Papers

The complex spacing ratio is the complex displacement from a reference eigenvalue to its nearest neighbor divided by the corresponding displacement to its next-to-nearest neighbor. Its statistics provide a useful diagnostic of spectral correlations and nonintegrability in open quantum systems. Here, starting from the exact joint eigenvalue probability densities, we derive finite-$N$ complex-spacing-ratio distributions for the Gaussian ensembles of non-Hermitian random matrices in classes AI$^{\dag}$ and AII$^{\dag}$, realized by complex symmetric and complex self-dual random matrices, respectively. In class AII$^{\dag}$, we obtain an exact algebraic expression for arbitrary $N$ and explicitly evaluate the distributions and representative moments for $N=3, 4, 5, 6$. In class AI$^{\dag}$, although the joint density retains a noncompact integral over nonunitary eigenvector degrees of freedom, we analytically derive a normalized one-dimensional integral representation for $N=3$ and determine the asymptotic behavior, including a logarithmic correction to the cubic level repulsion and a nonanalytic contribution to the angular density. We further confirm these analytical results through direct numerical diagonalization of non-Hermitian random matrices.

We study heterotic $G_2$ structures near a torsion-free standard embedding, to first order in $\alpha'$ and to all orders in formal deformations. The tangent connection is the Hull connection determined by the metric and flux. We construct a locally invertible family of differential operators relating the supersymmetry and instanton $BPS$ equations to the Euler-Lagrange equations of the heterotic superpotential. Using this comparison and the physical gauge symmetries, we construct a homological vector field locally formally conjugate to the Hamiltonian vector field of the superpotential. On a closed manifold in the fixed gerbe sector, the torsion equation implies vanishing of the leading flux. We prove ellipticity of the linearized deformation complex with mixed differential-order weights and construct a finite-dimensional cyclic $L_\infty$ model with a formal effective potential.

We consider the Sachdev-Ye-Kitaev model with $N$ Majorana fermions and interaction order $q$. For even $q\in \{2,4,...,N-2\}$, we give an elementary proof and show that the eigenvectors of the model are completely delocalized in all deterministic directions.

Medical image segmentation remains challenging in terms of accurate boundary localization and complex-structure preservation, as target regions often exhibit weak boundaries, fine-grained structures, and irregular shapes, while high-quality images and precise annotations are usually limited. Existing generative segmentation approaches based on Flow Matching mainly learn velocity fields in the state space but lack explicit constraints on the spatial regularity and distance-field properties of the recovered representation. This limitation may lead to spatial oscillations, boundary displacement, and discontinuities in fine structures. To address these issues, we propose an image-conditioned Geometrically Regularized Flow Matching framework, termed GR-FM, which formulates segmentation as a continuous probability transport process from an initial distribution to a target implicit representation distribution. Instead of directly modeling binary masks, GR-FM adopts the SDF to describe the target structure, where each pixel is represented by its signed distance to the object boundary, resulting in a continuous geometric field. Flow Matching and ordinary differential equations are then employed to achieve deterministic and efficient distribution transport. The training objective further incorporates a biharmonic regularization term and an Eikonal constraint to enhance the spatial smoothness, structural consistency, and distance-field characteristics of the recovered representation. Moreover, we analyze the continuous transport process under geometric constraints and investigate the evolution of the zero level set. Experiments conducted on the MoNuSeg, GlaS, and DRIVE datasets demonstrate that GR-FM achieves competitive performance in both region overlap and boundary accuracy, reduces performance variation, and maintains stable segmentation results with only a small number of integration steps.

