Papers

We show that the percolation model induced by the double random current representation of the Ising model on $\mathbb Z^d$, with $d\geq 3$, satisfies local uniqueness of the macroscopic cluster with high probability for every $\beta>\beta_c$, uniformly in boundary conditions (i.e. sources). Such a result is sometimes called supercritical sharpness in percolation theory and yields a detailed description of the model via standard renormalisation techniques. As applications, we show the exponential decay of truncated correlations for the Ising model and exponential mixing for the FK-Ising measure, results previously obtained by Duminil-Copin, Goswami and Raoufi. However, our proof is considerably simpler, as it replaces the use of a highly technical multi-valued map principle with a simple sprinkling argument.

Onsager solved the two-dimensional Ising model in zero field, but no solution is known in a magnetic field. We prove that in any dimension $d\ge2$, at every fixed dimensionless field outside a countable set, the free energy obeys no algebraic differential equation in the coupling, even locally. The obstruction arises from droplet geometry: the maximal droplet volume grows as the $d/(d-1)$ power of its boundary size, while an integer band in the pressure coefficients protects the extremal contributions against cancellation. Our result excludes Onsager-type solutions and closed forms in elliptic, hypergeometric, or Painlevé functions of the coupling at these fields.

We extend Hintz's separation of variables for radiative gravitational and electromagnetic perturbations from Kerr-Newman to Kerr-Newman-de Sitter black holes. On subextremal backgrounds with nonzero rotation and charge, an explicit algebraic matrix transformation yields coupled two-component radial and angular ordinary differential systems linked by a common separation constant. The transformation is independent of the frequency and azimuthal number and applies to both signs of spin weight. For real frequencies, we establish a complete angular mode decomposition. We also translate regularity at the future black hole outer and cosmological horizons into ingoing and outgoing boundary conditions, respectively, for the separated radial systems. These results show that exact separation of the coupled radiative equations extends beyond asymptotically flat Kerr-Newman spacetime.

We prove a new extension principle for the spherically symmetric Einstein–Maxwell–charged scalar field system. Rather than imposing conditions for mass ratio $2m/r$ as in Christodoulou's small-mass-ratio criterion or as in An–Chen–Singh's proof [1] with $2m/r<3/8$, here we work through a proposal by An-Tan [2] with finite Doppler exponent (blueshift). By conducting two successive blow-up analyses, followed by a stationary comparison and a rigidity argument, we establish the uniform boundness of $\alpha'$ with $\alpha'=\partial_v\alpha/\partial_vr$ and $\alpha=\partial_v(r\phi)/\partial_vr$. This provides the desired extension principle in An-Tan [2] and completes the picture of weak cosmic censorship developed therein. As we were about to release our preprint, Cicortas communicated to us that together with his collaborators, they concomitantly used a different approach to obtain the corresponding extension principle with finite blueshift (Doppler exponent) for the uncharged Einstein-scalar field system. Hence, we decided to put the current paper as an appendix to An-Tan [2], thereby completing the proof of the weak cosmic censorship for the Einstein-Maxwell-charged scalar field system. At the same time, we are working with Cicortas and his collaborators to write a new preprint containing both approaches for the Einstein-scalar field system.

We present a new quantitative, scale-critical $C^1$ extension principle for the spherically symmetric Einstein–scalar field system. We derive estimates for solutions in the causal past of a possibly singular endpoint $\mathcal{O}$ of the center. Assume that there is no trapped surface in this region and assume that the Doppler exponent (blue shift) along each incoming null cone is uniformly bounded. We then prove that the solution can extend regularly across $\mathcal{O}$ provided the mass ratio $\mu=2m/r$ satisfies $\sup\mu<\frac38$. This improves the small-mass-ratio condition in Christodoulou's extension principle by an explicit threshold of order one. Our proof has three new ingredients. First, we derive several new uniform scale-critical pointwise estimates for the scalar field and its derivatives. Second, we obtain refined estimates in an exterior self-similar domain within the triangular causal past region. Third, we establish a family of higher-order Morawetz-type identities and identify the ranges of $\mu$, with which there are coercive spacetime estimates. The unweighted identity yields the threshold $\mu\leq\frac14$, while employing extra self-similar weight raises the threshold to $\mu<\frac38$ and provides a weighted control, which implies that the solution's BV norm along an outgoing null cone is small, so that Christodoulou's theory of bounded variation solutions gives the extension.

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.

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.

