We present a unified theoretical framework for the mapping between diffusive and wave-like dynamics, formulated as a spectral integral operator acting on temporal fields. By introducing an analytic continuation in the complex frequency plane, we establish an explicit correspondence between thermal diffusion and a virtual wave field governed by a hyperbolic equation. This mapping is shown to define a causal, compact Fredholm operator that acts as a nonstationary low-pass filter, thereby revealing the intrinsic information loss of diffusive processes and the fundamental ill-posedness of the inverse reconstruction. Within this operator framework, we demonstrate that commonly used excitation schemes-including pulse, lock-in, chirped, and coded excitations-emerge as distinct projections onto subspaces of a single underlying transformation, corresponding to different sampling strategies of its spectral structure. This unifies previously disparate virtual wave formulations and provides a systematic interpretation of excitation design in terms of operator sampling and information encoding. The framework further generalizes to matrix-valued systems and suggests a spectral-geometric interpretation of temporal evolution across diffusive and propagative regimes.

Let $f$ be a holomorphic automorphism of a compact Kähler manifold $X$ with simple action on cohomology. We show that every ergodic measure with sufficiently large entropy is supported on the Julia set of $f$. In particular, when $X$ is a surface, any ergodic measure with positive entropy is supported on the Julia set. The proof relies on quantitative estimates for the speed of convergence towards the Green currents of $f$, with respect to a suitable norm on an adapted functional space of non-necessarily closed currents.

We consider some extremal combinatorial questions for bipartite graphs definable in stable one-based (and related) structures. We show that they satisfy both strong Erdős-Hajnal property and linear Zarankiewicz. We also show that the same is true for both collapsed and uncollapsed Hrushovski's ``ab initio'' constructions, and discuss some connections to Zilber's trichotomy principle. For strong Erdős-Hajnal, we show that in fact it holds in a more general class of $1$-semi-equational theories.

We present the Minimal Retroreflective Microfacet (MRM) model, which turns any existing microfacet BSDF into a physically plausible retroreflective one by a single substitution: replacing the view direction with its reflection about the surface normal before evaluating the standard model. Based on the previously published back-vector formulation, MRM requires only minimal code changes and has been adopted in the OpenPBR and MaterialX material standards. We prove reciprocity and energy conservation under the assumption of a reflection-symmetric normal distribution function (NDF), which holds for all commonly used distributions, and validate the model against measured retroreflective material data.

We show that if $L$ is an extremal Type II lattice of rank $72$, then $L$ is generated by its vectors of norm $10$. The proof determines the full inner product distribution between the shells of norms $8$ and $10$ using the spherical $11$-design property of the norm-$10$ shell.

We study degree-$2$ Siegel theta series of extremal even unimodular lattices from the genus-$2$ viewpoint initiated by Ozeki. Using Igusa's structure theorem, we define a depth filtration on genus-$2$ cusp forms, measured by the total degree in $χ_{10}$ and $χ_{12}$, and relate it to the vanishing of low Fourier--Jacobi coefficients forced by extremality. In ranks $48$, $72$, and $96$, this interaction closes exactly and yields a direct genus-$2$ proof that the degree-$2$ theta series is uniquely determined by extremality (conditional on existence in rank $96$). In rank $120$ (again conditional on existence), the same argument leaves a one-dimensional residual line spanned by $χ_{10}^6$: with $χ_{10}$ in the standard integral normalization, any two such degree-$2$ theta series differ by an integer multiple of $χ_{10}^6$.

Wave resistance is the drag generated by the wave radiation that a source moving at a fluid interface sustains. Under stochastic trajectories, the mean drag is controlled by the ensemble-averaged surface profile built from the trajectory history. We show that the result is a finite resistance below the deterministic radiation threshold and a regularization of the singular response at the minimum phase velocity of the capillary-gravity waves. We derive explicit scaling laws for drifted Brownian trajectories, including a universal high-diffusivity decay. For drifted Lévy flight, we find the mean wave resistance in closed-form, extending wave-drag theory to non-Gaussian trajectories.

In this paper, we propose a finite transposition method to solve backward stochastic differential equations (BSDEs, for short). Based on the transposition solution theory for BSDEs, our method offers a promising way of efficiently computing solutions, which can be regarded as an analogous method for BSDEs as the classical finite element method for partial differential equations. Our method has the advantage of easily computable conditional expectations.

