determined up to a bounded q-dependent term
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Local Cohomology Modules With Nonclosed Support
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The Modified Lyons–Sidorova Conjecture for Bounded-Variation Paths
This is the MODIFIED conjecture, not the original Lyons-Sidorova one, and it is proved for continuous bounded-variation paths. Prior work had a line-image result under the stronger assumption of infinite radius on every subinterval; this removes that assumption.
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Erdős Problem #106
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Lami-Regula Conjecture on Entanglement Irreversibility
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Written on the Wall II, Conjecture 284
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Fuglede's Conjecture on Square-Free Cyclic Groups With Rapidly Growing Primes
A partial result. Fuglede's conjecture remains open for finite cyclic groups generally; this settles an infinite family and, for the spectral-to-tiling direction, only under rapid growth of the prime factors.
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Optimal Partial Plank Coverings
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Huneke-Wiegand Conjecture
Verified by author of the conjecture
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Bosonic Quantum Communication Beyond the Thermal Threshold
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Twelve Common Flex Lines in a General Pencil of Cubics
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Completeness of Fixed-Order Atom-Centered Descriptors
The key ingredients were already in the literature, decades old and in distant fields; what was missing was anyone connecting them to this question.
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The Lukic Conjecture
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Optimal Exponent Relating Sumsets and Difference Sets
New arXiv preprint with an author-provided Lean formalization; not yet peer-reviewed.
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Sharp Bounds on the Ground State Energy of the SYK Model
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Mixed Partition Functions and Exponentially Bounded Edge-Connection Rank
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Maxwell's Conjecture on Point-Charge Equilibria
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Positivity on Deligne–Mumford Stacks Without the Torsion-Free Hypothesis
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Tournaments Determined by Three and Five Voters
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Improved Bounds for Distinct Multiples in Intervals
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Boucksom's Local Analytic Bertini Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Martinsson-Steiner Conjecture on Fractional Chromatic Number
girth >= 5 case; the triangle-free case remains open
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Stanley's Problem 4 on Differential Posets
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Written on the Wall II, Graph Conjecture 109
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
-
Bondal-Polishchuk Conjecture for a Smooth Projective Variety
first counterexample of the form D^b(X) for X smooth projective
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Existence of Bipartite Bound Information
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The 4^k Barrier for the k-Distinct Language
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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General Position for Planar Line Arrangements and $HD_2(p,3)$
improved bounds; the exact Hadwiger-Debrunner numbers remain open
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Monotone Slowdown of Turbulent Flame Speed under Curvature
within the curvature G-equation model, in three dimensions
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Kemeny Rank Aggregation for Three Voters
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Probabilistic Automatic Complexity Is At Most Three
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Stanley's Rankwise Lower-Bound Conjecture for Differential Posets
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Crouzeix's Conjecture
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 for 2-dilations - produced with ChatGPT 5.6 Pro exploring proof strategies. The entry's headline axes record Jin's proof; the earlier version of this entry recorded Lorist-Schwenninger's as primary while Jin's AI provenance was still unknown.
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KLS Conjecture for Quadratic Forms
with constant 2; also improves the global KLS bound to $O(\log^{1/4} n)$
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Unimodality of Kazhdan-Lusztig Polynomials of Matroids
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Infinitely Many Components in Auslander–Reiten Quivers over Perfect Fields
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Hall-Ho Heat Flow Conjecture for Random Matrices
Proves general cases of the conjecture rather than every case.
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Bellman's Lost-in-a-Forest Problem for the Golden Gnomon
Bellman's problem for general regions remains open
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Chafai-Dadoun-Youssef Questions on Logarithmic Energy Monotonicity
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Kinetic Trace Estimates in the Gaussian Model
the named open question is answered negatively; the paper's positive theory goes further
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Sharp Continuity Bound for Quantum Conditional Entropy
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Kuperberg's Six-Cylinder Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Feige's Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Brezis's Open Problem 5.6 on Universal Fourier Summation
negative below the 1/3 threshold; the endpoint case is still open
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Explicit Presentation of the 2-adic Absolute Galois Group
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Covering Number $C(12,6,4)$
closes a one-block gap in the covering tables; the analogous next case is not reachable by this method
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Powers of the Vandermonde Determinant Are Eventually Non-SNP
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Zero Forcing versus Independence in Subcubic Graphs
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Carrasco's Conjecture on the O'Shea-Zames-Falb Test
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Carlson's Associated-Prime Depth Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Djament's Problem on Locally Noetherian Grothendieck Categories
the locally noetherian case; whether such a category can fail to admit a projective generator is left open as Problem 1.3
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Erdős Problem #684
VibeMathed reports this item as retracted. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Černý Conjecture for One-Cluster Automata
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #768
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Graveyard Problem for Dissipative Barrier Truncations
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Rota's Unimodality Conjecture for Matroid Flats
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Strichartz-Tse $L^p$-Integrability on the Sierpinski Gasket
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #131
The new content is the upper bound; the matching N^(1/5) construction is prior work of Erdős and Csaba. erdosproblems.com has not accepted the claim
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Minimum Sparsity of S-Decoding Polynomials
conditional on a plausible number-theoretic conjecture; unconditional through s = 15
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Depth of the in-tree of $s$ under $q \mapsto q s q^{-1}$ on $n$-cycles
Opus 4.8 constructed a branch of the stated depth, giving a lower bound, and believed it had a matching upper bound; that proof was wrong and the statement stayed a conjecture. FABLE 5 later proved it. In the author's summary of the method: "The proof turns conjugation, near $s$, into base-$p$ arithmetic." A cycle near the fixed point splits into a coarse base permutation and a vector of carries in $\mathbb{Z}/p$, $D$ acts on the carries by the carrying of ordinary base-$p$ addition, and the depth comes out as the nilpotency length of a shift difference - exactly that for odd $p$, one less for $p = 2$. The single missing carry that odd primes absorb and $2$ cannot is what produces the two-branch answer. A companion survey paper covers the rest of the graph: the other periodic orbits, congruences on basin sizes, and a cyclic-sieving count. The depth theorem is the substantive part.
