Papers

We establish nonlinear stability of three-dimensional Couette flow for the incompressible Navier–Stokes equations on $\mathbb T\times\mathbb R^2$. Divergence-free perturbations satisfying $\|u_{\rm in}\|_{H^2}+\|u^\perp_{0,\rm in}\|_{\dot B^{-2}_{2,\infty}}\leq c_*\nu$ generate unique global $H^2$ solutions that remain close to Couette flow, where $u^\perp_{0,\rm in}$ denotes the streamwise average of the transverse velocity. This result retains the $O(\nu)$ stability threshold despite the absence of a spanwise spectral gap. The nonzero streamwise modes exhibit enhanced dissipation, their wall-normal component satisfies a viscosity-uniform time-integrated inviscid-damping estimate, and the transverse zero mode decays in $L^2$ at rate $(1+\nu t)^{-1}$. The endpoint space $\dot B^{-2}_{2,\infty}$ controls the transverse low frequencies, accommodates divergence-form nonlinear forcing, and is critical for linear lift-up from the wall-normal zero mode. The proof combines propagation of this endpoint norm, Fourier splitting, and a two-level roll-to-streak representation with an adaptation of the nonzero-mode framework of Wei–Zhang (CPAM, 2021).

We extend a result by Ikeda and Suriajaya (2025) to the average number of representations of an integer as a sum of an arbitrary number $j\ge 3$ of primes. We show that the corresponding asymptotic formula holds in the general case. In the proof we use new ingredients like averages for the singular series of the generalized Goldbach problem.

We study a positive radial graph evolving by $V=H^\alpha+b$, with $\alpha>0$ and $b<0$, in the unit cylinder and with $u_r(1,t)=u(1,t)$. We separate finite-time continuation estimates from the time-uniform interior estimates needed for convergence of normalized translates. On existing classical intervals we prove height and gradient bounds and an explicit conversion of velocity bounds into estimates for $H$, $u_r/r$, and $u_{rr}$. We derive the correct boundary defect for comparisons with finite-slope translators. Under independent interior velocity bounds and two-sided slope trapping, we prove compactness and identify every translated limit with the cup translator. We identify the radial zero-number comparison still required to establish that trapping. Stationary caps and power-law Robin laws are included.

We investigate a thermal transmission problem in a domain composed of a film region and a bulk region coupled through an interface where temperature continuity and heat-flux balance are imposed. The analysis addresses both the finite-thickness film configuration and the corresponding thin-film limit, with anisotropic, non-homogeneous thermal conductivity tensors. Letting the film thickness vanish directly leads to a degenerate problem due to the vanishing measure of the film region. To preserve the contribution of the film in the limit, we introduce suitable scaling assumptions for the normal heat fluxes, namely for the flux balance across the film-bulk interface and for the flux prescribed on the external boundary of the film region, together with the requirement that the volumetric source in the film be uniformly bounded with respect to the film thickness. Under these scaling assumptions, we derive a non-trivial thin-film limit formulation that takes the form of a coupled bulk-interface problem, where the effect of the collapsing film is represented through an additional balance relation posed on the interface. Well-posedness results are established for both the finite-thickness film and thin-film limit configurations, and a rigorous convergence analysis is carried out to justify the asymptotic limit. The main mathematical achievement consists in finding explicit convergence rates in suitable functional norms, relying only on the weak regularity of the finite-thickness problem due to the transmission conditions at the film interface. To complement the study, analytical solutions are derived for a simplified film-bulk geometry and are used to compare the thermal responses of the two configurations. These solutions also provide a quantitative assessment of the convergence rates established in the theoretical analysis.

