In the last decade, and even more so in the last few years, our knowledge of the internal structure of the proton has become more accurate and precise thanks to the large amount of data available and developments in theory and methodology. The reduction of the uncertainties associated to these developments has brought previously neglected effects into focus as their typical magnitude are competitive with the size of the uncertainties. One such effect is the inclusion of QED into PDF fits. Typically this is a percent effect, and thus while theoretically important, it has had a relatively limited impact on phenomenological studies up to this point. In this proceeding we study some of the effects which, while peripheral to the inclusion of QED in the proton, can considerably change the relative size and shape of the QCD+QED fit with respect to the QCD only determination. These may become important in the future as precision continues to increase. After a comparison of the QCD+QED PDFs with the QCD only PDFs of various global PDF fitting groups, we focus largely upon NNPDF4.0, which shows the biggest effect when including QED. Focusing largely on a single set of PDFs also enables more subtle effects to be analysed, making it an ideal candidate for this study.

For sequences of Poisson-Laguerre tessellations and their duals in $\mathbb{R}^d$, generated by Poisson point processes $(η_n)_{n\in\mathbb{N}}$ in $\mathbb{R}^d \times \mathbb{R}$, we prove limit theorems as $n\to \infty$. The intensity measure of $η_n$ has density of the form $(v,h)\mapsto f_n(h)$ with respect to the Lebesgue measure, where $v\in \mathbb{R}^d$ and $h\in \mathbb{R}$. Identifying a tessellation with its skeleton (the union of the boundaries of all its cells) we provide verifiable conditions on $(f_n)_{n\in\mathbb{N}}$ that ensure convergence either to the classical Poisson-Voronoi/Poisson-Delaunay tessellation or to another Poisson-Laguerre tessellation. We also discuss convergence of the corresponding typical cells. As a corollary, we show that the Poisson-Voronoi and the Poisson-Delaunay tessellations arise as limits of the $β$-Voronoi and the $β$-Delaunay tessellations, respectively, as $β\to -1$.

Galaxy clusters are tracers of the large scale structures of the Universe, making the time evolution of their mass function dependent on key cosmological parameters, such as the cosmic matter density or the amplitude of density fluctuations $σ_8$. Accurate measurements of cluster's total masses are therefore essential, yet they can be challenging, particularly for clusters with complex morphologies, as simple mass profiles are often adopted to fit the measurements. In this work, we focus on the Frontier Fields galaxy clusters: a sample of six extremely massive systems, that, in most cases, exhibit highly complex mass distributions. The BUFFALO survey extended the Hubble Space Telescope observations for the Frontier Fields galaxy clusters, providing high-resolution multi-band imaging within a few Mpc. Combining this high-quality imaging dataset with ancillary spectroscopy, we produce weak-lensing catalogues with very high source densities, about 50 sources/arcmin$^2$. This allows us to robustly estimate the individual weak-lensing cluster masses and quantify the sensitivity of these measurements on different factors, such as the cluster centring, the uncertainty on the redshift distribution or the foreground contamination and boost factor correction. This provides a data-driven analysis of the different sources of systematics that can impact such measurements. We find that the largest sources of systematic bias arise for the most disturbed clusters, such as the multi-modal, merging galaxy cluster Abell 2744. This analysis sets a comprehensive framework for assessing the impact of systematics on the weak-lensing estimates of cluster masses, and in particular, in the case of unrelaxed clusters. This can play a key role in forthcoming cosmological analyses based on wide-field surveys such as Euclid and the Legacy Survey of Space and Time of the Rubin Observatory.

Structural parameters of graphs, such as treewidth, play a central role in the study of the parameterized complexity of graph problems. Motivated by the study of parametrized algorithms on phylogenetic networks, scanwidth was introduced recently as a new treewidth-like structural parameter for directed acyclic graphs (DAGs) that respects the edge directions in the DAG. The utility of this width measure has been demonstrated by results that show that a number of problems that are fixed-parameter tractable (FPT) with respect to both treewidth and scanwidth allow algorithms with a better dependence on scanwidth than on treewidth. More importantly, these scanwidth-based algorithms are often much simpler than their treewidth-based counterparts: the name ``scanwidth'' reflects that traversing a tree extension (the scanwidth-equivalent of a tree decomposition) of a DAG amounts to ``scanning'' the DAG according to a well-chosen topological ordering. While these results show that scanwidth is useful especially for solving problems on phylogenetic networks, all problems studied through the lens of scanwidth so far are either FPT with respect to both scanwidth and treewidth, or W[$\ell$]-hard, for some $\ell \ge 1$, with respect to both. In this paper, we show that scanwidth is not just a proxy for treewidth and provides information about the structure of the input graph not provided by treewidth, by proving a fairly stark complexity-theoretic separation between these two width measures. Specifically, we prove that Weighted Phylogenetic Diversity with Dependencies is FPT with respect to the scanwidth of the food web but W[$\ell$]-hard with respect to its treewidth, for all $\ell \ge 1$. To the best of our knowledge, no such separation between these two width measures has been shown for any problem before.

