Papers

Motivated by the work of Ohira and the advantages of the offset linear canonical transform (OLCT) over the Fourier transform (FT), this paper proposes an OLCT based framework for solving a class of delay differential equations. By exploiting the operational properties of the OLCT, the original delay differ- ential equation is transformed into a Volterra-type delay integral equation in the transform domain and solved numerically using Brunners method of steps . An explicit analytical solution is derived for a special case to investigate the effect of the delay parameter. A unified transform-domain formulation is also established by relating the proposed OLCT approach to its Fourier transform counterpart. Numerical and graphical results demonstrate the accuracy and effectiveness of the proposed method and validate it through comparisons with the Fourier transform based formulation of Ohira . The proposed framework may provide a promising foundation for solving non linear delay differential equations involving transcendental terms, as both the OLCT and the resulting Volterra integral equation possess the essential properties required to handle non linearities and transcendental terms.

Oscillator Potts machines (OPMs) provide a physics-inspired, energy-minimization framework for solving combinatorial optimization problems described by the $q$-state Potts Hamiltonian. Although OPMs may be viewed as multistate extensions of oscillator Ising machines (OIMs), here, we show that they exhibit dynamical properties absent in the binary case. Specifically, we derive a configuration-dependent local-stability condition for a recently proposed multiharmonic OPM formulation and show that configurations with the same Potts energy need not be dynamically equivalent. In particular, for $q\geq4$, permutations of the Potts labels can alter the Jacobian spectrum and, consequently, the regularization strength required to locally stabilize a given Potts configuration. Thus, different phase encodings of the same Potts solution can exhibit different local stability properties despite having identical Potts energies.

We study how first-order differential observables affect the zero-frequency behavior of black-hole wave equations. Quasinormal modes and late-time tails do not always change in the same way. We give a simple condition for when an observable removes the leading threshold term of a Green function. For Schwarzschild Regge-Wheeler modes, the relevant operator is determined by the regular static solution. The transformation is nondegenerate at nonzero frequency but becomes globally degenerate at zero frequency. As a result, the nonzero quasinormal-mode problem is unchanged, while the leading fixed-radius late-time tail gains one extra inverse power of time.

We formulate a precise sense in which a BRST anomaly may possess a classical phase space shadow. It is the non-equivariance class of a Hamiltonian symmetry action on covariant phase space. To organize this statement, we introduce the Weil covariant phase space bicomplex, whose basic subcomplex controls equivariant Hamiltonian lifts. We then compare this purely classical class with local BRST descent. For a Hamiltonian admissible mixed transgression, the ghost-number-two descendant $a^{2}_{d-1}$ defines a deghostification map to the charge algebra, which is generated by the failure of the Cartan representative to be equivariant. The induced cohomology class is independent of Bardeen type changes of descent representative. Thus the classical anomaly is a shadow of the BRST anomaly in the precise sense that both are realizations of the same Weil transgression class, while remaining objects of different theories. As an explicit example, we analyze a five dimensional inflow transgression and show that the classical charge cocycle and the BRST descendant arise as two realizations of the same Weil transgression class.

We prove that the negative $r$-spin tautological relations of Chidambaram–Garcia-Failde–Giacchetto are a consequence of Pixton's $3$-spin relations, for every $r\ge2$, and deduce that they hold in the tautological Chow ring of $\overline{\mathcal{M}}_{g,n}$. In particular, our result for $r=2$ implies that Kazarian–Norbury $K$-relations hold in Chow. The proof factors through two essential parts. The first part shows that the tautological relations arising from Chiodo's class imply the negative $r$-spin relations. This requires an intricate study of limiting behavior of the Givental–Teleman graph contributions. In the second part, we show that the Chiodo relations are a consequence of Pixton's $3$-spin relations as an application of a generalization of Janda's work on tautological relations from CohFTs.

