Papers

This paper studies the element-wise convergence behavior of subspace iteration. All results are established for both \(\mathbb{F}=\mathbb{R}\) and \(\mathbb{F}=\mathbb{C}\). For a diagonal matrix \(\Lambda=\mathrm{diag}(\lambda_1,\ldots,\lambda_n)\) with \(|\lambda_1|>\cdots>|\lambda_n|>0\), we analyze the modulus of each entry of the iterative matrix sequence generated by subspace iteration, both without and with the Rayleigh–Ritz procedure. Under mild assumptions on the initial matrix \(X\in\mathbb{F}^{n\times m}\), we first derive exact asymptotic expressions for \(|Q_k(i,j)|\) in the subspace iteration without the Rayleigh–Ritz procedure: entries with \(i\neq j\) decay as \(|\lambda_{\max\{i,j\}}/\lambda_{\min\{i,j\}}|^k\), and the deviation of \(|Q_k(j,j)|\) from \(1\) decays as \(\max\{|\lambda_j/\lambda_{j-1}|,|\lambda_{j+1}/\lambda_j|\}^{2k}\), with explicit coefficients determined by the LU factorization of \(X\). Similar conclusions could also be established for upper triangular matrices. For the Rayleigh–Ritz variant, we obtain element-wise bounds for the normalized Ritz vectors \(Z_k\). Specifically, for \(|Z_k(i,j)|\), off-diagonal entries with \(i\geq m+1\) decay as \((|\lambda_i|/|\lambda_j|)^k\), off-diagonal entries with \(i\leq m\) decay as \((|\lambda_{m+1}|^2/(|\lambda_i||\lambda_j|))^k\), and the deviation of \(|Z_k(j,j)|\) from \(1\) decays as \((|\lambda_{m+1}|/|\lambda_j|)^{2k}\). These results give an element-wise description of the convergence behavior of subspace iteration. After an orthogonal or unitary change of basis, the results could be further generalized.

We prove that, for every positive integer $r$, the positive normalized diameters of isometric quotients of simply connected irreducible compact Riemannian symmetric spaces of rank $r$ have a uniform positive lower bound. The bound is independent of the dimension and of the acting group, which need not be connected. For the higher-dimensional Grassmannian families, we obtain an explicit bound for positive-dimensional acting groups by constructing nonconstant invariant functions of bounded trigonometric degree. The real and complex cases use moment maps. The quaternionic case follows from a quartic rigidity theorem proved by means of invariant four-forms and the classification of compact quaternionic symmetric spaces. We also give examples showing why the rank and the relative scales of product factors must be controlled.

We study Bousfield–Kan completions through the interaction of free simplicial resolutions, their filtration spectral sequences, and a noncommutative arithmetic square. For every free discrete simplicial group of finite type, we express its integral pronilpotent completion as the homotopy pullback of its rational prounipotent completion and the product of its pro-$p$ completions over an explicit adelic simplicial group. The adelic entry is formed by taking restricted products at finite nilpotent stages and then their inverse limit; no nilpotency assumption on the original group of components is required. Finite subpresentations of contractible presentations provide an explicit application of this construction. Independence of the specified relators makes the positive-degree terms of the rational and mod-$p$ filtration spectral sequences vanish, with convergence verified on the quotient towers. Continuous comparison of free simplicial resolutions then realizes, in characteristic zero, the equivalence with a constant free prounipotent group by morphisms and homotopies in that category. For the corresponding presentation complex $K$ we obtain $R_\infty K\simeq K(F_R(Z),1)$ for $R=\mathbb Q,\mathbb F_p,\mathbb Z$, where $Z$ indexes a complementary basis and $F_R(Z)$ denotes, respectively, the rational points of a free prounipotent group, a free pro-$p$ group, or a free pronilpotent group. Compatible contractions at the nilpotent stages identify all four entries of the arithmetic square in this case.

In cryptography, critical sets of Latin squares have particularly been implemented to design secret sharing schemes. A main problem in these cryptographic protocols arises from absent holders of pieces of information that are common to different critical sets, because they become indispensable to recover the secret. This paper solves this problem by making use of the orbits of entries described by the autoparatopism group of the Latin square under consideration. To this end, we introduce the more general problem of computing critical sets of Latin squares having a given paratopism in their autoparatopism group. These critical sets depend only on the conjugacy class of the autoparatopism and the main class of the Latin square under consideration. Based on this fact, as an illustrative example, we determine the smallest and largest sizes of critical sets associated with autoparatopisms of Latin squares of order up to six. We implement this approach in the design of a new secret sharing scheme.

