Papers

We present solutions to six problems concerning sets of nice recurrence and van der Corput (vdC) sets. We answer a question of Moreira, repeated by Fish and Skinner, by showing that every set of ergodic nice recurrence is a set of nice recurrence. We answer questions of Bergelson as well as Bergelson and Lesigne by showing that every set of nice recurrence is a nice vdC-set, that the family of nice vdC-sets is partition regular, and that the family of sets of nice recurrence is partition regular. We mention that Chapman independently proved that sets of nice recurrence are partition regular. We answer a question of Kelly and Lê, a special case of which was previously asked by Peres, and show that for every nontrivial compact metrizable group $K$, every $K$-vdC set is also a vdC set. Lastly, we answer a question of Farhangi, Rodríguez, and Tucker-Drob and show that for any countably infinite discrete group $G$, a vdC set in $G$ is a set of operatorial recurrence in $G$. The initial solutions to these problems were produced by an AI system. The authors checked their correctness, simplified and reformulated them as clear, human-readable proofs, and situated the resulting contributions within the existing literature.

We study higher-order hyperuniformity through the large-scale variance of local $k$-point patterns, and write $\HU_k$ for the resulting condition; $\HU_1$ is ordinary hyperuniformity. The conditions $\HU_k$ need not coincide: for every $k\geq1$ there exists an $\mathbb R$-invariant weakly mixing simple point process that belongs to $\HU_j$ for all $j\leq k$ but not to $\HU_{k+1}$. Randomly translated lattices belong to $\HU_k$ for every $k$, whereas sufficiently small non-degenerate iid perturbations of lattices and projection determinantal point processes already belong to $\HU_1\setminus\HU_2$. For regular Euclidean cut-and-project processes we give an exact criterion for $\HU_k$. For ball windows, $\HU_k$ is equivalent to $\HU_1$ in internal dimensions two and three, whereas in every internal dimension $m\geq4$ we construct ball-window examples in $\HU_1\setminus\HU_2$. Finally, the nonperiodic Kurasov–Sarnak Fourier-quasicrystalline point process is stealthy—its first-order spectrum has a gap at the origin—yet does not belong to $\HU_2$.

We study Kähler metrics with cone angle $2\pi\beta$ along a smooth divisor, in the range $1/2<\beta<1$. We show that bounded Riemann curvature near the divisor forces the normal bundle sequence to split holomorphically, confirming a condition proposed by Donaldson.

Writing error-free floating-point programs is a challenging task, especially for programmers who lack a strong background in numerical analysis and rounding-error propagation. State-of-the-art techniques typically aim to bound such errors using static or dynamic analysis. However, only a few tools explicitly address critical floating-point pitfalls such as absorption and catastrophic cancellation. These anomalies represent situations in which rounding errors are significantly amplified, causing the semantics of the finite-precision computation to deviate substantially from the real-number semantics. In this article, we present FPScan, a novel tool to formally define and detect both catastrophic cancellation and absorption in floating-point programs. Our approach starts with a custom static analyzer based on abstract interpretation to infer the order of magnitude of all program variables. This magnitude information is then used to build a set of first-order constraints that model error propagation and numerical precision within the program. Finally, we employ an off-the-shelf SMT solver to determine whether the program exhibits any of these critical numerical pitfalls. Experiments were conducted on FPBench, a well-known benchmark suite of floating-point programs, to evaluate the effectiveness of our tool. We also present a comparison with state-of-the-art tools regarding soundness and analysis time.