We show that for a horizontally stretched Bunimovich billiard, for almost every stretching parameter, there exists a sequence of eigenfunctions concentrating in the rectangular part. Thanks to the work of Markarian–Oliffson Kamphorst–Pinto de Carvalho, we know that for an open set of parameters such billiards are ergodic. The proof is a result of interaction with ChatGPT 6: we were curious if it could establish the existence of bouncing ball modes for Bunimovich or Sinai billiards. That was not successful until we suggested varying the wings. That produced an essentially complete argument for the existence of quasimodes. We then suggested using Hassell's parameter-variation argument which led to the result about eigenfunctions. The argument was then significantly simplified and clarified by the authors.

There are infinitely many functions that satisfy the Black-Scholes partial differential equation and the terminal condition corresponding to the European call option. This means that the Black-Scholes formula, which led to the award of the 1997 Nobel Prize in Economic Sciences, is not the unique solution, as was once assumed. Consequently, it violates the law of one price, one of the fundamental laws of economics and finance. In this article, we present a rigorous derivation of these solutions to the Black-Scholes boundary value problem.

I prove that all Mellin residues are determined exactly by integer samples known up to errors smaller than every algebraic order, for an explicit class of functions consisting of a slow component with a complete inverse-power expansion and countably many carriers $\exp(2\pi i\sigma 2^k a^x)$ with admissible envelopes. A single exceptional null set of parameters $a>1$ works for the entire function class. The key estimate is an eventual uniform lower bound for a polynomially weighted sampling Gram matrix on windows of length $N^{1/2}$, with $O(N^{1/4})$ active modes. I derive this bound from parameter-averaged oscillatory integrals and a finite trace-moment comparison with product Haar measure, retaining the exact resonances among dyadic frequencies. Summed Taylor estimates then identify every coefficient of the slow asymptotic expansion, and a Mellin continuation argument identifies these coefficients with the actual residues. I also give an alternative derivation from existing higher-correlation estimates, an infinite-mode example, and exact counterexamples at integer parameters. The result supplies an interpolation-invariance statement for a model class motivated by Mellin averaging in quantum gravity.

We prove that below the multicritical (Nishimori) point the ferromagnetic boundary of a biased Ising spin glass, argued by Nishimori (1986) to be exactly vertical, can be vertical and can also be reentrant, so that cooling at fixed disorder destroys order. It is vertical in the Sherrington-Kirkpatrick (SK) model with a Curie-Weiss bias $J_0$, and reentrant near the triple point of fully connected $p$-spin glasses and, at one disorder strength, on a decorated planar $\mathbb{Z}^2$-periodic lattice. For every $T<J$ the SK magnetization vanishes for $J_0\le J$, continuously at $J_0=J$, and is macroscopic for $J_0>J$. For every integer $p\ge3$, at each bias slightly larger than the triple-point bias, the $p$-spin magnetization is macroscopic at the Nishimori temperature and vanishes at an explicit lower temperature; this is Nishimori's recent replica prediction in two-temperature form. On the lattice ($\mathbb{Z}^2$ with each bond replaced by a fixed series-parallel gadget; maximum degree four), with iid $\pm J$ couplings at $\mathbb{P}(J_e=-1)=9/10000$, the Gibbs state is almost surely unique at high temperature, non-unique in a window containing the Nishimori temperature, and unique again at low temperature. As Dey and Kang observed, the SK boundary is set by the zero-field susceptibility together with an envelope bound on the free-energy gain in a field, which follows from the gauge inequality. We prove that this susceptibility is the curvature at the origin of the zero-field Parisi PDE solution, which Lopatto evaluated for every $\beta>1$ via his theorem that the support of the Parisi measure accumulates at the origin; we give a short alternative proof of that theorem. The $p$-spin and lattice results and the alternative proof are computer-assisted. The models are proxies for the nearest-neighbour $\pm J$ model on $\mathbb{Z}^2$ and say nothing about $\mathbb{Z}^d$.

