Papers

We provide a simple ansatz for finding integer spin $SO(3)\times\Z_2^T\times\Z_2^R$-invariant matrix product states representing a phase for each triple of $\Z_2$ topological indices studied in the literature by Tasaki \cite{tasakiheistop}\cite{tasakitop} and Ogata (corresponding to time-reversal and reflection symmetries) \cite{ogatatime}\cite{ogatareflection}. Every combination of trivial and non-trivial indices has a spin-$1$ $SU(2)$, time-reversal, and site-reflection symmetric MPS representative with injectivity length $l\leq 8$. We provide some exact examples for spin-$1$ and a Fortran program which will find representatives of any triple for any integer spin, as well as some data from random sampling for small spin. The existence of such states agrees with the classification of Chen, Gu, and Wen \cite{chen}, and with the recent results of Tasaki \cite{tasakinew} equating the hidden-order and one of Ogata's indices, wherein the Ogata index is that associated to the dihedral $D_2$ symmetry.

We provide a geometric description of Lee-Yang tensors, building upon several recent results of Bravyi, Gosset, Liu, and Wong. By applying tools from complex Perron-Frobenius theory, we generalize one of their key lemmas to bound the spectral ratio of non-normal operators with Lee-Yang radius greater than one. As a direct consequence, we provide a local-to-global principle for bounding spectral gaps of Suzuki-Fisher Hamiltonians, a family of $2$-local Hamiltonians, using only the Lee-Yang radius of each local operator to bound the gap from below. On the other hand, we develop a simple perturbative technique to show that the Gibbs states of $k$-local Hamiltonians, $k \geq 3$, cannot have Lee-Yang radius at least one for all inverse temperatures $\beta \geq 0$. We apply this same method to conclude that Suzuki-Fisher Hamiltonians are precisely those Hamiltonians whose Gibbs states are Lee-Yang at all inverse temperatures $\beta \geq 0$.

We prove a single-logarithm bound for the eigenvalues in the plunge region of Fourier concentration operators, removing the extra logarithmic factor in the Kulikov–Dam Larsen bound for two general admissible sets throughout each fixed exponential window. For bounded spatial and frequency sets $A,B\subset\mathbb{R}^d$ of positive measure with finite upper codimension-one Minkowski boundary content, the number $\Lambda_\varepsilon(cA,B)$ of eigenvalues in $(\varepsilon,1-\varepsilon)$ satisfies $\Lambda_\varepsilon(cA,B)\le C(d,A,B,\beta)c^{d-1}D(1+\log_+(c/D))$, where $D=\log(1/(\varepsilon(1-\varepsilon)))$, for every fixed $\beta>0$, $c\ge2$, $0<\varepsilon<1/2$, and $D\le\beta c$, with $\log_+t=\max(0,\log t)$. The argument uses positive band-limited smoothing and requires no symmetry, convexity, or connectedness. In dimension two, the corresponding exponential-window estimate has a constant uniform in the window parameter. We also determine the optimal coordinate-Lipschitz constant $d-1$ for recursive decompositions of balls and standard simplices in every dimension, and obtain box-frequency estimates with explicit geometric constants. For $C^1$ convex bodies in three dimensions, the optimal infimum of recursive coordinate slopes is $2$. Uniform real-order Bessel estimates give independent disk and ball bounds at every spectral threshold, including an explicit deep-tail term. For smooth strictly convex planar frequency bodies with positive curvature, we prove correlation estimates for aligned Gaussian packets at arbitrary relative scales and quantitative lower bounds on the cost of estimating spatial blocks separately. Together these results connect the global spectral bound with sharp geometric constructions and the cancellation retained by grouping boundary interactions.