We construct and analyze a renormalized three-dimensional massive Lee model; its renormalized Hamiltonian is defined intrinsically through an abstract construction which explicitly provides the operator domain, action and a Kre\u\i n-like resolvent formula. Conservation of the total excitation number reduces the spectral analysis of the full Hamiltonian to a sector-wise study of the spectral profile of an explicit operator pencil, playing the role of an operator-valued Weyl function. We construct essential spectrum components in a sector generated by lower excitation sectors and obtain the inclusion in the expected sector-wise HVZ type formula; the full formula is proved in the first two sectors and, under explicit conditions, in every sector. Below the free or one-excitation threshold, the point spectrum in every higher sector is described by a bounded self-adjoint nonnegative Birman–Schwinger operator. In every sector with at least two excitations we prove absence of sub-threshold spectrum for arbitrary coupling when the renormalized gap is not smaller than the boson mass, and under an explicit weak-coupling condition when it is smaller. In the latter case we also derive a sufficient condition for the existence of a simple bound state in the two-excitation sector.

We resolve an open problem posed by Frank–Loss. Let $n\ge 3$ and let $\varphi\in L^p(\mathbb{S}^n)$, with $\frac{n}{n-1}<p<\infty$, be a nontrivial Dirac zero mode on $\mathbb{S}^n$, i.e. a nonzero spinor satisfying \begin{equation*} D\varphi = iA\cdot\varphi, \end{equation*} where $D$ is the Dirac operator and $A$ is a vector field with $\mathrm{d} A^\flat\in L^{n/2}$. We prove the sharp lower bound \begin{equation*} \|\mathrm{d} A^\flat\|_{\frac{n}{2}} \ge 2\left[\frac{n}{2}\right]^{-\frac12}\frac{n-1}{n-2}S_n = \left[\frac{n}{2}\right]^{-\frac12}\frac{n(n-1)}{2}\omega_n^{\frac{2}{n}}. \end{equation*} Equality is attainable if and only if $n$ is odd; in that case, modulo conformal and gauge transformations, $\varphi$ is a Killing spinor and $A$ is a real multiple of the Reeb field associated with $\varphi$. The case $n=3$ was proved in our recent paper using a different method. In the paper we divide the remaining cases into 3 cases: i) $n\ge 5$ is odd, ii) $n\ge 5$ is even and iii) $n=4$. All these cases need to use different methods. For $n\ge 5$, the argument crucially reduces to an improved Sobolev inequality on $\mathbb{S}^n$ under a barycenter constraint. Specifically, for $u\in W^{1,2}(\mathbb{S}^n)$ we consider \begin{equation*} \mathfrak{a}_n := \inf\Bigg\{ \frac{ \int \Big(|\nabla u|^2 + \frac{n(n-2)}{4}u^2\Big) - \frac{n(n-2)}{4}\omega_n^{\frac{2}{n}}\|u\|_{\frac{2n}{n-2}}^2 }{ \omega_n^{\frac{2}{n}}\|u\|_{\frac{2n}{n-2}}^2 - \int u^2 } \,\Bigg|\, \int x|u|^{\frac{2n}{n-2}}=0,\ \omega_n^{\frac{2}{n}}\|u\|_{\frac{2n}{n-2}}^2 - \int u^2>0 \Bigg\}, \end{equation*} and we obtain the following universal estimate \begin{equation*} \mathfrak{a}_n > \frac{n(n-2)}{4(n^2-3n+1)}, \end{equation*} which is enough for our aim. Determining the exact value of $\mathfrak{a}_n$ remains open.

A compact formulation of magnetostatics is presented. Without using constitutive relations, including the aether relations, an expression is proposed for the potential energy of a region carrying a distribution of magnetic dipoles under a vector potential field, where the dipole distribution is specified by a current density potential field. Assuming that during a virtual motion of the dipole distribution the virtual work expended by the field is equal to minus the time derivative of the potential energy, an expression for the mechanical force functional is derived. In addition to the Lorentz and Kelvin force densities, the force functional contains an asymmetric active stress field, the skew-symmetric part of which is the mechanical couple density, and a pressure term.

We formulate scattering experiments in terms of parameterized finite measures on outcome spaces and develop the associated information geometry defined by the Fisher metric. The decomposition of a scattering measure into total mass and normalized shape separates Fisher information into contributions from changes in overall scattering strength and redistribution among outcomes. Measurement processes are represented by probability or sub-probability kernels acting on the scattering measure. Such kernels contract Fisher information, placing a broad class of experimental operations within a common information-theoretic framework. This formulation connects scattering experiments to the theory of optimal experimental design through the Fisher geometry of scattering measures. The same finite-measure structure arises naturally in quantum measurements with incomplete detection, clarifying its place in the hierarchy from quantum measurement to classical outcome statistics.

