Papers

We introduce a tensor envelope (TEE) for composite objectives $\varphi=f+g$, built from a $q$th-order Taylor model of the smooth term $f$ regularized by a $p$th power with $p>q$. To construct it, we first establish a tensor descent lemma and characterize its behavior in the three regimes $q<p<q+1$, $p=q+1$, and $p>q+1$. Under suitable prox-boundedness and parameter conditions, the associated tensor operator (TOP) has nonempty compact values, and TEE is finite-valued and continuous, preserves global minimizers, and inherits the level-boundedness of $\varphi$. We further show that TEE is locally Lipschitz continuous and directionally differentiable, characterize its Fréchet differentiability, and prove that it is $C^1$ whenever TOP is single-valued. As such, TEE preserves the essential optimization landscape properties of $\varphi$ while enjoying a richer analytical structure, which facilitates the design and analysis of tensor-based optimization methods. Finally, TEE serves as a Lyapunov function for the exact tensor iteration: every cluster point is critical, and, for bounded trajectories with $q<p\le q+1$, an $\epsilon$-stationary point is reached within $\mathcal{O}(\epsilon^{-p/(p-1)})$ iterations. To our knowledge, both the tensor descent lemma and this complexity bound are the first of their kind obtained without assuming global Lipschitz or Hölder continuity of the $q$th-order derivative of $f$.

For $0<\theta<\pi$, let $K_\theta=\{e^{is}:|s|\le\theta\}$ and $E_\theta=\operatorname{span}_{\mathbb C}\{1,z\}\subset C(K_\theta)$. We prove that every contractive linear map from $E_\theta$ into $\mathcal B(H)$ admits a factorization through a unitary operator whose spectrum is contained in $K_\theta$, and is therefore completely contractive. Consequently, $E_\theta$ has a unique operator space structure. For $0<\theta,\varphi<\pi$, we also show that $E_\theta$ and $E_\varphi$ are isometrically isomorphic if and only if $\theta=\varphi$, and that none of these spaces is isometrically isomorphic to $\ell_1^2$ or $\ell_\infty^2$. Thus we obtain uncountably many mutually non-isometric two-dimensional complex Banach spaces with a unique operator space structure, giving a negative answer to a question of Pisier.

We show that the abelian category of coherent toric sheaves is generated, as a Serre subcategory, by pushforwards of rank one reflexive sheaves supported on invariant subvarieties. As an application, we extend previous results by Klyachko on cohomology of toric vector bundles to coherent toric sheaves.

Best arm identification under differential privacy is a pure-exploration problem in which both statistical efficiency and privacy protection must be achieved simultaneously. We study fixed-budget best arm identification for bandits under pure $\epsilon$-differential privacy, where the learner must recommend an arm after a prescribed sampling budget while protecting the full transcript. We prove that the optimal exponential decay rate of the error probability is upper bounded by an instance-dependent privacy-aware transportation exponent that differs from the analogous quantity used to characterize the stopping time in fixed-confidence analysis by Jourdan and Azize [2025]. Guided by this exponent, we propose AO-Pri-BAI, an adaptive algorithm that maintains private running estimates through Laplace-tree mechanisms and learns a sampling design through a min–max interaction between hard alternatives and arm allocations. We prove that AO-Pri-BAI satisfies pure $\epsilon$-differential privacy. We also establish that the exponent of the failure probability of AO-Pri-BAI matches the privacy-aware benchmark. Numerical studies show that even in the non-asymptotic setting, AO-Pri-BAI outperforms benchmark algorithms on various instances, complementing the theoretical analyses.

In this paper we study the constant speed, biased random walk in dynamical random conductances. In certain regimes of the parameters of the model, we prove that the random walk is ballistic. We show annealed and quenched CLT in the full parameter range. Furthermore, as a consequence of the non-reversible nature of this model, we show that the classical Einstein relation does not hold.