Many machine learning algorithms rely on the Fourier transform, which is also the inner core of many analog computing devices and, specifically, of optical computing. The square of the modulus of the Fourier spectrum naturally maps onto pairwise interactions, as in the renowned Ising model, making it applicable to combinatorial optimization. Here we show how the nonlinear Fourier transform, related to integrable nonlinear partial differential equations, significantly widens the computational capability and also enables a wide variety of computing gates. We study specifically the case of the nonlinear Schrödinger equation; we introduce an input composed by “tokens”, i.e., batches of information encoded in a multi-step potential, and we show how the nonlinear spectral computing arising in the wave propagation not only generalizes combinatorial optimization to higher order problems, but also realizes boolean gates. In the weakly nonlinear regime, nonlinearity generates higher-order interactions as perturbative corrections to the pairwise terms. In the solitonic regime, the discrete nonlinear spectrum performs digital computation through soliton generation. By using four input tokens, we demonstrate the programmability of all 256 Boolean functions of three variables, including 3-SAT instances. Our results show that integrable nonlinear propagation provides an unprecedented single physical platform for higher-order and logical computation, opening the way to a rich variety of new algorithms for analog computing, machine learning, and classical and quantum information processing.

Let $P_h$ be a semiclassical pseudodifferential operator on a closed Riemannian manifold $(M,g)$, with principal symbol $p$. We consider perturbations $P_h+h^\alpha Q$, with $0<\alpha\le 1$, where $Q$ is a semiclassical pseudodifferential operator whose symbol oscillates at scale $h^\beta$, with $2\beta <\alpha$. Assuming that the classical dynamics of the unperturbed system on a regular energy surface $p^{-1}(E)$ is hyperbolic, we prove a Gutzwiller trace formula for the perturbed operator that holds up to times of order $|\log h|$. This formula relates the distribution of energy levels of the quantum system near the energy $E$ to the periodic orbits of its classical counterpart. The perturbation $h^\alpha Q$ affects only the phases appearing in the trace formula.

Exponential directed last passage percolation is a random maximization model connected to tandem queues and interacting particle systems. We study the line-to-point model of the percolation, in which the starting point is allowed to vary inside a line boundary, conditioned on an atypically large last-passage time to a fixed endpoint. Extending recent results for the point-to-point model where the starting point is fixed, we develop heuristic predictions for the conditional limit shape and prove corresponding fluctuation results in certain regimes through steepest-descent analysis of exact multi-point formulas. Our calculations suggest that the conditional fluctuations near diagonal are described by Brownian bridges coupled through an additional shared Gaussian component.

Experience with invariant finite-difference schemes for one-dimensional hydro\-dynamic-type equations for a function $u(t,x)$ suggests that nonlinear polynomial schemes on orthogonal uniform meshes with the 9-point stencil cannot simultaneously preserve difference analogues of the energy and momentum conservation laws, with the multipliers given by the standard difference approximations of $u_t$ and $u_x$. For the 9-point stencil this hypothesis is proved for polynomials of arbitrary degree. On larger stencils it is investigated by means of the difference analogue of the direct method: for polynomials of bounded degree, such schemes are found to exist only on sufficiently large stencils, and the second-order equations they approximate are of Monge–Ampère type. As a by-product, a scheme for the homogeneous Monge–Ampère equation is obtained that has five conservation laws, admits most of the symmetries of the differential equation, and is invariant under the discrete dihedral group $D_4$. Numerical tests on exact solutions confirm its second order of approximation and show that the conservation laws are satisfied on these solutions with the same order of accuracy.

We introduce novel scalar generating potentials to compute three and four point cosmological correlators in orthogonal Grassmannian space. These correlators are obtained by applying appropriate directional derivatives on these potentials, combined with helicity-dependent minors. We emphasize that, this is not a spin-raising analog, but a much more powerful construction. The four point generating potential is associated with the geometry of a projective triangle. First-order directional derivatives generate gluon correlator, while the Hessian of the same scalar potential produces the four-graviton correlators, thus realizing a differential double copy. Thus, double copy is a natural consequence of the underlying scalar potential as the origin of the gluon and graviton four point correlators. This framework extends naturally to any spin-$s$. We exhibit a geometric origin for the gluon colour identities: the projective homogeneity of the scalar potential implies Kleiss-Kuijf identity, while the kinematic Jacobi identity follows as a consequence of the closure of the triangle under directional derivatives. We also uncover a connection between the BCFW bridge deformation of the scalar potential and directional derivatives that produces the spin-s correlator. Thus the generating potential provides a universal geometrical interpretation for any spin-$s$ four point Grassmannian correlator, unlike \cite{Bala:2026lvw} where geometric interpretation was only shown for the gluons.

