Tracked problems: 719; 493 fully resolved · 0 under review
Combined years open: 9,936; before AI closed them
Lean-verified: 154; machine-checked
Community members: 432; 6 verified · 316 votes · 78 comments

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Rasmus Lindahl approved Bailey-Crandall's mutual-nondivisibility question for Stoneham numbers: $\alpha_{6,4}$ is not $6$ -normal, and $\alpha_{b,c}$ is $b$ -normal iff some prime of $c$ misses $b$ 4 hours ago
Rasmus Lindahl approved Upper bound $\Lambda \le 0.1787854$ on the de Bruijn-Newman constant4 hours ago
SilentIbis759 changed the statement and 4 other fields on Morris's asymptotic conjecture: $\mathrm{FC}(k,n)=\Theta_k(n^{k-2})$ to “Morris's Conjecture 2 asks whether, for every fixed integer …”2 days ago
nufrogcaca submitted Bailey-Crandall's mutual-nondivisibility question for Stoneham numbers: $\alpha_{6,4}$ is not $6$ -normal, and $\alpha_{b,c}$ is $b$ -normal iff some prime of $c$ misses $b$ 2 days ago
PluckyWombat559 submitted Upper bound $\Lambda \le 0.1787854$ on the de Bruijn-Newman constant2 days ago
Rasmus Lindahl approved Minimum reverse-complement-invariant decycling sets2 days ago
Rasmus Lindahl approved Erdős matching conjecture: the four-uniform case2 days ago
Oleksiy Babanskyy submitted Minimum reverse-complement-invariant decycling sets3 days ago
Oleksiy Babanskyy submitted Erdős matching conjecture: the four-uniform case3 days ago
Rasmus Lindahl approved Mahler's $Z$ -number problem in its generalized form3 days ago

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Erdős matching conjecture: the four-uniform case

ProvedPartialAI-assisted

(For r=4, the manuscript claims the sharp extremal value max\4s+34, n4--s4\ for every s1 and n4s+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 (s6961) 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.)

Erdős #1020 · Extremal set theory and hypergraph matchings

Let $r\geq2$ , $s\geq1$ and $n\geq r(s+1)$ be integers. If $\mathcal F\subseteq\binom{[n]}r$ contains no $s+1$ pairwise disjoint members, must
$|\mathcal F|\leq\max\left\{\binom{r(s+1)-1}{r},\binom nr-\binom{n-s}{r}\right\}?$
The two candidate extremal families are all $r$ -sets inside an $(r(s+1)-1)$ -set and all $r$ -sets meeting a fixed $s$ -set. This is the Erdős matching conjecture (1965), recorded as Problem #1020. This entry concerns only the four-uniform specialization, $r=4$ ; it does not claim to settle arbitrary uniformity. The tracker's forbidden matching parameter is $k=s+1$ .

Posed by Paul Erdős, A problem on independent r-tuples (1965); Erdős Problem #1020, 1965Open 61yModel GPT-5.6 Sol; Astra; GPT-5 Pro; GPT-6 Pro (OpenAI)Solved 2026-09

AnnouncedSignificance 40 Submitted by Oleksiy Babanskyy on 12 Sep 2026

Morris's asymptotic conjecture: $\mathrm{FC}(k,n)=\Theta_k(n^{k-2})$

ProvedUnder reviewAI co-developed

(It is proved that FC(k,n)=Θ_k(n^k-2) for every fixed k2, resolving Morris's Conjecture 2. The proof combines robust weighted certificates, pair lifting and extremal forcing with a self-contained lower-bound argument. For four-sets, n^2/4-O(n)(4,n)≤(18+o(1))n^2. More generally, for fixed k4, FC(k,n)≥^k-22(k-2)!-O_k(n^k-3). The sharp FC and robustness thresholds for sunflowers with a two-point core and disjoint pair petals are nine and ten petals. Exact finite certificates establish FC(4,9)=16 and refute Pulaj and Wood's lexicographic extremality conjecture.)

Extremal set theory; union-closed families

Morris's Conjecture 2 asks whether, for every fixed integer $k\ge2$ , $\mathrm{FC}(k,n)=\Theta_k(n^{k-2})$ as $n\to\infty$ . Here $\mathrm{FC}(k,n)$ is the least $m$ such that every collection of $m$ distinct $k$ -subsets of an $n$ -point set is Frankl-complete. A configuration $\mathcal G$ with support $U=\bigcup\mathcal G$ is Frankl-complete if every finite union-closed family $\mathcal F\supseteq\mathcal G$ has an element of $U$ belonging to at least half its sets. The extension may contain additional points.