We study quantum transport phenomena induced by anisotropic Lifshitz scale anomaly in a boundary Lifshitz field theory (BLFT) coupled to an external electromagnetic background. In this context, we obtain the anisotropic scale anomaly in Lifshitz field theories coupled to a background $U(1)$ gauge field and subsequently compute the anomaly induced near boundary current in a BLFT. Focusing on 5D BLFTs, we find that the temporal and spatial components of the induced current exhibit distinct power law dependencies on the distance from the boundary, reflecting the intrinsic time-space anisotropy of the theory. We further derive this anomalous current holographically from the bulk dual of BLFT and find that the temporal component is independent of the boundary conditions while the spatial component depends explicitly on them. The distance dependence is in exact agreement with the dual field theory result.

In 2001, Bhargava proved a composition law for $2 \times 2 \times 2$ integer cubes, which generalized Gauss composition of integral binary quadratic forms. Furthermore, he derived four new composition laws defined on the following spaces: 1) binary cubic forms with triplicate middle coefficients, 2) pairs of binary quadratic forms with duplicate middle coefficients, 3) pairs of quaternary alternating 2-forms and 4) senary alternating 3-forms. In each of the five cases, there is a natural group action on the underlying space with a unique polynomial invariant called the discriminant, and a notion of projectivity for the elements of the space. The strategy behind Bhargava's approach is to construct a discriminant-preserving bijection between the set of orbits under the group action and the set of (tuples of) suitable ideal classes of quadratic rings. The projective ideal classes are equipped with a natural group structure and hence we get a group structure on the spaces of equivalence classes of projective forms of fixed discriminant $D$. In each case the class group of projective forms of discriminant $D$ has a natural interpretation in terms of the narrow class group of the quadratic ring of discriminant $D$. The aim of this paper is to give explicit composition identities (similar to Gauss' formulation of composition of binary quadratic forms) for these higher composition laws.

The integer convex hull $I(H_N)$ of the set $H_N=\{(x,y)\in \mathbb{R}^2: xy\ge N\}$ is the convex hull of the lattice points in $H_N$. The vertices of $I(H_N)$ lie in the square $[1,N]^2$. Improving on a recent result of Alcántara et al. ~\cite{Santos} we show that the number of vertices of $I(H_N)$ is of order $N^{1/3}\log N$. We also show that the area of the part of $H_N \setminus I(H_N)$ that lies in the square $[1,N^{2/3}]^2$ is also of order $N^{1/3}\log N$.

Integrated frequency comb sources are a key enabling technology for frequency metrology applications. Their on-chip integration promises to bring metrology capacity outside of the lab, particularly since they can operate at low continuous-wave pump laser power in the dissipative Kerr soliton (DKS) regime. Yet, such small foot-print and low power comes at a cost: higher noise and overall lower comb power. In particular, this translates to highly challenging detection and locking of the carrier-envelope offset, necessary for complete stabilization of the comb. Recently, Kerr-induced synchronization (KIS) of a DKS to a reference laser has been demonstrated as a tool for passive all-optical stabilization of DKS microcombs, with fundamental modification to the DKS and microcomb properties. Here, we demonstrate that the combination of additional power from the reference laser (now part of the DKS) and the KIS phase locking that pins the repetition rate together fundamentally alter the DKS, forcing an energy redistribution to maintain its center of mass. We demonstrate this self-balancing effect theoretically, which in a pure quadratic dispersion resonator leads to reference-dependent recoil. With higher-order dispersion through which the DKS yields phase-matched dispersive waves (DWs), we demonstrate that self-balancing increases the DW radiation, experimentally showing a 22 dB increase of comb teeth at 780 nm in an octave-spanning microcomb for efficient deployable carrier-envelope offset detection.

