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Near-optimal rather than optimal: the bounds match up to logarithmic-type factors.
Four results, and the first is partly a refutation. Jakimiuk conjectured is optimal for every ; the paper proves that for and gives a counterexample for every , so the conjecture is false as posed and the corrected range is . The witness is the two-coordinate vector . The Baranski-Murawski-Nayar-Oleszkiewicz flat-po…
Nazarov conjectured that for the quadratic form of the spectral fractional Dirichlet Laplacian strictly increases under when changes sign. Proved and substantially generalized, with the same conclusion for the restricted form.
Pavez-Signe (2024) conjectured a Dirac-type condition for spanning -subdivisions and asked whether the subdivision paths can additionally be required to have similar lengths; Lee (2025) resolved the existence conjecture in the stronger digraph setting. Answered affirmatively with epsilon-room: for every there is such that every -vertex digraph with and minimum semi-deg…
Every purely-maximal ideal of a commutative ring is purely-prime, and the converse holds for several important classes of rings; Tarizadeh conjectured (Conjecture 5.8 of his earlier published paper) that in a commutative ring every purely-prime ideal is purely-maximal. False: there is a commutative ring with a purely-prime ideal that is not purely-maximal.
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 ever…
The claim is for every , reduced to three finite facts: the base value , and the unsatisfiability of a 321-position and a 521-position spoke template. The reduction is Lean-checked and holds for every ; 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 sc…
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.
Aldroubi, Cabrelli, Krishtal and Molter conjectured that for a bounded normal operator and any vector , the normalized orbit is never a frame. It can be: an explicit construction produces a normalized orbit that is a frame.
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.
Does a general pencil of plane cubics over have exactly common flex lines? Ciliberto, Miranda and Roé asked this in Remark 5.3 of their paper; the answer is yes.
For an irreducible crystallographic root system of rank with Coxeter number , the paper proves that Au's normalized Witten zeta function has a simple pole at and evaluates its residue in closed form in terms of the Cartan determinant, the Weyl group order and the invariant degrees.
Coble and Barg introduced binary Coxeter codes, the span of indicators of standard cosets of fixed rank in a finite Coxeter system, generalizing Reed-Muller codes, and proposed a conjectural value for the minimum distance of a general Coxeter code. The conjecture is true, and it yields a decoding consequence.
Gill introduced the probabilistic automatic complexity of a string: the least number of states of a probabilistic finite automaton for which is the unique most probably accepted string of its length. He asked whether is unbounded, no string with being known. The paper proves for every string over every finite alphabet, with an explicit three-state witness.
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 chain…
Can the minimum edge-outerplanarity of a finite loopless planar graph, minimized over all planar embeddings, be computed in polynomial time? Asked by Bentz in 2009.
Ross introduced -perfect numbers, integers expressible as over their proper divisors with coefficients in , and conjectured that they have the same density as the nondeficient numbers, plus a second conjecture relating odd nondeficient numbers to -perfection. Both are false.
22 conjectures of Cohen about cyclic numbers (integers with ) settled at once - 16 proved, 6 disproved - together with a complete resolution of a related OEIS problem on sequences whose running averages are Fibonacci numbers (Fried's Conjecture 2).
Does every nontrivial finite simple graph have noninteger Sombor energy? If are the eigenvalues of the Sombor matrix of a graph , its Sombor energy is The conjecture asserted that for every nontrivial graph. A connected graph on nine vertices is exhibited with , disproving the conject…
conditional on a plausible number-theoretic conjecture; unconditional through s = 15
Is the exact nonreal spectral region of the four-cycle family of row-stochastic nonnegative matrices determined by the Karpelevich constraint, as Ran and Teng conjectured in 2024?
For a bipartite graph without isolated vertices and with a unique minimum dominating set, does the proposed function bound the number of edges whenever and ? A -vertex bipartite graph with edges exceeds the conjectured maximum of .
Casalaina-Martin and Zhjeqi proved that the first Chern class of every torsion-free coherent quotient of a tensor power of the logarithmic cotangent sheaf is pseudo-effective, noting in Remark 4.5 that torsion-freeness was imposed only for technical reasons. Can it be dropped? Yes.
If a finite graph has girth at least five, must its minimum dual degree satisfy , where is the smallest eigenvalue of its distance matrix? The Hoffman-Singleton graph violates it: dual degree against eigenvalue bound .
For every , is the independence polynomial of real-rooted if and only if is even? Exact Sturm counts refute both directions.
WOW-284 asserts that the minimum dual degree of every connected graph of order at least three and girth at least five is at most the negative of its least distance eigenvalue. The paper refutes it with exact counterexamples of orders 38, 39, 40, 42 and 50, and develops a structural theory of the failure.
For the Fubini numbers , is for every , as conjectured on the OEIS in 2018?
Define and . Is , as conjectured by Büyükaşik and collaborators?
For every connected graph, is the variance of its positive adjacency eigenvalues at most its order divided by its average distance? Exact dumbbell-graph certificates refute the bound under both conventions for average distance.
For a simple -polytope with at least three faces of size at least , must ? Five minimal ten-face counterexamples refute the printed inequality.
If is connected, cubic and diamond-free, must the zero-forcing number satisfy ? A connected cubic triangle-free -vertex graph has and .
The directed five-dimensional torus has a Hamilton decomposition for every odd , extending the decomposition program for directed tori beyond the three-dimensional case.
This settles Steinerberger's third open question and separates the two models: in the disk, full chords cost you a factor growing like over what is achievable without the restriction. It also improves the Steinhaus-type 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 se…
A problem from Fajtlowicz's Graffiti program, studied by Erdős and Staton, on the Havel-Hakimi residue of common-divisor graphs. The paper resolves the problem and extends it, determining the residue's first-order scale and its nontrivial constant from the degree sequence.
Does a single ReLU neuron trained on modular addition align to one Fourier frequency? Open Problem MAIS-O60, itself posed by Claude Fable 5 under the direction of Lionel Levine, is answered negatively: an explicit construction reaches a frozen state whose Fourier energy is spread equally across all nonzero frequency classes, on an open set of initial conditions.
refuted under the standard-deviation reading; the statement is reading-sensitive and other readings remain open