We show that all normal covers of a fixed finite graph $G$ sitting between the universal and maximal abelian covers share the same atomic part of their spectral measures. We give an explicit double formula for the mass of atoms which generalises and unifies known results on universal covering trees and maximal abelian covers. It also proves an extension of a conjecture in physics literature on hyperbolic lattices. Moreover, we also establish the logarithmic Hölder regularity of the continuous part of the spectral measures. Our proof combines techniques from von Neumann algebras, matching theory and the monotone labelling method. In an appendix, we prove a converse generalised Gallai-Edmonds structural theorem which might be of independent interest.

The order $m$ antisymmetrised matrix product ensemble is represented by the product $X_1^T\cdots X_m^TJX_m\cdots X_1$, where $X_1,\ldots,X_m$ are independent real Ginibre matrices and $J$ is the elementary antisymmetric matrix, while the order $m$ Hermitised matrix product ensemble is represented by $X_1^\dagger\cdots X_m^\dagger HX_1\cdots X_m$, where $X_1,\ldots,X_m$ are now independent complex Ginibre matrices and $H$ is a Hermitian matrix drawn from the Gaussian unitary ensemble. These ensembles have recently been shown to be related to certain Muttalib–Borodin ensembles and integrals of Harish-Chandra–Itzykson–Zuber type, thereby motivating further investigation into their eigenvalue statistics. In this work, we construct ribbon graphs and constellations that are enumerated by the mixed cumulants of these ensembles and give loop equation characterisations for the generating functions of said cumulants when $m=1$.

We study the delayed one-dimensional swarmalator model when all units share a common intrinsic frequency {\omega}. Unlike the zero-frequency case, {\omega} cannot be removed by a rotating-frame transformation because delayed interactions retain the phase accumulated over the lag. The result is a two-order-parameter analogue of the delayed Kuramoto model: the asynchronous state acquires incoherence lobes, the synchronized state becomes a rotating branch stable even when both the spatial and phase couplings are repulsive, and the phase wave develops narrow resonant stability bands. These delay-selected windows overlap to produce broad branch coexistence, including a narrow window near K=-J where all three canonical states-asynchronous, phase wave, and synchronized-are simultaneously stable. In regions where no canonical branch is stable, the order parameters oscillate persistently.

Disk-level scattering amplitudes involving one Ramond–Ramond and two NSNS states on a D-brane obey nontrivial identities among scalar world-sheet integrals imposed by gauge and T-duality Ward identities. We investigate the cohomological organization of these relations using twisted de Rham cohomology and show that the Ward identities admit twisted-exact representatives. In the $I/J$ sector, the unique generic linear dependence among the Ward identities is realized one degree lower in the same twisted de Rham complex, revealing a common cohomological origin for the identities and their syzygy. We also identify independent pointwise algebraic relations among the integrands. An independent cohomological rank computation then establishes that the physically selected $I/J$ family spans a five-dimensional subspace of the top twisted cohomology at generic complexified kinematics. Extending the construction to the $K$-family, including the exceptional double-pole representative $K_3$, introduces no additional cohomology direction. Hence the complete $I/J/K$ family considered here spans the same five-dimensional subspace.

Our recent research on Langevin dynamics with self-harmonic drift and classical spins has revealed a strong connection to “Stochastic Localization," a concept attracting attention in the fields of mathematical geometry and data science. Roughly speaking, the former can be viewed as a version of the latter in which the $n$-dimensional Euclidean space - the domain of the probability distribution function appearing in the latter - is “compactified” to the surface of an $n$-dimensional sphere. By comparing the two frameworks, we argue that this compactification is a consequence of requiring the drift to be autonomous. The relationship between the two approaches may expand the scope of Stochastic Localization.

