Papers

We derive computable upper bounds on the maximum wave speed in the Riemann problem for the compressible Euler equations with a composite equation of state, in which the pressure is the sum of a convex hydrodynamical part and a barotropic correction. The correction may destroy the convexity of the equation of state, so that the Riemann solution can contain composite shock-rarefaction waves. The bounds are obtained by comparison with the auxiliary Riemann problem for the hydrodynamical equation of state alone. The bounds hold irrespective of whether the outer waves are elementary or composite and require no case distinction on the wave type.

For any category, there exists both a free dagger category and a cofree dagger category over it. A natural question to ask is: given a dagger category, how can we tell if it is free or cofree without specifying an external base category? In this paper, we provide characterizations of both free dagger categories and cofree dagger categories via internal dagger category structure. To characterize cofree dagger categories, we use rectangular bands and show that a dagger category is cofree if and only if it is enriched over rectangular bands. For free dagger categories, we define the notion of a zigzag dagger category, and then show that a dagger category is free if and only if it is a zigzag dagger category. We also show that free dagger categories can be characterized as the coalgebras of the induced comonad from the free dagger category adjunction, and similarly that cofree free dagger categories can be characterized as the algebras of the induced monad from the cofree dagger category adjunction.

For a vector bundle $E$ on a smooth projective curve $C$, one defines the Quot scheme $Q_d(E,C)$ parametrizing subsheaves of $E$ having length $d$ torsion quotients. If $E_i$ is a vector bundle of rank $r\geq 2$ on a smooth projective curve $C_i$ for $i=1,2$, isomorphisms between $\mathbb{P}_{C_i}(E_i)$ preserving the projective bundle structures induce isomorphisms of $Q_d(E_i,C_i)$, called natural isomorphisms. We show that for $d\geq 2$ all isomorphisms between $Q_d(E_i,C_i)$ are natural, except for a non-natural involution of $Q_2(\mathcal{O}_C^{\oplus r},C)$. This gives a complete description of the automorphism group of $Q_d(E,C)$, generalizing previous works of Biswas-Dhillon-Hurtubise and Gangopadhyay. This also shows that one can reconstruct the curve from the Quot scheme, a Torelli-type theorem. As a key step in our proof, we show that any isomorphism between symmetric powers of smooth projective curves preserving big diagonals is natural, which is interesting in its own right.

We prove that if $E$ is a UMD symmetric Banach function space on $(0,\infty)$, then $E(\mathcal{M},\tau)$ is UMD for every semifinite von Neumann algebra $\mathcal{M}$ equipped with a faithful normal semifinite trace $\tau$. This resolves in the affirmative an open problem that has circulated in the non-commutative world for more than four decades.

We prove that every $4$-partite $4$-uniform hypergraph $H$ with matching number $\nu(H)=2$ satisfies $\tau(H)\le 6$, where $\tau$ denotes the vertex-cover number. This confirms a claim made by Tuza in his 1979 manuscript but never published with a proof, and closes the case $(r,\nu)=(4,2)$ of Ryser's conjecture. The best previous bound was $\tau\le 7$, an integrality consequence of the theorem of Haxell and Scott (2012). The proof uses Gyárfás's intersecting-case theorem ($\tau\le 3$ for intersecting $4$-partite $4$-uniform families), a short projection lemma (four base-disjoint edges in an intersecting family force a two-element cover), and Kőnig's matching theorem.

We prove two complementary Lewy-type theorems for elliptic equations whose coefficients depend only on the gradient. First, let $\Omega\subset\mathbb{R}^3$ and let $u$ solve \[ a^{ij}(Du)u_{ij}=0, \] where $a$ is a smooth, symmetric, positive definite matrix field on an open set containing $Du(\Omega)$. We show that if the gradient map $Du$ is a local homeomorphism, then $\det D^2u$ never vanishes; hence $Du$ is a local $C^\infty$-diffeomorphism. Second, in every dimension, we consider capacitary solutions on convex rings. If the solution has no critical points and its level hypersurfaces are strictly convex, then ellipticity alone forces the Hessian to have one positive and $n-1$ negative eigenvalues, that is, inertia $(1,n-1)$. Consequently, the gradient is a global diffeomorphism onto a radially parametrized ring in gradient space. Both results apply to the $p$-Laplace and minimal surface equations. For the global minimal-surface result, existence of a smooth solution is assumed. Classical convex-ring results supply the noncriticality and strict level-set convexity needed in the global corollaries. Since the two coefficient matrices are real analytic on the relevant gradient ranges, the corresponding local and global gradient diffeomorphisms are real analytic.

