Papers

For the neighborhood seed associated with the four-vertex path $P_4$, the diamond operations do not recover all Boolean-lattice pseudo-roots. The corresponding question for unrestricted rational operations in the free skew field is subtler: the seed map has an invertible linearization and therefore a unique formal inverse near every generic scalar point. We prove that this formal inverse is not free rational. A symmetric one-parameter curve of $2 \times 2$ matrix outputs has a formal inverse whose coefficient field contains an element of degree three over $\mathbb{Q}(t)$. An exact elimination in a quadratic Pauli algebra produces the irreducible cubic. Its conjugate inverse branches are unramified, forcing the generic matrix degree of the seed map to be at least three in every size $n \ge 2$. This contradicts the degree-one consequence of any free rational inverse. The same matrix-degree argument, without specializing a hypothetical inverse, extends the obstruction to every graph containing an induced $P_4$.

The Reissner-Simo and Hodges models are two equivalent continuous descriptions of finite-strain beam dynamics. The Reissner-Simo formulation uses displacements and rotations, while the Hodges formulation is intrinsic and avoids both variables. Although equivalent in theory, the two approaches behave differently after discretization and offer distinct numerical advantages. In this work, we develop a structure-preserving discretization of the intrinsic planar beam formulation. Because the intrinsic equations involve linear differential operators, both kinematic and dynamic boundary conditions can be imposed naturally using mixed finite elements. The resulting formulation also enables multibody systems to be assembled without algebraic constraints, avoiding the stiff differential-algebraic equations typically introduced by kinematic constraints. We demonstrate the approach on different examples, also showing that closed kinematic loops can be modeled without algebraic constraints. The resulting interconnected systems retain a port-Hamiltonian structure, with all nonlinearities confined to the interconnection operator. This structure allows exact energy preservation when combined with implicit midpoint time integration. Furthermore the scheme appears to require less Newton iterations compared to existing energy preserving scheme.

In 1979, Jeltsch [Numer. Math. 32 (1979), 167-181] conjectured that every zero-stable Brown method is stiffly stable. For Brown $(K,L)$ methods, Meneguette subsequently reduced the relevant $A_0$-stability question to a zero-location property for a family of perturbed characteristic polynomials. A closely related perturbation question already appeared in his 1987 Oxford doctoral thesis, and he later posed a broader purely polynomial version of the problem in 1994 [SIAM Review 36 (1994), 656-657]. We provide a complete solution to Meneguette's polynomial problem.

While combinatorial optimization problems are central to many scientific and engineering applications, their solution remains challenging due to exponentially large search spaces. Variational quantum algorithms offer a promising route for tackling such problems, yet their practical performance is limited by repeated quantum circuit evaluations and classical parameter updates. In this work, we introduce DQAOA-GPT, a hybrid framework that integrates the distributed quantum approximate optimization algorithm (DQAOA), which decomposes a large optimization problem into smaller sub-problems, with GPT-based quantum circuit generation for solving those sub-problems. Rather than relying on iterative variational optimization, the proposed approach uses a trained generative model to directly generate high-quality quantum circuits for the decomposed sub-problems. As a benchmark, we evaluate DQAOA-GPT against conventional DQAOA on dense HUBO optimization problems with up to 100 decision variables. The results demonstrate that DQAOA-GPT significantly reduces computational cost while maintaining competitive solution quality, with larger acceleration observed for larger sub-problem sizes. Although this work focuses on benchmark-scale validation, the framework provides a promising foundation for larger-scale combinatorial optimization in hybrid HPC-QC environments through increased GPU resources and parallel computing capability.

We prove that the centro-affine Poincaré inequality holds with constant $n$ for every $C^{\infty}_+$ unconditional convex body and every smooth function with natural orthogonality conditions.

We investigate finite-index problems in skew braces. For every sub-skew brace \(A\) of a skew brace \(B\), we prove that finite additive index is equivalent to finite multiplicative index; whenever these indices are finite, they coincide. This answers Question~3.7 of \cite{CPV} affirmatively. We then construct a left brace with a strong left ideal of index \(3\) containing no finite-index ideal, giving a negative answer to Question~3.6 of \cite{CPV}. We also show that finite additive and multiplicative conjugacy classes, together with a finite \(\lambda\)-orbit, force an element to be an \((s)\)-element, thereby answering Question~5.21 of \cite{CPV}. Finally, for every \(n\geq 3\), a one-generated free right-nilpotent skew brace of class \(n\), introduced in \cite{Free}, has an index-\(2\) ideal that is not finitely generated as a skew brace, although it is finitely generated as an ideal.