In this paper, we revisit the work of Berkovich and Paule on variants of the Andrews-Gordon identities. We find a generalization of their principal polynomial identity and, as a consequence, obtain new $\vec M$ versions of the Andrews-Gordon identities. More precisely, for non-negative integers $M_1\ge M_2\ge M_3\ge\ldots\ge M_\nu$, we show that a broad class of multi-sums of the form $$ \sum\limits_{\mathbf n} \frac{ q^{ N_1^2+\cdots+N_\nu^2 - M_1N_1 - M_2N_2 - \cdots - M_\nu N_\nu } }{ (q)_{n_1}(q)_{n_2}\cdots(q)_{n_\nu} } $$ can be expressed as a sum of products. Above, we use standard notations for $q$-Pochhammer symbols and $N_i = \sum\limits_{k=i}^{\nu}n_k$ for $1\le i\le \nu$.

We first prove a version of the Demailly-Paun theorem for compact weakly Kaehler spaces. Secondly, for a big and nef Bott-Chern class on a compact normal Kaehler space, we prove that the restricted non-Kaehler locus, defined using Kaehler currents with weak analytic singularities, is equal to the usual non-Kaehler locus.

A family $\mathcal F$ of finite subsets of $\mathbb Z$ is packed into $[N]$ if suitable integer translates of its members are pairwise disjoint subsets of $[N]$. We study two prescribed-difference packing problems of Alon, Dębski, Grytczuk and Przybyło for the arithmetic progressions $A_d=\{d,2d,\ldots,\lfloor n/d\rfloor d\}$ and $B_d=\{d,2d,\ldots,nd\}$. Let $m(n)$ and $M(n)$ denote the corresponding minimum packing lengths, with subscripts indicating restrictions on the admissible differences, and let $\mathcal P(x)$ denote the set of primes at most $x$. The key ingredient is a residue-class selection scheme that encodes pairwise intersection constraints by cyclic intervals. A lattice-covering argument yields a simultaneous admissible choice, permitting substantial overlap of the containing intervals and providing the sharp upper bounds needed for the bounded-diameter family and the prime-difference equal-cardinality family. For the bounded-diameter family, we prove $m(n)\sim m_{\mathcal P(\sqrt n)}(n)\sim 4n^{3/2}/(3\log n)$. For the equal-cardinality family with prime differences, we prove $M_{\mathcal P(n)}(n)\sim n^3/(6\log n)$. For the full equal-cardinality family, we show $M(n)\ge \left(\frac{19}{108}-o(1)\right)n^3/\log n$. Together with corresponding estimates for restricted ranges of differences, these results prove several conjectures of Alon–Dębski–Grytczuk–Przybyło and disprove others.

As is already known, there is a correspondence between circle patterns and the dimer model, which is invariant under Euclidean transformations. We establish a new correspondence that is invariant under the group of Möbius transformations PO(3,1), which is the natural symmetry group of circle patterns. We show that Miquel dynamics preserve the partition function, and that convexity of the t-embedding guarantees positivity of the face weights. We also show that the correspondence has a somewhat hidden symmetry group PO(3,3), which relates to the Lorentz lift of the t-embedding. Finally, we consider isoradial graphs, Doyle spirals, the once-punctured torus and isothermic circle patterns.

The directed clique number $\vec{\omega}(T)$ of a tournament $T$ is the minimum, over all orderings of the vertices of $T$, of the clique number of the graph whose edges are the arcs that point backward with respect to the ordering. In this paper, we prove that, for every integer $k \geq 3$, deciding whether $\vec{\omega}(T) \leq k$ is NP-complete. This answers a question of Nguyen, Scott, and Seymour, and contrasts with the classical undirected setting, where deciding whether $\omega(G) \leq k$ is polynomial-time solvable for every fixed integer $k$. On the other hand, we give a polynomial-time algorithm distinguishing tournaments with $\vec{\omega}(T) \leq 2$ from those with $\vec{\omega}(T) > 100$. We also study the tournament analogue of the Gyárfás–Sumner conjecture. We construct new $\vec{\chi}$-bounding tournaments and thereby prove a conjecture of Aboulker, Aubian, Charbit, and Lopes stating that every class of tournaments with bounded twin-width is $\vec{\chi}$-bounded. We then exhibit new tournaments that are not $\vec{\chi}$-bounding, disproving another conjecture of Aboulker, Aubian, Charbit, and Lopes, as well as two conjectures of Kim. Finally, we present infinite families of 3-$\vec{\omega}$-critical and 4-$\vec{\omega}$-critical tournaments.

