ALT I investigate the average and variance of species abundance distributions for fixed-point solutions to generalized Lotka-Volterra ecosystems. Two different analytical methods are constructed. First, I build up on the perturbative Random Matrix Theory approach developed in my earlier work [Phys. Rev. E. 113, L062202 (2026)], and formalize this method diagrammatically. The abundance average is given by a sum of planar diagrams with non-crossing vertex pairings which can be encoded as Dyck paths. This allows me to express the combinatorial factor multiplying perturbative terms at each order as Catalan and Narayana numbers, following which the abundance average is given by the latter's generating functions. In this way, the infinite perturbation expansion is exactly resummed in a closed-form formula. I have not been able to find a similar resummation for the abundance variance, however diagrammatic Feynman rules are expressed, following which the perturbative series can be calculated to an

ALT We announce the solution of constructing perturbative quantum BCOV invariants on arbitrary Calabi-Yau manifolds. In this note, we outline the construction and present an explicit solution. The quantization includes all descendant fields and all loop orders, and expresses the quantum BCOV interaction by an explicit integration over the moduli space of marked Riemann surfaces. This canonical solution provides a mathematical construction of all higher genus BCOV invariants on compact Calabi-Yau manifolds, which are conjecturally equivalent to all higher genus Gromov-Witten invariants on the mirror.

ALT We prove a version of topological recursion for the correlation differentials of colored HOMFLY-PT polynomials of torus knots on the corresponding augmentation curves, following the proposal of Gu, Jockers, Klemm, and Soroush. To this end, we explicitly construct the spectral curve from the so-called auxiliary curve of Brini, Eynard, and Mariño and prove that it coincides with the augmentation curve for every torus knot with coprime (r,s). Finally, our version of topological recursion proves the higher-genus closed-string prediction of the Aganagic–Vafa conjecture for all torus knot augmentation curves.

ALT We study dynamical conservation laws and the propagation of thermodynamic structure in translation-invariant lattice quantum spin systems. Using Lieb-Robinson bounds [CMP 28 (1972), 251-257] in arbitrary dimension, and Araki's bounds [CMP 14 (1969), 120-157] in dimension one, we prove conservation of the specific entropy, specific energy, and specific relative entropy under suitable assumptions on the interactions. The transported local Hamiltonians admit surface energies that vanish per unit volume in the van Hove limit, and local observables remain a core for the transported generators at every finite time. Weak Gibbsianity is preserved for sufficiently small times in arbitrary dimension and for all times in dimension one. These results provide the analytical foundation for the structural theory of approach to equilibrium developed in a companion paper, and establish partial results toward the conjectures formulated there.

ALT We continue the development, started in [AHP 25 (2023), 715-749], of a structural theory of approach to equilibrium in translation-invariant quantum spin systems. Dynamical Steady States (DSS) are defined as w*-limit points of the time-averaged evolution of an initial state. We show that if a DSS is an equilibrium state of a sufficiently regular interaction, then the specific energy of this interaction is necessarily a constant of the motion. Two distinct mechanisms lead to this conclusion, one based on dynamical surface energies and the other on conservation of specific relative entropy. We conjecture that these properties hold for a natural class of interactions. For a DSS admitting a physical equilibrium description, the associated equilibrium states are characterized by a dynamical Gibbs variational principle: they are precisely the minimizers of the specific relative entropy from the initial state over the physical equilibrium states. The achieved minimum equals the increase of sp

ALT We study the two lowest Bloch bands of the infinite Chern band model in the semiclassical limit. When the Berry curvature parameter is an odd multiple of π, we derive an explicit asymptotic formula for their exponentially small splitting, uniformly over the Brillouin torus. The proof combines Agmon estimates and WKB approximations for a matrix Schrödinger operator whose lowest potential eigenvalue has two wells. Geometric properties of the Agmon distance allow us to compute the leading tunneling coefficients by the method of stationary phase. We provide numerical computations to illustrate the asymptotic formula and support the conjecture that the two bands remain separated for other values of the Berry curvature parameter.

