Papers

The code quality in the game domain is perceived as less structured and more convoluted than in other domains, which results in lower maintainability and higher bug frequency. In this work, we conduct a systematic literature review to address four main research questions: (1) how the incidence of code smells and technical debt differs in game development compared to other software domains; (2) what academic solutions are proposed to mitigate these code quality issues; (3) whether researchers collaborate effectively with developers to implement these solutions; and (4) if the proposed solutions are sufficient to tackle the diagnosed problems. Thirty-four works were mapped and analyzed. We found that the game domain is affected by a short-term mentality, continuous requirements changes, reduced code reuse, and minimal to no automated testing. As potential solutions, we identified tools and models to detect code smells, along with testing practices and product lines adapted for game development. Although these solutions exist, they are not widely adopted by practitioners, highlighting the need to investigate barriers to their integration.

Gaussian processes constitute a cornerstone of probabilistic machine learning, yet scaling them to large datasets typically forces a trade-off between computational efficiency and model fidelity. This work bridges this gap by presenting an adaptive multi-resolution Gaussian process framework that is both scalable and exact. Our key innovation is constructing a naturally data-sparse covariance matrix with adaptive multi-resolution basis functions. These basis functions are directly anchored to samples, eliminating the need for auxiliary points. By shrinking the support domains of multi-resolution basis, the matrix block sizes are limited, guaranteeing sparsity. The inverse of the data-sparse covariance matrix is computed exactly and efficiently via the sparse Cholesky inverse algorithm. To further improve predictive uncertainties, we construct an augmented basis function. Theoretical analysis and numerical experiments demonstrate that our model achieves exact inference with $\mathcal{O}(n \log^2 n)$ training cost and $\mathcal{O}(\log^d n)$ prediction cost, establishing a principled framework for scalable and high-fidelity Gaussian process regression.

We study the homology of percolation clusters on disk-shaped patches of the regular $\{3,q\}$ triangle tilings for $q\ge7$. Varying $q$ gives a one-parameter family of hyperbolic tilings, from mildly curved ($q=7$) to deeply hyperbolic ($q=20$), and this paper measures and derives loop content as a function of curvature. A single growth-rate parameter $\lambda(q)$, the asymptotic ratio of successive ring sizes, governs every result. Total persistent $H_1$ of the bond-percolation complex, per vertex, is measured across $q=7,\dots,20$ and found to be a nearly linear function of $1/\lambda(q)$. An exact identity reduces total loop persistence to the weight of the lattice's minimum spanning tree minus a boundary correction. The boundary correction follows from the ring recursion, and the spanning-tree weight is derived to first order in $1/\lambda(q)$ by attaching rings one at a time to a contracted interior, with a correction for loops that close through the ring outside. The resulting formula has no free parameters, gives the large-curvature intercept in closed form as $5/4-2\pi/(3\sqrt3)$, and agrees with the measurements to within the size of the next-order term.

Early string descriptions of doped antiferromagnets associate hole motion in a locally Néel-ordered background with a path-dependent antiferromagnetic string. We ask what becomes of this mechanism when long range Néel order is lost while short range antiferromagnetic correlations persist. In a singlet bond representation, the underlying basis states do not assign an oriented \(A/B\) spin registration to each bond. Therefore, any surviving long wavelength antiferromagnetic axis is naturally described as a director, \(\mathbf n\sim-\mathbf n\). Long range Néel order can nevertheless provide a global \(S^2\) lift of this director field; when that long range registration is lost, non liftable configurations of the \(RP^2\) manifold become physically accessible, opening a topologically nontrivial \(\mathbb Z_2\) sector. Its elementary defects are half vortices that globally minimize the \(RP^2\) \(\sigma\)-model energy within the nontrivial \(\mathbb Z_2\) sector. An a posteriori energetic comparison further shows that the projective half vortex texture is softer than the corresponding vector reference texture. A pair of half vortices with opposite windings is logarithmically confined, \(V(R) = \kappa \ln\left(R/r_{\text{core}}\right)\), \(\kappa = \pi \rho_s /2\). The resulting confinement is carried not by a one dimensional string but by an extended two-dimensional spin texture. Thus, the loss of long-range Néel order does not eliminate antiferromagnetic confinement but removes the global Néel lift. This activates a distinct projective mechanism in which a local path dependent string is replaced, at long wavelengths, by a global topological texture with logarithmic confinement.