Continuous-time affine frequency division multiplexing (AFDM) waveforms, constructed via frequency wrapping and phase correction, are known to be sample-wise equivalent to the widely adopted discrete AFDM framework. In this paper, we uncover a fundamental and previously overlooked flaw in this construction: its complex envelope is inherently discontinuous for generic chirp parameters. We show that these discontinuities are the direct cause of the high out-of-band emission (OOBE). To resolve this issue, we propose a fundamentally different continuous-time waveform, termed stepped frequency division multiplexing (SFDM). Unlike conventional approaches that allow continuous frequency variation, SFDM freezes the instantaneous frequency at the midpoint of the underlying chirp trajectory within each Nyquist sampling interval. This design yields a complex envelope that is strictly continuous over the entire symbol duration while preserving exact sample-wise equivalence with discrete AFDM. A unified spectral analysis reveals that the superior OOBE performance of SFDM stems from the absence of internal jump discontinuities, which otherwise dominate the far-out spectral roll-off. Numerical results confirm that SFDM consistently achieves significantly lower OOBE across a wide range of chirp rates.

In decentralized decision-making, several agents connected according to a network graph aim at solving a classification problem by collecting streaming observations. Due to decentralization, they run an iterative algorithm where, at each iteration, they can only exchange information locally with their neighbors. While decentralized estimation solutions have been shown to match the performance of optimal centralized systems, we show here that surprisingly this conclusion does not hold for decentralized decision-making. Specifically, we prove that the error probability for the best decentralized decision strategy exhibits an irreducible loss with respect to the optimal centralized classifier. This result establishes a fundamental limit for the performance of any decentralized decision strategy. We obtain an analytical relation showing that this limit is related to the interplay between decentralization and classification. The first aspect appears through the distances between the nodes in the graph, while the second aspect plays through the moment generating functions of the likelihood ratios that describe the decision problem. By applying the derived closed-form relation to different network topologies and inference problems, we observe some interesting and perhaps unexpected behavior emerging. In particular, we characterize the scaling law (with the network size) for the loss over popular network topologies, showing that the error probabilities might differ by orders of magnitude; and we examine how performance is affected by the relative distance between informative and uninformative agents over the graph.

We give direct combinatorial proofs that 1-semi-equational theories satisfy two combinatorial properties that imply the non-interpretability of certain fields. Our main result is that Boolean combinations of 1-semi-equations have almost linear Zarankiewicz bounds, and hence no infinite field is interpretable in a 1-semi-equational theory; this answers a question of Chernikov–Mennen and Chernikov–Starchenko, who had respectively established almost linear Zarankiewicz bounds for Boolean combinations of $(2,1)$-semi-equations and Boolean combinations of weakly normal relations. (This was recently and independently established by Gou, Mirabi, Mittal, Tran, and Yang, using different techniques and producing different bounds.) We also show that $(k,1)$-semi-equations satisfy the $\delta$-strong Erdős–Hajnal property with $\delta=1/6^{k-1}$; this had previously been established by Chernikov–Starchenko for an ineffective constant $\delta>0$.

This work presents a comparative analysis of Reduced-Order Models (ROMs) applied to incompressible fluid dynamics within geometrically parametrized domains. Two distinct reduced-order solution strategies are investigated and compared: a monolithic coupled solver and a segregated SIMPLE-based algorithm. Their performance is assessed on two steady, two-dimensional benchmark cases: a lid-driven cavity flow and a flow past a cylindrical obstacle. The two algorithms are compared in terms of fields evaluation and aerodynamic coefficients prediction. The computational results highlight a fundamental trade-off between accuracy and numerical efficiency. On the one hand, after an opportune supremizers enrichment, the coupled approach guarantees a faster convergence, despite the need of a larger number of degrees of freedom. On the other hand, the segregated SIMPLE algorithm yields superior reconstruction accuracy, particularly for lower-dimensional reduced spaces, at the cost of a slower convergence rate.