We show that every Anosov diffeomorphism is topologically expansive in the sense of [9]. We also prove that Lipschitz shadowing forces the stable and unstable bundles to be uniformly transverse, with a quantitative bound in terms of the shadowing constant. Finally, we give two examples on complete Riemannian manifolds: one is expansive but does not have the shadowing property, while the other has finite volume, bounded sectional curvature, Lipschitz shadowing, and orthogonal invariant bundles, but is not expansive with respect to its Riemannian distance.

The paper focuses on the following problem: given a generalized Bratteli diagram $B$, determine conditions under which the tail equivalence relation $\mathcal R$ does not admit a tail invariant probability measure on the path space $X_B$. We give several sufficient conditions, formulated in terms of the incidence matrices, that guarantee the nonexistence of such a measure. We also discuss a new spectral method for finding tail invariant measures for a class of stationary generalized Bratteli diagrams. This method allowed us to answer affirmatively an open problem formulated in arXiv:2212.13803.

We give a quantitative disorder comparison for the zero-field Sherrington–Kirkpatrick model at inverse temperature one. For any fixed number of replicas, replacing Gaussian couplings by independent couplings from a bounded symmetric variance-one law changes a bounded replica expectation by at most a constant times the Gaussian overlap second moment plus $N^{-1}$. A multiplicative comparison for nonnegative observables controls the tails. The proof uses Talagrand's positive-observable estimates and Yu-Ting Chen's fourth-order replica cancellation. Assuming the Gaussian critical overlap limit stated by Du and Huang, we obtain the same limit for the random quenched overlap measure in the iterated Wasserstein-2 topology. No separate moment assumption is needed for this transfer. Their additional exponential moment bound gives convergence of the rescaled spin-glass susceptibility in every finite Wasserstein distance. The coupling class includes fair signs, uniform disorder, and signs thinned at any fixed positive density.

We introduce a notion of nilpotency for surjective quandle homomorphisms, which relativizes the nilpotency of quandles recently defined by Darné. A surjective homomorphism is called nilpotent if its relative inner automorphism group is nilpotent relative to the inner automorphism group of the source. We show that this condition can equally be described through the relative transvection group and the relative adjoint group, and that it is equivalent to being a finite composite of covering homomorphisms. Extending the relative lower central series to transfinite ordinals, we then introduce hypocentral homomorphisms and characterize them as the surjections that are right orthogonal to the strongly connected homomorphisms. As a consequence, the strongly connected and the hypocentral homomorphisms form an orthogonal factorization system for the surjections in the category of quandles. Finally, we also introduce reduced and operator-reduced homomorphisms, which relativize reduced quandles, and give sufficient conditions for such a homomorphism to be nilpotent.

We study physics-informed neural networks (PINNs) for the Dirichlet boundary control of a semilinear parabolic equation with Tikhonov regularization. Two approaches are considered. A direct PINN parameterizes the state and control by separate networks and minimizes a penalized form of the tracking objective. An indirect PINN instead represents the state, adjoint, and control by unconstrained networks trained jointly to satisfy the first-order optimality system, with the state-control coupling and the homogeneous adjoint boundary and terminal conditions imposed as soft penalty terms rather than enforced architecturally. For the indirect formulation we develop an error estimation framework that decomposes the total error into approximation, optimization, quadrature, and soft boundary/terminal-constraint contributions. Under standing assumptions on optimal-solution regularity and compatibility, network approximability, uniform Hölder control of the soft-constraint residuals, and a local neighborhood of the reference optimality-system solution, we derive a quantitative linearized stability estimate and a conditional local nonlinear residual-to-error estimate, and construct a computable residual indicator with a conditional reliability bound. Numerical experiments on two manufactured test problems - a cubic reactiondiffusion equation and a linear equation with a nontrivial boundary control and adjoint - illustrate the behavior of the direct and indirect formulations.

