Papers

In this paper, we study Weil–Petersson circle homeomorphisms from the viewpoint of harmonic maps. We prove that a homeomorphism $\varphi:\mathbb S^1\to\mathbb S^1$ is Weil–Petersson if and only if its unique quasiconformal harmonic extension to the hyperbolic disk $\mathbb D$ has square-integrable Beltrami differential. Our approach is based on the anti-holomorphic $L^2$-energy of harmonic maps. We show that this energy is finite for the quasiconformal harmonic extension of every Weil–Petersson circle homeomorphism, and that, among suitable quasiconformal extensions, the harmonic extension minimizes this energy.

For $\boldsymbol{\alpha} = (\alpha_1, \dots, \alpha_k) \in \mathbb{F}_2^k$, an $\boldsymbol{\alpha}$-town is a set family in which every $i$-wise intersection has parity $\alpha_i$. Denote by $f_{\boldsymbol{\alpha}}(n)$ the maximum size of an $\boldsymbol{\alpha}$-town on $[n]$. The classical oddtown and eventown problems study the cases $\boldsymbol{\alpha} = (1, 0)$ and $(0, 0)$, respectively. We determine the sharp asymptotics of $f_{\boldsymbol{\alpha}}(n)$ for all $\boldsymbol{\alpha}$, answering questions of Johnston–O'Neill and Wei–Zhang–Ge. We also study a symmetric variant $g_{\boldsymbol{\alpha}}(n)$, in which $i$-wise intersection sizes $|F_1 \cap \dots \cap F_i|$ are replaced by $i$-wise intersection-union sizes $|F_1 \cap \dots \cap F_i| + |F_1 \cup \dots \cup F_i|$.

We introduce First-Order Trajectory Matching (FTM), a surrogate-modeling method that learns the first-order local transport of probability mass from trajectories of stochastic systems. By matching the symmetric first-order motion of trajectories, FTM learns the probability current velocity, whose flow preserves time marginals to match ensemble averages, while also capturing current-like trajectory quantities such as fluxes, circulations, and barrier-crossing currents. FTM learns the current velocity directly from trajectories, avoiding drift, diffusion, and score estimation. Our stability analysis separates discretization error from sampling variance and shows that the one-step simulation-free FTM loss is stable when temporal resolution and sample size are properly balanced. Across stochastic dynamical systems and PDE examples, we empirically demonstrate that FTM provides trajectory-aware ensemble predictions at low, deterministic-rollout cost.

Spherical $G$-parking functions are a distinguished subset of standard monomials, arising from the skeleton ideals of the $G$-parking function ideal. Explicit enumeration formulas for spherical $G$-parking functions are known only for a few classes of graphs. In this paper, we consider a family of graphs $G_{\ell}$ ($1\leq \ell \leq n-2$), obtained from the complete graph $K_{n+1}$ by deleting the $\ell$ edges joining vertex $1$ to the vertices in $F_{\ell}= \{n-\ell+1, \ldots, n\}$. The uprooted spanning trees of $G_{\ell}-\{0\}$ correspond to the set $\mathcal{U}_n^{1\not\sim F_{\ell}}$ of uprooted trees with vertex set $[n]$ in which vertex $1$ is not adjacent to any vertex in $F_{\ell}$, and we establish that $|\mathcal{U}_n^{1\not\sim F_{\ell}}| = (n-1)^{n-\ell-2}(n-2)^{\ell}(n-\ell-1)$. We derive this formula combinatorially and independently recover it as an application of the matrix tree theorem, obtaining some combinatorial identities as consequences. Finally, we determine the number of spherical $G_{\ell}$-parking functions as $|\mathrm{sPF}(G_{\ell})| = (n-1)^{n-3}(n-\ell-1)^2$.