For a prime number $p > 2$, let $\bm{0} \in D \subset \mathbb{Z}^n$ be a $p$-element digit set satisfying $ \mathcal{Z}(\widehat{\delta}_D) =\cup_{j=1}^{p-1}(\frac{j}{p}\bm{a}+\mathbb{Z}^{n}) $ for some \( \bm{a} \in \{ (i_1, \dots, i_n)^t : i_k \in [1, p-1] \cap \mathbb{Z}, 1\leq k\leq n \} \), where $\mathcal{Z}(\widehat{\delta}_D)$ is the zero set of the Fourier transform of $\delta_D$. Let $Q$ be an integer expansive diagonal matrix in $\mathbb{R}^{n}$, the self-affine measure $\mu_{Q,D}$ is defined by \[ \mu_{Q,D}(\cdot) = \frac{1}{\#D} \sum_{d \in D} \mu_{Q,D}(Q(\cdot) - d). \] In this paper, we first provide sufficient condition for a maximal orthogonal family $E_{\Lambda}=\{ e^{-2\pi i \langle \lambda, x \rangle} : \lambda\in \Lambda \subset\mathbb{R}^{n}\}$ to be an orthogonal basis of $L^2(\mu_{Q,D})$ when $Q = p\operatorname{diag}[q, \dots, q]$ with $q\geq 1$. Then we obtain necessary and sufficient conditions for the integer matrix $R$ such that $E_{\Lambda}$ and $E_{R\Lambda}$ are both orthogonal basis of $L^{2}(\mu_{Q,D})$. Furthermore, for $Q = p\operatorname{diag}[l_1, \dots, l_n]$ with $|l_1 \dots l_n| > 1$, we give a necessary and sufficient condition under which the real diagonal matrix is the second type spectral eigenmatrices of $\mu_{Q,D}$.

We introduce a cross-product Landau gradient that commutes with Gaussian mollification. For the hard-potential interaction kernels of the form $A(|v-v_*|)\sim \langle v-v_*\rangle^\gamma|v-v_*|^2$ with $\gamma\in(-\infty,1]$, this construction yields a variational characterisation of the gradient-flow structure of the spatially homogeneous Landau equation. The cross-product gradient induces the same gradient-flow geometry as that introduced by Carrillo, Delgadino, Desvillettes and Wu (2024), while providing a different representation of the Landau gradient. The main advantage of this formulation is that it requires only weighted $L^1$-control. We also discuss a similar variational characterisation for the GENERIC structure of the fuzzy Landau equations.

We study the Møller wave operators for scaling critical electromagnetic Hamiltonians in the plane. For smooth transverse magnetic and angular electric potentials, with nonnegative angular operator and magnetic flux outside $\frac12\Z$, we prove that the wave operators relative to $-\Delta$ exist, are unitary on $L^2$, and, together with their adjoints, are bounded on every $L^p$, $1<p<\infty$. We then specialize to the the Aharonov–Bohm model and we determine the exact ranges for the boundedness of their wave operators on weighted $L^p(\R^2,|x|^\beta\,dx)$ spaces, for both the Friedrichs and the Krein realizations. In the Friedrichs case, this gives in particular the already known boundedness on all $L^{p}$ spaces $1<p<\infty$, while boundedness fails at $p=1,\infty$. In the Krein case, both wave operators and adjoints are bounded precisely when $2/(2-\eta_\alpha)<p<2/\eta_\alpha$, where $\eta_\alpha=\max\{\alpha,1-\alpha\}$ (here $\alpha\in(0,1)$). Thus the boundary condition changes the admissible exponents.