Graph matrices encode dependencies in random matrices built from shared random variables and arise in spectral algorithms, sum-of-squares (SoS), and high-dimensional statistics. We determine how finite graph structure controls their sharp spectral growth. For every fixed simple graph shape in the dense Rademacher model, including overlapping or empty matrix boundaries, we prove $\mathbb E\|M_\alpha\|=\Theta_\alpha(n^{(v+h-s)/2}(\log n)^{a_*/2})$, where $v$ counts vertices, $h$ isolated summation vertices, $s$ the minimum boundary-separator size, and $a_*$ maximizes an active-component count over minimum separators. Thus two finite cut optimizations determine both the polynomial and logarithmic exponents. The formula closes the polylogarithmic gap in separator bounds, and an infinite family with identical coarse parameters but different norms shows that the logarithmic exponent records genuinely new structure. The proof controls all defect layers in growing trace moments by converting label loss into separator excess; conditional flattening and synchronized fluctuations yield matching lower bounds. We extend the analysis to specified independent-factor chaoses, local weights, unequal dimensions, bounded asymmetric noise, Gaussian inputs, and fixed-degree Hermite inputs. Applications include degree-four clique SoS feasibility for $9\le k\le c\sqrt n$ without an asymptotic logarithmic loss, and Gaussian random tensor networks: deviation thresholds, sharp expected scales, entropy estimates, and, for connected loopless equal-dimensional networks, convergence of the rescaled largest output eigenvalue to the exact right edge of the limiting law. These results connect finite structure to sharp growth scales, and additional algebraic and spectral structure to full feasibility and exact limiting constants.

For each fixed integer $K\ge2$, we consider the Enots–Wolley sequence in which every term after the initial $1,2$ is required to have between two and $K$ distinct prime divisors. We prove that every integer satisfying this restriction occurs. If an exact prime support $T$ were selected only finitely often, then after a finite cutoff the terms meeting $T$ would form short episodes, and every full term would force an earlier proper term at comparable numerical height. The case $|T|=K$ is then ruled out directly. In the remaining case $|T|<K$, at a record proper value $H$, greediness forces every unblocked rank-$K$ integer below $H$ containing exactly one prime of $T$ to have occurred earlier. Fixed-order Landau estimates show that this proper population has order $H(\log\log H)^{K-2}/\log H$, while the entire possible full population on the same scale has strictly smaller logarithmic order. This contradiction proves surjectivity. The theorem concerns each fixed rank cap and does not settle surjectivity of the unrestricted Enots–Wolley sequence.

We study the linear stability of compactly supported static spherical equilibria of the non-isentropic Euler–Poisson system. In the Schwarzschild-stable case, the linearized equations are realized as a separable Hamiltonian system on weighted spaces adapted to the physical vacuum. We prove stability against non-radial perturbations and show that the algebraic dimension of the radial unstable subspace is the Morse index of a density quadratic form subject only to the mass constraint. Thus the infinitely many linearized entropy constraints reduce, for radial perturbations, to one mass constraint and an explicit entropy reconstruction. We apply this criterion to two entropy prescriptions. No smallness condition is imposed on a fixed entropy–density relation: under the stated effective-pressure and branch hypotheses, a simple mass maximum is not a stability transition. By contrast, for a small entropy distribution fixed as a function of enclosed mass, the turning-point principle holds. In the Schwarzschild-unstable case, we construct the self-adjoint velocity operator from its quadratic form, prove a strictly negative spectral bottom and sharp exponential growth, and reconstruct energy solutions of the original first-order linearized system.

