Problems and conjectures

The 47 problems, conjectures and questions from the literature that results in this collection address, in order of significance, each with its source, its status and the result, followed by 7 further questions that the results study without a published source.

A status describes what the linked preprint claims. The papers have not been externally peer reviewed, no priority is claimed, and each formal check applies only to the stated scope.

Problems from the literature

Keller–Klein conjecture on Berry–Esseen bounds for Rademacher sums
Partially resolvedKeller and Klein, Advances in Mathematics 407 (2022), 108558, Conjecture 9.3 (in their proof of Tomaszewski's conjecture)
For X = Σ vᵢεᵢ with independent uniform signs, Σ vᵢ² = 1 and |vᵢ| ≤ a ≤ 7/100, P(X ≤ x) − Φ(x) ≤ Φ(a) − 1/2 for every x ≥ 0: the upper half of the Keller–Klein conjecture in this small-coefficient range.
The small-coefficient upper half of the Keller–Klein conjecture
Manuscript: The small-coefficient upper half of the Keller–Klein conjecture Lean proof: The small-coefficient upper half of the Keller–Klein conjecture Verification: The small-coefficient upper half of the Keller–Klein conjecture
Taylor–Hendrickx–Glineur Conjecture 3: last-iterate gradient norm of gradient descent
ProvedTaylor, Hendrickx and Glineur, Mathematical Programming 161 (2017), 307–345, Conjecture 3 (Section 4.1.3)
Gradient descent with step h/L, 0 ≤ h ≤ 2, on L-smooth μ-strongly convex f on any real Hilbert space satisfies ‖∇f(x_N)‖ ≤ L‖x₀ − x*‖ max{p_N(μ/L, h), |1 − h|^N} for every N, with equality attained in one dimension.
The exact last-gradient bound for constant-step gradient descent in the strongly convex case
Manuscript: The exact last-gradient bound for constant-step gradient descent in the strongly convex case Lean proof: The exact last-gradient bound for constant-step gradient descent in the strongly convex case Verification: The exact last-gradient bound for constant-step gradient descent in the strongly convex case
Glowinski's open question on the ADMM dual step size beyond the golden ratio
Partially resolvedFortin and Glowinski proved convergence for 0 < τ < (1+√5)/2 (1983); Tao and Yuan, J. Optim. Theory Appl. 179 (2018), 163–196, call the general convex case Glowinski's open question
Two-block ADMM with multiplier step τ converges for every 0 < τ ≤ 1.886795 on finite-dimensional convex problems with attained proximal steps and a saddle point; fixed polytopes in R^128 cycle for every τ ≥ 1.8868840…; scalar problems converge exactly for 0 < τ < 2
A narrow interval for the universal multiplier step of two-block ADMM
Manuscript: A narrow interval for the universal multiplier step of two-block ADMM Lean proof: A narrow interval for the universal multiplier step of two-block ADMM Verification: A narrow interval for the universal multiplier step of two-block ADMM
Finite collapse of the Lami–Mele–Regula upper PPT entanglement-cost hierarchy
Answered negativelyLami, Mele and Regula, Physical Review Letters 134, 090202 (2025), Supplemental Material (the finite-collapse question accompanying Conjecture S33); left open for the upper κ hierarchy by Brinster, Kondra, Kampermann, Bruß and Wyderka (arXiv:2609.20698)
For every q ≥ 1 a full-rank product-Bell-diagonal state in local dimension 2^((2q+4)4^q) has κ_{q+1}(ρ) < κ_q(ρ), so no fixed finite level of the upper PPT entanglement-cost hierarchy is exact for all states.
No universal finite collapse of the upper PPT entanglement-cost hierarchy
Manuscript: No universal finite collapse of the upper PPT entanglement-cost hierarchy Lean proof: No universal finite collapse of the upper PPT entanglement-cost hierarchy Verification: No universal finite collapse of the upper PPT entanglement-cost hierarchy
Nart–Novacoski conjecture on depth-one Artin–Schreier defect towers
ProvedNart and Novacoski, 'Depth of Artin–Schreier defect towers', J. Pure Appl. Algebra 230 (2026), 108184 (arXiv:2503.18827), Conjecture 4.1
Over a perfect henselian K ⊆ k((y^ℚ)) containing k(y) without finite accumulation points (k algebraically closed, char. p), each nontrivial finite Artin–Schreier tower of depth one is a compositum of Artin–Schreier extensions.
