Papers

Belief-propagation (BP) decoding for quantum low-density parity-check (QLDPC) codes is attractive due to its low complexity, but its performance is often limited by short cycles, degeneracy, and convergence failures. Recently, reinforcement-learning-based sequential variable-node (VN) scheduling (RL-S) was shown to improve BP decoding by learning state-dependent update orders. However, the VN-by-VN nature of that approach offers limited within-iteration parallelism, since only one VN is updated at a time. In this paper, we propose a cluster-based extension of RL-S for QLDPC codes. The VNs are partitioned into fixed clusters, and at each scheduling step the RL agent selects one cluster to update, after which all VNs in that cluster are updated in parallel using the same pre-update incoming messages. To keep the tabular state space practical for large cluster sizes, we introduce a permutation-invariant cluster state based on a normalized histogram of local mismatch weights, followed by quantization. This representation makes the number of cluster states depend on the quantization resolution rather than the cluster size. We also develop the corresponding cluster-level Markov decision process, reward function, and Q-learning update. Numerical results on representative QLDPC codes show that the proposed clustered learned scheduling preserves most of the error-rate benefit of VN-level learned sequential scheduling while substantially reducing the number of scheduling decisions per BP iteration, thereby providing an attractive latency-parallelism tradeoff.

In this article, we aim to solve a monotone inclusion problem involving the sum of a maximally monotone operator and a continuous operator. While several algorithms exist to solve this problem when the continuous operator is cocoercive or Lipschitz continuous, they typically require the estimation of the global Lipschitz constant, which can be computationally expensive and often imposes overly restrictive step-sizes. To avoid these limitations and to handle merely continuous operators, linesearch subroutines are employed. A popular method in this context is the forward-backward-forward (FBF) algorithm (also known as Tseng's splitting), which utilizes a linesearch to guarantee convergence. However, a drawback of FBF is that the continuous operator must be evaluated twice per iteration. On the other hand, the forward-reflected-backward (FRB) algorithm proposed by Malitsky and Tam (2020) requires only a single evaluation of the operator per iteration. Although a linesearch version of FRB exists, its convergence is guaranteed only for locally Lipschitz operators; in fact, we present an example demonstrating that this existing linesearch can fail to terminate when the operator is merely continuous. In this work, we propose a novel linesearch strategy for FRB that is well defined and guarantees convergence even when the operator is merely continuous. We also extend the proposed algorithm to handle additional cocoercive and Lipschitz continuous operators. Finally, we provide numerical experiments on saddle-point problems and image restoration. The numerical results show that FRB with the proposed linesearch can accelerate the numerical convergence even when the operator is Lipschitz continuous. In addition, these results show that the proposed method is competitive with linesearch FBF, offering considerable computational advantages in various scenarios.

We show that inner Lipschitz classification of real algebraic curves is equivalent to outer Lipschitz classification in $\mathbb{R}^3$ of real algebraic curves with LNE connected components. Moreover, for every real affine algebraic curve we find its LNE model which is real birational to it. Lastly, we show that this claim does not hold in the complex category.

The KP equation is a prototypical $(2+1)$-dimensional integrable PDE. Its soliton solutions are famously parametrized by the Sato Grassmannian. In seminal work, Kodama and Williams made the surprising discovery that the combinatorics of soliton solutions are intimately related to the combinatorics of the totally positive Grassmannian as pioneered by Postnikov. They introduced novel algorithmic methods inspired by polyhedral structures arising from tropical geometry. Soliton solutions to the 2D Toda lattice and the Davey–Stewartson equation, two closely related integrable systems with soliton solutions, are also classified by the Sato Grassmannian. Kodama suggested that the methods of his work with Williams could generalize to these two integrable equations. In this work, we show that this is indeed the case. We derive algorithms to produce contour plots from elements in the totally nonnegative Grassmannian in both cases. In the asymptotic setting, we recover and refine previous work of Biondini and Wang; as well as Biondini, Kireyev and Maruno.

