Edits, submissions and discussion
Most recent results
Most significant of the last seven days
Number theory - integer sequences
For every real the sequence of integer parts , , contains infinitely many composite numbers. Second, there is no infinite right truncatable prime in base~.
Extremal graph theory - digraphs
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-degree contains a spanning -subdivision whose path lengths differ by at most one, for every digraph with arcs and no isolated vertices.
Optimization - splitting methods
After Chen-He-Ye-Yuan's counterexample to direct three-block ADMM, the subclass in which the third constraint block is the identity matrix remained unresolved: the literature contained neither a convergence proof nor a counterexample. Disproved: an explicit rational counterexample exists in which the first two blocks are strongly convex quadratics and direct three-block ADMM produces a bounded nonconvergent orbit of period 66, verified by exact checks along a piecewise-affine reduction path.
Birational geometry (positive characteristic)
Extending the minimal model program beyond threefolds in positive characteristic is a standing goal of birational geometry. Assuming the log resolution conjecture for all log pairs birational to , the cone theorem holds for projective log canonical, -factorial fourfold pairs with , over bases of positive and mixed characteristic .
Discrete geometry
Kusner conjectured in 1983 that the maximum number of points in that are pairwise at -distance one is exactly for every , as in the Euclidean case. False: an explicit configuration of equilateral points exists for some exponent, placing the infimum of exponents at which the conjecture fails in . The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.
(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.)
Rado numbers / partition regularity
for every such that is divisible by 3, 4, 5, or 7 (covering of all ); the full conjecture (Myers 2015 Conj. 4.9, ABEMRS16 §5.5) reduces to prime cases , all smaller primes settled by SAT. Twenty-eight exact values, nineteen new primes , zero deviations from the conjectured line.
Tournament theory
A tournament orients every pair in a round-robin (winner → loser). The score sequence is the sorted win-count list. Reversing a directed 3-cycle never changes scores, so score-equivalent tournaments can look structurally different.
Question: Which linear combinations of induced k-subtournament type-counts are score-determined — identical across all tournaments sharing a score sequence, at any host size?
Answer: Exactly the linear combinations of degree-multiplicity counts m₀,…,m_{k−1}, where mᵣ counts how many of the k chosen vertices have exactly r internal wins. These k functions satisfy one linear relation, so score-determined statistics have dimension k−1.
Posets / hyperplane arrangements
Reading computed the order dimension of the poset of regions for most finite Coxeter arrangements, observed that an exceptional type whose dimension exceeds its rank would be the first known simplicial arrangement with that property, and recorded the general guess that every simplicial region poset has dimension equal to its rank (Problem 9.3 of his 2016 chapter); Segovia later asked the analogous question for oriented-poset lattices. False: the order dimension of the poset of regions can exceed the rank - the Coxeter arrangements and satisfy and .
Polynomial automorphisms
The Hessian conjecture asks whether every polynomial with has a polynomial gradient inverse. It is known for , false for , and open exactly in dimension four, where it implies the plane Jacobian conjecture. Proved for every quartic polynomial in dimension four: the quartic case reduces to with , and every constant-Hessian polynomial of this form has a polynomial gradient inverse.
(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.)
Numerical range / Crouzeix's conjecture
Lorist and Schwenninger prove Crouzeix's conjecture (arXiv:2608.03841, Lemma 1) by combining a lower bound (their inequality (4)) with an upper bound (inequality (5)). In Remark 2 they observe that (5) alone gives , so positivity of would prove the lemma outright. They write: "it is unclear whether in general."
Graph theory - chi-boundedness
Sivaraman asked whether perfect divisibility is characterized by its chromatic consequence: is a graph perfectly divisible if and only if for every induced subgraph of ? False: the Paley graph satisfies the chromatic bound hereditarily but is not perfectly divisible.
Commutative algebra
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.
(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=± 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.)
Mathematical relativity - Rainich geometrization, Kaluza-Klein
Is the EMD coupling square a function of the metric three-jet on an explicit active, non-null, simple-spectrum family of truncated Einstein-Maxwell-dilaton data, and can one more derivative recover it? Proved: no function of the common metric three-jet returns - the order-three ambiguity is exactly a free affine shear orbit () mixing with the phase gradient - while the fourth-order quotient recovers , with equality fiber exactly (). In particular Kaluza's and the control collide through metric order three.
Graph isomorphism
Among classes of tournaments for which neither hardness nor polynomial-time solvability of isomorphism was known, bounded VC dimension stood out as an open problem of Neuen and Grohe. Resolved: isomorphism of tournaments of VC dimension is decidable in time , so automorphism groups of bounded-VC tournaments are computable in polynomial time; isomorphism of tournaments of bounded chromatic number is also polynomial-time decidable.
(The new content is SOP_2 ⇒ SOP_3; the converse implication was known from the start. Dzamonja and Shelah asked whether either implication in SOP_3 ⇒ SOP_2 ⇒ 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 ≥ 3 remains, as does everything above it.)