Depth-one Artin–Schreier defect towers are composita
Manuscript: Depth-one Artin–Schreier defect towers are composita Lean proof: Depth-one Artin–Schreier defect towers are composita Verification: Depth-one Artin–Schreier defect towers are composita
Dodig–Stošić conjecture on bounded rank perturbations of matrix pencils
ProvedDodig and Stošić, 'Is the bounded rank perturbations problem for matrix pencils just a completion problem?', Electron. J. Linear Algebra 41 (2025), 491–510, Conjecture 1
Over any infinite field, pencils A, B have δ(A,B) ≤ x exactly when K_{c,d}(A) is a constant full-rank compression of K_{c,d}(B) ⊕ (t − d)I_x; a cited linearization theorem then gives Conjecture 1 over algebraically closed fields.
Matrix-pencil rank distance as an explicit completion problem
Manuscript: Matrix-pencil rank distance as an explicit completion problem Lean proof: Matrix-pencil rank distance as an explicit completion problem Verification: Matrix-pencil rank distance as an explicit completion problem
Vandaele–Gillis–Glineur–Tuyttens conjecture rank₊(M ⊗ M) ≥ rank₊(M) rank(M)
DisprovedVandaele, Gillis, Glineur and Tuyttens, 'Heuristics for exact nonnegative matrix factorization', J. Global Optim. 65 (2016), 369–400, Conjecture 1 (Section 6.1)
For every k ≥ 0 an explicit rational nonnegative matrix C of order k + 11, rank k + 10 and nonnegative rank k + 11 has rank₊(C ⊗ C) ≤ (k + 10)² + 9, at least k + 1 below the conjectured bound rank₊(C) rank(C).
Counterexamples to the Kronecker-square lower bound for nonnegative rank
Manuscript: Counterexamples to the Kronecker-square lower bound for nonnegative rank Lean proof: Counterexamples to the Kronecker-square lower bound for nonnegative rank Verification: Counterexamples to the Kronecker-square lower bound for nonnegative rank
Real-symmetric case of the Bapat–Lal q-permanent monotonicity conjecture on [−1,1]
Disproved for real symmetric positive-definite matricesBapat and Lal, Linear Algebra Appl. 197–198 (1994); extended by da Fonseca (2010, Conjecture 1; 2018, Conjecture 2)
An explicit integer positive-definite matrix of order 896 has a q-permanent that strictly decreases between two explicit rationals in (0,1); counterexamples exist in every order N ≥ 896.
Finite real counterexamples and spectral obstructions for q-permanent monotonicity
Manuscript: Finite real counterexamples and spectral obstructions for q-permanent monotonicity Lean proof: Finite real counterexamples and spectral obstructions for q-permanent monotonicity Verification: Finite real counterexamples and spectral obstructions for q-permanent monotonicity
Lei Yu's problem on the level-1 Fourier weight W(1/8) of Boolean functions of density 1/8
Determined exactlyLei Yu, 'On Average Distance, Level-1 Fourier Weight, and Chang's Lemma', arXiv:2504.02593 (v2, 2026 revision), Section VII, Problem 2
Every Boolean function of mean 1/8 on {−1,1}^n has level-one Fourier weight at most 3/64, attained by a codimension-three subcube for n ≥ 3; equivalently, density-1/8 sets have average Hamming distance ≥ (n − 3)/2.
The sharp level-one Fourier weight at density one eighth
Manuscript: The sharp level-one Fourier weight at density one eighth Lean proof: The sharp level-one Fourier weight at density one eighth Verification: The sharp level-one Fourier weight at density one eighth
Boundedness of Nesterov ODE trajectories and iterates below critical damping r < 3
Partially resolvedRecorded as open by Jang and Ryu, 'Point convergence of Nesterov's accelerated gradient method: an AI-assisted proof', arXiv:2510.23513 (version 2, 2026); the equation is due to Su, Boyd and Candès
On ℓ²(ℕ;ℝ²), Nesterov flows from rest and iterates with momentum k/(k+r) stay bounded for all convex objectives with 1-Lipschitz gradient and attained minimum exactly when r ≥ 3; a planar potential escapes for 0 < r ≤ 2.
Escaping Nesterov flows and iterates below critical damping
Manuscript: Escaping Nesterov flows and iterates below critical damping Lean proof: Escaping Nesterov flows and iterates below critical damping Verification: Escaping Nesterov flows and iterates below critical damping
Leung–Wang question on the rate of general universal entanglement embezzling families
Answered positivelyLeung and Wang, 'Characteristics of universal embezzling families', Physical Review A 90, 042331 (2014), Section 6
For each L ≥ 1 one target-independent catalyst of local dimension n = 2^(4L) lets local basis permutations reach root fidelity at least 1 − 125m/L² = 1 − 2000m/(log₂ n)² for every Schmidt probability vector on m entries.