Let $k\ge 3$ and let $n=km$. A $K_k$-factor in an $n$-vertex graph is a collection of $m$ vertex-disjoint copies of $K_k$ that covers the entire vertex set. We determine the maximum adjacency spectral radius of an $n$-vertex graph containing no $K_k$-factor when $m\ge 2k-1$. More precisely, we prove that every such graph $G$ satisfies \[ \rho(G)\le \rho(H_{n,k}), \qquad H_{n,k}=K_{k-2}\vee\bigl(K_{n-k+1}\cup K_1\bigr), \] with equality if and only if $G\cong H_{n,k}$. Equivalently, the unique extremal graph is obtained from $K_{n-1}$ by adding one vertex adjacent to exactly $k-2$ vertices of the clique. Our proof combines a decomposition lemma for sparse complements, derived from the Hajnal–Szemerédi theorem, with the Motzkin–Straus inequality and spectral estimates based on quotient matrices and the Rayleigh quotient.

We prove that on a compact locally flat HKT manifold the quaternionic Monge–Ampère equation always has a unique continuous solution. This provides new evidence for the Conjecture of Alesker and Verbitsky.

In this article we claim that we are going to give a priori and a posteriori error estimates for a Crank Nicolson type scheme. The problem is discretized by the finite elements in space. The main result of this paper consists in establishing two types of error indicators, the first one linked to the time discretization and the second one to the space discretization.

A state may need to be compressed before its exact downstream task is known. For a finite family of linear task operators $\{T_u\}_{u\in\U}$, we characterize exactly how much state must survive when only one of $K$ coarse task messages is available at compression time: $$p^*(K)= \min_{\substack{\Pcal\text{ partition of }\U\\|\Pcal|\le K}} \max_{C\in\Pcal}\rank(T_C),$$ where $T_C$ stacks the tasks that remain unresolved within cell $C$. Thus task information reduces retained state precisely by separating tasks whose joint task-visible subspace is expensive. We derive the dual bit frontier, an irreducible common-core floor, and a sharp singular-value characterization for nonzero tolerance. For overlapping task families, a cumulative overlap deficit places the exact frontier within an additive $\Delta$ of weighted load balancing. Optimal partitioning is strongly NP-hard, yet direct-sum families admit classical approximation guarantees and a PTAS. We stress-test the theory on learned task operators from CIFAR-10, Fashion-MNIST, SVHN, and Burgers observables. For Burgers with $K=2$ and $\eps=0.35$, a constructive partition attains $p=4$, while balanced random partitions require median $p=6$, with held-out worst standardized RMSE $0.0368$. The results distinguish uniform task sufficiency from distributional and decision-level performance.

We first prove that every smooth boundary metric on a compact manifold extends to a metric with any prescribed positive lower bound for Ricci curvature, whereas extensions under stronger positive \(k^{\mathrm{th}}\)-intermediate Ricci curvature conditions may fail due to local obstructions. For negative sectional curvature, we identify a global obstruction to extension. We then consider the class of compact manifolds defined by the existence of a metric with negative sectional curvature and boundary index at most one. On every manifold in this class, we prove that any smooth boundary metric extends to a metric with negative sectional curvature and strictly convex umbilical boundary. We further show that every such initial metric admits a complete asymptotically hyperbolic isometric extension with the same curvature condition and any prescribed conformal infinity. This class of manifolds is closed under boundary connected sums. As a geometric application of these extension results, every smooth metric on \(\mathbb S^n\) admits a strictly convex isometric embedding into \(\mathbb R^{n+1}\) equipped with a complete metric of negative sectional curvature. The proofs combine neck constructions with corner smoothing for upper curvature bounds.

We give a continuous version of the comparison theorem for free simplicial resolutions in the categories of prounipotent groups over a field of characteristic zero and of pro-$p$ groups. The resolutions carry free bases compatible with degeneracies; their ranks may be infinite. In characteristic zero the lifting step follows from continuous linear splittings and the universal property of completed free Lie algebras. In the pro-$p$ case it follows from the classical projectivity of free pro-$p$ groups. A relative lifting argument and an explicit simplicial cylinder establish homotopy uniqueness. We formulate the free bases using actual surjections in the simplex category, so degeneracy words related by simplicial identities are identified from the outset, and we spell out the full matching objects used in the lifting argument. The question arose naturally in our preceding work on Bousfield–Kan completions of subcontractible presentations.