ALT We prove Poisson local eigenvalue statistics for one-dimensional Schrödinger operators with nondegenerate Bernoulli potentials at any fixed nonzero coupling. After unfolding by the integrated density of states (IDS), the local eigenvalue process converges to a Poisson point process of unit intensity at every energy in the interior of the almost-sure spectrum. The theorem also holds for bounded nonconstant iid potentials, provided the IDS is continuous and strictly increasing near the reference energy. If the IDS also has a finite positive derivative at that energy, the usual linear normalization gives the same limit. The proof uses a phase coordinate with exact IDS drift to obtain a uniform exponential crossing law and joint Poisson counts in disjoint energy windows.

ALT We give a new proof that translation-invariant free Fermion frustration-free models have zero Hall conductance. Our proof is analytic in nature and it uses the bulk-edge correspondence. The central argument, which holds beyond the translation-invariant situation, is that in frustration-free models, edge modes cannot merge into the ground state band. We give an example of a gapped frustration-free model that is nonetheless gapless on the half-plane.

ALT Jellium, a system of electrons in a uniform neutralizing background, and the uniform electron gas (UEG), a system of electrons with constant density and no background, are two standard models of the electron gas. We prove that the ground-state energy per unit volume of Jellium, as defined by Lieb and Narnhofer, coincides with the UEG energy defined by Lewin, Lieb and Seiringer, i.e. eℏJel(ρ₀)=eℏ_UEG(ρ₀). This equality was previously established in the classical case by Lewin, Lieb and Seiringer and independently by Cotar and Petrache.

ALT We establish topological recursion for the spin enumeration of hypermaps, or, in the dual language, spin dessins d'enfants. This problem has remarkable features distinguishing it from other applications of topological recursion to enumerative combinatorics. First, the spectral curve is elliptic rather than rational, as in most cases studied before. Second, it contains degenerate ramification points, and one has to apply a version of degenerate topological recursion. Finally, the meromorphic correlator differentials have "moving" poles on the antidiagonals between pairs of distinct arguments, and one has to adjust the loop equations to this kind of degeneracy as well. The computations involve concepts of different kinds, such as BKP integrability, Gaussian integrals, the x-y swap, and symplectic duality.

ALT We review the role of relative entropy in quantum field theories, where the local von Neumann algebras are generically of type III. Consequently, only few exact results are known, and even bounds are difficult to obtain. As a new result, we prove that the relative entropy between a locally squeezed state and the Minkowski vacuum is finite for free fermions.

ALT A recent graph-theoretic criterion for a Hamiltonian to exhibit a free-parafermion spectrum is that its frustration graph be an indifference graph. We show that an arbitrary finite indifference graph can be realized as a frustration graph of the quantized six-vertex model at odd roots of unity, by a suitable choice of the defining wiring diagram and parameters. This provides a concrete realization of the criterion through the quantum Frobenius property of the quantized six-vertex model, recently established from its three-dimensional integrable structure.

ALT In this paper, we solve the direct monodromy problem for the Borel-summed 0-parameter solution of Painlevé I. Using the auxiliary time equation, we first construct formal WKB wave matrices whose amplitudes are regular at the double turning point. Rigorous Borel summation then yields exact WKB matrices in each Stokes region. Uniform estimates on moving action strips provide a common exact wave matrix in a full neighbourhood of the double point, proving that all four incident jumps are identities for consistent column normalisations. We formulate the resulting normalised Riemann–Hilbert problem and prove its existence and uniqueness on the Borel domains of the construction. Finally, after Painlevé scaling, the five Borel-summed solutions coincide with the five rotated tritronquée solutions of Painlevé I.

ALT Inserting a resolution of the identity between two operators turns a matrix element of their product into a phase-space integral. For unbounded operators the product need not exist on the vector at hand, while the integral may still converge. We follow this gap through the chain eμᵃ²eλᵃ^⁽†²) of two-photon exponentials, where it can be traced completely. Five regions of parameter space appear: where the Bochner conditions of the strong formulation hold, where the product exists as a vector, where it exists as a sesquilinear form, where the coherent-state integral converges, and where the Fock-space series converges. The first three are strictly nested inside the intersection of the last two, and these two overlap without either containing the other. A single Hermite-type function agrees with the matrix element where it exists and with the integral and the series wherever they converge, so the integral carries the analytic continuation of the matrix element beyond the domain of the produ