Complex engineering optimization problems are often characterized by multimodality, high dimensionality, and nonlinear constraints, posing significant challenges for efficient and reliable computation. To address these challenges, this paper develops a hybrid crossover-based optimization framework that enhances population interaction and improves search efficiency. The proposed framework integrates two complementary mechanisms, namely a Levy long jump crossover strategy for global exploration and a horizontal-vertical crossover strategy for effective information exchange and local refinement, thereby improving convergence behavior and robustness. To support scalable computation, the method is implemented within a unified multi-backend computational framework based on FEALPy, enabling consistent and efficient execution across heterogeneous platforms, including NumPy and PyTorch on both CPU and GPU. This design enhances portability, reproducibility, and computational efficiency in large-scale optimization tasks. Extensive experiments on the IEEE CEC2022 benchmark suite demonstrate that the proposed framework achieves competitive or superior performance compared with several representative metaheuristic algorithms, as validated by Wilcoxon rank-sum and Friedman statistical tests. In addition, the method shows strong performance on constrained engineering design problems. Finally, the proposed framework is applied to UAV path planning, formulated as a constrained optimization problem, demonstrating its effectiveness and scalability in complex engineering scenarios.

We prove the global well-posedness and nonlinear stability of classical solutions to the three-dimensional compressible viscous, non-resistive MHD system on $\mathbb{T}^3$ near the equilibrium $(1,\mathbf{0},\mathbf{e}_3)$, for small $x_3$-symmetric perturbations and without any Diophantine condition on the background magnetic field. The central obstruction is the $x_3$-independent sector, in which the density and the magnetic field possess no dissipation, no damping, and no decay. We overcome it by exhibiting a hidden wave structure for the exact total pressure $\mathcal{D}=P(1+a)+B_3+\frac12|\mathbf{B}|^2$, the averaged pair $(\overline{\mathbf{u}},\overline{\mathcal{D}})$ obeys a closed system, and $\overline{\mathcal{D}}$ satisfies a strongly damped wave equation whose principal wave part propagates at the fast magnetosonic speed and whose non-parabolic branch damps at a rate that involves no gain of derivatives. This hidden damping substitutes for the missing magnetic dissipation, and simultaneously absorbs the magnetic pressure $\frac12\nabla|\mathbf{B}|^2$, the main obstruction created by compressibility. Together with a damped wave structure for the oscillatory sector and space-time weighted energy functionals with shifted time weights, this yields global existence, uniform stability, and explicit polynomial decay rates.

We study classical Maxwell fields with non-trivial sources in four-dimensional Minkowski spacetime. In particular, contact and symplectic geometric aspects are discussed. As a point of departure, it is shown that a few classes of solutions with sources are constructed from the so-called Beltrami fields, where these fields are contact forms on three-dimensional Riemannian manifolds. In addition, several applications in these classes are provided. One is to give a solution to the London equations for superconductors. Another one is that, if there is a non-trivial conserved quantity along the magnetic vector field, then the Maxwell system can be viewed as a Liouville integrable Hamiltonian system on a four-dimensional symplectic manifold, where this manifold is obtained by a symplectization of the contact manifold. As examples, Maxwell fields with the Lundquist model, a restricted case of the ABC flow model, and so on, are shown to be integrable in the above sense.

Let $K(n)$ be the generalized Kronecker quiver, and let $(T(n),\Omega)$ be the universal covering of $K(n)$, where $n\geq 3$ and $\Omega$ is a bipartite orientation. We introduce three different types of representations for quiver $(T(n),\Omega)$ by $K(n)$: maps are injective in a representation maps are all surjective in a representation, and representations are of constant rank. We use $\Inj, \Sur$ and $\GR$ to denote the corresponding subcategories containing these representations, respectively, and we show that \begin{center} $\GR= \Inj\cup\Sur$. \end{center}