Kemp and Ramgoolam proposed a basis of monomials for a commutative semisimple algebra with generators \(C_1,\ldots, C_L\), constructed from a degeneracy graph which records how the eigenspaces of \( C_1\) are successively split by \( C_2,\ldots, C_L\). They showed that the basis property is equivalent to the non-vanishing of the determinant of the matrix of values of these monomials on the projector basis, and conjectured a factorised formula for this determinant, as a product of powers of eigenvalue differences. We prove this formula for any number of generators. The proof uses graph analogues of Newton polynomials. Their evaluation matrix is not triangular in general, but expanding the rows of the children of each node of the penultimate layer in a Newton basis of their eigenvalues makes it block triangular, with diagonal blocks given by smaller degeneracy graphs in which the final layer is removed. The formula then follows by induction on the number of layers. We show that the exponents in the determinant are overlap volumes of higher-dimensional Young diagrams attached to the nodes of the graph: intervals, ordinary Young diagrams, plane partitions and their higher analogues. Finally, the block triangular structure gives an explicit recursive inversion of the evaluation matrix, expressing the projectors in the monomial basis.

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.

We revisit the effective radial Schrödinger equation for a charged spinless particle in collinear static electric and magnetic fields that was formulated in our earlier work within the tridiagonal representation approach (TRA). In that model the angular factor of the electric coupling is replaced by the orientation parameter $C=\cos\theta$, the linear Zeeman term is diagonal in the azimuthal quantum number, and the effective field-modified central potential takes the form $V(r)=\Vt(r)+qECr$. Here we show that the same equation can be solved exactly without the TRA, by the Nikiforov–Uvarov (NU) method, once the potential component $\Vt(r)$ is prescribed. Three examples are worked out in closed form: a Kratzer-type component, which yields a linear-plus-Coulomb-plus-inverse-square central potential; a pseudoharmonic-type component, which yields a linear-plus-harmonic-plus-inverse-square potential; and a Hulthén-type component with a repulsive core (Manning–Rosen form), which yields a screened confining potential with a finite number of $s$-wave bound states. For each case we obtain the energy spectrum, the normalized eigenfunctions in terms of Laguerre or Jacobi polynomials, the location of the potential minimum, and closed-form radial moments. The spectra are independent of the orientation $\theta$ and of the electric field, in agreement with the structure found by the TRA, while the field enters through the shape of $V(r)$, the radial moments, and the expectation value $\langle V\rangle_k$. All analytical results are confirmed by an independent finite-difference diagonalization.

Let $W$ be a complex reflection group and $X$ the De Concini-Procesi wonderful model of its reflection arrangement. Inside Etingof's sheaf of Cherednik algebras on $X$ we define a subsheaf generated by the Dunkl operators that extend the standard ones on the hyperplane complement. We conjecture that at $t=0$ the image of the logarithmic vector fields in a natural module over this subsheaf is the sheaf of Gaudin algebras on $X$ defined by Aguirre, extending a construction of Bonnafé and Rouquier from the hyperplane complement to $X$. We prove the conjecture for $W=S_{3}$.

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.

We study the relation between the exact-WKB Delabaere–Dillinger–Pham Stokes jumps and the multi-instanton expansion of a one-dimensional potential with $N$ equal-depth minima. For a single barrier this relation is the Bogomolny–Zinn-Justin cancellation between the instanton–anti-instanton ambiguity and the Borel ambiguity of perturbation theory, and we extend it to every consecutive barrier interval and to the integration contours themselves. In the dilute large-$T$ regime at fixed nonperturbative grade, we derive the quasi-zero mode integral over the instanton separations from the Euclidean path integral and reduce it to Gamma and Hankel periods. With the local Weber level function of each well and one normalization per well fixed by the local Weber connection matrix, these periods reproduce the two lateral connected exact-WKB coefficients. The two lateral contours of each instanton–anti-instanton separation differ by exactly one Hankel cycle, and this linear Picard–Lefschetz relation reproduces the Stokes jump of the connected coefficient. A numerical solution of the Schrödinger equation for a symmetric triple well confirms the resulting two-barrier level splitting. At higher fixed grade, the map from ordered event sequences to normalized scalar coefficient products is not one-to-one. At leading dilute order the scalar weight of a closed event sequence depends only on the numbers of barrier crossings, $\mathcal I\bar{\mathcal I}$ pairs, and $\mathcal I\mathcal I$/$\bar{\mathcal I}\bar{\mathcal I}$ pairs. We construct two inequivalent grade-six triple-well event sequences with the same product of local Weber amplitudes, which embed in every potential with $N\ge3$ wells. Our results hold within the regular dilute, one-loop, local Weber approximation.