Posed by Robert Morris (2006), “FC-families and improved bounds for Frankl’s conjecture,” European J. Combin. 27, 269–282; Conjecture 2 in arXiv:math/0702348v1., 2006Open 20yModel GPT-6 Astra; Fable 5.1 (OpenAI; Anthropic)Solved 2026-09

PreprintSignificance 15 Submitted by SilentIbis759 on 11 Sep 2026

Bailey-Crandall's mutual-nondivisibility question for Stoneham numbers: $\alpha_{6,4}$ is not $6$ -normal, and $\alpha_{b,c}$ is $b$ -normal iff some prime of $c$ misses $b$

DisprovedAI-discovered

(Mutual nondivisibility is not enough: b = 6, c = 4 is a counterexample, since every prime dividing 4 divides 6, and then the zero digit of α_6,4 in base 6 has limiting frequency one. The exact criterion is that α_b,c is b-normal if and only if some prime dividing c does not divide b. Sufficiency applies Bailey and Crandall's own exponential-sum estimate to the part of c coprime to b; the repository's prior-art note records that Kano's 1993 theorem on Korobov-type constructions already reaches some non-coprime cases, α_4,6 among them, so the classification rather than the method is what is new. Reversing the parameters, α_4,6 is 4-normal, is also formalised.)

Normal numbers and digit expansions

For integers $b, c \ge 2$ let $\alpha_{b,c} = \sum_{k \ge 1} \dfrac{1}{c^k\, b^{c^k}}$ . Bailey and Crandall proved in 2002 that $\alpha_{b,c}$ is normal in base $b$ whenever $b$ and $c$ are coprime, and asked whether coprimality can be weakened to the condition that neither of $b$ , $c$ divides the other: is $\alpha_{b,c}$ then still $b$ -normal?

Posed by David H. Bailey and Richard Crandall, Random Generators and Normal Numbers, Experimental Mathematics 11 (2002), Section 7 "Conclusions and open problems", p. 544, 2002Open 24yModel GPT-6 Astra (OpenAI)Solved 2026-09-12

Lean-verifiedSignificance 18 Submitted by nufrogcaca on 13 Sep 2026

Mahler's $Z$ -number problem in its generalized form

ProvedPartialAI-discovered

(Theorem E gives Z_3/2(27,57)=: a window of length 37=0.428571… at base 3/2, beating the 31/81 of [Dub19, Thm. 1.2], and with no assumption on the arithmetic nature of , 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.)

Distribution mod one; Mahler's Z-numbers

Let $p>q>1$ be coprime integers and let $Z{p/q}(s,s+t)$ be the set of $\xi>0$ whose fractional parts $\{\xi(p/q)^{n}\}$ all lie in $[s,s+t)$ . The question for which parameters $Z$ is empty remains open.

Posed by Mahler, 1968Open 58yModel Fable 5, Opus 5 (Anthropic)Solved 2026-09-12

Lean-checked, statement unauditedSignificance 30 Submitted by LucidKestrel185 on 12 Sep 2026

Nontriviality of number-restricted arithmetic over subDMQ

DisprovedAI-discovered

(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.)

Paraconsistent logic and foundations

Weber's programme of paraconsistent mathematics keeps unrestricted comprehension and revises the logic of inference so that contradictions do not make every statement provable. Ripley and Weber's 2026 proposal restricts induction to numbers as part of a strategy for blocking paradox, and Ripley's presentation of it records that the resulting theory is not known to be trivial, the programme working "without a net: no nontriviality proofs". Is number-restricted arithmetic over subDMQ nontrivial: does it, together with naive comprehension and induction over the combined language, avoid proving everything?

Posed by Ellie Ripley and Zach Weber, 2026Open 0yModel GPT-6 Astra (OpenAI)Solved 2026-09-10

PreprintSignificance 10Submitted by BraveEgret318 on 11 Sep 2026

Normality of a sum of two Stoneham constants

ProvedAI-discovered

(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.)

