We study Nash equilibrium problems with mixed-integer variables in which each player solves a mixed-integer optimization problem parameterized by the rivals' strategies. We distinguish between standard Nash equilibrium problems (NEPs), where parameterization affects only the objective functions, and generalized Nash equilibrium problems (GNEPs), where strategy sets may additionally depend on rivals' strategies. We introduce a branch-and-cut (B&C) algorithm for such mixed-integer games that, upon termination, either computes a pure Nash equilibrium or decides their non-existence. Our approach reformulates the game as a bilevel problem using the Nikaido--Isoda function. We then use bilevel-optimization techniques to get a computationally tractable relaxation of this reformulation and embed it into a B&C framework. We derive sufficient conditions for the existence of suitable cuts and finite termination of our method depending on the setting. For GNEPs, we adapt the idea of intersection cuts from bilevel optimization and mixed-integer linear optimization. We can guarantee the existence of such cuts under suitable assumptions, which are particularly fulfilled for pure-integer GNEPs with decoupled concave objectives and linear coupling constraints. For NEPs, we show that suitable cuts always exist via best-response inequalities and prove that our B&C method terminates in finite time whenever the set of best-response sets is finite. We show that this condition is fulfilled for the important special cases of (i) players' cost functions being concave in their own continuous strategies and (ii) the players' cost functions only depending on their own strategy and the rivals' integer strategy components. Finally, we present preliminary numerical results for two different types of knapsack games, a game based on capacitated flow problems, and integer NEPs with quadratic objectives.

As the number of Common Vulnerabilities and Exposures (CVE) continues to grow exponentially, security teams face increasingly difficult decisions about prioritization. Current approaches using Common Vulnerability Scoring System (CVSS) scores produce overwhelming volumes of high-priority vulnerabilities, while Exploit Prediction Scoring System (EPSS) and Known Exploited Vulnerabilities (KEV) catalog offer valuable but incomplete perspectives on actual exploitation risk. We present Vulnerability Management Chaining, a decision tree framework that systematically integrates these three approaches to achieve efficient vulnerability prioritization. Our framework employs a two-stage evaluation process: first applying threat-based filtering using KEV membership or EPSS threshold $\geq$ 0.088), then applying vulnerability severity assessment using CVSS scores $\geq$ 7.0) to enable informed deprioritization. Experimental validation using 28,377 real-world vulnerabilities and vendor-reported exploitation data demonstrates 18-fold efficiency improvements while maintaining 85.6\% coverage. Organizations can reduce urgent remediation workload by approximately 95\%. The integration identifies 48 additional exploited vulnerabilities that neither KEV nor EPSS captures individually. Our framework uses exclusively open-source data, enabling immediate adoption regardless of organizational resources.

Volume comparison theorem is a type of fundamental results in Riemannian geometry. In this article, we extend the volume comparison result in \cite{Besse2008} to the comparison of total $σ_l$-curvature with respect to $σ_k$-curvature ($l<k$). In particular, we prove the comparison holds for metrics close to strictly stable positive Einstein metric with $l<\frac{n}{2}$. As for negative Einstein metrics, we prove a similar comparison result provided certain assumptions on sectional curvature holds for the manifold.

In this paper, we investigate integrated sensing and communication (ISAC) in a cell-free (CF) multiple-input multiple-output (MIMO) network, where each access point functions either as an ISAC transmitter or as a sensing receiver. We devote into the ISAC sensing metric using the discrete-time signal-based Cramer-Rao lower bounds (CRLBs) for joint location and velocity estimation under arbitrary power allocation ratios under the deterministic radar cross section assumption (RCS). Then, we consider the power allocation optimization problem for the CF MIMO-ISAC as the maximization of the communication signal-to-interference-plus-noise ratio (SINR), subject to CRLB-based sensing constraints and per-transmitter power limits. To solve the resulting nonlinear and non-convex problem, we propose a penalty function and projection-based modified conjugate gradient algorithm with inexact line search (PP-MCG-ILS), and an alternative method based on a modified steepest descent approach (PP-MSD-ILS). We show that the proposed algorithms are scalable and can be extended to a broad class of optimization problems involving nonlinear inequality constraints and affine equality constraints. In addition, we extend the PP-MCG-ILS algorithm to the pure sensing scenario, where a penalty function-based normalized conjugate gradient algorithm (P-NCG-ILS) is developed for sensing power minimization. Finally, we analyze the convergence behavior and qualitatively compare the computational complexity of the proposed algorithms. Simulation results confirm the accuracy of the derived CRLBs and demonstrate the effectiveness of the proposed power allocation strategies in enhancing both sensing and overall ISAC performance.