Universal entanglement embezzlement with inverse-quadratic logarithmic infidelity
Manuscript: Universal entanglement embezzlement with inverse-quadratic logarithmic infidelity Lean proof: Universal entanglement embezzlement with inverse-quadratic logarithmic infidelity Verification: Universal entanglement embezzlement with inverse-quadratic logarithmic infidelity
Spectral independence of the hard-core model at the uniqueness threshold λ_c(Δ)
Partially resolvedBencs, Goldberg, Jenssen, Jerrum, Pete, Regts and Yin, arXiv:2608.09729, Section 5 (critical two-spin question)
For every Δ ≥ 3, every n-vertex graph of maximum degree at most Δ, every activity 0 < λ ≤ λ_c(Δ) and every feasible pinning, each absolute row sum of the hard-core influence matrix is O_Δ(√n log n), with an explicit constant.
Critical hard-core influence bounds via nonbacktracking collisions
Manuscript: Critical hard-core influence bounds via nonbacktracking collisions Lean proof: Critical hard-core influence bounds via nonbacktracking collisions Verification: Critical hard-core influence bounds via nonbacktracking collisions
Gyires's permanent inequality for doubly stochastic matrices (Cheon–Wanless Conjecture 20)
Partially resolvedGyires (1978); listed as Conjecture 20 in Cheon and Wanless (2005)
Proves Gyires's permanent inequality for nonnegative circulants of orders 5, 6 and 7, permutation polynomials aI+bQ+cQ² and four-block matrices in all orders, one-exceptional-column/row matrices, and rank ≤ 2 through order 15.
Gyires's permanent inequality for structured matrices
Manuscript: Gyires's permanent inequality for structured matrices Lean proof: Gyires's permanent inequality for structured matrices Verification: Gyires's permanent inequality for structured matrices
Small-extrapolation convergence region of the Chambolle–Pock (PDHG) method
CounterexampleLeft open in the conclusion of Upadhyaya, arXiv:2604.06423v2 (improving the small-θ region)
For unmodified Chambolle–Pock, every step product κ = τσ‖L‖² > 2 admits polytopes in ℝ⁴ with periodic nonconvergent iterates for all small θ ≥ 0, so any universal condition κ < c(θ) has limsup c(θ) ≤ 2 as θ ↓ 0.
Periodic obstructions to small-extrapolation Chambolle–Pock
Manuscript: Periodic obstructions to small-extrapolation Chambolle–Pock Lean proof: Periodic obstructions to small-extrapolation Chambolle–Pock Verification: Periodic obstructions to small-extrapolation Chambolle–Pock
Classify the connected graphs with exactly five Steinerberger boundary vertices
Solved with a computer-assisted proof: 227 reduced graphs; subcubic six-boundary graphs and minimum-boundary two-trees also classifiedChiem, Dudarov, Lee, Lee and Liu, A characterization of graphs with at most four boundary vertices, J. Combin. 15 (2024), Problem 6.1; boundary defined by Steinerberger, Discrete Appl. Math. 338 (2023)
A connected graph has exactly five boundary vertices if and only if it is a bridge subdivision of one of 227 reduced graphs; two-connected graphs with at most five boundary vertices have diameter at most two.
Graphs with five harmonic boundary vertices, subcubic graphs with six, and minimum-boundary two-trees
Manuscript: Graphs with five harmonic boundary vertices, subcubic graphs with six, and minimum-boundary two-trees Lean proof: Graphs with five harmonic boundary vertices, subcubic graphs with six, and minimum-boundary two-trees Verification: Graphs with five harmonic boundary vertices, subcubic graphs with six, and minimum-boundary two-trees
Boege's selfadhesivity question: structural vs numerical selfadhesive polymatroids
Answered negativelyBoege, IJAR 2023 (arXiv:2205.07667), Question 4.18 (4.14 in the PDF version)
An explicit five-variable conditional-independence model with 26 of the 80 elementary CI statements is structurally selfadhesive at every overlap, yet no real polymatroid inducing it is selfadhesive in Matúš's numerical sense.
Structural selfadhesivity does not imply a numerically selfadhesive realization
Manuscript: Structural selfadhesivity does not imply a numerically selfadhesive realization Lean proof: Structural selfadhesivity does not imply a numerically selfadhesive realization Verification: Structural selfadhesivity does not imply a numerically selfadhesive realization
Berge-path saturation conjecture (English–Graber–Kirkpatrick–Methuku–Sullivan, Conjecture 2.4)
Disproved for three-uniform hypergraphs; exact formula givenEnglish, Graber, Kirkpatrick, Methuku and Sullivan, Saturation of Berge hypergraphs, arXiv:1710.03735v1, Conjecture 2.4
Two loose 8-cycles plus two isolated vertices give a Berge-P₁₀-saturated 3-graph on 34 vertices with 16 edges, not 17; the exact 3-uniform formula holds for all m ≥ 10, n ≥ 2a_m + 1.