Topological signals are dynamical variables supported on higher-order structures such as simplicial or cell complexes. In this work, we investigate the impact of time-delayed interactions on the emergence of global synchronization of topological oscillators, with application to the Stuart-Landau system. We first examine the case where signals are supported on simplexes, or cells, of a given dimension and coupled through the Hodge-Laplace matrix. We derive the Master Stability Function and we prove that for suitable discrete delay values, the stability of the synchronous solution becomes analytically tractable and can be solved by using the Lambert W-function. This analysis yields explicit spectral conditions for Global Topological Synchronization to emerge and shows that admissible delays may either promote or suppress synchronization. We then study delayed interactions between signals supported on simplexes, or cells, of different dimensions coupled by the Dirac operator. Under analogous admissible time delays the variational problem, still related to the Master Stability Function, can again be simplified, however in this case we prove that Global Topological Dirac Synchronization cannot emerge. Numerical simulations on simplicial and cell complexes corroborate and complement these findings.

We define the classical BV pushforward and establish its two main properties. First, we prove that it maps solutions of the classical master equation (CME) on the total space to solutions of the CME on the smaller space. Second, we prove that, for isolated critical points, it maps BV canonically equivalent solutions to BV canonically equivalent solutions. We illustrate the construction with examples and discuss its applications.

We distinguish two Casimir observables associated with a conformal surface defect. A circular cylinder embedded in flat space has a logarithmic contribution to its Casimir energy, $-d_1\log(R/\epsilon)/(24R)$, fixed by the displacement anomaly. A great cylinder obtained by radial quantization of the ambient theory instead has Casimir energy $-b/(12R)$ in the conformal defect vacuum with a fixed local anomaly convention. Free-field and holographic calculations provide checks of the proposed relations in both embeddings.

We study the Heisenberg dynamics of continuum fermions in $\IR^d$ interacting through a pair potential. For a large class of nonconstant pair potentials and under suitable assumptions on the dispersion relation and external potential, we prove that the dynamics fails to be pointwise-norm continuous even on the gauge-invariant CAR algebra. An automatic-continuity argument then shows that neither the CAR algebra nor its gauge-invariant subalgebra is invariant under the dynamics, with non-invariance occurring for almost all times.

We establish an exact correspondence between renormalization-group (RG) evolution and real-time quantum dynamics in the anisotropic current-current perturbed $SU(2)_1$ Wess-Zumino-Witten (WZW) model. Using its integrable fermionic representation, we construct the exact time-dependent many-body wavefunction by means of the generalized Bethe ansatz and show that periodic boundary conditions lead to quantum Knizhnik-Zamolodchikov equations associated with the XXZ trigonometric $R$-matrix, which corresponds to the quantum affine algebra $\mathcal{U}_q(\widehat{\mathfrak{sl}_2})$ evaluated in its spin-$1/2$ evaluation representation. Consistency of these equations constrains the temporal evolution of the longitudinal and transverse interaction strengths. In the universal regime, these integrability conditions are exactly equivalent to the Berezinskii-Kosterlitz-Thouless RG flow equations upon identifying physical time with the logarithmic RG scale. Consequently, the RG limit cycles of the anisotropic current-current perturbed $SU(2)_1$ WZW model are realized as periodic real-time evolution of its interaction strengths. Our results provide an exact dynamical realization of RG limit cycles and establish a direct connection between cyclic RG flows, quantum integrability, and periodically driven interacting quantum field theories.

Nakayama found the way to limit the contexts in the Döring-Isham topos quantum theory and unified such limitation and the definition of probability as a single choice of topology. We explain how this unification supports the Bayesian perspective where the probability changes as data is obtained. We describe this change as a local shift of the applicability of a predictive theory and apply it to quantum computing.