We show that infinitely many cubic fields have class group of 2-rank 1.

This study examines the rare decay $B \to K_0^*(1430)\,\ell^+ \ell^-$ as a possible probe for new physics beyond the standard model (SM). We first analyze this channel within the SM and then include scalar leptoquark (LQ) contributions. We provide predictions for key observables, like differential decay rate, branching ratio, ratio of branching fractions at different channels, forward-backward asymmetry and different lepton polarizations, and assess their sensitivity to leptoquark scenarios, highlighting $q^2$ regions less affected by the long-distance charmonium effects. The results can be useful for future Belle II and LHCb measurements.

We establish an ideal-theoretic rigidity principle for quadratic distance images over integer residue rings. Specifically, we prove that near-extremal collapse of the distance set in $\mathbb{Z}_n^d$ forces strong algebraic structure supported on annihilator submodules arising from the arithmetic of $n$. As a consequence, we obtain the first inverse theorem for the Falconer distance problem over $\mathbb{Z}_n$ for composite moduli. We show that if a set $E \subset \mathbb{Z}_n^d$ of size $|E| \asymp n^{(d+1)/2}$ determines only $O(n)$ distinct squared distances, then $E$ must be supported on a coset of an annihilator submodule on which the distance form is algebraically degenerate. The proof introduces a divisor-depth decomposition intrinsic to $\mathbb{Z}_n$, together with a lifting mechanism that transfers local degeneracies at prime moduli into global ideal-theoretic constraints. This yields a complete classification of near-extremizers for the Falconer distance problem in the ring setting, revealing a rigidity phenomenon with no analogue over fields.

The discovery of 2024 YR4 presented the planetary defense community with the most significant impact threat in almost two decades, reaching level 3 on the Torino scale. The community, now mature and well-organized, responded with a global observational effort. Astrometric measurements, forming the basis for orbital refinement and impact prediction, were a central component of this response. In this paper, we present the astrometric data collected by the international community, from the time of discovery until the object became too faint for all existing observational assets, including JWST. We also discuss the coordination role played by the International Asteroid Warning Network, and the importance of publicly available image archives to enable precovery searches.

Large language models (LLMs) are increasingly deployed in security-sensitive applications, yet remain vulnerable to backdoor attacks. However, existing backdoor defenses are difficult to operationalize for Backdoor Defense-as-a-Service (BDaaS), as they require unrealistic side information (e.g., downstream clean data, known triggers/targets, or task domain specifics), and lack reusable, scalable purification across diverse backdoored models. In this paper, we present PROTOPURIFY, a backdoor purification framework via parameter edits under minimal assumptions. PROTOPURIFY first builds a backdoor vector pool from clean and backdoored model pairs, aggregates vectors into candidate prototypes, and selects the most aligned candidate for the target model via similarity matching. PROTOPURIFY then identifies a boundary layer through layer-wise prototype alignment and performs targeted purification by suppressing prototype-aligned components in the affected layers, achieving fine-grained mitigation with minimal impact on benign utility. Designed as a BDaaS-ready primitive, PROTOPURIFY supports reusability, customizability, interpretability, and runtime efficiency. Experiments across various LLMs on both classification and generation tasks show that PROTOPURIFY consistently outperforms 6 representative defenses against 6 diverse attacks, including single-trigger, multi-trigger, and triggerless backdoor settings. PROTOPURIFY reduces ASR to below 10%, and even as low as 1.6% in some cases, while incurring less than a 3% drop in clean utility. PROTOPURIFY further demonstrates robustness against adaptive backdoor variants and stability on non-backdoored models.

Homophily based on observables is widespread in networks. Therefore, homophily based on unobservables (fixed effects) is also likely to be an important determinant of the interaction outcomes. Failing to properly account for latent homophily (and other complex forms of unobserved heterogeneity) can result in inconsistent estimators and misleading policy implications. To address this concern, we consider a network model with nonparametric unobserved heterogeneity, leaving the role of the fixed effects unspecified. We argue that the interaction outcomes can be used to identify agents with the same values of the fixed effects. The variation in the observed characteristics of such agents allows us to identify the effects of the covariates, while controlling for the fixed effects. Building on these ideas, we construct several estimators of the parameters of interest and characterize their large sample properties. Numerical experiments illustrate the usefulness of the suggested approaches and support the asymptotic theory.