Normal numbers and digit expansions

For coprime $b,c\ge 2$ the Stoneham constant $\alpha_{b,c}=\sum_{k\ge1}1/(c^{k}b^{c^{k}})$ is known to be $b$ -normal. Bailey and Borwein asked in 2012 what happens to a sum of two of them sharing the base: with $b,c_1,c_2\ge2$ , $(b,c_1)$ and $(b,c_2)$ coprime, is $\alpha_{b,c_1}+\alpha_{b,c_2}$ normal in base $b$ ? In their words, "it is not known at the present time whether the sum $\alpha_{b,c_1}+\alpha_{b,c_2}$ is $b$ -normal".

Posed by David H. Bailey and Jonathan M. Borwein, "Nonnormality of Stoneham constants", Ramanujan J. 29 (2012) 409-422, Section 4 (repeated in Section 6), 2012Open 14yModel ChatGPT-6 Astra (OpenAI)Solved 2026-09-10

Lean-verifiedSignificance 25 Submitted by nufrogcaca on 12 Sep 2026

The Hilbert transform-UMD dependence problem

ProvedAI co-developed

(Neither exponent can be lowered. Theorem 1.1 constructs explicit 2^n-dimensional spaces X_n and Y_n with _2,X_n n, _2,X_n√ n and _2,Y_n n, _2,Y_n√ 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<∞ 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.)

Banach space geometry and harmonic analysis

For fixed $1<p<\infty$ and a Banach space $X$ , Burkholder and Bourgain showed in 1983 that the UMD property is equivalent to boundedness of the Hilbert transform on $L^p(\mathbb{R};X)$ , with the quadratic comparisons $\hbar_{p,X}\lesssim\beta_{p,X}^{2}$ and $\beta_{p,X}\lesssim\hbar_{p,X}^{2}$ between the Hilbert transform constant and the UMD constant. What is the optimal dependence between $\hbar_{p,X}$ and $\beta_{p,X}$ , uniformly over all UMD spaces $X$ : can either quadratic bound be improved, in the best case to a linear one?

Posed by Implicit in Burkholder's and Bourgain's 1983 quadratic bounds; stated as an open problem in Burkholder's 2001 Handbook survey, 1983Open 43yModel GPT-6 Astra (OpenAI)Solved 2026-09-09

Lean-verifiedSignificance 25 Submitted by RustyKestrel290 on 10 Sep 2026

Binary normality of a localized logarithm value

ProvedAI-discovered

(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: α_b,p=_k1p^-kb^-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^a3^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 , 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.)

Normal numbers and digit expansions

Let $\xi=\sum_{a,c\ge 0}\frac{1}{2^{a}3^{c}\,2^{2^{a}3^{c}}}$ , the value at $1/2$ of Dibag's logarithm localized at the primes $2$ and $3$ ; summing the same terms over every positive integer rather than only the $3$ -smooth ones gives $\log 2$ . Is $\xi$ normal in base two, that is, does every finite binary word of length $l$ occur in its binary expansion with limiting frequency $2^{-l}$ , counting overlapping occurrences?

Posed by —Open —Model ChatGPT-6 Astra (OpenAI)Solved 2026-09-09

Lean-verifiedSignificance 15 Submitted by nufrogcaca on 9 Sep 2026

An improved lower bound for the Shannon capacity of $C_{11}$

ProvedPartialAI co-developed

(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 Θ(C_11) ≥ 5.295526013632343, from an independent set in C_11^ 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.)

Zero-error information theory; graph capacity; combinatorics

Determine the Shannon capacity of the eleven-cycle $C_{11}$ , or improve its best explicit lower bound. The preceding BPZ construction, updated on 10 August 2026, gives an independent set of cardinality $N_0$ in dimension 207 and the lower bound $\Theta(C_{11}) \ge N_0^{1/207} = 5.29549231578462014255\ldots$ . The exact capacity remains open.

