Papers

We construct a compressible isotropic energy on $\mathrm{GL}^+(3)$ combining polyconvexity and global true-stress-true-strain monotonicity (TSTS-M++), answering a question of Wollner, Holzapfel and Neff. For every $\mu>0$ and $k>0$, the energy $W_k(F)=\frac{\mu}{2k}[\exp(k(\lVert F\rVert^2+3J^{-1}+J-7))-1]$, $J=\det F$, is polyconvex and strictly rank-one convex. Its Cauchy-stress response satisfies TSTS-M++ globally if and only if $k\ge1/(8\sqrt3)$. The reference state is stress free with positive infinitesimal shear and bulk moduli. Stress bijectivity has the smaller sharp threshold $k_{\rm B}\approx0.0083$: at equality the stress map is a global homeomorphism with a nondifferentiable inverse, and above it a global smooth diffeomorphism. Thus, for $k_{\rm B}\le k<1/(8\sqrt3)$, the Cauchy-stress map remains globally bijective while TSTS-M++ fails at finite strain. In the TSTS-M++ regime, each prescribed inner radius of a finite plane-strain annulus with a traction-free outer wall has a unique radial equilibrium, and the inner pressure increases smoothly and strictly from zero to infinity. Comparisons with other energies distinguish the roles of polyconvexity, TSTS-M++, reduced convexity and endpoint growth. Under incompressible spherical inflation, global pressure inversion holds for the Demiray family with $b\ge b_*$, including $W_k|_{J=1}$ for $k\ge k_*$ and the stress-free $I_1$-exponential energy, and for the exponentiated Hencky energy when $k_{\mathrm{eH}}>3/8$. Incompressible neo-Hooke instead has a limit point. A further compressible $I_1$ energy retains the global constitutive properties and reference moduli of $W_k$ but exhibits two pressure turning points under incompressible spherical inflation, showing that these properties allow a descending pressure branch.

We record two results on additive diameters of group representations. First we give a corrected version of Proposition 5.3 from the first arXiv version of arXiv:2504.07573. Let $0<\varepsilon<1/3$, let $n>9/\varepsilon^2$, and let $U\leq\mathfrak{sl}_n(\mathbb{C})$ with $\dim U>\varepsilon n^2$. Then $\operatorname{diam}^{\mathrm{SL}_n(\mathbb{C})}_{+}(\mathfrak{sl}_n(\mathbb{C}),U)\leq 32/\varepsilon+8$. The proof uses an averaging argument for the action of the symmetric group on the off-diagonal matrix positions. This argument was subsequently developed further in arXiv:2609.03882. We also prove the right-hand inequality in Question 6.7 of arXiv:2504.07573. If $G$ is a complex algebraic group, $V$ is a finite-dimensional $G$-module, $U\leq V$, and $\mathfrak g=\operatorname{Lie}(G)$, then $\operatorname{diam}^{G}_{+}(V,U)\leq\operatorname{diam}^{\mathfrak g}_{+}(V,U)$. No irreducibility or connectedness assumption is needed.

In this note, we revisit a Liouville-type theorem of Chae and Wolf for stationary Navier-Stokes equations in $\mathbb{R}^3$ [J. Differential Equations 261 (2016) 5541-5560]. We show that their logarithmic improvement of the classical $L^{9/2}$ condition is part of a substantially broader weighted framework. More precisely, we prove that a solution $u\in\dot H^1(\mathbb{R}^3)$ is necessarily trivial whenever $$ \int_{\mathbb{R}^3}|u(x)|^{9/2}\,\omega(|u(x)|)\,dx< +\infty, $$ for every positive, nondecreasing and bounded weight $\omega$ satisfying a mild growth condition near the origin. This structural condition encompasses the logarithmic weight due to Chae and Wolf, as well as a hierarchy of iterated-logarithmic weights and (genuinely) non-logarithmic examples, including a dyadic weight. Our result identifies a broader class of weighted integrability conditions under which the triviality of stationary Navier-Stokes solutions follows, and shows that the mechanism underlying the Chae-Wolf improvement is not intrinsically tied to a specific logarithmic weight or to a single logarithmic scale.

We prove that continuous $A$-bilinear pairings on the ring of Laurent series that are invariant under continuous automorphisms coincide, up to a constant, with the pairing given by the residue of a differential form over any commutative associative ring with identity.