Based on our previously proposed direct approach to central configurations in the full configuration space, we study the degeneracy of planar four-body kite central configurations with arbitrary positive masses both in kite symmetry restricted subspace and in full configuration space. We derive a factorization of the determinant of the Jacobian that substantially simplifies the subsequent analysis. The methods used to study the singularity of the Jacobian include rigorous interval arithmetic, the Krawczyk operator, a quantitative implicit function theorem, and computer-assisted proofs. Both convex and concave kite central configurations are considered. In the convex case, we prove the nondegeneracy in both the restricted subspace and the full configuration space. In the concave case, there is a unique smooth degeneracy curve in the restricted subspace; an additional degeneracy curve emerges in the full space setting, which is hidden in the subspace of kite configurations and makes it unavoidable to study the degeneracy in the full configuration space. We also establish several properties of these degeneracy curves. As an illustration to this complete characterization, we exam the well-known central configuration of equilateral triangle with equal masses at the vertices and a fourth mass at the center, which reveals additional features of degeneracy captured by our full-space framework.

Let $\Omega\subset\mathbb C^n$, $n\ge2$, be a bounded connected pseudoconvex domain whose boundary contains a smooth strongly pseudoconvex point. We prove that if the Bergman metric of $\Omega$ is extremal, then its scalar curvature is identically $-n$, and that constant scalar curvature forces the Bergman metric to be Kähler–Einstein. Consequently, extremality, constant scalar curvature, and the Kähler–Einstein condition are equivalent in this setting. A key analytic ingredient is a local unique-continuation theorem for the Bergman Laplacian at an ACH boundary, proved using Hörmander's Carleman estimate. We also obtain an extension for possibly unbounded pseudoconvex domains: if the Bergman metric is cscK or extremal near such a boundary point, then it is well-defined and Kähler–Einstein throughout the locus where $K_\Omega(z,z)>0$.

We prove a simultaneous Williamson's normal form for two symplectically commuting positive operators $A, B$ on a real symplectic Hilbert space $(\mathbf{H}, J)$. Under a natural commutativity condition, there exists a (generally unbounded) operator $\Phi$ that simultaneously diagonalizes $A$ and $B$ in the symplectic setting as $A = \Phi^t M_A \Phi$, $B = \Phi^t M_B \Phi$, and maps $J$ to a canonical symplectic operator. The result generalizes Williamson's normal form to infinite dimensions and yields a complete spectral invariant for the pair $(A,B)$.

Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. We prove that \begin{align*} \sum_{n\leq x^{1/c}}d\left(\left[\frac{x}{n^c}\right]\right)=d_cx^{1/c}+O_{\varepsilon,c}(x^{\theta_c+\varepsilon}), \end{align*} where $d_c$ is a suitable constant, $\theta_c<2/(3c+2)$ for $0<c<2/11$. This result constitutes an improvement upon that of Y. Feng (2024).

An $\mathcal H^2$ matrix represents well-separated interactions using nested low-rank bases while retaining nearby interactions explicitly. We consider the problem of constructing an explicit, strongly admissible $\mathcal H^2$ approximation using only products with the matrix and its adjoint, and develop two randomized recovery algorithms on an adaptive geometric hierarchy. Carried recovery begins with a single pair of dense Gaussian test matrices, without hierarchy-specific probing, and updates the resulting forward–adjoint sketches across levels. All operator samples can therefore be acquired in one batch. Resketched recovery applies the same recovery procedure using fresh Gaussian test matrices at each level. For this variant, we prove a global quasi-optimal bound on the expected Frobenius error relative to the best approximation with the prescribed hierarchy, rank bounds, and retained near-neighbor pattern. Carried recovery requires only $O(k+p)$ applications of the matrix and its adjoint, independent of tree depth, whereas resketched recovery requires $O((k+p)\log(N/k))$ on trees of logarithmic depth. Here $N$ is the matrix dimension, $k$ the maximum prescribed box rank, and $p$ the oversampling parameter. These bounds assume bounded near-neighbor counts and $O(k)$ local dimensions. Experiments on Laplace, Dirichlet-to-Neumann, and Helmholtz operators in two and three dimensions with up to $1.28$ million unknowns show that, with sufficient oversampling, carried recovery achieves nearly refinement-independent error and accuracy comparable to resketched recovery while using many fewer operator applications. Its compressed application cost scales linearly for a fixed rank.