The prediction-powered inference (PPI) proposed by Angelopoulos et al. (2023) is a popular method that leverages a small number of labeled samples and machine learning predictions for semi-supervised inference. While several variants of PPI have appeared in the literature, its rigorous statistical theory has not been fully developed. In this paper, we study the statistical optimality of PPI. Our contributions span both foundational theory and new methodology. First, we frame PPI as an M-estimation problem, revealing a link between the bias-corrected PPI estimating equation and the ideal full-data estimating equation. This connection leads to the consistency and asymptotic normality of the PPI estimator under simple random sampling without replacement. Next, we identify the efficient influence function and prove that PPI can attain the semiparametric efficiency lower bound when the predictor is score-calibrated, that is, when the predictor's output aligns with the true conditional expectation of the estimating function. Finally, for learned prediction rules, we develop asymptotic theory for cross-fitting and for a single-fit variant with variance correction in the special case of semiparametric mean estimation. Simulation experiments and a real-data application support these findings.

The aim of steganography is to hide secret information inside ordinary media so that the existence of communication is hidden rather than encrypted. In audiovisual context, the availability of audio and video streams creates an opportunity to split a payload across these two modes thus, reducing the embedding burden on any single carrier. This paper evaluates whether such split-payload audiovisual steganography can help evade unimodal and multimodal steganalysis under synchronized and asynchronous embedding settings. We create audiovisual samples where the hidden message is divided between the audio and video tracks, and then test how well different detectors can identify them. The single mode detectors performs close to random guessing, thus showing the benefit of this hiding mechanism, while the multimodal model initially appears more effective. However, further checks show that this improvement mostly comes from the video stream, not from a true combined audio-video signal. Overall, the results suggest that splitting the payload across modalities can make detection harder, but multimodal detectors must be evaluated carefully to ensure they are learning the intended signal.

Recently the poset of ranked cactuses $(\mathfrak{P}(X),\preceq)$ was introduced. For a finite set $X$, this poset consists of a set $\mathfrak{P}(X)$ of certain collections of ordered pairs of subsets of $X$ together with an ordering $\preceq$ that is similar to the refinement ordering of partitions of a finite set. In addition, the maximal chains in this poset correspond to binary ranked cactuses, a fact which can be used to construct the so-called space of equidistant cactuses. In this paper, we show that the poset of ranked cactuses is EL-shellable. As a consequence we also show that the proper part of the link of the origin of the space of equidistant cactuses has the homotopy type of a wedge of spheres.

A sufficient account of how the neocortex learns must meet three criteria: 1. Computationally, it must approximate a powerful, general-purpose learning algorithm known to scale to human-level intelligence; 2. Algorithmically, it must be implementable using known, well-established neural circuits within the neocortex and associated brain structures; 3. Implementationally, there must be a detailed account for how all of the algorithmic mechanisms actually function at a neurochemical level. At present, there is only one framework that meets all of these criteria: error-driven predictive learning via temporal derivatives, driven by corticothalamic circuits, based on competitive kinase synaptic plasticity induction mechanisms. This has been implemented in the Axon neural simulation framework using spiking neurons, and demonstrated to learn across a wide range of challenging cognitively motivated tasks.

We propose a theoretical mechanism to indirectly detect the small parity-violating energy difference (PVED) between chiral enantiomers through a macroscopic enantiomeric excess observed in an ultracold gas. We consider that chiral molecules are formed resonantly via ultracold collisions of achiral diatomic molecules, with PVED inducing a slight asymmetry in the resonance energies of right- and left-handed configurations. After formation, chiral molecules evolve within a Bose-Einstein condensate (BEC), incorporating nonlinear interactions, tunneling between enantiomeric states, intrinsic PVED, and thermal conversion rates. These collective dynamics enable amplification of the microscopic bias into a global population imbalance. Using coupled rate equations, we show that, under realistic regimes, a complete enantiomeric excess can be achieved even for extremely small intrinsic asymmetries. We illustrate the model with concrete examples (HSOH, H$_2$Se$_2$, H$_2$Te$_2$), predicting observable enantiomeric excesses under plausible experimental conditions. We also consider non-PVED effects that could be amplified under the proposed mechanism, including electric and magnetic fields as well as thermal fluctuations, the latter being illustrated through the aforementioned molecular examples. Overall, our results suggest that ultracold physics could provide a new pathway to probe molecular parity violation, a fundamental weak effect that remains experimentally undetected.