Let $k$ be an algebraically closed field. The generalized or $n$-Kronecker quiver $K(n)$ is the quiver with two vertices, called a source and a sink, and $n$ arrows from source to sink. We use $\modd\cK_n$ to denote the category of the finite-dimensional modules of the path algebra $\cK_n=kK(n)$. There exist a function called dimension vector $\dimu: \modd \cK_n\rightarrow \mathbb{Z}^2; M \mapsto (\dim_k M_1, \dim_k M_2 )$, and two functors $\sigma: \mathbb{Z}^2 \rightarrow \mathbb{Z}^2; (x,y)\mapsto (nx-y,x)$, and $\delta: \mathbb{Z}^2\rightarrow \mathbb{Z}^2;(x,y)\mapsto (y,x)$. Then the set $\mathbf{F}=\{(x,y)\mid \frac{2}{n}x\leq y\leq x\}$ is the fundamental domain of the dimension vectors of regular modules under the action of $\delta$ and $\sigma$ in $\modd\cK_n$. We call a regular module $M\in \modd\cK_n$ elementary if there does not exist a short exact sequence $(0)\rightarrow L\rightarrow M\rightarrow N\rightarrow (0)$ with $L,N$ being non-zero regular. In this note, we focus on the set $\mathbf{F}$, and when $x<n$, we establish a one-to-one correspondence between elementary modules and spaces of fixed rank.

We study the deterministic first-order oracle complexity of smooth nonconvex-concave minimax optimization over a bounded convex dual domain under two notions of stationarity: optimization stationarity (OS) and game stationarity (GS). OS concerns stationarity of the value function, viewed as a structured nonsmooth objective obtained by maximizing over the dual variable. GS concerns the joint first-order optimality conditions for both primal minimization and dual maximization, each with the other variable held fixed. Existing upper bounds on the oracle complexity of first-order methods for finding $\epsilon$-OS and $\epsilon$-GS points have different dependence on $\epsilon$, but whether this difference reflects an intrinsic separation in computational hardness remains unresolved. We resolve this question by establishing worst-case lower bounds for arbitrary deterministic first-order methods: $\Omega(\epsilon^{-3})$ oracle queries for finding an $\epsilon$-OS point and $\Omega(\epsilon^{-5/2})$ for finding an $\epsilon$-GS point, with the remaining problem parameters fixed. These bounds match existing upper bounds up to logarithmic factors. To close the remaining logarithmic gap, we introduce Tracked-FOAM, a new first-order algorithm whose oracle complexity matches our lower bounds for both OS and GS. Together, these results establish an intrinsic separation between OS and GS in the optimal accuracy dependence of deterministic first-order oracle complexity.

We study lower bounds on the first-order oracle complexity of smooth nonconvex-concave minimax optimization. We consider objectives $f$ that are jointly $L$-smooth in the primal and dual variables $(x,y)$, concave in $y$, and whose primal value function $\Phi(x) := \max_{y\in\mathcal Y} f(x,y)$ satisfies the initial-gap condition $\Phi(0)-\inf_{x\in\mathcal X}\Phi(x)\le \Delta_\Phi$, with a bounded dual domain satisfying $\operatorname{diam}(\mathcal Y)\le D_{\mathcal Y}$. We measure stationarity by the norm of the gradient of the Moreau envelope of $\Phi+\iota_{\mathcal X}$ with parameter $1/(2L)$. We prove that any deterministic zero-respecting first-order algorithm requires $\Omega\left(L^2D_{\mathcal Y}\Delta_\Phi\epsilon^{-3}\right)$ oracle evaluations to find an $\epsilon$-stationary point. Under an unbiased stochastic first-order oracle with bounded variance, any stochastic zero-respecting algorithm requires $\Omega\left(L^3D_{\mathcal Y}^2\Delta_\Phi\epsilon^{-6}\right)$ oracle evaluations. The same lower bounds hold when $\Delta_\Phi$ is replaced by the initial primal-dual gap $\mathcal G_0$. These deterministic and stochastic lower bounds match the corresponding upper bounds of [14] and [29], respectively, up to a logarithmic factor in the deterministic setting.

