Papers

Foundation models for ground states in spin-1/2 systems are a promising method for problems ranging from quantum chemistry to identifying new phase diagrams. Nearly all such models are currently pure-states that condition on the Hamiltonian's parameters, whose Monte Carlo samples give energy estimates according to the variational principle. In this contribution, we show that this representation is topologically obstructed. For any gapped Hamiltonian family whose ground-state bundle is non-trivial, every continuous normalized state-vector model has zero fidelity with the ground state at some parameter value in the Hamiltonian family. For that value, the energy is at least one spectral gap, $\Delta$, with an $O(\Delta)$ gap in an open-neighbourhood of that point. We show that this is a sufficient no-go also in the case of degenerate ground-state manifolds, time dynamics, and periodic systems with mixed space-time topology, demonstrating these obstructions on one- and two-qubit systems. We discuss how this causes a spike in the fidelity susceptibility, giving a numerical signature of a phase-transition where there is none. We then show that operator-valued models canonically avoid these obstructions and preserve topological information, implying a structural necessity in representation for foundation neural quantum states.

We consider $\nabla\varphi$ height functions with even and convex interaction energy $W$ on the lattice $\mathbb{Z}^d$, which are restricted to take values in the set $\{-S, \ldots, S\}$ for some integer $S \ge 1$. We study the effect of entropic repulsion, which tends to push the spin values to the middle layer. We prove that the model has a unique Gibbs measure, with exponential decay of correlations, in the cases: (a) Dimension $d=2$ at all temperatures. (b) Dimensions $d\ge3$ at all temperatures, for a wide class of $W$ with non-increasing second derivative, including the family $W(x)=|x|^p$ for $p\in[1,2]$. (c) Dimensions $d\ge 3$ at both low and high temperatures, $T\in (0,\frac{W(1)d}{4(\ln d+\ln 8)})\cup(d W(2S),\infty]$, with the normalization $W(0)=0$. At low temperatures, our proof provides an alternative to Pirogov–Sinai methods. Conversely, we exhibit a class of even and convex interaction energies $W$ which, in high dimensions and suitable temperature regimes, have multiple Gibbs measures. Though uniqueness may fail, we show that the magnetization of every Gibbs measure lies in $(-\frac{1}{2},\frac{1}{2})$. This implies the delocalization of the model restricted to take values in $\{0,1,\ldots\}$ (i.e., conditioned to lie above a floor) for all dimensions, any even and convex $W$, and all temperatures. Our methods extend to additional setups: We prove that height functions taking values in the real interval $[-1,1]$ always have a unique Gibbs measure, a result previously proved only for the quadratic interaction. For height functions taking values in $\{-S+\frac{1}{2}, \ldots, S-\frac{1}{2}\}$, $S \ge1 $ integer, we prove that the magnetization of every Gibbs measure lies in $(-1,1)$. The special case $W(x)=x^2$ of our results addresses questions left open in the work of Bricmont–El Mellouki–Fröhlich (1986).

We study the one-dimensional time-independent discrete Dirac equation on graphs with discrete edges. Both continuous and discrete formulations of the graphs are considered, with particular focus on star graphs with three edges as a simple example. Exact solutions for eigenstates and spectra are derived and compared with the continuum case, showing good agreement in tabulated results. This approach offers a systematic framework for analyzing relativistic quantum transport on branched structures relevant to nanostructures and quantum device design.

As is already known, there is a correspondence between circle patterns and the dimer model, which is invariant under Euclidean transformations. We establish a new correspondence that is invariant under the group of Möbius transformations PO(3,1), which is the natural symmetry group of circle patterns. We show that Miquel dynamics preserve the partition function, and that convexity of the t-embedding guarantees positivity of the face weights. We also show that the correspondence has a somewhat hidden symmetry group PO(3,3), which relates to the Lorentz lift of the t-embedding. Finally, we consider isoradial graphs, Doyle spirals, the once-punctured torus and isothermic circle patterns.