Symmetry-protected topological (SPT) phases in quantum spin chains are distinguished by topological indices. Ogata's operator-algebraic indices apply to a broad range of symmetry classes without requiring continuous symmetry, whereas the elementary twist index of Tasaki requires $\mathrm{U}(1)$ symmetry. We prove their equivalence in the common setting of integer-spin chains with on-site $\mathrm{U}(1)\rtimes\mathbb{Z}_2$ symmetry and a symmetric locally-unique gapped ground state, without assuming translation invariance or a matrix product representation. A charge-fluctuation estimate yields a strong-limit construction of half-chain rotations with their phases fixed. An exact group commutator identity then identifies the Ogata index with the limit of local twist expectations and gives an explicit error bound. This provides a computable formula for the index when a lower bound on the gap and the relevant ground-state expectations are available. Combining the equivalence with a recent boundary-field gap theorem of OpenAI and our earlier twist-index calculation, we obtain a spin-rotation-invariant locally-unique gapped ground state of the infinite $S=1$ antiferromagnetic Heisenberg chain with both indices equal to $-1$, without assuming infinite-volume uniqueness. The author formulated the problem and proposed the basic strategy, while ChatGPT Astra made substantial contributions to the development of the proof.

We give a dynamical characterization of the Chandrasekhar mass for cold white dwarfs governed by the radial Euler–Poisson equations with the exact Chandrasekhar equation of state. This mass is the infimum of masses for which a finite-energy classical solution with a physical vacuum can contract its entire support to a point in finite time. Below it, conservation of energy prevents complete collapse. For every mass in a sufficiently small interval above it, we construct a solution whose support shrinks to the origin and whose density concentrates there with the full mass of the star. The interior approaches a collapsing Goldreich–Weber profile. Near the vacuum boundary, a different rescaling gives a limiting profile that connects the high- and low-density regimes of the pressure law. We determine this profile and show that, after adding the electron rest energy, the conserved energy equals that of the Goldreich–Weber reference solution. The proof combines matched asymptotic expansions with local existence and energy estimates that extend solutions backward from times approaching collapse to a common earlier time.

We study the monic recurrence coefficients of the generalized Charlier polynomials associated with the discrete weight \[ \rho_\mu(k) = \frac{\mu^k}{(k!)^2}, \qquad k\in\mathbb N_0, \qquad \mu>0. \] The coefficients $\beta_n$ and $\gamma_n$ in the recurrence relation \[ P_{n+1}(x) = (x-\beta_n)P_n(x)-\gamma_nP_{n-1}(x) \] are described through complementary moment, determinant, discrete dynamical, and continuous integrable formulations. We give a consistent normalization of the Markov function and its Jacobi continued fraction, and express the recurrence coefficients in terms of Hankel determinants, using a bordered determinant for the diagonal coefficient $\beta_n$. Starting from the original Laguerre–Freud equations, we derive the local recursive system used for computation and relate it to the Toda deformation. We also recall the known Painlevé~V representations associated with the two-parameter generalized Charlier family, in a normalization consistent with the present specialization. Moment/Hankelconstructions provide complementary moment-theoretic realizations of the recurrence coefficients, while the local Laguerre–Freud recursion provides an independent dynamical construction. Their comparison is used for numerical cross-validation. The numerical results are consistent with the leading large-degree behavior \[ \beta_n\sim n, \qquad \gamma_n\sim\mu. \]

