Papers

We construct area-preserving irrational pseudo-rotations of the closed disk that fail to be $C^0$-rigid. For every Brjuno rotation number we obtain $C^1$ examples, and they may be chosen arbitrarily $C^1$-close to the rigid rotation with the same rotation number. For $0<\beta<1$, a weighted Brjuno condition gives $C^{1,\beta}$ examples that are arbitrarily close in the $C^{1,\beta}$ topology; this weighted range contains every rotation number with denominator-growth exponent strictly smaller than $1/\beta$ and also some numbers at the critical exponent. For a suitable Brjuno–Liouville rotation number, we also obtain an example whose derivative growth is not bounded by any polynomial.

We give a combinatorial criterion for detecting roots of Bernstein-Sato polynomials of ideals. As applications, we prove the strong monodromy conjecture for determinantal and Pfaffian ideals.

Bacterial co-infection, particularly with Streptococcus pneumoniae, is a major driver of morbidity during influenza epidemics, and this burden falls disproportionately on sub-Saharan Africa, where both pathogens circulate year-round at high levels and pneumococcal carriage remains common and persistent across the region. Yet existing mathematical models rarely track a host's bacterial carriage status jointly with their full influenza history through recovery. We develop a twelve-compartment ODE model that classifies each host by viral status (susceptible, infected, recovered) and bacterial status (susceptible, carrier, actively infected, recovered), allowing carrier-to-active progression to depend on prior influenza exposure. We establish positivity, boundedness, and a positively invariant region, then derive the basic reproduction number $\mathcal{R}_0$ and quasi-endemic equilibria of a single disease, together with invasion reproduction numbers $\mathbf{Inv}^{\mathbf{B}}$ and $\mathbf{Inv}^{\mathbf{V}}$ quantifying each pathogen's ability to invade a population where the other is already endemic. Center-manifold analysis shows that the bacterial subsystem and the full coexistence equilibrium undergo a backward bifurcation precisely when carrier transmission exceeds the carrier's total exit rate, $\beta_c>\kappa_S+\theta+\mu$, so that reducing $\mathcal{R}_0$ below one is not sufficient for elimination, while the viral subsystem bifurcates forward. A global sensitivity analysis (Latin Hypercube Sampling with partial rank correlation coefficients) shows that the viral and bacterial thresholds are governed by largely separate parameters, while the bacterial invasion potential is strongly shaped by viral transmission and recovery, giving a quantitative signature of cross-pathogen synergy.

Prunescu, Sauras-Altuzarra, and Shunia asked whether addition, totalized integer division, and base-two exponentiation form a substitution basis for the Kalmár elementary functions. We answer this problem negatively, even after adjoining multiplication. On independent powers of two, every fixed term built from addition, multiplication, and integer division takes only polylogarithmically many distinct values below a given bound, and the same restriction survives arbitrary nesting of $2^x$. No such term recovers $n$ from $2^n$, so neither $\lfloor\log_2 x\rfloor$ nor integer remainder is representable. The obstruction is not special to the base two. For every fixed integer $b\geq 2$, the same sparsity bound holds with constants independent of $b$, and $\langle x+y,\lfloor x/y\rfloor,b^x\rangle$ remains a proper subclass of the elementary functions. Variable exponentiation reverses the conclusion: addition, integer division, and $x^y$ do form a substitution basis, $$ \langle x+y,\ \lfloor x/y\rfloor,\ x^y\rangle=\mathcal{E} .$$ Once the base may depend on the input, integer remainder and truncated subtraction become representable. The incomplete base-two class is nevertheless complete for truth values. Every elementary characteristic function belongs to it, as does every elementary function with a fixed finite range.

We give a deterministic polynomial-time algorithm for the Bilu–Linial signing problem on bipartite graphs. For every finite simple bipartite graph of maximum degree at most an integer $\Delta\ge3$, the algorithm assigns signs to its edges so that the signed adjacency matrix has operator norm strictly less than $2\sqrt{\Delta-1}$. Our algorithm builds on the randomized recursive repair framework of Jadbabaie, Saberi, and Sra~\cite{JSS26}, with deterministic rules for sign selection and vertex deletion.