We develop a framework for conformal prediction in dyadic regression problems under complex missingness mechanisms. At the theoretical level, we develop general technical tools for establishing finite-sample validity of conformal prediction under distributional invariance conditions weaker than exchangeability. A key result handles the case where the sample itself is a random subset of the index set, a setting not covered by existing theory. In addition, we propose conformal prediction procedures for jointly exchangeable arrays, including full conformal, split conformal, a row-column approach exploiting similarities within rows and columns, and a selective conformal procedure achieving mask-conditional validity. For missing elements, we establish asymptotic validity of a weighted conformal procedure under a nonparametric graphon model for the missingness mechanism. We further establish conditional validity results for both continuous and discrete responses; to the best of our knowledge, this is the first formal proof of asymptotic conditional validity for weighted conformal prediction under a missing-not-at-random assumption. The proposed methods are illustrated on synthetic and real network data.

Indicial umbral calculus offers an effective operational framework for manipulating transcendental functions, yet its analytic foundations have long remained only partially understood. In this work, we provide a rigorous analytic realisation of the theory grounded in Mellin-Barnes integrals, Borel-Laplace summation, and resurgent analysis. By elevating umbral operators from formal algebraic symbols to continuous linear functionals, we establish a topological duality akin to Gelfand-Shilov theory. Formal substitutions are thereby replaced by well-defined, continuous pairings between geometric kernels and suitably topologised admissible ground states. Within this framework, divergent umbral evaluations acquire a precise meaning: they emerge as sectorial asymptotic expansions of analytic functions reconstructed exactly by Mellin-Barnes integrals. The associated Stokes phenomena are natively encoded by jump functions in the spectral variable. This leads to a central spectral transmutation law relating entire and rational kernels directly through Gamma regularisation, proving that classical algebraic umbral identities are merely local expansions of a global analytic correspondence. The construction is systematically extended to general umbral functionals via the Pólya representation of entire functions of exponential type. Explicit examples - including Hankel contours, genuinely Barnes-type integrals, and Lerch transcendents - demonstrate the exact geometric resolution of formal algebraic obstructions. Ultimately, this approach embeds indicial umbral calculus within a unified functional-analytic and resurgent framework, where formal series, analytic continuation, and topological spectral data are intrinsically linked.

In this article we consider a system of coupled partial differential equations that mirrors the interconnection between the evolution of an electromagnetic field – described by the Maxwell's system – and that of the temperature distribution in a bounded region in three-dimension. A discussion of the mathematical model is provided. We establish the well-posedness of the initial-boundary value problems associated with the thermoelectromagnetic system, in an appropriate functional-analytic framework. Then, our investigation and main results pertain to the long-time behaviour of the solutions. It is shown that either exponential stability or convergence to stationary solutions hold true, according as the conductivity is positive definite or semidefinite, respectively; in the latter case, strong stability is attained in specific topological or analytical settings. This complements and expands earlier results obtained for the uncoupled Maxwell's system.

We propose a model-independent axiomatic framework for the derived skein theory of oriented 3-manifolds with coefficients in a ribbon tensor category, especially focusing on the case where the input category is the category of finite-dimensional representations of a quantum group with quantum parameter not a root of unity. The axioms are designed so that the 0th homology recovers the ordinary skein module and gluing is governed by a bar construction. We establish several relationships between the derived skein theory and the ordinary skein theory. We show that this framework yields computable formulas in terms of ordinary internal skein modules and internal skein algebras. We also prove a Hochschild formula for Sigma x S^1. We give the first computations of derived skein modules and establish finiteness properties for generic parameters using deformation quantization methods.

We study a dynamic assortment problem on a two-sided service platform with incomplete information and heterogeneous customers in a discrete-time setting. In each period, a customer arrives seeking service, and the platform chooses an assortment of sellers to display. The customer then proposes a transaction to at most one seller in the assortment according to a multinomial logit choice model. After a fixed number of periods, sellers review the proposals they have received and each chooses at most one customer according to another multinomial logit choice model, after which the cycle repeats. A key challenge is that the platform does not know the choice-model parameters of either customers or sellers in advance. To our knowledge, this is the first study of a dynamic assortment problem in which both sides' choice parameters are unknown. We develop a data-driven algorithm that learns these parameters while optimizing the platform's objective over time. We evaluate performance using regret, which measures revenue loss relative to a clairvoyant benchmark that knows all parameters and customer arrivals in advance. We show that the algorithm's worst-case regret grows polylogarithmically over time, and we derive a matching lower bound, establishing its rate optimality.