We give an elementary proof that if $f$ is a locally constant function on a topologically mixing subshift of finite type, then the equilibrium measures for $tf$ converge as $t\to\infty$, a result due to Brémont. The proof shows that the limit measure assigns an algebraic number to every cylinder.

We study the linear energetic stability of non-Abelian vortices in an $SO(3)$ gauge-Higgs theory with residual gauge group $A_4$. In our previous work, we identified two families of such vortices associated with order-two and order-three elements of $A_4$, and found Bogomol'nyi–Prasad–Sommerfield-like behavior at special ratios of the Higgs and gauge-boson mass scales. Here, we investigate their stability against $SO(3)$ gauge and Higgs perturbations by analyzing the second order variation of the vortex tension using analytical and numerical methods. In particular, when all relevant mass scales are equal, we prove linear stability for all vortices in the order-three family. By contrast, at the same mass point, we prove that a vortex in the order-two family is unstable. Thus, although both vortex families exhibit Bogomol'nyi–Prasad–Sommerfield-like behavior in their reduced equations at this mass point, their stability properties in the full $SO(3)$ theory are qualitatively different.

Scattering of sound waves propagating in a compressible fluid is linked to the quantum dynamics generated by a class of non-Hermitian Hamiltonian operators. The scattering amplitude is expressed in terms of the corresponding evolution operator. This is used to provide a new proof of the reciprocity theorem in acoustic scattering, conduct analytic low-frequency acoustic scattering calculations, and study the scattering of sound waves by a finite planar array of delta-function point scatters embedded in an infinite acoustic slab. A detailed discussion of acoustic scattering properties of a homogeneous acoustic slab including a pair of delta-function point scatters on one of its faces is provided.

Computational compressive imaging, such as single-pixel imaging and lensless imaging, is a promising imaging technique, enabling the reconstruction of objects from under-sampled measurements through compressed sensing algorithms or deep learning methods. However, the demand for electronic computational resources grows rapidly when recovering high-resolution and multi-dimensional images and training neural networks with numerous parameters, which has become a bottleneck for the widespread adoption of this technique. Here, we propose a physics-driven reconstruction framework that formulates imaging inverse problems as Ising models and directly recovers target objects through the self-evolution of non-von Neumann Ising machines toward the ground state of the Hamiltonian. We employ a simulated-annealing-based Ising solver to demonstrate this proof-of-concept by reconstructing objects in both single-pixel imaging and a lensless diffuser camera at sampling rates ranging from 1% to 25%. By further incorporating a foreground-sparsity constraint and a smoothness constraint, the reconstruction quality of our method is comparable to or even better than that of mainstream algorithms. This approach has the potential to substantially reduce the electronic computational burden of inverse reconstruction, paving the way for real-time, low-power computational imaging.

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 $e^{\mu a^2}e^{\lambda a^{\dagger2}}$ 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 product and beyond the convergence of its Fock series. On the real slice of parameter space, an iterated form of the integral also reaches every point where the series converges. The analysis rests on a weak composition lemma whose hypotheses concern only the bra, the ket and the resolving family. For bounded chains the same identity is unconditional, and worked examples of that kind are given in a companion paper.

We study random walks with geometric or log-gamma increments, reweighted by certain exponential functionals. We establish precise asymptotics for all moments of exponential functionals, revealing a phase transition as the drifts of the walks vary. To describe the large time limit of reweighted random walks, we introduce a discrete $T$ transformation on random walk paths and establish Girsanov relations. As a corollary, we show that the stationary measures of geometric last passage percolation and the log-gamma polymer on a strip converge to their half-space counterparts as the strip width tends to infinity. In the log-gamma case, this provides an alternative proof of a recent result in arXiv:2607.17030.

By means of $W$-representation, we investigate the Wishart-Laguerre $\beta$-ensemble and present a compact expression for Catalan polynomials. For a special Nekrasov partition function associated with the Wishart-Laguerre $\beta$-ensemble, we show that it can be expressed as a exponential form of the special Catalan polynomials. Furthermore, we make a rule to calculate the Catalan polynomials by means of Feynman diagrams and present another expression for Catalan polynomials. We also establish the connection between certain Catalan polynomials and knot theory.

