Moraga and Yeong conjectured that for a smooth complex projective variety $X$ of dimension $n$, an ample line bundle $A$ on $X$ and an integer $m \ge 3 n + 1$, very general elements of the adjoint linear system $|ω_{X} \otimes A^{\otimes m}|$ are algebraically hyperbolic. We prove the conjecture for spherical varieties with smooth orbit closures. As a corollary, we conclude that the conjecture holds for horospherical varieties, and for toroidal spherical varieties. Furthermore, for any spherical variety, we show that the conjecture holds modulo the complement of an open dense orbit.

We explore a cosmological framework in which a Gauss-Bonnet (GB) coupled scalar field, acting as dark energy, interacts with a fermionic dark matter field through a coupling obtained from the point of view of particle physics. This setup is inspired by string/M-theory, and two representative scalar field potentials are investigated: exponential and power-law. A distinctive feature of the GB-coupled models is their potential to alter the propagation speed of gravitational waves (GWs), a property with significant implications in light of recent multi-messenger astrophysical observations. To account for this, we analyze models under two scenarios: one where the GW speed differs from that of light and the other where they are equal, but all consistent with current observational constraints. The dynamical evolution of the system is investigated by reformulating the field equations into an autonomous dynamical system, enabling a detailed analysis of the Universe's long-term behavior, including the radiation-, matter- and dark energy-dominated epochs. We constrain the model parameters using a broad set of recent observational data, including mock high-redshift measurements from the Roman Space Telescope. Our findings indicate that both potentials yield cosmologies that are in excellent agreement with current data, closely tracking the expansion history predicted by the standard \(Λ\)CDM model, while still allowing room for subtle deviations that could be tested by future observations.

Information about peers' performance is pervasive in workplaces, yet its effects on worker behavior are mixed. We show that a key reason is that workers differ in how they value such information. In a real-effort experiment with 793 workers, we elicit willingness-to-pay for peer information delivered either before or after the task. We document substantial heterogeneity in demand for peer information: some workers are indifferent, some prefer to avoid it before the task, and others value it more as their relative performance increases. These differences strongly predict effort responses to peer information. Notably, 15% of workers would pay to avoid information ex ante due to stress and exhibit no productivity gains from it. We further show that uniform feedback policies can impose welfare losses on such workers, while tailoring the timing of peer information increases welfare by up to 48%. Our results highlight the importance of accounting for heterogeneous information preferences when designing workplace feedback policies.

For an $S$-valued function $f$ of $m \geq 1$ variables we consider the dynamical process in which the output $f(\overline{v})$ replaces exactly one entry of the input $\overline{v} \in S^m$ at each step. This can be viewed as a special case of multivariate polynomial semigroup dynamics, and our study focuses on periodic vectors with respect to this process. We define a stratification of periodic vectors according to their type, and characterize types for which the determination of periodic vectors comes down to dynamics of univariate polynomials. We then restrict to the case of a diagonal binary quadratic form $f$ over $\mathbb{Q}$, and classify rational periodic vectors for all types of period up to $5$. This includes two types, of periods $4$ and $5$, which do not arise from the univariate case, and we prove that there are no periodic vectors over the rationals of the single non-univariate type of period $4$.

Using the fact that Hopf-Galois structures on separable extensions and skew bracoids are both intrinsically connected to transitive subgroups of the holomorph of a finite group, we present algorithms to classify and enumerate these objects for small degree, and apply them to obtain significant extensions to existing results. We also explore the classifications of these structures of degree $2pq$, where $p$ and $q$ are distinct odd primes. We conclude with some enumeration-inspired observations and a conjecture.

Automatic vision inspection holds significant importance in industry inspection. While multimodal large language models (MLLMs) exhibit strong language understanding capabilities and hold promise for this task, their performance remains significantly inferior to that of human experts. In this context, we identify two key challenges: (i) insufficient integration of anomaly detection (AD) knowledge during pre-training, and (ii) the lack of technically precise and context-aware language generation for anomaly reasoning. To address these issues, we propose ADSeeker, an anomaly task assistant designed to enhance inspection performance through knowledge-grounded reasoning. ADSeeker first leverages a curated visual document knowledge base, SEEK-M&V, which we construct to address the limitations of existing resources that rely solely on unstructured text. SEEK-M\&V includes semantic-rich descriptions and image-document pairs, enabling more comprehensive anomaly understanding. To effectively retrieve and utilize this knowledge, we introduce the Query Image-Knowledge Retrieval-Augmented Generation Q2K RAG framework. To further enhance the performance in zero-shot anomaly detection (ZSAD), ADSeeker leverages the Hierarchical Sparse Prompt mechanism and type-level features to efficiently extract anomaly patterns. Furthermore, to tackle the challenge of limited industry anomaly detection (IAD) data, we introduce the largest-scale AD dataset, Multi-type Anomaly MulA, encompassing 72 multi-scale defect types across 26 categories. Extensive experiments show that our plug-and-play framework, ADSeeker, achieves state-of-the-art zero-shot performance on several benchmark datasets.