Exact three-uniform Berge-path saturation numbers and a counterexample to a conjectured formula
Manuscript: Exact three-uniform Berge-path saturation numbers and a counterexample to a conjectured formula Lean proof: Exact three-uniform Berge-path saturation numbers and a counterexample to a conjectured formula Verification: Exact three-uniform Berge-path saturation numbers and a counterexample to a conjectured formula
Berge saturation numbers of trees and the Berge analogue of the minimum-tree theorem of Faudree, Faudree, Gould and Jacobson
Exact formula proved in every uniformity; minimality among trees proved for three-uniform hypergraphsEnglish, Graber, Kirkpatrick, Methuku and Sullivan, Saturation of Berge hypergraphs, arXiv:1710.03735; graph case: Faudree, Faudree, Gould and Jacobson, Saturation numbers for trees, Electron. J. Combin. 16 (2009), R91
sat_r(n, Berge-B_a) = ⌊a(n + 1)/((r − 1)a + 1)⌋ for a, r ≥ 3 and n ≥ (r − 1)a + 1; for a ≥ 5, B_a is the unique tree on a + 2 vertices with the least three-uniform saturation coefficient.
Exact Berge saturation numbers of the subdivided star in every uniformity and its minimality among trees
Manuscript: Exact Berge saturation numbers of the subdivided star in every uniformity and its minimality among trees Lean proof: Exact Berge saturation numbers of the subdivided star in every uniformity and its minimality among trees Verification: Exact Berge saturation numbers of the subdivided star in every uniformity and its minimality among trees
Leighton–Moitra entropy conjecture for rankings: an explicit quartic ε⁴ entropy deficit
SharpenedLeighton and Moitra (qualitative conjecture, proved by Acan, Devlin and Kahn); a Cε⁴ deficit is proposed in Devlin's research statement
Every probability distribution on rankings of n labels in which each pairwise comparison has probability at distance at least ε > 0 from 1/2 has entropy at most (1 − ε⁴/144) log(n!), for every n.
A quartic entropy deficit for random rankings
Manuscript: A quartic entropy deficit for random rankings Lean proof: A quartic entropy deficit for random rankings Verification: A quartic entropy deficit for random rankings
Ferroni–Morales–Panova conjecture on positivity of order polynomials of width-two posets
Partially resolvedFerroni, Morales and Panova, Skew shapes, Ehrhart positivity and beyond, arXiv:2503.16403v3, Conjecture 7.1; Proceedings of the London Mathematical Society 133 (2026), e70204
If a finite poset P is the union of two chains, one with at most five elements and the other arbitrary, then |P|! Ω_P(t) has nonnegative coefficients, and [t]Ω_P(t) ≥ (a − 1)!(b − 1)! m/(a + b)!.
Order-polynomial positivity for width-two posets with a chain of at most five elements
Manuscript: Order-polynomial positivity for width-two posets with a chain of at most five elements Lean proof: Order-polynomial positivity for width-two posets with a chain of at most five elements Verification: Order-polynomial positivity for width-two posets with a chain of at most five elements
Three-uniform Berge-C₄ saturation number (English–Graber–Kirkpatrick–Methuku–Sullivan)
Linear coefficient bounded between 5/8 and 3/4; exact values for n ≤ 13English, Graber, Kirkpatrick, Methuku and Sullivan, Saturation of Berge hypergraphs, arXiv:1710.03735v1, Theorem 5.7 (upper bound n − 2)
For n ≥ 9, (5n − 3623)/8 ≤ sat₃(n, Berge-C₄) ≤ ⌈(3n − 9)/4⌉; saturated cones are classified and the values for n ≤ 13 are exact.
Three-uniform Berge-C₄ saturation: linear bounds with coefficients 5/8 and 3/4, the cone spectrum and exact small values
Manuscript: Three-uniform Berge-C₄ saturation: linear bounds with coefficients 5/8 and 3/4, the cone spectrum and exact small values Lean proof: Three-uniform Berge-C₄ saturation: linear bounds with coefficients 5/8 and 3/4, the cone spectrum and exact small values Verification: Three-uniform Berge-C₄ saturation: linear bounds with coefficients 5/8 and 3/4, the cone spectrum and exact small values
Galvin–Zhang conjecture on ultra-log-concave domination sequences of path and cycle powers
ProvedGalvin and Zhang, arXiv:2408.12731, Conjecture 1.8; also Zhang's 2025 dissertation, Conjecture 4.1.8
For every n ≥ 0 and power ℓ ≥ 1, the number d_k of size-k dominating sets of P_n^ℓ and of C_n^ℓ satisfies (k+1)(n−k+1)d_{k−1}d_{k+1} ≤ k(n−k)d_k², so d_k/binom(n,k) is log-concave.