Fundamental logic (Holliday 2023) is a non-classical logic based only on the introduction and elimination rules for conjunction, disjunction, and negation in a Fitch-style natural deduction system, while a preconditional (Holliday 2025) is a binary operation on a bounded lattice satisfying five natural axioms and subsuming Heyting implication, the Sasaki hook on ortholattices, and Lewis-Stalnaker-style conditionals satisfying flattening. We combine the two by giving a consequence-relation presentation $\mathsf{K}$ whose algebras are exactly Holliday's bounded lattices with a preconditional, and then studying two natural extensions, $\mathsf{T}$ and $\mathsf{F}$, the latter being fundamental propositional logic with a preconditional. For $\mathsf{T}$ and $\mathsf{F}$, we prove strong completeness with respect to a purely relational semantics, using a canonical model whose points are pairs of theories, and establish the finite model property and hence decidability. Finally, following Holliday and Massas (2026), we adapt their GMT- and Goldblatt-style embeddings to fundamental logic with a preconditional. The resulting translations are full and faithful into ortho-$\mathsf{S4}$ and intuitionistic $\mathsf{KTB}$, respectively. The new conditional clauses send the preconditional to a boxed Sasaki hook on the former side and to a strict intuitionistic conditional on the latter whose classical $\mathsf{KTB}$ reading is equivalent to the Goldblatt translation of the Sasaki hook. The frame constructions follow the reduct-and-companion pattern of Holliday and Massas; the essential additional ingredient is the semantic transfer calculation for the preconditional.

We give a direct construction of the spectral projectors of a complex square matrix, based on explicit interpolation polynomials previously introduced. This yields a spectral resolution $A=\sum_{i=1}^r(\lambda_iP_i+N_i)$ from which the Primary Decomposition Theorem, the Cayley–Hamilton theorem, criteria for diagonalizability, and the spectral theorem for normal matrices are proved by short formal arguments. A weaker form of the construction, extends to any perfect fields via a Galois invariance argument and produces the Jordan–Chevalley decomposition.

We show that there exist graphons that interpolate between Riemannian manifolds and weighted geometric graphs. Specifically, the graph-to-manifold approximation used in manifold learning can be regarded as the composition of a graph-to-graphon convergence and a graphon-to-manifold convergence in a certain sense. Furthermore, we establish a monotonicity inequality which reveals an implicit relationship between numerous combinatorial parameters and geometric quantities on graphons. Using this inequality, we find relations among conductance, maxcut problem, capacity, and packing radius, as well as their limiting behaviors under graph-to-graphon and graphon-to-manifold convergences; some of these relations are novel even for simple graphs and closed manifolds. We also find that in the graphon-to-manifold limiting process, the purely combinatorial parameters diverge, leading to a “blowup” phenomenon; and continuous versions of some graph parameters cannot be realized via Riemannian manifolds, but can be realized via graphons.

We derive an almost sharp lower bound on the Casson invariant of positive braid knots. As a consequence, we show that the $\Delta$-unknotting number is quasi-additive on the set of positive and negative braids on a fixed number of strands.

Rational points on weighted projective spaces are sparse: a point of $\mathbb{P}^n(\mathbb{Q})$ lifts along the Veronese morphism only when a Kummer condition holds at every prime (arXiv:2509.02319). We ask what this viewpoint says about the Jacobian Conjecture. The map recently announced as a counterexample is equivariant for the grading $\mathrm{wt}(x,y,z)=(1,-1,-2)$, and we show that the sign pattern of such a grading decides everything. If the weights are all positive, the setting of weighted projective spaces, an equivariant Keller map is always an automorphism, so no counterexample can be graded that way. In dimension two the same holds for every sign pattern. The Keller condition itself descends to the quotient, where it says that the Jacobian of the quotient map vanishes to order two along the contracted locus. Over $\mathbb{Q}$ the image of the counterexample is a thin set. But along the line where the group action has stabilizer $\mu_2$, the two preimages on the contracted locus are rational exactly when $-a$ is a square, so the Kummer condition of the positive-weight theory reappears on the stacky stratum. We propose a classification of equivariant polynomial maps by the signature of the weights, with weighted projective geometry as the positive case.