Posed by Claude Shannon (1956), underlying capacity problem; Buys, Polak and Zuiddam (2026), preceding C11 record, 1956Open 70yModel Astra 6 Pro; Codex GPT-6 Astra Extra-High (OpenAI)Solved 2026-09-08

Lean-checked, statement unauditedSignificance 14 Submitted by Matthew Protti on 8 Sep 2026

Sharp Finite Markov Order in Intrinsic Sofic Dimension

ProvedUnder reviewAI-discovered

(For a stationary finite-alphabet law , let n=_ R H_p be the intrinsic real Hankel dimension of its cylinder-probability function. If has any finite Markov order, then ()≤ 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 ^2 turns this into vanishing of products on a space of dimension n2; a uniform nilpotence argument then forces vanishing after n2 steps. Sharpness is attained for every n2. The construction gives a stationary sofic process with a nonnegative rational presentation of minimal dimension n and exact order n2. One realization uses n2-1+n^2 output symbols.)

Sofic measures

Let $n$ be the real Hankel dimension of the cylinder probabilities of a stationary finite-alphabet process. If its Markov order is finite, then $\operatorname{ord}(\mu)\le \binom n2.$ For every $n\ge2$ , there exists a stationary sofic process with a nonnegative rational presentation of minimal real dimension $n$ and exact Markov order $\binom n2.$ Thus the bound is sharp when the alphabet is allowed to grow.

Posed by Béal, Jugé, Mairesse and Perrin, 2026Open 0yModel GPT 6 Astra (OpenAI)Solved 2026-09-08

Lean-checked, statement unauditedSignificance 12 Submitted by VibeGene on 8 Sep 20262

Erdős Problem #1221

ProvedUnder reviewAI co-developed

Discrepancy

Posed by de Bruijn, Erdos (1949), 1949Open 77yModel GPT-Astra (OpenAI)Solved 2026-09-08

UnreviewedSignificance 10 Submitted by GoldenMongoose827 on 9 Sep 2026

Finite-time blowup for the IPM equation with a uniformly space-time smooth force

ProvedAI co-developed

(Yes. Theorem 2.1: there are a smooth odd initial density _in∈ C^∞( T^2) of zero spatial mean, an odd force F∈ C^∞([0,1]× T^2), and a solution smooth on [0,T] for every T<1 with (t)→_* in C^ for every 0≤<1, yet ‖(t)‖_∞ and ‖D_xu_ T((t))‖_∞ both diverging as t1. The advance over Córdoba and Martínez-Zoroa is precisely the force class, from L^∞_t C^∞_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".)

Fluid dynamics; incompressible porous media equation

Córdoba and Martínez-Zoroa proved finite-time singularity formation for the two-dimensional incompressible porous media equation from smooth initial data with a force smooth in space but merely bounded in time, that is in $L^\infty_t C^\infty_x$ . Their Remark 1 anticipates joint smoothness in space and time but does not prove it. Can the force be taken uniformly smooth in space and time?

Posed by Diego Córdoba and Luis Martínez-Zoroa, Remark 1 of arXiv:2410.22920, where joint space-time smoothness is anticipated but not part of the theorem, 2024Open 2yModel Claude, Codex with GPT-5.6 Sol (Anthropic / OpenAI)Solved 2026-09-08

AnnouncedSignificance 25

The ceiling of the single-letter entropy method for the union-closed sets conjecture

ProvedAI-discovered

(Two unconditional ceilings. Every single-letter certificate whose classes contain product laws certifies at most 1-h(1/√2)/√2 = 0.383099…. 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…. So the entropy method, in every form used since 2022, cannot reach 1/2: Frankl's conjecture is out of its range, and only 1.8·10^-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.)

Extremal set theory / entropy method

Every lower bound on the union-closed constant $c_0$ since Gilmer (2022) — including $(3-\sqrt5)/2$ , Sawin–Yu–Cambie's $0.3823455$ and Liu's $0.382709$ — is certified by the same single-letter inequality: a memoryless protocol coupling two conditional bits, together with a class of admissible joint laws of the two prefix-conditional probabilities. It was not known how far this framework can go: whether it could be pushed to $1/2$ and settle Frankl's conjecture, or whether it has a hard ceiling strictly below it.

Posed by Raised in the entropy-method literature after Gilmer (2022): Cambie (2022) studies the approach's boundaries, Liu (2023) asks whether other couplings can improve it, 2022Open 4yModel Fable 5.1Solved 2026-09-08

Lean-checked, statement unauditedSignificance 18 Submitted by CobaltPanther851 on 10 Sep 2026

Reciprocal Fermat Constant is Nonnormal

ProvedAI-discovered

(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.)