The wave-particle decomposition of kinetic relaxation equations is extended to the full Boltzmann equation. An identity transformation writes the collision term as a relaxation toward the local Maxwellian plus a conservative remainder. Along characteristics, the equilibrium integral over a local kinetic horizon defines the wave, and its exact complement, the particle, carries the remainder. The moments of the wave and particle equations are extended Navier-Stokes equations, and the total conservation law and the particle equation form the wave-particle multiscale equations, a closed system, exact for every horizon, that spans a continuous spectrum from the Boltzmann equation to Navier-Stokes hydrodynamics, parametrized by the horizon-to-relaxation ratio. A coupled implicit wave-particle iteration solves this system, updating the conservative variables with the particle fixed and the particle with the predicted wave and an endpoint traction from the macroscopic prediction. Under discrete compatibility assumptions the iteration preserves the discrete Boltzmann solution as its fixed point, and its near-continuum convergence factor is bounded by a particle transport share decaying exponentially with this ratio. Tests on a normal shock, Couette and cavity flows, and hypersonic cylinder and Apollo flows agree with the conventional iterative scheme in rarefied and transition regimes and with Navier-Stokes solutions near the continuum limit. There the wave-particle iteration needs about three to more than four orders of magnitude fewer iterations than this scheme or, where it did not converge, its Shakhov version, and the Boltzmann solution costs one to four times the wall time of a Navier-Stokes solver on the same mesh.

The one-dimensional swarmalator model is usually understood through three macroscopic states: synchrony, the phase wave, and asynchrony. We show that the identical model also contains a much larger set of hidden stable equilibria - families of q-twisted and m-clustered states - which random initial conditions almost never reach, because many are embedded in a degenerate neutral manifold of vanishing global basin. They are nonetheless locally robust: under Gaussian perturbations of per-coordinate amplitude {\epsilon} their survival radii shrink only algebraically, as 1/q and 1/m, in sharp contrast to the Gaussian basins and winding ceiling of twisted states on a nearest-neighbour Kuramoto ring, both removed by mean-field coupling. A block-circulant Jacobian fixes the stability in closed form, and far-kick experiments indicate the basins are compact cores rather than tentacled sets. Several patterns reported in extended swarmalator models thus already exist, stable or neutrally stable, in the minimal one.

By homogenizing high-contrast periodic metric graphs, we create continuous media that admit bound states produced by local defects at energies embedded in the continuous spectrum. This is achieved by fitting two copies of a lattice graph simultaneously into space and coupling them by connecting edges. The homogenization of an ODE on a graph to a PDE in space is achieved by representing the associated operators in a common functional space through interpolation by Fourier analysis. The two coupled graphs result in an effective medium with internal degrees of freedom that allow a decomposition into two orthogonal spaces of hybrid states. Critical high contrast retains microscopic resonances in the effective medium and allows a spectral band of one hybrid state space to overlap a spectral gap of the other. This serves as the mechanism for creating bound states at frequencies embedded in the continuous spectrum of the effective medium. These bound states in the continuum persist for $\varepsilon$-scale metric graphs due to $O(\varepsilon)$ operator-norm estimates for reduced resolvents on the defect region.

We present an expository excursus on modulation spaces, from Feichtinger's original construction on locally compact abelian groups to recent symplectic formulations. We trace the same local-global principle through bounded uniform partitions of unity, the short-time Fourier transform, coorbit methods, and Gabor coefficients, while distinguishing equivalent descriptions from the weighted, quasi-Banach, and Gelfand-Shilov extensions of the underlying functional setting. We also discuss Gabor matrices, convolution and embedding estimates, and an application to a nonlinear dispersive equation. The final part is devoted to metaplectic Wigner distributions $W_{\mathcal A}$. In the shift-invertible case these representations are, up to chirps and a linear change of phase-space variables, rescaled short-time Fourier transforms. This structural fact explains both their characterization of modulation and Wiener amalgam spaces and their discretization by metaplectic Gabor frames, and places the recent symplectic theory naturally within Feichtinger's modulation-space framework.

The quantum data processing inequality asserts that quantum relative entropy is monotonically non-increasing under quantum channels. Here, we promote this fundamental inequality to an exact equality via boundary and surface Poisson integral representations across a complex strip. The relative-entropy loss is resolved into two independent, nonnegative physical mechanisms: an operational Petz recovery component governing state reconstructibility, and a boundary term measuring the mismatch of relative modular dynamics. We further express the loss as the measured relative entropy between the original state and its averaged Petz recovery state plus a nonnegative variational remainder. For the Belavkin–Staszewski (BS) loss, analogous integral representations yield a decomposition into negative log Uhlmann fidelity with an averaged reconstruction plus a nonnegative variational remainder. Furthermore, we establish recovery bounds at any prescribed modular parameter, including the canonical unrotated Petz map, with optimal square-root scaling in the relative entropy loss. The prefactors depend on distinct-eigenvalue counts or logarithmic spectral quantities, improving upon the inverse-power dependence on small eigenvalues in previous bounds.