We compute the higher-rank translation ($\mathcal{T}$) and rotation ($\mathcal{R}$) matrices associated with a genus-one spectral curve. Using these data, we reconstruct the worldsheet correlators of vertex operators in the corresponding supersymmetric Landau–Ginzburg (LG) theory. This provides a generalization of the genus-zero, one-cut spectral-curve construction to higher-genus geometries. The matrix-model correlators are mapped to integrals over the moduli space of Riemann surfaces for the symmetric sextic double-well potential, which has a double-cut solution. We further identify the rational and elliptic contributions to the worldsheet integrated vertex operators and show that each of the sector precisely matches the corresponding correlators of single-trace operators in the matrix model. Altogether, our construction showcases an explicit dictionary between the higher-genus spectral-curve data, the associated Cohomological field theory (CohFT), and the worldsheet description of the corresponding matrix-model observables.

Emergent generalized symmetries and the holographic principle play very important roles in understanding quantum phase transitions. In recent years, tensor-network renormalization with generalized symmetry and the corresponding fixed-point tensor formulation have been proposed as a discrete spacetime framework for reformulating conformal field theory. Nevertheless, a complete holographic formulation within this framework is still lacking, and the current formulation does not directly accommodate sector partition functions that are modular covariant in general. In this work, we formulate $\mathcal{R}$-TNR, which builds the categorical constraints of a fusion category $\mathcal{R}$ into local tensor spaces and preserves these constraints under Loop-TNR. For suitable choices of $\mathcal{R}$, the initial tensors constructed here flow under $\mathcal{R}$-Loop-TNR to stable gapless fixed points. In the Ising, tricritical Ising, and three-state Potts conformal field theory examples, a broad range of initial parameters leads to the same gapless fixed point in each example. The method resolves conformal towers by anyon sector and yields scaling dimensions, conformal spins, and magnitudes of sector-resolved structure constants with extremely high accuracy compared to previous methods. These results realize a topological bootstrap based on stable symmetry-topological-order in which finite categorical and geometric input, together with an initial tensor that need not be fine-tuned, suffices to recover the infrared conformal field theory and its sector-resolved conformal data.

A gap between achievable rates and converse bounds persists for energy-constrained bosonic communication in the presence of excess noise. We derive a unified adaptive weak-converse bound for thermal attenuation, noisy amplification, and additive Gaussian noise under an unconditional mean transmitted-photon-number constraint. The bound applies to arbitrary adaptive protocols with quantum memories and constrains the two-way quantum, entanglement-distribution, private, and secret-key capacities. It vanishes throughout the entanglement-breaking region and approaches the corresponding PLOB bound at infinite energy. At finite energy, it improves the evaluated Gaussian squashed-entanglement and PLOB bounds in relevant parameter regimes, while comparison with hashing rates quantifies the remaining gap to achievability. The result provides a common energy-dependent benchmark for noisy bosonic Gaussian channels and extends finite-energy converse methods beyond the quantum-limited setting.

We solve a longstanding problem in Bernoulli percolation: we derive sharp Ornstein–Zernike asymptotics for the probability that two distant points are connected by a finite cluster in the whole supercritical regime. Our proof has two main ingredients. First, the renewal structure from Fridbergh and Hammond (2024) yields an a priori bound on cluster volume. Second, we adapt the coarse-graining and irreducible-cluster framework of Companion, Ioffe, Velenik (2008) to finite supercritical connections. The main distinction from the subcritical analysis is that our argument does not require a separation of masses between irreducible and unrestricted finite connections. The volume bound and a modified coarse-graining construction instead provide stretched-exponential tails for the effective random-walk increments, which suffice to obtain the sharp asymptotics.