We continue the study of the effective content of $K$-theory for C*-algebras, with a focus on AF algebras. We show that from a c.e. presentation of an AF algebra it is possible to compute a representation of the algebra as an inductive limit of finite-dimensional algebras. Using this, and an analogous result for dimension groups, we show that the computable $K_0$ functor provides a computable equivalence of categories between c.e. presentations of AF algebras and c.e. presentations of unital (scaled) dimension groups, giving an effective version of Elliott's classification theorem. We use our results to determine the complexity of the index set and isomorphism problems for various classes of AF algebras.

The correlation diagrams of vibrational energy levels considering the Planck constant as a variable parameter have proven as a very useful tool to study vibrational molecular states, and more specifically in relation to the quantum manifestations of chaos in such dynamical systems. In this paper, we consider the highly nonlinear K-CN molecule, showing how the regular classical structures, i.e., Kolmogorov-Arnold-Moser tori, existing in the mixed classical phase space appear in the quantum levels correlation diagram as emerging diabatic states, something that remains hidden when only the actual value of the Planck constant is considered. Additionally, a quantum transition from order to chaos is unveiled with the aid of these correlation diagrams, where it appears as a frontier of scarred functions.

The LISA mission will be the first observatory to detect gravitational waves from space within the millihertz frequency band. Magnetic forces have an important impact on the instrument's sensitivity below the millihertz. Hence, monitoring the magnetic environment within each of the LISA spacecrafts is of utmost importance. In this Letter we present the characterization of the coils that were used in LISA Pathfinder (LPF) when operating as magnetic sensors in the audio frequency band. The necessity of implementing this type of magnetometer is presented in order to monitor high frequency magnetic signals from the electronics on-board. We show that the LPF coils have a performance one order of magnitude better than the current requirements set by the LISA mission at the low end of the audio-band frequency. The LPF coils are able to measure a magnetic noise level of 1.45 $\rm pT/\sqrt{\rm{Hz}}$ at 50 Hz and 0.17 $\rm pT/\sqrt{\rm{Hz}}$ at 500 Hz. Additionally, the LPF coils can reach a magnetic noise floor of 0.1 $\rm pT/\sqrt{\rm{Hz}}$ at frequencies above 1 kHz.

For $t > 0$, let $\{\mathcal{H}^{(t)}_n, n \in \mathbb{N}\}$ be the KPZ line ensemble with parameter $t$, satisfying the homogeneous $\mathbf{H}$-Brownian Gibbs property with $\mathbf{H}(x) =e^x$. We prove quantitative concentration estimates for the $n$th line $\mathcal{H}^{(t)}_n$ which yield the asymptotics $\mathcal{H}^{(t)}_n = n \log n + o(n^{3/4 + ε})$ as $n \to \infty$. A key step in the proof is a general integration by parts formula for $\mathbf{H}$-Brownian Gibbs line ensembles which yields the identity $\mathbb{E} \exp(\mathcal{H}^{(t)}_{n + 1}(x) - \mathcal{H}^{(t)}_n (x)) = n t^{-1}$ for any $n, t, x$.

Let $G$ be a graph and let $\mathrm{cl}(G)$ be the number of distinct induced cycle lengths in $G$. We show that for $c,t\in \mathbb N$, every graph $G$ that does not contain an induced subgraph isomorphic to $K_{t+1}$ or $K_{t,t}$ and satisfies $\mathrm{cl}(G) \le c$ has bounded treewidth. As a consequence, we obtain a polynomial-time algorithm for deciding whether a graph $G$ contains induced cycles of at least three distinct lengths.

This paper studies the integration problem in differential fields that may involve quantities reminiscent of the Weierstrass $\wp$ function, which are defined by a first-order nonlinear differential equation. We extend the classical notion of special polynomials to elements of Weierstrass-like extensions and present algorithms for reduction in such extensions. As an application of these results, we derive some new formulae for integrals of powers of $\wp$.