We rigorously analyze the non-equilibrium thermodynamic behavior of various formulations of quantum Brownian motion (QBM) using the framework of stochastic thermodynamics. While the widely used Caldeira-Leggett master equation exhibits desirable thermodynamic features, such as the fulfilment of a detailed balance, it fails to ensure complete positivity. In contrast, several completely positive and trace-preserving (CPTP) extensions turn out to be thermodynamically controversial. We show that such extensions introduce anomalous phase-space structures that violate detailed balance at the steady state, leading to non-vanishing entropy production and effective non-equilibrium current of unclear physical origins. Our results highlight a fundamental tension between quantum consistency and thermodynamic equilibration in open quantum systems.

The Maxey-Riley-Gatignol equation for the flow around a sphere at low particle Reynolds number tells us that the fluid-particle interaction force decomposes into a contribution from the local flow disturbance caused by the particle's boundary -- consisting of the drag, virtual-mass, and history forces, and their Faxén corrections -- and another contribution from the stress of the background flow, termed generalized-buoyancy force. There is also a consensus that, for general disperse two-phase flow, the interfacial force density, coupling the average fluid phase and dispersed-phase momentum balances, decomposes in a likewise manner. However, there has been a long-standing controversy about the physical closure separating the generalized-buoyancy from the interfacial force density, especially whether or not pseudo-stresses, such as the Reynolds stress, should be attributed to the background flow. Furthermore, most existing propositions for this closure involve small-particle approximations. Here, we show that all existing buoyancy closures are inconsistent with particle-resolving numerical simulations and/or at least one of two simple thought experiments designed to determine the roles of pseudo-stresses and small-particle approximations. We then derive the unique consistent closure. It requires no approximation and implies that all stresses and pseudo-stresses in the average fluid phase momentum balance, except the Reynolds stress, fully contribute to the background flow responsible for buoyancy. Remarkably, it exhibits a low-pass filter property, attenuating buoyancy at short wavelengths, that prevents it from causing Hadamard instabilities, constituting a first-principle-based solution to the long-standing ill-posedness problem of two-fluid models. When employing the derived closure, even simplistic two-fluid models are linearly well-posed.

Runtime efficiency and termination are crucial properties in the studies of program verification. Instead of dealing with these issues in an ad hoc manner, it would be useful to develop a robust framework in which such properties are guaranteed by design. This paper introduces a new imperative programming language whose design is grounded in a static type system that ensures the following equivalence property: All definable programs are guaranteed to run in polynomial time; Conversely, all problems solvable in polynomial time can be solved by some programs of the language. The contribution of this work is twofold. On the theoretical side, the foundational equivalence property is established, and the proof of the equivalence theorem is non-trivial. On the practical side, a programming approach is proposed that can streamline program analysis and verification for feasible computations. An interpreter for the language has been implemented, demonstrating the feasibility of the approach in practice.

In the present paper, we use discrete Morse theory to provide a new implementation of torsion subcomplex reduction for arithmetic groups. This leads both to a simpler algorithm as well as runtime improvements. To demonstrate the technique, we compute the mod 2 Farrell-Tate cohomology of PSL(4,Z).

We construct two representations of the twin group $T_n, n\geq 2$, namely $η_1: T_n \rightarrow \text{Aut}(\mathbb{F}_n)$ and $η_2: T_n \rightarrow \text{GL}_n(\mathbb{Z}[t^{\pm 1}])$, where $\mathbb{F}_n$ is a free group with $n$ generators and $t$ is indeterminate. We then analyze some characteristics of these two representations, such as irreducibility and faithfulness. Moreover, we prove that both representations can be extended to the virtual twin group $VT_n$ in the $2$-local extension way, for $n\geq 2$, and we find their $2$-local extensions. On the other hand, we obtain a different result for the welded twin group $WT_n$. More deeply, we show that $η_1$ cannot be extended to $WT_n$ in the $2$-local extension way, for $n\geq 3$, while $η_2$ can be extended to $WT_n$ in the $2$-local extension way, for $n\geq 2$, and we find its $2$-local extensions.