Heterogeneous exascale systems are reshaping computational fluid dynamics, yet rewriting mature high-order solvers for accelerators can fragment code bases and weaken reproducibility. We present a single-source OpenMP target modernisation of UCNS3D, a high-order unstructured finite-volume solver for compressible flows. The common CPU/GPU implementation preserves its numerical formulation, Fortran code base, and workflows. Rather than porting isolated kernels, we enable the complete explicit time-advancement path: high-order reconstruction, gradient evaluation, inviscid and viscous fluxes, boundary treatment, halo exchange, and solution update. The implementation uses persistent target data regions, flat run-time arrays, compile-time sizing of temporary storage, explicit local algebra, target-callable routines, and GPU-aware MPI with device-resident buffers. These choices address the irregular stencils, complex data structures, and substantial temporary storage of high-order unstructured CFD. Verification using the compressible Taylor-Green vortex shows CPU and GPU dissipation histories agreeing to machine precision on LUMI and with published reference data. A single-node run on the ARCHER2 GPU platform provides an independent portability check. End-to-end performance is evaluated on fully populated nodes using Taylor-Green vortex, LM1021 sonic-boom, and NASA high-lift CRM benchmarks. Relative to the previous production implementation, the refactored CPU path is 1.27-1.67 times faster, while GPU offload delivers same-node speed-ups of 2.71-4.05 and 84-101 percent strong-scaling efficiency. The results show that standards-based OpenMP provides portable, production-scale acceleration without sacrificing numerical fidelity, CPU performance, or software sustainability.

Genuine multipartite Bell nonlocality (GMNL) aims at capturing correlations between many distant observers that are globally Bell nonlocal. A commonly used definition for GMNL is based on biseparable models constructed from non-signalling correlations. Here we uncover an effect of superactivation of GMNL in this framework. Specifically, by locally wiring together two copies of a biseparable correlation, i.e. not GMNL, one can obtain another correlation that is GMNL. We show that this is possible for any number of parties, and for quantum-realizable correlations. Finally, we show that superactivation of GMNL is also possible at the single-copy level, via a catalytic protocol involving only local wirings. These results question the operational meaning of this definition of GMNL.

For a symmetric weight $w(x)$ on a finite discrete lattice and its Christoffel transforms $x^{2}w(x),x^{2}(x^{2}w(x)),\dots$, we prove that, at each Christoffel step, the conjugated projection associated with the transformed Christoffel–Darboux kernel differs from the original orthogonal projection by a rank-one operator on the subspace of functions vanishing at the origin. This provides a general mechanism for transferring local asymptotic results from an orthogonal polynomial ensemble to its Christoffel-transformed counterpart as the lattice size tends to infinity. As a main application, we study local fluctuations of random Young diagrams arising from skew $(\mathrm{Sp}_{2n},\mathrm{Sp}_{2k})$ Howe duality. The corresponding particle ensemble is obtained from the Krawtchouk orthogonal polynomial ensemble on a quadratic lattice by a Christoffel transform. We identify four asymptotic regimes of local fluctuations in the limit $n,k\to\infty$ with $n/k\to c\in(0,\infty)$. Besides the universal bulk fluctuations governed by the discrete sine kernel and universal Airy fluctuations at the right edge of the limit shape, we obtain the discrete Hermite kernel in the critical regime $(k-n)/\sqrt{n+k}\longrightarrow r\in\mathbb{R}$, and the discrete hard-wall sine kernel at the left corner.

The Julia set is defined by the closure of repelling periodic points in chaotic systems. Why does this structure not appear in integrable systems? In this paper, we address this question by demonstrating the existence of the closure of divergences of the periodic equations, which we designate as the "anti-Julia set." We also call the sets before taking the closures the pre-Julia and pre-anti-Julia sets, respectively. We illustrate the transition mechanism by considering a complex map that interpolates between integrable and non-integrable dynamics, by introducing a real deformation parameter $a$. For $0<a\le 1/2$, we show that the Julia set and the anti-Julia set coincide in the complex plane, although the pre-Julia and pre-anti-Julia sets remain completely disjoint. At the integrable limit $a\to 0$, these two dual structures undergo a critical collision and subsequent annihilation, reminiscent of a cosmological Big Crunch. When $1/2<a<1$, on the other hand, the boundary of the pre-anti-Julia set includes the Julia set. We analytically characterize these phenomena, focusing in particular on the asymptotic behavior of the pre-anti-Julia set as it approaches the integrable limit, and provide numerical visualizations that elucidate the underlying mechanisms of this Big Crunch phenomenon.

This paper derives a multiscale Bakry-Émery criterion for an upper bound on the Araki relative entropy of perturbations of an equilibrium state in relativistic QFT. Using stochastic positivity, affiliated perturbations of the relativistic bosonic field can be put in correspondence with perturbations of the massive Euclidean free field. This gives a Feynman-Kac formula, which in turns is used to derive a probabilistic expression for the Araki relative entropy. The description of the massive Euclidean free field as an abstract Wiener space provides the infinite-dimensional framework to derive a Polchinski equation. Extending the methods of Bauerschmidt, Bodineau, and Dagallier to this context yields a probabilistic bound on the Araki relative entropy, which can be formulated entirely in operator-algebraic terms. Using the criterion, it is shown that the Lorentzian massive Sine-Gordon model in the ultraviolet finite regime satisfies the relative entropy bound.