We introduce a sampling approach for energy- and score-based generative models that requires no gradient evaluations of the model. Replacing the drift term that would normally contain the score $\nabla_\mathbf{x} \log p_\theta(\bf{x})$ with a high-frequency dithered cosine of the model's value, $\sqrt{\alpha\omega}\,\cos(\omega t + k \log p_\theta(\bf{x}))$, produces, in the high-frequency averaging limit, Langevin Markov chain Monte Carlo for energy-based models and the reverse-time SDE of score-based diffusion. We prove that trajectories of the dithered Itô SDE converge to those of the target SDE, driven by the same Brownian motion, uniformly on compact time intervals in probability, by an averaging argument that extends bounded extremum seeking (ES) to Itô processes, with an explicit $O(\omega^{-1/2})$ mean-square rate under global bounds. The approach is not confined to smooth targets: it extends to $C^{1,1}$ energies with discontinuous curvature (without ellipticity requirement) and to Sobolev energies whose Hessians exist only off measure zero sets; for Lipschitz energies with gradient kinks the averaged limit remains well posed; the Krylov-Röckner integrability class is the boundary of provability. The approach provides a hard a priori bound on the per-step update rate and applies to explicitly time-varying targets on finite horizons. Gradient-free pixel-space sampling is not competitive with well-tuned backpropagation-based samplers at practical evaluation budgets; the regime where the approach offers an advantage is latent-space sampling when the model is a black box and the target drifts in time. We demonstrate latent-space tracking for time-varying images on CelebA-HQ ($256{\times}256$) from limited 1D projection measurements and latent-space EBM sampling on CIFAR-10.

For positive matrices $A,B$ and $0<t<1$, Bourin asked whether the crossed-Heinz expression \[ C_t(A,B)=A^tB^{1-t}+B^tA^{1-t} \] satisfies \[ \UIN{C_t(A,B)}\le \UIN{A+B} \] for every unitarily invariant norm. The most direct route to this inequality is to compare $C_t(A,B)$ with the ordinary Heinz expression \[ H_t(A,B)=A^tB^{1-t}+A^{1-t}B^t, \] because the classical Heinz inequality gives $\UIN{H_t(A,B)}\le\UIN{A+B}$. This route, however, encounters a genuine dimensional obstruction: the stronger comparison $\UIN{C_t(A,B)}\le\UIN{H_t(A,B)}$ is known to fail for positive definite $3\times3$ matrices, and hence, by block-diagonal embedding, in every dimension $n\ge3$. We first determine the exact threshold for this obstruction. For every pair of positive $2\times2$ matrices and every $0<t<1$, we prove the weak-majorization relation \[ s(C_t(A,B))\prec_w s(H_t(A,B)). \]

We establish exact limits for the normalized temporal power variations of every fixed real order \(\gamma\geq2\) for the mild solution to a one-dimensional space–time fractional stochastic heat equation driven by multiplicative space–time white noise. The results include the exact normalized quadratic variation and the critical power variation of order \(2\alpha/(\alpha-\beta)\) as special cases. We also obtain the corresponding spatially averaged temporal variations. As applications, we construct consistent estimators for the drift parameter and for the ratio \(\beta/\alpha\).

We study strictly convex central configurations of the planar Newtonian five-body problem with one dominant mass and four satellites of arbitrary positive masses. We prove that a single positive bound on the total satellite-to-primary mass ratio guarantees existence and uniqueness for every prescribed directed order of the five bodies on the convex hull, up to translations, rotations and positive scaling. The bound is independent of all satellite mass ratios, including ratios arbitrarily close to the boundary of the mass simplex. The constrained Hessian is nondegenerate on the six-dimensional shape space. We first prove uniqueness and nondegeneracy for the convex coorbital limit by an angular Hessian estimate. We then classify the possible collision limits as the satellite mass ratios vary and construct regularized local continuations for pairs, triples and pairs within triples. An exact factorization of the signed areas preserves strict convexity on independent parameter neighborhoods. These continuations connect every sufficiently small-mass configuration to the unique coorbital configuration and yield the common bound.

In this paper, we focus on the existence of both global and non-global nonnegative solutions for the abstract semilinear heat equation $u_t - Lu = \sum_{i=1}^k \ell_i(t) f_i(u)$ in $L^p(X, \mu)$. Here $(X,\mu)$ represents a $\sigma$-finite measure space, $L$ is the infinitesimal generator of a sub-Markovian strongly continuous semigroup of bounded linear operators in $L^p(X,\mu)$, and for each $i = 1, \ldots, k$, the functions $\ell_i: [0, \infty) \rightarrow [0, \infty)$ and $f_i: [0, \infty) \rightarrow [0, \infty)$ are continuous. We utilize our findings to apply to the Laplacian operator on sub-Riemannian manifolds and locally finite graphs, obtaining Fujita-type results. Our results extend the recent work of Lenz, Schmidt, and Zimmermann, "Blow-up of nonnegative solutions of an abstract semilinear heat equation" (Calc. Var. Partial Differential Equations, 2023).

