Papers

We study the problem of constructing highly nonlinear vectorial maps $F: \mathbb F_2^n \to \mathbb F_2^m$. Concretely, we want an $F$ and an $A = A(m,n)> 0$ as small as possible, so that for every affine map $L: \mathbb F_2^n \to \mathbb F_2^m$ (of the form $L(x) = M x + b $) we have: $$\mathrm{agree}(F, L) := |\{ x \in \mathbb F_2^n \mid F(x) = L(x) \}| \leq A.$$ Such questions have been studied by Nyberg (1991,1993), Carlet and Ding (2004,2007), Liu, Mesnager and Chen (2017), Nagy (2025), and Biryukov, Turecek, and Udovenko (2026). There is a classical method of constructing such functions from bent-functions and Fourier analytic ideas; the best bound achievable by this method is: $$ A(m,n) = \Theta(2^{n-m} + 2^{n/2}),$$ and in particular, is never smaller than $2^{n/2}$. In this work, we show how to construct highly nonlinear functions beyond this Fourier bound. Concretely, we show how to construct for every $\gamma>0$, a function $F: \mathbb F_2^n \to \mathbb F_2^m$ with $m = O_{\gamma}(n)$, achieving $$ A(m,n) \leq (1 + \gamma)^n.$$ Surprisingly, we even achieve the same quantitative behavior for the much harder question of having low agreement with $m$-tuples of degree $d$ polynomials $Q: \mathbb F_2^n \to \mathbb F_2^m$, with $m = O_{\gamma, d}(n)$. Here the previously best bounds were of the form $A(m,n) = O( 2^{-\frac{n}{2^{d+1}}} \cdot 2^n )$ of Ben-Sasson and Kopparty (2010), based on Gowers-norm-type arguments. All our results generalize to all finite fields $\mathbb F_q$ in place of $\mathbb F_2$. Our methods are based on a new connection to classical results on counting solutions to systems of polynomial equations via algebraic methods. This connection brings us to basic questions in combinatorics, about graphs and hypergraphs with simultaneously a small number of edges and independent sets.

Del Pezzo orbifolds are del Pezzo surfaces equipped with a strict normal crossing divisor, called boundary, such that the log anticanonical divisor is ample. We classify del Pezzo orbifolds of degree less than 6 in terms of components of the boundary, in the sense of Campana. This completes our previous work on classification of del Pezzo orbifolds with irreducible boundaries.

We study positive solutions of a discrete Hénon equation on the integer lattice graph and on the half-space, with zero Dirichlet boundary conditions in the half-space case. The nonlinearity is governed by a Hénon-type potential with positive exponent. For sufficiently large nonlinearity exponents, we prove the existence of positive solutions both on the whole lattice in dimensions at least three and on the half-space in dimensions at least two. Thanks to the Hénon potential, the standard variational compactness is unavailable to obtain the positive solution. Our proofs combine weighted Banach spaces, two-sided Green kernel estimates and the Schauder fixed point theorem.

We study the equational theory and subvariety structure of the ai-semiring variety $\V_\infty$ generated by all flat semirings $S(a_1\cdots a_k)$, where the letters \(a_i\) are pairwise distinct. Using interval semirings of posets, we characterize its subdirectly irreducible members and describe variety membership in terms of jointly separating families of strict order-preserving maps. We obtain explicit finite identity bases for \(\V_\infty\) and each $\V_k$ generated by $S(a_1\cdots a_k)$. Consequently, every flat semiring \(S(W)\) associated with a nonempty set \(W\) of linear words is finitely based. This yields finitely based ai-semirings with exactly \(k\)-nilpotent multiplicative reduct for each \(k\geq 1\). For each \(k\geq 1\), let \(\B_k\) be the subvariety of \(\V_\infty\) defined by the \((k+1)\)-nilpotent identity. We prove that each \(\B_k\) is generated by a finite interval semiring and that every proper subvariety of \(\V_\infty\) is contained in some \(\B_k\). The variety \(\B_3\) is a Cross variety with exactly \(11\) subvarieties, whereas \([\V_k,\B_k]\), \([\V_k,\V_{k+1}]\), and \([\B_{k-1},\B_k]\) each contain continuum many subvarieties for every $k\geq 4$. In particular, this provides infinitely many finitely based finite semirings $S(a_1\cdots a_k)$ whose generated variety has continuum many subvarieties for every \(k\geq 5\).