Affine frequency division multiplexing (AFDM) has emerged as a promising waveform for high-mobility communications owing to its ability to exploit multipath diversity. However, channel state information (CSI) acquisition remains challenging in channels with fractional delay and Doppler and an unknown number of propagation paths. In this paper, we investigate the channel estimation for AFDM systems. We first analyze the AFDM response in the presence of fractional-delay-induced frequency wrapping. The analysis reveals that fractional delay may displace the dominant extremum and generate informative secondary extrema. Then, we develop a model-order-adaptive channel estimator within an improved space-alternating generalized expectation-maximization (SAGE) framework. A dual-pilot reference symbol is employed for path and delay initialization, while pilot observations across multiple AFDM symbols provide temporal information for Doppler estimation. New paths are identified through statistically controlled residual tests, whereas unsupported or redundant paths are removed by conditional support pruning. The delay, Doppler frequency, and complex gain of each retained path are subsequently estimated through a coarse-to-fine procedure and refined using the exact AFDM likelihood. Simulation results demonstrate that the proposed method achieves lower normalized mean squared error (NMSE) and bit error rate (BER) than representative SAGE, sparse-recovery, and Bayesian benchmarks. It also provides reliable path detection and model-order estimation and maintains robust performance at terminal velocities of up to 600~km/h.

Universal maps (maps whose renormalized iterations approximate every map in a given class) are locally generic in several spaces of diffeomorphisms [Bonatti–Díaz 2003, Turaev 2015]. In fluid dynamics, steady Euler flows whose Poincaré maps are universal are also known to be locally dense [Berger–Florio–PeraltaSalas, 2023]. Motivated by Arnold's vision [Arnold, 1966] that the complexity of orbits in celestial mechanics and that in fluids should be comparable, in the present paper we address the question of the existence of universal maps in celestial mechanics. We give a partial answer by proving a weak, finite-dimensional form of universal dynamics. More concretely, we show that every symplectic embedding of the disk into $\mathbb{R}^2$ can be approximated, with arbitrary precision, by a renormalization of a Poincaré map of a restricted planar circular $(n+1)$-body problem, for suitable $n$ and appropriate choice of the masses of the primaries. The proof relies on Gonchenko–Shilnikov–Turaev theory [Turaev 2003, Gonchenko–Turaev–Shilnikov 2007] and requires control of the dynamics near homoclinic tangencies of arbitrarily high order; the techniques developed in [Garrido–Martín–Paradela, 2025] are essential for such control.

Using the recent proof of the Cassels–Swinnerton-Dyer conjecture for cubic surfaces, we finish the classification of $G$-unirational complex cubic threefolds. In particular, we prove that the Klein cubic threefold is $\mathsf{PSL}_2(\mathbb{F}_{11})$-unirational, establishing that the essential dimension of $\mathsf{PSL}_2(\mathbb{F}_{11})$ is $3$. This disproves a conjecture of Dolgachev that the essential dimension of a group is at least its Cremona dimension.

We give an elementary, straightforward proof of Ovenhouse's interpretation of the denominator and the numerator of a (right) q-deformed rational in terms of weighted lattice paths on a ”snake graph”, and we give an analogous result for the left q-deformed rational introduced by Bapat, Becker, and Licata. The key ingredient is a simple formula for the q-deformation of the matrix of the composition of two homographies of the form z $\rightarrow$ a + 1/z.

We study entropy realization for continuous actions of infinite countable amenable groups. We introduce an approximate product property for amenable group actions that permits a small proportion of tracing mistakes on prescribed pairwise disjoint, sufficiently invariant finite sets. We prove that this property implies entropy-denseness and almost entropy-approximability of every invariant measure. The construction combines zero-entropy exact tilings with a finite-block estimate for the complexity of tracing mistakes. Under asymptotic entropy expansiveness, almost entropy-approximability upgrades to entropy-approximability. For every $0\leq\alpha<h(X,G)$, ergodic measures of entropy $\alpha$ then form a residual subset of the invariant measures whose entropy is at least $\alpha$. In particular, the set of ergodic measure entropies equals $[0,h(X,G)]$.37A35, 37B40, 37B05

In this note, for every topological colored nonsymmetric operad with CW spaces in each arity we construct a homotopy isomorphism with a CW homotopy operad (i.e. all higher homotopies are cellular with respect to the product structure and CW structure on cubes) not changing its objects. By using the two-sided bar construction, we rectify this to a homotopy isomorphism with a strict CW operad. This applies more generally to topological algebras over colored operads in sets. In the case of diagrams of finite CW complexes over finite posets, the resulting diagram consists of finite CW spaces enhancing the Quillen adjunction between simplicial sets and topological spaces. As a byproduct, we produce an embedding of topological homotopy algebras over operads into the presheaf category over the branch resolution for the category of $P$-trees generalizing the known case of topological operads.