We study isomorphism decision and basis construction for finite Abelian groups of known order $n$. We measure complexity by the number of additions performed in the groups and by the total running time. We give a randomized isomorphism decision algorithm that performs $\tilde O(n^{1/4})$ group additions and runs in $\tilde O(n^{1/4})$ time. This improves the $\tilde O(\sqrt n)$ upper bound of Chen and Fu and matches Bshouty $\Omega(n^{1/4})$ lower bound up to polylogarithmic factors. We also prove that finding a basis with success probability at least $2/3$ requires $\Omega(\sqrt n)$ group additions in the worst case. This matches the $\tilde O(\sqrt n)$ upper bound of Chen and Fu up to polylogarithmic factors. These results separate isomorphism decision from basis construction in finite Abelian groups: their optimal worst-case complexities are $\tilde\Theta(n^{1/4})$ and $\tilde\Theta(\sqrt n)$, respectivel

We prove the localization dichotomy conjecture for non-periodic Hall insulators on the lattice. That is, we show that the Fermi projector $P$ of a gapped exponentially local Hamiltonian on $l^2(\mathbb{Z}^2) \otimes \mathbb{C}^n$ admits an exponentially localized Wannier basis if and only if $P$ has vanishing Chern number.

We study the graphical forced mean curvature flow in a planar strip subject to Robin boundary conditions. The normal velocity is prescribed by $V=H+b$, where $b>0$ is constant and $H$ denotes the curvature of the graph. We consider positive even solutions and reduce the problem to the half interval. The main results describe the degeneration produced by the Robin condition: the height, the instantaneous velocity, and the spatial gradient diverge, while the normalized slope $u_x/u$ converges to one away from the symmetry axis. The normalized velocity $u_t/u$ converges to the forcing parameter $b$. The curvature converges to zero in the interior in an integral sense, whereas its horizontal curvature measure concentrates at the symmetry axis.

We consider site percolation on general graphs and give a sufficient condition on the escape probability of a certain random walk with degree-dependent conductances, such that the number of minimal vertex cutsets of size $n$ separating a given vertex from infinity is bounded above exponentially in $n$. This result extends an analogous theorem of Easo, Severo and Tassion under slightly stronger assumptions. Moreover, we give an alternative sufficient condition in terms of the isoperimetric dimension or, more specifically, an isoperimetric-type inequality. Furthermore, our theorem is sufficient to extend the results of Panagiotis and Severo, and show that there exists a universal positive constant $\varepsilon_1$ such that site percolation on every infinite, connected, locally finite, vertex-transitive graph satisfies $p_c = 1$ or $p_c \leq 1-\varepsilon_1$.

The strong universality conjecture of Bobrowski and Skraba asserts that, under a prescribed data-dependent additive centering, the empirical laws of log-log transformed persistence ratios of random geometric complexes share a universal limit across sampling models, dimensions, filtrations, and homological degrees. They conjectured a left-skewed Gumbel limit and used it as a null law for testing topological significance. We show that, when the uncentered empirical laws and the centerings have deterministic limits, the conjecture forces a second-order quantity of each model's limiting law, determined by ratios close to one and unchanged by additive shifts, to agree across models and dimensions. To compute this quantity for degree-one Čech persistence, we bypass the global persistence pairing with a local proxy based on edges and their first incident triangles in the equivalent alpha filtration. In dimensions two and three, a Palm-Mecke calculation evaluates the proxy intensity exactly for stationary Poisson processes, and two-sided topological bounds show that it agrees with the true normalized persistence intensity through second order. After transfer to finite samples, the quantity equals $\pi^2/32$ and $4/\pi^2-1/4$ for the limiting laws of independent uniform samples from the unit square and the unit cube, respectively. The centered empirical laws converge weakly in probability to distinct deterministic limits, and their mean laws, the expectations of these random laws, also converge weakly to these limits. Hence the conjecture fails even for mean laws. Neither limiting law belongs to the left-skewed Gumbel location-scale family.