Fractional partial differential equations couple a memory-dependent time derivative with a nonlocal fractional Laplacian, and their repeated solution under varying fractional orders, initial data, or diffusivity fields is computationally expensive. Neural operators offer a fast surrogate, but standard architectures must relearn the dominant linear fractional evolution from data. We propose the Fractional Fourier Neural Operator (FrFNO), a resolution-robust conditional operator. It injects the analytic Mittag–Leffler response of the frozen linear part as a once-precomputed, resolution-independent propagator table, and trains a spectral convolutional network only on the residual induced by variable coefficients and nonlinear advection, conditioned continuously on both fractional orders. The central theoretical result is that under zero-shot super-resolution the injected propagator fills the out-of-band modes that a standard Fourier neural operator sets to zero, replacing the out-of-band full-field tail by the smaller residual tail (a $K$-independent constant-factor reduction under weak perturbation). The residual and the full solution carry the same Sobolev order, so the advantage is amplitude reduction rather than a steeper tail. Mesh refinement drives the error to a residual-controlled floor. On a nonlinear two-dimensional space–time fractional Burgers problem FrFNO achieves the lowest relative $L^2$ error among six baselines (FNO, PINO, PDNO, CNO, DeepONet, U-Net) at the training resolution and under zero-shot super-resolution, yields the smallest spectral phase error, and remains the best at the integer-order limit and for long integration windows. The same construction extends to coupled fractional systems such as fractional Allen-Cahn and Navier–Stokes equations. Code, training scripts, and reference outputs are available at https://github.com/Derek2021Pang/FrFNO.

For finite measures with positive densities on convex Euclidean supports, we prove that $MCP(\kappa,N)$ passes with unchanged parameters to almost every isometric leaf of an arbitrary nonexpansive map. The proof rests on a sharp contraction inequality for geometric conditional densities, with exponent equal to the leaf codimension. The inherited dimension parameter is optimal. A total-variation limit on resolvent graphs extends the result to inverse fibres of maximal monotone relations, including convex gradient fibres. We also disprove Klartag's curvature-dimension inheritance conjecture by a firmly nonexpansive example in dimension three and a gradient example in dimension four. In codimension one, affinity of the geometric density yields curvature-dimension inheritance. The first example also gives failure on monotone fibres. Both constructions admit arbitrarily large curvature loss, including for a fixed Gaussian ambient measure on families of leaves of positive quotient measure.

We investigate the emergence of rigid polycrystalline structures in atomistic ionic particle systems as low-energy configurations. The interaction between particles of opposite charge is modeled by hard spheres that interact when they are tangential. The interaction between particles of same charge is modeled as a hard repulsion that forces a minimal distance between them. The atomistic energy is frame invariant, and no underlying reference lattice is assumed on the ionic configurations. The asymptotic behavior of configurations with finite surface energy scaling is identified by means of $\Gamma$-convergence. The related continuum theory is described by piecewise constant fields that encode the local orientation of the configuration. The limiting energy is local and concentrates at grain boundaries, which correspond to the boundaries of the regions where the underlying configuration has a constant orientation. The limiting energy density is anisotropic and depends on the relative misorientation of the two grains, their translation misfit, and the normal to their interface. Furthermore, we perform a fine analysis of surface energies for solid-solid and solid-vacuum phase transitions and determine energetically favorable orientation mismatches. This relies on a structure result for our grain boundaries, which shows that, due to the rigid setup, interpolating layers near the grain interface are energetically not favorable.

This study investigates the diametral metric dimension of generalized corona graphs, where the main graph is connected graph and the branch graphs form a sequence of connected graphs. The concept of diametral metric dimension is an extension of the metric dimension concept, requiring the resolving set to contain all diametral vertices. To determine the diametral metric dimension of a generalized corona graph, one must first identify the distance of each vertex in the graph and the graph's diametral set. Subsequently, a resolving set containing the diametral set is determined. The results show that the diametral metric dimension of the generalized corona graph depends on the diameter of the main graph and the metric dimensions of the graphs in the sequence. These concepts provide theoretical insights into resolving structures in the planning infrastructure and motivate further studies on other graph families and graph operations.

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $S$ be a minimal smooth projective surface of general type over $k$. If the second Chern class satisfies $c_2(S)<0$, then the order of the automorphism group satisfies $|\mathrm{Aut}_k(S)|<2598(K_S^2)^4$, and the order of every abelian subgroup $G\subset \mathrm{Aut}_k(S)$ satisfies $|G|<81(K_S^2)^3$.