We propose a novel method for extracting non-singlet (NS) fragmentation functions (FFs) of light charged hadrons from charge asymmetries measured in hadron fragmentation, using data from both single-inclusive electron-positron annihilation and semi-inclusive deep-inelastic scattering processes. We determine the NS FFs for pions and kaons at next-to-next-to-leading order in Quantum Chromodynamics, including a comprehensive uncertainty analysis. The extracted FFs reveal a scaling index of about 0.7 at large momentum fractions and low energy scales, a strangeness suppression factor of about 0.5, and universality in fragmentation of light mesons. Our findings provide a valuable benchmark for testing non-perturbative QCD models and Monte Carlo event generators, and serve as crucial input for future electron-ion colliders.

Dark sectors with confining gauge interactions can provide both simple dark matter candidates and striking signals at colliders. We recast Large Hadron Collider searches for two different signatures of dark mesons that arise from a strongly-coupled theory with vector-like dark quarks that are in some non-trivial representation of Standard Model SU(2)$_L$. For any such electroweak representation, there is a 3-plet of dark mesons whose charged components are long-lived, and we reinterpret searches for disappearing tracks to place a lower bound on their mass of $\sim 1.2$ TeV. When the dark quarks are in SU(2)$_L$ representations larger than the fundamental, there is also a 5-plet of dark mesons that interacts with the electroweak gauge bosons via a chiral anomaly. We show that the 5-plet is the unique non-trivial meson multiplet with this anomaly and recast searches for the resulting diboson resonances to place bounds on model parameters. With additional measurements, the anomaly also enables one to reconstruct some ultraviolet parameters (the numbers of dark flavors and colors) while only measuring states in the infrared. Each of these signals represents an exciting opportunity for future searches using higher luminosity.

Let K be the unit-cube in Rn and f\,: K $\rightarrow$ R^n be a Morse function. We assume that the function f is given by an evaluation program $Γ$ in the noisy model, i.e., the evaluation program $Γ$ takes an extra parameter $η$ as input and returns an approximation that is $η$-close to the true value of f . In this article, we design an algorithm able to compute all local minimizers of f on K . Our algorithm takes as input $Γ$, $η$, a numerical accuracy parameter $ε$ as well as some extra regularity parameters which are made explicit. Under assumptions of probabilistic nature -- related to the choice of the evaluation points used to feed $Γ$ --, it returns finitely many rational points of K , such that the set of balls of radius $ε$ centered at these points contains and separates the set of all local minimizers of f . Our method is based on approximation theory, yielding polynomial approximants for f , combined with computer algebra techniques for solving systems of polynomial equations. We provide bit complexity estimates for our algorithm when all regularity parameters are known. Practical experiments show that our implementation of this algorithm in the Julia package Globtim can tackle examples that were not reachable until now.

We study the topological properties of the leaves of the singular foliation induced by a closed 1-form of Morse type on a compact orbifold. In particular, we establish criteria that characterize when all such leaves are compact, when they are non-compact, and how both types may coexist. As an application, we extend to the orbifold setting a celebrated result of Calabi, which provides a purely topological characterization of intrinsically closed harmonic 1-forms of Morse type.

We describe large deviations for normalized multiple iterated sums and integrals of the form $\bbS_N^{(ν)}(t)=N^{-ν}\sum_{0\leq k_1<...<k_ν\leq Nt}ξ(k_1)\otimes\cdots\otimesξ(k_ν)$, $t\in[0,T]$ and $\bbS_N^{(ν)}(t)=N^{-ν}\int_{0\leq s_1\leq...\leq s_ν\leq Nt}ξ(s_1)\otimes\cdots\otimesξ(s_ν)ds_1\cdots ds_ν$, where $\{ξ(k)\}_{-\infty<k<\infty}$ and $\{ξ(s)\}_{-\infty<s<\infty}$ are centered bounded stationary vector processes whose sums or integrals satisfy a trajectorial large deviations principle.