In this work we present the following general principle: every positive homogeneous functional on an abstract convex cone satisfying a suitable Brunn-Minkowski inequality gives rise to an information functional -defined as the quotient of the functional by its relative surface area- that satisfies a linear Brunn-Minkowski inequality. We also establish the dual counterpart of this principle for functionals satisfying an analogous convexity property, yielding reverse linear Brunn-Minkowski inequalities for the associated information functionals. Furthermore, we show that homogeneity is necessary within this principle by providing a counterexample to the expected result for the information functional associated to the standard Gaussian measure. As applications, we recover, via an alternative first-order argument, a classical result by Giannopoulos, Hartzoulaki and Paouris (2002) for the ratio of two consecutive quermassintegrals, together with a characterization of the equality case. The same argument extends to the setting of mixed discriminants of symmetric positive definite matrices. Finally, our approach yields new extensions of the Brunn-Minkowski inequality of Dembo, Cover and Thomas, for the classical information functional, both to the $L_p$ setting ($p \geq 1$), where the relative surface area is expressed in terms of Lutwak's first $L_p$ quermassintegral, and to the measure-theoretic framework, for measures with homogeneous densities satisfying some concavity. Moreover, we prove that our principle applies also in the context of the Blaschke addition, both in the volume and in the (mixed) affine surface area cases, obtaining new inequalities as an application of it.

With help of group characters we enumerate orientably-regular maps of a given type with automorphism group isomorphic to $\PGam(2,2^p)$ for prime $p$.

We prove that the gradient graph of any $W^{2,1}$ semi-convex viscosity solution to the three-dimensional subcritical special Lagrangian equation is area-minimizing and smooth. Furthermore, we prove the smoothness of $W^{2,1}\cap C^{1,1/3+}$ semi-convex viscosity solutions. This yields progress on the conjecture proposed by Wang–Yuan [WY13]. The corresponding results in higher dimensions are also established.

For a simple graph $G$ with $n$ vertices, write its chromatic polynomial in the rising factorial basis as $$ \chi_G(x)=\sum_{i=0}^{n}(-1)^{n-i}c_i(G)\langle x\rangle_i,$$ where $ \langle x\rangle_i=x(x+1)\cdots(x+i-1).$ The associated $\tau$-polynomial $$ \tau_G(x)=\sum_{i=0}^{n}c_i(G)x^i $$ was defined and systematically investigated by Brenti in 1992. In this paper, we prove that if the $\tau$-polynomials of two vertex-disjoint simple graphs $G$ and $H$ have only real zeros, then the $\tau$-polynomial of their join $G\vee H$ has only real zeros. This settles a conjecture posed by Brenti, Royle and Wagner since 1994.

Kechris, Solecki and Todorčević's $G_0$-dichotomy characterizes Borel graphs that admit a Borel measurable coloring with countably many colors. We show that the analogue to the $G_0$-dichotomy for generalized Cantor spaces fails at the lowest possible complexity, namely for closed graphs.

We establish free-energy fluctuation limits for the Ising Sherrington–Kirkpatrick model in the nonzero parts of its critical window. For fixed $b\ne0$ and $\beta_N=1+bN^{-1/3}\sqrt{\log N}$, our main Gaussian orthogonal ensemble (GOE) result is \[ \sqrt{\frac6{\log N}}\left(F_{N,\beta_N}-N\,\mathrm{FE}(\beta_N)+\frac{\log N}{12}\right)\xrightarrow{d}G+\sqrt{\frac32}\,b_+TW_1, \] where $G$ is standard Gaussian, $TW_1$ has the real Tracy–Widom law, and $G$ is independent of $TW_1$, and $\mathrm{FE}$ denotes the limit of spherical Sherrington–Kirkpatrick free-energy. Additionally, we show that in a moderately supercritical regime \[ \frac{2}{N^{1/3}(\beta_N-1)}\left(F_{N,\beta_N}-N\,\mathrm{FE}(\beta_N)+\frac{\log N}{12}\right)\xrightarrow{d}TW_1. \] We also show that the above results remains valid for independent, not necessarily identically distributed, disorder matrices whose first three moments match the Gaussian law and whose fourth moments satisfy an averaged bound.