Theory of Normal Numbers and Digit Expansions

Posed by —Open —Model ChatGPT-6 Astra (OpenAI)Solved 2026-09-08

Lean-checked, statement unauditedSignificance 18Submitted by nufrogcaca on 8 Sep 2026

Finite-time blowup for the inviscid Boussinesq system with smooth forcing

ProvedAI co-developed

(Yes. Blowup for the inviscid Boussinesq system on R^2 with forces in C^∞( R^2×[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 ‖(t)‖_∞→∞ and the vorticity norm has infinite limsup as t 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".)

Fluid dynamics; singularity formation for incompressible flow

Does the inviscid Boussinesq system on $\mathbb R^2$ admit finite-time blowup from smooth data with forces that are smooth in both space and time? Córdoba, Laín-Sanclemente and Martínez-Zoroa obtained finite-time singularity for the two-dimensional Boussinesq equation with a force only of class $C^{1,\sqrt{4/3}-1-\epsilon}\cap L^2$ , leaving the smooth-force case open.

Posed by Diego Córdoba, Antonio Laín-Sanclemente and Luis Martínez-Zoroa, whose multiscale construction reached a force of limited Hölder regularity, 2025Open 1yModel Claude, Codex with GPT-5.6 Sol (Anthropic / OpenAI)Solved 2026-09-08

Lean-verifiedSignificance 45

Finite-time blowup for the 3D incompressible Euler equations with smooth forcing

ProvedAI co-developed

(Yes. Theorem 1.1: for every r_0>0 and z_0 there are a time T_*>0, a divergence-free axisymmetric u_0∈ C_c^∞ supported in a fixed solid torus with nonzero swirl and zero meridional velocity, and an axisymmetric force f∈ C^∞( R^3×[0,T_*]) supported in that torus, with a solution smooth on [0,T_*) for which circulation and meridional velocity stay bounded while ‖(t)‖_∞ and ‖(t)‖_∞ tend to infinity and _0^T_*‖(t)‖_∞ dt=∞, 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".)

Fluid dynamics; singularity formation for incompressible flow

Can a solution of the three-dimensional incompressible Euler equations on $\mathbb R^3$ , started from smooth data and driven by a force that is smooth in space and time up to and including the blowup time, lose regularity in finite time? Finite-time singularity formation from genuinely smooth data is the central open question for the equations. Elgindi obtained blowup for $C^{1,\alpha}$ solutions in 2021, and Córdoba and Martínez-Zoroa built a multiscale program producing forced blowup for related equations with forces of limited regularity, but no construction reached three-dimensional Euler with a space-time smooth force. Note on the forced formulation, since it is easy to misread: alternatives (C) and (D) of Fefferman's Clay problem description do permit a smooth force obeying rapid space-time decay, so forcing is not a dodge and the forced route is a genuine path to the prize. It is a path for Navier-Stokes with viscosity, however, and not for Euler, which Fefferman's description excludes from the prize list.

Posed by Leon Lichtenstein (1925) and Nikolai Gunther (1927), whose local existence left global regularity open; Elgindi states the smooth-force form as his Question 1.1, 1925Open 101yModel Claude, Codex with GPT-5.6 Sol (Anthropic / OpenAI)Solved 2026-09-08

Lean-verifiedSignificance 70

Minimum-Rank and Mortality Bounds for Finite Real Matrix Monoids

ProvedPartialAI-discovered

(For any family of real n× n matrices generating a finite entire product monoid of minimum rank s, some word attains rank s within B(n,s)=n2^n-s-(n+1)2+(s-1)2 letters. The same holds over Q, without requiring a finite generating alphabet. For mortality, s=0, giving a bound of order n2^n. Over 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.)

Linear algebra

Let $n>0$ , and let a family of real $n\times n$ matrices generate a finite entire product monoid $S$ . If $s=\min_{X\in S}\operatorname{rank}X$ , then some word attains rank $s$ with length at most $B(n,s)=n2^{n-s}-\frac{n(n+1)}2+\frac{s(s-1)}2.$ In particular, if $S$ contains zero, a zero word has length at most $B(n,0)=n2^n-\frac{n(n+1)}2=\Theta(n2^n).$ Over the rationals, these bounds improve Kiefer–Ryzhikov's (2026) $3^{n^2}$ bounds for mortality and minimum-rank diameter. The mortality bound also improves Almeida–Steinberg's (2009) universal rational bound $(2n-1)^{n^2}-1$ for $n>1$ . No finiteness assumption on the generating alphabet is needed. The empty word is allowed and suffices when $s=n$ .

