Papers

We consider aggregation-diffusion models for a density of mass in $\mathbb R^d$, $d\ge 2$, where the diffusion can be either linear or of porous medium type, while the aggregation effect is governed by the attractive Newtonian or Bessel potential. It is well known that the dynamics can be interpreted as Wasserstein gradient flow of the natural free energy of the system, and that, under suitable assumptions on the diffusion exponent and the mass of the initial datum, solutions exist globally in time. In these regimes, we perform the analysis of the discrete variational approach by means of the JKO scheme. We establish a sharp $L^\infty$ smoothing effect, which proves to be of same rate as that of the porous medium equation. Thanks to this estimate, we obtain uniqueness of global gradient flow solutions for initial data of finite energy and, in the diffusion dominated regime, for measure data having finite second moment. We prove the energy dissipation equality and characterize the gradient flow in terms of suitable evolution variational inequalities. In the fair competition regime, we show uniform extinction of solutions for large time under smallness conditions on the mass.

Let $\omega$ be a radial $\widehat{\mathcal D}$-weight and $u$ be a bounded function on the unit disk $\mathbb D$. We prove that the Toeplitz operator \(T_{\omega,u}\) is compact on \(A_\omega^2\) if and only if its Berezin transform vanishes at the boundary. Our approach is based on a polynomial frame for $A_\omega^2$ and a detailed localization analysis of the resulting infinite matrix representation of $T_{\omega,u}$. Even in the unweighted Bergman space \(A^2\), our argument is new and does not rely on the classical translation operators. We further show that this Axler–Zheng compactness characterization does not extend, in general, to products of Toeplitz operators on \(\widehat{\mathcal D}\)-weighted Bergman spaces, and hence to the corresponding Toeplitz algebra generated by bounded symbols. More precisely, we construct a radial log-subharmonic \(\widehat{\mathcal D}\)-weight \(\omega\) and bounded symbols \(u,v\) such that the product \(T_{\omega,v}T_{\omega,u}\) is noncompact, whereas its Berezin transform vanishes at the boundary.

We report our findings and the results of our experiment of autoformalizing the homotopy type theoretic computation of $\pi_3(S^2)$ in agda-unimath using Codex.

The human diaphragm is vital for respiration and its simulation can aid in further understanding complications arising in individuals that have been subjected to mechanical ventilation. The diaphragm geometry is very thin and hence its numerical simulation poses a challenge for most numerical methods. In this paper, we compute the deformation of real diaphragm geometries which are modelled as linearly elastic and are subject to realistic boundary conditions. The method used is the unfitted least-squares radial basis function partition of unity method, which is well suited to solve problems on thin geometries given its unfitted nature and the shape and refinement of the cylindrical patches that are used. We derive a theoretical proof showing that the method converges with high order to the solution of this problem.

A sixer is a configuration of six pairwise skew lines on a smooth cubic surface, equivalently a choice of six exceptional curves defining a blow-down to $\mathbb{P}^2$. We study the $72$ sixers on the Fermat cubic surface $x_0^3+x_1^3+x_2^3+x_3^3=0$. We show that these sixers split into two orbits under the automorphism group of the Fermat cubic, of sizes $18$ and $54$. We give a geometric interpretation of this decomposition through the corresponding plane blow-up models: representatives of the two orbit types determine six-point configurations in $\mathbb{P}^2$ whose projective automorphism groups have orders $36$ and $12$, respectively. These groups identify with the stabilizers of the corresponding sixers and recover the two orbit sizes. We then compute the projective groups associated with representatives of the two orbits over $K=\mathbb Q(\omega)$, where $\omega^2+\omega+1=0$, and distinguish them arithmetically by the determinant square-class character $\delta_K:\mathrm{PGL}_2(K)\longrightarrow K^*/(K^*)^2$. Its images have $\mathbb F_2$-dimensions $1$ and $2$ for the orbits of sizes $18$ and $54$, respectively. Modulo $13$, the corresponding finite images are $\mathrm{PSL}_2(\mathbb F_{13})$ and $\mathrm{PGL}_2(\mathbb F_{13})$, respectively, and the determinant-character distinction persists for all choices of normalization triple.