Petabytes of archival high time resolution observations have been captured with the Murchison Widefield Array. The search for Fast Radio Bursts within these using established software has been limited by its inability to scale on supercomputing infrastructure, necessary to meet the associated computational and memory requirements. Hence, past searches used a coarse integration time, in the scale of seconds, or analysed an insufficient number of hours of observations. This paper introduces BLINK, a novel radio interferometry imaging software for low-frequency FRB searches to be run on modern supercomputers. It is implemented as a suite of software libraries executing all computations on GPU, supporting both AMD and NVIDIA hardware vendors. These libraries are designed to interface with each other and to define the BLINK imaging pipeline as a single executable program. Expensive I/O operations between imaging stages are not necessary because the stages now share the same memory space and data representation. BLINK is the first imaging pipeline implementation able to fully run on GPUs as a single process, further supporting AMD hardware and enabling Australian researchers to take advantage of Pawsey's Setonix supercomputer. In the millisecond-scale time resolution imaging test case illustrated in this paper, representative of what is required for FRB searches, the BLINK imaging pipeline achieves a 3687x speedup compared to a traditional MWA imaging pipeline employing WSClean.

The factorization proposal claims that the co-dimension one "pinning defect", on which a local relevant operator is integrated, factorizes the space into two halves in general conformal field theories in the infrared limit. In this letter, we study a two-dimensional long-range Ising model at criticality with a line defect or an interface, which physically corresponds to changing the local temperature on it. We show that in the perturbative regime, it is not factorizing even in the infrared limit. An intuitive explanation of the non-factorization is that the long-range Ising model is equivalent to a local conformal field theory in higher dimensions. In this picture, the space is still connected through the "extra dimension" across the defect line.

Latency is a key indicator of Internet service performance. Continuously tracking the latency of client requests enables service operators to quickly identify bottlenecks, perform adaptive resource allocation or routing, and mitigate attacks. Passively measuring the response latency at intermediate vantage points is attractive since it provides insight into the experience of real clients without requiring client instrumentation or incurring probing overheads. This paper presents PIRATE, a passive approach to measure response latencies when only the client-to-server traffic is visible, even when transport headers are encrypted. PIRATE estimates the time gap between causal pairs - two requests such that the response to the first triggered the second - as a proxy for the client-side response latency. Our experiments with a realistic web application show that PIRATE can estimate the response latencies measured at the client application layer to within 1 percent. A PIRATE-enhanced layer-4 load balancer (with DSR) cuts tail latencies by 37 percent.

We prove that unital graph C*-algebras often admit a convenient decomposition into amalgamated free products. We use this to give a complete characterization of when a unital graph C*-algebra is residually finite-dimensional and when it is operator norm stable (that is, matricially semiprojective).

Machine learning (ML) models hold the promise of transforming atomic simulations by delivering quantum chemical accuracy at a fraction of the computational cost. Realization of this potential would enable high-throughout, high-accuracy molecular screening campaigns to explore vast regions of chemical space and facilitate ab initio simulations at sizes and time scales that were previously inaccessible. However, a fundamental challenge to creating ML models that perform well across molecular chemistry is the lack of comprehensive data for training. Despite substantial efforts in data generation, no large-scale molecular dataset exists that combines broad chemical diversity with a high level of accuracy. To address this gap, Meta FAIR introduces Open Molecules 2025 (OMol25), a large-scale dataset composed of more than 100 million density functional theory (DFT) calculations at the $ω$B97M-V/def2-TZVPD level of theory, representing billions of CPU core-hours of compute. OMol25 uniquely blends elemental, chemical, and structural diversity including: 83 elements, a wide-range of intra- and intermolecular interactions, explicit solvation, variable charge/spin, conformers, and reactive structures. There are ~83M unique molecular systems in OMol25 covering small molecules, biomolecules, metal complexes, and electrolytes, including structures obtained from existing datasets. OMol25 also greatly expands on the size of systems typically included in DFT datasets, with systems of up to 350 atoms. In addition to the public release of the data, we provide baseline models and a comprehensive set of model evaluations to encourage community engagement in developing the next-generation ML models for molecular chemistry.