We estimate the Toeplitz covariance matrix of a centered Gaussian distribution from data that are both coarsely quantized and sparsely sampled. Only the coordinates of a sparse ruler are recorded, and each recorded value is kept as a single bit: the sign of its comparison with a fixed threshold. Such data arise in low-precision sensing front ends and sparse sensor arrays. Because the threshold is nonzero, every bit has a common mean. Each bit is also reused across many of the products that build the covariance, so one bit's error enters many of them. Centering removes the shared error. We prove a Gaussian variance contraction theorem for products of a centered, bounded nonlinearity of a Gaussian vector, the non-smooth one-bit sign included; it sets each lag's variance by how well the ruler covers that lag. The resulting estimator needs neither the signal scale nor the bit mean in advance, since the nonzero threshold makes both identifiable from the marginal bits. A matching minimax lower bound shows the resulting coverage rate is optimal up to constants over a neighborhood of white noise; the bound holds even for the unquantized real-valued samples, so one-bit quantization costs only a constant factor.

A signed graph is a graph with signatures ($+1$ or $-1$) on its edges. A cycle is called positive if the product of its edge signatures is positive, and a signed graph is called balanced if each cycle in it is positive. The frustration index is the minimum number of edges whose deletion makes the signed graph balanced, which is considered to be a measurement of the imbalance of the signed graph. In this paper, we compare the frustration index of the all-negative signature with the maximum frustration index of all possible signatures on the unsigned graph. We classify some families of graphs into three scenarios: the all-negative signature does not maximise the frustration index, the all-negative signature maximises the frustration index non-uniquely, and the all-negative signature maximises the frustration index uniquely. For all three scenarios, we can exhibit chordal and non-chordal graphs alike. The classes we consider include apex trees, fan graphs, wheel graphs, and complete split graphs. Moreover, for the families of fan graphs and wheel graphs, we fully characterise and count the signatures maximising the frustration index. Throughout our study, we exhibit different classes of signed graphs for which the frustration index equals the number of edge-disjoint negative triangles. Moreover, as part of our study, we are able to refute three conjectures of Zaslavsky on the frustration index.

We investigate limitations of learning $\tanh$ neural networks from point evaluations under finite-precision computations and $L^p$ accuracy guarantees, building on Berner, Grohs, and Voigtländer (2023). Our approach is based on a novel construction of sharply localized bump functions via iterated $\tanh$ activations. Using this mechanism, we show that, in a finite-precision setting, no adaptive randomized algorithm based on $m$ samples can achieve a convergence rate higher than the Monte Carlo rate $O(m^{-1/p})$ in the $L^p$ norm, unless the sampling budget grows exponentially with the size of the network parameters and architecture. The results reveal fundamental limitations imposed by finite precision on the learnability of classes containing localized bump functions, extending previous results for ReLU networks to the $\tanh$ setting.

Let $0<\lambda_1<\cdots<\lambda_n$. Motivated by a problem related to Lyapunov equations we consider a class of Cauchy-like matrices whose elements have the form $C_{ij}=\frac{r_i(k,l)+r_j(k,l)}{\lambda_i+\lambda_j},$ where for any pair $1\le k,l\le n$, $r_i(k,l)$ are functions of $\{\lambda_1,\cdots,\lambda_n\}\setminus \lambda_i$. We show that these matrices are positive semidefinite for every pair $1\le k,l\le n$. After passing to the reciprocal variables $x_i=1/\lambda_i$, the problem is reduced by a diagonal congruence to the positivity of a two-parameter family $A_n^{(p,q)}(x)$. The proof introduces a singular augmented matrix $\mathcal H_n^{(p,q)}(x)$, proves its singularity by Cauchy-kernel generating function identities, and then proves positive semidefiniteness by induction on $n$ using the principal-minor criterion.