We investigate uniqueness and non-uniqueness of least energy normalized and least energy nodal normalized solutions for nonlinear Schrödinger equations on compact metric graphs. We first prove existence of least energy nodal normalized solutions in the $L^2$-subcritical regime and, at the critical exponent, below a graph-dependent threshold. We then establish a conditional non-uniqueness result for slightly $L^2$-subcritical powers and identify broad classes of graphs for which the required condition either holds or fails. In particular, we prove uniqueness of least energy nodal normalized solutions on the interval for every mass and every $p\in(2,6)$. Finally, using ODE and phase-plane techniques, we show that least energy normalized solutions on the interval are unique for $p$ sufficiently close to $2$. Overall, the results reveal a strong dependence of the uniqueness picture on the nonlinearity power, the topology, and the metric of the graph, and a structural difference between the constant sign and the sign-changing settings.

We exhibit 2-step Carnot groups with Dehn function $n^2 \log n$, sharpening Wenger's superquadratic lower bound for these groups. Some of these groups admit lattices, yielding finitely generated nilpotent groups with Dehn function $n^2 \log n$. This gives the first exact computation of a Dehn function which does not grow like $n^\alpha$ for any $\alpha \geq 1$ among nilpotent groups and, more generally, among finitely presented solvable groups with subexponential Dehn function.

The $F$-symbols of an anyon theory encode the associativity of fusion and constitute some of the theory's most fundamental and, simultaneously, subtle data. Much of the subtlety lies in the gauge-dependence of the $F$-symbols. Despite their generic complexity, many braided anyon theories admit gauges in which all $F$-symbols are real, a phenomenon for which no general organising principle has been known. We identify a physical mechanism underlying this reality. For a unitary ribbon fusion category admitting an appropriate braided charge-conjugation symmetry, we show that the complex-conjugated $F$-symbols are related to the original ones by a gauge transformation. Finding a real gauge is thereby reduced to a condition on these transformations. When the charge-conjugation symmetry is suitably "flat,” or equivalently when the associated "twisted” Frobenius-Schur (or "generalized” Kawanaka-Matsuyama) data respects a grading, these local transformations can be trivialised and a real gauge exists. This framework unifies a broad range of previously disparate examples, including families of Chern-Simons theories whose $F$-symbols are difficult to directly compute. Finally, we exhibit a unitary ribbon category that realizes a novel obstruction to the existence of a real gauge and therefore has inherently complex $F$-symbols.

We study the one-sided time evolution of thermofield double (TFD) states in random matrix theory, where the Hamiltonian is taken to be a $D\times D$ random matrix drawn from a unitarily invariant emsemble of Hermitian matrices. We argue that the Krylov basis with the maximally entangled state taken as the initial vector gives a semiclassical Hilbert space description of these TFD states in random matrix theory at any $O(1)$ temperature and time in the $D\to \infty$ limit, very analogous to the “chord Hilbert space” construction in the double-scaled SYK (DSSYK) model. We study this semiclassical description in detail for ensembles where the spectral density in the $D\to \infty$ limit is even, compactly supported on an interval and has square root edges. With a few more conditions on the analytic structure of the spectral density, we observe that the semiclassical Hamiltonian has the same asymptotic behavior at large Krylov depth as that of DSSYK, with the corresponding parameter $\mathfrak{q}=e^{-\lambda}$ being related to the location of the nearest zero of the spectral density away from the spectral cut. Furthermore, in a large class of models corresponding to ultraviolet deformations of the DSSYK spectral density, i.e., where the spectrum in the UV is modified while leaving the near-edge behavior unchanged, we show that the semiclassical effective Hamiltonian in the Krylov basis reduces to the Liouville Hamiltonian of JT gravity in a low-energy, continuum limit. This suggests that our semiclassical Hilbert space should be interpreted as the bulk Hilbert space of a dual gravity description.

This comment gives dimension-independent analytic proofs of the local inequalities arising in the odd- and even-dimensional constructions of higher-dimensional partition charge functions. The prior works established these inequalities by exhaustive computation in low dimensions and tested them numerically in selected higher dimensions.

We discuss the famous conjecture {$\theta(p_c,{\mathbb Z}^d)=0$} for the Bernoulli site percolation model on $\Z^d$ with $d\geq 3$. We prove that, for $d\geq 3$, in the Bernoulli percolation model on ${\mathbb Z}^d$, at the critical point $p_c$, either there is no infinite cluster or the tail of the finite cluster distribution is not a stretched exponential.

This reports on an experiment in autoformalization of arxiv:2606.15977, an inverse problems result for piecewise polynomial anisotropic conductivities, using Isabelle/HOL. The proof of the result is relative to a number of external results which were deemed to be well known and generally accepted to be true. The formalization files are made available at a GitHub repository. Translation issues are discussed.