We study disorder-free localization in a spin-coupled quantum walk on the hypercube within the quantum Bernoulli noises framework. We derive a general criterion for disorder-free localization: it occurs if and only if some spectral subspace of a reduced unitary evolution contains a vector whose position marginal probability is non-uniform. For the Grover walk with distance-dependent coupling \(\phi_\sigma=|\sigma|\pi/(n+1)\) in dimension \(d=4\), we explicitly construct a non-uniform fixed point in the all \(+1\) spin configuration. This implies that disorder-free localization occurs in our model.

We study finite-rank multiplicative deformations of rotationally invariant non-Hermitian random matrices. More precisely, we consider models of the form $\mathbf{A}(\mathbf{I}+\mathbf{T})$, where $\mathbf{A}$ is a large rotationally invariant non-Hermitian random matrix, $\mathbf{T}$ is a finite-rank normal perturbation, and $\mathbf{I}$ denotes the identity matrix. We characterize the emergence of outlier eigenvalues, their fluctuations, and the associated eigenvector overlaps. Our results provide a multiplicative non-Hermitian counterpart to the classical Baik–Ben Arous–Péché framework.

We present a data-driven approach to identifying linear index-1 differential-algebraic pH systems (pH-DAEs) based on input-output measurements. In comparison to the identification of port-Hamiltonian (pH) systems, the algebraic constraint and the index condition pose additional challenges. First, we establish a structure-preserving formulation of the considered pH-DAE class and derive an implicit midpoint discretization that preserves the algebraic constraints and discrete dissipation inequality. We formulate the identification problem as a regularized least-squares minimization problem subject to the pH-DAE dynamics. Exploiting the index-1 structure, we reduce the constrained problem to an unconstrained optimization problem over the system parameters while preserving the port-Hamiltonian structure. Next, we derive an adjoint-based formulation to efficiently evaluate the gradient of the resulting reduced cost functional. This enables us to use gradient-based optimization methods for parameter estimation. Under suitable assumptions on the admissible parameter set, the existence of a minimizer is established. Numerical experiments demonstrate that the proposed approach can identify surrogate pH-DAE systems that accurately reproduce the input-output behavior of reference systems. Further investigations show the approach's potential for identifying reduced-order surrogate models. Cross-validation with independent input signals confirms the predictive capability of the identified models.

We generalize the concept of Bott connection to the de Rham mapping cone situation. As the main results, when the given closed smooth manifold is foliated, we have the mapping cone version of the Bott vanishing theorem and an adiabatic limit of a family of mapping cone connections on the orthogonal complement of the foliation. The second result shows the necessity of a refined metric if we want to see more interactions between geometry and topology in the mapping cone situation.

We give a new proof of the existence of absolutely continuous spectrum for the weakly disordered Anderson model on the Bethe lattice. The argument follows a general cyclicity criterion for Anderson-type Hamiltonians and reduces the problem to showing that the spectral measures of two independent copies of the rooted tree are not mutually singular. This is detected by their Hellinger affinity, and the required weak-disorder estimate follows from a short harmonic and compactness argument. We also formulate an analogous quantitative cyclicity criterion for the Anderson model on $\mathbb{Z}^d$, expressed through the Schur complement of the Poisson transform of a $2\times2$ matrix spectral measure.

In the recent work of Barbier, Ko, and the second present author on sublinear-rank symmetric matrix factorization [Math. Stat. Learn. 9 (2026), 1-68], a key result is that, in the Bayes-optimal setting, the large-size limit of the free entropy of the finite-rank spiked Wigner model is the same as in the rank-one case when the signal has centered i.i.d. entries. In this paper, we show that this rank-one equivalence result extends to the case of finite-rank, even-order, symmetric tensor factorization. Moreover, we give a natural reformulation of a hypothesis that was stated in the aforementioned work to be necessary for this result. As in the matrix case, we use information-theoretic identities and replica symmetry to reduce a known multi-dimensional variational formula for the limiting free entropy to its one-dimensional analog. The novelty stems from the fact that said formula involves a replica symmetric potential containing Hadamard (entrywise) powers, rather than squares, of the matrix-valued variational parameter, so the eigenvalue-based approach used in the matrix case must be adjusted.