The Fuel Optimizer is an inverse design framework for sustainable aviation fuels that starts from a user-defined merit function and identifies the optimal combination of chemical species or hydrocarbon families that optimize a combination of targets. As a case study, a database of fuel blends meeting selected property standards was simulated in a reactor model to obtain pollutant emissions at cruise conditions. A surrogate model was developed to reduce the computational cost of evaluating candidate blends, taking fuel composition as input and predicting emissions as output. A genetic algorithm was used to optimize fuel formulation according to two merit functions designed to break the nitrogen oxides - CO trade-off, and minimize pollutant emissions over the landing-and-take-off cycle. Constraints included selected property standards (including seal swelling) and composition limits. Fuel candidates from the framework outperformed the training database across all merit functions, and the optimal candidates were validated through reactor simulations.

A long-standing conjecture in the emerging discrete Bakry-Émery theory asserts that bounded-degree graphs satisfying $\mathrm{CD}(0,\infty)$ have polynomial growth. In the present paper, we prove this conjecture for all edge-regular graphs, and even obtain a volume doubling estimate with a constant that depends only on the degree. This is made possible thanks to the discovery of a surprising self-improvement phenomenon, which seems of independent interest: any edge-regular graph satisfying $\mathrm{CD}(\kappa,\infty)$ for some $\kappa\in\mathbb R$ must in fact satisfy $\mathrm{CD}(\kappa,n)$ for some explicit, universal and optimal dimension parameter $n$.

In this paper, we study the Cauchy problem for a class of semilinear evolution equations with scale-invariant time-dependent dissipation \begin{equation*} \begin{cases} u_{tt} + L_{w^2}u + \dfrac{\mu}{1+t}u_t = \Delta^{\theta} f(u), & t>0,\ x\in\mathbb{R}^n,\\ u(0,x) = 0,\qquad u_t(0,x) = u_1(x), & x\in\mathbb{R}^n, \end{cases} \end{equation*} where $\mu>0$, $f(u)=|u|^\alpha$ with $\alpha>1$, $\theta\in\{0,1\}$, and the operator $L_{w^2}$ is defined on the Fourier transform by multiplication by $w(\xi)^2$. We prove the global (in time) existence of small data solutions for $\alpha>\alpha_{\mathrm{crit}}$, where the critical exponent $\alpha_{\mathrm{crit}}$ depends on the choice of the operator $L_{w^2}$, the parameter $\mu$, and the nonlinear term. In particular, we consider two model cases. For Boussinesq-type operators with $w(\xi)=\sqrt{|\xi|^2+|\xi|^4}$, combined with the derivative-type nonlinearity $\Delta |u|^\alpha$, we obtain a Strauss-type critical exponent. On the other hand, for plate-type operators with $w(\xi)=|\xi|^\sigma$, $\sigma\geq2$, and power-type nonlinearity $|u|^\alpha$, the critical exponent is of Fujita type.

I give a short proof that the border determinantal complexity of $F_n=x_1^n+\cdots+x_n^n$ over $\mathbb{C}$ is at least $(n-1)^2/(4e)$, and at least $(n-1)^2/(2e)$ in the symmetric model. The same argument shows that the border determinantal complexity of $x_1^d+\cdots+x_N^d$ is $\Theta(Nd)$ for all $N\ge3$ and $d\ge2$. The earlier article arXiv:2606.13628, by the same author, introduced the first superlinear lower bounds and includes verification by Lean formalization. However, the AI written exposition was hard to follow. Kumar and Volk (arXiv:2609.34462) give a short proof of the exact ordinary case, but do not address the border or symmetric cases. In this article, I provide a greatly simplified exposition of the border and symmetric cases, using the implicit function theorem and an explicit section of simple solutions in place of the conormal specialization arguments of arXiv:2606.13628. I choose an explicit polynomial system with $n(n-1)^{n-2}$ simple solutions for $F_n$. I then use the implicit function theorem to ensure that each point persists under small perturbations. Finally, kernel equations and the multihomogeneous Bézout theorem bound the number of these points for a small determinant. The exposition is due to the human author. The mathematical simplification was obtained in discussion with ChatGPT Astra and refined in conversation with Claude Opus 5.5.