We introduce the notion of a Cartan inverse semigroup and an (algebraic) semi-Cartan subalgebra in an $R$-algebra A, providing a purely algebraic analogue of Cartan semigroups and semi-Cartan subalgebras in a C*-algebra. This extends the theory of quasi-Cartan subalgebras, shifting the focus from designated subalgebras to designated subsemigroups. Our main result characterises twisted Steinberg algebras in terms of the existence of a Cartan inverse semigroup: we show that an $R$-algebra admits a Cartan inverse semigroup if and only if it is isomorphic to a twisted Steinberg algebra. Moreover, when our construction is applied to a twisted Steinberg algebra over an ample Hausdorff groupoid, the constructed twist is naturally isomorphic to the original one.

Function-correcting codes, recently introduced by Lenz et al.\ (2023), provide a framework for the receiver to provide a higher level of protection against channel errors than the level of protection for messages. In this paper, we derive a generalised Plotkin-type bound on the redundancy of FCCs under the Hamming metric for arbitrary functions over finite fields. The proposed bound recovers the classical Plotkin bound for classical error-correcting codes as a special case. We further obtain explicit Plotkin-type bounds for linear functions, Hamming-weight functions, and Hamming-weight threshold functions. In addition, we study FCCs for monomial mappings $f(x)=x^n$ over $\mathbb{F}_{q^k}$, which are widely utilised in coding theory, cryptography, and finite geometry, and derive an explicit Plotkin-type redundancy bound.

This paper presents a nonlinear extension of Density-Driven Optimal Control (D2OC) for multi-agent spatial coverage with prescribed density distributions. Rather than assigning individual target locations, D2OC drives the collective spatial distribution of agents toward a desired density through a Wasserstein-based objective. We extend this framework to multi-step finite-horizon control for discrete-time control-affine nonlinear systems using sequential convex programming. At each control update, the nonlinear dynamics are locally linearized over the prediction horizon, yielding a strictly convex quadratic program that preserves the Wasserstein barycentric structure while directly incorporating input constraints. We further characterize the effect of constrained control deviations and nonlinear Taylor remainders on the accuracy of the local linear prediction, establishing an explicit finite-horizon error bound and a two-step specialization for receding-horizon implementation. The resulting method retains the decentralized, distribution-driven nature of D2OC while providing a computationally efficient optimization procedure for nonlinear multi-agent systems. Simulations with unicycle and quadrotor teams show coverage performance comparable to nonlinear model predictive control, while substantially reducing computation time. These results demonstrate a tractable and theoretically characterized framework for density-driven spatial coverage under nonlinear dynamics.

We establish cellwise pressure monotonicity for a mixed finite element discretization of the rotation-based Biot system. The spatial discretization is based on compatible discrete de Rham complexes, using standard lowest-order elements for rotation, displacement, Darcy flux, and pressure. For the temporal discretization, we employ the backward Euler method. Exactness of the discrete complexes reduces the mechanics contribution to the pressure Schur complement to a scaled pressure mass matrix, yielding a positive diagonal mechanics capacity. For the Darcy part, positive circumcentric facet weights yield a weighted cell graph Laplacian after flux elimination, while the mass-lumped product remains uniformly equivalent to the standard Darcy product. The diagonal mechanics capacity and the weighted cell graph Laplacian make the condensed pressure matrix Stieltjes. Consequently, the pressure propagator is entrywise nonnegative, which yields a cellwise discrete maximum principle. Numerical experiments confirm the predicted convergence rates and show that the proposed discretization suppresses spurious cellwise pressure oscillations.

We study density regularity and optimal Gaussian approximation for complex multiple Wiener-Itô integrals. We first give a kernel criterion for absolute continuity and derive explicit formulas for the density and its derivatives, together with finite-order Sobolev estimates. Our main quantitative result shows that the sum of an intrinsic non-Gaussian term and the covariance mismatch gives the optimal rate simultaneously in every fixed Sobolev norm and in total variation, Kolmogorov and $1$-Wasserstein distances. The intrinsic term is determined by two complex third-order moments and the complex fourth-moment defect, and is equivalently described by complex kernel contractions. No additional Malliavin non-degeneracy assumption is imposed in these asymptotic results. As an application, we obtain explicit optimal rates for the normalized numerator in the least-squares estimator of a complex fractional Ornstein–Uhlenbeck process.