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A Five-Variable Counterexample to the Hessian Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Ghasemi-Kopparty Problem on Sparse $S$-Decoding Polynomials
the PIR consequence is conditional on a number-theoretic conjecture implied by either the generalized repunit conjecture or Schinzel's hypothesis H, and is unconditional for s <= 15
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Reiner's Conjecture on Higher Bruhat Orders in Corank 3
corank 3; the general conjecture remains open, corank 2 being McConville's case
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Monical's Saturated Newton Polytope Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Two-Copy Distillability of Werner States
Two-copy distillable if and only if already one-copy distillable. Four independent papers settled this within five days of each other, and no single one of them is the account of record. Fu, Gao and Park posted first on 23 July (arXiv:2607.21367), followed by Song and Chen on 26 July (arXiv:2607.23416), then on 27 July both Fraser, Huber, Pozsgay and Vona (arXiv:2607.24309) and Bharti, Gajjala and Haug (arXiv:2607.24479). The headline axes here follow the first posting, which is a filing convention and not a claim about who solved it. Pozsgay has stated publicly that his group had their AI-assisted proof before the first paper appeared. This site cannot verify a private completion date, so that is recorded as his account rather than as a finding. All four are linked below.
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Araujo-Piga-Schacht Question on Tight Hamilton Cycles
the question is answered negatively and the correct threshold is determined
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Commutator Relators Do Not Force Hopficity, Residual Finiteness or Automaticity
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Petersen Coloring Conjecture
This preprint was not the first disproof. A 68-vertex counterexample was posted to X on 23 July 2026 by @NeuralReformist, credited to GPT-5.6 Sol Ultra, sixteen days earlier. This site decoded that sparse6 string and checked it independently: 68 vertices, 102 edges, simple, cubic, connected, bridgeless, girth five, and no Petersen coloring under the same encoder used for the 112-vertex graph. Whether the two are independent is unknown - the preprint does not cite the post. The headline axes still record the preprint, the only complete writeup with certificates. The implication runs one way: the Petersen coloring conjecture implies Berge-Fulkerson and the 5-cycle-double-cover conjecture, so refuting it leaves both of those open. The paper does not claim 112 is minimum, and it supplies a second, nonisomorphic $D_3$-symmetric 112-vertex counterexample. Combined with a theorem of Ma, Mattiolo, Steffen and Wolf, one counterexample yields infinitely many.
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Erdős Problem #593
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Four-Terminal Planar Case of the Dinitz-Garg-Goemans Cost Conjecture
restricted planar four-terminal case
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Stanley's Claw-Free Schur-Positivity Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1177
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Unconditional One-Bit Unclonable Encryption
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Uniqueness of the Faber–Krahn Position of Convex Bodies
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Liu-Morin Extension Conjecture
Han's conjecture itself remains open; this settles the Liu-Morin extension case
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Dual Sequential Fat-Shattering and Tight Threshold Extraction
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Target-Free Clique Conjecture for Threshold-Linear Networks
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Separable Jacobian Conjecture in Characteristic 2
the counterexample to the original Jacobian conjecture does not specialize to characteristic 2; this is a modification that does
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WOWII Conjecture 72: Two induced trees pin down tree($ G $)
The evenly-divided reading of WOWII Conjecture 72 holds: $ \lceil(A + L)/3\rceil \le t, $ where $ t = $ tree($ G $) (order of a largest induced tree), $ A = $ average eccentricity and $ L = $ maximum neighbourhood independence number. A stronger reading that divides only $ L $ by three is false. The argument rests on two elementary observations (a diametral path is chordless and therefore induces a tree on $ D+1 $ vertices; a maximum independent set in a neighbourhood induces a star on $ L+1 $ vertices). The original conjecture’s precise wording is not yet pinned in a public formal repository, so statement fidelity remains to be audited.
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Spectral Edge of the Quartic SYK Model
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Dinitz-Garg-Goemans Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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An Explicit Counterexample to the Rank-Two Poisson Conjecture
The paper constructs explicit $R,T,D,S\in\mathbb Q[x,q,p,z]$ defining a Poisson endomorphism of the canonical rank-two Poisson algebra $\mathcal P_2$ that is not an automorphism. Its associated polynomial map preserves the canonical symplectic form, has Jacobian determinant $1$, and has an explicit fiber of exactly three points. Thus $\mathrm{PC}(2)$ is false, and stabilization gives failure of $\mathrm{PC}(n)$ for every $n\ge2$. An appendix further constructs an explicit nonautomorphic endomorphism of the fourth Weyl algebra, proving $\mathrm{DC}(4)$ false.
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Stability Radius of the Lamplighter Group
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Polynomial-Time Low-Degree Conjecture
the construction is probabilistic; an explicit uniformly samplable example remains open
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Improved Bound for Colorings Without Symmetrically Colored k-APs
Improves the exponent rather than answering the asked question: whether an $N^{o(1)}$ colouring exists remains open.