In this paper, we develop a method to extract the Bekenstein-Hawking entropy of $D$-dimensional black holes using the entanglement entropy of a lower-dimensional conformal field theory (CFT). This approach relies on two key observations. On the gravitational side, the near-horizon geometry of extremal black holes is AdS$_{2}$, and the Bekenstein-Hawking entropy is entirely determined by this two-dimensional geometry. Moreover, the higher-dimensional spherical part of the black hole metric is absorbed into the $D$-dimensional Newton's constant $G_{N}^{\left(D\right)}$, which can be effectively reduced to a two-dimensional Newton's constant $G_{N}^{\left(2\right)}$. On the field theory side, the entanglement entropy of two disconnected one-dimensional conformal quantum mechanics (CQM$_{1}$) can be calculated. According to the Ryu-Takayanagi (RT) prescription, this entanglement entropy computes the area of the minimal surface in the AdS$_{2}$ geometry. Since the near-horizon region of the black hole and the emergent spacetime derived from the entanglement entropy share the same Penrose diagram -- with both the black hole event horizon and the RT surface corresponding to specific points on this diagram -- the Bekenstein-Hawking entropy can be probed via entanglement entropy when these points coincide. This result explicitly demonstrates that the entanglement across the event horizon is the fundamental origin of the Bekenstein-Hawking entropy.

Signed networks capture the polarity of relationships between nodes, providing valuable insights into complex systems where both supportive and antagonistic interactions play a critical role in shaping the network dynamics. We propose a separable temporal generative framework based on multi-layer exponential random graph models, characterised by the assumption of conditional independence between the sign and interaction effects. This structure preserves the flexibly and explanatory power inherent in the binary network specification while adhering to consistent balance theory assumptions. Using a fully probabilistic Bayesian paradigm, we infer the doubly intractable posterior distribution of model parameters via an adaptive Metropolis-Hastings approximate exchange algorithm. We illustrate the interpretability of our model by analysing signed relations among U.S. Senators during Ronald Reagan's second term (1985-1989). Specifically, we aim to understand whether these relations are consistent and balanced or reflect patterns of supportive or antagonistic alliances.

Genes are connected in complex networks of interactions where often the product of one gene is a transcription factor that alters the expression of another. Many of these networks are based on a few fundamental motifs leading to switches and oscillators of various kinds. And yet, there is more to the story than which transcription factors control these various circuits. These transcription factors are often themselves under the control of effector molecules that bind them and alter their level of activity. Traditionally, much beautiful work has shown how to think about the stability of the different states achieved by these fundamental regulatory architectures by examining how parameters such as transcription rates, degradation rates and dissociation constants tune the circuit, giving rise to behavior such as bistability. However, such studies explore dynamics without asking how these quantities are altered in real time in living cells as opposed to at the fingertips of the synthetic biologist's pipette or on the computational biologist's computer screen. In this paper, we make a departure from the conventional dynamical systems view of these regulatory motifs by using statistical mechanical models to focus on endogenous signaling knobs such as effector concentrations rather than on the convenient but more experimentally remote knobs such as dissociation constants, transcription rates and degradation rates that are often considered. We also contrast the traditional use of Hill functions to describe transcription factor binding with more detailed thermodynamic models. This approach provides insights into how biological parameters are tuned to control the stability of regulatory motifs in living cells, sometimes revealing quite a different picture than is found by using Hill functions and tuning circuit parameters by hand.

Problems related to Perron-Frobenius operators (or transfer operators) have been extensively studied and applied across various fields. In this work, we propose neural network methods for approximating solutions to problems involving these operators. Specifically, we focus on computing the power series of non-expansive Perron-Frobenius operators under a given $L^p$-norm with a constant damping parameter in $(0,1)$. We use PINNs and RVPINNs to approximate solutions in their strong and variational forms, respectively. We provide a priori error estimates for quasi-minimizers of the associated loss functions. We present some numerical results for 1D and 2D examples to show the performance of our methods. We also demonstrate the applicability of our methods by approximating interior densities in a two-cavity system.

In this paper, we show that for a graph $Γ$ from a class named H-rigid graphs, its subgraph ${\rm Int}(Γ)$, named the internal graph of $Γ$, is an isomorphism invariant of the graph product of hyperfinite II$_1$-factors $R_Γ$. In particular, we can classify $R_Γ$ for some typical types of graphs, such as lines, cyclic graphs and infinite regular trees. As an application, we also show that for two isomorphic graph products of hyperfinite II$_1$-factors over H-rigid graphs, the difference of the radius between the two graphs will not be larger than 1. Our proof is based on the recent resolution of the Peterson-Thom conjecture.