We study the Hamiltonian structure of the Flaschka-Newell Painlevé II hierarchy via symmetry reduction of the associated space of meromorphic connections. Building on the realization of this hierarchy as a reduction of the Painlevé IV hierarchy via a $\mathbb{Z}_2$-symmetry, we construct a set of Darboux coordinates adapted to the involution. After a suitable change of trivialization, the symmetry acts diagonally in these coordinates, allowing the fixed-point locus to be explicitly described as a symplectic submanifold. This enables us to derive the reduced Hamiltonians after symmetry, thereby obtaining explicit expressions for the Hamiltonians and the Lax matrices of the Flaschka-Newell Painlevé II hierarchy. This strategy also illustrates how symmetry-adapted canonical Darboux coordinates enable explicit reductions of isomonodromic systems at the level of their underlying symplectic geometry.

Lifting obstructions for group actions, cocycle actions, and $\Gamma$-kernels admit a cohomological description via topological crossed modules, as recently developed by Izumi, Giron-Pacheco, and the first named author. For a strongly self-absorbing $C^*$-algebra $A$, we show that the classifying spaces $\mathcal{B}^D\mathcal{G}_A$ and $\mathcal{B}^DP\mathcal{G}_A$ of the respective crossed modules governing cocycle actions and $\Gamma$-kernels, respectively, carry infinite loop space structures induced by the tensor product; the same holds for the crossed module $B^D\tilde{\mathcal{G}}_A$ whenever $U(A)$ is connected. This confirms a conjecture from the aforementioned work and extends to $\Gamma$-kernels and cocycle actions the connection with stable homotopy theory. For the proof we construct $\mathbb{I}$-FCPs from the relevant crossed modules, pass to $\Gamma$-spaces, and apply the May-Thomason infinite loop space machine. Consequently, the natural transformation $H^1(\Gamma,\mathcal{G}) \to [\mathcal{B}\Gamma,\mathcal{B}^D\mathcal{G}]$ takes values in cohomology groups.

We study the planar relativistic two-centre problem at fixed energy, including a class of perturbations of the Keplerian potential. Up to a reparametrisation of time, the relativistic dynamics is equivalent to a classical two-centre system with critical strong-force singularities. This reduction allows us to apply variational methods based on the Maupertuis functional. We construct collision-free relativistic trajectories with prescribed symbolic itineraries and obtain a coding of the dynamics by admissible sequences. In particular, we prove the existence of bounded orbits, scattering solutions with prescribed asymptotic directions, and trapped trajectories that are asymptotically free in one time direction and exhibit prescribed symbolic behaviour in the other.

We study spectral estimates for polyharmonic Schrödinger operators $-\Delta^l-\mu$ in the subcritical regime $2l<\mathbf{N}$. The measure potential $\mu$ is assumed to satisfy a capacitary smallness condition which guarantees that the corresponding operator is semibounded and self-adjoint. With such a measure $\mu$ we associate an Otelbaev function, which reflects both the local concentration and the spatial distribution of the potential. In terms of this function, we obtain two-sided estimates for the distribution function of the negative eigenvalues, and derive a sufficient condition and a necessary condition for the discreteness of the negative spectrum. As an application, we establish two-sided estimates of Lieb-Thirring-type, improving the classical ones.

We consider supercritical branching random walks (BRW) on countable groups $G$ and we prove that the asymptotic entropy of the empirical distributions of the BRW has a phase transition at $\rho_* = e^{h(\mu)}$, where $h(\mu)$ is the asymptotic entropy of the underlying random walk on $G$ with step distribution $\mu$. Below this value $\rho_*$, the asymptotic empirical entropy of BRW equals the logarithm of the exponential growth rate of the population. Above this value, it is constantly equal to the asymptotic entropy of the underlying random walk. In particular, this answers questions from Kaimanovich-Woess [MR4663513, Section 6.3] about the existence and the behavior of the asymptotic entropy.