Isogeometric analysis (IGA) solves partial differential equations accurately on exact NURBS geometry, whereas neural solvers are mesh-free but often orders of magnitude less accurate and typically trained by non-convex optimization without error control. We propose IGA-KAN, which uses local Kolmogorov-Arnold networks, fitted in closed form, to improve the IGA solution instead of replacing it. An IGA Galerkin solve produces u_h; on every knot-vertex patch a Kolmogorov-Arnold ridge model is fitted to the strong form of the equation, the exact boundary data and u_h, and the models are blended by IGA hat functions. With fixed inner functions the fit is one batched linear least-squares problem, without optimizer, learning rate or initialization. An a posteriori safeguard, motivated by a maximum-principle bound, decides where local models are used, keeping the IGA solution elsewhere. On eight benchmarks with exact solutions, five from the literature and one also posed on a domain fitted to a brain slice from MRI, the method reduces the error of IGA, at an unchanged number of Galerkin unknowns, by factors of 4.2 to 90 in L^2 and 4.1 to 220 in H^1 on the reference meshes, and its L^2 error is 6 to 6x10^4 times smaller than that of the best Kolmogorov-Arnold network trained from scratch on the same equations with a fixed budget. In an inverse problem it recovers an unknown constant source from one noise-free observation 167 times more accurately than IGA. The gain is attributed to the superconvergence of local averages of the Galerkin solution.

In the continuity of [1, 2, 3], we study a class of constrained mean-field Langevin dynamics formulated as a Wasserstein gradient flow with an energy constraint. We show that the constrained evolution therefore defines a well-posed dissipative dynamics in the Wasserstein space $\mathcal{P}_2(\mathbb{R}^d)$. We prove that the flow preserves probability mass and the energy constraint, dissipates the free energy monotonically, and converges exponentially to a unique constrained equilibrium under geodesic convexity assumptions. Using synchronous coupling, under Lipschitz assumptions, we demonstrate the quantitative $L^2$-propagation of chaos with a Fournier-Guillin bound, and under conditions of sufficient dissipativity, we derive an explicit time-uniform bound. We also demonstrate exponential controls in the $\mathcal{W}_2$ Wasserstein metric and the contraction of the invariant measure map. Using log-Sobolev inequalities, we show that the mean-field entropies decay exponentially along the constrained flow. The results are applied to several concrete and original examples. Keywords and phrases: Wasserstein gradient flows $\bullet$ mean-field Langevin dynamics $\bullet$ constrained nonlinear Fokker-Planck equations $\bullet$ Lagrange multiplier $\bullet$ JKO scheme $\bullet$ propagation of chaos $\bullet$ synchronous coupling $\bullet$ exponential ergodicity $\bullet$ functional inequality $\bullet$ log-Sobolev inequality $\bullet$ transport inequality $\bullet$ Talagrand T 2 -inequality

We study the Euclidean Varadhan formula (EVF) and path laws for diffusions with critical divergence-free drifts in dimension $d\geq2$. For drifts given by smooth bounded skew-symmetric stream matrices, we prove a quantitative lower bound under a sector condition with a lower-order term. Its constants depend only on the dimension and the sector condition data. By approximation, the locally uniform Euclidean lower bound holds for all divergence-free $L^\infty_t{\rm BMO}^{-1}_x$ drifts. However, a time-dependent counterexample shows that EVF can fail even for a drift in $L^\infty_tL^d_x$. A complementary upper bound is given under a structural decomposition condition. In particular, for every time-independent $L^{d,\infty}$ drift (without any smallness assumption), and the decoupled diffusion associated with the two-dimensional vorticity equation, the EVF holds. Finally, we construct a time-independent $L^d$ drift whose SDE has a unique strong solution from the origin and whose path law is mutually singular with the corresponding driftless Brownian law. The same diffusion satisfies EVF, showing that Brownian short-time endpoint costs can persist despite singularity of the path law.