In this work, we utilize the Hermite expansion to approximate the distributions of the sum and maximum of independent random variables. We model distributions with a three-segment representation, where the left and right tails are respectively modeled as combinations of Hermite functions, and the intermediate segment is approximated by piecewise polynomials. This approximation admits rigorous $L^2$- and pointwise convergence properties supported by classical results. We develop an algorithmic framework for applying our model to the graph delay propagation problem, where sum and max operations are performed on the proposed model structure. Numerical experiments demonstrate that our model can capture the quantile values with high accuracy compared to Monte Carlo simulation results, significantly outperforming classical Gaussian-based models.

We construct a weighted Riemannian manifold $(\mathbb R^2,g,\mu)$ satisfying $\mathrm{CD}(1/2,\infty)$, the curvature-dimension condition, with the following property: if $\gamma$ denotes a centered Gaussian measure on $\mathbb R^2$, then there is no Lipschitz map $T:(\mathbb R^2,\|\cdot\|) \to (\mathbb R^2,g)$ satisfying $T_\#\gamma=\mu$. Building on this, we prove a Weyl-type asymptotic law for the eigenvalues of the weighted Laplacian $-\Delta_{g,\mu}$ and show that they are asymptotically negligible when compared to the eigenvalues of $-\Delta_{\gamma}$. These results give strong counterexamples to two questions of E. Milman and complement the recent counterexample of Aryan.

We define an infinite family of quiver representation-valued invariants of classical and virtual knots associated to a choice of data vector consisting of a biquandle, abelian group, set of biquandle arrows weights with values in the abelian group, coefficient ring and set of biquandle endomorphisms. As an application we extract four new polynomial invariants as decategorifications. We provide examples to show that these invariants are proper enhancements of the biquandle counting invariant and biquandle coloring quiver.

Let $m$ be a positive integer and $\mathbb{Z}_m$ the residue class ring modulo $m$. The Ruzsa number $R_m$ is defined to be the least integer $r$ such that there is a subset $\mathcal{A}$ of $\mathbb{Z}_m$ satisfying $ 1\le \sigma_{\mathcal{A}}(n)\le r $ for any $n\in \mathbb{Z}_m$, where $$ \sigma_{\mathcal{A}}(n) =\#\big\{(a,a')\in\mathcal A^2: a+a'\equiv n\pmod{m}\big\}. $$ Motivated by a 2024 conjecture of Ding and Zhao, we prove $ | R_{m+1}-R_m|\le 144. $ Let $\mathcal{A}$ be a subset of $\mathbb{Z}_m$ satisfying $1\le \sigma_{\mathcal{A}}(n)\le R_m$ for any $n\in \mathbb{Z}_m$. We also give nontrivial bounds for the size of $\mathcal{A}$. Additionally, we provide exact values of $R_m$ for all $m\le 100$, which substantially extends the table of values given by Sándor and Yang in 2017. Finally, we pose several related problems and prove some partial results.

We study the frog model on $\mathbb Z$ with particle-wise random discrete Weibull lifetimes and biased nearest-neighbour random walks. Each particle has an independent survival parameter $\pi\in(0,1)$. Conditionally on $\pi=p$, its lifetime $\Xi$ satisfies $$ \mathbb P(\Xi\ge k\mid \pi=p)=p^{k^\gamma}, \, k\in\mathbb{N}_0, $$ where $\gamma>0$. The distribution of $\pi$ is assumed to have right-edge density $$ f_\pi(u)\sim (1-u)^{\beta-1} L\left(\frac1{1-u}\right), \, u\uparrow1, $$ where $\beta>0$ and $L:(0,\infty)\to(0,\infty)$ is a slowly varying function at infinity. The main step is to estimate the tail of the maximal displacement of a single particle before death. If $\tau_n$ denotes the time needed by the underlying walk to reach distance $n$, then $$ \mathbb P(D^*\ge n)=\mathbb E[G(\tau_n)], \, G(k):=\mathbb P(\Xi\ge k). $$ Since $$ G(k)\sim \Gamma(\beta)k^{-\gamma\beta}L(k^\gamma), $$ and the biased nearest-neighbour random walk has linear hitting-time scale, the off-critical threshold is $\beta_c=1/\gamma$. If $\beta>\beta_c$ and the initial number of particles per site has finite mean, the model dies out almost surely. If $\beta<\beta_c$ and the initial configuration is not almost surely empty, the model survives with positive probability in the direction of the drift.