We introduce an optimization approach for constructing spherical quadrature rules on arbitrarily scattered data. Rather than designing node placements, the new approach focuses on optimally computing the weights for fixed configurations. Motivated by Pólya's necessary and sufficient conditions for quadrature convergence in 1933, we argue that pursuing weight positivity and high algebraic exactness for scattered data approximation is not necessary. To align the quadrature design with the underlying theory of approximation, we construct convex optimization models with suitable objective functionals by examining the accuracy of numerical integration with reproducing kernels of Sobolev spaces and the performance of hyperinterpolation with Marcinkiewicz-Zygmund (MZ) inequalities. The resulting optimization models encode the spatial distribution of the scattered sites and the analytic properties of the target function spaces. The proposed approach enables the derivation of rigorous theoretical stability bounds, and the resulting quadrature weights are efficiently computable by modern convex optimization techniques. Numerical results are reported to demonstrate the performance of the optimization approach for fundamental approximation tasks such as numerical integration and hyperinterpolation for scattered spherical data.

We develop the boundary regularity theory for solutions to linear kinetic Fokker-Planck equations with Maxwell boundary conditions. These conditions interpolate between diffuse and specular reflection via an accommodation coefficient $\alpha \in [0,1]$. While existing literature is restricted to the extreme cases $\alpha = 0$ and $\alpha = 1$, we resolve the entire intermediate regime $\alpha \in (0,1)$. Specifically, we show that solutions are Hölder continuous if the coefficients are merely uniformly elliptic. Furthermore, for sufficiently smooth coefficients, we establish boundary regularity of order $\frac{3}{\pi} \arccos(\frac{\alpha}{2}) - 1$ up to the grazing set. This exponent is optimal. Beyond Maxwell conditions, we develop a unified approach that extends to a broad class of reflection boundary conditions, including super-elastic collisions.

The metric theory of spacetimes studies Lorentzian manifolds using tools of metric geometry. This is achieved via the null distance, which is a definite distance constructed from a time function on a spacetime. This enables the study of Gromov-Hausdorff-type convergence of spacetimes, a program recently initiated by Sakovich and Sormani. In this paper we study such notions of convergence for cosmological spacetimes with compact slices, i.e., $(a,b)\times M$ endowed with a Lorentzian metric $-dt^2+h_t$, where $h_t$ is a family of Riemannian metrics on the compact manifold $M$. Assuming mild extension properties of $h_t$, we first establish that these spacetimes are causally-null compactifiable and future developed. We then study monotone sequences with a uniform upper bound on the spatial diameter, obtaining uniform convergence of the null distances, as well as convergence of the associated timed metric spaces in the future developed Gromov-Hausdorff sense. Finally, we prove that causally-null compactifiable spacetimes satisfying a mild causal accessibility condition are causally-null, and relate the causally-null distance induced by the limit distance with the null distance induced by the (possibly non-smooth) limit metric tensor. Examples are provided to motivate the necessity of our hypotheses.

We prove the Gromov–Ros conjecture for every rank-one symmetric space of noncompact type: finite-perimeter isoperimetric regions are precisely the geodesic balls, up to ambient isometry and null sets. The proof is organized uniformly in the root multiplicities $(p,q)$. We introduce the volume radius \[ R(r)^n=n\int_0^r\sinh^{n-1}s\,\cosh^q s\,ds, \] which converts the polar volume form into $R^{n-1}dR\,d\sigma$. Hence radial–angular stretches $R\mapsto e^{t\varphi(\theta)}R$ have the exact Jacobian $e^{nt\varphi}$. Starting from a volume geometric median, a log-partition correction produces an exactly volume-preserving family of global bi-Lipschitz stretches. The reduced-boundary area formula and the ambient cofactor yield a universal pointwise trace identity depending only on $(p,q)$ and the horizontal and vertical components of the measure-theoretic normal. Its specializations for $q=1,3,7$ admit explicit strict sign certificates, forcing the normal of an isoperimetric region to be radial almost everywhere. Isotropy invariance in BV and a one-dimensional weighted endpoint comparison then force a single ball. For complex hyperbolic space we additionally prove that every smooth bounded fixed-volume stable critical domain is a geodesic ball. The resulting complex-hyperbolic isoperimetric theorem removes the geometric hypothesis in several sharp weighted-Bergman contraction, Faber–Krahn, and Lieb–Wehrl inequalities, and their high-weight scaling limit recovers holomorphic Gaussian hypercontractivity.