Ultra-log-concavity of domination sequences of powers of paths and cycles
Manuscript: Ultra-log-concavity of domination sequences of powers of paths and cycles Lean proof: Ultra-log-concavity of domination sequences of powers of paths and cycles Verification: Ultra-log-concavity of domination sequences of powers of paths and cycles
Multi-marginal converse of Minty's theorem (Bartz–Bauschke–Phan–Wang, Remark 2.6)
Answered negativelyBartz, Bauschke, Phan and Wang, Math. Program. 185 (2021), Remark 2.6
For three marginals on ℝ² with cost Σ⟨xᵢ, xⱼ⟩, an explicit three-point c-monotone set admits no compatible zero-sum tuple, so every maximally c-monotone set containing it has a nonconvex sum image omitting 0.
The multi-marginal converse of Minty's theorem fails in the plane
Manuscript: The multi-marginal converse of Minty's theorem fails in the plane Lean proof: The multi-marginal converse of Minty's theorem fails in the plane Verification: The multi-marginal converse of Minty's theorem fails in the plane
Saturation spectrum of the Berge triangle in triple systems (Bushaw–English–Heath–Johnston–Rombach)
Partially resolvedBushaw, English, Heath, Johnston and Rombach, The saturation spectrum of Berge stars, arXiv:2502.17686 (2025), Section 6
For n ≥ 8 every saturated Berge-triangle-free triple system with more than ⌊(n−1)²/8⌋ edges is a paired product F(n,p), so the spectrum above this sharp threshold is exactly {p(n−2p)}.
The upper saturation spectrum of Berge triangles in triple systems
Manuscript: The upper saturation spectrum of Berge triangles in triple systems Lean proof: The upper saturation spectrum of Berge triangles in triple systems Verification: The upper saturation spectrum of Berge triangles in triple systems
Blocking sets with bounded line intersections in finite projective planes (Erdős Problem #1159), Singer-symmetric case in PG(2,13)
Determined exactlyP. Erdős; Erdős Problem #1159, erdosproblems.com
Every blocking set of PG(2,13) invariant under the fixed-point-free subgroup of order three of a Singer group meets some line in at least five points, and the 21-point Blokhuis–Brouwer–Wilbrink set attains five.
Blocking sets of PG(2,13) invariant under a fixed-point-free Singer subgroup meet some line in five points
Manuscript: Blocking sets of PG(2,13) invariant under a fixed-point-free Singer subgroup meet some line in five points Lean proof: Blocking sets of PG(2,13) invariant under a fixed-point-free Singer subgroup meet some line in five points Verification: Blocking sets of PG(2,13) invariant under a fixed-point-free Singer subgroup meet some line in five points
Daans–Kala–Man–Widmer–Yatsyna Question 3.4 on represented unit square classes
Answered negativelyDaans, Kala, Man, Widmer and Yatsyna, PNAS 122(20), e2419414122 (2025), Question 3.4
A totally positive definite quadratic lattice of rank n over any totally real algebraic field, with integral norms but not necessarily classical, represents at most n totally positive unit square classes, so n + 1 cannot occur.
A rank bound for represented totally positive unit squareclasses
Manuscript: A rank bound for represented totally positive unit squareclasses Lean proof: A rank bound for represented totally positive unit squareclasses Verification: A rank bound for represented totally positive unit squareclasses
Wan–Wright–Wu log₂ k variance bound for read-k decision-tree covariance
DisprovedWan, Wright and Wu, ITCS 2014, Section 2.4; called a conjecture by Shalev, arXiv:1806.03646v2, Section 6.2
Read-k decision trees with no repeated coordinate on a path have tree covariance at most I(f) log₂ k, sharp for k = 2^d, while exact read-3 trees refute the proposed bound Var(f) log₂ k.
Logarithmic influence bounds and variance counterexamples for Boolean decision-tree covariance
Manuscript: Logarithmic influence bounds and variance counterexamples for Boolean decision-tree covariance Lean proof: Logarithmic influence bounds and variance counterexamples for Boolean decision-tree covariance Verification: Logarithmic influence bounds and variance counterexamples for Boolean decision-tree covariance
Lachaume's diagonal concavity conjecture for sums of elementary symmetric polynomials
DisprovedLachaume, arXiv:1712.10327v1, Conjecture 1
For f = 1 + Mσ₁ + σ₂ + Mσ₃ + σ₄ in four variables with M > 0, f^(1/4) is concave on the positive diagonal exactly when M ≥ 1 and on the whole positive orthant exactly when M = 1, so every M > 1 is a counterexample.
Diagonal concavity does not imply concavity for sums of elementary symmetric polynomials
Manuscript: Diagonal concavity does not imply concavity for sums of elementary symmetric polynomials Lean proof: Diagonal concavity does not imply concavity for sums of elementary symmetric polynomials Verification: Diagonal concavity does not imply concavity for sums of elementary symmetric polynomials
Beirami et al.'s best-of-n win-rate versus KL divergence conjecture (Conjecture 7.2)
DisprovedBeirami et al., ICML 2025 (PMLR 267), Conjecture 7.2 and Eq. (28)
For every budget n ≥ 2 and cap δ > 0, a finite reference law with all probabilities ≤ δ has a best-of-n policy with D < ℓ(W), violating W ≤ ℓ⁻¹(D); for n = 2, p = (2/5,1/5,1/5,1/5) gives W = 163/250.