We present Cadence, an error-bounded lossy compressor for numeric time series pairing a 330M-parameter time-series foundation model (Google TimesFM-3) with an adaptive arithmetic coder, guaranteeing $|\hat{x}_t-x_t|\le\tau$ on every sample. One negative result constrains the design space: for lossless coding a foundation model is worth nothing, because bits saved are logarithmic in predictor accuracy, $\Delta b=\log_2(\mathrm{MAE_{old}}/\mathrm{MAE_{new}})$. So the $1.51\times$ advantage TimesFM-3 holds over a 32-tap linear predictor buys 0.60 bits of 20.28, a median gain of +0.03%. Error-bounded coding escapes this at one point: once a forecast lands inside the band the residual index is zero and the sample nearly free. Cadence contributes: (1) an adaptive range coder with context-modelled binarization, beating xz/zstd on real indices by 9.7% (15/15) and reversing a finding from a general-purpose back end; (2) a determinism result – predictions are not bit-identical across batch sizes, and no PyTorch configuration repairs this, forcing group size and execution device into the container format; and (3) domain localization on corpora postdating any plausible training cutoff. On 49 EIA-930 balancing-authority demand series (2026) Cadence gains 13.3% over the best of six classical predictors, and 28.3% on 50 MTA ridership series (2026): 21.4% median over 297 series-tolerance pairs, winning all 297. Against downsampling, what time-series databases deploy for retention, its guaranteed worst-case error is $28$–$56\times$ tighter at equal size. End-to-end, once the context bootstrap is paid for, gains run from 6.8% at six months of hourly data to 15.1% asymptotically. Attempting to falsify the domain claim on SDRBench, theory predicts failure and delivers: -0.8% median, 0 of 27 pairs gaining. Three further negative results and eight retracted claims are reported in full.

In this paper, we prove Shokurov's conjecture in relative dimension $1$-case in sufficiently large characteristics. Depending on this result, we prove the boundedness of Stein degrees for horizontal boundaries in a log Calabi-Yau fibration in sufficiently large characteristics.

Let $M$ be an $n$-dimensional complete connected self-shrinker in $\mathbb{R}^{n+p}$. Denote by $\operatorname{Ric}$ and $\mathbf{H}$ the Ricci curvature tensor and the mean curvature vector of $M$, respectively. We prove that if the self-shrinker $M$ satisfies $\operatorname{Ric}\ge(\frac{n-2}{n^2}+\varepsilon_n)|\mathbf{H}|^2g$, then $M$ is either a linear subspace or a round shrinking sphere, where $\varepsilon_n$ is an explicit positive constant equal to $\frac{1}{3n^2}+O\left(\frac{1}{n^5}\right)$. Moreover, we obtain a rigidity theorem that is sharp in codimension two for the self-shrinker satisfying $\operatorname{Ric}\ge\frac{n-2}{n^2}|\mathbf{H}|^2g$.

We construct an endpoint map with no regular values, disproving the sub-Riemannian Sard conjecture. Remarkably, in our construction the minimizing Sard property remains valid.

We study randomized constructions of quasi-uniform designs on a compact metric-measure space satisfying two-sided polynomial ball-growth conditions. Given integers $N\le M$, we first draw $M$ independent candidate points and then retain the first $N$ points of a farthest-point traversal of the candidate set. We prove probabilistic non-asymptotic bounds on the mesh ratio and show that $M=\Theta(N\log N)$ is the sharp order of the candidate-pool size required for bounded mesh ratio: a sufficiently large multiple of $N\log(N/\delta)$ suffices with probability at least $1-\delta$, whereas, if $M=o(N\log N)$, the mesh ratio diverges in probability for every procedure that selects $N$ points from the same independent candidate pool. If $M/(N\log M)\to\infty$, the upper bound on the mesh ratio for the exact farthest-point sampling (FPS) tends to $2$. We also quantify the effect of approximate FPS and analyze direct and fast implementations in spaces of bounded doubling dimension. Finally, using a single infinite stream of candidate points and increasing candidate budgets, we construct an almost surely quasi-uniform nested sequence; under supercritical oversampling and exact FPS, or more generally when the approximation factors tend to $1$, its mesh-ratio limit superior equals $2$ for compact positive-volume subsets of Euclidean space.