Broadly studied origami crease patterns such as the Miura-ori are capable of rigid folding that maintains their spatial periodicity. However, periodic crease patterns generically require additional creases and fold into quasi-cylindrical shapes. Here, using group theory, we show that the folding properties of such sheets are largely determined by their spatial symmetries, known as layer groups. Within each symmetry class, the allowed linear shape-periodic modes of sheets are constrained and classified by the irreducible representation they belong to. Such classification further gives a sufficient condition, containing a symmetry constraint and a topological constraint, for the existence of a nonlinear strain mode. In this way, familiar special cases are placed within a unified framework that connects symmetry, network topology, and deployability. Finally, we turn our classification results into a predictive design principle that enables the rapid design of novel sheets with prescribed properties.

We study entropy production for stationary diffusions on separable Hilbert spaces with possibly degenerate, state-dependent trace-class covariance. We first define the reversible–irreversible drift decomposition to provide a candidate drift for the reversed dynamics. Then by working directly with the invariant measure, we establish a lower bound in terms of the extended stationary Cameron–Martin energy of the irreversible drift, without requiring a diffusion representation of the stationary reversal. The bound implies infinite entropy production when the energy is infinite and we establish sufficient conditions for equality in the finite-energy regime. We also develop complementary criteria for identifying the stationary reversal as a Hilbert-space diffusion with constant or continuous state-dependent diffusion coefficients, respectively. Two nonlinear infinite-rank examples are provided to show that finite entropy production does not require the irreversible drift to lie in the covariance range, whereas entropy production may be infinite even with globally Lipschitz coefficients and injective covariance.

We develop a deformation-theoretic framework for algebraic superstacks and apply it to the construction of perfect obstruction theories on moduli spaces of stable supermaps. In order to achieve this we employ the tools and concepts from derived algebraic geometry, extending them to the supergeometric setting. Our main technical contributions are (i) the notions of derived superstacks, (ii) their cotangent supercomplexes and (iii) perfect obstruction theories for algebraic superstacks. We prove that a quasi-smooth derived enhancement of an algebraic superstack always yields a perfect obstruction theory: this generalizes a well-known principle from classical deformation theory. We then turn to the case of the Deligne–Mumford superstack $\boldsymbol{\mathcal{M}}^{\mathsf{st-}\mathrm{SUSY}}_{g,\mathfrak{n}}(\boldsymbol{Y})$ of stable supermaps of fixed genus, homology class and numbers of Neveu–Schwarz and Ramond–Ramond punctures, with target a smooth projective superscheme $\boldsymbol{Y}$. We exhibit a natural derived enhancement $\mathbb{R} \boldsymbol{\mathcal{M}}^{\mathsf{st-}\mathrm{SUSY}}_{g,\mathfrak{n}}(\boldsymbol{Y})$ and compute its cotangent supercomplex; we prove that the induced obstruction theory on $\boldsymbol{\mathcal{M}}^{\mathsf{st-}\mathrm{SUSY}}_{g,\mathfrak{n}}(\boldsymbol{Y})$ is perfect. Finally, we compute its virtual dimension by super Grothendieck–Riemann–Roch. The resulting formula gives, in particular, a deformation-theoretic proof of the virtual dimension formula previously conjectured for stable supermaps and specializes to the usual virtual dimension of the classical moduli stack of stable maps.