Counterexamples to a best-of-n win-rate bound at every budget
Manuscript: Counterexamples to a best-of-n win-rate bound at every budget Lean proof: Counterexamples to a best-of-n win-rate bound at every budget Verification: Counterexamples to a best-of-n win-rate bound at every budget
Da Fonseca's q-permanent (μ-permanent) monotonicity conjecture
DisprovedDa Fonseca (2010), Conjecture 1; restated as Conjecture 2 in Da Fonseca (2018)
A real symmetric positive-definite 5×5 matrix has q-permanent 8,968,822,249 at q = 14 and 7,441,960,000 at q = 15; for each real q₀ > 1 a positive-definite matrix (size depending on q₀) has negative derivative at q₀.
Real positive-definite counterexamples to extended q-permanent monotonicity
Manuscript: Real positive-definite counterexamples to extended q-permanent monotonicity Lean proof: Real positive-definite counterexamples to extended q-permanent monotonicity Verification: Real positive-definite counterexamples to extended q-permanent monotonicity
Letouzey–Li–Steiner late-iterate ordering conjecture for generalized Hofstadter functions
DisprovedLetouzey, Li and Steiner, Journal of Integer Sequences 29 (2026), Article 26.3.3, conjecture before Lemma 7.6
For every k ≥ 4 some input n has Fₖ^(2k+1)(n) > Fₖ₊₁^(2k+2)(n), contradicting the conjectured order Fₖ^j ≤ Fₖ₊₁^(j+1) for j > 2k; an explicit witness is F₄⁹(132) = 8 > 7 = F₅¹⁰(132).
Counterexamples to a late-iterate ordering conjecture for generalized Hofstadter functions
Manuscript: Counterexamples to a late-iterate ordering conjecture for generalized Hofstadter functions Lean proof: Counterexamples to a late-iterate ordering conjecture for generalized Hofstadter functions Verification: Counterexamples to a late-iterate ordering conjecture for generalized Hofstadter functions
Convergence of two-block ADMM with multiplier (dual) step size τ in (0, 2)
CounterexampleQuestion commonly associated with Glowinski; discussed by Tao and Yuan, J. Optim. Theory Appl. 179 (2018)
For two-block ADMM with penalty one and multiplier-only step τ = 199/100, two explicit compact convex polytopes in ℝ⁴ give a nonconstant period-five orbit with squared primal residual at least 500000, so ADMM fails to converge.
A rational five-cycle for two-block ADMM with multiplier step below two
Manuscript: A rational five-cycle for two-block ADMM with multiplier step below two Lean proof: A rational five-cycle for two-block ADMM with multiplier step below two Verification: A rational five-cycle for two-block ADMM with multiplier step below two
Zhao's root log-concavity conjecture for alternating sums of Motzkin numbers
ProvedFeng-Zhen Zhao, INTEGERS 26 (2026), A55, Conjecture 1
With Tₙ the alternating sum of the Motzkin numbers M₀,…,Mₙ, the sequence Tₙ^(1/n) is strictly log-concave from n = 5 and the inequality reverses at center 5, so 5 is optimal; this contains Zhao's tail from n = 9.
Sharp root log-concavity of alternating Motzkin sums
Manuscript: Sharp root log-concavity of alternating Motzkin sums Lean proof: Sharp root log-concavity of alternating Motzkin sums Verification: Sharp root log-concavity of alternating Motzkin sums
Bounded-forcing log-concavity for Motzkin-type recurrences and alternating Motzkin sums
New theoremMotivated by Zhao, INTEGERS 26 (2026), A55, Conjecture 1 (p. 8); the extensions answer no posed question
Real aₙ with (m+2)aₘ = m(2aₘ₋₁+3aₘ₋₂)+εₘ, |εₘ| ≤ 2 for m ≥ 14, a₁₂ > 1800 and L₁₂ < a₁₃/a₁₂ < U₁₂ satisfy aₙ > 8(n+3)² and a ratio log-concavity margin 800aₙ⁴/(n+3)³ for all n ≥ 12.
Sharp log-concavity for alternating Motzkin sums
Manuscript: Sharp log-concavity for alternating Motzkin sums Lean proof: Sharp log-concavity for alternating Motzkin sums Verification: Sharp log-concavity for alternating Motzkin sums
Coalesced clause-mass certificate for resolution trees (Kunisky's Komlós approach)
Determined exactlyAttributed to Kunisky, arXiv:2111.02974v1, Conjecture 4.4 and Eq. (41); literal correspondence unverified
For every n ≥ 3, the coalesced clause-mass certificate over finite unrestricted resolution refutations of the formula of all 2ⁿ full signed clauses has exact supremum n + 1; a three-variable witness exceeds 61/25 > 1 + √2.