Understanding what limits many-body quantum entanglement is a central problem in physics. Spatial locality has long provided a fundamental mechanism: correlations between a region and its complement must be mediated through a small spatial interface, thereby constraining their entanglement. Here we show that universal constraints on entanglement can persist even when interactions are strongly nonlocal, through collectivity: many weak interactions generate fluctuations controlled by their square-summed, rather than total, strength. We establish this mechanism rigorously for generic gapped Hamiltonians with Kac-normalized power-law interactions $r^{-\alpha}$ on a $D$-dimensional lattice. For arbitrary bipartitions, we prove that the ground-state entanglement scales at most logarithmically with system size for $\alpha<D/2$ and subextensively for $D/2<\alpha<D$, due to suppressed collective fluctuations around individual sites. For spatially regular bipartitions with codimension-one boundaries, we use a renormalization-group construction to extend this suppression to larger length scales, yielding polylogarithmic scaling for $D/2 < \alpha < (D+1) / 2$ and parametrically stronger subvolume bounds for $(D+1)/2<\alpha<D$. We also establish the corresponding optimality results: for arbitrary bipartitions, the logarithmic scaling for $\alpha<D/2$ is optimal and $\alpha=D/2$ marks the optimal threshold for universal logarithmic bounds, while for regular bipartitions the threshold at $\alpha=(D+1)/2$ is likewise optimal for universal polylogarithmic bounds when $D\ge2$. Together, these results reveal collectivity as a fundamental mechanism for constraining many-body entanglement alongside spatial locality.

We present a high-order finite element arbitrary Lagrangian-Eulerian (ALE) method for hydrodynamics with slip-wall boundary conditions on analytically curved domains. The method combines distinct Lagrangian, remesh, and remap phases while preserving the geometric features of curved boundaries and allowing tangential motion of boundary nodes. In the Lagrangian phase, we use weak boundary condition enforcement to impose the slip-wall condition without freezing tangential motion. In the mesh optimization phase, we adapt TMOP (target-matrix optimization paradigm) to analytic walls by replacing selected boundary coordinate degrees of freedom with curve or surface parameters. The Newton solve then moves boundary nodes tangentially with respect to the prescribed geometry while carrying the drift introduced by the Lagrangian phase. In the remap phase, we transfer the solution from the old mesh to the optimized mesh by a conservative advection-based remap over pseudo-time. We present numerical results in two and three dimensions demonstrating the method on high-order curved meshes and showing that it preserves boundary fidelity while producing the expected hydrodynamic solution features.

In a restricted untwisted sector, Diaconescu, Moore & Witten (2000, 2003) gave a quantum M-theoretic consistency check, at the level of partition functions, of what one may call "Hypothesis K", namely that magnetic and electric type IIA brane charges are jointly quantized in K-theory. We show that the more recently discussed "Hypothesis H" — that magnetic and electric charges in 11D are jointly quantized in coHomotopy — together with a topological smallness condition on the M-theory circle, systematically implies that IIA charges are quantized in a nonabelian deformation of twisted K-theory by the previously neglected non-linear NS1-brane charge. This is essentially a derivation of twisted K-theory from IR-completed 11D SuGra, and conversely a further consistency check on Hypothesis H from Hypothesis K. We motivate and precisely state this result, discuss its structural implications, and outline the elaborate algebro-topological proof.

Several physically interesting Lindbladians, such as those describing generic open quantum system models as well as engineered state preparation processes, are often rapidly mixing, that is, they relax to steady states in time scaling polylogarithmically with the system size. Owing to the short mixing times, it is physically expected that local perturbations do not substantially increase the mixing time. In this work we question this expectation and ask whether rapid mixing survives. We provide counterexamples showing that rapid mixing is not stable in general, even when the perturbation does not change the steady state. We also show that arbitrarily weak local Hamiltonian perturbations can destroy rapid mixing. Complementing these results, we develop an array of sufficient conditions which imply stable rapid mixing, including several classes of quasifree fermionic systems, commuting Lindbladians, and local spin systems. We also identify conditions under which modified logarithmic Sobolev inequalities (MLSIs), a stronger form of convergence, lead to stable rapid mixing. As a further implication of our results, we show that rapid mixing does not imply a positive MLSI constant.