In this paper we construct new examples of diameter graphs and Reuleaux polyhedra in $\mathbb{R}^3$, obtaining a full characterization of their combinatorial structure. For a finite set of points $X\subset\mathbb{R}^d$, its diameter graph is the graph on vertex set $X$ where pairs forming a diameter pair are connected by an edge. Grünbaum, Heppes and Straszewicz independently proved that the diameter graph of $X\subset \mathbb{R}^3$ has at most $2|X|-2$ edges, answering a question of Vázsonyi. Their proof relied on ball polytopes. The ball polytope $\mathcal{B}(X)$ is the intersection of the unit balls centered at the points of $X$. We call a ball polytope a Reuleaux polyhedron if the centers form a family with $2|X|-2$ diameter pairs. Kupitz, Martini and Perles showed that the skeleton of a Reuleaux polyhedron must be a 2-connected strongly involutive self-dual graph. They conjectured that in the simple 3-connected case this is also sufficient. We not only confirm this conjecture, but we show that any 2-connected (not necessarily simple) strongly involutive self-dual graph arises as the skeleton of a Reuleaux polyhedron. To construct the new Reuleaux polyhedra we construct new diameter graphs. It was known that any 3-dimensional diameter graph is a subgraph of a non-bipartite quadrangulation of the projective plane. We show that the reverse holds. That is, for any such graph we construct a diameter realization. This also confirms and strengthens a conjecture of Montejano, Pauli, Raggi, Roldán-Pensado on metric embeddings of strongly involutive self-dual graphs. The construction relies on ideas from rigidity theory. We also discuss a number of applications of these results, such as the construction of bodies of constant width and connections to Steinitz's theorem and Borsuk's conjecture.

This paper investigates the preservation of three classes of Ehrhart properties under various operations on lattice polytopes. These operations include Cartesian products, lattice joins, lattice pyramids, free sums, Minkowski sums, Cayley sums, reflexive polarity, and integral dilations. Specifically, we consider the following properties frequently studied in Ehrhart theory: (i): Positivity of Ehrhart coefficients, including Ehrhart positivity and magic positivity. In particular, we present lattice point counting formulas for the polytopes generated by these operations. (ii): Coefficient properties of the $h^*$-polynomial, including symmetry, unimodality, log-concavity, ultra log-concavity, real-rootedness, and $\gamma$-positivity. (iii): Geometric properties, including the spanning property, the integer decomposition property, very ampleness, and the existence of unimodular triangulation, regular unimodular triangulation, and quadratic triangulation. We determine which properties are preserved under these eight operations, establishing preservation theorems or constructing explicit counterexamples. Furthermore, when a property is not preserved in general, we investigate sufficient or equivalent conditions for its preservation.

We prove the LeBrun–Salamon conjecture for positive quaternionic Kähler manifolds of quaternionic dimension n <= 14. To do so, we first show that the isometry group Isom(M) of any minimal counterexample M has rank at most one, hence dimension at most three. On the other hand, we consider a characteristic number which equals dim Isom(M) - (n+3) if n is even and dim Isom(M) - (n+1) if n is odd, and show that this index is a sum of positive quantities when n <= 14, yielding a contradiction. The toolset used to show positivity enlarges in stages. Projection curvature moments suffice through n=8; positive-semidefinite curvature moments extend the argument through n=10; and signed scalar orbital moments give the new inequalities needed for n=11,12. Scalar orbital moments alone fail at n=13; combining them with Hodge–Riemann positivity for squares of cohomology classes extends the proof to n=13,14. We give exact separating obstructions explaining the changes of method.

This paper introduces a robust strategy to establish large deviation principles for mean field interacting particle systems in T d with vanishing noise. Viewed as a Freidlin-Wentzell type problem, our approach relies on recent techniques for partial differential equations (in short PDE) on the space of probability measures [5]. We first outline the proof strategy for the PDE approach to large deviations in a self-contained manner. As the number of particles goes to infinity, we then show that the transfer function of the system converges to the unique viscosity solution of a limiting Hamilton-Jacobi equation on the space of probability measures. Finally, we explicitly derive the rate function by formulating the solution of this limiting PDE as the value function of an optimal control problem.