We derive a residual-based $hp$-a posteriori error estimator for hybrid high-order (HHO) methods on simplicial meshes applied to the biharmonic problem posed on two- and three-dimensional polytopal Lipschitz domains. The a posteriori error estimator hinges on an error decomposition into conforming and nonconforming components. To bound the nonconforming error, we use a $C^1$-partition of unity constructed via Alfeld splittings, combined with local Helmholtz decompositions on vertex stars. For the conforming error, we design two residual-based estimators, each associated with a specific interpolation operator. In the first setting, the upper bound for the conforming error involves only the stabilization term and the data oscillation. In the second setting, the bound additionally incorporates bulk residuals, normal flux jumps, and tangential jumps. Numerical experiments confirm the theoretical findings and demonstrate the efficiency of the proposed estimators.

This paper establishes an explicit obstruction to constructing algebraic cycles from automorphic cohomology classes on Shimura varieties. We produce a rational Hodge class $Ω_E$ in the intersection cohomology of the Baily-Borel compactification of a Shimura variety for $\text{SO}(2,26)$, arising from a stable residual automorphic representation via theta lift from the weight-$2$ newform of conductor $11$. While $Ω_E$ is automorphic and of pure Hodge type, we prove it is non-interior and hence cannot be obtained from special cycles, theta lifts, endoscopic transfers, or boundary pushforwards, all of which yield interior classes. The result is unconditional, relying only on Arthur's classification, Vogan-Zuckerman theory, the fundamental lemma, and the Zucker conjecture (proven by Looijenga-Saper-Stern), and it highlights a fundamental asymmetry between automorphic cohomology and geometric access to algebraic cycles, refining the Hodge conjecture from a question of existence to one of constructive tractability.

We establish some sufficient conditions for the Lagrangian skeleton of the affine complement of an effective ample Q-divisor in a smooth rationally connected projective variety to be a Lagrangian barrier in the sense of Biran, and establish bounds on the Gromov width of the complement of the skeleton. We particularly focus on hyperplane arrangements in projective space, where we obtain tight bounds in two dimensions when the divisor is a generic collection of at least three lines.

We study a fermionic dark matter candidate that couples to the standard model particles exclusively through electric and magnetic dipole operators mediated by a massive dark photon. Such dipole portals naturally arise in dark sectors where the dark matter is neutral under a hidden $U(1)_D$, and they lead to phenomenology distinct from conventional vector-current interactions. We consider the direct-detection signals arising from dark matter-nucleus scattering including the Migdal effect, dark matter-electron scattering, and semiconductor targets, which allow sensitivity to sub-GeV dark matter masses, together with the cosmological bounds from such as thermal relic abundance, cosmic microwave background, big-bang nucleosynthesis, and cosmic-rays. We find that the dark dipole coupling can be largely constrained by direct detection (in particular, electric dipole coupling). However, the cosmological observations have already constrained most of the parameter space, in particular for magnetic dipole interactions of $U(1)_D$ for sub-GeV dark matter. For the dark matter mass below 10 MeV, the semiconductor (in particular, using skipper-CCD) experiments can play a crucial role in probing the dark dipole interactions: future low-threshold experiments utilizing the semiconductor targets can further extend the constraints. Our results have demonstrated that the sub-GeV dark matter with dark dipole interactions can be still safe from the direct-detection constraints, and the future low-threshold semiconductor experiments may play a significant role in constraining the dark dipole interactions.

Range queries are simple and popular types of queries used in data retrieval. However, extracting exact and complete information using range queries is costly. As a remedy, some previous work proposed a faster principle, {\em approximate} search with range queries, also called single range cover (SRC) search. It can, however, produce some false positives. In this work we introduce a new SRC search structure, a $c$-DAG (Directed Acyclic Graph), which provably decreases the average number of false positives by logarithmic factor while keeping asymptotically same time and memory complexities as a classic tree structure. A $c$-DAG is a tunable augmentation of the 1D-Tree with denser overlapping branches ($c \geq 3$ children per node). We perform a competitive analysis of a $c$-DAG with respect to 1D-Tree and derive an additive constant time overhead and a multiplicative logarithmic improvement of the false positives ratio, on average. We also provide a generic framework to extend our results to empirical distributions of queries, and demonstrate its effectiveness for Gowalla dataset. Finally, we quantify and discuss security and privacy aspects of SRC search on $c$-DAG vs 1D-Tree, mainly mitigation of structural leakage, which makes $c$-DAG a good data structure candidate for deployment in privacy-preserving systems (e.g., searchable encryption) and multimedia retrieval.