Model theory
The classes of SOP_2 and SOP_3 first-order theories coincide. This answers a question of Džamonja and Shelah from 2004.
(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.)
Permutational Ramsey theory
Sixteen previously unknown exact values, plus three that confirm the sibling theorem entries' predictions computationally, across five ordered-pattern families (, , , , ) under dihedral and reflective group actions - each closing one open cell of Damnjanovic-Dordevic (arXiv:2607.06817, Tables 3-13). Five sit in cells the paper left without a conjecture. Full per-value table with regeneration commands, certificate hashes and referee verdicts in the evidence repo.
(The dihedral case only, for every a ≥ 4 and b ≥ 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 ≥ 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 ≥ 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 _m [P(m)+Q(m)] ≥ 2|E(G)|, from which the theorem falls out by averaging and a pivot decomposition.)
Permutational Ramsey theory
for all , — the slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817). Combined with the case (see sibling entry), this resolves Conjecture 4.9 in full for .
(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.)
Functional analysis
Banach asked in 1932 whether a real Banach space X whose ndimensional subspaces, for some fixed 1 < n < dim X, are all isometric must be a Hilbert space. Gromov proved the conjecture for even n, and subsequent work settled several odd-dimensional cases. We prove the conjecture for every odd n, including all previously unresolved cases. Together with Gromov’s even-dimensional result, this completes Banach’s isometric conjecture in the real case. The proof combines bundle topology with Brouwer degree theory.
(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.)
Entanglement theory
We prove that every PPT linear map has finite entanglement-breaking index, thereby establishing the eventual entanglement-breaking property of PPT channels in full generality. Furthermore, by utilizing completely positive maps with low entanglement dimensionality, we show that a large family of PPT maps, which strictly containing the class of 2-superpositive maps, has entanglement-breaking index bounded above by 3, uniformly in the dimension. In particular, these results provide strong evidence that the PPT-cubed conjecture may hold in full generality.
(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√(41))/32=1.13974707…, 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 C2, 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^*2.5652…, twice the refined general lower bound 1.28260069….)
—
For a single-source unsplittable flow, find the optimal universal additive constant s.t. every feasible fractional flow with arc costs should admit an unsplittable routing with and on every arc. We provide several new results on :
(1) record lower bound for planar instances (against known ceiling 2):
(2) local envelope ladder (proved): , , , attained by the counterexamples from our previous work; record constants of our previous work are now exact local envelopes of the general theory;
(3) global results: every exact-two-path instance with rows touching at most three terminals satisfies (first unconditional constant for an unbounded class); interaction arity m gives ;
(4) classes closed exactly: out-trees 0; two-layer hubs 1; outerplanar two-exit interval spines 1 (sharp); series-parallel ;
(5) band merger constant (twice the general lower bound ).
(Gromov's 1986 question has two independent proofs one day apart, and the first one public is not the AI-assisted one. Jian Ge posted "Heat kernel geometry and Gromov's volume growth conjecture" to arXiv on 13 August 2026, proving the same theorem by a different route - heat-kernel Fisher metric and Nash entropy. This paper appeared on 14 August, its author stating he learned of Ge's work only after his own manuscript was complete. The headline axes record this paper because it is the one with an AI in the loop. What it adds beyond Gromov's case is the uniform family: for every 0 ≤ m ≤ n-2, nonnegative Ricci plus a positive lower bound on the (m+1)-intermediate curvature forces at most m-dimensional volume growth. That interpolates between Ricci-type and scalar hypotheses, gives linear growth under positive biRicci curvature in every dimension at m=1, and yields a noncollapsed Urysohn-width bound. The author states this extension is his own contribution, not the model's.)
Differential geometry
In 1986 Gromov asked whether every complete -dimensional Riemannian manifold with and satisfiesfor every and every . The three-dimensional case had been settled, and higher dimensions were known only under extra hypotheses such as nonnegative sectional curvature, noncollapsing or an injectivity-radius bound.
This paper answers the question affirmatively, as the case of a uniform family: for every , if and the -intermediate curvature of Brendle-Hirsch-Johne is at least 1, then . At this gives linear volume growth under positive biRicci curvature in every dimension.
(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.)
Complex analysis
Let be a complex polynomial of degree whose zeros all lie in the closed unit disk. For every zero of , there is a critical point satisfying , except when and is a nonzero scalar multiple of .
(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.)
Calculus of variations and geometric measure theory
For a measurable set , letwhere is De Giorgi perimeter, and setThe conjecture asks for the complete fixed-volume minimization picture. Chodosh and Gianocca prove that, for every , balls of volume uniquely minimize among all measurable with , up to translation and null sets; for , no minimizer exists. Consequently,with equality exactly for translates, modulo null sets, of the ball of volume , equivalently radius .
(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.)
Operator algebras
(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.)
Combinatorial design theory
There exists a Hadamard matrix of order : a matrix
such that
Equivalently, the rows of are pairwise orthogonal.