We study the equations $\Delta_p^H u=u^{-q}$, $u>0$, and $-\Delta_p^H u=e^u$ in $\mathbb{R}^N$, where $\Delta_p^H u=\operatorname{div}(H(\nabla u)^{p-1}\nabla H(\nabla u))$ is the Finsler $p$-Laplacian associated with a uniformly convex norm $H$, and $p\ge2$. We prove nonexistence of solutions stable outside a compact set for the negative-power equation when $p\le N<\frac{p(p+3)}{p-1}$ and $q$ lies above the sharp critical threshold, and for the exponential equation when $p<N<\frac{p(p+3)}{p-1}$. Consequently, neither equation admits solutions of finite Morse index in these ranges. In the borderline case $N=p$, we prove that the solutions of $-\Delta_N^H u=e^u$ that are stable outside a compact set are precisely the finite-mass solutions classified by Ciraolo and Li. A main ingredient is independent of stability: if $N>q\tau$, where $\tau=\frac{p}{q+p-1}$, a positive exterior solution of $\Delta_p^H u=u^{-q}$ cannot remain above $\theta$ times the singular solution near infinity for any $\theta>1$. An analogous obstruction holds for the exponential equation. Both obstructions are proved by comparison with explicit radial barriers having frozen right-hand sides. We also prove sharpness when $N>p$, and for the negative-power equation when $N=p>2$. Finally, for operators $\operatorname{div}(B'(H(\nabla u))\nabla H(\nabla u))$ with $p$-growth, we establish the integral estimates needed for the stability arguments with fully justified truncation and absorption procedures. These arguments also repair gaps in earlier proofs for the corresponding isotropic and anisotropic problems.

We study theta series with half-integral characteristics attached to odd positive-definite unimodular lattices. In even rank, we get non-vanishing theta series with odd characteristics. We emphasize ranks divisible by four, where the involved rational lattices are integral unimodular neighbors.

We study strong completeness and the existence of a set attractor for the stochastic flows induced by a class of two-dimensional stochastic differential equations resembling a planar Ornstein-Uhlenbeck process with an additional radius-dependent rotational drift term. Our main results give both sufficient and necessary conditions for the stochastic flows to be strongly complete and for the existence of set attractors. In particular, we demonstrate that if the derivative of the angular velocity $\rho(r)$ with respect to the radius $r$ satisfies $|\rho'(r)| \geq K_2\, r^3$, for large $r$ and some universal constant $K_2$, the diameter of a compact set can grow exponentially fast with positive probability. Furthermore, we show that for $|\rho'(r)|\geq r^{3+\varepsilon}$, $\varepsilon>0$, with probability one, exceptional initial conditions diverge to infinity in finite time, ruling out the existence of a global stochastic flow.

We investigate the conditions under which a Hessian operator satisfies the integral comparison principle. We establish a rather unexpected connection with the notion of a null Lagrangian from the calculus of variations. Under mild conditions we obtain a complete classification, showing that the only operators for which the integral comparison principle holds are linear combinations of $k$-Hessians and, in the homogeneous case, precisely the $k$-Hessians themselves. An analogous result is established for the complex Hessian operator, revealing in particular substantial obstructions to the development of a genuine pluripotential theory for the $\mathcal{J}$-equation.

Based on the CRT for polynomial rings, Yang, Zhu, Fu and Xia (ISIT 2026) proposed a compartmented secret sharing scheme for the compartmented access structure with lower bounds, claiming it can be ideal. We exhibit three families of unauthorized subsets that reconstruct the secret. The scheme is therefore insecure, except for a degenerate choice of parameters in which the only authorized subset is the whole participant set. The flaw in the security analysis is that it does not take all published polynomials into account. Similar flaws appear in the same first author's hierarchical schemes (ISIT 2024, 2026).

In PDE-constrained optimization a control is applied within a partial differential equation (PDE), e.g., as a force field, to achieve a desired configuration of the PDE solution. The objective is usually to find an optimal balance between control cost and deviation of the controlled state from the desired configuration. We consider this problem class in a setting, where the PDE is given by a geometric variational equation and the state is a mapping into a nonlinear manifold. We define a Lagrangian function corresponding to this geometric optimal control problem and derive first-order optimality conditions. Moreover, we present a Lagrange-Newton method for their numerical solution. We introduce an implementable algorithm using a local splitting technique and show its equivalence to Newton's method applied to the derivative of the Lagrangian. The theoretical framework is illustrated by numerical examples, including the optimal control of elastic geodesics and surfaces of minimal energy by force fields and the control of inextensible rods and shells.

We study steady solutions of the two dimensional incompressible Euler equations in the exterior of a compact domain. We assume that fluid enters the exterior domain through every point of the obstacle boundary. Under suitable assumptions on the direction and decay of the velocity at infinity, we prove that the flow is irrotational if and only if the velocity has no stagnation points.