Posed by Jorge Almeida and Benjamin Steinberg, 2009Open 17yModel GPT-6 Astra (OpenAI)Solved 2026-09-08

Lean-checked, statement unauditedSignificance 12 Submitted by VibeGene on 8 Sep 20262

The Erdos-Borwein Constant is 2-Dense

ProvedUnder reviewAI-discovered

(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.)

Analytic number theory; digit distribution of constants

Is the Erdos-Borwein Constant 2-Dense?

Posed by Richard E. Crandall, 2002Open 24yModel ChatGPT-6 Astra (OpenAI)Solved 2026-09-07

Lean-checked, statement unauditedSignificance 25 Submitted by nufrogcaca on 7 Sep 2026

Koizumi–Liu Eventual Sign Alternation Conjecture

DisprovedAI co-developed

(Koizumi constructs a representable simple rank-6 matroid M for which (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)=(M;-t), the higher-order pole at t=i forces the Taylor coefficients of A_M(t) to fail eventual nonnegativity. Consequently, (-1)^[q^](M;q)<0 for infinitely many . The matroid is representable by twelve vectors in R^6, so it yields an actual real central hyperplane arrangement and therefore directly disproves Koizumi–Liu's conjecture for real arrangements.)

Matroid theory

Koizumi and Liu conjectured that for every real hyperplane arrangement $\mathcal A$ , the coefficients of $\operatorname{Mag}(\mathcal A;-t)$ are eventually nonnegative, equivalently that the coefficients of $\operatorname{Mag}(\mathcal A;q)$ eventually alternate in sign.

The conjecture is false. There exists a rank- $6$ real hyperplane arrangement $\mathcal A$ such that $(-1)^\ell[q^\ell]\operatorname{Mag}(\mathcal A;q)<0$ for infinitely many $\ell$ .

Posed by Junnosuke Koizumi and Ye Liu, 2026Open 0yModel GPT-5.6 Sol; GPT-6 Astra (OpenAI)Solved 2026-09-07

PreprintSignificance 12Submitted by VibeGene on 9 Sep 2026

Reciprocal-triple-free sets: the finite plateau at 732

ProvedPartialAI-discovered

(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.)

Extremal sets avoiding distinct reciprocal triples

Let f(n) be the maximum size of a subset of {1,...,n} containing no three distinct a,b,c satisfying 1/a = 1/b + 1/c. A published finite frontier for Erdős problem 302 asks whether f(732) is 606 or 607, using the existing OEIS value f(731)=606.

Posed by Finite frontier: Erdős Frontier Atlas P302, recorded July 2026. Parent problem: Erdős and Graham (1980); the parent asymptotic problem is not solved., 2026Open 0yModel OpenAI ChatGPT/Codex (exact model identifier unavailable) (OpenAI)Solved 2026-09-07

Lean-checked, statement unauditedSignificance 4Submitted by ZestyDingo473 on 7 Sep 2026

Lower bound for the complex Grothendieck constant

ProvedPartialAI-discovered

(The paper proves K_G^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_∞-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 K_* denotes the supremum obtainable within that framework, it proves 1.35584631827168<K_*<1.35584697425050. The upper endpoint here is a limitation of this specific criterion, not an upper bound on K_G^C itself.)

Functional analysis

What is the exact value of the complex Grothendieck constant $K_G^{\mathbb C}$ , the least $K$ such that $\bigl|\sum_{i,j}a_{ij}\langle x_i,y_j\rangle\bigr|\le K\max_{|\varepsilon_i|=|\delta_j|=1}\bigl|\sum_{i,j}a_{ij}\varepsilon_i\delta_j\bigr|$ for every complex matrix $(a_{ij})$ and all unit vectors $x_i,y_j$ in any complex Hilbert space? Grothendieck proved it finite in 1953. Before this work the best bounds were Davie's lower bound of about $1.33807$ and Haagerup's 1987 upper bound of about $1.40491$ ; the value is open, and this entry records progress on the lower bound.