Let \(F\subseteq\binom{V}{q}\) be a \(q\)-uniform family on a finite vertex set \(V\). Write \(s_r(F)\) for the sum of the \(r\) largest eigenvalues of its simplicial up-Laplacian and \(d_F(v)\) for the degree of \(v\in V\). Then $D_r(F)=\sum_{v\in V}\min\{d_F(v),r\}$ is the \(r\)-th partial sum of the conjugate degree sequence of \(F\). The majorization assertion in the Duval–Reiner conjecture [Trans. Amer. Math. Soc., 2002] states that \(s_r(F)\le D_r(F)\) for every \(q\)-uniform family \(F\) and every \(r\ge1\). We disprove this assertion in two complementary senses: for every \(r\ge5\), there is a strict counterexample at index \(r\) in some uniformity, while every uniformity \(q\ge3\) admits a strict counterexample at some index \(r\ge5\). In contrast, we prove the universal inequality \(s_2(F)\le D_2(F)\) and classify all equality cases. The counterexamples are obtained from two \(3\)-uniform seeds with explicitly computed characteristic polynomials through defect-preserving ridge-whiskering and set-complement duality. For the second partial sum, core completion reduces the problem to the boundary matrix of a complete simplex, where Ky Fan variational and compression arguments yield both the inequality and its equality classification.

We study a specific polynomial mapping G: \mathbb{C}^5 \to \mathbb{C}^5 induced by a homogeneous polynomial F of degree 6 consisting of four tridiagonal harmonic blocks. We prove that while the Jacobian matrix of this mapping is unipotent at every point (implying det J_G(x) \equiv 1), the mapping itself is not globally injective. We construct explicit algebraic sparse pairs of distinct points a \neq b that map to the identical image G(a) = G(b) = \vec{0}. Furthermore, we perform a comprehensive classification of the zero-set structure of the corresponding gradient field, uncovering a total of 37 precise analytical complex solutions split across six distinct geometric series.

A unified algebraic framework for two different six-parameter families of bivariate $q$-Racah type functions is given using the representation theory of the quantum loop algebra $\mathcal{L} U_q sl_2$ of $sl_2$. The starting point of the analysis is two left and right coideal subalgebras of $\mathcal{L} U_q sl_2$ and six commutative subalgebras, built from eight elements in $\mathcal{L} U_q sl_2 \otimes \mathcal{L} U_q sl_2$. The eight elements depend on two scalars $a,b \in {\mathbb C}^*$, and are diagonalized on a finite-dimensional vector space. The bivariate $q$-Racah type functions are interpreted as the overlap coefficients relating six `distinguished' eigenbases parametrized by $a,b$ of the tensor product (evaluation) representations of $\mathcal{L} U_q sl_2$ labeled by the evaluation parameters $u_1,u_2$. Upon certain conditions on $a,b,u_1,u_2$, it is shown that a subset of pairs of elements act as tridiagonal pairs of type I (also called $q$-Racah type). For $u_1/u_2=1$, another subset of pairs of elements are conjectured to act as factorized Leonard pairs. Thus, in both cases corresponding overlap coefficients relating the various eigenbases associated with different pairs are obtained. Some of their properties are also discussed, as well as their relation with known bivariate polynomials of Tratnik type and the rank 2 Askey–Wilson algebra.