We exhibit a family of smooth incompressible velocity fields on the two-dimensional unit sphere such that the time evolution of any mean-free initial data passively advected by any of them is mixed exponentially fast. In the presence of molecular diffusivity, we show that the solution to the associated advection-diffusion equation experiences enhanced dissipation with optimal decay rates. Each member of this family is an alternating combination of two Rossby-Haurwitz flows with random amplitudes and constitutes a spherical analogue to the sine shear-alternating example of Pierrehumbert.

We prove an explicit formula for the generic rank of hypergeometric perverse sheaves on higher dimensional tori over a finite field, as defined by Gabber and Loeser. This allows to give an explicit estimate for the corresponding hypergeometric character sums for almost all primes $p$.

We investigate the evolution of thick points under two natural dynamics for the Gaussian free field (GFF) in dimension 2. The first dynamic we analyze is the Ornstein-Uhlenbeck GFF. We prove that, simultaneously for all points, the evolution of their thickness is continuous. Additionally, we characterize all deterministic functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that there are points whose thickness function is $f$. The second dynamic we study is the stationary solution of the additive stochastic heat equation. In this case, the thickness of points is not continuous. Moreover, this rougher dynamic generates super-thick points, namely points with thickness greater than $2$. As a function of $\gamma > 2$, we identify infinitely many phase transitions corresponding to the existence of exceptional times where at least $N$ points are $\gamma$-thick. These phase transitions, occurring at $\gamma^2 = 8, 6, 16/3, \dots$, converge to $4$ as $N \to \infty$. Mapping to the critical FK-model parameter via $q=4\cos^2(4\pi/\gamma^2)$, these critical values correspond to the Beraha numbers, which are precisely the points at which the CFT for critical FK percolation should be minimal

Cellular generation, which generalises cofibrant generation, is an important categorical smallness condition on a class of morphisms. A general challenge is to determine whether a given class of morphisms $\mathcal{M}$ is cellularly generated, in which $\mathcal{M}$-effective squares are often useful. These are commuting squares consisting of morphisms in $\mathcal{M}$, so that the induced morphism from the pushout square is also in $\mathcal{M}$. When we drop the requirement that the vertical morphisms in the square are in $\mathcal{M}$ we obtain the weaker notion of $\mathcal{M}$-quasieffective square. We prove that, in a locally presentable category, $\mathcal{M}$ is cellularly generated if and only if $\mathcal{M}$ is almost everywhere quasieffective. The latter is a set-theoretic condition stating that for almost every partial elementary set-theoretic subuniverse $\mathfrak{N}$, we have that restricting any morphism in $\mathcal{M}$ to $\mathfrak{N}$ yields an $\mathcal{M}$-quasieffective square. For locally finitely presentable categories this yields an additional categorical characterisation in terms of filtrations of $\mathcal{M}$-quasieffective squares. If we additionally assume that $\mathcal{M}$ is continuous (i.e., the corresponding wide subcategory is closed under directed colimits) then we obtain a stronger characterisation of cellular generation in terms of accessibility of the category of $\mathcal{M}$-effective squares. This improves on a theorem by Lieberman, Vasey, and the third author.