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The Minimal Distance Problem
also disproves a separate finite-field conjecture of Hunter, Pohoata, Verstraete and Zhang
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Written on the Wall II, Graph Conjecture 103
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Levit–Mandrescu Unimodality Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Graffiti Conjecture 284
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Nikolov-Ullman Pure-DP Query Release Conjecture
information-theoretic; a polynomial-time implementation remains open
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$e$-Log-Concavity of Chromatic Quasisymmetric Functions
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Dittert's Conjecture in Dimension 16
Partial: this settles n=16 only. Combined with Pang's n>=17 the conjecture holds for all n>=16, leaving the small cases open.
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Strong Log-Concavity of Chernoff's Density
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Norine's Antipodal-Colouring Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2)
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Pak-Slonim Conjecture on Stretched Schubert Structure Constants
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Johnson-Freyd-Ostrik-Yu Question on Categorical Cocycles
answered for 3-cocycles, inside a broader partial classification of twisted Deligne products
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Written on the Wall II, Graph Conjecture 143
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Boots-Royle/Cao-Vince Conjecture on Planar Spectral Radius
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #469
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The mod 4 Kawauchi Conjecture
the mod 4 form; the general conjecture is false by Ermotti, Hongler and Weber
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Batyrev's Stringy Hodge Number Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Kourovka Problem 21.150 - Rank Inequality for p-Group Extensions
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Universal Multiplicative FDR Bound for Benjamini-Hochberg
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Kourovka Problem 20.125 - Noninjective Rota-Baxter Operator
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Kourovka Problem 21.24 - Cograph Power Graphs Are Chordal
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Virtual Surjection Conjecture for Discrete Groups
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Kourovka Problem 3.46 - Maximal Locally Soluble Normal Subgroups
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Kourovka Problem 21.8 - Horizontal Class Transpositions
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Optimality of Greedy for Single-Pass Semi-Streaming Matching
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Full-RSB in the Sherrington–Kirkpatrick spin glass
For the Sherrington-Kirkpatrick model with no external field, at zero temperature $\beta=\infty$, the paper proves that the zero-temperature Parisi minimizer is absolutely continuous with a smooth density and has support $[0,1)$ - full replica symmetry breaking, confirming the Parisi picture at the ground state. It also shows $q_\beta\to1$ as $\beta\to\infty$. The positive-temperature input is Lopatto's: for every $\beta>1$ the Parisi measure is supported on the closed interval $[0,q_\beta]$, with a smooth density on $[0,q_\beta)$ and a single atom at the right endpoint $q_\beta$. That endpoint atom is what the paper's own quantitative estimates target, so it is not incidental. This entry is cited as Theorem 1.1 rather than reproved. Remark 1.4 is worth reading beside the support claim: the half-open interval is essential, because the zero-temperature functional cannot see an endpoint atom at all, so there is no canonical mass there to converge to.
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Kourovka Problem 21.147 - Relatively Convex Subgroups
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Gaussian Moments Conjecture
Explicit counterexamples in dimensions 3 and 4, so GMC(n) fails for every n >= 3; GMC(1) was already known, and a separate human-authored preprint claims the remaining n = 2 case affirmatively
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Kourovka Problem 18.50 - Prescribed Permuted-Product Cardinality
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #424
Proves positive lower density. The Formal Conjectures encoding asks for Set.HasPosDensity, a density that exists and is positive; erdosproblems.com says Erdos most likely meant lower density.
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Gaussian product inequality conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Kourovka Problem 19.25 - Totient Sums and Simplicity
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Jacobian Conjecture
n ≥ 3; plane case open
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Completeness of Canonical Closure Representations Is coNP-Complete
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Toroidal Elton–Odell Theorem
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Primariness of the Mixed-Norm Space $L_p(L_1)$
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Weakly Compact Factorization Through a Space With a Basis
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Strict Cosingularity and Adjoints for Separable Range
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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A Unital Banach Algebra That Is Not a Calkin Algebra
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #390: the second-order constant for $f(n)-2n$
The headline is the constant, and its two halves have different histories. The lower bound, $\liminf (f(n)-2n)/(n/\log n) \ge 4029639598/25970038185$, is not new here: it is Mausberg's thirteen-layer valuation cut, posted to the erdosproblems.com forum in May 2026 and credited as such in the paper. Its author wrote there that it "does not prove an upper bound, nor does it prove that an asymptotic constant exists." The novelty is the matching upper bound, so the claim is that the thirteen-layer bound is exactly tight. It is assembled from an exact cofactor-allocation certificate, central-binomial anchors, a guarded rough-signature selector, a friable-number covariance bridge, a finite-band tangent correction, and column-sparse rounding. That construction is what a reader should scrutinize; everything else is inherited or machine-checked. erdosproblems.com still lists #390 as open, and the paper calls itself a proposed solution.
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Kaul-Mudrock Conjecture on the Unlabeled List Color Function
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Ji-Zhang Question on the Power Set of a Quasinilpotent Operator
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Online Spencer Vector-Balancing Question
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Complexity of Terminal-Only Manhattan Prim-Dijkstra Routing
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Fulek's Question on the Extremal Function of $L_3$
the companion pattern Fulek proposed alongside L_3 is not covered by this method
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The Homogeneous Polynomial Lyapunov Converse Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Feige's Hypergraph Moore-Bound Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Belinskaya's Theorem for Measure-Preserving Flows
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Abbott-Hanson Recurrence for Schur Numbers
improves the classical recurrence; the exact Schur numbers beyond S(5) remain unknown
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The Order of Long Rainbow Arithmetic Progressions
the true order is determined, and it is not the conjectured one
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Erdos-Graham Question on Averages of Unit Fractions
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Gao-Huo-Ma Question on Cycle Lengths in Critical Graphs
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Small Davenport Constant of the Heisenberg Group of Order 125
Settles the case p=5. The posed formula for every odd prime remains open.