We study the $d$-dimensional generalization of Models II ($d=2$) and V ($d=3$) for liquid crystals introduced by Heilmann–Lieb (1979), which consists of dimers on the hypercubic lattice at chemical potential $\mu$, interacting via a hard-core repulsion and a lateral attraction between parallel dimers with coupling constant $b>0$. For $d\in\{2,3\}$, these models were conjectured to exhibit liquid-crystalline order at low temperatures provided that $\mu+2(d-1)b>0$. We establish nematic order for all $d\ge 2$ under the additional condition $(4d-6)b>\mu$, which corresponds to the physical regime in which misaligned dimers are rare on scales comparable to the correlation length of the one-dimensional reference system. While the treatment of orientational order here follows the approach of our recent work on Model I, the proof of the absence of translational order relies on a custom implementation of cluster swapping in the sense of Sheffield (2005). Consequently, we deduce uniqueness of Gibbs measure and anisotropic exponential decay of correlations within each oriented phase.

We prove asymptotic completeness of the wave operators in time-dependent scattering theory which intertwine the ordinary or fractional Laplacian and the corresponding Hardy operator, i.e., the fractional Laplacian with an added subcritical or critical Hardy potential, whenever the Hardy potential is short-range. Moreover, we prove asymptotic completeness of the wave operators intertwining Hardy operators with and without external short-range perturbations. We also show the nonexistence of the considered wave operators when the Hardy or external potentials are long-range. We arrive at these results using three different approaches. The first approach uses an abstract semigroup framework for asymptotic completeness of wave operators for Fourier multipliers with short-range potentials. The second approach uses Kato smoothness theory. In the third approach, we develop and apply new propagation estimates for Hardy operators with and without perturbations.

Local-basis Shannon entropies of critical wave functions contain universal subleading information, but the Shannon point of the critical transverse-field Ising chain is singular in the conventional Rényi approach. We treat it directly at $n=1$ by representing the complete computational-basis Born distribution as the odd-degree boundary of independent long-range Bernoulli edges. The Shannon chain rule separates the entropy into explicit independent-edge and conditional cycle-space terms. For the periodic chain this isolates the analytic edge anomaly and organizes the remaining constant by linked vertex support. Conditional on the stated finite-part matching and forest assumptions, every nonvanishing linked coefficient is represented by an explicit finite-dimensional integral, giving an all-orders analytic hierarchy; the four-vertex term is evaluated in closed form, and the first few linked sectors already nearly saturate the established periodic constant, which is also reconstructed independently from the exact finite-size distribution. For an interval, the exterior reduces exactly to one ghost vertex and the explicit squared logarithms cancel, leaving the analytic edge contribution $\gamma_E=\log2/4-1/16$. The conditional cycle term has an exact all-support decomposition into physical-line and ghost-linked sectors; conditional on closure of the boundary forest subtraction, these sectors define an all-orders hierarchy of renormalized line and monomer–dimer boundary periods for the remaining logarithmic coefficient. The first complete augmented linked coefficient is evaluated in closed form, with an exact cancellation between its line and ghost boundary-layer anomalies; an independent reconstruction from exact finite-size probabilities recovers the established coefficient $\gamma_1=0.060020(3)$.

We characterize when a banded totally positive matrix admits a positive bidiagonal factorization. In finite dimension such a factorization always exists. For semi-infinite matrices of arbitrary finite bandwidth, existence is equivalent to convergence of a finite family of positive series determined by the Metelmann factors. Their reciprocals are the limiting values of scalar Jacobi continued fractions. We also prove that every bounded banded totally positive matrix can be approximated in operator norm by matrices admitting positive bidiagonal factorizations, through perturbations confined to a fixed initial block. For banded Toeplitz matrices, the negative roots of the polynomial symbol give a global factorization with bounded factors. Ordered partitions of these roots determine explicit mixed-type multiple orthogonal polynomials. Applying the existing spectral Favard theorem, we compute their matrix of positive measures: its support is one interval, and the measure is purely absolutely continuous with an algebraic density of rank one. These formulas also give transition and first-return laws for the associated killed random walks.