We determine the spectrum of the predual generator of an irreducible Gaussian quantum Markov semigroup on trace-class operators, the natural space for density operators. The drift matrix governs the linear part of the Gaussian evolution and organizes the spectral classification. When the linear flow generated by this matrix is not periodic, the spectrum is the closed left half-plane. If all drift eigenvalues have negative real part, every interior point is an eigenvalue. There are no trace-class eigenvectors if some drift eigenvalue has positive real part or all have zero real part. For periodic flow, the spectrum consists of unions of horizontal half-lines or filled parabolic regions. Examples show that irreducibility cannot in general be omitted. We apply the results to oscillators, free particles, and boson chains.

Building on the dagger Bergner and dagger Joyal models constructed in Part I, we develop simplicial-space models for dagger $(\infty,1)$-categories. We first construct a homotopically constant resolution model and a model structure for unitarily complete dagger Segal spaces. A change-of-index adjunction compares the two presentations of dagger simplicial spaces. We then prove a dagger analogue of the Joyal–Tierney Quillen equivalence. Together with Part I, these results give a chain of Quillen equivalences between the dagger Bergner, dagger Joyal, and dagger Rezk models.

This work presents a generalization of the commutative factorization framework for a broad class of second-order nonlinear ordinary differential equations. The main objective was to extend the commutative factorization procedure in order to use the Riccati-Bernoulli equation and the Bäcklund transformation. This provides a systematic route for constructing both particular and general solutions of the original second-order nonlinear equation. The theoretical development is illustrated through several nonlinear models of physical and mathematical interest, including equations arising in classical mechanics and nonlinear oscillatory systems. The results demonstrate that commutative factorization constitutes an effective analytical tool for solving nonlinear differential equations.

We solve in closed form a one-dimensional relativistic system: two masses interacting only through elastic collisions with a massless mediator bouncing between them. Momenta, times, and positions are hyperbolic functions of the collision index. The mediator energy, interpreted as the pair's effective potential, obeys an exact discrete Coulomb law, $V\propto 1/r$, with a Lorentz-invariant action as coupling. A massive Newtonian mediator instead transmits a $1/r^{3}$ force; one adiabatic invariant traces both laws to the mediator's dispersion relation. Continued to negative mediator energy, the closed forms turn trigonometric, binding a one-dimensional mechanical analog of the Coulomb atom.

Based on the representation theory of the $q$-CCR algebra, we develop a representation-theoretic construction of the $q$-deformed polyanalytic Fock spaces, their reproducing kernels, and the associated $q$-deformed polyanalytic Bargmann transforms. The associated determinantal point processes provide $q$-deformations of the pure and full polyanalytic Ginibre ensembles. Moreover, we determine their limiting density and establish central limit theorems for the fluctuations of their linear statistics as the number of particles tends to infinity and the deformation parameter $q$ tends to 1 simultaneously. Remarkably, although the $q$-deformation changes the macroscopic density and the size of the droplet, the limiting covariance structures coincide, after spatial rescaling, with those of the classical polyanalytic Ginibre ensembles.

Heterogeneous operators often organize response through hidden barriers, wells, and weakly communicating compartments that are not faithfully described by Euclidean distance or raw graph connectivity alone. We study this structure through the localization landscape, obtained from a single source solve, and the associated effective potential $W_{\mathrm{eff}} = 1/u$. Starting from both discrete Schrödinger-type operators and heterogeneous reaction-diffusion operators, we show how $W_{\mathrm{eff}}$ induces effective wells, basin partitions, and Agmon-type neighborhoods that serve as an operator-aware geometry for localization and transport. This geometry explains the spatial support of low-energy eigenmodes, produces adaptive receptive fields on a fixed graph, sharpens locality in attention-like interactions, and yields bottleneck-sensitive modulation of diffusion. We also include a finite-element reaction-diffusion example with strongly compartmentalized coefficients, demonstrating that the same framework extends beyond simple grid operators to heterogeneous continuum systems. The results support a precise conclusion: localization-landscape geometry is a strong inductive bias when relevance is controlled by confinement, barriers, and low-energy accessibility, but its benefits are task-dependent rather than universal.