We study deterministic and randomized midpoint discretizations of Langevin dynamics for a target $\pi \propto e^{-V}$, where $0 \prec \alpha I\preceq\nabla^2V\preceq\beta I$ and $\kappa=\beta/\alpha$. To achieve $\sqrt\alpha\,W_2\leqslant\varepsilon$, we show that deterministic Heun uses at most $\widetilde O(\kappa^{4/3}d^{1/3}\varepsilon^{-2/3})$ gradient queries, and underdamped exponential midpoint uses $\widetilde O(\kappa^{5/4}d^{1/4}\varepsilon^{-1/2})$. The proofs exploit cancellation at stationarity and smoothing using techniques from Malliavin calculus, outperforming previous upper bounds based on standard couplings. At bounded condition number, a lower bound matches the $d$ and $\varepsilon$ powers of both deterministic methods. To contrast, for the randomized midpoint methods and Poisson midpoint with at least two grid points (both overdamped and underdamped variants), a simple Gaussian calculation yields a lower bound $d^{1/3}\varepsilon^{-1/3}$ to get an $\varepsilon$-close sample despite starting at a benign initialization. This shows surprisingly that in high dimensions, deterministic discretizations can outperform their random counterparts.

Biological agents, from regulatory modules and microbial strains to organisms and populations, act on a common environment while each responds to its own costs, benefits, and limits. A dynamic game makes such interactions explicit by assigning who controls which decision, what each participant values, and which restrictions constrain unilateral deviations. We present a library of 25 biological dynamic games with 2 to 10 players spanning molecular, cellular, organismal, and population scales and competitive, antagonistic, mutualistic, and regulatory interactions. Instances cover single seasons, cycles that return to their initial state, or episodic tasks; seven include duration choice and some players' feasible sets are coupled. For each game we provide biological motivation, an executable Julia specification with player-owned controls, objectives and restrictions, a parameter instance, and a reference open-loop generalized Nash equilibrium candidate with interpreted trajectories. The solver CorleoneGame.jl computes candidates by sequential optimal-control best responses and audits them for feasibility and profitable unilateral deviations on a coarse and a refined control grid. All 25 reference computations satisfy the audit tolerances on both grids. Using a plant–herbivore game as a running example, we show how an executable specification yields an audited reference candidate that qualitatively reproduces a published prediction, how biological hypotheses become instance variants with different candidates, and how the candidates generate synthetic data for inverse games.

We consider a general smooth bounded strictly convex domain $\Omega\subset\mathbb{R}^d$ of dimension $d\geq2$ and describe dispersion for the semiclassical Schrödinger equation with Dirichlet boundary condition. Our results hold more generally on compact smooth Riemannian manifolds with smooth strictly convex boundary. More specifically, we construct a sharp local in (semiclassical) time parametrix and then proceed to obtain dispersion estimates: our fixed-time decay rate for the Green function exhibits a loss of $1/4$ in the exponent of $h/t$ with respect to the boundaryless case. The loss is sharp for dispersion, as shown earlier by the first author in the case of a model convex domain. On compact three-dimensional manifolds with smooth strictly convex boundary, the resulting spectrally localized Strichartz estimates yield global well-posedness in the energy space for the defocusing cubic nonlinear Schrödinger equation, thus matching the corresponding result for boundaryless manifolds due to Burq-Gérard-Tzvetkov. Moreover, we extend bounds on the time growth of higher Sobolev norms from Planchon-Visciglia-Tzvetkov to our setting.

In his Tau Manifesto, Michael Hartl presents a formula, credited to Jeff Cornell, for the volume of the unit $n$-ball in terms of the quarter-turn constant $\eta=\pi/2$. We read the formula as a set of clues to a geometric proof, leading through orthant geometry and a higher-dimensional version of Lambert's equal-area projection. We conclude with some additional thoughts on the circle constant.

We construct a nonmicroscopic $G_\delta$ subset of the real line all of whose compact subsets are microscopic, answering a question of Zindulka and Nowakowski. We first obtain such a set in the standard Cantor space by coding branches of a finitely branching tree together with suitable choices of marked nodes. The coding of the branches ensures that the resulting set is not microscopic, while the marked nodes allow us to construct covers showing that every compact subset is microscopic. The example is then transferred from the Cantor space to the real line by the binary-expansion map.