Guy Brousseau began developing the theory of situations in the 1960s, with the ambition of building an experimental epistemology of mathematics. Over the decades, the theory has been structured, consolidated and extended, drawing on the contributions of numerous researchers. It is now recognized as one of the ”genuine ground theories” in mathematics education. In this chapter, we will describe the foundations and reasons for the existence of this theory, its systemic approach to teaching and learning issues, its current architecture, which articulates a theory of mathematical situations and a theory of didactic situations, and its key concepts, as well as its functions, particularly its predictive and explanatory functions. We will illustrate this presentation with an example in geometry that also highlights the role played by the didactic engineering methodology in this theory and related work.

We prove that the combinatorial type of a slice of the cube by a hyperplane is completely determined by its graph, within the class of cube slices. We also show that such a result cannot extend to slices of arbitrary simple polytopes by constructing a simple polytope that has two combinatorially distinct slices with isomorphic graphs.

Reversible Jump Markov Chain Monte Carlo (RJMCMC) is a principled framework for Bayesian model comparison, but its practical use is often limited due to the difficulty of designing between-model proposals that reach regions of high posterior probability. In addition, its inherently sequential nature limits efficient use of modern GPU architectures. We address both challenges by introducing Anchored Reversible Jump Sequential Monte Carlo. Our approach combines RJMCMC with Sequential Monte Carlo (SMC) to explore the model space in parallel. Importantly, the particle population enables effective between-model proposals without problem-specific knowledge: particles are used to construct kernel density approximations of the target distributions within each model, which are then used to generate trans-dimensional proposals. Numerical experiments show that the proposed method is computationally efficient and competitive with state-of-the-art RJMCMC approaches, demonstrating its potential for scalable Bayesian model comparison.

We report on an exhaustive census of all 1,006,700,565 connected graphs on $11$ vertices to find the minimal examples of graphs whose independence complexes are Cohen-Macaulay but not shellable, or shellable but not vertex decomposable. In particular, there are exactly two graphs $G$ where ${\rm Ind}(G)$ is Cohen-Macaulay but not shellable, when the characteristic is not two, and exactly six graphs where ${\rm Ind}(G)$ is shellable but not vertex decomposable.

Let $E$ and $F$ be separable symmetric Banach sequence spaces, and let $C_E$ and $C_F$ be the corresponding ideals of compact operators on a separable Hilbert space. We prove that if $C_F$ is isomorphic to a complemented subspace of $S_E$, then $F=\ell_2$. As a consequence, if $C_E$ is isomorphic to $C_F$, then $E=F$ with equivalent norms. This answers Pełczyński's 1979 question concerning the uniqueness of symmetric structure. We also strengthen Arazy's result by showing that if $E$ has finite cotype and $C_F$ embeds isomorphically into $S_E$, then $F=\ell_2$. For reflexive $F\ne\ell_2$ with non-trivial Boyd indices, the embedding theorems of Lindenstrauss and Szankowski yield a separable reflexive symmetric sequence space $E$ with trivial cotype such that $C_F$ embeds isomorphically into $S_E$. This gives negative answers to several questions raised by Arazy in 1981.

Hessian spectra at trained models in deep learning exhibit a persistent pattern: eigenvalues organize into distinct clusters, including a large bulk near zero and a few isolated outliers. This paper shows that a natural account of these spectral phenomena emerges when the original setting is understood as a departure from a nearby, otherwise hidden, highly symmetric reference. Modifications, including changes to the architecture, data distribution, or parameter metric, expose a nearby reference configuration whose Hessian exhibits rich invariances-ones not accounted for by weight symmetries. There, symmetry enables a precise description of the spectra, forcing high-dimensional kernels and eigenvalues of large multiplicity. Returning to the original configuration breaks the Hessian symmetry and thereby produces the observed hierarchy of clusters and outliers. The framework is developed in some generality, with a detailed analysis of three-layer ReLU networks and applications to convolutional, graph, and transformer models, as well as to the NTK. The same mechanism is further shown to yield analogous spectral structures in layerwise Hessians and the Gauss-Newton matrix.