The familiar wave kinetic theory revolves around turbulent energy cascades characterized by random phases of the individual normal modes. By contrast, recent years have revealed turbulent cascades where the transfer of energy is mediated by phase-coherent dynamics. Such coherent cascades have been previously established in deterministic systems with a tightly controlled algebraic structure of the mode interactions, such as nonlinear Schrödinger equations. Here, we show that the presence of phase-coherent cascades is robust with respect to substantial alterations of the mode couplings. We prove that, in infinite-dimensional resonant Hamiltonian systems with fully resonant normal mode spectra and random nonlinear couplings, a structured subset of couplings — an infinitesimal fraction of the total — suffices to support coherent cascades characterized by phase alignment, the formation of power-law spectra, and either unbounded growth of Sobolev norms or a finite-time blow-up. Numerical simulations further demonstrate the emergence of coherent cascades from random initial data through a process of phase locking, suggesting a mechanism for their emergence in broader classes of initial data and systems. We thus show that disordered mode interactions can sustain, rather than block, coherent energy transport across arbitrarily distant scales.

We determine asymptotic global fidelities of cloning generic mixed states with simple (non-degenerate) full-rank spectra. In fixed dimension $d$, let $n$ inputs and $m_n$ outputs satisfy $m_n/n\to\gamma>1$. For a known simple full-rank spectrum $p_1>\cdots>p_d>0$, we prove that the optimal root fidelity between the entire output and the product target state is $$ F_\rm{orb}(\gamma,p) =\prod_{i<j} \frac{\sqrt{\gamma}+\sqrt{q_{ij}(\gamma-1+q_{ij})}} {\gamma+q_{ij}}, \qquad q_{ij}=\frac{p_j}{p_i}. $$ When the spectrum is unknown, one spectrum-independent cloner attains $$ F_\rm{univ}(\gamma,p) =\left(\frac{2\sqrt\gamma}{1+\gamma}\right)^{(d-1)/2} F_\rm{orb}(\gamma,p), $$ uniformly on compact subsets of the simple full-rank chamber and is minimax optimal. The two factors quantify spectral and eigenbasis uncertainty. Our channels combine classical processing of Schur–Weyl labels with a covariant Cartan-sector channel. Local asymptotic normality and a Gaussian fidelity bound against arbitrary channels yield the converses. For flat projector states $\rho=P/r$, with $P$ a rank-$r$ orthogonal projector, we separately prove the optimum $\gamma^{-r(d-r)/2}$. Furthermore, we estimate the exact asymptotic fidelity of purify–clone–trace (PCT), and show that our cloners strictly outperform PCT at every fixed gain for simple full-rank spectra and rank-$r$ projector states with $1<r<d$.

We study the topological band theory of quantum Hall systems in which the adiabatic flux parameters of an external gauge potential form a high-dimensional manifold. This situation arises both for strongly correlated matter on spatial Riemann surfaces of genus $g > 1$, as well as in the formal study of matter on compact oriented manifolds of dimension $4\ell+2 = 2p$ coupled to $p$-form gauge potentials. In these cases, the underlying topological invariants defined over the parameter space are significantly richer than the low-dimensional (i.e., two-dimensional) case and involve both the curvature (i.e., first Chern class) as well as higher order curvature invariants. These higher curvature invariants correspond to topologically protected contributions to Kubo-like formulae, constructed from correlation functions of the physical current operators. We illustrate these general considerations with an explicit example from hyperbolic band theory based on the genus-two Bolza Riemann surface, a case which has recently been simulated on a synthetic-dimension platform.

We prove that the Gibbs state of any spin chain can be exactly decomposed into a mixture of product states over contiguous blocks of qubits with bounded length, at any constant temperature. As a consequence, any such Gibbs state can be prepared by local unitary circuits of constant depth, with efficient classical preprocessing. We place strong limits on the spatial structure of thermal entanglement: both the entanglement depth and entanglement width (measures of multiparticle entanglement) are bounded by constants, while the localizable entanglement (the bipartite entanglement that can be generated with the aid of local measurements and classical communication) vanishes exactly beyond a constant threshold distance. Our result also has implications for quantum thermalization: for any translation-invariant spin-chain Hamiltonian with nondegenerate spectral gaps, typical low-complexity pure states drawn from a maximally entropic ensemble at any constant effective temperature become locally indistinguishable from the Gibbs state upon unitary evolution. Additionally, we obtain an analogous decomposition for fermionic systems: the Gibbs state of any 1D local fermionic Hamiltonian is exactly a mixture of constant-depth local fermionic circuits acting on Gaussian states.