In this paper we consider the general $n-2$ dimensional linear section $X$ of the Grassmannian $\mathbb G(1,n)$ of lines in $\mathbb{P}^n$, that is a Fano variety of index 1. We first prove that for $n$ odd $X$ is birational to a hypersurface defined by a pfaffian determinant of order $n+1$ of linear forms in $\mathbb{P}^{n-1}$, and we study this birational transformation in some detail. Then for $n$ even, we prove that $X$ is unirational and birational to a hypersurface defined by a pfaffian determinant of order $n$ of linear forms in $\mathbb{P}^{n-1}$. Moreover we prove that the hypersurface of $\mathbb{P}^n$ described by the lines corresponding to the points in $X$ has degree $n-1$ and is rational, and we find a rational parametrization of it.

Residential heating accounts for a large share of building energy use, and predictive control strategies can reduce it by anticipating rather than reacting to the room temperature alone. The usual manner of implementing predictive control, model predictive control (MPC), requires an accurate model of each individual residential unit for heating control. Obtaining this model is time-consuming to obtain manually or even impossible, and such models do not transfer across a heterogeneous building stock. Data-enabled predictive control (DeePC) removes this modeling step by designing the controller directly using measured trajectories of the system. The foundations of DeePC, however, are built on the class of deterministic linear time-invariant systems, which is an unrealistic assumption for residential heating. This work applies three recently proposed nonlinear extensions of DeePC, collectively referred to as data-driven predictive control (DPC): Select-DPC, gain-scheduling DPC, and linear parameter-varying DPC. These are compared against standard DeePC and the hysteresis controller, the industry standard in residential heating. The comparison runs over a full heating season on a calibrated digital twin of an occupied research unit, the NEST research building in Switzerland, assessed by heating energy and comfort-band violation. Moreover, gain-scheduling DPC was deployed on the real residential apartment and validated the conclusions from the simulations. Each of the DPC controllers consumes significantly less energy than the hysteresis controller, amounting to roughly $11\%$ over the season, validating the use of DPC. The nonlinear methods further outperform linear DeePC in all assessed scenarios, and we analyze the relative costs and benefits of each nonlinear method.

The $3$-link snake robot is an example of a non-holonomic mechanical system with rank $2$ distribution in a $5$-dimensional configuration space. It is one of $(2,3,5)$-geometries, and as such, it admits a description by a parabolic geometry of type $(G_2,P)$. Another example of $(2,3,5)$-geometry was well-studied years ago, and it is known that for balls rolling one over the other without slipping or twisting, if the ratio of ball radii is $1:3$, then it is locally isomorphic to the flat model in sense of $(G_2,P)$ parabolic geometries. Answering a question by P. Nurowski, we are looking for parameters of the $3$-link snake robots yielding a locally flat $(2,3,5)$-geometry. We extend the observation of a previous paper that the distributions of the snake robots contain bases generating finite dimensional Lie algebras. We exploit this observation to simplify the exterior calculus of the robots' geometry. This allows us to implement an effective normalization procedure and we obtain an explicit binary quartic invariant of the robot. Finally, we show that it does not vanish for any of the parameters. Therefore, the locally flat model cannot be achieved for these types of snake robots.

Let $f \colon S \rightarrow B$ be a fibred surface over an algebraically closed field $k$ of characteristic $p \geq 5$, where $S$ is a minimal smooth projective surface of general type and $B$ is a smooth projective curve of genus $b\geq 2$. We prove that the group of fibration-preserving automorphisms of $f$ has order at most $15658(K_S^2)^{4}$. Furthermore, we provide an example to show that the exponent $4$ of the polynomial bound is sharp.

Let $a,b$ be multiplicatively independent integers. We prove that if $\mu$ is a zero-entropy non-atomic $\times a$-invariant measure on $[0,1]$, then $\mu$-a.e.~$x$ has dense $\times b$-orbit. Related statements follow for many closed $\times a$-invariant sets, and in particular we conclude the existence of irrational numbers with minimal complexity in base $a$ and maximal complexity in base $b$. We extend~these results to commuting endomorphisms of the torus under an irreducibility and hyperbolicity condition.