Posed by Alexandre Grothendieck, Résumé (1953); the value of the complex constant an explicit open question at least since Haagerup's 1987 upper bound, 1953Open 73yModel Odin Automatic AI Research AgentSolved 2026-09-07

PreprintSignificance 30 Submitted by VibeGene on 12 Sep 2026

A 44-vertex triangulation of $\mathbb{RP}^{6}$

ProvedAI-assisted

(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.)

Combinatorial topology; triangulations of manifolds; convex polytopes

Does real projective 6-space admit a simplicial triangulation with fewer
than 45 vertices? This is Question 3.2 of Guyer, Steinerberger and Yang,
“An Efficient Triangulation of RP^5”, arXiv:2603.07808 (2026).

The question asks for an improvement on their 45-vertex construction,
not for the exact minimum number of vertices.

Posed by Dan Guyer, Stefan Steinerberger, Yirong Yang — Question 3.2, arXiv:2603.07808, 2026Open 0yModel ChatGPT (OpenAI, model version unstated) (OpenAI)Solved 2026-09-06

Site-confirmedSignificance 10Submitted by cheptil on 6 Sep 2026

Optimality of the added-vector code in the 19-dimensional kissing construction

ProvedAI co-developed

(The manuscript proves the upper bound |A|1280 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.)

Coding theory; algebraic graph theory; discrete geometry

Let $D\subseteq\mathbb F_2^{19}$ be the fixed 4096-word ambient binary linear code used in Ho’s 19-dimensional improvement of the Cohn–Li kissing construction. Is every subset $A\subseteq D$ with minimum Hamming distance at least $5$ of size at most $1280$ ? No linearity assumption is imposed on $A$ .

Posed by Gonzalez, Conjecture 16 (preprint, version 3, 2026), after Ho's 1280-word construction, 2026Open 0yModel ChatGPT (OpenAI, model version unstated) (OpenAI)Solved 2026-09-06

Site-confirmedSignificance 8Submitted by cheptil on 6 Sep 2026

The Ibragimov–Iosifescu conjecture for φ-mixing sequences

DisprovedUnder reviewAI-discovered

(Astra constructs a strictly stationary real process X_t=_t+K(_t-1,_t-2,…), where the innovations _t are i.i.d. Gaussian variables convolved with a symmetric rare-spike law, and K is bounded, continuous, and odd. The process is -mixing, centered, square-integrable, and satisfies (S_n)→∞. Nevertheless there are times n_j→∞ such that _n_j√()(S_n_j)0 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.)

Stationary stochastic processes

Posed by I. A. Ibragimov, 1971Open 55yModel GPT-6 Astra (pre-release) (OpenAI)Solved 2026-09-05

Lean-checked, statement unauditedSignificance 40 Submitted by VibeGene on 6 Sep 2026

Bollobás–Nikiforov conjecture

ProvedUnder reviewAI co-developed

(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 λ_1(G)^2+λ_2(G)^2 ≤ 2(1-1/(G))|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) ≤ (1-1/(G))‖B‖_F^2. The human-readable manuscript is still preliminary and is being rewritten and polished.)

Spectral graph theory

Let $G$ be a finite simple graph with $m=|E(G)|$ edges and clique number $\omega(G)$ , and let $\lambda_1(G)\ge\lambda_2(G)\ge\cdots\ge\lambda_n(G)$ be the eigenvalues of its adjacency matrix. Bollobás and Nikiforov conjectured in 2007 that every non-complete graph satisfies $\lambda_1(G)^2+\lambda_2(G)^2\le 2\Bigl(1-\frac{1}{\omega(G)}\Bigr)m.$ Before this work it was known for triangle-free graphs (Lin, Ning and Wu), regular graphs (Zhang), graphs with few triangles, complete multipartite graphs, and asymptotically almost surely for random graphs, and open in general.

Posed by Béla Bollobás and Vladimir Nikiforov, “Cliques and the spectral radius”, 2007Open 19yModel GPT-6 Astra; Grok 4.6; Claude Fable 5.1 (OpenAI; xAI; Anthropic)Solved 2026-09-05

Lean-verifiedSignificance 30 Submitted by StormyCobra617 on 11 Sep 2026

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Finite-time blowup for the 3D incompressible Euler equations with smooth forcing

ProvedAI co-developed

Fluid dynamics; singularity formation for incompressible flow

Lean-verifiedSignificance 70

Math problems solved with AI

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