This paper studies the value distribution of the Riemann zeta-function under general shifts. We establish discrepancy estimates for the Bohr–Jessen limit theorem for a broad class of shifts. We also prove a universality theorem for logarithmic-power shifts, which are not covered by previous universality results for general shifts. Furthermore, we obtain a quantitative Bohr–Jessen limit theorem for logarithmic-power shifts.

In this paper, we introduce the $(r,d)$-Stirling numbers of the second kind, a new generalization of the $r$-Stirling numbers obtained by imposing a distance restriction: for any two elements $i$ and $j$ within the same subset, we require that $|i-j|>d$. We establish recurrence relations, an explicit formula, ordinary and exponential generating functions, a reduction identity connecting these numbers to the standard $r$-Stirling numbers, and several combinatorial identities.

In this paper, we study the Eisenbud–Green–Harris (EGH) conjecture for ideals generated by quadrics. We establish a sharp lower bound for the dimension of the cubic component of an ideal generated by a regular sequence of six quadrics and two additional quadrics in six variables. Furthermore, we prove the Eisenbud–Green–Harris conjecture for almost complete intersections of quadrics in six variables.

We study the use of runlength-limited (RLL) block codes in bit-interleaved coded modulation (BICM) systems. In this setting, the RLL code acts as the symbol mapper, whose assignment between input bits and RLL symbols is critical for performance. In this work, we aim at optimizing the assignment scheme of RLL codes. One of the main applications of RLL codes is the mitigation of intersymbol interference (ISI) in systems with coarse quantization. However, channel memory and quantization nonlinearity complicate information theoretic analysis. To enable analytical treatment, we consider a block channel with 1-bit analog-to-digital conversion, modeling the transmission of a single RLL code block. For this channel, we investigate the relationship between the achievable rate in BICM systems-termed BICM capacity-and the RLL code's assignment scheme. Focusing on low signal-to-noise ratios (SNRs), we derive the optimization problem yielding the optimal assignment scheme. By looking at asymptotically large block-lengths, we infer a practical optimization strategy for RLL codes with finite block-length and channels with inter-block interference. Further, we extend this optimization to two-state RLL (TS-RLL) codes, which offer higher code rates than state independent RLL codes. We demonstrate that optimized TS-RLL codes exhibit significant performance improvements over literature counterparts.

We study the Boltzmann equation in the whole space under the radial anharmonic confining potentials $\Phi(x)=|x|^p/p$ for $p>2$ and $\Phi(x)=\langle x\rangle^p/p$ for $1<p<2$. We first classify all positive finite-mass-and-energy entropy-invariant, equivalently zero-entropy-production, solutions. For $p>2$, the nonlinear equilibrium manifold is parametrized by mass, temperature, and the three components of angular momentum; for $1<p<2$, integrability excludes rotating equilibria and only mass and energy remain as equilibrium parameters. We then identify the five-dimensional stationary space of the equation linearized about an arbitrary equilibrium and construct an explicit projection determined by the conserved moments. After normalization, the collision term takes the form $C_Me^{-\widetilde\Phi(x)}\mathsf L$, so its microscopic coercivity degenerates at spatial infinity. A far-field weight-transfer estimate compensates for this degeneracy. After subtracting the stationary projection, the corresponding semigroup solution converges algebraically in exponentially weighted $L^2$ spaces. The rate is governed by the growth exponent $p$ and the gap between the two weights, up to an arbitrarily small loss. For $1<p<2$, the mismatch between the two-dimensional nonlinear equilibrium manifold and the five-dimensional linear stationary space yields a conditional obstruction to nonlinear asymptotic attraction for perturbations carrying nonzero angular momentum.

We classify the left-invariant complex structures on the real Lie group SL(2,C) up to Lie group automorphism. The classification consists of the bi-invariant complex structure, a family parameterised by the unit disk, and one new, non-regular complex structure. The topology of the space of left-invariant complex structures (modulo automorphisms) is not Hausdorff. Finally, we show that the corresponding complex manifolds always admit left-invariant balanced metrics.