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Cutoff for Kac's Walk on the Sphere
the conjectured cutoff location of 2n log n is wrong
-
Quantum Memory Advantage for Process Tomography
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
-
The Weak Simplex Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Record Compositions of Alternating Permutations
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Oracle-Complexity Gap in Derivative-Free Convex Optimization
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
-
A Universal Leading-Residue Formula for Witten Zeta Functions
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Sabidussi's Compatibility Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Brualdi's Question on Hamiltonicity of Interchange Graphs
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Thakur's Conjecture on Carlitz-Wieferich Primes
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #254
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #584
the literal wording is refuted; the intended variant remains open
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Benjamini-Hochberg FDR Under Correlated Gaussian Tests
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #769
the conjectured lower bound is disproved; good bounds for c(n) remain open
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Erdős Problem #394
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #130
the infinite-chromatic subquestion is proved; the rest of the problem remains open
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Erdős Problem #959
superlinear lower bound M(n) ≥ n^{1 + 1/(50000 log log n)}, improving Ω(n log n); the exact order remains open
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Erdős Problem #709
upper bound improved to f(n) ≤ 14n^{3/7} with an explicit logarithmic lower bound; matching bounds remain open
-
Erdős Problem #538
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Optimal Chain Density and Space-Time Tradeoffs for the TSP
The authors describe the tradeoff as nearly settled rather than settled.
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Krauth-Mezard Storage Capacity of the Ising Perceptron
the last conditional step in the Ding-Sun and Huang bounds is discharged
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Erdős Problem #267
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1189
exact largest modulus 3·2^{k-3} for k ≥ 5, near-linear least maximum, reciprocal mass Θ(log k), and an infinite divisor family; the counting asymptotic rests on the cited BBMST theorem
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Erdős Problem #450
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #489
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Two Counterexamples in the Geometry of Numbers
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Ross's Two Conjectures on Nondeficient Numbers
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #336
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1188
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1186
exact k = 3 constant established, settling the Parrilo-Robertson-Saracino conjecture for 3-APs; general k remains open
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Type-D ASEP Tracy-Widom Marginals
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #796
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #662
natural readings of the ambiguous historical statement are disproved
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Tightness of the Cohn-Elkies Bound in Dimension 36
rules out the two-point LP method in this dimension; the optimal packing in dimension 36 remains unknown
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The Classical Smith-Ward Problem
Harris had settled the generalized problem in dimension four; this reaches dimension three
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Counting Fixed Cycles in Graphs with Bounded Circumference
the odd-cycle half; the even-cycle half is a companion paper by the same authors
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The Coxeter Code Minimum Distance Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Elizalde-Luo Pattern-Avoidance Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Grothendieck's Finite Flat Group Scheme Order Question
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Klopp-Zadik Question on Polynomial-Time Node-Private Recovery
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Worst-Case Complexity of Shellsort with Tokuda's Gap Sequence
a lower bound for Tokuda's sequence; the general Shellsort complexity question stays open
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New Bounds for Double Covers of the Discrete Box
Improved bounds rather than a settled question: the asked-for inequality is not established in general.
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A Counterexample to Nivat's Conjecture for a Non-Convex Window
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Cycle Double Cover Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Kotzig's Perfect 1-Factorisation Conjecture, Asymptotically
asymptotic form only; Kotzig's conjecture itself remains far from solved
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Minimum Edge-Outerplanar Embedding
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Purdy's Inequality for Hyperplane Arrangements
the refined form for essential arrangements in projective three-space
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Strichartz's Question on Fourier Frames for the Cantor Measure
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Kurkov's Fubini-Number Sum Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #866
h₄(n) = 4 for every n ≥ 331,777, with improved global bounds; the broader problem remains open
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The Quantum Wasserstein Semidistance Is Not a Distance
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Remodeling for the Affine Binary Dihedral Calabi–Yau Threefold
A variant rather than the BKMP conjecture itself: the target lies outside the toric setting the conjecture addresses.
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Stable Phase Retrieval for Spans of Independent Random Variables
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Faber-Harris Conjecture on the Isolation Lemma
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Amdeberhan-Medina-Moll Arctangent Sum Conjecture
density-one set of n; the conjecture itself remains open
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Counterexample to the Odd-Dimensional Rank Bound for Abelian p-Group Actions
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Optimal Online Discrepancy in Linear Time
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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K-Polystable Toric Fano Varieties With Small Alpha Invariants
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Signed BAR Conjecture for Reflected Brownian Motion
uniqueness proved for stable Harrison-Reiman systems with a nonsingular M-matrix reflection; an infinite-dimensional obstruction is shown in the larger completely-S class
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The Tree Product Conjecture
disproved at d = 4; the conjecture for smaller d is untouched
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The Signed BAR Uniqueness Problem
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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White's Conjecture on Matroids
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Log-Concavity of Flats of Matroids
Refuting log-concavity of the flat counts is weaker than refuting their unimodality, since log-concavity is the stronger property. A counterexample to unimodality followed three weeks later and is tracked separately as Rota's Unimodality Conjecture for Matroid Flats; this paper came first.