We consider approximate message passing (AMP) with discontinuous denoisers, such as hard slicers, quantizers and hard thresholding, for which the classical state evolution (SE) analysis does not apply. It is first shown that the standard Onsager coefficient; i.e., the average pointwise derivative of the denoiser, misses a boundary term, namely, the sum over the jumps of the jump size times the density of the denoiser input at the jump. Then, for real independent and identically distributed (i.i.d.) Gaussian matrices, a positive noise variance, a fixed number of iterations and a prior with finite second moment, we transfer the Lipschitz SE theorem directly to denoisers, fixed in advance, that are continuously differentiable with bounded derivative except at finitely many jumps, by bounding the distance between the iterates of AMP and of AMP on Lipschitz ramps of the denoisers through the fraction of inputs that cross a jump. It is thereby proven that AMP follows SE, also for test functions such as symbol errors, if its Onsager coefficient contains the boundary term, evaluated either along SE or, as we propose, at the noise level estimated from the residual, with the density computed from the known prior. Simulation results with 4- and 8-level pulse amplitude modulation (PAM) demonstrate that AMP with a hard slicer, whose standard coefficient is identically zero, fails with this coefficient, while with the proposed one, it follows SE and comes close to Bayes-optimal AMP.

We develop a theory of distributional Riesz fractional gradients in variable exponent Lebesgue spaces and introduce the associated nonlocal Sobolev spaces. Moreover, Sobolev, Hardy, Morrey, and Trudinger-type inequalities are established, and a fractional fundamental theorem of calculus is derived. As an application, we study nonlinear variational problems with nonstandard growth and establish existence and Euler–Lagrange equations for minimizers.

We describe a Segre-Veronese construction of polynomial solutions to the interpolation conditions of the numerators of tree-level MHV gravity amplitudes given by Hodges' formula. The five- and six-point Hodges numerators arise from Segre determinants, while for every $n\geq7$ suitable Segre determinants lead to solutions linearly independent of the Hodges numerator. The existence of such independent elements for $n=7,8$ was recently established by Mai and Zhang.

We obtain recurrence relations for formal characters of Feigin-Stoyanovsky type subspaces of standard $B_2^{(1)}$-modules at fixed level. The recurrences are derived directly from the combinatorial bases described by difference and initial conditions. Unlike in type $A_\ell^{(1)}$, the right side in general involves several shifted characters. We give explicit polynomial coefficients and prove that the resulting system, together with the constant term normalization, determines all formal characters at fixed level uniquely. At level $1$ we solve the recurrence system explicitly and obtain fermionic formulas for all three characters. We also relate these characters to the multigraded generating functions for exponent triples parametrizing the Meurman-Primc monomial basis of standard $A_1^{(1)}$-modules.

We study $\varepsilon$-relative approximation of mixed volumes of a fixed number $k$ of full-dimensional convex bodies in $\mathbb{R}^n$, given membership oracles and a known bound $B_n\subseteq K_i\subseteq R_0B_n$. We present a randomized algorithm that estimates any prescribed mixed volume within relative error $\varepsilon$ with probability at least $1-\delta$, using polynomially many oracle calls and bit operations in $n$, $\log R_0$, $\varepsilon^{-1}$, and $\log\delta^{-1}$ for fixed $k$.

A program for fluctuating kinetic theory is proposed in this paper to study nonlinear Lévy-type fluctuation corrections to kinetic PDEs in a unified way. The Vlasov–Fokker–Planck (VFP), Landau, and Boltzmann equations can each be decomposed into a kinetic transport term and a generalized gradient structure generated by a nonlinear dual pair of dissipation potentials. Going beyond the linear Onsager structure and the classical fluctuation–dissipation relation, we propose a new nonlinear Lévy-type fluctuation–dissipation relation that identifies the thermal fluctuations associated with this generalized gradient structure. On a regular tilted class, the canonical action of the generalized gradient structure agrees with the Freidlin–Wentzell-type action predicted by the resulting SPDE. This consistency gives thermodynamic support for the new nonlinear fluctuation–dissipation relation and leads to a master fluctuating kinetic equation. We further establish a sufficient condition on the nonlinear dual dissipation potential under which its centered cumulant admits a Lévy–Khintchine representation, and show that the resulting Lévy current can be realized through Gaussian white noise and a compensated Poisson random measure. As an application of this framework, an \(\alpha\)-stable Lévy-type Dean-Kawasaki equation is derived. Furthermore, by rescaling the fluctuating Boltzmann equation under a distinguished relation between the noise intensity and the hydrodynamic scaling parameter, we prove that its Hamiltonian converges to the Hamiltonian of the Landau–Lifshitz–Navier–Stokes equations. This reveals a scaling relation between kinetic fluctuations and fluctuating hydrodynamics.

