Rasmus Lindahl (mrconter1)

1.17353 planar lower bound and exact local envelopes for cost-preserving single-source unsplittable flow

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For a single-source unsplittable flow, find the optimal universal additive constant $C$ s.t. every feasible fractional flow $x$ with arc costs $c$ should admit an unsplittable routing $y$ with $c^\top y \le c^\top x$ and $y_a \le x_a + C \cdot D$ on every arc. We provide several new results on $C$: (1) record lower bound for planar instances (against known ceiling 2): $$C \ge \frac{58676765987259}{50000000000000} = 1.17353531974518;$$ (2) local envelope ladder (proved): $E(2) = 1$, $E(3) = 9/8$, $E(4) = (299 - 41\cdot\sqrt{41})/32 = 1.13974707\ldots$, 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 $C \le 2$ (first unconditional constant for an unbounded class); interaction arity m gives $\lceil\lfloor 3m/2\rfloor /2\rceil \cdot D$; (4) classes closed exactly: out-trees 0; two-layer hubs 1; outerplanar two-exit interval spines 1 (sharp); series-parallel $\le 1$; (5) band merger constant $K^* \ge 2.5652\ldots$ (twice the general lower bound $1.2826\ldots$).

1.28249... Lower Bound and Partial Upper Bounds for Cost-Preserving Single-Source Unsplittable Flows

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For a single-source unsplittable flow, find the optimal universal additive constant $C$ s.t. every feasible fractional flow $x$ with arc costs $c$ should admit an unsplittable routing $y$ with $c^\top y \le c^\top x$ and $y_a \le x_a + C \cdot d_{\max}$ on every arc. Goemans conjectured $C=1$; this was disproved in July 2026 by a separate seven-vertex counterexample with critical constant $16/15$ (see the Dinitz–Garg–Goemans entry), leaving the optimal $C$ open. Lower bound: a seventeen-terminal common-point interval instance certifies $$C\ \ge\ \frac{1282494797984843521}{10^{18}}=1.28249\ldots$$ Upper bounds: the paper proves the first unconditional ceiling below 2, but for the codimension-two case only, at complement mass $q=2$. The record cells lie outside it, the $k=17$ instance having $q=11$, so that ceiling does not bound the record ladder. Two figures are conjectures rather than results: $4/3$ as the supremum of critical constants over common-point cells, approached but not attained and not an extrapolation from the ladder (Conjecture 1.1, Theorem 5.1), and $2$ for the universal constant itself (Conjecture 1.2). The proved gap remains $[1.28249\ldots,\ 2]$.

Han's Conjecture

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For a finite-dimensional algebra $A$, finite global dimension forces $\mathrm{HH}_n(A) = 0$ for all large $n$. Han conjectured the converse: eventual vanishing of Hochschild homology should detect homological smoothness. Disproved by an explicit finite-dimensional $\mathbb{C}$-algebra with $\mathrm{HH}_n(A) = 0$ for every $n \geq 1$ and $\mathrm{gldim}\, A = \infty$, built by transporting Krah's phantom into a singularity category via one-periodic folding.

A counterexample to the Howland-Kato problem for positive commutators

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The Howland-Kato conjecture that every nonzero positive commutator $i[f(P),g(Q)]$ must arise from functions in appropriate Kato classes is false: $i[\arctan(P),\arctan(Q)]$ is nonzero and nonnegative.

The Inverse Generator Problem on Hilbert Spaces

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If $A$ generates a bounded $C_0$-semigroup on a Hilbert space and has dense range, does $A^{-1}$ also generate a bounded $C_0$-semigroup? Posed by deLaubenfels in 1988. Answered negatively: Lorist, Meyries and Veraar construct a bounded operator with dense range generating a bounded, strongly stable semigroup whose inverse generates no $C_0$-semigroup at all. The counterexamples come from one explicit finite-dimensional construction, using bases of $\mathbb{C}^{2n}$ with uniformly bounded partial-sum projections but unconditionality constants growing like $n^\alpha$.

Exhaustive AdaBoost Cycling Question

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Does exhaustive AdaBoost always converge to a finite cycle of weak classifiers and weight vectors on every finite training set? A finite instance whose orbit never becomes periodic answers no.

Albertson–Berman Induced Forest Conjecture

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Albertson and Berman conjectured that for every simple planar graph $G$ on $n$ vertices, the largest vertex set inducing a forest has size at least $n/2$. The standing lower bound since the same year has been Borodin's $2n/5$, from his acyclic five-colour theorem. False: there is an explicit $31$-vertex simple $3$-connected maximal planar graph $T$ whose largest induced forest has exactly $15$ vertices, and an infinite family $M_k$ on $31k$ vertices with induced-forest number exactly $15k$, giving the ratio $15/31 < 1/2$ even for triangulations of minimum degree five.

The Aluffi-Chen-Marcolli Real-Rootedness Conjecture

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Aluffi, Chen and Marcolli conjectured that the Poincare polynomial of the Deligne-Mumford moduli space $\overline{\mathcal{M}}_{0,n}$ of stable $n$-pointed rational curves has only real roots. True, with simple roots and strict interlacing between consecutive $n$.