In 2006 Neeb asked whether the Hausdorff completion of a continuous inverse algebra must again be a continuous inverse algebra. The noncommutative case remained open, while the commutative case was known to be true. We give a negative answer after twenty years. We construct a Hausdorff metrizable locally m-convex complex continuous inverse algebra whose completion is a Fréchet locally m-convex algebra, where inversion stays continuous but the set of invertible elements is not open. The construction uses finite-support sequences in a dense nil subalgebra of a Jacobson-semisimple Banach algebra. Finite support makes every element nilpotent, giving continuous inversion without a locally uniform bound on nilpotence indices. In the completion, shifting a fixed noninvertible element to later and later coordinates produces noninvertible elements converging to the identity. Thus completion destroys exactly the local spectral stability at the identity. The counterexample is necessarily noncommutative and marks the precise boundary of the commutative completion theorem.

A \(\mathbb{T}\)-gain digraph is a directed graph with complex unit gains on its arcs, allowing for no restrictions on oppositely directed arcs. This framework unifies various graph types, such as signed graphs, mixed graphs, complex unit gain graphs, digraphs, and signed digraphs. The gain adjacency, Laplacian, and signless Laplacian matrices are generally non-Hermitian with complex spectra. We develop the spectral theory of these matrices, extending classical results from signed digraphs and complex unit-gain graphs. For the gain adjacency matrix, we establish determinant and characteristic polynomial formulas, characterize cycle balance through switching equivalence and cospectrality with the underlying digraph. We further bound the spectral radius in terms of the underlying digraph and its maximum out-degree, with equality characterized by $\mu$-balance. As a consequence, we determine the spectra of $\mathbb{T}$-gain unicyclic digraphs. Moreover, we characterize cycle balance and antibalance for the Laplacian and signless Laplacian matrices, respectively, through the presence of a zero eigenvalue.

For integers $k,g \ge 3$ let $n_g(k)$ denote the minimum order of a graph with chromatic number $k$ and girth at least $g$. Exoo and Goedgebeur (DMTCS 2019) proved $26 \le n_6(4) \le 66$; their 66-vertex witness has remained the smallest known 4-chromatic graph of girth 6. We improve both bounds to $29 \le n_6(4) \le 64$. The upper bound is witnessed by an explicit 4-chromatic graph of girth 6 on 64 vertices with 152 edges; it is vertex- and edge-critical, and its automorphism group is cyclic of order 8 and acts semiregularly. The lower bound is an exhaustive isomorph-free computation in the SAT modulo symmetries framework with co-certificate learning, driven by the Liu-Postle edge-density bound for 4-critical graphs of girth five; it re-derives $n_6(4) \ge 26$ by a disjoint method and is validated on the known values $n_4(4)=11$ and $n_5(4)=21$. We complement the bounds with structural obstructions: no smaller witness arises from either known witness by local modifications; no 4-chromatic Cayley graph of girth 6 exists on 54-63 vertices (for orders 59 and 61 no vertex-transitive witness exists at all); and no witness on at most 63 vertices admits a semiregular automorphism group with two or three vertex orbits, for any finite group. Since every known witness of an $n_g(4)$ record with $g \ge 6$ is a lift of a small base graph along a semiregular action, these results close the most symmetric part of that regime below 64 vertices. All properties of the new graph are verified by independent programs and formally certified in the Lean 4 proof assistant: the non-3-colourability is established inside Lean by a formally verified checker that re-validates a 219,532-node refutation certificate, with a machine-checked soundness theorem.

We construct the category of overt discrete quasi-Polish spaces and the category of compact Hausdorff quasi-Polish spaces (and some of their subcategories) as internal categories of the category of effective quasi-Polish spaces and computable maps. To demonstrate that the constructions are natural, we show that Stone duality is computable, in the sense that the dual contravariant functors and the natural transformations demonstrating their adjointness are computable.