We study the Iwahori Gaiotto conjecture for cotangent types and its application to the representation theory of the (mixed, small, and Lusztig) quantum supergroups. Using local and global models and SW-Zastava spaces, we construct factorization functors and compare the Iwahori Gaiotto category with the category $O$ of the mixed quantum supergroup. We then study the images of intersection cohomology sheaves and the metaplectic Hecke action. In type A, we describe the finite and affine monodromic categories, the corresponding Yokonuma–Hecke actions, and their role in the Kazhdan–Lusztig algorithm. The last section treats the pointwise purity.

HOMFLY-PT polynomials describe analytic continuation of Wilson averages in Chern-Simons theory to arbitrary complex values of two independent parameters $q$ and $A$. They also depend on the knot and representation of the gauge group. However, when $q^2$ is a primitive $m$-th root of unity, e.g. $q=\pm e^{i\pi/m}$, these polynomials exhibit a universal (knot-independent) factorization: $H_R$ becomes the product of the polynomials for the $m$-core of the Young diagram $R$ and of one and the same factor $H_{[m]}$ for every $m$-ribbon of $R$. In all the examples which we computed this factor is just the special polynomial at $A^m$. At $q=\pm1$ factorization follows from the singularity of the cabled HOMFLY, which is the same for all representations. At $m>1$ the relevant singularity is in the projector, and what survives of it is a finite sum of braids $X_m$ – the $m$-th power sum inside the cable, which can be freely moved through the other strands. For the proof we use the Murnaghan-Nakayama rule and the Adams operation. Discovery of this hidden structure explains the old factorization puzzle.

We prove that for every prime power $q$ and every integer $r\geq 3$ there exists a maximum $2$-scattered $\mathbb{F}_q$-subspace of $V(r,q^6)$, that is, a $2r$-dimensional $\mathbb{F}_q$-subspace meeting every $2$-dimensional $\mathbb{F}_{q^6}$-subspace in dimension at most $2$. Since an example in dimension three is known and direct sums preserve the property, it suffices to treat $r=4$ and $r=5$. For $r=4$ we study the subspaces $U_b=\{(x,x^q+bx^{q^5}):\mathrm{Tr}_{q^{12}/q^4}(x)=0\}$ of $\mathbb{F}_{q^{12}}^2$, $b\in\mathbb{F}_{q^{12}}$. We describe the parameters $b$ for which $U_b$ is not maximum $2$-scattered in terms of trace conditions and Dickson determinants, and deduce that at least $\frac{1}{2}q^{12}-q^{11}+O(q^{10})$ parameters work; this number is positive for $q\geq 3$, and a computer search settles $q=2$. For every $r\geq 5$ coprime to $6$ we show that $U_s=\{x+\varepsilon s\,x^{q^r}:x\in\mathbb{F}_{q^{2r}}\}\subseteq\mathbb{F}_{q^{6r}}$ is maximum $2$-scattered for all $\varepsilon\in\mathbb{F}_{q^6}\setminus\mathbb{F}_{q^2}$ and $s\in\mathbb{F}_{q^r}\setminus\mathbb{F}_q$. Consequently, $\mathbb{F}_{q^6}$-linear $[2r,r,4]_{q^6/q}$ MRD codes exist for every $r\geq 3$; for $r$ coprime to $6$, the codes obtained from $U_s$ are equivalent to their duals.