We prove the future-completeness conjecture of Gödeke and Rendall for expanding spatially homogeneous vacuum spacetimes in four spatial dimensions. More generally, an expansion-energy estimate gives future timelike and null geodesic completeness for homogeneous Einstein equations coupled to maps into complete Riemannian targets, through nine spatial dimensions for vanishing potential. In arbitrary dimension we obtain an affine-length criterion which includes positive potential floors and potentials that decay sufficiently slowly along the scalar trajectory. For future-global homogeneous solutions, the critical nine-dimensional bound also holds for a general stress tensor in an explicit normal energy-condition window, including a broad range of perfect fluids. At the opposite time end, we establish a quotient-stable $C^0$-inextendibility criterion for two-step nilpotent cosmologies. Its proof combines boundary localization with an optimal, intrinsically defined allocation of the central correction in timelike homotopies, and continues to apply when the diameter of compact spatial slices collapses. Exact Heisenberg solutions realize both conclusions. On every fixed compact Bianchi~II quotient, the expanding invariant vacuum data split into an open dense set with globally $C^0$-inextendible developments and a codimension-two locally rotationally symmetric locus with analytic compact Cauchy horizons. We determine the chronology-violating region of the exceptional extensions and quantify the exponentially small proper-time scale on which transverse anisotropy replaces the horizon by a singular end.

Three-dimensional bioheat simulation aims to predict transient temperature distributions in biological tissue and is commonly modeled using the Pennes bioheat equation, which combines thermal diffusion, perfusion-mediated heat loss, and external heat generation. In this study, we consider a controlled 3D Pennes bioheat simulation under a localized heat-source condition inspired by microwave ablation (MWA). To evaluate neural approximation performance, we compare three neural partial differential equation (PDE) solvers under the same controlled simulation: a spatial Fourier-feature physics-informed neural network (PINN), a generic PINNMamba temporal subsequence model, and Pennes Physics-Informed Mamba (PPIM). PPIM builds on the temporal subsequence model by incorporating conditioned heat-source input and Pennes-aware state-space model (SSM) decay initialization. All three neural models are trained under the same conditions with the same Pennes residual, and an explicit finite-difference method (FDM) solution is used only as the numerical reference. In a representative 600~s run, PPIM achieved the lowest MAE, relative $L_1$ error, and relative $L_2$ error among the evaluated neural solvers. Error maps further showed that the remaining PPIM errors were more concentrated near the heat-source region than across the rest of the domain. These results indicate that PPIM is effective for approximating the FDM reference final temperature field in this controlled simulation. The source code is available at https://github.com/muvYun/PPIM.

We prove that, for all $\beta>0$, the Dobrushin-Lanford-Ruelle (DLR) equations for the unit-intensity $\mathsf{Sine}_\beta$ point process have a unique stationary solution under a particular finite electric energy condition. Uniqueness fails if this condition is dropped. This answers a question of Dereudre-Hardy-Leblé-Maïda and gives a canonical statistical physics characterisation of $\mathsf{Sine}_\beta$. The main technical ingredient is a relative entropy estimate which allows us to compare the conditional distribution in a finite box of a DLR solution to a corresponding circular $\beta$ ensemble.

This paper presents a study of the Weber-Schafheitlin integral and its connections to random walk theory. The Weber-Schafheitlin integral, defined as the improper integral of a product of Bessel functions, arises naturally in the evaluation of probability distributions for multidimensional random walks. This work bridges pure special function theory with applied probability.

We study Maxwell-Stefan diffusion with additive friction coefficients $f_{ij}=g_i+g_j$. In mass fractions, the system isolates the constrained pair-friction block; in mole fractions, it is the classical ideal isothermal/isobaric Maxwell-Stefan system at constant total molar concentration. Additivity makes the constrained pair-friction dissipation species-diagonal; conversely, species-diagonality on one interior barycentric constraint space forces a pair-sum representation. At operator level, the positive constrained relaxation operator is a scalar shift of a compression of $G=diag[g_1,\ldots,g_N]$. Its scalar resolvent yields both an explicit constrained inverse and interlacing spectral roots, which form global real-analytic coordinates on the open simplex and whose differentials are left eigen-covectors. In root coordinates the principal part is diagonal, no self-square gradient term occurs, and scalar comparison yields invariant rectangles and separation from the simplex boundary. For regularity we introduce entropy-stabilized one-sided multi-EPD truncations: Euler-Poisson-Darboux entropies cancel mixed quadratic production, while for $N\ge4$ a truncation-weighted mixing-entropy correction supplies transverse coercivity. Caccioppoli and logarithmic estimates, shrinking, and critical mass yield Hölder continuity up to the Neumann boundary. The mixing entropy also symmetrizes the moment system; frozen conormal estimates give spatial Lipschitz bounds. Together with time Hölder control and short-interval maximal regularity, this yields global strong solvability on bounded $C^{2+\alpha}$ domains with $0<\alpha<1$, for each $N\ge2$, $d\ge2$, and $p>d+2$, for all uniformly positive, compatible initial concentrations in the natural trace class. Solutions become classical for positive times and converge exponentially to equilibrium in relative entropy, $L^2$, and $C^1(\bar\Omega)$.