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North-East Lattice Paths with Few Collinear Vertices
Both bounds move, and the gap stays enormous: the lower bound rises from $\exp(\Omega(\log^2 k))$ to $\exp(\Omega(k^{1/3}))$ and the upper falls from $\exp(O(k^4))$ to $\exp(O(k^2))$, so $A(k)$ is still undetermined between an exponent of $k^{1/3}$ and one of $k^2$. The paper's own closing discussion argues its lower-bound construction is near the limit of the method and that beating it needs additional randomness, a sharper line-counting step, or a different model entirely.
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Erdős Problem #123
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1038
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #119
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #321
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #793
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #320
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Bertoin-Yor Moment Determinacy Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Ziegler's Cross-Polytope Conjecture (simplicial 0/1-polytopes)
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The da Silva Machado-Seade Conjecture
an independent human proof of the same conjecture appeared the same week
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Seymour's Second Neighborhood Conjecture
minimum out-degree 7; the conjecture is open in general
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Rectangles versus Isosceles Triangles in Lattice Sets
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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FGG Conjecture for QAOA on the Ring of Disagrees
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Mossel-Peres Multivariable Bernoulli Factory Claim
the published multivariable claim fails; the single-variable theorem stands
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The Erdos-Sos Pairwise-Sums Problem
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdos Problem #731
resolved under an explicit formalization of 'reasonable', not in full generality
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Erdos's Question on the Independence Ratio of Unit-Distance Graphs
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Conjecture on $k$-Antichains in the Unit Cube
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Lassak's Area Bound for Reduced Planar Bodies
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1061
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdos Problem #1061
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Quadratic-Time Hardness of Furthest Pair in Superconstant Dimension
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Existence Problem for Regular Gabor Frames
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdos Problems #593 and #1177
settles two numbered Erdos problems at once
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The Erdos-Lovasz Cover Number Problem
an improved lower bound; the true order of g(r) remains open
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Erdos Problem #768
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Kontsevich's Asphericity Conjecture for Strata of Differentials
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #865
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Finite Field Restriction Problem for the Paraboloid
a record exponent; the conjectured range is not yet reached
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Erdős Problem #671
Both parts claimed answered affirmatively, with a Lean formalization; two proof claims are filed on erdosproblems.com but the problem is still listed open
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Erdős Problem #550
Claimed proved in a preprint of E. Li; erdosproblems.com still lists the problem open pending human review
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Baker's Question on Smooth Hyperplane Sections over Finite Fields
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Kannan–Tetali–Vempala conjecture (bipartite/binary-matrix case)
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1197
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #176
a polynomial bound for N(k,2), stronger than the exponential bound asked for; the two-parameter problem remains open
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Zhi-Wei Sun's Conjecture 3.4 on a Truncated Legendre-Symbol Determinant
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #346
The problem statement is ambiguous: the limit-exists reading is claimed proved (Lean), while the convergence-from-hypotheses reading was disproved by a Lean-checked construction of Price that the community classes as a variant
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Erdős Problem #948
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #306
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Conjecture 3 of the Dynamical Sampling Survey
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Ramachandra-Natarajan Pairwise Independent Correlation Gap Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Sabok's S-Prime Simplex Questions
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #451
the conjectured superpolynomial growth is established; the sharper order remains open
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Erdos's Conjecture on Consecutive Integers Free of Certain Prime Factors
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #986
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Yun-Sra-Jadbabaie SS-RS-GD Inequalities
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Erdos-Hajnal High-Girth Subgraph Conjecture
in polynomial edge-density regimes; the general question remains open
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Erdos's Question on Shifted Pairwise-Coprime Reciprocal Sums
the average order; the uniform bound Erdos asked about is not settled
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Wegner's Piercing Conjecture for Rectangles
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Vinzant's Conjecture on Phase Retrieval Injectivity
part (1) of the conjecture
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The Near-Quadratic Elekes-Ronyai Expander Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Brezis-Mironescu Open Problems 23 and 24 on Minimizing Maps to the Circle
in the range s in (1/4,1); Brezis's Problem 5.4 outside that range is untouched
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The Matrix Spencer Conjecture for C*-Algebra Contractions
a structured special case, proved the same week as the independent group version
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Erdős Problem #942
lower bound improved to ≫ log n/(log log n · log log log n) infinitely often; the extremal order remains open
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Erdős Problem #326
Affirmative answer claimed, contrary to the negative answer Erdős and Graham conjectured; erdosproblems.com still lists the problem open
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The Erdos-Graham Semiprime Unit Fraction Problem
the omega = 2 integer case, the one Butler, Erdos and Graham left open
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Koch-Narayan Conjecture 1
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Papamanthou-Tollis Conjecture on Parameterized st-Orientations
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Graffiti Conjecture 143
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Sun's Conjecture 4.6(ii) on Trigonometric Permanents
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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TxGraffiti-Davila Conjecture 9
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Quantum Pyramids Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Pandey Parity Conjecture for Generalized Petersen Graphs
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Graffiti Conjecture 154 (Standard-Deviation Reading)
refuted under the standard-deviation reading; the statement is reading-sensitive and other readings remain open
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IRIS Conjecture 6.1 on Simple 3-Polytopes
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Matrix Spencer Conjecture for Finite Groups
the group case; the full Matrix Spencer conjecture remains open
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The Schwartz Quadratic Meander Number Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #539
main exponent determined; sharper subpolynomial factors remain open
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The Sharp Threshold for the Dyn-Farkhi Conjecture
pins the sharp threshold; the conjecture itself was already known false above dimension two
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Erdős Problem #619
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Free Fermions in Disguise without Exponential Degeneracies
An existence question settled by exhibiting an object, not a general theorem: one model in the family has no exponential degeneracies for generic couplings, and nothing here says which others do. The route is worth recording because it is not the one anyone was looking down. The author had tried and failed to find such a model directly. It surfaced instead from an unrelated classification of medium-range spin chains, where the AI's computations produced a list of integrable Hamiltonians and one of them combined two terms of a standard XY chain with two of Fendley's FFD model, a combination nobody had considered. The author noticed it in the list; the rest followed from asking the model a sequence of increasingly specific questions.