We study a bilinear singular integral operator obtained by taking rough averages of certain directional variants of the triangular Hilbert transform. This operator can be interpreted as the twisted paraproduct with a rough homogeneous kernel. Under a balance condition on the $L^q$ integrability of the kernel on the unit sphere, we establish a range of $L^{p_1} \times L^{p_2} \rightarrow L^p$ bounds for this operator. Our results are optimal within the known boundedness range for the twisted paraproduct with a smooth kernel.

Low-rank adaptation (LoRA) has become a widely used parameter-efficient fine-tuning method for large language models. Since different modules and layers may contribute unequally to downstream adaptation, allocating rank resources under a fixed parameter budget is an important problem for balancing efficiency, expressiveness, and generalization. Existing adaptive rank methods address this problem mainly through carefully designed importance scores constructed from gradient-derived sensitivity and uncertainty measures, without an explicit statistical interpretation. In this paper, we formulate LoRA rank allocation as a statistical hypothesis testing problem and propose StatLoRA, a statistical inference-based rank allocation method. StatLoRA associates each LoRA component with a test statistic and uses estimated p-values to determine which components should be retained or pruned under a prescribed rank budget. The proposed testing procedure is supported by our central limit theory for stochastic optimizer trajectories. In particular, we establish asymptotic normality for a broad class of commonly used optimizers in deep learning, including AdamW, and derive the corresponding asymptotic distributions for the proposed component scores used in hypothesis testing. We evaluate StatLoRA on LoRA fine-tuning of DeBERTaV3-base, BART-Large, and Qwen2.5-7B across natural language understanding, natural language generation, and question answering tasks. Experiments show that StatLoRA achieves comparable or better performance than vanilla LoRA, AdaLoRA, and IGU-LoRA under matched rank budgets. Sensitivity analyses and empirical diagnostics further support the stability of the proposed hypothesis-testing-based allocation rule and provide empirical evidence for the asymptotic theory of component scores.

This article concerns sonic-supersonic jet flows issuing from a two-dimensional straight nozzle described by the steady compressible Euler system under the van der Waals equation of state. The flow state is prescribed at the nozzle exit, while the surrounding medium is assumed to be either a vacuum or a static atmosphere with lower pressure. When the flow reaches the sonic state at the nozzle exit, the governing equations become degenerate hyperbolic and the resulting jet flow problem can be formulated as a degenerate free boundary problem. The analysis is further complicated by singular behavior near the endpoints of the nozzle exit. By employing characteristic decompositions and suitable a priori estimates, we establish the global existence of locally Lipschitz continuous sonic-supersonic jet flows expanding into a vacuum. Moreover, in the presence of a static atmosphere with pressure lower than that at the nozzle exit, we prove the local existence of sonic-supersonic jet flows near the nozzle exit.

This paper investigates the fundamental limits of MIMO integrated sensing and communications (ISAC) systems, specifically comparing sparse arrays (SAs) against conventional uniform linear arrays (ULAs). A unified theoretical analysis of ergodic channel capacity and the Cramér$\unicode{x2013}$Rao bound (CRB) for angle estimation is developed while accounting for the array geometry. Utilizing the framework of stochastic majorization, the study reveals that SAs consistently outperform ULAs by creating a more $\unicode{x201C}$uniform$\unicode{x201D}$ spatial eigenvalue distribution, which decorrelates the multipath environment and increases communication capacity. For sensing, the paper proves that the angle CRB is inversely proportional to the array's second-order central moment of antenna positions asymptotically, demonstrating that SAs achieve superior accuracy$\unicode{x2014}$improving by up to the square of the number of antennas$\unicode{x2014}$due to their increased physical aperture. These analyses and conclusions are demonstrated to be also valid for MIMO ISAC systems employed with the modern waveforms OTFS and OFDM, suggesting that spatial geometry, rather than waveform, is the primary driver of fundamental performance gains in the spatial dimension.

We study machine unlearning: the removal of memorized training data from a trained model. Specifically, we investigate the algorithmic complexity of certified unlearning from an optimization perspective. We formalize the goal of an unlearning algorithm as simultaneously achieving certified unlearning and optimization accuracy. Utilizing the notion of uniformly convex regularizers, we prove new bounds on the distance between initial and unlearned models using a novel substitute for generalization error. Thus we theoretically demonstrate that if the removed data is well-predicted by the unlearned model, the corresponding optimization problem is simple. Furthermore, we develop a new second-order unlearning algorithm with an anisotropic Gaussian mechanism and state-of-the-art global convergence. We prove fast rates for our method in achieving certified unlearning for linear models with quasi-self-concordant losses. As a direct application, our theory covers unlearning for logistic and exponential regressions and shows a provable benefit of utilizing second-order information compared to first-order unlearning methods.