In this paper, we study strong uniqueness for smooth instantaneously complete Chern-Ricci flows on noncompact complex manifolds. Under suitable reference metric and positivity assumptions, we establish strong uniqueness for initial Hermitian metrics which are Kähler outside a compact set. The initial metric may be incomplete, and no curvature, metric comparison, or growth bounds are imposed on the evolving solutions. In particular, we obtain strong uniqueness from complete initial metrics in this class with bounded Chern-Ricci curvature. We also prove stationarity of flows from the Euclidean metric on $\mathbb{C}^n$ and uniqueness of the instantaneously complete flow from the Euclidean metric on the complex unit ball for all positive time. Finally, on complex surfaces, we show that a Ricci flow remaining Hermitian for a fixed complex structure is Kähler at all times if it is Kähler at one time, and derive a corresponding strong uniqueness result for Ricci flows in this class.

NVFP4 is an attractive datatype for large language model (LLM) inference, offering compact storage and native tensor-core acceleration. However, preserving accuracy using NVFP4 requires careful quantization. In this work we develop a novel quantization scheme called Optimized Smoothing and Scaling for NVFP4 (OSFP4). For each linear projection it uses a diagonal smoothing matrix whose entries are optimized to minimize the squared matrix-product quantization error under NVFP4, taking into account the rounding procedure that is used (either round-to-nearest, or GPTQ-style successive interference cancellation). This requires performing joint optimization on the smoothing entries as well as the block scales, which is facilitated by analyzing a multiplicative-dither FP4 quantizer instead of the fixed deterministic one. Experiments show that OSFP4 achieves the highest average accuracy among the evaluated competitors in the corresponding quantization settings, while retaining approximately 94-97% of vendor NVFP4 prefill throughput on the measured workloads. Our code is available in https://github.com/neriahbd/OSFP4

We show that any integral cubic form in ten or more variables has a nontrivial integer zero. Like earlier authors on this topic, we use geometric methods to produce a zero in some cases, and the circle method in all other cases. New ingredients are required on both fronts: in the first case, bounds on the dimension of singular loci and incidence varieties, and in the second case, a stratified geometric sieve, together with stratification results for point counts over finite fields and a cube-root Hensel lifting analysis.

The mean exit time of a Brownian motion from a tube around a closed, minimal embedded \(n\)-dimen\-sional submanifold of the \((n+m)\)-dimen\-sional sphere is studied, establishing two-sided bounds via radial transplant. In accordance with known results when \(n = 0\) or \(m = 1,\) the mean exit time goes to infinity as \(n \to \infty\) and to \(0\) as \(m \to \infty.\) Simultaneously, bounds for the volume of such tubes are derived and compared to Weyl's tube formula.

We study the truncation $(-)^{\leq0}$ of dg categories. We first show that given a dg category $\mathscr{C}$ with shifts, for example a pretriangulated dg category, the canonical functor $\mathscr{C}^{\leq0}\to\mathscr{C}$ is a localization whose kernel is compactly generated by the $0$-th cohomology. Next we demonstrate that the derived category of the truncation serves as a triangulated analogue of the Auslander's category of coherent functors over abelian categories. We give a description of the Auslander-Reiten-Serre duality in terms of the inverse dualizing bimodule of the truncation. Building on truncations, we introduce the notion of connective resolutions of dg categories, defined as a localization functor from a connective dg category. This notion encompasses non-commutative resolutions of algebras in module categories, cluster tilting objects in triangulated categories, and also the truncation as the universal connective resolution. We give a sufficient condition for an object in a triangulated category to give a connective resolution of its enhancement in terms of finiteness of resolution dimension. Furthermore, we show that every proper dg module over a connective dg algebra is a direct summand of a dg module giving a connective resolution. Consequently, for every proper connective dg algebra, its bounded dg derived category has a connective resolution. Finally, we give some explicit description of the truncation for some categories including the derived and cluster categories of a Dynkin quiver, and the Yoneda category of a finite dimensional algebra.