We investigate the properties of the vacuum state for the Proca field in the geometry of two parallel plates on background of (D+1)-dimensional Minkowski spacetime. The two-point functions for the vector potential and the field tensor are evaluated for higher-dimensional generalizations of the perfect magnetic conductor (PMC) and perfect electric conductor (PEC) boundary conditions. Explicit expressions are provided for the vacuum expectation values (VEVs) of the electric and magnetic field squares, field condensate, and for the VEV of the energy-momentum tensor. In the zero-mass limit the VEVs of the electric and magnetic field squares and the condensate reduce to the corresponding expressions for a massless vector field. The same is the case for the VEV of the energy-momentum tensor in the problem with PEC conditions. However, for PMC conditions the zero-mass limit for the vacuum energy-momentum tensor differs from the corresponding VEV for a massless field. This difference in the zero-mass limits is related to the different influences of the boundary conditions on the longitudinal polarization mode of a massive vector field. The PMC conditions constrain all the polarization modes including the longitudinal mode, whereas PEC conditions do not influence the longitudinal mode. The vacuum energy-momentum tensor is diagonal. The normal stress is uniformly distributed in the region between the plates and vanishes in the remaining regions. The corresponding Casimir forces are attractive for both boundary conditions.

We consider the wave scattering and inverse scattering in an inhomogeneous medium embedded a homogeneous droplet with a small size, which is modeled by a constant mass density and a small bulk modulus. Based on the Lippmann-Schwinger integral equation for scattering wave in inhomogeneous medium, we firstly develop an efficient approximate scheme for computing the scattered wave as well as its far-field pattern for any droplet located in the inhomogeneous background medium. By establishing the approximate relation between the far-field patterns of the scattered wave before and after the injection of a droplet, the scattered wave of the inhomogeneous medium after injecting the droplet is represented by a measurable far-field patterns, and consequently the inhomogeneity of the medium can be reconstructed from the Helmholtz equation. Finally, the reconstruction process in terms of the dual reciprocity method is proposed to realize the numerical algorithm for recovering the bulk modulus function inside a bounded domain in three dimensional space, by moving the droplet inside the bounded domain. Numerical implementations are given using the simulation data of the far-field pattern to show the validity of the reconstruction scheme, based on the mollification scheme for dealing with the ill-posedness of this inverse problem.

Cold atom experiments show that a mobile impurity particle immersed in a Bose-Einstein condensate forms a well-defined quasiparticle (Bose polaron) for weak to moderate impurity-boson interaction strengths, whereas a significant line broadening is consistently observed for strong interactions. Motivated by this, we introduce a phenomenological theory based on the assumption that the most relevant states are characterized by the impurity correlated with at most one boson, since they have the largest overlap with the uncorrelated states to which the most common experimental probes couple. These experimentally relevant states can however decay to lower energy states characterised by correlations involving multiple bosons, and we model this using a minimal variational wave function combined with a complex impurity-boson interaction strength. We first motivate this approach by comparing to a more elaborate theory that includes correlations with up to two bosons. Our phenomenological model is shown to recover the main results of two recent experiments probing both the spectral and the non-equilibrium properties of the Bose polaron. Our work offers an intuitive framework for analyzing experimental data and highlights the importance of understanding the complicated problem of the Bose polaron decay in a many-body setting.

We construct an action of the positive Witt algebra on the categorified quantum group associated to a simply-laced Lie algebra. In the type A case, we show that this action induces an action of the positive Witt algebra on $\mathfrak{gl}_n$-foams, recovering the action of Qi, Robert, Sussan, and Wagner. We also show that this construction is compatible with the trace decategorification, inducing the action of the positive Witt algebra on the current algebra.

Reconstructing quantum states from measurement data represents a formidable challenge in quantum information science, especially as system sizes grow beyond the reach of traditional tomography methods. While recent studies have explored quantum machine learning (QML) for quantum state tomography (QST), nearly all rely on idealized assumptions, such as direct access to the unknown quantum state as quantum data input, which are incompatible with current hardware constraints. In this work, we present a QML-based tomography protocol that operates entirely on classical measurement data and is fully executable on noisy intermediate-scale quantum (NISQ) devices. Our approach employs a variational quantum circuit trained to reconstruct quantum states based solely on measurement outcomes. We test the method in simulation, achieving high-fidelity reconstructions of diverse quantum states, including GHZ states, spin chain ground states, and states generated by random circuits. The protocol is then validated on quantum hardware from IBM and IonQ. Additionally, we demonstrate accurate tomography is possible using incomplete measurement bases, a crucial step towards scaling up our protocol. Our results in various scenarios illustrate successful state reconstruction with fidelity reaching 90% or higher. To our knowledge, this is the first QML-based tomography scheme that has been implemented on real quantum processors using exclusively classical measurements. This work establishes the feasibility of QML-based tomography on current quantum platforms and offers a scalable pathway for practical quantum state reconstruction.