Let $q=p^k$, and let $A_1,\ldots,A_{r-1}$ be invertible $n\times n$ matrices over ${\mathbb F}_q$. We prove that, if $k\ge r$, there is a vector $x$ for which \[ x,A_1x,\ldots,A_{r-1}x \] are all nowhere zero. For $r=2$ this recovers the theorem of Alon and Tarsi over nonprime finite fields. The proof tracks one monomial in the product of the coordinate forms. Frobenius powers keep every exponent below $p^r$, and finite-field polynomial nonvanishing supplies the required vector. The same method treats rectangular matrices with independent rows and prescribed forbidden values. The method also gives an extension-degree criterion for representable matroids and clarifies an unpublished prime-field conjecture of M. J. Moghaddamzadeh. Projective-geometric examples explain why the analogous field-size statement fails over proper extensions and, translated back to matrices, give lower bounds for the large-field problem.

We prove a new effective bound for cubic Thue–Mahler equations with power-saving dependence on the regulator. This has some applications. First, we improve Stark's bound $\log \max\{|x|,|y|\} \ll_\epsilon |k|^{1+\epsilon}$ for the integer solutions of Mordell's equation $y^2=x^3+k$ ($k$ a non-zero integer) by reducing the exponent $1+\epsilon$ to $1/2+\epsilon$; this is the first power-saving improvement without restrictions on $k$ in more than 50 years. Secondly, for integer squares and cubes of size $\asymp T$ we improve the known unconditional separation lower bound $(\log T)^{1-o(1)}$ obtained by Stark in 1973 to $(\log T)^{2-o(1)}$. Finally, we obtain a power-saving improvement in the conductor aspect of the strongest currently available bounds for Frey's height conjecture (a strengthening of Szpiro's conjecture) in the case of elliptic curves over $\mathbb{Q}$ with integral $j$-invariant.

Discrete diffusion models offer a promising alternative to autoregressive generation by enabling parallel updates, but their sampling efficiency can depend strongly on the choice of the forward process and the sampler. For the uniform forward process, existing lower bounds for the standard $\tau$-leaping sampler scale linearly with the ambient dimension $d$, raising the question of whether this dependence is intrinsic to the forward process. We answer this question in the negative. We consider a first-order sampler based on the leave-one-out denoiser for uniform and remasking processes whose coordinate updates can be performed in parallel. In both cases, the sampler can correct denoising mistakes during the sampling process, which becomes necessary when many coordinates are updated together. Our main result establishes an adaptive sampling guarantee: up to logarithmic factors, $N = O(\mathrm{DTC}(X_0) / \varepsilon)$ discretization steps suffice to achieve sampling error $O(\varepsilon_{\mathrm{score}}+\varepsilon)$, where $\varepsilon_{\mathrm{score}}$ is the error in score estimation. Thus, the sampling complexity is governed by the intrinsic dependence structure of the target distribution, as measured by its dual total correlation $\mathrm{DTC}(X_0)$, rather than directly by the ambient dimension $d$. Our analysis proceeds through a Bayes-optimal auxiliary sampler that separates discretization error from score-estimation error. We also derive an exact information-theoretic representation of the discretization error in terms of the mutual information between different coordinates of the forward process at different times. This representation applies to general forward processes and, in the uniform and remasking cases, can be controlled by $\mathrm{DTC}(X_0)$. Numerical experiments on structured synthetic distributions illustrate the predicted dimension-adaptive behavior.

In this paper, we introduce the Inertial Manifold Neural Operator (IMNO) for solving dissipative time-dependent partial differential equations (PDEs). The long-time dynamics of such systems often exhibit an effective low-dimensional structure due to dissipation. Unlike standard neural operator architectures such as the Fourier Neural Operator (FNO), IMNO explicitly leverages the low-dimensional structure to achieve better physical interpretability, accuracy, and stability in long-horizon autoregressive training and prediction for nonlinear dissipative PDEs. For shift-equivariant PDEs, we further introduce a shift-equivariant variant (IMNO-SE) of the proposed neural operator, ensuring that a spatial shift in the input induces the same spatial shift in the output. This symmetry-preserving inductive bias substantially improves its performance in shift-equivariant PDEs. Extensive benchmark experiments are presented to evaluate IMNO's performance numerically.

In the Eulerian walker model (also known as rotor walk), each site of the square lattice begins with an arrow pointing to one of its four neighbors. A walker that starts at the origin repeatedly turns the arrow at its current site clockwise by $90^\circ$ and steps in the new direction. Priezzhev, Dhar, Dhar, and Krishnamurthy (1996) introduced this as a model of self-organized criticality and conjectured that, for independent uniform initial directions, the region explored in the first $t$ steps has radius of order $t^{1/3}$. We establish this conjecture and further show that the walker visits every lattice site infinitely often, and that the region it has visited by time $t$, rescaled by $t^{1/3}$, converges to a convex body.