A topological group is functionally balanced if every bounded real-valued left uniformly continuous function is right uniformly continuous. We prove that every Hausdorff functionally balanced group has coinciding left and right uniformities, giving an affirmative answer to the Itzkowitz problem. Consequently, the classes of SIN, SFSIN and FSIN groups coincide.

Physics-informed neural networks (PINNs) represent a growing frontier in using artificial intelligence to solve partial differential equations (PDEs). Automatic differentiation (AD) plays a central role in this paradigm, which is mesh-free and replaces traditional iterative solvers with gradient-based optimization in continuous space. However, the inherent limitations of AD, particularly in handling higher-order derivatives and discontinuous solutions, pose significant challenges for complex problems. This has motivated a growing number of researchers to explore discretization-based constraints as an alternative path. Yet, the respective applicability of these two paradigms remains largely unexplored. In this work, we conduct systematic experiments across a wide spectrum of problems, from simple linear Poisson to high-Mach hypersonic flows with strong discontinuities. Through a rigorous decomposition of approximation, optimization, and truncation errors, we systematically elucidate the fundamental trade-offs and error-governing mechanisms of both paradigms, as well as two representative network architectures: multi-layer perceptron (MLP) and graph neural network (GNN). Our results reveal a consistent trend: as nonlinearity strengthens, the accuracy advantage of discretization-based constraints becomes increasingly pronounced, with smaller optimization errors compensating for the truncation errors. Moreover, the more complex the nonlinearity and boundary conditions, the greater the advantage of GNN over MLP. These insights offer a robust practical guideline for configuring neural PDE solvers in demanding engineering applications. Our source data and code are available at https://github.com/guoxing0809/neuropde_analysis.

This paper establishes the fully discrete stochastic maximal $L^p$-regularity and the accompanying sharp maximal estimate for numerical approximations of parabolic stochastic partial differential equations. We consider the spatial finite element discretization $A_h$ of a general second-order elliptic operator $A=-\nabla \cdot a\nabla +b\cdot \nabla +c$ with Dirichlet boundary conditions on a smooth, bounded, convex domain in $\mathbb{R}^3$, coupled with a broad class of temporal schemes, including rational approximations and the exponential Euler method. To obtain these optimal discrete regularity results, we establish a bounded $H^\infty$-calculus for the discrete spatial operator $A_h$, uniformly in the mesh size $h$. As a direct byproduct, we also establish the discrete-in-space stochastic maximal regularity for the corresponding spatial semi-discretizations. As an application, we establish the pathwise uniform convergence of a fully discrete approximation of the three-dimensional stochastic Allen–Cahn equation with variable diffusion coefficients and multiplicative noise.

Marginal Mann-Whitney effects are widely used across various fields of research, and extensions of this estimand have been developed in many directions in statistical methodology. In this paper, we focus on an extensions for repeated measurements and factorial designs subject to randomly missing data. In a previous work by Rubarth et al. (2022a), asymptotically correct tests were developed under the assumption of deterministic missing indicators. In contrast, the approach in the present paper accounts for the stochastic nature of missing values under realistic mechanisms. Thus, the involved covariance matrix incorporates the true variability of missing data. The combination with a quasi-randomization procedure using random permutations within each data point yields asymptotically exact tests and a generally improved type-I error control. Additionally, the tests control the type-I error for finite sample sizes in the special case of exchangeable sampling distributions. Simulations across a wide range of settings demonstrate the benefits of the proposed method in small samples, also for different missingness mechanisms. A real data analysis about school children learning math illustrates several practical aspects of the tests' application.