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The Odd Area Conjecture for Unit Disks
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Espuny Diaz-Lichev-Wesolek Conjecture on Dirac Subgraphs
asymptotic in k; the paper also shows the analogous statement is false for d = 2
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Erdős Problem #696
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #623
Resolved (if correct) by an independence result rather than a proof or disproof in ZFC: consistency of the positive answer is equivalent to a measurable cardinal, of the negative to ZFC alone
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FullRSB Jamming Identity a + b = 1
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Divisibility Set of the Generalized Euler Totient
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Darboux injections from closed manifolds: Banakh–Banakh Problems 1.7 and 1.8
Both problems are answered affirmatively, in every dimension at once: every connectedness-preserving injection from a connected closed $n$-manifold into an $n$-manifold is a homeomorphism onto a component, so in particular every Darboux self-bijection of $\mathbb S^n$ and of every closed manifold is a homeomorphism. This removes the finite-$H_1$ hypothesis of the 2020 theorem for 3-manifolds and extends it above dimension 3. Compactness of the source is essential: the companion preprint (Zenodo 10.5281/zenodo.22346412) shows the corresponding statement fails for $\mathbb R^n$, $n\ge2$. The note does not address noncompact sources, manifolds with boundary, or targets of different dimension.
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Erdős Problem #477
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Gaussian Mass Maximality of the Integer Lattice
integral unimodular lattices of rank at most 32; the general conjecture is open
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The Thin Matching Problem
up to polylogarithmic factors
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Zhang's Question on Howson and Strongly Howson Groups
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Steurer's Conjecture on Vectors with Small Average Correlation
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Unit-Area Triangles in Planar Sets of Large Measure
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Arborescence-Sampling Barrier for Eulerian Tours
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Aluffi-Chen-Marcolli Real-Rootedness Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Sum-Product Conjecture over the Reals
the model's contribution is one simplifying lemma; the authors state the main ideas are human
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Bombari's Question on Sign-Quantized Linear Maps
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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No-Go Theorems for Poisson Certificates of Gaussian Mass Maximality
a barrier result about one proof strategy, not the conjecture itself
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Borsuk Conjecture lowest-ever counterexample (N=63)
Priority: the result was first obtained by Max Grinsztajn with GPT-5.5 Pro assistance, published 26 May 2026 and recorded as the current best bound on Tao's optimization-problems ledger. The same construction was found again independently in August 2026 by Nicholas Konz working with Claude, with a different derivation and a fuller AI disclosure; the two efforts were evidently unaware of each other, and the submitter of this entry surfaced the earlier work themselves after publication. Dimension 63 is the current record; whether Borsuk's conjecture fails for any dimension in 4..62 remains open.
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The Frankl-Peng-Rodl-Talbot Question on Turan Density Intervals
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Arithmeticity of Degree-Six Symplectic Hypergeometric Monodromy Groups
two of the three remaining cases; one is still unclassified
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Strong Convergence of Cesaro Means of Firmly Nonexpansive Iterates
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Growth Constants for Lipschitz Functions on Sparse Random Graphs
Resolved the sharp constant (w.h.p.) for random graphs G(n, d/n)
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Positivity of Chern Classes of Symmetric Powers
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Open Problems in Commutative Algebra Resolved by Rethlas
a batch of problems from published lists, resolved in one paper
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Brezis's Problems on Degenerate Constants in Degree Inequalities
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Simon's Extendable Shellability Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Optimal Vector Balancing for Zonotopes
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Binary Digits of the Erdős-Borwein Constant
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Log-Concavity of Codimension-Three Pure O-Sequences
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Shokurov's Global Index Conjecture for Foliations
proved in dimension at most three
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Han and Jiang's Question on klt Type in Flat Families
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #12
parts (i) and (ii) resolved - a near-linear-density construction exists, refuting the N^{1-c} decay; the reciprocal-sum part remains open
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Four-Particle Monochromatic Quantum Graphs
the N = 4, D ≥ 4 family is fully ruled out; the general two-parameter problem remains open
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Mauri and Moraga's Question on Log Calabi-Yau Pairs with Big Boundary
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Optimal Approximation Ratio for Permanents of PSD Matrices
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #138
the stronger question $W(k)^{1/k} \to \infty$ remains open
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Written on the Wall II, Graph Conjecture 2
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Monochromatic Quantum Graphs in the Diagonal Family
the diagonal family is ruled out; the broader two-parameter problem remains open
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Ben Green's Open Problem 57
intended complex form disproved, with a certified strict support-function gap
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Erdős Problem #387
false in general; the positive direction holds for k large relative to n
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The Ballantine-Beck-Feigon-Maurischat Conjectures on Subsum Polynomials
six of the ten conjectures proved; one was found false as printed and its corrected form remains open
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Kollár and Kovács's Question on Cohomology of Fibers
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Ciliberto et al. Question on Effective Divisors of Positive Self-Intersection
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Conjectures of Hopkins, Sagan-Wilson and Defant et al. on Lattices, Parking Functions and the Plactic Monoid
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Polylogarithmic Full-Chord Buffon Discrepancy
This settles Steinerberger's third open question and separates the two models: in the disk, full chords cost you a factor growing like $\log L$ over what is achievable without the restriction. It also improves the Steinhaus-type $O(L^{1/3})$ to polylogarithmic within the full-chord class. It does not settle the Buffon discrepancy problem itself. Steinerberger's first question - whether every convex body admits a set of discrepancy $O(1)$, and if not what the truth is - is untouched, and the paper's closing line names it as the natural next question. Inside the full-chord model the order is pinned only between $\Omega(\log L)$ and $O\left((\log L)^{3/2}\right)$, and the lower bound is proved for the disk alone. The paper says the exponents are unlikely to be sharp. The upper bound is an existence statement: it inherits the Aistleitner-Bilyk-Nikolov theorem, which is proved by transference and supplies no explicit construction.