This paper studies the two-way wiretap channel (TW-WC) with an external eavesdropper under strong one-sided (mixed) secrecy: only User~1's message is required to be secure from the eavesdropper, while no secrecy constraint is imposed on User~2's message. Secrecy is measured by the information leakage of User~1's message to the eavesdropper. Applying a non-adaptive construction and a one-sided reduction of the key-exchange construction, we obtain exponential error and leakage bounds for every fixed number of adaptive rounds. A quantitative one-time-pad argument propagates the leakage of the key used in the next round, and an explicit padding construction removes the initialization-rate loss for all sufficiently large blocklengths. We derive the corresponding strong mixed-secrecy achievable regions. The non-adaptive construction recovers the previously reported weak one-sided single-letter inner bound under the stronger secrecy criterion. The overall adaptive achievable region includes the non-adaptive construction as a fallback and also contains a key-exchange subregion that can be strictly larger.

A second order finite volume scheme rests on two local quantities: a gradient reconstructed in each cell, and a limiter which scales it down where the reconstruction would overshoot. Both are set by fixed formulas, and on coarse unstructured meshes a small network can supply better values. But a network is free to output anything, and the usual safeguard is a penalty in the training loss, which discourages inadmissible states without preventing them. We replace the penalty by a hard constraint. The network still sets both quantities, and every value it can produce lies inside safe bounds: its stencil weights cannot cancel a neighbour, and its limiter is capped by the local flow. The flux, the wall treatment and the time step are not learned and carry their own guarantees. Admissibility therefore holds for every value of the weights rather than as an outcome of training, and no negative density or pressure occurred in any computation reported here. Because the scheme is safe whatever the network does, we could ask what the network contributes. We test it on supersonic channel flow over an obstacle, including the forward facing step of Woodward and Colella. Learning lowers the error by 38% on an unseen geometry and 29% on an unseen obstacle topology, measured against the same scheme with the network switched off. The method aims at the accuracy of a fine mesh for the cost of a coarse one, and refining once improves the error fourfold while multiplying the run time by eight. Learning secures half of this improvement for a sixth of this time. All of this comes from one of the two quantities the network sets. The gradient reconstruction reproduces the full effect on its own, and the limiter accounts for about a tenth as much. This also explains why the gain fades beyond the Mach numbers the weights were trained on.

In a recent work [Adv. Math. 401 (2022), Paper No. 108318], a central limit theorem was established for the linear combinations of the iterates of a non-rotational inner function fixing the origin. In this paper, we prove the law of iterated logarithm (LIL) in the same setup, with a very mild condition on the coefficients. We also identify the full set of subsequential limit points at the LIL scale. Using the Aleksandrov–Clark decomposition and measure-preserving properties of the inner functions, one can construct a reverse martingale that is close to the linear combinations of inner functions. We prove the LIL for the partial sums of reverse martingale differences under a Feller-type assumption, which then transfers to the linear combinations of iterates of the inner functions.

Let $X$ be a Riemann surface of genus $g \geqslant 2$ and let $\sigma : X \to X$ be an antiholomorphic involution on $X$. Let $\mathcal{N}(r,d)$ be the moduli space of semistable vector bundles of rank $r$ and degree $d$ on $X$, with the induced real structure. Using a gauge-theoretic approach, we determine the number of connected components of the real locus of $\mathcal{N}(r,d)$ for general $r$ and $d$. We show in particular that, when the base curve has real points, quaternionic vector bundles can exist for even rank and degree but that the number of connected components of $\mathbb{R}\mathcal{N}(r,d)$ is still equal to that of $\mathbb{R}\mathrm{Pic}_d$. In contrast, when the base curve has empty real locus and $r$ and $d$ are not coprime, the number of connected components of $\mathbb{R}\mathcal{N}(r,d)$ can be smaller than that of $\mathbb{R}\mathrm{Pic}_d$. We then generalize these results to real loci of moduli spaces of Higgs bundles and apply them to the study of the topology of certain $(A,A,A)$ and $(A,B,A)$ branes in the associated hyperkähler quotient.