Unbounded coalesced-mass certificates for resolution trees
Manuscript: Unbounded coalesced-mass certificates for resolution trees Lean proof: Unbounded coalesced-mass certificates for resolution trees Verification: Unbounded coalesced-mass certificates for resolution trees
Ordered-instance reduction for allocating indivisible goods with extra copies
New theoremAttributed to Akrami et al., arXiv:2502.09377v3; source conventions unverified
Any set-valued ordered allocation under nonnegative additive valuations transfers to named goods with every agent's utility weakly higher, the same total number of extra copies, the same per-good cap, and every good allocated.
Ordered-instance allocation transfer with exact extra-copy preservation
Manuscript: Ordered-instance allocation transfer with exact extra-copy preservation Lean proof: Ordered-instance allocation transfer with exact extra-copy preservation Verification: Ordered-instance allocation transfer with exact extra-copy preservation
Burcroff–O'Brien leaf-deletion mode question for tree domination polynomials
CounterexampleAttributed to Burcroff and O'Brien, Discrete Math. 346 (2023), 113508, Question 8.1 (arXiv version), and Zhang, arXiv:2111.00641v2, Conjecture 4.1; numbering unverified
A tree on 7284 vertices has a unimodal domination polynomial with unique mode 3978, while deleting one specified leaf gives a 7283-vertex tree whose domination polynomial is unimodal with unique mode 3976.
Leaf deletion can move the unique mode of a tree domination polynomial by two
Manuscript: Leaf deletion can move the unique mode of a tree domination polynomial by two Lean proof: Leaf deletion can move the unique mode of a tree domination polynomial by two Verification: Leaf deletion can move the unique mode of a tree domination polynomial by two
Ptaszyński's conjectured interval −1/16 ≤ C₃/C₁ ≤ 1 for unicyclic Markov currents
ProvedK. Ptaszyński, Phys. Rev. E 106, 024119 (2022), Eq. (21)
For the net crossing count of a marked edge on every finite bidirectional Markov ring with n ≥ 3 states, positive rates and unequal forward and reverse rate products, −1/16 ≤ C₃/C₁ ≤ 1, and both constants are sharp.
A sharp universal skewness bound for unicyclic currents
Manuscript: A sharp universal skewness bound for unicyclic currents Lean proof: A sharp universal skewness bound for unicyclic currents Verification: A sharp universal skewness bound for unicyclic currents
Power ranks of constant-recursive sequences: Rowland–Sistos Barrón finiteness conjecture
CounterexampleRowland and Sistos Barrón, 'Complexity of powers of a constant-recursive sequence' (arXiv:2501.14643), unnumbered fixed-rank finiteness conjecture on page 2 of the author-hosted preprint
s(n) = 1 + 2^n + 3^n(1 − 2^n)² + n9^n(1 + 2^n)³ has rank 13 and power ranks P(M) − 2Q(M), not eventually quasipolynomial; a companion family gives infinitely many power-rank sequences at each rank r ≥ 3.
A positive constant-recursive sequence with nonquasipolynomial power ranks
Manuscript: A positive constant-recursive sequence with nonquasipolynomial power ranks Lean proof: A positive constant-recursive sequence with nonquasipolynomial power ranks Verification: A positive constant-recursive sequence with nonquasipolynomial power ranks
Higuchi–Sudbery problem: maximal average two-qubit marginal entropy of four qubits
ProvedHiguchi and Sudbery, 'How entangled can two couples get?', Physics Letters A 273 (2000), 213–217, which proposed the candidate maximizing state
For every pure four-qubit state the average von Neumann entropy of the three two-qubit marginals ρ_AB, ρ_AC, ρ_AD is at most log 2 + (1/2) log 3 (natural logarithms), and the Higuchi–Sudbery state attains it.
A sum-of-squares proof of the four-qubit Higuchi–Sudbery entropy bound
Manuscript: A sum-of-squares proof of the four-qubit Higuchi–Sudbery entropy bound Lean proof: A sum-of-squares proof of the four-qubit Higuchi–Sudbery entropy bound Verification: A sum-of-squares proof of the four-qubit Higuchi–Sudbery entropy bound
Fannes–de Melo–Roga–Życzkowski conjecture: root-fidelity matrix entropy bounds Holevo χ
ProvedFannes, de Melo, Roga and Życzkowski, 'Matrices of fidelities for ensembles of quantum states and the Holevo quantity', Quantum Inf. Comput. 12 (2012), 472–489, Conjecture 1, eq. (18)
For any three density matrices ρᵢ of finite dimension and probabilities pᵢ, the Holevo quantity S(Σ pᵢρᵢ) − Σ pᵢS(ρᵢ) is at most the entropy of the 3 × 3 matrix C_ij = √(pᵢpⱼ) ‖√ρᵢ √ρⱼ‖₁.