Amdeberhan-Medina-Moll Arctangent Sum Conjecture

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Let $x_n = \tan\left(\sum_{k=1}^{n} \arctan k\right)$. Amdeberhan, Medina and Moll conjectured that $x_n \notin \mathbb{Z}$ for every $n \ge 5$. Any integer value $x_n = m$ must satisfy $|m| \ge e^{(1/2+o(1)) n \log n}$, which forces $\#\{1 \le n \le N : x_n \in \mathbb{Z}\} = O(\log N)$. The conjecture therefore holds for a density-one set of $n$, improving on the previously known density of $120/817 \approx 0.147$.

Absolutely Maximally Entangled States in Five Open Cases

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A pure state of $n$ parties with $q$ levels each is absolutely maximally entangled, written $\mathrm{AME}(n,q)$, when every subsystem of at most $\lfloor n/2 \rfloor$ parties is maximally mixed. These are the perfect tensors, and existence is a parameter-by-parameter problem: some $(n,q)$ admit one, some provably do not, and a maintained table records which cells are still unknown. This paper settles five of them. It exhibits Hermitian self-dual MDS codes $[12,6,7]_{25}$, $[18,9,10]_{121}$ and $[18,9,10]_{169}$, from which the stabilizer construction gives $\mathrm{AME}(12,5)$, $\mathrm{AME}(18,11)$ and $\mathrm{AME}(18,13)$, and projecting one party gives $\mathrm{AME}(17,11)$ and $\mathrm{AME}(17,13)$.

Whether Amenability of an Algebra Depends on the Ground Field

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Cornulier asked, in a MathOverflow discussion, whether amenability of a module over an associative algebra depends on the ground field. It does not: the notion is invariant under change of base field.

The Thin Matching Problem

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Anari, Charikar and Ramakrishnan asked whether every fractional perfect matching admits a perfect matching that is $\alpha$-thin with respect to it, meaning it crosses every cut at most $\alpha$ times the fractional amount. Resolved up to polylogarithmic factors.

Last-Iterate Rate for Anchored Gradient Descent-Ascent

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For smooth convex-concave min-max problems, can anchored gradient descent-ascent be scheduled so that its exact last-iterate squared-gradient residual is $O(1/t)$, closing the gap left by the 2019 analysis?

Anderson's Quasi-Completeness Question

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Is every weakly quasi-complete Noetherian local ring quasi-complete? Asked by D. D. Anderson in 2014. The ring $A = k^p[[X, Y]][k]$ with $k = \mathbb{F}_p(u_1, u_2, \dots)$ is weakly quasi-complete but not quasi-complete.

The Anstee-Sali Conjecture on Forbidden Configurations

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For a forbidden configuration F, the Anstee-Sali conjecture predicts that forb(m, F) is Theta(m^(X(F)-1)), where X(F) comes from an explicit product construction. Disproved: the 4-uniform family on six vertices formed by a two-vertex core joined to the edges of a 4-cycle has X(F) = 4, so the conjecture predicts Theta(m^3), while a random-alteration argument gives Omega(m^(10/3)).

Araujo-Piga-Schacht Question on Tight Hamilton Cycles

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Araujo, Piga and Schacht asked whether density and codegree both above $1/4$ force a tight Hamilton cycle in a linearly quasirandom 3-graph. No: the threshold is $p_0 = \max_{0 \le x \le 1}\min\{x^3, 1-x\} \approx 0.3177$, and below it there are dense 3-graphs with large codegree and no tight Hamilton cycle. For every $p > 1/3$ the asymptotically sharp minimum-codegree threshold is determined.

Sum-Difference Exponents for Boundedly Many Slopes

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The arithmetic Kakeya conjecture asserts the infimum of sum-difference exponents is 1, which would imply the Kakeya conjecture in all dimensions. In the bounded-slope-count regime, Tao establishes that the exponents converge to 2 at a rate controlled by a new notion of rational complexity - mapping where the conjectured route cannot succeed.

Asymptotically attaining the Moore bound

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For positive integers $d$ and $k$, let $n_k(d)$ be the maximum order of a graph of maximum degree at most $d$ and diameter at most $k$. It is shown that $$\lim_{d \to \infty}\frac{n_k(d)}{d^k} = 1$$ for every fixed $k$, thereby resolving the asymptotic degree-diameter problem for fixed diameter. Also proved a similar lower bound on the edge-variant of the problem, and a tight asymptotic for the bipartite variant of the edge problem.

Completeness of Fixed-Order Atom-Centered Descriptors

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Pozdnyakov, Willatt, Bartók, Ortner, Csányi and Ceriotti showed in 2020 that the 2-, 3- and 4-point correlations of an atomic neighbour density are incomplete: noncongruent environments can share them exactly. Every degeneracy found since was dissolved by going to a higher correlation order, leaving open whether the trispectrum (5-body correlations), or any fixed finite order, separates all noncongruent environments. It does not. There are noncongruent three-dimensional environments agreeing on every cluster of up to seven neighbours, and for each finite correlation order and angular cutoff there are continuous families of noncongruent environments with identical retained features.

Infinitely Many Components in Auslander–Reiten Quivers over Perfect Fields

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For a perfect field $k$ and a representation-infinite finite-dimensional $k$-algebra $A$, the Auslander–Reiten quiver of $A$ has infinitely many connected components. This establishes a conjecture of Auslander, Reiten and Smalø, for finite-dimensional algebras over perfect fields.