We develop a sparse domination framework adapted to nested families in general measure spaces, possibly including point masses. We show that the Carleson condition is equivalent to a slight generalization of the usual sparse condition, and use this characterization to obtain one- and two-weight estimates for the associated averaging operators, together with converse testing conditions. We apply our general results to integral operators on locally finite trees. Trees come naturally equipped with two nested geometries: sectors and pruned sectors. We prove pointwise sparse domination of natural versions of the harmonic Bergman projector by the corresponding averaging operators, depending on whether the measure on the tree is doubling or not, and show that the distinction is necessary in general. For the positive Bergman projector, we obtain pointwise comparability. As a consequence, we obtain new weighted inequalities and boundedness results for the Bergman projector and a class of Toeplitz-like multipliers.

For a given data distribution $(X_t)_{t \in \mathbb{N}} \sim Q$ i.i.d., we investigate the hypothesis testing problem: $H_0: Q = P_0$ vs. $H_1: Q = P_1$, for two different model probability distributions $P_0$ and $P_1$. In contrast to the standard setting, where analytic densities $p_0$ and $p_1$ are given, here, we consider the density-free setting, where we only have access to i.i.d. simulations $(Z^0_t)_{t \in \mathbb{N}} \sim P_0$ and $(Z^1_t)_{t \in \mathbb{N}} \sim P_1$. For this simulation-based hypothesis testing setting, we construct an e-test martingale, resulting in a sequential test with anytime-valid type-I error guarantees, approximate growth optimality, geometrically decaying type-II error bounds, and asymptotic power one. Most ingredients used in our constructions are variants of well known concepts. The value of this paper lies in the compact presentation of an effective, anytime-valid solution for the density-free simulation-based sequential hypothesis testing case.

This is a course about Ethics in Mathematics. That is, about the ethical issues that arise in mathematics itself, in mathematical work, and in the actions of mathematicians, broadly interpreted to cover all mathematically-powered technologies. This is a novel course. Traditionally, ethics has seldom been considered by, let alone taught within, the mathematics discipline. This course seeks to change that, by bringing to light the specific ethical issues relevant to mathematics, in a way that is clear and understandable to mathematicians. This course is written by mathematicians, for mathematicians, and about mathematicians. It presents new and original material, in the form of case studies covering harmful and unethical behaviour, analysis of the causal chain of events leading to such harm, and examination of the surrounding processes that drive such actions. All this is fully cited, and supported with interactive activities to aid learning. It develops much of the foundational theory of ethics in mathematics: what can go wrong, why it goes wrong, and how to address it. Students and practitioners alike take away an understanding of the context of their mathematical work, and how to change their actions to ensure more appropriate and useful outputs. In contrast to more traditional philosophy teaching, this course does not delve into the depths of ethical theory. Rather, it presents a practical, hands-on, pragmatic approach to ethics in mathematics, treating it as a branch of applied ethics. It uncovers the very particular ethical issues that arise from mathematical technologies, as well as show how general principles from ethics and responsible development apply even to mathematical work. As such, this makes it highly relevant to adjacent areas such as computer science, physics, engineering, and any other numerate discipline.

In this paper, we investigate Lindelöf-type properties of spaces of Baire-one functions equipped with the topology of pointwise convergence. In particular, we investigate the properties of normality, extent and realcompactness for the space $B_1(X)$ of Baire-one real-valued functions. We also study when $B_1(X)$ is realcompact: we show that this is the case for every hereditarily Baire (in particular, every Polish) separable metrizable space $X$ and for every Luzin set $X$. The paper provides answers to two Teng's questions. The paper raises open questions.