This paper explores the measurement requirements for signal recovery in compressed sensing, comparing the performance of shearlet frames with traditional wavelet systems. Directional representation systems such as shearlets are known for their ability to sparsely represent images with anisotropic features, which allows for efficient nonlinear approximations. The central question we address is whether this difference in sparsity allows for a proportional reduction in the number of Fourier measurements needed for successful reconstruction. On the theoretical front, we study the obstacles encountered when trying to apply standard frame-based recovery results to shearlet systems. First, we show that the (optimal) local coherence between Fourier measurements and cone-adapted shearlets decays more slowly than the corresponding local coherence for wavelets. Second, managing the sparsifying system's redundancy relies on evaluating a localization factor, which requires lower frame bound estimates that can become exponentially small. Thus, even under optimal theoretical conditions, the number of samples required for shearlets scales quadratically with sparsity, which offers no substantial theoretical reduction over the standard wavelet benchmark. These theoretical limitations are assessed through a series of numerical experiments on a dataset of piecewise smooth images. While empirical observations confirm that shearlets can accurately represent these images using fewer coefficients than wavelets, phase diagrams indicate that this advantage in sparsity does not yield a proportional reduction in required Fourier samples. Ultimately, we conclude that despite the superior nonlinear approximation rates of shearlets, their practical sample complexity in compressed sensing scenarios with subsampled Fourier measurements remains comparable to that of traditional wavelets.

In this paper, we focus our attention on the following inequality, which holds for any integrable log-concave function with positive integral $f$, and gives an inclusion between the $n$-th Ball body of the covariogram function of $f$ and its polar projection body, providing a functional version of Zhang's projection inequality in the setting of log-concave functions: $$ n\int_0^\infty r^{n-1}\int_{\mathbb{R}^n}\min\{f(z),f(z-re_n)\}dzdr\leq n!\frac{\left(\int_{\mathbb{R}^n}f(x)dx\right)^{n+1}}{\left(\int_{e_n^\perp}P_{e_n^\perp}f(y)dy\right)^n}. $$ Here $(e_i)_{i=1}^n$ denotes the canonical basis in $\mathbb{R}^n$ and $P_{e_n^\perp} f$ denotes the projection of $f$ onto the hyperplane orthogonal to $e_n$ given, for $y\in e_n^\perp$, by $P_{e_n^\perp} f(y)=\sup_{\lambda\in\mathbb{R}}f(y+\lambda e_n)$. We will prove a discrete version of this inequality in which we will consider discrete measures involving the lattice point enumerator measure, rather than the Lebesgue measure. In order to prove it, we will obtain a discrete version of the functional version of Berwald's inequality. We will also provide a discrete version of the functional Rogers-Shephard inequality. All the discrete versions of the inequalities considered will imply their continuous counterparts.

We study one-dimensional half-harmonic maps from the real line into the circle with prescribed exterior data. We show that, for every positive integer $k$, if two disjoint intervals are sufficiently close, there exist at least $k$ distinct non-constant half-harmonic maps with constant exterior data. More generally, we establish a multiplicity result for boundary data with energy below the critical threshold $2\pi$ by introducing a local relative degree and proving corresponding degree-jump estimates. Working on a single interval and for boundary data arising as traces of finite Blaschke products, we investigate the existence and non-existence of energy minimizers in prescribed degree classes.

In this paper we apply the newly developed theory of extended graphons to a certain class of non-exchangeable, non-conservative problems. We establish the mean-field limit by proving the convergence of the associated generalized weighted empirical measures.