ALT We introduce a new Euclidean quantum model with zero-spin scaled scalar fields on a spacetime hypercubic lattice, which we refer to as the hybrid model. The model coupling parameters are the hopping parameter κ²>0 and the field interaction coupling α>0. Using the technique of lattice scaled fields, with a noncanonical scaling that corresponds to a partial renormalization (wave function renormalization), we show that the partition function of this model satisfies both thermodynamic and ultraviolet stability bounds, with constants which are uniform in the lattice volume Λₛ and the lattice spacing a∈(0,1]. As a consequence, the model has a finite free energy per degree of freedom in the thermodynamic limit Λₛ↗∞, at least in the subsequential sense. Subsequently, we can also take the ultraviolet or continuum limit a↘ 0. Besides the finiteness property of the free energy, for sufficiently small values of the model coupling parameters κ² and α, we show the model satisfies Osterwalder-Schrade

ALT We study helical shear of a thick-walled tube made of an incompressible, isotropic hyper-anelastic solid carrying a radially symmetric distribution of axial screw dislocations. Through the Bilby–Kröner–Lee decomposition, the dislocations are encoded in a Weitzenböck material manifold, whose Riemannian metric alone enters the elastic response. Equilibrium reduces at each radius to an algebraic system for the azimuthal and axial shear strains, whose cross terms are proportional to the dislocation strength and to the dependence of the strain energy on the second invariant; the two shear modes therefore couple only in the presence of dislocations. Exact solutions are obtained for neo-Hookean, Fung–Demiray, Gent, Knowles, and Mooney–Rivlin materials, and numerical ones for Gent–Gent materials. The universal relations of Horgan and Saccomandi are extended to dislocated generalized neo-Hookean tubes. Dislocations stiffen the tube when the strain energy stiffens with the first invariant, coupl

ALT For a Chern nontrivial Bloch bundle over the three torus, we construct an orthonormal Wannier basis with the optimal pointwise decay O(|x|⁻⁵/²). All but one of the generators decay exponentially. This resolves the question left open in arXiv:2505.01999.

ALT Motivated by recent advances in the synthesis of effective one-dimensional mechanical metamaterials, we develop a direct theory of strain-gradient special Cosserat rods based on the principle of virtual work. The internal virtual work is formulated intrinsically in terms of variations of the classical Cosserat strains and their spatial derivatives, without requiring a three-dimensional parent material or a stored energy. Objectivity and insensitivity to rigid virtual displacements characterize the admissible internal-work density, while localization yields the dynamic field equations, higher-order contact interactions, natural boundary conditions, and balances of linear and angular momentum. For inextensible and unshearable rods, the constrained principle determines a unique multiplier functional, although its representation by reactive force and double-force fields possesses a gauge freedom absent from classical rod theory. We also formulate objective elastic, hyperelastic, and viscoe

ALT Symmetries of a quantum field theory or a quantum lattice model are expected to be described not by a group but by a homotopy type, whose homotopy groups are the higher symmetry groups and whose Postnikov invariants encode 't Hooft anomalies. We propose a homotopy-theoretic definition of 't Hooft anomalies based on the structure of localized symmetries in the operator-algebraic approach. The input is a crossed precosheaf: a collection of groups attached to regions of space, equipped with pairings recording that the commutator of two transformations localized in two regions is localized in their intersection. To a crossed precosheaf and a finite family of regions we attach a pointed connected homotopy type, the Cech-Loday classifying space. As the family of regions ranges over finite covers of space, the classifying spaces form a pro-object in the homotopy category. There is a canonical map from the ordinary classifying space of the symmetry group to this pro-type. Composing this map wi

ALT We extend Mark Kac's kinetic framework from single-species Euclidean spaces to a two-species, non-linear interaction space on the open cone (0, ∞)² governed by the Matsumoto–Yor rational collision involution. We identify a non-linear conserved energy E(x,y) = x + y + 1/x and an invariant conic metric measure dμ=dx dy/x(x+y). By establishing a two-species H-theorem, we prove that entropy production vanishes if and only if post-collision velocities are statistically independent (I(U;V)=0). In contrast to the transform-domain proof of Letac and Wesołowski, which relies on joint Laplace transforms of general measures, our approach yields a direct physical derivation in state space via the transformation law for logarithmic gradients and separation of variables. This uncouples the functional-differential system to uniquely characterize the Generalized Inverse Gaussian (GIG) and Gamma distributions as the non-equilibrium thermodynamic equilibria.