Convex optimization on polytopes arises in many areas of science. When the polytope is given implicitly or has exponentially many vertices and facets, standard methods may not apply or be ineffective. This is the case for moment polytopes, such as the entanglement polytopes, which play a foundational role in quantum information and algebraic complexity. They give rise to important entanglement measures and tensor parameters such as the quantum functionals, yet general effective methods for computing these quantities have been elusive. In this paper we address this challenge. We develop a first-order framework called Hadamard mirror descent to optimize suitable convex functions over moment polytopes and, more generally, the gradient sets of geodesically convex functions. It operates locally and does not rely on any explicit description of the polytope. Our framework extends mirror descent, an effective and widely used framework for convex optimization, from the Euclidean setting to Hadamard manifolds, and is motivated by a recent work by Hirai, which we interpret as a Hadamard version of mirror flow. Applying the framework to entanglement polytopes yields the first efficient first-order algorithms to compute the quantum functionals, the symmetric quantum functional, and the G-stable ranks, as well as a new direct algorithm for the non-commutative rank.

The first goal of this paper is to give a new family of special function related to the Debye functions. By using these function and generating function method, we derive many novel formulas and relations involving the Debye functions, multiple Hurwitz Lerch zeta function which interpolate the Apostol-Bernoulli numbers of higher order at negative integers. Proof method of theorems in this paper are different from those of theory of analytic numbers. The second goal of this paper is to show that special values and integral presentations of this function are closely related to unification of the Debye functions, the moment generating function for the negative binom distribution, the generating functions for the Apostol-type numbers of higher-order, and the Frobenius-Euler numbers, the Bernoulli numbers, and the Stirling numbers etc. Moreover, we give recurrence relations, Maclaurin's series expansion, and approximation formula for unification of the Debye functions. Finally, we give further remarks, observations and applications in mathematical physics on the results of this paper.

A time fractional extension of the Hunter Saxton equation is examined, in which the temporal derivative of u_x is replaced by a Caputo derivative of order 0 < alpha <= 1. This modification introduces memory effects into a model traditionally associated with director field dynamics in nematic liquid crystals. By employing a fractional separation of variables strategy together with the exponential spatial profile phi(x) = exp(b+x), which cancels the nonlinear structure exactly, the governing nonlinear partial differential equation is reduced to a fractional relaxation ODE, whose closed form solution is the one-parameter Mittag Leffler function. The exact analytic solution is derived algebraically from the separation process. This appears to be the first exact closed form solution of the Caputo time fractional Hunter Saxton equation. The result is validated symbolically using the MathHandbook computer-algebra system. Quantitative analysis in both the fractional (0 < alpha < 1) and classical (alpha -> 1) limits demonstrates how fractional-order memory slows temporal relaxation relative to the exponential baseline. The solution provides a reliable benchmark for numerical schemes and clarifies how fractional dynamics influence nonlinear wave propagation in orientationally ordered fluids.

We consider a system of $N$ interacting bosons in a rotating harmonic trap in $\mathbb{R}^2$, where the two-body interaction is attractive and scaled as $N^{2\alpha}U(N^\alpha x)$ with $0<\alpha<1/12$, and the three-body interaction is repulsive and scaled as $N^{4\beta}W(N^\beta x,N^\beta y)$ with $0<\beta<1/24$. In the mean-field limit, the ground state energy is effectively described by a rotating cubic-quintic nonlinear Schrödinger functional. We analyze the collapse regime where the two-body coupling $a$ approaches the critical value $a_*$ and the three-body coupling $b$ tends to zero. The NLS ground states blow up with a universal profile given by the optimizer of the Gagliardo-Nirenberg inequality, and the energy satisfies $E^{\mathrm{NLS}} = (1-\zeta/4+o(1))\mathcal{Q}{\rm{pot}}\ell_n^2$ with $\zeta\geqslant0$ determined by the relative rates of $a_n\to a_*$ and $b_n\searrow0$. From the many-body theory, we rigorously justify this effective description: the quantum ground state energy converges to the NLS energy with the same asymptotic expansion in the collapse regime, and the many-body ground states exhibit complete Bose-Einstein condensation onto the universal blow-up profile.