A subset of the $d$-dimensional lattice cube $[n]^d$ is sum-free if it contains no solution to the equation $x+y=z$. We study the total number of such subsets. For $d=1$, Cameron and Erdős conjectured that the number of sum-free subsets of $[n]$ is $O(2^{n/2})$, and this was proved independently by Green and Sapozhenko. A recent work by Ghosal solved the case $d = 2$. In this paper, we consider all remaining dimensions and prove that for every fixed integer $d \geqslant 3$, the number of sum-free subsets of $[n]^d$ is $2^{M([n]^d) + O_d(n^{d-1})}$, where $M([n]^d)$ is the maximum possible size of a sum-free subset of $[n]^d$. This verifies a conjecture of Elsholtz and Rackham. Our proof combines the dual weights constructed by Keevash and Lim in their work for $M([n]^d)$, a one-dimensional counting estimate due to Ghosal, a bipartite swapping lemma of Zhao, and a strong fractional entropy inequality of Madiman and Tetali, and it avoids the use of the container lemma or deriving a stability theorem first.

We study extremal questions on the average length and the number of subpaths in a graph. In particular, we prove the questions of Jamison (from $1983, 1984$) for the analogous concept of the average length of a subpath. Among other results, we prove that $K_n$ maximizes the average path length.

We establish the existence of a stochastic flow on $L^{\infty} (\mathbb{T})$ for the stochastic Allen-Cahn equation with multiplicative noise \[ (\partial_t - \partial_x^2) u = u - u^3 + \sigma (u) \xi \quad \text{on} \quad \mathbb{R}_+ \times \mathbb{T}, \] where $\xi$ is space-time white noise and $\sigma : \mathbb{R} \rightarrow \mathbb{R}$ is sufficiently smooth, bounded, and has bounded derivatives. Our strategy is to obtain pathwise a priori estimates via regularity structures. In fact, we consider a general singular multiplicative equation with superlinear damping, driven by noises of parabolic regularity $\alpha - 2$, for ${\alpha \in (0, 1)}$, which can be lifted to a weakly admissible model. We show that the required estimates hold whenever \[ m > \frac{2 - \alpha}{\alpha} \varepsilon_{\alpha}, \quad \text{where} \quad \varepsilon_{\alpha} = 1 - \alpha \left( 1 - \frac{2}{3 - \alpha} \right) \in (0, 1) . \] Thus the strength of the damping needs to be chosen only as a function of the regularity of the driving noise. Under an additional smoothness assumption on $\sigma$, we show that the stochastic flow is differentiable with respect to its initial condition.

The boxicity of a graph $G$, denoted by $box(G)$, is the minimum integer $d\geq 0$ such that $G$ is the intersection graph of axis-parallel boxes in $\mathbb{R}^d$. The class of zero divisor graphs introduced by Beck (1988) is a popular class of graphs and has been studied extensively by several researchers. Suppose $Z(R)$ is the set of zero divisors of a ring $R$. The zero divisor graph $\Gamma(R)$ for a ring $R $ is defined as the graph with the vertex set $V(\Gamma(R))=Z(R)$ and $E(\Gamma(R))=\{\{x,y\}\colon x,y\in Z(R)\text{ with }x\neq y\text{ and }x y=0\}$. One can define an equivalence relation $\sim$ on $V(\Gamma(R))$ such that for vertices $x$ and $y$, one has $x\sim y$ if and only if $x$ and $y$ have the same annihilator, i.e., $Ann(x)=Ann(y)$. The compressed zero divisor graph $\Gamma_E(R)$ for a ring $R$ is the simple graph obtained from $\Gamma(R)$ by retaining exactly one vertex from each equivalence class induced by $\sim$. In this paper, we completely answer two open questions posed in Discrete Applied Mathematics 391 (2026), pp. 127-136. Let $N=\prod_{i=1}^a p_i^{n_i}$ be the prime factorization of a positive integer $N$ and let $\mathbb{Z}_N$ be the ring of integers modulo $N$. We determine the exact boxicity of the compressed zero divisor graph $\Gamma_E(\mathbb{Z}_N)$. We show that when $a\geq 2$, $box(\Gamma_E(\mathbb{Z}_N))= a-1$ if and only if one of the following is true: $(i)$ $a\geq 2$ and $N$ is the product of two coprime integers $x$ and $y$ such that $x$ is a square-free integer and $y$ is the cube of a prime number; $(ii)$ $a\geq 3$ and $N$ is square-free; $(iii)$ $a\geq 2$, $N$ is cube-free, not square-free, and contains at least one prime divisor $p_i$ such that $n_i=1$. If $a=2$ and $n_1=n_2=1$, then $\Gamma_{E}(\mathbb{Z}_N)$ is a clique, and so, $box(\Gamma_{E}(\mathbb{Z}_N))=0$. In all other cases, $box(\Gamma_{E}(\mathbb{Z}_N))=a$.