We make use of the free data formalism developed in \cite{CK25} to construct solutions of the Einstein vacuum constraint equations by integrating in the forward direction. This, together with a new gauge condition based on effective uniformization, allows us to construct general solutions with limited decay at spacelike infinity. In particular, we construct solutions with minimal and even borderline decay, as considered in \cite{Shen23}, \cite{Shen24} in connection with the stability of the Minkowski space. In a forthcoming paper, we make use of the techniques we develop here to identify and construct a general class of short-pulse Cauchy data that lead to the formation of trapped surfaces, extending the well-known result of \cite{LiYu}.

The $k$-center problem is one of the best-studied and most intuitive clustering formulations. It asks, given a set of $n$ points in a metric space, for $k$ of the points to be designated as cluster centers, so that the maximum distance of an input point to its nearest center is minimized. Gonzalez's greedy algorithm from 1985 is a simple and efficient way to find a $2$-approximate solution. The algorithm has the attractive feature of \emph{incrementality}: it outputs the centers one by one, with a guaranteed $2$-approximation for every prefix of the obtained sequence of centers. Incrementality imposes a geometric constraint on how solutions can be built, and it is natural to ask whether this comes at a price in the quality of the solution. It is known that in polynomial time, the approximation ratio of $2$ is best possible, assuming $P \neq NP$. In this paper we show that even with \emph{unlimited} computational power, the factor $2$ cannot be improved, if the solution is required to be built incrementally. The lower bound construction imposes a tradeoff between all $n$ levels of the clustering simultaneously; it was obtained with the help of ChatGPT, an aspect we discuss in Section 3 of the paper.

Correcting numerical dispersion artifacts from Finite Difference solvers is a well-identified challenge in computational wave physics, but existing approaches evaluate only a restricted family of CNN-based architectures and have been applied exclusively to the elastic wave equation. We instantiate the Deep Finite Difference framework on two-dimensional anisotropic acoustic wave propagation, pairing a fourth-order Finite Difference proxy with a Pseudo-Spectral reference over 27,000 heterogeneous velocity fields. We benchmark twelve correction architectures, from linear regression to Fourier Neural Operators, under a unified 10-fold cross-validation protocol.

Let \(V\) be an \(n\)-dimensional vector space over the finite field \(\Fq\), and let \(\Gr{V}{k}\) denote the family of all \(k\)-dimensional subspaces of \(V\). A family \(\cF\subseteq\Gr{V}{k}\) is called intersecting if \(\dim(F\cap F')\ge1\) for all \(F,F'\in\cF\). For a point \(P\le V\), let \(d_P(\cF)\) denote the number of members of \(\cF\) that contain \(P\), and order the point-degrees as \(d_1(\cF)\ge d_2(\cF)\ge\cdots\ge d_{\points{n}}(\cF)\), where \(\points{m}=(q^m-1)/(q-1)\) is the number of points in an \(m\)-dimensional subspace. Recent work of Frankl and Wang~\cite{FW2025} and of Huang and Rao~\cite{HR2026} established that for \(k\)-uniform intersecting families \(\cF\subseteq\binom{[n]}{k}\) with \(n\ge2k+1\), the bound \(\binom{n-2}{k-2}\) governs the order statistic \(d_{2k+1}(\cF)\). We prove that every intersecting family \(\cF\subseteq\Gr{V}{k}\) with \(n\ge2k+1\) satisfies \(d_{\points{k}^{2}}(\cF)\le\qbinom{n-2}{k-2}\). The naive \(q\)-analog of the Huang--Rao \((k+2)\)-th degree theorem fails, as a vector-space Hilton--Milner construction has \(\points{k+1}\) points of degree larger than \(\qbinom{n-2}{k-2}\); we prove the corrected bound \(d_{1+\points{k+1}}(\cF)\le\qbinom{n-2}{k-2}\) for fixed \(q\), sufficiently large \(k\), and \(n>3k\), using a structural theorem of Ihringer and Kupavskii~\cite{IK2026}. For larger degree indices \(i\), a saturated Frankl--Hilton--Milner family of Ihringer and Kupavskii identifies the conjectural sharp bound on \(d_{\points{i+1}}(\cF)\), and we prove two necessary conditions that any strict counterexample to this conjecture must satisfy.

