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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
Question 8 of the First Proof experiment (Abouzaid et al.) asks whether a polyhedral Lagrangian surface with exactly four faces meeting at every vertex necessarily admits a Lagrangian smoothing. The research report assembles ChatGPT-suggested constructions into an affirmative argument for orientable surfaces in (R4,ω): smooth the edges, verify the vertex links are unknots with rot 0 and tb -1, and…
Let A(x) count n≤x such that every prime p∣n has a divisor d>1 of n with d≡1(modp). Erdos asked whether A(x)/x=exp(−(c+o(1))logxloglogx). It does, with c=1/(2log2).
What is the largest possible measure of a subset of a radius-R disk in R2 containing no pair of points at a positive integer distance? A Poisson-Bessel kernel argument gives M(R)≪R1/2; with Sárközy's lower construction, M(R)=R1/2+o(1).
Prim-Dijkstra routing interpolates between a minimum spanning tree and a shortest-path tree, and has been used and improved in VLSI physical design since the early 1990s, but the complexity of the terminal-only Manhattan decision problem was never settled. It is weakly NP-complete. A continuous cost-radius tradeoff with a balanced (2,2) guarantee accompanies the classification.
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.
Let S(x) count ordered pairs (a,b) with a+b≤x and σ(a)+σ(b)=σ(a+b). Erdos asked whether S(x)∼cx. The opposite extreme holds: for every R>0, S(x)/(x(logx)R)→∞, so the count beats every fixed logarithmic scale.
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
A subset A of the pointwise-ordered cube [0,1]n is a k-antichain when it meets every chain in at most k points. The conjecture concerns the largest possible (n−1)-dimensional Hausdorff measure of such a set; it is settled here, following work of Janzer.
Dogon, Levit and Vigdorovich asked for an explicit upper bound on the stability radius of an infinitely presented group. The lamplighter group provides the first: explicit polynomial bounds on both its Hilbert-Schmidt stability rate and its stability radius, obtained through approximately invariant measures and an effective marker construction.
A paper torus is an embedded polyhedral torus isometric to a flat torus. Schwartz proves no paper torus with 7 vertices exists and constructs one with 8, settling the minimum-vertex question in the flat-torus embedding tradition of Császár-torus combinatorics and the Lazarus-Tallerie universal triangulation.
For fixed d, can every d-dimensional feasible solution of the triangle-strengthened Max-Cut SDP be rounded in polynomial time with ratio strictly larger than αGW? A rounding achieving αGW+2−O(d) answers yes.
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
Sharp global and almost-everywhere convergence rates for periodic homogenization of viscous quadratic Hamilton-Jacobi equations, settling the sharpness question left open by the first-order theory.
A graph on n vertices is very well-covered if every maximal independent set has size n/2. Levit and Mandrescu conjectured that the independence polynomial i(G,x) of every very well-covered graph is unimodal, i.e. its coefficient sequence is nondecreasing and then nonincreasing.
For Pn(z)=∑k=0nεkzk with independent uniform signs, does the number Rn of roots in ∣z∣≤1 satisfy Rn/(n/2)→1 almost surely? The manuscript proves the strong law with Rn=n/2+Oω(n149/150).
What is the minimax optimal error rate for density estimation when observations are perturbed by Wasserstein-bounded contaminations? Chao and Dobriban's 2023 preprint left a gap between upper and lower bounds; the sharp rate is now derived, closing the problem.
Must every sufficiently large node set admit bounded labels that force any polynomial fitting almost all labels at degree below (1+ε)n to have arbitrarily large uniform norm? Claimed via Beurling density for Bernstein spaces.
For S(x)=#{(a,b):a+b≤x,σ(a)+σ(b)=σ(a+b)}, is S(x)∼cx? The preprint claims S(x) grows faster than x(logx)R for every fixed R, ruling out the linear asymptotic.
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.
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