Azuma, Kim, and Yamashita formulated the verification problem for one-layer ReLU networks as a quadratically constrained quadratic program and established tight semidefinite relaxations for the edgeless case and for one-unit settings. In this work, we represent the sharing pattern of undecided ReLUs over a box input set through an input-sharing graph and focus on the case where this graph is a matching. We then derive an explicit, checkable sufficient condition for the tightness of the order-$2$ block-sparse SOS relaxation associated with the connected-component decomposition of this graph. Under the matching assumption, the global problem decomposes into isolated-vertex blocks and single-edge blocks. The key difficulty, which is absent from the edgeless case, is establishing tightness for a two-unit edge block. For regular rank-one edges, we show that the convex hull of each two-unit local set can be described exactly by two reduced one-unit hulls coupled through a common shared scalar. Combining the one-unit tightness result of Azuma et al. with Farkas' lemma and affine elimination, we obtain a local order-$2$ certificate for each edge block. Isolated-vertex blocks reduce to one-unit problems over box input sets and are therefore handled at the same order. We prove that, when the input-sharing graph is a matching and every edge satisfies the regular rank-one condition, the order-$2$ block-sparse SOS relaxation is tight. This extends the tight sparse relaxation result for the edgeless case to the first sparse setting with a nontrivial two-unit interaction.

Let $\Gamma$ be a straight-line crossing-free drawing of a tree $T$. A bad pair in $\Gamma$ is a pair of non-adjacent vertices of $T$ whose Euclidean distance in $\Gamma$ is smaller than the length of the longest edge in the path connecting them in~$\Gamma$. When $\Gamma$ has no bad pairs, $\Gamma$ is a Euclidean Minimum Spanning Tree of its vertex set (or EMST-drawing for short). Deciding whether a tree of maximum degree at most six admits an EMST-drawing is known to be \NP-hard. In contrast, we characterize those caterpillars that admit an EMST-drawing. The characterization gives rise to a linear-time algorithm that decides if a caterpillar admits an EMST-drawing, and in the affirmative case, computes such a drawing. For caterpillars of maximum degree six, we further present a linear-time algorithm to compute a crossing-free straight-line drawing with the minimum number of bad pairs. For $n$-vertex trees with maximum vertex degree $\Delta$, we prove the $\Delta^2n\log n$ upper bound on the minimum number of bad pairs. In the special case of stars, we construct a drawing with the minimum number of bad pairs.

We give another proof of the classic result of Dickson that determines the finite subgroups of 2-dimensional projective general linear groups. In fact we go a little further and give an abstract characterization of those subgroups in terms of permutation groups.

We certify, exactly, the coordinate fields of a faithful planar realization of two neighbouring triangle-free Exoo-Ismailescu unit-distance graphs (UDGs), and show they realize the two opposite extremes of the constructibility hierarchy. The 17-vertex graph EI17 (House of Graphs 51375) is the smallest triangle-free UDG with chromatic number 4; the 19-vertex graph EI19 (HoG 51376) is its state-of-the-art origami neighbour. In both, fixing a rational base edge, the remaining vertices are intersections of unit circles – each on the radical axis of its two neighbours, a tower of square roots over the free angles – and a small closure system locks the realization. For EI19 the base lies in Q(sqrt 2, sqrt 5, sqrt 7) and a single free angle has an irreducible degree 12 = 2^2*3 minimal polynomial with Galois group the solvable transitive group 12T236 (order 2304 = 2^8*3^2): not ruler-and-compass, but origami-constructible (the cubic Beloch fold O6 necessary, in casus irreducibilis). For EI17 two free angles are locked by two closures, whose resultant is irreducible of degree 20 = 2^2*5 with Galois group the full symmetric group S20 (a Frobenius census exhibits a 17-cycle, forcing A20 by Jordan, and an odd 20-cycle, raising it to S20): non-solvable, so the coordinates are not expressible by radicals – neither compass nor origami of any fold order. Thus the smallest triangle-free 4-chromatic UDG is the generic, maximally exotic case, the exact opposite of its origami neighbour. We give the full certification pipeline as explicit algorithms, record two methodological pitfalls, and read the pair through a conjectural bridge between the Laman number and the Galois group.