We answer two questions about the heavy-ball method on smooth strongly convex functions. The first concerns Polyak's tuning. Its dimension-uniform worst-case exponential rate equals the largest growth factor of an interpolable mixture of rotating geometric sequences, and we determine this factor for every condition number. The optimal mixtures form four algebraic families; the classification combines exact polynomial certificates, a rational parametrization, an exact solution in a quintic number field at the critical three-cycle, and verified interval continuation. Two frequencies suffice, one function in dimension five attains the rate at every horizon, the upper and lower bounds differ by constant factors, and the method converges on the whole class if and only if $\kappa<9+4\sqrt5$, the threshold located by Badithela and Seiler. The second question concerns arbitrary step sizes and momenta, in particular the regions of the parameter plane between the computed Lyapunov and cycle regions. We describe the phase diagram through the angle $\Theta(\omega)$ between the characteristic polynomials of the two extreme quadratics. The method converges when $\Theta(\omega)>\omega/2$, has a $k$-periodic orbit when $\Theta(2\pi j/k)\le\pi/k$, and the remaining regions are rotation-number gaps organized by Farey fractions. A single strict certificate at one growth exponent bounds the rate, so convergence can be certified at the threshold exponent. In the first level of the gaps, fans of Farey fractions accumulating at a resonant endpoint carry periodic orbits with several harmonics, explicit fan-lag inequalities certify convergence, and both behaviors occur in the regions left open before.

We propose a novel unfitted finite element method based on polynomial splines over hierarchical T-meshes (PHT-splines) for elliptic partial differential equations (PDEs) posed on domains whose boundaries are described implicitly by a level-set function. Exploiting the enhanced smoothness of the spline space, we formulate the method by minimizing the $L^2$ residual of the PDE over an extended neighborhood of the physical domain. This formulation allows the method to be implemented entirely on the background mesh using standard element-wise quadrature. It yields a symmetric positive-definite linear system and avoids the geometric construction of curved cut subregions in boundary elements, as well as the associated nontrivial numerical integration. We establish well-posedness of the numerical scheme and derive optimal a priori error estimates in the $H^1$ norm. In addition, we employ a residual-based a posteriori error indicator to guide local adaptive refinement of the hierarchical T-meshes. Numerical experiments corroborate the theoretical results and demonstrate the accuracy and efficiency of the proposed approach.

We study normalized ground states for the nonlinear Schrödinger energy on two-dimensional square grids with unbounded defects. We first show that unbounded defects do not necessarily destroy the dimensional crossover, proving its persistence on the half-grid and the quarter-grid. We then consider a prototypical defected grid where a macroscopically two-dimensional region coexists with an unbounded one-dimensional channel. On this graph, normalized ground states do not exist for sufficiently small or sufficiently large masses, showing that the one-dimensional component can dominate the variational problem and induce loss of compactness. Finally, we show that this mechanism is sensitive to the local geometry of the graph: suitable additional defects preserve small-mass nonexistence but restore the existence of ground states for large masses. Our analysis combines small-mass energy asymptotics with a large-mass blow-up argument.

We study the relation between viscosity solutions and weak solutions of the equation $$\partial_t\left(|u|^{p-2}u\right)\,=\, \operatorname{div}(|\nabla u|^{p-2}\nabla u).$$ We prove that at least for non-negative functions these are equivalent concepts.

For every quartic Salem number $\beta\in(1,2)$, we prove that the Bernoulli convolution $\nu_{\beta^{-1}}$ with equal weights has correlation dimension strictly less than one. In particular, neither of the two corresponding measures has an $L^2$ density. We relate correlation dimension to exact collision probabilities and express these probabilities as positive integrals over a four-dimensional torus associated with the defining polynomial. The unit-circle conjugates produce a persistent oscillatory term; using carefully chosen return times of the corresponding rotation, we take differences that make this term small while controlling the length and cost of the resulting configurations. Many well-separated placements of these configurations then give a collision probability above the critical scale, which yields the strict dimension drop. The qualitative argument is independent of the computer-assisted estimates. Separate computer-assisted arguments give \[ 0.99999<D_2(\nu_{\beta_1^{-1}})<1-10^{-65}, \qquad 0.99999<D_2(\nu_{\beta_2^{-1}})<1-10^{-75}, \] where $\beta_1<\beta_2$ are the two quartic Salem numbers in $(1,2)$.