The weighted $k$-matroid intersection problem asks for a maximum-weight set that is independent in each of $k$ matroids on a common ground set. The natural LP relaxation optimizes over the intersection of the $k$ matroid independent set polytopes. It is conjectured that this LP has integrality gap at most $k-1$. The conjecture is known for $k\le3$, but for $k\ge4$ the best general upper bound was $k$. We improve this bound to $k-1+1/k$. More generally, we prove that the natural LP of a $p$-matchoid has integrality gap at most $p-1+1/p$, with a deterministic LP-relative algorithm attaining the same factor. The matchoid extension resolves the $p$-matchoid part of a conjecture of Lee, Sviridenko, and Vondrák; projective planes give explicit tight instances whenever one of order $p-1$ exists.

We give a construction of admissible partitions that yields the Bernoulli decomposition theorem

We prove linear stability of all steady and expanding gradient Kähler-Ricci solitons. In the expanding case, we prove strict linear stability under very general assumptions. In particular, every asymptotically conical expanding gradient Kähler-Ricci soliton is strictly linearly stable.

Let $(\mathcal{M},d_{\mathrm{GH}})$ denote the Gromov-Hausdorff space of isometry classes of nonempty compact metric spaces. We prove that every nonempty compact metric space is isometrically embeddable into $(\mathcal{M},d_{\mathrm{GH}})$. More precisely, for every $D>0$ and every nonempty compact metric space $K$ with $\operatorname{diam} K\le D$, we realize the space of all $1$-Lipschitz functions on $K$ with values in $[0,D]$ as a family of metrics on a fixed Cantor space. Under this realization, the Gromov-Hausdorff distance agrees exactly with the uniform distance between functions, and each resulting metric space has diameter at most $76D$. We also construct finite approximations for which the Gromov-Hausdorff distance is given by an exact formula, together with a uniform approximation estimate.

Soft-input detection over a discrete constellation \(\mathcal{M}\) of cardinality \(M\) requires computing a posterior whose mean and mode are respectively given by the minimum mean square error (MMSE) and maximum a posteriori (MAP) estimates, both of which incur a computational cost of order \(\mathcal{O}(M)\) per symbol. We show that this cost is reduced to \(\mathcal{O}(L)\), where \(L \le M\) is the number of distinct amplitudes (rings), once the discrete prior is replaced by its maximum-entropy counterpart subject to the same radial marginal. This orbital prior, which is a mixture of uniform circular shells, is obtained by maximizing a mixed discrete-continuous entropy. We prove in this paper that such a distribution is the only distribution on \(\mathbb{C}\) that preserves the amplitude statistics of \(\mathcal{M}\) exactly while remaining maximally noncommittal in phase. Under the additive white Gaussian noise (AWGN) channel, the orbital prior induces a closed-form posterior that factors into a softmax over the \(L\) rings and a von Mises phase distribution whose concentration is supplied entirely by the observation, yielding closed-form orbital MMSE and MAP detectors of the discrete symbol at \(\mathcal{O}(L)\) cost. The resulting hierarchical rule selects the ring by posterior mass and the phase by conditional mode. We compare the pairwise ring boundary with that of the joint posterior-density and quantify the leading-order outward shift at high signal-to-noise ratio (SNR). Numerical results using standard constellations confirm that the orbital detectors maintain similar symbol error rate (SER) performance to optimal detectors, at a fraction of the complexity.

For a matrix $\gamma\in\mathrm{SL}_2({\mathbb Z})$, we define \[ {\mathcal R}(\gamma)=a_1^2+a_2^2+a_3^2+a_4^2, \qquad \text{where} \ \gamma=\begin{pmatrix}a_1&a_2\\ a_3&a_4\end{pmatrix}, \] and let $S_{\mathrm{sq}}(X)$ count the number of matrices $\gamma$ with $\|\gamma\|_\infty = \max\{|a_1|, |a_2|,|a_3|,|a_4|\} \leq X$ and such that ${\mathcal R}(\gamma)$ is squarefree. We prove that \[ S_{\mathrm{sq}}(X) = {\mathfrak S}_{\mathcal R}^{\mathrm{sq}}N(X) +O(X^{19/10+o(1)}), \quad \text{as}\ X\to \infty, \] where $N(X)=\#\{\gamma\in{\mathrm{SL}}_2({\mathbb{Z}}):\|\gamma\|_\infty\leq X\}$ and ${\mathfrak{S}}_{\mathcal{R}}^{\mathrm{sq}}$ is an explicit positive Euler product of local $p^2$-densities. The proof combines the $\delta$-method for small moduli with a sum-of-two-squares estimate for large square divisors. This complements a result of J. B. Friedlander and H. Iwaniec (2009) on prime values of ${\mathcal R}(\gamma)$, which, however, is conditional on a very strong form of the Elliott–Halberstam conjecture. We also show that ${\mathcal R}(\gamma)$ is squarefree and has at most $9$ prime divisors for at least $cN(X)/\log X$ matrices $\gamma\in\mathrm{SL}_2({\mathbb Z})$ with $\|\gamma\|_\infty\le X$, where $c>0$ is an absolute constant.