The classical voter model is dual to coalescing random walks, in much the same way that the Kingman coalescent describes genealogies through binary mergers. Motivated by the passage from the Kingman to the~\(\Lambda\)-coalescent, where multiple mergers are allowed, we introduce the~\(\Lambda\)-broadcast voter model, in which a voter may transmit its opinion simultaneously to several neighbors according to a mechanism governed by a finite measure~\(\Lambda\) on~\([0,1]\); the classical voter model is recovered when~\(\Lambda=\delta_0\). We show that our model is well-defined on general graphs with bounded degrees, and we construct a family of stationary measures~\({\mu_{\alpha,\Lambda}}\), parametrized by bounded harmonic functions, that characterize the extremal stationary measures according to the finite or infinite collision property of the underlying random walk. We then study the dependence of the stationary measures on the broadcasting measure $\Lambda$ through a finite number of its moments. We give general conditions under which different moment profiles yield different stationary measures and obtain, in particular, a complete characterization on trees with bounded degrees and on nearest-neighbour~\(\mathbb Z^d\),~\(d\ge3\). We also prove continuity of the stationary measures with respect to the moment profile. Our main tool is a dual system of coordinated coalescing random walks, whose individual particles are simple random walks but which may jump and coalesce simultaneously; a coupling with independent random walks allows us to relate their long-time behaviour to ordinary random walk collision properties.

Godbersen's long-standing conjecture states that the mixed volumes of any convex body $K \subset \mathbb R^n$ satisfy $V(K[k], -K[n-k])\le \binom{n}{k} \operatorname{vol}(K)$ for any $k \in \{1, \dots, n - 1\}$, with equality if and only if $K$ is an $n$-simplex. Kotrbatý and Mouamine recently proved the inequality in full generality, and showed that if $K$ is a (full-dimensional) polytope attaining equality, it must be a simplex. We complete the characterization of the equality case: if $K \subset \mathbb R^n$ is a convex body for which $V(K[k], -K[n-k]) = \binom{n}{k} \operatorname{vol}(K)$ for some $k \in \{1, \dots, n -1\}$, then $K$ is an $n$-simplex. By a similar method, we characterize the equality case in the higher-order Godbersen inequality conjectured by Schneider and by Kotrbatý.

Integrated sensing and communication (ISAC) has emerged as a promising approach for vehicle-to-everything (V2X) systems by enabling communication and sensing over shared radio resources without additional installation of dedicated sensors. However, candidate resources may experience different communication qualities due to varying channel conditions and resource contention, which should be considered when designing sensing resource patterns. In this letter, we propose a quality-of-service (QoS)-constrained resource pattern selection framework that jointly considers communication reliability and sensing performance. The sensing objective is formulated based on the Fisher information matrix for joint range and velocity estimation, while a minimum packet reception ratio (PRR) is imposed as the communication QoS constraint. To avoid the complexity of exhaustive search, a greedy selection algorithm with one-swap refinement is developed. Simulation results show that the proposed method improves sensing performance over conventional resource patterns while satisfying the PRR requirement and achieves performance close to the exhaustive-search optimum and sensing-only optimum.

This paper presents a locally conservative Enriched Linearized Neural Network (ENN) method for the Darcy flow model. A linearized shallow ReLU$^k$ network, whose hidden-layer parameters are fixed on a quasi-uniform set of the sphere, carries the approximation, and piecewise constant functions on an auxiliary subdivision of the domain are added to ensure local mass conservation. The subdivision may consist of general, possibly curved, elements. Since neural network functions do not satisfy the inverse inequality, the stability and error analysis of the standard discontinuous Galerkin interior penalty formulation do not apply. The missing inverse inequality is remedied by an edge identity combined with a Galerkin least-squares (GLS) term, together with a single neural approximant that is optimal in the $L^2$, $H^1$ and $H^2$ norms simultaneously. We use the nonsymmetric interior penalty (NIPG) form, which is coercive for every positive penalty parameter. With $n$, the width of the shallow neural network, used in ENN, we prove an error estimate in the EG norm that is of the optimal $H^1$ order, i.e., $O(n^{-(r-1)/d})$ for solutions in $H^r$, $r\ge2$, and $O(n^{-(s-1)/d})$ for solutions in $H^s$, $\frac32<s\le2$, for any shape-regular subdivision, when the GLS parameter satisfies $\min\{h,n^{-1/d}\}^2\lesssim\tau\lesssim n^{-2/d}$ and the edge terms are scaled with $\min\{h_e,n^{-1/d}\}$, i.e., by the resolution of the network rather than of the mesh. A numerical Darcy flux is constructed, which is locally conservative, regardless of the size or the number of the elements. Sample numerical results demonstrate the correctness of the theory. In particular, a fixed $4\times4$ subdivision gives the same accuracy as a subdivision refined with the network, and the mass loss stays at round-off unlike a non-conservative neural network method.