We study a one-dimensional scalar model for an effective-mass quasiparticle field coupled to a continuum of two-level atoms with at most one excitation present. We prove a general theorem on the existence of bound states under sign-definite spatially localized perturbations of constant atomic density and analyze the bound states for several exactly solvable examples.

Conventional full counting statistics assigns commuting phases to integrated stochastic currents and therefore resolves net transport but not, in general, the temporal ordering of events associated with different cycles. We introduce a non-Abelian counting construction in which crossings of two fundamental cycles rotate an auxiliary spin about different axes. The resulting ordered trajectory statistic has an exact finite-dimensional evolution equation for its first moment. The two rotation angles form a counting torus, and whenever the mean spin is nonzero its normalized direction defines a map $\mathbb T^2\to\mathbb S^2$, equivalently a complex eigenline bundle with a Chern number. Our main analytical result is a Chern–parity correspondence. Reflection symmetry equips this eigenline with a real structure and expresses $C_T\bmod2$ through first Stiefel–Whitney classes on the circles fixed by reflection. When the transverse polarization has no additional zeros along the reflection-fixed circles, these classes reduce to ordinary current-parity statistics at the four high-symmetry counting points. For a five-state nonequilibrium figure-eight network, varying only the observation time produces four polarization-gap closings and the reentrant sequence $C_T=0\to-1\to0\to-1\to0$. Every transition occurs at $(\pi,\pi)$ and coincides with a sign reversal of $\mathbb E[(-1)^{Q_1+Q_2}]$, while the full integer Chern number is obtained independently from the two-dimensional spin texture. Finite observation time can therefore organize a fixed stochastic process into distinct topological sectors of its ordered path ensemble.

We prove Dennin's conjecture (Conjecture 8.9 of arXiv:2506.21052) that his variant of dual RSK correspondence is symmetric when restricted to biGrassmannian permutations. For a binary matrix $A$, let $A^\dagger$ denote its transpose-complement, and let $\operatorname{ins}(A)$ and $\operatorname{rec}(A)$ denote its insertion and recording tableaux. We prove that $\operatorname{ins}(A^\dagger) = \overline{\operatorname{rec}(A)}$, where the bar denotes the natural complement of the recording tableau. Our proof uses rectification of super pipe dreams and a downward induction on suffixes of $A$.

For every integer $d\geq 1$, we construct a countably infinite set $X_d\subset\mathbb{R}^d$ in affine general position, with all pairwise distances rational. When $d$ is odd, $X_d$ may also be chosen so that no $d+2$ points lie on a common sphere. The construction is uniform in $d$: positive Chebyshev square decompositions produce harmonic curves on spheres whose points corresponding to rational parameter values have pairwise rational distances. A divided-difference factorization of the affine determinant shows that sufficiently short arcs are locally convex, and stereographic projection produces the odd-dimensional examples. We also construct infinite rational distance sets in $\mathbb{Q}^d$ in affine general position for every even $d$, and in general position for every $d\equiv 1\pmod 4$. For every $d\geq 1$ and $n\geq d+1$, taking and rescaling suitable finite subsets gives $n$-point integral point sets in affine general position. A suitable ordered choice yields integral-distance realizations of all cyclic polytopes. In dimension three, we give an explicit rational parametrization and obtain infinitely many pairwise non-similar primitive $n_3$-clusters for every $n\geq 4$.

We investigate which 3-manifolds bound 4-manifolds that are homeomorphic but not stably diffeomorphic, where stabilising means taking connected sum with copies of $S^2\times S^2$. We show that every closed, orientable 3-manifold admits such fillings, as do certain families of nonorientable 3-manifolds. In contrast we show that for a 3-manifold containing a 2-sided $\mathbb{RP}^2$, any two smooth, homeomorphic fillings are stably diffeomorphic.