The three-state root-fidelity entropy bound
Manuscript: The three-state root-fidelity entropy bound Lean proof: The three-state root-fidelity entropy bound Verification: The three-state root-fidelity entropy bound
Ferguson–Iosevich Question 8.1: subspace tiling complements of spectral sets in F₂⁹
CounterexampleFerguson and Iosevich, arXiv:2610.36417, Question 8.1 (subspace-complement part) and its equivalent Question 8.2
An explicit 64-point set E ⊂ F₂⁹ is spectral and tiles F₂⁹, but its tiling complements are exactly the 512 translates of one aperiodic 8-point set, so no subspace complements E and F₂⁹ fails the periodic tiling property.
A 64-point spectral tile in F₂⁹ with no periodic complement
Manuscript: A 64-point spectral tile in F₂⁹ with no periodic complement Lean proof: A 64-point spectral tile in F₂⁹ with no periodic complement Verification: A 64-point spectral tile in F₂⁹ with no periodic complement
Ptaszyński's heat-current kurtosis bound C₄/C₂ ≥ 0 for thermal Markov systems
CounterexamplePtaszyński, Physical Review E 106, 024119 (2022), Eq. (19)
An explicit three-state, two-bath thermal rate model with local detailed balance has spectral heat-cumulant ratio C₄/C₂ = −3470947982622836/176627429595846691 < 0, contradicting Eq. (19) in its tilted-generator form.
A three-state thermal counterexample to a heat-current kurtosis conjecture
Manuscript: A three-state thermal counterexample to a heat-current kurtosis conjecture Lean proof: A three-state thermal counterexample to a heat-current kurtosis conjecture Verification: A three-state thermal counterexample to a heat-current kurtosis conjecture
Zhang–Zhao DomiRank entropy monotonicity conjecture for disassortative graphs
DisprovedZhang and Zhao, arXiv:2610.00107v4, Conjecture 4.9
A connected graph with 14 vertices and 32 edges has negative degree–log-degree edge covariance and positive DomiRank scores on [251/1000, 63/250], yet its entropy increases by more than 1/10000 nats across that interval.
An entropy increase for a structurally disassortative DomiRank graph
Manuscript: An entropy increase for a structurally disassortative DomiRank graph Lean proof: An entropy increase for a structurally disassortative DomiRank graph Verification: An entropy increase for a structurally disassortative DomiRank graph
Projection AM–GM inequality in Loewner order for symmetrized products of projections
CounterexampleMotivated by Sun, Luo and Ye, Math. Oper. Res. 45(1) (2020); exact match with their published conjecture is conditional
Three rational rank-one orthogonal projections in ℝ³ have symmetrized product G ⋠ M³, M their mean (quadratic form −8/243), and for every real C some such triple with positive-definite mean has G ⋠ C·M³.
An unbounded obstruction to projection AM–GM in Loewner order
Manuscript: An unbounded obstruction to projection AM–GM in Loewner order Lean proof: An unbounded obstruction to projection AM–GM in Loewner order Verification: An unbounded obstruction to projection AM–GM in Loewner order
Pure-state ECQC: matched-basis mutual-information inequality with mutually unbiased bases
CounterexampleAttributed to Iqbal, Quantum Information Processing 25 (2026), Article 250, Conjecture 3.1; correspondence unverified
A normalized pure state in local dimension 5 with six fixed mutually unbiased bases, used identically by both parties, has five-setting classical information sum 5 log 5 versus quantum mutual information 2 log 5 (ratio 5/2).
A pure-state matched-basis information inequality violation in dimension five
Manuscript: A pure-state matched-basis information inequality violation in dimension five Lean proof: A pure-state matched-basis information inequality violation in dimension five Verification: A pure-state matched-basis information inequality violation in dimension five
Kovačec's polygon enclosure conjecture for determinants of sums of normal matrices
CounterexampleKovačec, Universidade de Coimbra preprint 26-11 (April 2026), Conjecture 7; literal statement unverified
Normal 3×3 matrices with spectra (0, 1, i) and (−1−i, 1, −1+i) give det(A+B) = 2/3 + 2i/9, outside every bounded component closure of ℂ∖Γ, Γ the exterior boundary of the transposition graph, yet inside the permutation-value hull.
A polygon enclosure fails for sums of normal matrices
Manuscript: A polygon enclosure fails for sums of normal matrices Lean proof: A polygon enclosure fails for sums of normal matrices Verification: A polygon enclosure fails for sums of normal matrices

Problems from the literature

Other questions studied