Spectral zeta functions provide a standard tool for regularizing determinants of differential operators in quantum physics. In a previous paper (J. Math. Phys. 66: 083505, 2025), we introduced a $p$-adic spectral zeta function for discrete spectra via a locally analytic interpolation function. In this paper we extend the framework to continuous spectra using the inverse Stieltjes transform. Starting from the resolvent of a bounded operator, we construct a generalized distribution, called the $p$-adic spectral distribution, and define bosonic and fermionic zeta functions and functional determinants. We establish their analytic properties, special value formulas, and Stirling expansions, and show that our previous framework is embedded into the present one in the discrete case. In this approach, the bosonic and fermionic cases exhibit an interesting symmetry. As an application, we consider the position operator in $p$-adic quantum mechanics on an arbitrary compact subset of $\mathbb{C}_{p}$.

Reduced descriptions of unresolved fluctuations are commonly based on Gaussian processes, although many realistic forcings have finite correlation times and a renewal structure. For memoryless step (Kubo–Anderson) noise, we show that the reduced dynamics admit two exact and complementary descriptions: a kernel representation governed by the Boolean cumulants of the jump distribution, and a frozen-noise representation adapted to stationary probability densities. For exponentially distributed waiting times, the totally time-ordered $G$-cumulants coincide with the Boolean cumulants of the jump law, and the memory kernel is resummed exactly as the Boolean generator $\eta$ evaluated on a resolvent operator. Second-order closure is exact if and only if the jump law is symmetric Bernoulli; otherwise, the leading closure error is controlled by $(b_4/b_2)\lambda^2$. The Boolean hierarchy, however, acts on the kernel, not on the stationary measure. Stationary densities are approximated by replacing the jump law with its $N$-point Gauss quadrature: each surrogate preserves all multi-time correlations up to order $2N-1$ for any waiting-time law, its kernel is a Padé resummation of the Boolean one, and it is itself an exactly solvable renewal problem. For the linear system with linear multiplicative interaction (LIMI/CAM), the frozen-noise representation reduces the dynamics to a random affine recursion and yields exact support boundaries, singularity exponents and Kesten tail indices, confirmed numerically. Rates, moments and closure errors are governed by the Boolean hierarchy, whereas stationary densities are determined by the geometry of the jump distribution. For memoryless renewal noise, the combinatorial structure controlling finite-correlation reductions is Boolean.

We develop a Hopf-algebraic theory of the graph coefficients arising in the quantum Magnus expansion. At the classical level, the graph expansion is governed by directed trees, whereas its quantum counterpart involves loop graphs, multiple edges, and different types of edge data. We introduce a contraction Hopf algebra of mixed quivers containing these physical diagrams and closed under contraction. On the Hopf subalgebra spanned by quivers, we define a character $e$ from normalized linear-extension data and a tadpole prescription, and let $\omega$ be its convolution inverse. Our main result is a universal closed formula for the connected function $\omega_c$ on every finite quiver. The formula is a finite sum over ordered set partitions of the vertex set and is valid without acyclicity or simplicity assumptions. We also give a local characterization: a two-vertex contraction identity, together with tadpole factorization, vanishing on disconnected graphs, and boundary data, determines $\omega_c$ uniquely. For acyclic quivers, the general formula reduces to a permutation formula whose coefficients depend only on descent numbers. We derive the same formula independently from the quantum Magnus expansion via operator products and Wick contractions, showing that its physical graph coefficients are governed by the convolution structure of the contraction Hopf algebra. As further consequences, we obtain a refined quantum Murua formula and orientation-sum identities related to Tutte and chromatic polynomials.