We developed a surface-electrode ion trap with a square hole measuring $40\,\mathrm{μm}$ for atomic loading. The hole was fabricated using anisotropic etching of a silicon substrate and was designed to minimize potential distortion in the trapping region. By introducing the atomic beam through the hole, we achieved enhanced isotope selectivity and experimentally demonstrated the selective trapping of calcium isotope ions using an atomic oven. We successfully prepared isotope ion pairs directly from the oven via sympathetic cooling at a rate comparable to that achieved using ablation loading. The sympathetic cooling process occurred on the order of a few seconds. We demonstrated the direct generation of an ion chain above the through-hole. This approach can be applied for trapping a wide range of ion species using a remarkably simple experimental setup, making it desirable for several applications such as quantum-charge-coupled-device (QCCD) architectures and precision measurements of isotope shifts.

In his foundational study of $p$-adic Hodge theory, Faltings introduced the method of almost étale extensions to establish fundamental comparison results of various $p$-adic cohomology theories. Scholze introduced the tilting operations to study algebraic objects arising from $p$-adic Hodge theory in mixed characteristic via the Frobenius map. In this article, we prove a few results which clarify certain ring-theoretic or homological properties of the tilt of an extension between perfectoid rings treated in the construction of big Cohen-Macaulay algebras.

A novel framework is introduced to formalize identifiability in well-specified but ill-posed linear regression models. The framework is distribution-free and accommodates highly correlated features that may or may not relate to the response, reflecting typical real-data structures. First, the identifiable parameter is defined as the least-squares solution obtained by regressing the response on the largest subset of relevant features whose condition number does not exceed a specified threshold, and the relative risk incurred by using this predictor instead of the optimal one is quantified. Second, simple, verifiable conditions are provided under which a broad class of linear dimensionality reduction algorithms can estimate identifiable parameters; algorithms satisfying these conditions are termed statistically interpretable. Third, sharp high-probability error bounds are derived for these algorithms, with rates explicitly reflecting the degree of ill-posedness. With heavy-tailed features and sufficiently low effective rank, these algorithms achieve convergence rates that improve upon both the minimax least-squares rate and lower bounds for sparse estimation under sub-Gaussian features. Results are illustrated via simulations and a real-data application, in which effective rank grows logarithmically with dimension. The framework may extend to algorithms modeling nonlinear response-feature dependence.

Simplex slicing (Webb, 1996) is a sharp upper bound on the volume of central hyperplane sections of the regular simplex. We extend this to sharp bounds in the probabilistic framework of negative moments, and beyond, of centred log-concave random variables, establishing a curious phase transition of the extremising distribution for new sharp reverse Hölder-type inequalities.

In this work we consider domain walls and end-of-the-world branes in AdS/CFT, holographically dual to codimension-one conformal defects and conformal boundaries respectively. In this setting there is an analogue of the ``bulk point'' singularity in boundary correlation functions, which we use to compare top-down and bottom-up constructions of these systems. For example, for a range of parameters the D3/D5 boundary CFT cannot be imitated by a tensionful end-of-the-world brane coupled to Einstein gravity, and in another range it can be modeled with a negative tension brane. Along the way we compute the central charge $b$ for the M2/M5 boundary CFT.

Given an integer $n \ge 2$, its prime factorization is expressed as $n= \prod_{i=1}^s p_i^{a_i}$. We define the function $f(n)$ as the smallest positive integer such that $f(n)!$ is divisible by $n$. The main objective of this paper is to derive an asymptotic formula for both sums $\sum_{n \le x} f(n)$ and $\sum_{n \le x, n \in S_k} f(n)$, where $S_k$ denotes the set of all $k$-free integers.