We study the numerical approximation of high-dimensional integrals using randomized quasi-Monte Carlo (RQMC) methods, with a focus on scrambled Sobol' sequences. While asymptotically faster than Monte Carlo, classical RQMC suffers from error bounds that grow exponentially in the dimension $s$, and its root mean squared error (RMSE) convergence rate is generally no better than $O(N^{-3/2})$. To overcome these limitations, we combine two recent developments: Suzuki's coarse scrambling and the median trick. Coarse scrambling randomizes Sobol' sequences according to their generating base polynomials and significantly reduces the maximal gain coefficient. By appropriately choosing the base polynomials, we show that the coefficient can be made uniformly bounded in $s$, yielding an $O(N^{-1/2})$ RMSE for $L^2$ integrands with a dimension-independent constant. The median trick then enables near-optimal convergence for function classes beyond $L^2$: for $L^p$ integrands with $p\in(1,2)$, the median of independent coarsely scrambled estimates achieves an error of $O(N^{-1+1/p})$ with high probability; for integrands in the Haar wavelet space $\mathcal H_{\mathrm{wav},\alpha,s,p,q}$ with $\alpha_p:=\alpha-(1/p-1/2)_+>0$, we prove a high-probability error bound of $O(N^{-\alpha_p-1/2+\varepsilon})$ for any $\varepsilon>0$; the bound improves to $O(N^{-r-\alpha_p-1/2+\varepsilon})$ when the integrand has dominating mixed derivatives of order $r\ge1$ that belong to $\mathcal H_{\mathrm{wav},\alpha,s,p,q}$. The latter two bounds are uniform in $s$ under suitable conditions on the ANOVA components of the integrands. Numerical experiments confirm the predicted convergence rates and demonstrate the robustness of the proposed approach.

Since Young's 1936 theorem, irregular integration has been organized around the threshold 1/2: above it the path determines the integral, while below it higher-order data are needed. For fractional Brownian motion, H = 1/4 is the threshold for the canonical Gaussian enhancement, although geometric rough lifts exist for every H > 0. We prove that H = 1/4 is instead the exact threshold for measurable locality. For d-dimensional fractional Brownian motion with independent components, d at least 2, if 0 < H <= 1/4, no positive-measure Borel set supports even one finite off-diagonal second-level coordinate satisfying Chen's relation and measurable interval by interval from path increments. No moment, Holder, geometricity, stationarity, or scaling assumption is imposed. If 1/4 < H <= 1/2, every full-law local rough-path lift in the standard Holder range is automatically square-integrable and hence classified without an assumed L2 condition; for H > 1/2, every finite-step enhancement with natural graded Holder bounds is uniquely the Young signature. We also introduce an exact scale calculus below classical differentiability. Matched dyadic differences recover normalized derivatives with sharp O(epsilon^2) error, exact localized inversion, and lossless reconstruction. Multiplication and smooth functional calculus transport exactly to scale coordinates, while the corona quotient yields an exact universal derivation. Reinserting the Holder amplitude and using Fourier-normal ordering produces strong geometric lifts for every positive input regularity and every lower rough exponent, with explicit ultraviolet rates and stability. Thus rough lifts exist below one quarter, but no measurable lift on a positive-measure domain can be interval-local there. The results separate existence from local recoverability and show that nondifferentiability does not destroy exact differential information.

We establish a Cheng-Yau type estimate for positive biharmonic functions on complete Riemannian manifolds with Ricci curvature satisfies $Ric_g \ge -(n-1)K g$. If $u$ is a positive biharmonic function in $B_{2R}(p)$, then $$ -\frac{\Delta_g u}{u} +\frac{1}{8n}\frac{|\nabla u|_g^2}{u^2} \le C_n\left(R^{-2}+K\right) \quad\text{on }B_R(p). $$ Further, if the Ricci curvature is nonnegative, every global positive biharmonic function satisfies $\Delta_g u\equiv c$ and the sharp estimate $|\nabla u|_g^2\le2cu$ for some nonnegative constant $c$. We also show that every positive $k$-polyharmonic function has nonnegative constant $(k-1)$-st Laplacian and growth of order at most $2k-2$.

We prove that no product of positive-dimensional compact complex manifolds admits a strict locally conformally Kähler (lcK) metric. This solves a long-standing conjecture in lcK geometry, previously established only under certain conditions.

We construct the Cartan calculus of differential cubical forms on an object $X$ of a cartesian tangent category with a scalar multiplication by a commutative ring object $R$. We prove that the differential cubical forms are naturally equipped with the structure of a commutative differential graded algebra. Then we show that, if $R$ has no $2$-torsion, this CDGA together with the Lie algebra of vector fields and the inner derivatives constitutes a Cartan calculus, which lies between the initial Cartan calculus of algebraic Kaehler forms and the terminal one of Lie-Rinehart forms. We give a number of examples, including affine schemes and elastic diffeological spaces, and show that we recover the established notions of differential forms in all cases.