Predicting the output of quantum circuits is a central bottleneck for verifying quantum processors because a generic wavefunction grows exponentially with system size. We introduce a neural statebank that learns this wavefunction along the circuit trajectory. Each layer is stored as an autoregressive Transformer checkpoint trained from local gate updates to the preceding checkpoint, producing a compact neural representation that can evaluate amplitudes and generate independent samples. On long-range circuits combining entanglement, magic, and non-diagonal branching, a 0.3-million-parameter statebank reaches $\sim 10^{-2}$ infidelity at 34 qubits, outperforming the other tested approximate simulators while using far less memory than exact state-vector evolution. The same architecture accurately simulates quantum approximate optimization, Clifford+$T$, and Clifford circuits.

These lecture notes are based on the lectures on \emph{Statistical Mechanics of Random Matrices} delivered at the Spring College on the Physics of Complex Systems, held at the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, from 19 February to 15 March 2024. Their aim is to present a statistical-mechanics route to the spectral theory of sparse and diluted random matrices, with emphasis on cavity and replica methods, resolvent techniques, population dynamics, typical spectral densities, spectral-count fluctuations and large deviations, conditioned spectra, and non-Hermitian extensions. The written form of the notes has been deliberately expanded beyond the material actually covered during the lectures. This is partly because a set of lecture notes can afford a more systematic development than a sequence of blackboard lectures, and partly because several natural continuations of the material become clearer once the central methods have been introduced. Consequently, not every topic discussed here was presented during the College. The additional material is included to give a more coherent account of the subject and to indicate directions that, hopefully, can be covered in greater detail in future lectures or schools. Since these are lecture notes rather than a state-of-the-art review, the choice of topics is necessarily selective and is naturally tilted towards the author's own work and collaborations on this subject. I have nevertheless tried, within the limits of this format, to place the material in contact with the broader literature and to represent the surrounding state of the art as fairly as possible. Inevitably, some relevant contributions may be missing or treated too briefly; such omissions are unintentional and reflect the pedagogical scope of the notes rather than a judgement on their importance.

We develop a regularised arbitrary-gauge formulation of nonrelativistic quantum electrodynamics and use it to compare Coulomb and multipolar descriptions with a Lorentzian form factor. We analyse the effect of regularisation in perturbation theory, including alternative partitions of the Hamiltonian into free and interaction parts, and the limits of the electric dipole approximation. The regularised multipolar gauge exhibits a cut-off-dependent trade-off between the strength of individual interaction terms and the localisation of material subsystems that suppresses direct inter-atomic interactions. We discuss the implications of the framework for short-range phenomena, including Dicke criticality.

Telehealth, the delivery of medical care remotely, is hoped to increase access to specialty services or decrease health care utilization. Physicians can provide telehealth to each other or to patients. Specialists often treat complex patients who can be adequately cared for only in academic hospitals, suggesting that providing specialty services via telehealth will reallocate rather than reduce system utilization. Here I use agent-based modeling to investigate telehealth's effects on clinical outcomes and system utilization in medical toxicology. I found that physician-physician telehealth increased patient health but system utilization did not change. The effects were more pronounced as clinical complexity increased. Physician-patient telehealth increased cost and system utilization but not clinical outcomes. Within the limitations of our approach, these results suggest that telehealth is more cost-effective for improving generalist access to specialist knowledge than in providing care to the public.

This paper is devoted to simultaneously recovering multiple parameters from internal measurements for nonlocal diffusion equations. The uniqueness of the inverse problem is established by employing the asymptotic behavior of solutions, analytic continuation, the Laplace transform, and properties of analytic functions. For numerical reconstruction, we apply the Levenberg-Marquardt method to obtain a stable approximate solution of the inverse problem. Numerical examples are provided to demonstrate the efficiency of the proposed algorithm and to validate our theoretical findings.

We formulate the quotient admission problem for finite graph-window rows. The input is a finite row set, an admissible evidence map, semantic labels, witness-support hypergraphs, and atom-level admissibility predicates. The output is a quotient decision on evidence atoms, with possible decisions certificate, residual, low-confidence, or blocked. The problem asks for the maximal guard-respecting atom-level decision map that uses no refinement beyond the admissible evidence partition. We prove an atom-union characterization of identifiable classes, give a witness-support hypergraph guard for certificate admission, characterize projected-label conflicts as blocked atoms, and present quotient admission algorithms with correctness, maximality, and complexity guarantees. With explicit evidence vectors and hyperedges, the algorithms run in expected O(B + I + n) time and space by hashing and deterministic O(B + I + n log n) time by sorting under a key-linear comparison model, where n is the number of rows, B is the total evidence encoding length, and I is the total hyperedge incidence size. We also prove a magnitude-only indistinguishability lower bound: any evaluator that observes only residual magnitudes fails on instances whose evidence atoms require different residual decisions after the magnitudes collapse them.