We present a new constraint on the primordial helium abundance, $Y_\mathrm{P}$, based on Subaru observations. A major source of uncertainty in previous $Y_\mathrm{P}$ determinations is the lack of extremely metal-poor galaxies (EMPGs; $0.01-0.1\,Z_\odot$), which have metallicities a few to ten times lower than the metal-poor galaxies (MPGs; $0.1-0.4\,Z_\odot$) predominantly used in earlier studies, requiring substantial extrapolation to zero metallicity. Here, we perform Subaru near-infrared spectroscopy of 29 galaxies, including 14 EMPGs. By incorporating existing optical spectra, we derive He/H for each galaxy using photoionization modeling of helium and hydrogen emission lines, including the He \textsc{i} 10830Å\, line to break the density--temperature degeneracy. After carefully selecting galaxies with robust He/H determinations, and adding 58 galaxies from previous studies, we obtain $Y_\mathrm{P} = 0.2402^{+0.0040}_{-0.0040}$. This $Y_\mathrm{P}$ value is $\sim1σ$ lower than most of the previous estimates, but agrees with recent determinations using EMPGs and the CMB constraint from the Atacama Cosmology Telescope (ACT) experiment. Our result indicates $N_\mathrm{eff} = 2.54^{+0.20}_{-0.25}$, showing a mild ($\sim2σ$) tension with the Standard Model and Planck results. These tensions may suggest a nonzero lepton asymmetry $(ξ_\mathrm{e}\neq0)$, which would alleviate the tension with $ξ_\mathrm{e} = 0.05^{+0.02}_{-0.03}$. More observations of EMPGs and further assessments of systematic uncertainties are essential to test the potential tension more rigorously.

In this manuscript, we present a general and exact method for classicalizing the dynamics of any $N$-level quantum system, transforming quantum evolution into a classical-like framework using the geometry of complex projective spaces $\mathbb{CP}^{N-1}$. The method can be expressed as five-step algorithmic procedure to derive a classical Hamiltonian and a symplectic structure for the Poisson brackets, yielding $N-1$ Hamilton's equations that precisely replicate the quantum dynamics, including complex phenomena like entanglement. We demonstrate the method's efficacy by classicalizing two interacting qubits in $\mathbb{CP}^3$, exactly reproducing quantum observables such as quantum probabilities, quaternionic population differences and the concurrence, capturing entanglement dynamics via a classical analog.

This review examines data from hadron-nucleus collisions, primarily focusing on hard processes like Drell-Yan, heavy flavor and quarkonium production. It highlights observed modifications of particle yields as functions of momentum and rapidity, aiming to clarify the underlying QCD effects in cold nuclear matter (CNM). The paper outlines strategies for future experiments, including the Electron-Ion Collider (EIC), to distinguish between these effects. Key questions address the universality of suppression mechanisms and the role of non-perturbative physics, providing a road map for upcoming nuclear data.

The interplay between superconductivity and charge correlations in the kagome metal CsV$_3$Sb$_5$ can be tuned by external perturbations such as doping or pressure. Here we present a study of charge correlations and superconductivity upon hole doping via Ti substitution on the V kagome sites in CsV$_{3-x}$Ti$_x$Sb$_5$ via synchrotron x-ray diffraction and scanning SQUID measurements. While the superconducting phase, as viewed via the vortex structure, remains conventional and unchanged across the phase diagram, the nature of charge correlations evolves as a function of hole-doping from the first superconducting dome into the second superconducting dome. For Ti doping in the first superconducting dome, competing $2\times 2 \times 2$ and $2\times 2 \times 4$ supercells form within the charge density wave state and are suppressed rapidly with carrier substitution. In the second superconducting dome, no charge correlations are detected. Comparing these results to those observed for CsV$_3$Sb$_{5-x}$Sn$_x$ suggests important differences between hole doping via chemical substitution on the V and Sb sites, particularly in the disorder potential associated with each dopant.