For an Alexander quandle on an abelian group $G$ with automorphism $\phi$, we identify its augmentation powers and nonunital commutator subalgebra with explicit ideals in the ordinary group ring over any commutative unital coefficient ring. Each left-normed augmentation power is also the span of all products with the same number of factors, independently of parenthesization. Over $\mathbb{Z}$, we determine every successive augmentation quotient for Alexander quandles on nontrivial finite cyclic groups. For an even dihedral quandle of order $m \ge 4$, every quotient in degree $r \ge 2$ is $(\mathbb{Z}/(m/2)\mathbb{Z})^2$; this proves the proposed replacement for the disproved order-$m$ prediction. For the integral quandle ring of $\operatorname{Core}(\mathbb{Z})$, we describe all augmentation powers, determine the nonunital commutator subalgebra, and prove that the difference of the basis elements indexed by 1 and 0 does not lie in the square of the augmentation ideal. When $(1-\phi)G$ is a finite $p$-group, the integral filtration is separated, so every nonzero integral idempotent has augmentation one. Fourier analysis shows that finite odd-order commutative Alexander quandles have only zero and the basis elements as integral idempotents. Combined with the midpoint description of medial commutative quandles, this gives the same classification for every finite medial commutative quandle. Finally, the complex quandle algebra of the core of any nontrivial finite abelian group of odd order has right Peirce spectrum $\{0,1,-1\}$, proving the conjectured odd-order dihedral spectrum.

This work studies non-linear function computation broadcast (NFCB), in which a sender with access to $N$ datasets $(X_1,\dots,X_N)$ broadcasts a common message to $K$ users, each possessing side information and requesting a function of the datasets. The goal is to minimize the rate required for asymptotically lossless recovery of all demands. We introduce a broadcast graph that jointly captures the source distribution, side information, and demanded functions. Using Körner's characteristic-graph framework, we develop an achievable scheme for arbitrary $K$, general source distributions, and general finite-field demands, including linear, non-separable, and non-linear functions, without restricting the encoding or decoding operations to be linear. We also present a side-information-assisted independent-set scheme and characterize the optimal graph-based achievable rate for compatible functions. For the converse, we derive a multi-letter response-profile bound that strengthens a basic side-information converse, zero-error and asymptotically lossless clique-entropy bounds based on the operational block broadcast graph, and a genie-aided lower bound. For binary NFCB with $N=K$ and side information $X_i$ at user $i$, we characterize the optimal rate in several special cases and bound the worst-case and average additive gaps between the proposed achievable rate and the genie-aided converse for $K=3$ and $K=4$. Finally, three-user examples with Boolean and linear demands illustrate the proposed bounds and show that non-linear encoding can strictly outperform the best scalar and vector linear schemes.

Consider the $\mathbb Z_2$ synchronization problem \[ \boldsymbol{Y} = \frac{\lambda}{\sqrt n} \theta\theta^{\top} + \boldsymbol{Z}, \] where $\theta$ is uniform on $\{-1,1\}^n$ and $\boldsymbol{Z}$ is an independent Gaussian Wigner matrix with off-diagonal variance one. We give a polynomial-time posterior sampling algorithm for every fixed $\lambda>1$, for which the conditional output law converges to the posterior in total variation, in expectation over the observation. The construction combines sequential TAP proposals with an independence Metropolis correction. The key is to control signed overlaps after logarithmic pinning and conditional TAP approximations along a random revealing path, which give an efficiently evaluable proposal with a polynomial density-ratio bound outside a set of vanishing posterior mass. To the best of our knowledge, this is the first polynomial-time posterior sampler for $\mathbb Z_2$ synchronization with a total-variation guarantee throughout the supercritical regime. For comparison, the diffusion-based sampler of \cite{montanari2023posterior} gives normalized Wasserstein guarantees at sufficiently large fixed signal-to-noise ratio.