Let $G$ be a connected simple vertex-transitive graph on $n$ vertices with degree $d$, and let $G_p$ be the random spanning subgraph obtained by retaining each edge of $G$ independently with probability $p$. Put $q:=1-p$. Motivated by a conjecture of Bedert, Draganić, Müyesser, and Pavez-Signé on Hamilton cycles in percolated Cayley graphs, we establish the corresponding matching and $2$-factor statements uniformly over the larger class of all connected vertex-transitive host graphs. For every $A>0$, if $q^d\le n^{-(5A+250)},$ then, with probability at least $1-n^{-A}$, the graph $G_p$ has a perfect matching when $n$ is even and is factor-critical when $n$ is odd. Separately, if $0<\epsilon<1$ and $ \epsilon^2pd\ge64(A+6)\log(2n), $ then, with probability at least $1-n^{-A}$, the graph $G_p$ contains at least \[ \left\lfloor\frac{(1-\epsilon)pd}{2}\right\rfloor \] pairwise edge-disjoint spanning $2$-factors. Moreover, if $pd/\log n\to\infty$, then \[ \nu_2(G_p)=(1+o(1))\frac{pd}{2} \] with high probability, which is asymptotically optimal, where $\nu_2(G)$ is the maximum number of pairwise edge-disjoint spanning 2-factors in $G$. Thus logarithmic-order percolation already forces these two factor-theoretic consequences of Hamiltonicity beyond the Cayley setting.

In this work, we propose high-order boundary-preserving numerical schemes for the strong approximation for some scalar stochastic differential equations with invariant domains being open and bounded intervals. The proposed methods involve using the Lamperti transform to map the SDE to another SDE with additive noise with a trivial invariant domain. Then, by imposing regularity assumptions on the original coefficient functions, we can guarantee that the drift coefficient function of the transformed SDE is regular, and known high-order schemes can be used to achieve the desired convergence order. We confirm the theoretical results with numerical experiments.

We study Gaussian averaging as a smooth surrogate for quantized neural models. Under bounded local oscillation, we derive a local dimension-dependent bound on |f-g|, linking Gaussian smoothing to the stability analysis of discontinuous networks. We compute closed-form Gaussian averages of the rectified linear unit (ReLU) and sign activation functions, and illustrate the mechanism on a high-dimensional binary perceptron, where layer-preactivation aggregation under an explicit quantization-noise surrogate yields the Gaussian envelope used in inference-side smoothing and training-side smooth surrogate gradients.

To derive dimension-free convergence rates of empirical measures for Markov processes on a Banach space, we adopt the sliced Wasserstein distance (SW distance) induced by a probability measure with full support on the unit ball in the dual space. This distance refers to the convergence in finite dimensions, and is topologically equivalent to the classical Wasserstein distance when the Banach space is finite-dimensional. With the SW distance we derive dimension-free convergence rates for the empirical measures of ergodic Markov processes on Banach space, which can be sharp as illustrated by concrete examples. The study provides an efficient way to simulate infinite-dimensional distributions using sample trajectories of Markov processes, avoiding $``$the curse of dimensionality" in the classical Wasserstein distance. The main results apply to a broad class of infinite-dimensional models, and are illustrated by partially dissipative SPDEs in the end of the paper.

In Proposition II of his manuscript, Galois writes the well-known remark: Il y a quelque chose à compléter dans cette démonstration. Je n'ai pas le temps. Although Galois did not complete the proof, it is possible to reconstruct the essential content of Proposition II and to supply the missing arguments. As usual, V denotes the linear form ax1+bx2+cx3+..., where x1,x2,x3,... are distinct roots of a separable polynomial. Let L be the corresponding splitting field over a ground field of characteristic zero. On the one hand, Proposition II concerns the factorization of the minimal polynomial g(x) of V over an intermediate field M contained in L; on the other hand, it concerns the factorization of the same polynomial into irreducible factors whose coefficients belong to the intermediate fields conjugate to M. It is precisely this latter aspect that constitutes the central theme of Proposition II. The notions of 'groupe de permutations' and 'groupe de substitutions' are of fundamental importance in the Mémoire. We provide a characterization of these notions in modern terminology. Associated with every 'groupe de permutations' is a polynomial whose coefficients are invariant under the corresponding 'groupe de substitutions'. Moreover, a 'groupe de permutations' determines a partition of the Galois group, making it possible to factor the minimal polynomial g(x) into factors whose coefficients belong to the intermediate fields corresponding to the associated 'groupes de substitutions'. Finally, we prove that the substitutions of the Galois group are field automorphisms of the splitting field L. This establishes the connection between Galois' original formulation and the modern formulation of Galois theory.