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Log-Convexity of Fisher Information Along Heat Flow
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Integral Local Invariant Cycles in Degree One
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1039
order of magnitude determined; the exact asymptotic constant remains open
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Maz'ya's Question on Distinguishing Two Maximal Operators
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Existence of $t$-Edge-Balanced Graphs for $t \ge 3$
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Total Variation for the Lamplighter Walk on Z
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Return Probability for the Lamplighter Walk on a Tree
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Kinoshita Conjecture and Kirby Problem 4.37
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Ben Green's Open Problem 90
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Maximum Entropy of Sums of Independent Ternary Random Variables
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Gaussian Completely Monotone Conjecture
also refutes the McKean and Toscani conjectures
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Talagrand's Convexity Problem
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Jauslin-Kreiss-Moser Vanishing-Viscosity Selection Problem
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Hamaker-Reiner Conjecture on ASM Weak Order Intervals
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #690
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Escobar-Klein-Weigandt Conjecture on Cohen-Macaulay ASM Varieties
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1032
a new density-degree inequality gives δ(G) ≤ (3/10 + o(1))|V(G)|, improving 0.328; existence of a linear construction remains open
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Erdős Problem #7
VibeMathed reports this item as retracted. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Odifreddi's Problem 3 on Irreducible m-Degrees
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #750
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Litvak's Conjecture on Gaussian Minima
the paper proposes that the cosine matrix is the true minimizer for all p and n, and proves a stronger stochastic domination statement conditional on a new volumetric extension of Fejes Toth's zone conjecture
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Erdős Problem #351
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #283
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #870
A total refutation is claimed for all k>=3, building on the Larsen-Larsen resolution of problem #868; erdosproblems.com still lists the problem open
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Erdős Problem #694
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Banks-Martin Conjecture on Primitive Sets
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Largest Sum-Free Subset of the Lattice Cube
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Carbery's Almost-Orthogonality Inequality in Lp
exponent 2 fails for every p > 2; the sharp exponent p' form is proved for integer p ≥ 2
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Erdős's Planar Unit Distance Conjecture
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1201
As Tao notes on the problem page, the claim establishes natural LOWER density at least 1-eta but not that the natural density exists, so the problem as stated remains technically open
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The Proportion of Permutations Fixing a k-Set
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1151
An elementary solution via a primitive-row decomposition of the Chebyshev-node measures; the main theorem is formalized in Lean, but erdosproblems.com still lists the problem open
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Garamvölgyi-Jackson-Jordán Conjecture on Cliques in Minimally Globally Rigid Graphs
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1133
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Hamilton Decompositions of the Directed 5-Torus, Odd Modulus
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Kirby Problem 5.16 for Noncommutative Semifree DGAs
resolves the noncommutative half of the Kirby-list problem
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Erdős Problem #1092
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1101
a subexponential good sequence is constructed; the polynomial-growth question remains open
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Erdős Problem #43
both proposed bounds fail
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Erdős Problem #953
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #42
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Mixing Time of Kac's Walk on the Rotation Group
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #896
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #906
Two independent affirmative claims (Adriano's, posted first, and a GPT-5.5 Pro note); Erdős himself wrote in 1982 that the problem had been solved affirmatively long before, without a locatable reference
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Erdős Problem #888
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #38
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1138
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Optimal Strategies in the All-Heads Coin Game
VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #330
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1014
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1190
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #202
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #863
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Parity obstruction in the minimum-determinant problem for Latin squares
For even $n$, let $q(L)=\det(L)/b_n$. The work proves that $q(L)$ is even exactly when the stronger centered divisibility $n^2\mid\det(E_{\mathrm{std}})$ holds. For $n\equiv2\pmod4$, this is equivalent to $\operatorname{rank}_{\mathbb F_2}(A\bmod2)<n-1$; for $n\equiv0\pmod4$, it is equivalent to $\operatorname{adj}(A\bmod2)\mathbf1=0$. An explicit family gives odd $q(L)$ for every $n\equiv2\pmod4$, $n\ge6$. This removes a universal extra-factor-two obstruction, but it does not prove $|q(L)|=1$. Exact minimum attainment and the separate singularity question remain open.
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Erdős Problem #996
Answered negatively in a preprint that also settles the p=2 case of problem #995; erdosproblems.com still lists the problem open
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Erdős Problem #610
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #603
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Sharp Convergence Rates for Viscous Hamilton-Jacobi Homogenization
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Density Thresholds for Large Dilates of Point Configurations
Near-optimal rather than optimal: the bounds match up to logarithmic-type factors.