We give a systematic creative-microscoping treatment of truncated basic hypergeometric sums attached to three classical third-order mock theta denominator patterns, represented by $\phi(q)$, $\omega(q)$, and $\nu(q)$. The main conceptual point is that these denominator patterns naturally admit exact finite theta evaluations at $a=q^{\pm n}$, microscopic $q$-supercongruences, and cyclotomic/$p$-adic consequences within a single framework. For \[ S(q,a):=\sum_{n\ge 0}\frac{q^{2n+1}(aq,q/a;q^2)_n}{(-q^2;q^2)_n}, \] we prove, for odd $n$, a finite theta evaluation at $a=q^{\pm n}$, equivalently a microscopic $q$-supercongruence. We prove an analogous finite theta evaluation for the $\nu$-type denominator \[ N_m(q,a):=\sum_{j=0}^{m} q^{2j+1}\frac{(aq,q/a;q^2)_j}{(-q;q^2)_{j+1}}. \] We then introduce the one-parameter denominator family \[ \T(q;b):=\sum_{n\ge 0} q^{2n}\frac{(q;q^2)_n^2}{(q^2;q^2)_n(b;q^2)_n}, \] whose specializations include the $\omega$-type shifted denominator ($b=q^3$) and the two-color family ($b=q^{2k}$). For its truncated $a$-extension we obtain an exact evaluation at $a=q^{\pm n}$, deduce microscopic congruences modulo $(a-q^n)(1-aq^n)$, and derive cyclotomic consequences, including zero congruences in the $q^{2k}$-subfamily and nontrivial $p$-adic versions for the cases $b=q$ and $b=q^3$.

We prove that every smooth vector field on a closed smooth manifold is a single Lie bracket. The proof combines a relative transport construction on the complement of a ball with an exact extension across transverse graph disks. For the local extension, we first arrange that an auxiliary bracket is transverse to the disks, straighten it by its flow, and then remove the resulting scalar discrepancy. The argument applies without an orientability assumption. It also implies that every smooth fiberwise linear function on the cotangent bundle is a Poisson bracket of two such functions.

Mixed-integer programming (MIP) is a cornerstone in applied optimization, both in industry and academia. Recently, there has been increased attention to finding strong primal solutions quickly. This is reflected, for example, in the development of the NVIDIA cuOpt solver and, most recently, in the new MIPFEAS benchmark, which has a tight time limit of 600 seconds and evaluates solvers based on how quickly they find high-quality primal solutions. This article introduces a MIP portfolio parallelization scheme, focusing on efficiently exchanging information between its workers. We present two implementations of this scheme: one built directly into the open-source MIP solver SCIP, and an external one, which we call ReXi. ReXi is currently the fastest non-commercial solver in the MIPFEAS benchmark, followed by the SCIP-integrated implementation. Moreover, we present new versions of both implementations that considerably outperform their predecessors on the MIPFEAS benchmark.

We prove that the gradient range of every nonzero compactly supported $C^1$ function on $\mathbb{R}^n$ is the closure of its interior, for every $n\geq1$. More generally, for an arbitrary $C^1$ function, the closure of its gradient range is the union of the closure of the range's interior and the set of asymptotic gradient values. We also prove that compact smooth Lagrangian submanifolds without boundary in $\mathbb{R}^{2n}$ and their images under symplectic homeomorphisms have regular closed momentum projections.

We construct four infinite families of cyclotomic fields and show that each member of these families has class number a multiple of 3. In fact, we show that the class number of the corresponding maximal real subfields has class number divisible by 3. A construction by Kishi and Miyake [11] and a result of Yamaguchi [13] help us achieve this target. Finally, we produce computational evidence corroborating our results.

In this paper, we are concerned with the fine properties of entropy solutions of the Cauchy problem for the one-dimensional pressureless Euler system, wherein the initial density $\rho_0$ is a locally finite Radon measure and the initial velocity $u_0\in L^\infty_{\rho_0}$. We employ the solution formula introduced by [F.M. Huang and Z. Wang, Comm. Math. Phys. 222(1) (2001), 117–146.] for this Cauchy problem to analyze the entropy solutions and obtain various new fine properties of entropy solutions; these can be summarized in four aspects: (i) Characteristics and initial waves for the Cauchy problem; (ii) Fine local structures of entropy solutions; (iii) Divides and global structures of entropy solutions; (iv) Invariants and asymptotic behaviors of entropy solutions including the asymptotic profile and the corresponding decay rates. Through these results (i)-(iv), we establish the global dynamic patterns of entropy solutions of the Cauchy problem for the $1$-D pressureless Euler system with general initial data $\rho_0$ being locally finite Radon measures and $u_0\in L^\infty_{\rho_0}$.