In this paper we introduce the model of Percolative MDLA (PMDLA) on $\mathbb{Z}^2$ which engulfs the entire occupied percolation component of each particle that reaches the aggregate. We prove a phase transition of finite time explosion, as a function of the particle density $\lambda$, where the critical value equals $\lambda_c=-\log(1-p_c^{\rm site}(\mathbb{Z}^2))$. Below $\lambda_c$, the process is nonexplosive and has a linear upper bound on its radius. For $2/3<\lambda<\lambda_c$, it also has a linear lower bound on the growth rate. Above $\lambda_c$, an infinite component is swallowed in finite time a.s. At $\lambda_c$, every update is finite, but the updates accumulate in finite time, causing a blowup. Excluding criticality, the results are generalized to all higher dimensions.

We show that limit-periodic Jacobi matrices in the weak exponential class $EC(\eta)$ have purely absolutely continuous spectral type when $\eta$ is sufficiently large, depending only on the $R$-boundedness constant. The proof first establishes homogeneity of the spectrum from the weak exponential approximation rate. A closure theorem for reflectionless Jacobi matrices then transfers reflectionlessness from the periodic approximants to the limit, and a theorem of Poltoratski and Remling rules out singular spectrum on a homogeneous set.

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 $\kappa^2>0$ and the field interaction coupling $\alpha>0$. 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 $\Lambda_s$ and the lattice spacing $a\in(0,1]$. As a consequence, the model has a finite free energy per degree of freedom in the thermodynamic limit $\Lambda_s\nearrow\infty$, at least in the subsequential sense. Subsequently, we can also take the ultraviolet or continuum limit $a\searrow 0$. Besides the finiteness property of the free energy, for sufficiently small values of the model coupling parameters $\kappa^2$ and $\alpha$, we show the model satisfies Osterwalder-Schrader positivity and construct the underlying physical quantum mechanical Hilbert space, and, using a polymer expansion, we show correlations also remain bounded in the thermodynamic limit. Finally, we perturbatively compute an infinite volume truncated four-point correlation showing it is nonzero. Noting that, for fixed $a$, perturbation series in $\kappa^2$ and $\alpha$ converges, these results are sufficient to show that our hybrid model makes sense as a lattice quantum field theory in spacetime dimension $d=3,4$, with possible extensions to other higher values of $d$.

Generators of continuous-time quantum Markovian evolution, known as Lindblad operators, play a key role in the theory of open quantum systems. The question of how to define and characterize random Lindblad operators has attracted considerable attention in recent years. Purely dissipative random Lindbladians have been extensively studied using toolboxes of free probability and non-Hermitian random matrix theories, which helped, among other results, to uncover a characteristic lemon-like spectral support. In this work, we focus on random Lindbladians with a Hamiltonian component. We analyze three ensembles of increasing complexity, designed to illustrate the joint effects of dissipative and Hamiltonian dynamics. While the first two ensembles allow for an exact analytic treatment with standard probabilistic and free-probability methods, the generic case of independently random Hamiltonian and dissipative components requires a novel framework that combines operator-valued free probability with sequential noise averaging. For this ensemble, our approach preserves the underlying tensor-product structure of the Lindblad operator and derives a general equation for the spectral boundary. We characterize the effect of Hamiltonian-induced spectral squeezing and explain how the spectral support changes with the relative Hamiltonian strength and the rank of the dissipative component.

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, couple the two shear modes chirally when it depends on the second, and may restrict admissibility: they cap the dislocation strength in Gent and Gent–Gent materials, and contract the admissible traction region in Knowles materials with power-law index below n = 1/2. A weakly nonlinear analysis to third order in the applied tractions and dislocation strength shows that, at first order (linear elasticity), the dislocations leave the shear response unaffected while carrying a residual stress. The chiral coupling enters at second order, through the dependence of the strain energy on the second invariant, so that every dislocated material couples the modes as the Mooney–Rivlin solid with the same second- and third-order moduli. At third order, no new coupling arises; the modes interact through stiffness corrections involving the fourth-order modulus, which the dislocations further correct mode by mode.