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Nathanson's Problems on Product Intersection Sets
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #522
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1195
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #670: Diameter with Separated Distances
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Wickstead's Conjecture on Positive Projections
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #741
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1217
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Avidor-Zwick Question on Low-Dimensional Max-Cut SDP
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #856
Identifies the exponent as a variational sunflower-capacity constant, sharpening the Tang-Zhang bounds; the value of that constant itself remains open, as does site acceptance
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Erdős Problem #858
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #258
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #990
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #987
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #960
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1091
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1141
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Exhaustive AdaBoost Cycling Question
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Thiele Rules on the Voter Interval Domain
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #26
The question as posed was implicit in Davenport–Erdős (1951); the AI result settles Tenenbaum's open variant negatively
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Equality in Fill's Spectral Gap Problem
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Anderson's Quasi-Completeness Question
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Last-Iterate Rate for Anchored Gradient Descent-Ascent
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #152
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Bipartite Exact Matching in P
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1202
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1196: Primitive Sets
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #514
an escape path dominating every power of |z| with bounded initial length is constructed, and universal positive-power lower bounds are ruled out; the broader variant remains open
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Erdős Problem #997
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Counting Partial Hadamard Matrices in the Cubic Regime
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Sárközy's Conjecture on Sums and Products Modulo a Prime
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #380
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Improved Approximation Ratios for Multiway Cut
Record bounds on the ratio; the exact approximability of Multiway Cut remains open.
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Erdős Problem #125
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #369
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Lattice Triangle Problem in the Hard Obtuse Window
A density-1 obstruction, not a full resolution: the conjecture that the hard window contains no lattice triangles remains open on a density-0 set.
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Hamilton Decompositions of the Directed 3-Torus
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Ramsey-Style Hypergraph Partition Bound H(n)
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1153
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Lower Bounds for Lebesgue Constants and an Erdős-Turán Interpolation Problem
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Simplicity of the Hodge Bundle
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Manin's Question on R-Equivalence for the Diagonal Cubic
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1148
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Equilibria of the Vlasov-Maxwell-Landau System
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Pach's Tangency Conjecture: Improved Bounds
Exponent improvements toward Pach's conjecture, which remains open.
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Kissing Number in 19 Dimensions
A record lower bound; the kissing number in dimension 19 remains unknown.
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Nine Ramsey Number Lower-Bound Records
Records, not resolutions: the exact values of these Ramsey numbers remain unknown.
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Transposition is Nearly Optimal for IID List Update
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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A 24-Vertex Triangulation of Real Projective 5-Space
A record, not an endpoint: whether fewer vertices suffice is posed as an open question in the same paper.
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Erdős Problem #650
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #457
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Uniform Witnesses for Uniform Set Systems: the k=3 Question
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #966
Erdős reported in 1975 that Spencer had shown existence but gave no reference; no proof was on record before the AI solution
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Erdős Problem #846
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Ran-Teng Conjecture 20 on 4-Cycle Stochastic Matrices
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Leading Constant for Large-Order Davenport-Schinzel Sequences
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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First Proof Question 8: Smoothing Polyhedral Lagrangians
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Single-Minus Gluon Tree Amplitudes
nonzero on half-collinear complex kinematics, with a closed formula
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Erdős Problem #851
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Almost All Primes are Partially Regular
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #347
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Chen-Gendron Spin-Parity Identity for k-Differentials
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Fel's Conjecture on Syzygies of Numerical Semigroups
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1089
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #1040
capacity alone does not determine the invariant; the zero-measure clause is a separate open question
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Erdős Problem #654
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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Erdős Problem #1051
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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A Generalization of Boppana's Entropy Inequality
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #543
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #281
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Courtade and Kumar's Coordinate-wise Mutual Information Question
Fully resolves the posed coordinate-wise question; the main Courtade-Kumar conjecture itself remains open outside the extended high-noise range.
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Erdős Problem #659
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #401
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #729
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #205
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #397
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #728: Factorial Divisibility
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #871
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Large Hypercube Intervals in Bruhat Order
Asymptotically optimal for powers of 2; the exact extremal answer for general n stays open.
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Erdős Problem #897
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #333
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős-Herzog-Piranian Distance Products: Improved Lower Bound
An improved lower bound on the maximal product; the sharp maximizer for the 1958 question remains unknown.
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Erdős Problem #1026: Monotonic Subsequence Sums
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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The Lonely Runner Conjecture for Nine and Ten Runners
Settles 9 and 10 runners only; the general conjecture remains open.
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Minimax Rate for Density Estimation under Wasserstein Contamination
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #848
Resolved for all sufficiently large N via a stability theorem; small N remain a finite computation (erdosproblems.com marks the problem DECIDABLE)
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Sum-Difference Exponents for Boundedly Many Slopes
Charts the bounded-slope regime of the arithmetic Kakeya program; the conjecture itself remains open.
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Point Convergence of Nesterov's Accelerated Gradient Method
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Curto et al.'s Minimality Conjecture for Threshold-Linear Networks
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Erdős Problem #707: Sidon Sets and Perfect Difference Sets
Hall's 1947 counterexample predates the problem itself; this paper's counterexample is independent, smaller, and Lean-certified.
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Cohen's 22 Conjectures on Cyclic Numbers
VibeMathed reports this item as retracted. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Babai and Frankl's Oddtown Question for Composite Moduli
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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Vertex-Minimal Paper Tori
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.