For over four decades, Gribov's no-pole condition has been almost exclusively explored in Euclidean space, where the elliptic nature of the Faddeev-Popov operator provides a clear spectral boundary. In physical Minkowski spacetime, however, this operator becomes a hyperbolic wave operator, and the Euclidean positivity criterion collapses. We show that the natural Lorentzian replacement is a real-time boundary-value problem: a gauge configuration remains inside the first Gribov region as long as the Faddeev-Popov wave equation admits no source-free solutions obeying the Feynman boundary condition. For backgrounds localized in time, this condition translates to the injectivity of the negative-frequency block of the classical ghost scattering map. For stationary backgrounds, it becomes a spatial bound-state or threshold-resonance problem under Fourier transformation. Using a Wronskian identity, we prove that pure frequency mixing in stable self-adjoint time-dependent channels is structurally protected and cannot by itself produce the obstruction. While static chromomagnetic backgrounds reproduce the familiar zero-frequency horizon crossing, static chromoelectric potentials reach the horizon at finite, non-vanishing frequency - a uniquely Lorentzian phenomenon arising because $A_0$ couples directly to the ghost time derivative. We also cast the condition in Fredholm form and show that the exact restriction is a functional determinant, which the naive local continuation of the Gribov-Zwanziger action fails to reproduce, leaving the construction of a genuine local real-time action as the central open problem.

We investigate entanglement transfer in a multi-giant-atom waveguide system. By tailoring chiral spontaneous emission and exploiting dark-state dynamics, the setup enables perfect, unidirectional sequential or selective transfer of quantum states and their associated entanglement. The distance between two entangled atoms, i.e., the entanglement length, can be dynamically adjusted, allowing robust conversion between long-range and short-range entanglement during propagation. The system inherently converges to a dark state, guaranteeing high-fidelity directional transfer. When the additional phase is modulated as a periodic piecewise function, spatially separated giant atoms exhibit stable, nearly lossless state exchange and maintain steady entanglement even under non-Markovian conditions. This behavior mimics conventional braided architectures without suffering from propagation delays or spatial restrictions. Our proposal offers a scalable pathway for continuous long-distance entanglement transport and resilient state exchange in quantum networks.

Heterogeneous susceptibility models for epidemic dynamics preferentially assume that individual susceptibility follows a gamma distribution, which permits analytical reduction to a low-dimensional system. However, the true empirical distributional form in any given population is unknown. Here we investigate the consequences of misspecifying the susceptibility distribution by comparing gamma and lognormal specifications in a Susceptible-Exposed-Infectious-Removed (SEIR) framework. When both distributions are matched on mean and coefficient of variation ($ν$), we find that their epidemic trajectories diverge once heterogeneity is moderate or high ($ν\gtrsim 1$), with the lognormal producing a later, larger peak and a greater final size. We then assess the impact of distributional misspecification on statistical inference. Using synthetic datasets, we fit correctly specified and misspecified models by maximum likelihood. In a default scenario, where inference is based on simulated data for a single epidemic, both models can reproduce the data by compensating through correlated shifts in heterogeneity and intervention parameters. When inference is based on two simulated epidemics, however, this compensation may be reduced by known constraints of how parameters are related across epidemics. In these cases, the correctly specified model recovers all parameters accurately, while the misspecified model tends to give biased estimates. These inference biases propagate into forecasts, but predictions remain relatively accurate when compared to homogeneous models which more than double peak incidences in scenarios where $ν\approx 1$, for instance. We conclude that deviations resulting from the susceptibility distribution misspecifications assessed here are minor and encourage the adoption of heterogeneous models in future epidemic forecasting.