In this series of articles, we study two graph classes and their various geometric representations. The first class consists of strongly involutive self-dual graphs (SISD graphs); the second consists of non-bipartite quadrangulations of the projective plane (NBQP graphs). These classes naturally arise in many geometric problems. For example, NBQP graphs are connected to diameter graphs in $\mathbb{R}^3$, tangency graphs of families of pairwise intersecting circles and pseudocircles, generalized thrackles, and graphs formed by the main diagonals of negatively self-polar polytopes. On the other hand, SISD graphs correspond to the skeletons of negatively self-polar polytopes, self-dual cones, and extremal ball-polytopes. In Part I we show a natural one-to-one correspondence between non-bipartite quadrangulations of the projective plane and 2-connected strongly involutive self-dual maps. In particular, we show that the reduced vertex-face incidence structure of a 2-connected strongly involutive self-dual map is a non-bipartite quadrangulation of the projective plane, and every non-bipartite quadrangulation of the projective plane arises that way. This allows us to translate natural properties from one class to the other. These results lay the foundation for later parts of the series.

Physics-informed machine learning as recently emerged as a new paradigm for the resolution of partial differential equations, including functionalities for parametric modeling and inverse design, using neural networks to represent the solution fields. However, several difficulties have been reported concerning the training of such networks, resulting in slow convergence, low accurate solutions and sensitivity to the choice of hyper-parameters. In order to better understand the mechanisms underlying these difficulties, the present work proposes a rigourous comparison of neural networks and B-Splines representations within the same physics-informed learning framework. Four problems with closed-form solutions are numerically studied to establish a rigorous analysis of the accuracy and efficiency of the two representations, in terms of asymptotic convergence rates, sampling requirements, loss function minimization and arithmetic precision needed.

We investigate two learning procedures and their relationship to the formation of equilibrium in potential mean field games (MFGs). The first one is a version of the fictitious play in which players observe directly the actions (controls) of other players and learn the distribution of the MFG through an update rules on controls. This leads to a new iterative procedure which converges for a wide class of potential MFGs of controls, even in the presence of common noise. On the other hand we also consider the learning procedure of a principal agent trying to solve a mean field control problem with only local gradient information. We prove that under very general assumption, the gradient descent of such agent converges to a solution of the associated potential MFG.

Let \(\mathbb{F}_q\) be a finite field with \(q>7\). Zhi-Wei Sun conjectured that for every \(a_0\in\mathbb{F}_q\), there is a circular permutation \((a_1,\dots,a_{q-1})\) of non-zero elements of \(\mathbb{F}_q\) such that \(a_0+a_i a_{i+1}\) is primitive for \(1\le i\le q-1\), where \(a_q=a_1\). In this paper, we confirm this conjecture for \(q>18\,888\,871\). For \(a_0=0\), the conclusion holds for all \(q>4\). For \(a_0\ne0\), we apply the Chvátal–Erdős theorem to obtain a Hamilton cycle.

In this article, we study the viscoelastic wave equation with a localized history memory term, and without frictional damping. Under the geometric control condition on the memory region, we prove exponential decay of the viscoelastic energy using the Gearhart-Prüss-Huang Theorem. We prove in high frequency that a loss of derivative is compensated by a decay in frequency.

We give a criterion ensuring that sufficiently high sublevel sets of a discrete Morse function on a $\mathrm{CAT}(0)$ cube complex have polynomial integral homological filling functions. As one application, we verify this criterion for the Stein–Farley complexes of the Higman–Thompson groups $F_{d,r}$, $T_{d,r}$, and $V_{d,r}$. Consequently, these groups have polynomial homological filling functions in every degree $k\geq2$.

The binary perceptron model studies the set of solutions obtained by intersecting random Gaussian half-spaces with the discrete hypercube {-1,+1}^n. Pioneered in the work by Krauth and Mézard in the 1980s [KM89], one of the most striking physics predictions for this model is strong freezing, which says that typical solutions are isolated. Moreover, the conjecture was sharpened later to assert that typical solutions are at Hamming distance of order $n$ from every other solution [HWK13],[HK14]. This phenomenon has attracted a lot of attention over the years, partly due to its intricate connections to statistical physics and computational complexity. We prove that the binary perceptron model exhibits strong freezing with high probability at every fixed positive constraint density and every fixed margin. Our proof is based on the planted model and a restricted partition function that counts solutions having nonnegative overlap with the planted solution. Unlike the full partition function, this restricted one is monotone in the planted direction, so the FKG inequality decouples it from isolation, while the restriction costs only a factor of n+1. Together, these allow us to transfer the isolation of the planted solution to typical solutions, which completes the proof of strong freezing.