We develop an exact posterior-sampling framework for Bayesian inverse problems when selected bounded forward observables can be accessed through Bernoulli events. A Bernstein–Poisson construction converts these forward coins into scaled Gaussian likelihood coins, and thinning an inflated prior Poisson point process yields posterior atoms that are iid conditional on their number. For independent Gaussian observations, we derive an exact mean-work identity for the implemented early-stopped factory and sharp small-noise complexity laws governed by local prior-predictive mass near the exact-fit set; a factorial-moment construction extends the likelihood factory to correlated Gaussian errors. The posterior algorithm is model-agnostic once Bernoulli access is available. As one continuum realization, we use Feynman–Kac sampling, Poisson killing, and lazy random-series evaluation for a bounded elliptic resolvent problem with a function-valued coefficient. Numerical experiments validate the forward and likelihood coins, support the predicted work regimes, and demonstrate posterior sampling without deterministic spatial discretization or fixed parameter truncation in the target. A matched-accuracy benchmark against finite-difference prior rejection illustrates how deterministic discretization bias changes the posterior accuracy–cost balance.

We study the distribution of the number of coprime lattice points in randomly translated disks. We prove that its spatial variance is asymptotic to $\sigma^2r\log r$ and establish a central limit theorem under the corresponding normalization, where $\sigma^2$ is an explicit positive constant. By truncating the numerators of rational frequencies in lowest terms, the proof reduces higher-moment estimates to counting squarefree denominators in zero-sum relations.

We prove the remaining even-dimensional case of the Lambrechts–Stanley "uniqueness" conjecture for 1-connected differential Poincaré duality models. Together with our previous odd-dimensional result, this shows that any two such algebras weakly homotopy equivalent as PDGAs admit direct PDGA quasi-isomorphisms into a common connected differential Poincaré duality algebra. We use our previous method of obtaining differential Poincaré duality models as nondegenerate quotients of Hodge extensions, the new ingredient being a Hodge extension in the middle degree.

Let $(X,r)$ be a finite square-free distributive set-theoretic solution of the Yang-Baxter equation, and let $\mathcal{A}(X,r)$ be its associated Yang-Baxter algebra. We prove that $\mathcal{A}(X,r)$ is isomorphic to a skew polynomial algebra and explicitly compute its Nakayama automorphism. We study the natural faithful action of the permutation group $\mathcal{G}(X,r)$ on $\mathcal{A}(X,r)$ by graded algebra automorphisms. We characterize when $\mathcal{G}(X,r)$ is a reflection group and, in that case, describe the invariant subalgebra $\mathcal{A}(X,r)^{\mathcal{G}(X,r)}$ together with its Jacobian, reflection arrangement, and discriminant. We further establish a necessary and sufficient condition for the noncommutative Auslander theorem to hold for the permutation group action on a distributive Yang-Baxter algebra. Finally, for a class of non-trivial distributive Yang-Baxter algebras, we obtain a lower bound for the pertinency of the permutation group action.

We resolve a conjecture on the law of large numbers for coherent measures on Young diagrams associated with Macdonald-nonnegative specializations of the algebra of symmetric functions for all $0\le q,t<1$. We prove that row and column lengths divided by the diagram size converge to the corresponding specialization parameters. The proof combines a monomial estimate for skew Macdonald polynomials with exponential tilting and binomial decompositions obtained from the branching rule for the polynomials.

We study simultaneous recovery of the (variable) mass density, initial displacement, and initial velocity for the three-dimensional isotropic elastic wave equation with known constant Lamé parameters. The data are the complete displacement trace on an enclosing boundary. We first assume that the density-weighted initial displacement and velocity have fixed known profiles in one spatial direction. The $s^0$ and $s^1$ coefficients of the zero-frequency Laplace expansion identify these weighted states. The $s^2$ and $s^3$ coefficients then give static Lamé orthogonality identities for the density difference. The exact difference expansion through order $s^3$ has an $O(s^4)$ remainder, uniformly on bounded spatial sets and bounded density-contrast and weighted-state classes. Under alignment of the two initial states and a nonzero moment of the density-weighted initial velocity, the two density identities reduce to a constant-vector static transform. Two opposite elastic null phases give uniqueness for one or two fixed vertical density profiles when the profile family is two-sided Laplace nondegenerate. When $\lambda+\mu\ne0$, each nonzero normal root has partial multiplicities $2$ and $1$ and admits a length-two Jordan chain. The associated polynomial–exponential Lamé mode produces derivatives of the bilateral profile transforms. The resulting Hermite–Laplace system gives uniqueness for aligned classes with up to four fixed vertical profiles and independent horizontal coefficients, provided that the profile system is Hermite–Laplace nondegenerate. Distinct translations of one compactly supported profile provide an explicit four-profile class. We also prove rigidity of the alignment reduction and exhibit an infinite-dimensional kernel for the reduced transform on unrestricted densities.