We develop Delsarte theory for quantum association schemes, allowing both composition and the quantum Schur product to be noncommutative. A MacWilliams identity and positivity of the inner and dual distributions yield conic upper bounds for codes and lower bounds for designs in terms of effective size, which extends subset cardinality. These bounds recover classical linear programs for symmetric association schemes and semidefinite programs for homogeneous coherent configurations. We also quantize Schurian schemes using finite-group representations and characterize quantum block designs in quantum Johnson schemes arising from irreducible stabilizer representations by averaging over subgroups fixing prescribed points individually. In a quantum Johnson scheme on $M_2(\mathbb{C})^{\oplus10}$, we prove that the minimum effective size of a nonzero positive element satisfying the strength-two design condition is $20$, attained by complementary projections outside the center of the underlying algebra. Every such element in the center has effective size $40$, while the conic lower bound is $40/3$, strictly below the actual minimum.

We present a scalable method for computing persistence pairing graphs from large cubical filtrations and use it to test whether graph organization among persistent classes contains information beyond persistence intervals alone. The method cancels equal-value pairs before global reduction, constructs the compressed complex in streaming form, and recovers graph transitions from one persistence reduction through projected relations and triangular event solves. We apply the method to 119 signed distance volumes from the DRP-372 porous media collection for permeability prediction. Each 256-cubed volume induces more than 135 million cubical cells before contraction, whereas the median compressed complex contains 37,471 generators. In validation grouped by material family, adding persistence pairing graph descriptors improves prediction over persistence summaries under both Ridge regression and partial least squares regression. In the stricter experiment that holds out one source project at a time, Ridge retains a modest improvement over persistence alone, although additional benefit beyond geometry and persistence is not consistent across projects. Two null models show that the observed persistence and graph coupling and edge target organization are strongly nonrandom, but neither uniquely explains the predictive improvement; the useful signal appears to reside mainly in coarser graph and event organization. Spatial analyses likewise distinguish persistence magnitude from graph prominence and reveal nonlocal organization among algebraically related persistence events. These results show that persistence pairing graphs provide structurally informative summaries beyond the barcode while exposing domain dependence and representative sensitivity that motivate more invariant relational descriptors.

Inverse optimization estimates the weights of an objective function that explain observed decisions as optimal solutions, and is used in a variety of fields. For mixed-integer linear programs (MILPs), existing methods aim to reproduce the observations as optimal solutions, and thus learn compromise weights when the observations are suboptimal. We propose outperformance inverse optimization, which instead seeks weights that induce, at each state, an optimal solution outperforming the observed action in every component. We give a loss function that can be evaluated with forward-problem oracles alone and is thus applicable to MILPs, together with gradient-based and DC optimization algorithms for minimizing it. For weights inducing a unique outperforming optimal solution at all observations, we prove that the probability of failing to induce such a solution at a new state (the generalization error) is bounded by a quantity inversely proportional to the number of observations, and that this bound is tight in the number of observations up to logarithmic factors. In experiments on synthetic and real data, the proposed methods improve the prediction of solutions outperforming the actions over existing methods.

Human learning is a dissipative dynamical process: mastery accumulates through practice, decays through forgetting, and propagates across interdependent concepts. We model it as a nonlinear dissipative system of ordinary differential equations whose parameters are mechanistically meaningful (a transfer-of-practice matrix, per-concept forgetting rates, and a saturating practice-response gain), and we study when those parameters can actually be recovered from data. For two concepts, we prove a constructive structural identifiability theorem for the associated inverse problem under explicit excitation conditions, together with monotonicity, robustness and L-stability results. We derive an implicit L-stable scheme and a batched solver numerically equivalent to the per-trajectory formulation yet two orders of magnitude faster. The empirical study is two-sided. Under the theorem's excitation conditions, synthetic recovery is exact. On large observational benchmarks it is not. An apparently strong recovery, with forgetting rates correlating with topic error rates at Spearman $\rho = 0.83$, is refuted by four controls: it survives destroying the temporal order of the data, is matched by a classical Bayesian baseline, and is unaffected by removing real timestamps. A single shared forgetting rate attains the same accuracy, and replaying the real schedule with identical true rates reproduces most of the correlation. A positive control shows that the controls detect forgetting where it is observable. We trace the effect to the stationary structure of the model and show that it is consistent with the degeneration anticipated by the theory in the absence of designed excitation.