In this paper, we investigate the realizability problem, which asks whether prescribed group actions on graded modules can be realized by the groups of self-homotopy equivalences of CW-complexes acting on their homology. We provide a partial answer in the case of arbitrary groups and permutation modules concentrated in certain degrees. As a consequence, we realize every group as the group of self-homotopy equivalences of an $R$-local CW-complex with arbitrary prescribed connectivity, where $R$ may be chosen with $\rho(R)$ sufficiently large. In particular, this provides a complete answer to Kahn's realizability problem.

In this paper, we characterize fake quadrics $X$ satisfying the bounded cohomology property. Namely, there exists a constant $c_X>0$ such that $h^1(\mathcal O_X(C))\le c_X h^0(\mathcal O_X(C))$ for every curve $C$ on $X$.

Let $(\mathfrak g, \mathfrak k)$ be the complexication of a Hermitian symmetric pair and let $\gamma_1 <\cdots<\gamma_r$ be the Harish-Chandra strongly orthogonal roots, with $\gamma_1$ the lowest non-compact root, and let $\zeta$ be the normalized central character of $\mathfrak k$. We consider the unitary highest weight representations $L(\lambda(x))$ of $\mathfrak g $ of highest weights of the form $\lambda(x)=-k\gamma_1 -x\zeta$, $x\in\mathbb{R}$. In particular, we are interested in the case $x=a$, the last reduction point in the analytic continuation in $x$. We prove that $L(-k\gamma_1-a \zeta)$, $k\ge 1$, is $K$-multiplicity free and we describe all its K-types.

In this article, we study a weighted analogue of the $k$-plane transform, where the weight on the $k$-dimensional measure is given by a radial power of distance. Particularly, we first consider the existence of such operators on Lebesgue spaces imbibed with radial power weights. Then, we proceed to see the mapping properties of the said transform when it acts on radial functions. In this regard, we prove certain weighted $L^p$-improving boundedness and end-point Lorentz space estimates. Finally, we look at the weighted $L^p$-$L^p$ boundedness of the operator acting on general functions, and evaluate its operator norm.

In this contribution we consider the singular periodic problem \[ u''+k\,u=g(u)+e(t),\quad u(0)=u(2\pi),\,\,u'(0)=u'(2\pi), \] where $k\in\R,$ $e:\J\to\R$ is Lebesgue integrable and $g:\I\mapsto\R$ is continuous, but can have a singularity at $x=0.$ In particular, I wish to revisit our earlier result regarding the existence of at least two its solutions and present a proof that is somewhat more transparent than the original one.

We study how the ranks of factors in a positive semidefinite slack factorization constrain its size. We bound the number of low-rank vertex factors in any fixed factorization, and use this to determine the asymptotic order of the minimum factorization size under any fixed bound on vertex-factor ranks: $\Theta(n\log n)$ for permutahedra $\Pi_n$ and $\Theta(\log N)$ for regular $N$-gons. In the rank-one case this gives the order of the minimum dimension of a space of real functions on the vertices in which every facet slack is a sum of squares, with no symmetry or degree restriction. For unrestricted lifts, the bound gives $\mathrm{xc}_{\mathrm{PSD}}(\Pi_n)\ge n+\log_3 n-O(1)$, and shows that a permutahedron factorization of linear size would need vertex factors of rank $\Omega(\log n)$ at all but a vanishing fraction of vertices. We also prove that every polytope of real positive semidefinite rank at most four has at most twelve vertices, and construct explicit size-four lifts for every square-symmetric octagon, so eight is attained. The upper bound combines a count of the polygon corners reached by curves of rank-one factors with incidence constraints on factor ranks. Whether eight is the maximum remains open.