An $n$-vertex graph is degree 3-critical if it has $2n - 2$ edges and no proper induced subgraph with minimum degree at least 3. In 1988, Erdős, Faudree, Gyárfás, and Schelp asked whether one can always find cycles of all short lengths in these graphs, which was disproven by Narins, Pokrovskiy, and Szabó through a construction based on leaf-to-leaf paths in trees whose vertices have degree either 1 or 3. They went on to suggest several weaker conjectures about cycle lengths in degree 3-critical graphs and leaf-to-leaf path lengths in these so-called 1-3 trees. We resolve three of their questions either fully or up to a constant factor. Our main results are the following: - every $n$-vertex degree 3-critical graph has $Ω(\log n)$ distinct cycle lengths; -every tree with maximum degree $Δ\ge 3$ and $\ell$ leaves has at least $\log_{Δ-1}\, ((Δ-2)\ell)$ distinct leaf-to-leaf path lengths; - for every integer $N\geq 1$, there exist arbitrarily large 1-3 trees which have $O(N^{0.91})$ distinct leaf-to-leaf path lengths smaller than $N$, and, conversely, every 1-3 tree on at least $2^N$ vertices has $Ω(N^{2/3})$ distinct leaf-to-leaf path lengths smaller than $N$. Several of our proofs rely on purely combinatorial means, while others exploit a connection to an additive problem that might be of independent interest.

The analysis of data from multiple experiments, such as observations of several individuals, is commonly approached using mixed-effects models, which account for variation between individuals through hierarchical representations. This makes mixed-effects models widely applied in fields such as biology, pharmacokinetics, and sociology. In this work, we propose a novel methodology for scalable Bayesian inference in hierarchical mixed-effects models. Our framework first constructs amortized approximations of the likelihood and the posterior distribution, which are then rapidly refined for each individual dataset, to ultimately approximate the parameters posterior across many individuals. The framework is easily trainable, as it uses mixtures of experts but without neural networks, leading to parsimonious yet expressive surrogate models of the likelihood and the posterior. We demonstrate the effectiveness of our methodology using challenging stochastic models, such as mixed-effects stochastic differential equations emerging in systems biology-driven problems. However, the approach is broadly applicable and can accommodate both stochastic and deterministic models. We show that our approach can seamlessly handle inference for many parameters. Additionally, we applied our method to a real-data case study of mRNA transfection. When compared to exact pseudomarginal Bayesian inference, our approach proved to be both fast and competitive in terms of statistical accuracy.

This paper addresses the well-known local maximum problem of the expectation-maximization (EM) algorithm in blind intersymbol interference (ISI) channel estimation. This problem primarily results from phase and shift ambiguity during initialization, which blind estimation is inherently unable to distinguish. We propose an effective initialization refinement algorithm that utilizes the decoder output as a model selection metric, incorporating a technique to detect phase and shift ambiguity. Our results show that the proposed algorithm significantly reduces the number of local maximum cases to nearly one-third for a 3-tap ISI channel under highly uncertain initial conditions. The improvement becomes more pronounced as initial errors increase and the channel memory grows. When used in a turbo equalizer, the proposed algorithm is required only in the first turbo iteration, which limits any complexity increase with subsequent iterations.

MnTe has recently emerged as a canonical altermagnet, a newly identified class of magnetism characterized by compensated antiferromagnetic order coexisting with spin-split electronic bands, traditionally considered exclusive to ferromagnets. However, the extent to which altermagnetism persists as altermagnets are thinned to the two-dimensional (2D) limit remains unexplored. Here, we investigate the magnetic behaviour of 2D MnTe, specifically atomically-thin monolayers (MLs) and bilayers (BLs) grown on graphene/Ir(111) substrate, by combining experimental scanning tunneling microscopy, x-ray photoelectron spectroscopy, x-ray absorption spectroscopy and x-ray magnetic circular dichroism with density functional theory calculations. We find that while ML and BL MnTe adopt atomic structures with symmetries incompatible with altermagnetism, they exhibit intriguing magnetic phases: the BL forms a highly-robust layered antiferromagnet with in-plane spin anisotropy, whereas the ML exhibits a spin-glass-like behavior below its freezing temperature, a phenomenon not previously observed in an atomically thin material. These findings highlight how reduced dimensionality can promote the emergence of unusual magnetic structures distinct from those of their three-dimensional counterparts, providing new insights into low-dimensional magnetism.

Non-Hermitian (NH) Hamiltonians describe open quantum systems, nonequilibrium dynamics, and dissipative processes. Although a rich range of single-particle NH physics has been uncovered, many-body phenomena in strongly correlated NH systems have been far less well studied. The Kondo effect, an important paradigm for strong correlation physics, has recently been considered in the NH setting. Here we develop a NH generalization of the numerical renormalization group (NRG) and use it to solve the NH Kondo model. Our non-perturbative solution applies beyond weak coupling, and we uncover a nontrivial phase diagram. The method is showcased by application to the NH pseudogap Kondo model, which we show supports a completely novel phase with a genuine NH stable fixed point and complex eigenspectrum. Our NH-NRG code, which can be used in regimes and for models inaccessible to, e.g., perturbative scaling and Bethe ansatz, is provided open source.