In this paper, we explore how large a subset of a finite group can be without containing non-trivial solutions to linear equations of the form $aX+bY=cZ+dU$ (where $a+b=c+d$). These equations naturally generalize the classic concept of Sidon sets. Using Tao's slice rank method, we first establish strict upper bounds for the size of such solution-free sets in the vector space $\mathbb{F}_{p}^{n}$. Next, we turn to cyclic groups $\mathbb{Z}_{m}$ and construct surprisingly large sets that have no solutions for the asymmetric equation $X+5Y=3U+3Z$. These constructions achieve a logarithmic density of roughly $0.5283$, breaking the expected $0.5$ barrier for any sufficiently large modulus $m$. Finally, we show that this high-density behavior extends to a wider family of equations whose coefficients follow simple rules modulo $8$.

We study sampling from the Gibbs distribution of the Sherrington-Kirkpatrick (SK) model with Glauber dynamics. For every fixed inverse temperature $0\leq\beta<1$ and every fixed $M,D>0$, we give a polynomial-time simulated annealing algorithm whose output distribution is within $n^{-M}$ total-variation distance of the Gibbs distribution, with probability at least $1-n^{-D}$ over the interaction matrix. The algorithm starts from uniform product spins and uses Glauber dynamics along an increasing inverse-temperature schedule. The same approach also yields partition-function estimates with relative error $n^{-M}$. The key ingredient is a quantitative weak Poincaré inequality. Building on the stochastic-localization approach to weak Poincaré inequalities [Davies, Lee, Sandhu, and Shi, arXiv:2607.08160, 2026], we use Gaussian isoperimetry to directly compare exact stochastic localization paths. Our comparison tolerates failures of local Lipschitz continuity of the posterior mean by controlling their accumulated cost. The local Lipschitzness comes from approximating the posterior means by stationary points of the Thouless-Anderson-Palmer (TAP) free energy in locally strongly convex regions located by approximate message passing, a paradigm introduced by algorithmic stochastic localization [El Alaoui, Montanari, and Sellke, 2025; Celentano, 2024]. The approximation errors are roughly characterized by the validity of the TAP gradient equations, but in a different coordinate. To show that the errors rarely accumulate too much, we need a strong concentration control. A direct argument would require solving a variant of an open problem by Talagrand [2010, Research Problem 1.7.9]. We bypass this open problem with an intermediate conditioning step and transfer the control back by establishing the stability of the TAP gradient under deletion of coordinates.

We investigate the spectral statistics of random perturbations of semiclassical Schrödinger operators on compact manifolds, near energy levels corresponding to chaotic classical dynamics. The prototypical example is that of an operator $h^2\Delta+V(x)$ that is perturbed by a smooth potential $h^\alpha V_\omega$ with $\alpha\in (0,1)$, and $V_\omega$ is a random potential that decorrelates on distances $h^\beta$, with $0<\beta<2\alpha$. We show that for a generic perturbation, the spectral fluctuations of the smoothed counting function of eigenvalues obey a universal behavior at a certain mesoscopic scale, which is coherent with the predictions of random matrix theory.

We construct categorical models and cluster characters for skew-symmetrizable cluster algebras admitting good ramified realizations. Goodness is equivalent to nondegenerate realizability. When a reddening sequence is given, it is enough to check sign coherence of the arrow types along that sequence. Triangular extensions with arbitrary connecting bimodule types preserve goodness when both component matrices admit reddening sequences. Consequently, every matrix obtained from one vertex by triangular extensions and mutations admits a good Jacobi-finite nondegenerate realization for every compatible positive integral symmetrizer. For a good Jacobi-finite nondegenerate realization, the monodromy $F$-polynomials satisfy mutation for all finite decorated representations, without a rigidity hypothesis. A full initial order determines valued universal presentations and their cluster characters. Under the reachability condition, these characters form a basis of the middle cluster algebra for arbitrary geometric coefficients, with frozen variables inverted, and contain all cluster monomials. The same family is an upper-cluster-algebra basis when the middle and upper algebras coincide. For a fixed Jacobian algebra and $\delta$-vector, the Newton polytope of the generic monodromy $F$-polynomial is independent of the order.