Bühlmann--Straub (B-S) credibility assigns each account a weight $Z_i = E_i/(E_i+K)$, where $K$ is a single portfolio-wide ratio. The formula assumes $K$ is the same for every account regardless of size, history length, or volatility, and that recent and older years carry equal weight. On a held-out US commercial auto dataset these assumptions fail: standard B-S applied to 96 companies produces a calibration slope of 29 for small accounts, a signature of severe under-crediting. We propose a joint framework that retains B-S interpretability while addressing these limitations. The credibility weight $Z_i$ is modelled as a logistic function of account characteristics; historical experience is discounted by an EWMA decay parameter $λ$ estimated from the data; and $Z$, $λ$, and the complement are optimised in a single likelihood pass. The framework formally nests Bühlmann--Straub as a special case, admitting a likelihood-ratio test for any proposed extension. On a two-year held-out test set the proposed model restores calibration (slope = 1.00) and reduces exposure-weighted prediction error by 38% (90% bootstrap interval: 26%--50%). A size gradient in the decay rate emerges ($\hatλ\approx 0.6$, $0.84$, $0.13$ for Small, Mid, Large) and replicates qualitatively on Other Liability. A simulation study confirms the mechanisms. The model requires only account-year summaries and delivers three transparent outputs: credibility weight, complement, and recommended renewal rate.

Optimization of continuous-variable objectives on standard gate-based quantum computers via variational algorithms such as QAOA is typically approached by first discretizing each decision variable into a finite binary representation. This increases qubit requirements and restricts solution precision through fixed-resolution encodings. We propose a qubit-efficient variational framework for continuous optimization that instead encodes each decision variable into the squared amplitude or equivalently, the measurement probability of a single qubit state. This removes explicit discretization from the variable representation while remaining entirely within the standard qubit circuit model unlike methods like CV-QAOA employing qumode based hardware to achieve the same. To read out encoded variables, we propose using amplitude estimation rather than naive sampling or tomographic reconstruction, with the goal of improving precision scaling for continuous-value recovery. We outline how amplitude-estimation error propagates to decision-variable error and then to objective-value error under standard regularity assumptions, suggesting a distinct width-versus-precision tradeoff relative to discretized approaches. In particular, the framework replaces the logarithmic increase in qubits needed for finer binary precision with a constant cost of one qubit per decision variable, while shifting accuracy requirements into the estimation procedure. We position this approach relative to traditional discretized variational formulations, and argue that it provides a promising new direction for continuous optimization on standard qubit architectures.

Dynamical mean-field theory (DMFT) reduces dense high-dimensional disordered dynamics to a self-consistent effective stochastic process. For sparse and heterogeneous networks, however, local fields contain finitely many strong inputs, so the Gaussian closure mechanisms of dense DMFT need not apply. We develop a continuous-time cavity derivation of sparse-network DMFT at the level of path measures for stochastic dynamics with general pairwise interactions on sparse random graphs. The cavity equations are exact on trees and yield the finite-time thermodynamic description on locally tree-like graphs. They show explicitly how reciprocity changes dynamical closure: fully directed graphs recover the sparse directed path-probability equation, whereas reciprocal or bidirected edges require conditional path kernels driven by the imposed history of the receiving node. Ensemble averaging gives laws over path-probability messages, with barycenters and higher-message moments closing by multilinearity and independence of incoming branches. A causal discrete-time derivation yields the corresponding population-dynamics representation, distinguishing trajectory populations for directed graphs from conditional branch-law or finite-depth tree populations for reciprocal graphs. We also formulate finite-memory numerical closures and test them in an additive-input recurrent neural network specialization. Finally, high-connectivity limits are obtained as projections of the sparse path-measure theory, clarifying when dense drift, noise, and response channels reduce to standard low-dimensional DMFT and when path-level descriptions remain essential.

The phase prediction of chemically disordered alloys poses a significant computational challenge due to the combinatorial complexity of such materials. The high-throughput compositional exploration of chemically disordered alloys, including high-entropy alloys, requires an approach to efficiently explore the potential energy landscape of such complex materials. Additionally, a metric to quantify the potential energy landscape explored for phase prediction of the compositions needs to be defined. We propose an information-theoretic approach to phase prediction in chemically disordered alloys in the present work. We demonstrate the applicability of alchemical Monte Carlo sampling using an efficient Graph Convolutional Neural Network-Based machine learning model. We additionally demonstrate the applicability and limitations of the Bond Disproportion Vector (BDV) as a low-computational-cost descriptor and benchmark it against the state-of-the-art Smooth Overlap of Atomic Positions (SOAP) descriptor. We show the applicability of an information entropy-based metric for the phase prediction of binary (CoNi, MoW, FeNi and TaW), ternary (CoCrNi, CrFeNi), quaternary (CoCrFeNi) and quinary ($\mathrm{Al_x(CoCrFeNi)_{1-x}}$) alloys. Information entropy-based phase prediction can be applicable in challenging cases where conventional approaches are not feasible.