We study $SO(2n)$ gauge theories with a degenerate curvature kinetic operator defined by an orthogonal complex structure. The kinetic image is $\su(n)$, while the adjoint vacuum has stabilizer $U(n)$. We distinguish fixed projectors, unrestricted adjoint Higgs fields, and Higgs fields constrained exactly to the orbit $SO(2n)/U(n)$. For the constrained theory with a positive Higgs kinetic coefficient, the coset connection is auxiliary and the central connection is null. A nonlinear Dirac analysis establishes a regular weak-curvature branch with $2(n^2-1)$ physical degrees of freedom in four dimensions, locally equivalent to $SU(n)$ Yang–Mills theory. Normal Higgs displacements in the unrestricted theory can change the kinetic rank, so this count does not extend to that field space. We also characterize possible algebraic branches. We express the Stelle–West Higgs–gravity action through a common Clifford involution, and introduce projected Pontryagin theories and a full-rank equivariant completion.

Whether the spin-1/2 Aharonov-Bohm Hamiltonian supports bound states has remained controversial since Hagen concluded that no such states exist for a Dirac particle. Here, we show that Hagen's solution corresponds to a particular member of the one-parameter family of self-adjoint extensions that describe the singular Zeeman interaction, and we identify which member the microscopic physics selects. Matching the extension parameter to a finite-radius flux tube yields $\nu=0$ precisely at $g=2$, and it does so in every flux sector. The critical $g$ factor is therefore universal, $g_c=2$, and depends on neither the flux sector nor the regularization radius. The flux sector controls the depth of the bound state energy, and we obtain a closed-form expression showing that binding deepens markedly with the integer part of the flux. Hagen's conclusion is thus confirmed and sharpened: it is not a statement about one arbitrary member of a family of extensions, but about the member that the physics selects when the magnetic moment takes its Dirac value, and an anomalous moment is a necessary and sufficient condition for binding in the singular channel.

We construct magnetohydrostatic equilibria in rotationally symmetric toroidal domains whose magnetic lines define an $(m,n)$ Seifert fibration for arbitrary coprime integers $m,n$.

For the critical random-cluster model on $\mathbb Z^2$ with cluster-weight $q=4$, we prove the convergence of loops to CLE(4) in the scaling limit. Conformal invariance of the four-state Potts model is then deduced via the Edwards-Sokal coupling.

We propose an unified view to three well known quadratic Hamiltonians widely considered in quantum mechanics, i.e. those of the harmonic oscillator, of the inverted harmonic oscillator, and the Berry-Keating Hamiltonian. We describe an unified approach to determine the discrete spectrum of these operators, and to deduce their (generalized) eigenstates. We also look at these Hamiltonians as the generators of $\mathfrak{su}(1,1)$, and of a generalized version of it relevant in presence of non Hermitian generators.

Stability of matter is the property that the ground state energy of $N$ charged quantum particles is bounded below by $-CN$, with $C$ independent of $N$. We ask whether this holds for colour-charged fermions interacting with a classical non-abelian gauge field. The field is a dynamical variable on the same footing as the matter, and the energy is minimized over both. The charges are matrices, and every pair of particles has an attractive colour channel. Non-relativistic spinless fermions are stable for every value of the coupling constant $\alpha$. Relativistic spinless fermions and non-relativistic spin-$\frac12$ fermions are stable for small $\alpha$ and unstable for large $\alpha$. For Dirac fermions the choice of positive-energy subspace decides: with that of the full Dirac operator they are stable for small $\alpha$, with the free one unstable for every $\alpha$. Two species of bosons are unstable. In every case the outcome is the same as for electric charges, although essential steps of the abelian proofs fail. The key tool is an electrostatic inequality for matrix-valued charges, uniform in the field, which replaces the uniformly charged sphere of Newton's theorem by the surface charge of a covariant harmonic extension. The stability results hold for every compact gauge group, and most of them for non-smooth fields of finite energy.

We prove the localization dichotomy conjecture for non-periodic Hall insulators on the lattice. That is, we show that the Fermi projector $P$ of a gapped exponentially local Hamiltonian on $l^2(\mathbb{Z}^2) \otimes \mathbb{C}^n$ admits an exponentially localized Wannier basis if and only if $P$ has vanishing Chern number.