Sharp high-entropy lower bounds for the entropy of a sum were known for identically distributed summands in torsion-free abelian groups and in prime cyclic groups. For arbitrary independent summands, Gavalakis, Goh and Kontoyiannis obtained an additive constant of $1/8$ and conjectured that the sharp constant is $1/2$. We prove that independent discrete random variables $X,Y$ with finite Shannon entropies satisfy $H(X+Y)\ge (H(X)+H(Y))/2+1/2-o(1)$ in every torsion-free abelian group as $\max{H(X),H(Y)}\to\infty$. The same conclusion holds in the prime cyclic group $\mathbb F_p$ when both $\max{H(X),H(Y)}$ and $\log_2 p-\max{H(X),H(Y)}$ tend to infinity. We give explicit error bounds in both settings. The proof extracts a component with paired point probabilities while controlling the entropy of the remainder independently of its support. Discrete rearrangement and uniform perturbation then transfer the continuous entropy power inequality to this component. In prime cyclic groups, an additional estimate controls the entropy lost under modular reduction. Binomial distributions show that the constant $1/2$ is optimal.

Let $L_v=-\Delta-2v\cdot\nabla$ on $L^2(\mathbb R^n,e^{2v\cdot x}\,dx)$, where $v\in\mathbb R^n\setminus\{0\}$ is a constant drift vector. We prove that the full vector Riesz transform $\nabla L_v^{-1/2}$ satisfies a weak-type $(1,1)$ estimate for real-valued functions with constant at most $2$, uniformly in both the dimension and the drift vector. Our proof extends the fractional obstacle method of Ouyang, Spector and Stockdale (https://arxiv.org/abs/2608.18068) to the exponentially weighted setting. Interpolation then yields strong $L^p$ bounds for $1<p\le 2$, with constants depending only on $p$.

We study the large-time behavior of solutions to the aggregation–diffusion equation $$ u_t=\Delta u+\nabla\cdot\bigl(u\,\nabla K*u\bigr) \qquad \text{in } \mathbb{R}^d. $$ Our main results identify a transition, governed by the regularity and singularity of the interaction kernel, between Gaussian and nonlinear self-similar asymptotics. We prove that for regular interaction kernels, as well as for sufficiently mild singular repulsive kernels, the nonlinear drift is asymptotically negligible and solutions exhibit the same Gaussian large-time behavior as solutions of the linear heat equation. In contrast, for the critical logarithmic kernel $ K(x)=-\log |x|, $ the interaction persists at the diffusive scale and leads to genuinely nonlinear asymptotics. More precisely, for every mass \(M>0\), we prove the existence and uniqueness of a self-similar solution of mass \(M\) and show that every solution with mass $M$ and with finite second moment converges in all $L^p$-norms to this self-similar solution as \(t\to\infty\).

We study the problem of tiling an $N\times N$ square with axis-aligned squares whose side lengths are integers strictly less than $N$, with the objective of minimizing the number of tiles. The standard placement-based exact-cover MILP contains $\Theta(N^5)$ placement-to-cell nonzero coefficients and is also affected by the dihedral symmetry of the square domain. We introduce a Corner-Difference MILP (CD-MILP) that represents each selected square by at most four signed corner coefficients. Two-dimensional prefix reconstruction shows that the resulting constraints are equivalent to the original cell-cover equations, while reducing the coverage-related nonzero count to $\Theta(N^3)$. We also introduce lightweight corner-ordering constraints that retain at least one representative from every $D_4$ orbit, although ties may leave residual symmetry. Computational experiments on nine prime-size instances compare the baseline, CD-MILP, the baseline with corner ordering, and their combination using Gurobi and COPT. Under the stated experimental settings, the combined formulation solves eight instances with Gurobi and seven with COPT, compared with four and five, respectively, for the baseline. Its average presolved matrixs contain approximately $10^5$ nonzero coefficients on average, compared with approximately $10^7$ for the baseline.