We introduce Solver Agent, an AI framework based on large language models for calculations and proofs in mathematics and theoretical physics. The solution process is tracked through a persistent ledger that records assumptions, derivations, and computations. A central agent delegates tasks to specialized sub-agents, while independent agents verify both intermediate steps and the final result. This setup improves the traceability, reproducibility, and verification of computer-assisted calculations. Applying Solver Agent, we study global F-theory uplifts of Type IIB orientifolds and their S-fold generalizations. We establish sufficient conditions for Weierstrass models over projective threefolds with terminal $\mathbb{Z}_k$ quotient singularities ($k\in\{2,3,4,6\}$) to give $\mathbb{Q}$-factorial projective elliptically fibered Calabi-Yau fourfolds with isolated Gorenstein terminal quotient singularities. These geometries realize O3-planes and S-folds, where local D3-brane probes of the latter yield four-dimensional $\mathcal{N}=3$ superconformal field theories. Using stringy invariants, we derive fixed-point contributions to Hodge data and Euler characteristics, and show that these Euler corrections determine the localized D3-brane charges required for tadpole cancellation. We illustrate these results using toric hypersurface constructions, where a single three-dimensional polytope determines both the Type IIB Calabi-Yau threefold and the F-theory base; here, the orientifold double cover naturally forms a bisection of an alternative genus-one-fibered uplift with discrete $\mathbb{Z}_2$ gauge symmetry. Finally, we provide methods for toric computations and four-form flux analysis in four-dimensional $\mathcal{N}=1$ compactifications with non-abelian gauge sectors.

Lattice gauge theories are an important class of physical models with high-dimensional structured distributions that underpin first-principles calculations in particle, nuclear, and condensed-matter physics. Recent advances in generative modeling have opened new avenues for sampling Boltzmann distributions in lattice gauge theories, offering the potential to alleviate limitations of traditional Monte Carlo methods, including critical slowing down and topological freezing. However, generative samplers for lattice gauge theories are typically constructed in link space, where correlations become increasingly long-ranged toward weak coupling, posing a challenge for learning. Plaquettes provide a more natural representation, as the action is local in these variables. However, exact Bianchi constraints restrict them to a lower-dimensional manifold, complicating direct generative modeling. We introduce a multilevel normalizing-flow construction that samples directly in plaquette space while satisfying these constraints exactly. The key idea is a coarse-to fine factorization that transforms a globally coupled constraint problem into a sequence of local solves: at each refinement, every determined plaquette depends on at most four newly generated variables, while the size of the constraint problem remains independent of the lattice size. We validate the construction for $U(1)$ in two and four dimensions and for $SU(2)$ in two dimensions. Our multilevel Plaquette-Space Sampler (PSS) substantially outperforms link-space baselines, with the advantage increasing toward weak coupling, where the gauge coupling becomes small. This regime is particularly challenging for generative sampling and, in asymptotically free gauge theories, is relevant to continuum studies.

We introduce power variations of critical Gaussian multiplicative chaos (GMC) along refining partitions in arbitrary dimension. Under suitable renormalisation, we prove uniform fractional moment bounds via Laplace transform estimates as well as stable convergence to supercritical GMCs. Together, these results provide a unified approach to regularity and spectral questions for critical chaos, yielding: (i) the sharp exponent for the logarithmic modulus of continuity, (ii) the existence of non-empty essential spectrum for critical Liouville quantum gravity surfaces; and (iii) an almost-sure quantitative Fourier decay.