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

We analyze the KdV–ghost system arising from the BRST formulation of BF-type topological field theory, its symmetries, Lax pair, spectral problem, zero-curvature formulation and Darboux formalism. The system is invariant under shifts of the ghost field, whose generator does not commute with the BRST symmetry of the KdV–ghost system. Their generators allow the introduction of a systematic procedure for generating new solutions from a given one, and the corresponding expressions in the Grassmann algebra are obtained explicitly. We introduce a Grassmann-valued Lax pair and prove an isospectral theorem relating the spectral problem of the spatial Lax operator for the KdV equation to that of its KdV–ghost extension. We also construct a triangular zero-curvature representation and show that the Grassmann-valued spatial Lax operator is related to the KdV spatial Lax operator by a nilpotent conjugation. The natural variable for a local Darboux formulation is the differentiated ghost $v=C_x$, for which the ghost equation coincides with the Fréchet linearization of the KdV flow. We then construct a nilpotent lift of the scalar KdV Bäcklund–Darboux transformation. Using Riccati variables and off-shell factorization identities for the KdV residue and its Fréchet derivative, we prove that the transformation closes on the local KdV–ghost system. Successive Darboux transformations yield a recursive ghost extension of the bosonic Darboux chain, with a new odd homogeneous contribution introduced at each step. For the solitonic examples, the resulting local ghost fields are naturally identified with tangent directions associated with the soliton phase parameters. The construction provides a Darboux-theoretic interpretation of the ghost sector while remaining distinct from the standard supersymmetric KdV hierarchies.

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 viscoelastic constitutive laws, extend transverse hemitropy, transverse isotropy, and flip symmetry to strain-gradient rods, and derive invariant representations of symmetric stored energies. As an application, we study a planar inextensible cantilever under a compressive terminal force. We prove that the eigenvalues of the linearized problem are simple and coincide precisely with the loads at which bifurcation from the straight state occurs. In addition, each bifurcation point generates a connected global branch of nonlinear equilibria that is locally analytic near its origin.

Compartment models of infectious diseases are often linear in coefficients that are physical parameters or products or ratios of them. We exploit this with closed-form least squares and equip the estimates with distribution-free uncertainty. Physics-informed regression (PIR) fits the parameters to finite-difference derivatives of the data; we introduce physics-informed weak regression (PIWR), which integrates the model in time instead, needs no derivatives of the data, estimates the initial state jointly and is high-order accurate. To quantify uncertainty we wrap both estimators in split conformal prediction, with a calibration population generated by the physics-based model itself, giving simultaneous intervals for all parameters with a finite-sample coverage guarantee. A residual-adaptive factor, computed without ground truth, scales the intervals to each data set (PIWR-RACC). On the eleven benchmark models of disease-informed neural networks (DINNs), PIWR or PIR-FD6 is one to four orders of magnitude more accurate than the reported DINN results wherever the sampling resolves the dynamics, in about a millisecond per fit instead of minutes to hours of training, and the 90% intervals achieve 87-92% coverage in all 88 settings. On the S3I3R model of the original PIR study, PIWR lowers the reported PIR errors by about three orders of magnitude. On real data from the 1978 boarding-school influenza outbreak, a variable-projection variant gives calibrated 95% intervals that overlap those of a Bayesian analysis. On a time-varying transmission benchmark, a B-spline expansion of beta(t) keeps PIR and PIWR linear; they are more accurate than physics-informed neural networks in three of four cases and comparable in the noisiest, with simultaneous bands for beta(t). The complete conformal calibration takes less than two minutes for every benchmark.

We study reversible Higgs bundles on a compact Riemann surface $\Sigma$ of genus $g\geq 2$, that is, $G$-Higgs bundles $(E,\Phi)$ with $(E,\Phi)\cong(E,-\Phi)$. For a complex reductive group $G$ realized as a self-adjoint subgroup of $GL(n,\mathbb{C})$, we show that if $(E,\Phi)$ corresponds to a representation $\rho$ of $\pi_1(\Sigma)$, then $(E,-\Phi)$ corresponds to $(\overline{\rho}^{\,t})^{-1}$. As a consequence, the moduli space of reversible Higgs bundles is the fixed locus of an involution of the character variety and carries the structure of a real affine algebraic set. For a connected semisimple group and every topological type, we prove that the reversible locus in the smooth moduli space is a $(B,A,A)$-brane. We then describe reversibility through spectral data: in even rank, for a smooth spectral curve, reversible Higgs bundles correspond to line bundles invariant under the involution $\eta\mapsto-\eta$; in odd rank, under natural genericity assumptions for which the spectral curve is nodal, we give a gluing criterion together with an explicit polystability condition. Finally, for $SL(2n,\mathbb{C})$ we construct a relative Fourier–Mukai transform over the regular locus of the Hitchin base and prove that it maps the structure sheaf of the reversible locus to an object whose support is exactly the regular degree-zero $PSp(2n,\mathbb{C})$-locus in the Langlands dual $PGL(2n,\mathbb{C})$-moduli space. This support is of type $(B,B,B)$.