Projections of finite dimensional sets and their measures are investigated in infinite-dimensional power measure spaces. The starting point is the known algebraic formula, expressing \ the $y$-projection of a finite-dimensional set $a$ as a Boolean supremum of certain finite geometrical transformations of $a$ in the infinite-dimensional power space. This Boolean supremum somewhat unusual in classical measure theory because, it is different, in general, from the usual union of sets. The paper investigates the problem whether the power measure in the infinite-dimensional measure space is continuous with respect to the forementioned Boolean supremum. If so, then this continuity leads to a simple formula for calculating the measure of the projection of $a.$ It is shown that the answer concerning this continuity is affirmative for discrete measures but false for the Lebesgue measure, for example. However, it is proved that if the concept of the $y$-projection of $a$ is replaced by that of the so-called \textit{strong} $y$-\textit{projection} of $a,$ then the Lebesgue measure becomes continuous in this context and the value of the corresponding real supremum is exactly the measure of the foregoing strong $y $-projection. In this paper, the tools of the classical analysis are adapted to handle measures on Boolean algebras.

We describe families of probe D$q$-brane embeddings in the extremal black D$p$-brane backgrounds of type IIA and type IIB supergravity, specified by an arbitrary holomorphic function of a complex coordinate on the worldvolume of the D$q$-branes. These embeddings preserve one-quarter of the supersymmetry of the D$p$-brane background, or sometimes one-half of the supersymmetry when $p = q$. We discuss the holography of two example families of holomorphic probe branes in the near-horizon limit of the D3-brane background. The first is probe D5-branes, dual to defect hypermultiplets with a holomorphic mass, which in the infrared flow to Wilson lines located at the zeros of the mass. The second is probe D3-branes, holographically dual to states in the presence of Gukov--Witten surface defects in the dual $\mathcal{N}=4$ supersymmetric Yang--Mills theory.

Motivated by the sharp constants in the $L^2$-Caffarelli--Kohn--Nirenberg (or $L^2$-CKN for short) inequalities on Euclidean spaces, we study, in a unified framework, a sequence of $L^2$-CKN inequalities on metric measure spaces. On a general metric measure space, this sequence implies a reverse volume comparison of Günther type. Moreover, on a subclass of spaces admitting the measure contraction property, we show that this sequence of $L^2$-CKN inequalities are valid if and only if the spaces are volume cones. We also provide a stability result for inequalities of this type on volume cones.

The detection of black hole scalar hair, a possible deviation from general relativity's "no-hair" theorem, requires sensitive probes beyond conventional methods. This study proposes chaotic dynamics as a novel indicator for scalar hair in Horndeski gravity. We investigate the motion of a spinning test particle in a static, spherically symmetric hairy black hole spacetime. Our results show that increasing scalar hair systematically suppresses orbital chaos, as evidenced by regularized precession, reduced Lyapunov exponents, and contracted Poincare sections. Furthermore, scalar hair enhances the correlation between the two gravitational wave polarization modes, restoring phase coherence. These findings demonstrate that chaotic observables and gravitational wave signatures can jointly serve as sensitive probes for black hole hair, offering a complementary approach to testing gravity in strong-field regimes.

The realization of distributed quantum neural networks (DQNNs) over quantum internet infrastructures faces fundamental challenges arising from the fragile nature of entanglement and the demanding synchronization requirements of distributed learning. We introduce a Consensus-Entanglement-Aware Scheduling (CEAS) framework that co-designs quantum consensus protocols with adaptive entanglement management to enable robust synchronous training across distributed quantum processors. CEAS integrates fidelity-weighted aggregation, in which parameter updates are weighted by quantum Fisher information to suppress noisy contributions, with decoherence-aware entanglement scheduling that treats Bell pairs as perishable resources subject to exponential decay. The framework incorporates quantum-authenticated Byzantine fault tolerance, ensuring security against malicious nodes while maintaining compatibility with noisy intermediate-scale quantum (NISQ) constraints. Our theoretical analysis establishes convergence guarantees under heterogeneous noise conditions, while numerical simulations demonstrate that CEAS maintains 10-15 percentage points higher accuracy compared to entanglement-oblivious baselines under coordinated Byzantine attacks, achieving 90 percent Bell-pair utilization despite coherence time limitations. This work provides a foundational architecture for scalable distributed quantum machine learning, bridging quantum networking, distributed optimization, and early fault-tolerant quantum computation.