Normalized implied-volatility coordinates reveal no-arbitrage structure that is less transparent in strike space. This paper centers on a single quantity that gives those coordinates a direct probabilistic interpretation.

We introduce a graph invariant obtained from Pauli decompositions of Hamiltonians derived from local graph neighborhoods. Given a rooted h-hop neighborhood, we construct a local adjacency operator, embed it into a common Hilbert space dimension, and define its Pauli-support set as the collection of Pauli strings appearing with nonzero coefficients. For periodic graphs, we prove invariance under lattice-compatible graph equivalence and establish converse results under root-separation hypothesis. The framework naturally extends to Lie closures, commutants, and Cartan-type structures generated by the associated Pauli supports. We further discuss extensions to aperiodic graphs with finite local complexity and make connections to cut-and-project models of quasicrystals and notably, Penrose tilings. From a quantum information perspective, the resulting invariants provide a operator-theoretic description of graph-based Hamiltonians and offer a new mechanism for comparing graph-based quantum systems through Pauli-support data.

A minimal surface in $\mathbb{R}^3$ is $\textit{separable}$ if it is the zero set of $f(x)+g(y)+h(z)$, and $\textit{isotropic}$ if moreover $f=g=h$. We announce that there are exactly two non-planar isotropic separable minimal surfaces up to homothety and rigid motion, that they are conjugate, and that they are Schwarz's diamond surface $\mathrm{D}$ and Schwarz's primitive surface $\mathrm{P}$. In lattice-normalized coordinates their implicit equations are \[\mathrm{D}:\; \mathcal{D}(x)\mathcal{D}(y)\mathcal{D}(z)=1,\] \[\mathrm{P}:\; \mathcal{C}(x)+\mathcal{C}(y)+\mathcal{C}(z) =3\,\mathcal{C}(x)\mathcal{C}(y)\mathcal{C}(z),\] where $\mathcal{D}(t)=\frac{\operatorname{sn}\operatorname{dn}}{\operatorname{cn}}(K[\frac{1}{4}]t,\frac{1}{4})$ and $\mathcal{C}(t)=\operatorname{cn}(2K[\frac{3}{4}]t,\frac{3}{4})$. The first equation is classical – it is due to Schwarz, later Cayley, and appears in Nitsche's $\textit{Lectures}$ – we identify Nitsche's transcendental function exactly as $\operatorname{sn}\operatorname{dn}/\operatorname{cn}$ at parameter $\frac{1}{4}$. The second appears to be new; it is the symmetric member of the two-parameter family of Kim and Ogata (2024), and it proves their assertion that the family contains $\mathrm{P}$. Both surfaces are governed by one differential equation with two signs, \[\varphi'^{\,2}=1+2\cosh 2\varphi\ \ (\mathrm{D}),\] \[\varphi'^{\,2}=1+2\cos 2\varphi\ \ (\mathrm{P}),\] the two signs of the separation constant of the classical reduction; and in that common normalization the two lattice periods are $2K[\frac{1}{4}]$ and $2K[\frac{3}{4}]$, whose ratio is Schwarz's 1866 constant $K'[\frac{1}{4}]/K[\frac{1}{4}]=1.2792615...$, the necessary scale ratio for a conjugate pair. Proofs will appear in [D, P].