We study three particles on the square lattice with contact interactions of infinite strength, for all total quasimomenta, for three statistics: three identical bosons, two identical fermions plus a third particle, and two identical bosons plus a third particle. Previous work treated only the trimer and zero quasimomentum. Here the atom-dimer sector is analyzed on the whole Brillouin zone. A Feshbach-Schur reduction onto the infinitely degenerate eigenvalue 1 of the pair-contact operator gives explicit effective Hamiltonians with proved first-order error. The threshold problem reduces to a 4x4 Hermitian matrix criterion built from the two-point resistances of the square lattice, and hence to a quartic polynomial. This yields the complete atom-dimer zone and the critical mass ratio on the Brillouin zone, with bound between 1 and pi/(pi-2), and closed forms at all symmetry points. We prove: the zones in closed form; the critical mass ratio with exact value pi/(pi-2) at zero quasimomentum, reproducing a numerical value; at most two atom-dimer states for every quasimomentum and mass ratio; the onset laws, an essential singularity on the zone boundary and a linear-over-logarithm onset in the pure p-wave; two exact dualities showing that a change of statistics is equivalent to a shift of quasimomentum by pi; and the global range of the second fermionic threshold. The onset laws admit a renormalization-group reading: the inverse lattice Green function is a marginal s-wave coupling, binding energies arise by dimensional transmutation with lattice scale 16, and the zone boundary is of marginal (BCS/Kondo) type rather than of Berezinskii-Kosterlitz-Thouless type. All constants are verified independently by lattice quadrature, computer algebra, exact diagonalisation, and infinite-lattice solvers.

Two irreducible representations of $U(d)$ whose Young diagrams differ by a single box are connected by four covariant quantum channels: the box can be removed or added, and one keeps either the new diagram or the box ($\mathbb{C}^d$ or its dual). We prove that for each of these channels the output of a coherent state majorizes every other output, so that coherent states minimize the output entropy, and we compute the optimal output explicitly in terms of hook lengths. For the two channels that keep the box we also determine all minimizers; these need not be coherent, even when they are pure. As an application we consider $N$ particles with a given permutation symmetry and find sharp bounds on the spectrum of the reduced density matrix of a single particle, as well as the exact set of spectra that mixed states can reach (for fermions these are the Pauli principle and Coleman's theorem).

We prove Landau-Ginzburg mirror symmetry conjecture for ADE singularities with the nontrivial symmetry groups. In particular, we consider $(f,G)$ with $f \in \mathbb{C}[z_1,z_2,z_3]$ defining ADE singularity and $G$ — nontrivial group of diagonal symmetries of $f$. We show that orbifold Saito theory of $(f,G)$ is isomorphic to the Fan-Jarvis-Ruan-Witten theory of its Berglund-Hübsch-Henningson dual pair $(\widetilde f, \widetilde G)$.

Criticality in the Ising model is conventionally generated by Hamiltonian dynamics and controlled by temperature. Here we show that critical like behavior can emerge in an Ising system whose dynamics is completely independent of the Ising Hamiltonian and contains no temperature like parameter. We introduce an adaptive cluster dynamics in which spin connectivity is controlled by a local quiet time variable, the time elapsed since a spin was last updated. The Ising Hamiltonian enters only as an observable characterizing the resulting configurations. As the parameter $\alpha$, controlling the connectivity strength, is varied, the system undergoes spontaneous symmetry breaking accompanied by strong collective fluctuations. The cluster size distribution develops an extended power law regime, terminated by a small finite size hump at the largest scales, while the Ising energy exhibits a singular response to the variation of $\alpha$. These results show that memory dependent connectivity alone can generate collective critical behavior, revealing a route to nonequilibrium criticality without energy based dynamics.

We prove that the simple affine vertex algebra $L_k(\mathfrak{sl}_2)$ is classically free at the boundary admissible level, which confirms a conjecture of Andrews, van Ekeren, and Heluani.