A program for fluctuating kinetic theory is proposed in this paper to study nonlinear Lévy-type fluctuation corrections to kinetic PDEs in a unified way. The Vlasov–Fokker–Planck (VFP), Landau, and Boltzmann equations can each be decomposed into a kinetic transport term and a generalized gradient structure generated by a nonlinear dual pair of dissipation potentials. Going beyond the linear Onsager structure and the classical fluctuation–dissipation relation, we propose a new nonlinear Lévy-type fluctuation–dissipation relation that identifies the thermal fluctuations associated with this generalized gradient structure. On a regular tilted class, the canonical action of the generalized gradient structure agrees with the Freidlin–Wentzell-type action predicted by the resulting SPDE. This consistency gives thermodynamic support for the new nonlinear fluctuation–dissipation relation and leads to a master fluctuating kinetic equation. We further establish a sufficient condition on the nonlinear dual dissipation potential under which its centered cumulant admits a Lévy–Khintchine representation, and show that the resulting Lévy current can be realized through Gaussian white noise and a compensated Poisson random measure. As an application of this framework, an \(\alpha\)-stable Lévy-type Dean-Kawasaki equation is derived. Furthermore, by rescaling the fluctuating Boltzmann equation under a distinguished relation between the noise intensity and the hydrodynamic scaling parameter, we prove that its Hamiltonian converges to the Hamiltonian of the Landau–Lifshitz–Navier–Stokes equations. This reveals a scaling relation between kinetic fluctuations and fluctuating hydrodynamics.

HOMFLY-PT polynomials describe analytic continuation of Wilson averages in Chern-Simons theory to arbitrary complex values of two independent parameters $q$ and $A$. They also depend on the knot and representation of the gauge group. However, when $q^2$ is a primitive $m$-th root of unity, e.g. $q=\pm e^{i\pi/m}$, these polynomials exhibit a universal (knot-independent) factorization: $H_R$ becomes the product of the polynomials for the $m$-core of the Young diagram $R$ and of one and the same factor $H_{[m]}$ for every $m$-ribbon of $R$. In all the examples which we computed this factor is just the special polynomial at $A^m$. At $q=\pm1$ factorization follows from the singularity of the cabled HOMFLY, which is the same for all representations. At $m>1$ the relevant singularity is in the projector, and what survives of it is a finite sum of braids $X_m$ – the $m$-th power sum inside the cable, which can be freely moved through the other strands. For the proof we use the Murnaghan-Nakayama rule and the Adams operation. Discovery of this hidden structure explains the old factorization puzzle.

We prove that, on a Galton-Watson tree with almost surely finite offspring and infinite offspring mean, the random interchange process has critical parameter zero for infinite cycles, conditionally on survival. Combined with the finite-mean strict inequality established in arXiv:2503.03319, this gives a dichotomy for locally finite Galton-Watson trees with offspring mean in $(1,\infty]$: the loop and link critical parameters coincide exactly in the infinite-mean case. The proof reduces the tree problem to a finite permutation estimate derived from peeling estimates for random surfaces.

We consider the adjacency matrix of a uniformly random simple $d$-regular graph on $N$ vertices. For every fixed $d\geq3$ and every fixed bulk energy, we prove that the rescaled eigenvalue point process converges to the $\mathrm{Sine}_1$ process with intensity $1/\pi$. We also establish universality of consecutive gaps at deterministic bulk labels and of fixed-energy correlation measures. The eigenvector input consists of polynomial estimates for mixed fourth moments on a uniformly sampled star. These estimates prevent loss of variance in the qualitative joint Gaussian limits of Backhausz–Szegedy, yielding independent variance-one Gaussian waves for fixed tuples at nearby energies. Exact switching identities then give a stationary marked flow and the microscopic loop hierarchy, which identifies the point-process limit. The appendices prove the stronger correlation and gap statements using counting-moment bounds and conditional log-gas laws for tagged limits.