In this paper, we investigate the difference between ordinary and symbolic powers of edge ideals on the Castelnuovo-Mumford regularity and the depth. As a main theorem, we prove that the regularities of ordinary and symbolic powers coincide for edge ideals of simplicial graphs, as a partial result of Minh's conjecture. For the depth, we first show that, for each $k\geq2$, the inequality $\operatorname{depth} S/I^{(k)}\geq\operatorname{depth} S/I^{k}$ does not hold for squarefree monomial ideals in general. On the other hand, we prove that it holds for edge ideals when $k=2,3$. We also study the symbolic-ordinary discrepancy module $I(G)^{(k)}/I(G)^{k}$ of the edge ideal $I(G)$ of a graph $G$ and give a graph-theoretic formula for its Krull dimension in terms of induced odd cycles of $G$, thereby answering a question and a problem posed by Ha and Minh.

The quasi-geostrophic equation is well known as a model for predicting potential temperature around low-pressure systems in high-latitude regions. When diffusion effects are added to this equation, it serves as a model for potential temperature at the sea surface. In either case, the nonlinear term represents the effect of the Coriolis force. This paper yields nonlinear profile in solutions to the surface quasi-geostrophic equation. This profile is determined uniquely from the perspective of large-time behavior of solutions. Components of solutions are arranged sequentially from slow to fast decay based on the parabolic scale. Then such an expansion is determined uniquely. Since the equation is subcritical in the context of large-time effects, the main components exhibit linear features. Nonlinear characteristics appear in components that have a smaller effect on solutions. From the perspective of the correlation between spacetime variables, nonlinear profiles exhibit characteristics that are distinctly different from those of linear profiles. Moreover, the nonlinear effects of this equation are expected to be weaker than those of other equations with the same scale due to their spatial structure. The main theorem publishes minute, but unique rotational flow arising from the Coriolis force.

We construct good quotients for equivariant actions of group homomorphisms on morphisms of schemes. Using Geometric Invariant Theory, we obtain explicit open semistable loci in the source with Hilbert-Mumford descriptions admitting good quotients relative to a given good quotient of the target. In particular, we obtain quotients for reductive groups acting on projective-over-affine morphism. In the non-reductive case, we consider equivariant actions on affine morphisms which are 'graded' by a multiplicative group and satisfy certain unipotent stabiliser assumptions. This recovers known results in projective non-reductive GIT as a special case, which also proves the Hilbert-Mumford criterion in that setting. As applications, we consider moduli of unstable objects, representations of quivers with multiplicities and jets.

In this article, we define classical and non-classical Besov and Triebel–Lizorkin spaces associated with the Grushin operator with or without drift. We establish various characterisations of these spaces, and study their complex interpolation, embedding properties, and fractional Leibniz rules.

We introduce a novel structure-preserving numerical method for a first-order hyperbolic approximation system, which approximates the solutions of general fourth-order partial differential equations. By employing an implicit-explicit (IMEX) splitting between the stiff and non-stiff terms, we rigorously prove that the proposed scheme is energy-consistent and positivity-preserving under a CFL-type condition. Furthermore, we show that the method remains robustly stable in asymptotic regimes, with a time-step restriction that is entirely independent of the relaxation parameters. Finally, we present a series of numerical examples for thin film equations to validate the theoretical properties and efficacy of the scheme.

We summarize the classical passage from the chromatic specialization of Negami's polynomial to the Hilbert series of a broken-circuit Stanley–Reisner ring, and record formulas for several graph families with their connectivity hypotheses made explicit. Our main construction in this revision is the associated graded algebra of the squarefree edge algebra under the graphic-matroid rank filtration, for an arbitrary finite loopless graph. Its basis is indexed by all edge subsets, and its multiplication is governed by disjointness and rank additivity. Its bigraded Hilbert polynomial recovers the full three-variable Negami polynomial. We prove that its socle has a basis indexed by edge subsets of full matroid rank; for a connected graph, these are precisely the connected spanning subgraphs. A vertex-degree refinement counts connected regular spanning subgraphs. This finite-dimensional algebra is distinct from a broken-circuit Stanley–Reisner ring. We distinguish these elementary structural results from the unresolved question of whether abstract algebra isomorphism can distinguish graphs with the same Negami polynomial; no priority claim is made for the construction or its Hilbert identity.