Boolean networks model gene regulatory networks, but extracting quantitative dynamics requires translating their logical rules into differential equations, and the sigmoidal kernel chosen carries direct biological consequences. The near-universal choice, the Hill function, sets production to exactly zero when an activator is absent, creating a spurious absorbing off-state with no biological counterpart. We develop a product-of-logistics framework in which increasing logistic functions represent activation, decreasing logistic functions represent repression, and a recursive De Morgan product formula translates an arbitrary Boolean rule into a continuous regulatory function. The translation is automatic, confines every regulatory function to the unit interval, and retains a strictly positive basal rate. Our central result is a recovery theorem: every steady state of the Boolean network reappears, for sufficiently steep response, as an exponentially stable equilibrium of the continuous model, so the translation provably refines rather than distorts the Boolean analysis. We establish global well-posedness, forward invariance, and an explicit Lipschitz constant, and prove, for the two canonical two-gene motifs, global asymptotic stability of the negative-feedback oscillator and a closed-form bistability threshold for the toggle switch. Every threshold remains a positive, measurable concentration, unlike weighted-sum logistic formulations that place repressor thresholds at meaningless negative values. The eleven-gene Traynard mammalian cell-cycle network is translated automatically: in the proliferative regime its trajectories settle onto a sustained limit cycle reproducing the Boolean cyclic attractor. Because it is purely structural, the translation applies unchanged to existing Boolean models and supports exact feedback linearisation for control.

Our paper shows that if a matroid contains the Fano plane or its dual as a minor, then its toric ideal does not have any quadratic Gröbner basis. More than 25 years ago, Hibi, Herzog, and Sturmfels established a direct connection between the existence of quadratic Gröbner bases and regular unimodular flag triangulations. Our paper solves a famous question posed by Herzog and Hibi on a polyhedral reformulation for the existence of quadratic Gröbner bases: we show that the base polytopes of the Fano plane and its dual do not have regular unimodular flag triangulations which implies the main result on Gröbner bases. Our proof relies on several novel tools: a lemma that connects the $1$-skeleton of a lattice polytope to the lattice points in its dilations, an encoding with Boolean formulas and SAT solvers, and symmetry-breaking arguments.

Let $a_2''(n)$ and $b_2''(n)$ be the signed partition functions introduced by Andrews and El Bachraoui for interval-restricted partitions whose parts greater than $1$ are controlled by the smallest even part and by the number of ones. We prove two conjectures for these functions. The first gives the generating function for $a_2''(n)$ as an elementary rational term plus a false theta series with periodic signs; the second asserts that the companion coefficients $b_2''(n)$ take only the values $-1,0,1,2$. The central structural result introduces an auxiliary variable $z$ recording the number of non-compulsory parts greater than $1$. We obtain closed forms for the two resulting generating functions and prove the master identity $(1+q^2)\mathcal B(z,q)-(1+q)\mathcal A(z,q)=-q^4/(1-q^3)$ using both analytic and combinatorial techniques. At $z=-1$, this identity, together with a Rogers–Fine evaluation, gives the false theta formula for $a_2''(n)$ and an explicit generating function for $b_2''(n)$. The latter formula implies the asserted coefficient range and leads to an exact coefficient description of $b_2''(n)$. We also include a direct Heine–Rogers–Fine proof of the false theta formula, ordinary and fixed-refinement consequences of the master identity, and the resulting quantum modular interpretation.

Let $G$ be a finite abelian group of order $n$, and let $\mathsf M(G,m)$ denote the set of zero-sum sequences over $G$ of length $m$. We introduce the zero-sum polytope $\mathcal P_G$, a rational polytope of dimension $n-1$, whose lattice points encode zero-sum sequences: \[ |\mathsf M(G,m)|=|m\mathcal P_G\cap \mathbb Z^n|. \] This naturally realizes the enumeration of zero-sum sequences as a problem in rational Ehrhart theory, which leads to a combinatorial reciprocity theorem identifying the negative evaluations of the corresponding counting quasipolynomial with zero-sum sequences of full support. Our main results establish a face-stratified rigidity for zero-sum polytopes: whenever two such polytopes have equal total lattice point counts at specific dilations, the dimension-wise open-face strata are equinumerous. Moreover, we study the natural $\operatorname{Aut}(G)$-action on $\mathcal P_G$, derive equivariant generating functions and reciprocity formulas, and obtain cyclic sieving phenomena for natural cyclic actions.