We prove that for every choice of four positive masses and every cyclic ordering of the bodies there is exactly one strictly convex planar central configuration with that ordering, up to similarity. This answers Problem 10 in the list of Albouy, Cabral and Santos, which Santoprete calls the Simó-Yoccoz conjecture. The main step is a uniform lower bound for the Hessian at convex central configurations: it is at least one quarter of its radial part, which vanishes only on translations and rotations. Dziobek's relations turn the indefinite part of the Hessian into a negative multiple of a square, and a Cauchy-Schwarz argument bounds this term by the radial part times the trace of an explicit $2\times 2$ matrix in which the masses do not appear. We prove that this trace is less than 3/4 by interval arithmetic on the three-dimensional set of normalized convex central configurations, in the coordinates of Corbera, Cors and Roberts. Uniqueness then follows by a covering argument from the case of four equal masses. The computation has been repeated with a second, independently written program. The Hessian bound and the uniqueness theorem, including the computation, have been formalized in the Lean proof assistant and checked by its kernel, using only the standard axioms. As consequences, every degenerate four-body central configuration is concave, the convex central configuration depends analytically on the masses, and the known symmetry theorems for kites, isosceles trapezoids and rhombi follow in a few lines. The analytic dependence and the symmetry theorems are also formalized.

Pólya and Szegő conjectured in 1945 that the disk achieves the largest electrostatic capacity among all planar sets with given logarithmic capacity. This question generalizes naturally to balls and Newtonian capacities in higher dimensions. The conjecture remains open. This paper establishes a conditional result: the conjecture would follow if among all lower dimensional sets in a horizontal strip with Neumann boundary conditions, the transverse component of the Dirichlet integral of the equilibrium potential could be shown minimal for the lower dimensional ball. As supporting evidence toward this latter property, the paper shows extremality of the ball for $L^p$-norms of the potentials, via Baernstein star-function symmetrization techniques.

Connecting mathematical formalism in open quantum systems to its underlying physics necessitates the notion of a dilation, a way to bridge stochastic dynamics with deterministic Schrödinger evolution on a larger space. Although dilations are well known in the literature for states, channels, and quantum combs, there is a substantial gap when it comes to the fully general setting of non-Markovian dynamics on a continuous interval. In an accompanying work, we introduce a continuous process tensor (cPT) framework wherein these processes are represented by vectors in multi-species bosonic Fock spaces of $L^2$ functions. This accommodates all multi-time statistics of a quantum system on an interval. Here, we present a corresponding dilation theorem and show that all cPTs can be approximated arbitrarily well by an explicit closed description with system and environment coupled by a bounded Hamiltonian. As well as its physical appeal and demonstrating completeness of the framework, we show that such regular representations actually underpin the formalism, defining a dense subspace of all cPTs that is easy to handle. This leads to a continuum-suitable version of Choi duality, and permits the rigorous treatment of continuum controls.

We study the edge scaling limit of the logarithms of the squared singular values of multiplicative Brownian motion on the general linear group $\mathrm{GL}_N(\mathbb{C})$ with general initial conditions. Building on earlier work [arXiv:2605.06429], we prove that, for every initial condition in the enhanced state space, the limiting dynamics are determinantal and derive an explicit double-contour formula for their extended multitime correlation kernel. The dependence on the initial state, including information not determined by the particle coordinates, is encoded by the associated Laguerre–Pólya entire function. We further extend the corresponding infinite-dimensional stochastic differential equation description to arbitrary initial configurations, including those with coinciding coordinates.

Zero-knowledge proofs (ZKPs) enable the verification of a statement without revealing any information beyond its validity and constitute a fundamental primitive in cryptography and information theory. However, existing constructions rely on computational assumptions and are predominantly confined to bipartite settings, leaving their information-theoretic realization in bipartite or network scenarios largely unexplored. Here we develop a framework for zero-knowledge verification based on the indistinguishability of quantum states under operational constraints. Exploiting the fundamental limitations imposed by local operations, we show that a verifier is inherently restricted from extracting information about the underlying state while retaining the ability to verify correctness. We construct explicit protocols for multiparty quantum networks that achieve information-theoretic security, ensuring that no subset of collaborating parties can gain knowledge beyond the validity of the statement, independent of their joint computational power.