For interface tracking of an arbitrary number of materials in two dimensions, we propose a multiphase cubic MARS method that (a) represents the topology and geometry of the interface via graphs, cycles, and cubic splines, (b) applies to any number of materials with arbitrarily complex topology and geometry, (c) maintains an $(r,h)$-regularity of the interface so that the distance between any pair of adjacent markers is within a user-specified range, (d) distributes the markers adaptively along the interface so that arcs with high curvature are resolved by densely populated markers, and (e) achieves fourth-, sixth-, and eighth-order accuracy both in time and in space.} In particular, all possible types of junctions, which pose challenges to VOF methods and level-set methods, are handled with ease. Results of a variety of benchmark tests confirm the analysis and demonstrate the superior accuracy, efficiency, and versatility of the proposed method.

Personalized recommendation requires capturing the complex latent intents underlying user-item interactions. Existing structural models, however, often fail to preserve perspective-dependent interaction semantics and provide only indirect supervision for aligning user and item intents, lacking explicit interaction-level constraints. This entangles heterogeneous interaction signals, leading to semantic ambiguity, reduced robustness under sparse interactions, and limited interpretability. To address these issues, we propose DMICF, a Dual-Perspective Disentangled Multi-Intent framework for collaborative filtering. DMICF models interactions from complementary user- and item-centric perspectives and employs a macro-micro prototype-aware variational encoder to disentangle fine-grained latent intents. Interaction-level supervision enforces dimension-wise alignment between user and item intents, grounding latent factors and enabling their collaborative emergence. Importantly, each component is architecturally flexible, and performance is robust to specific module instantiations. We offer a theoretical analysis to help explain how prototype-aware conditioning may alleviate posterior collapse, while the reconstruction objective promotes intent-wise contrastive alignment between positive and negative interactions. Extensive experiments on multiple benchmarks demonstrate consistent improvements over strong baselines, with ablations validating each core component.

Metriplectic dynamical systems consist of a special combination of a Hamiltonian and a (generalized) entropy-gradient flow, such that the Hamiltonian is conserved and entropy is dissipated/produced (depending on a sign convention). It is natural to expect that, in the long-time limit, the orbit of a metriplectic system should converge to an extremum of entropy restricted to a constant-Hamiltonian surface. In this paper, we discuss sufficient conditions for this to occur. Then, we construct a class of metriplectic systems inspired by the Landau operator for Coulomb collisions in plasmas, which is included as special case. For this class of brackets, checking the conditions for convergence reduces to checking two usually simpler conditions, and we discuss examples in detail. We apply these results to the construction of relaxation methods for the solution of equilibrium problems in fluid dynamics and plasma physics.

Linear discriminant analysis (LDA) is a fundamental classification and dimension reduction method that achieves Bayes optimality under Gaussian mixture, but often struggles in high-dimensional settings where the covariance matrix cannot be reliably estimated. We propose LDA with gradient optimization (LDA-GO), which learns a low-rank precision matrix via scalable gradient-based optimization. The method automatically selects between a Gaussian likelihood and a cross-entropy loss using data-driven structural diagnostics, adapting to the signal structure without user tuning. The gradient computation avoids any quadratic-sized intermediate matrix, keeping the per-iteration cost linear in the number of dimensions. Theoretically, we prove several properties of the method, including the convexity of the objective functions, Bayes-optimality of the method, and a finite-sample bound of the excess error. Numerically, we conducted a variety of simulations and real data experiments to show that LDA-GO wins a majority of settings among other LDA variants, particularly in sparse-signal high-dimensional regimes.

We propose a novel theoretical framework in which proton decay is induced by the dark matter. While proton decay requires violation of the $B+L$ symmetry, dark matter stability often relies on the presence of an unbroken symmetry. These seemingly distinct phenomena are unified through the global $U(1)_{B+L}$ symmetry inherent in the Standard Model. Its spontaneous breaking leads to a residual $Z_4$ symmetry, which ensures dark matter stability and forbids proton decay at tree level. Consequently, proton decay occurs at the one-loop level, mediated by dark sector particles. The proton lifetime is linked with the dark matter, the heavier dark matter mass enhancing proton stability, and vice versa. The $\mathcal{O}$(TeV) masses of the mediators remain consistent with current proton lifetime limits, making them accessible to experimental searches. In particular, the leptoquark mediating proton decay, carrying exotic $B+L$ charges, leads to a distinctive signature in collider searches. By intertwining proton decay, dark matter stability, and collider phenomenology, this framework offers distinctive signatures that can be probed in current and future experiments.