We introduce notions of integrality and full integrality over one-sided ideals, together with associated radical constructions, and develop a general framework for their study. As applications, we extend several classical results concerning Jacobson radicals to the one-sided setting. In particular, we prove that the Jacobson radical of a left ideal is integral over that ideal in left artinian rings and in algebraic algebras, obtaining a one-sided analogue of a theorem of Amitsur. We also show that Jacobson radicals of graded left ideals are graded and derive a corresponding extension of Amitsur's theorem on Jacobson radicals of polynomial rings.

For an abelian group $\Gamma$ of order $n$, a graph $G$ of order $n$ is $\Gamma$-distance magic if it admits a bijection $V(G)\to\Gamma$ whose open-neighborhood sums are constant, and group distance magic if this holds for every such $\Gamma$. More generally, for any finite abelian $\Gamma$, a $\Gamma$-magic map is a map $f\colon V(G)\to\Gamma$ with constant open-neighborhood sums; we call $f$ generating if its labels generate $\Gamma$, and affinely generating if its pairwise differences do. Let $G$ be regular, let $\Gamma\cong\mathbb{Z}/d_1\oplus\cdots\oplus\mathbb{Z}/d_r$ with $d_1\mid\cdots\mid d_r$, put $B_\Gamma=\bigoplus_{i<r}\mathbb{Z}/d_i$, and let $\overline{A}_m$ be the adjacency operator induced on $(\mathbb{Z}/m)^{V(G)}$ modulo constants. We prove that $G$ admits a generating $\Gamma$-magic map iff $B_\Gamma\hookrightarrow\ker\overline{A}_{d_r}$, and an affinely generating one iff $\Gamma\hookrightarrow\ker\overline{A}_{d_r}$; when $|V(G)|=|\Gamma|$, it is $\Gamma$-distance magic iff the latter embedding has vertex-separating image. If the reduced adjacency operator is nonsingular over $\mathbb{Q}$, these become subgroup conditions in the reduced adjacency Smith group. Cichacz and Froncek conjectured that every distance magic graph is group distance magic. Using the affine criterion we construct a $6$-regular distance magic graph of order $27$ admitting no affinely generating $(\mathbb{Z}/3)^3$-magic map, so the conjecture fails even for affine generation. We propose the generating group distance magic conjecture, and prove it for regular distance magic graphs whenever $\Gamma$ is $2$-generated, hence for cube-free order. Further applications concern Cayley graphs on elementary abelian $p$-groups, Hamming relation graphs, strongly regular graphs, and symmetric designs.

We study finite-sample rigidity for left translates of similarity orbits over commutative rings. Let \(R\) be a commutative ring with \(2\in R^\times\), and let \(C\in\operatorname{Mat}_2(R)\) be cyclic. Given matrices \(G_i=FK_i\), where \(F\in\operatorname{GL}_2(R)\) is common and the underlying points \(K_i\) lie in \(\operatorname{Orb}(C)\), we determine when a finite translated sample has exactly the same multiplier ambiguity as the complete translated orbit. The key structural result is that the orbit differences span \(\mathfrak{sl}_2(R)\), which yields a scalar description of the left stabilizer. We then give a determinantal saturation criterion, expressed directly in the matrices \(G_i\), under which finite-sample consistency reduces to trace equations and one determinant condition. Off-diagonal quadratic operators on pairs of free rank-two modules realize the translated-orbit model exactly. Over fields we obtain sharp sample thresholds under the saturation criterion, and over finite fields we derive exact saturation probabilities for regular semisimple and nonzero nilpotent orbits. Torsion restrictions of elliptic isogenies provide a natural arithmetic realization of the rank-two module framework.

Schultz [Comm.~Pure~Appl.~Math., 1998] first established dispersive estimates for the fundamental solution of the discrete wave equation on lattice graphs $\Z^d$ for $d=2,3$, which can be seen as a discrete analogue of the classical dispersive estimate for the Euclidean wave equation. His result was extended by Bi, Cheng and Hua to $d=4$ and $5$. In this paper, we give complete answers to all the remaining dimensions, including $d=5$. Our estimate is sharp for every $d\geq 5$. Moreover, the proof remains valid for the upper bound of the case for $d=4$.