For any linear real reductive group or metapletic group, this article gives a geometric classification of special unipotent representations in the weak Arthur/Adams-Barbasch-Vogan (ABV) packets attached to quasi-distinguished nilpotent orbits in the Langlands or metaplectic dual Lie algebra in terms of their associated cycles. We provide a uniform construction of the Harish-Chandra modules of these representations via deformation quantization of admissible vector bundles over certain Lagrangian subvarieties of the affinizations of the universal covers of special nilpotent orbits in question, which aligns with the coadjoint orbit method philosophy of Kirillov, Kostant, and Vogan. As consequences, all such Harish-Chandra modules have irreducible associated cycles, and are unitarizable by results from the theory of mixed Hodge modules. Conjecturally, the unitarity of all ABV packets can be reduced to the case when the dual orbits are distinguished. Our approach highlights the application of Lusztig's conjecture on the geometry of special pieces, which has been proven for classical Lie algebras by Kraft and Procesi, and for all cases by Juteau, Levy, Sommers and the author.

Let $G$ be a graph with chromatic number $\chi(G)$, clique number $\omega(G)$ and zero forcing number $Z(G)$. We establish new lower bounds on $Z(G)$ in terms of induced triangle-free subgraphs. In particular, we show that if a graph $G$ contains an induced triangle-free subgraph $H$ with minimum degree $\delta(H) \ge 3$, then $Z(G)\ge\delta(H)+1$. As consequences, we prove that every triangle-free graph satisfies $\chi(G)\le\max\{3,Z(G)\}$ and obtain an application to planar graphs. Moreover, we prove that $\chi(G)\le \frac{Z(G)}{2}+2$ for every triangle-free graph.

We establish explicit polynomial bounds on the stability rate and radius of the lamplighter group. This provides the first example of an infinitely presented group with an explicit upper bound on the stability radius, answering a question of the first author, Levit and Vigdorovich. Our approach is based on new dynamical notions, including the analysis of approximately invariant measures. We establish an effective continuous tower decomposition procedure for approximately invariant measures with presence of periodic points. To achieve polynomial bounds we appeal to recent techniques from descriptive combinatorics and distributed LOCAL algorithms.

This paper studies a coupled nonsmooth dynamical system in which a semilinear evolution equation is coupled with an implicit algebraic law governed by the normal cone to a time-dependent convex set. The main difficulty is that the algebraic variable is not given by an explicit feedback, but must be recovered from a state-dependent normal cone relation. Using a transformed variable, we recast each frozen algebraic law as a variational inequality over a convex set. Under a strongly pseudomonotone, Lipschitz continuous hypothesis, this frozen problem has a unique algebraic response and admits a sensitivity estimate for the state-to-control map without a projection-contraction argument. We also give verifiable sufficient conditions for this hypothesis, including a weighted strongly monotone construction and a standard Lipschitz-smallness condition, and point out that the framework allows strongly pseudomonotone nonmonotone frozen operators. The coupled system is then reduced to a semilinear evolution equation with a single state variable. By combining $C_0$-semigroup estimates with a Bielecki fixed-point argument, we prove global existence and uniqueness of mild solution pairs on finite time intervals and establish Hadamard well-posedness through explicit continuous-dependence estimates. We further derive an incremental exponential stability criterion in the dissipative semigroup regime and prove continuity of the parameter-to-solution map with respect to the initial datum and an external parameter. A reduced contact-mechanics example illustrates the variational inequality formulation and an explicit projection residual that can be used as a starting point for projection- and Newton-type inner solvers after discretization.

Tamari lattices and weak Bruhat order are classical posets on the vertices of associahedra and permutohedra. These posets are particular instances of the flip order on the vertices of graph associahedra. We present here a generalisation of the flip order to all the faces of nestohedra, encoded in terms of hypergraphs. For right-filled hypergraphs, we characterise our order in terms of generalised inversions. As an application, we describe shuffle products on faces of suitable nestohedra as sums over intervals in this order.