The importance of Fisher information is increasing in nonequilibrium thermodynamics, as it has played a fundamental role in trade-off relations such as thermodynamic uncertainty relations and speed limits. In this work, we investigate temporal Fisher information, which measures the temporal information content encoded in probability distributions, for both classical and quantum systems. We establish that temporal Fisher information is bounded from above by physical costs, such as entropy production in classical Langevin and Markov processes and the variance of interaction Hamiltonians in open quantum systems. Conversely, temporal Fisher information is bounded from below by statistical distances (e.g., the Bhattacharyya arccos distance), leading to classical and quantum speed limits that constrain the minimal time required for state transformations. We perform numerical simulations on two quantum dot models to validate the obtained bounds. Our work provides a unified perspective on speed limits from the point of view of temporal Fisher information in both classical and quantum dynamics.

Inspired by recent advances in the manipulation of superconducting circuits coupled to mechanical modes in the quantum regime, we propose a protocol for generating superpositions of orthogonally squeezed states in a quantum harmonic oscillator. The protocol relies on a quadratic coupling between the oscillator and a qubit, and is conceptually similar to methods used for preparing cat states in qubit-oscillator systems. We numerically evaluate the robustness of the state-preparation scheme in the presence of decoherence, considering environmental coupling for both the harmonic oscillator and the qubit. As a potential application, we introduce a quantum error-correcting code based on conditionally-squeezed states and analyze its error-mitigation properties.

We study community recovery in the planted partition model in regimes where the number and sizes of communities may vary arbitrarily with the number of vertices. In such highly unbalanced settings, standard accuracy or overlap-based metrics become inadequate for assessing recovery performance. Instead, we propose the correlation coefficient between partitions as a recovery metric, which remains meaningful even when the number or sizes of communities differ substantially. We then analyze a simple common-neighbor-based clustering rule which groups two adjacent vertices if they share more than one common neighbor. We establish explicit recovery conditions under sparse inter-community connectivity, without requiring prior knowledge of the model parameters. In particular, in graphs of size $n$, this algorithm achieves exact recovery for communities with sizes $Ω(\log n)$, almost exact recovery for sizes $ω(1)$ and weak recovery for sizes $Ω(1)$. In contrast to most existing results, which assume (nearly) balanced communities, our method successfully recovers small and heterogeneously-sized communities, and improves existing guarantees even in some balanced settings. Finally, our results apply to community sizes that follow a power-law distribution, a characteristic frequently found in real-world networks.

Surface release of radiatively active particles, with high infrared- (IR-)to-visible extinction ratios, has been proposed as a method of warming Mars. However, to warm Mars using aerosols, particles released locally must disperse globally. Here we provide an initial reference study in a plume tracking, dry Martian atmospheric model to address this question. The winds that transport aerosols respond to the aerosol's IR forcing, implying strong radiative-dynamical feedbacks (RDF). We investigate RDF from surface release of two particle compositions: carbon (graphene) and metal (Al). Self-lofting helps particles rise and spread locally and regionally, and the Hadley cell strengthens under warming, aiding latitudinal mixing. Within our model, Mars RDF enable engineered-aerosol warming. Warming is slightly greater for three-dimensional vs. 1D-models and also depends on spectral resolution of radiative transfer. We assess implications for Mars warming. Many open atmospheric science questions remain, including the role of agglomeration, dry-deposition rate uncertainty, and modeling water cycle feedbacks.

Controlled stacking of van der Waals materials is a powerful tool for exploring the physics of quantum condensed matter. Given the small binding between layers, exploitation for engineering will require a breakthrough in stacking methodology, or an ability to take advantage of thicker defective stacks. Here we describe computational groundwork for the latter, using -- on account of its promise for cold memory applications -- 1$T$-TaS$_2$ as a model system. Comparing recursive Hendricks-Teller calculations and Monte Carlo simulations to published X-ray diffraction data, we obtain the key parameters describing the random stacking in mesoscopic flakes. These then regulate the electronic structures via specification of the random stacks in dynamical mean-field theory simulations. Hubbard repulsion induces strongly correlated metallic, band and Mott insulating layers, providing compelling evidence that electronic properties follow from the coexistence of more than the metallic and insulating planes associated by ordinary band theory.