We study the time-optimal control of a controllable linear system on a time horizon [0,T], focusing on the asymptotic switching density for large T. When the system matrix has only real eigenvalues, it is well-known that the number of switches is upper bounded uniformly in T; when it has complex eigenvalues, no such uniform bound exists, and the switching count instead typically grows with T. We characterize this growth for a system matrix with an arbitrary spectrum, allowing simultaneously for real eigenvalues, complex eigenvalues, and a non-trivial Jordan structure. If the dominant mode is complex, the number of switches grows at least linearly in T, with an explicit lower bound expressed via the mean motion problem and the Bohl-Weyl-Wintner formula. We illustrate the theory on a linearized aircraft pitch/altitude model, showing close agreement between the predicted asymptotic switching rate and numerically computed time-optimal controls.

The Group Epimorphism Problem (GpEpi) asks, given two finite groups $G_1$ and $G_2$, whether there exists a surjective group homomorphism, or epimorphism, from $G_1$ to $G_2$. When the input groups are given by their multiplication (Cayley) tables, the problem admits a quasipolynomial-time algorithm in general, but little is known about its complexity for structured classes of finite groups. In this paper, we study the computational complexity of GpEpi for several well-studied classes of finite groups. Our main results are polynomial-time epimorphism tests for several classes of groups for which polynomial-time isomorphism testing was previously known: Groups with Abelian normal Hall subgroups with cyclic complement; Groups with (product of) elementary Abelian normal Hall subgroup with elementary Abelian complement; and Groups with some constraints on their Abelian chief factors.

This paper presents a new algorithm addressing the problem of stochastic optimization where the cost function depends on a vector of uncertain parameters with known statistics. The algorithm is parameterized so as to address various stochastic formulations spanning from Expectation-focused to Value-at-Risk (VaR) as well as Conditional-Value-at-Risk (CVaR)-focused formulations. The algorithm leverages a recently proposed gradient-based Search & Accelerate algorithm which is originally dedicated to deterministic optimization problems. The approach is based on a sequence of warm-started solutions of instances of the problem. These solutions together with a samples of other solutions belonging to the convex hull of the first ones constitute the set of admissible candidates. Among this discrete set of candidates, the optimal solution is selected with regards to a sample-based approximation of the targeted criterion. The relevance of the algorithm and its efficiency are discussed and shown using a tailored illustrative example.

We prove Yau's conjecture $\lambda_{1}=2$ for the stacked Clifford tori of Wiygul: for all integers $N\ge2$, $k,\ell\ge1$ and every sufficiently large $m$, every closed embedded minimal surface arising from Wiygul's construction, of genus $k\ell m^{2}(N-1)+1$ in the round three-sphere and resembling $N$ parallel copies of the Clifford torus joined by small catenoidal tunnels, has first Laplace eigenvalue $2$. For $N\ge3$ these surfaces are chains rather than doublings, and the even–odd decomposition on which all previous verifications for gluing constructions rest is not available. The reflection lemma of Choe and Soret reduces the problem to the sector of functions invariant under the symmetry group of the construction, and we show that the lowest nonzero eigenvalue of that sector equals $4+O(m^{-1})$. The value $4$ is the outcome of an exact identity in the limiting weighted graph model: the limiting waist ratios of the construction form the Perron vector of the adjacency operator of the line graph of a path, so that the spectral gap of the path cancels against the total conductance of the tunnels prescribed by the balancing conditions, and what survives is the coefficient of the Jacobi operator of the Clifford torus. The analytic input consists of a conformally invariant channel inequality on a cylinder and of a Poincaré inequality on a periodically perforated torus. The properties of the construction on which the argument rests are isolated in a reduction theorem for closed surfaces decomposed into blocks joined by families of thin channels along the edges of a finite graph, subject to a symmetry assumption and to Poincaré and trace inequalities on the blocks.