Cloud-security investigation is dominated by population tasks: which identities can read a data store, how many resources fail a control, what is reachable from another account. These resolve against a complete inventory, not a named object, so a partial answer to one is not a partial result but a different one. Coding agents can now be given read-only cloud credentials and asked to investigate directly, which raises the question of what a purpose-built security context layer still contributes. We evaluate the Sola Security Brain, a security intelligence layer whose relational substrate is resolved offline and whose security logic is evaluated against it at query time, against two coding agents operating the same live AWS environment through a read-only CLI, Claude Code and OpenAI Codex, over 28 investigation tasks. All three are scored by a blinded, tier-weighted, grounding-gated relative recall over one joint claim pool, so the scores share a denominator. The Sola Security Brain reaches 0.549 +/- 0.012 coverage against 0.340 +/- 0.011 for Claude Code and 0.281 +/- 0.006 for Codex, with the ordering identical in every grading draw. It leads 24 of 28 tasks from the cheapest model tier, at 17.8x and 20.8x lower cost per task than Claude Code and Codex. Beyond the aggregate, we describe an answer-level pattern we term sample-and-generalise: both agents enumerate a fraction of a large population, assert an unhedged universal negative, and disclose the sample size only in answer metadata rather than in the answer. In one task Claude Code reported that no bucket policies exist after checking four bucket families, in a sweep that sampled 40 of roughly 5,000 buckets, in an account where 65 buckets carry a wildcard-principal read grant.

Modern sensing records the motion of physical systems, but often leaves the forces and mechanical response governing that motion unobserved. Inferring these quantities from discretely sampled trajectories is especially difficult at coarse time scales, when mechanical response evolves between observations and interactions propagate across the system. Here we introduce Newmark-e̱ta}-DGN, a graph neural network-based framework that combines two structures inspired by computational mechanics. First, a semi-implicit update inspired by the Newmark-e̱ta} method uses learned momentum fluxes and matrix-valued response operators to advance the state over each observed interval. Second, an operator-weighted virtual hub provides system-wide coupling through a sparse set of connections. The learned quantities thus determine the predicted motion and remain accessible for mechanical analysis. Across a deformable beam, human motion and protein dynamics, Newmark-e̱ta}-DGN supports long-horizon prediction at time steps for which explicit learned simulators deteriorate. Without force, moment or constitutive relation supervision, forces inferred from walking kinematics track independently derived hip and knee joint moments, while response operators learned on the beam recover the relative spatial and directional structure of its finite-element stiffness tangent. Newmark-e̱ta}-DGN therefore links coarse-step prediction to the inference of mechanical quantities that were never observed during training.

For a (quasi)projective variety $X$ and an automorphism $f$ of $X$, we define moduli spaces of twisted endomorphisms of sheaves on $X$. If the automorphism $f$ has finite order, we construct a sheaf of noncommutative algebras $\mathcal{A}_0(f)$ and show that sheaves with a twisted endomorphism can be described as $\mathcal{A}_0(f)$-modules. We compute the topological Euler characteristics of Hilbert schemes of points of the algebra $\mathcal{A}_0(f)$, generalizing the answer for the case where $f$ is the identity automorphism.

iKFAD is a recently proposed optimiser that replaces adaptive learning rates with adaptive friction in the momentum dynamics, yet performs as well as Adam. Its limitation is that the full friction tensor $\xi\in\mathbb{R}^{m\times n}$ carries the same $\mathcal{O}(mn)$ memory overhead per layer as Adam's second-moment buffer. Here we replace iKFAD's friction tensor $\xi$ with a rank-1 outer-product factorisation built from row and column momentum statistics, resulting in Rank-1 iKFAD (R-iKFAD). This reduces the friction memory footprint from $\mathcal{O}(mn)$ to $\mathcal{O}(m+n)$ per layer, which approximately halves iKFAD's total optimiser state. Despite this reduction, R-iKFAD maintains parity in performance with iKFAD: experiments on GPT2-Nano, TinyViT, DistilBERT and GPT2-S confirm that it matches or exceeds iKFAD while nearly halving the memory footprint and remaining comparably robust to hyperparameters. We analyse the continuous-time dynamics in two damping regimes. For linear damping ($\gamma>0$) we prove exponential convergence under strong convexity. For $\gamma=0$, the preferred option in our experiments, the friction is generated entirely from past momentum and switches off as the momentum vanishes, so geometric convergence cannot be shown. We nonetheless prove convergence to the minimiser, together with matching upper and lower bounds on the energy: of order $t^{-1}$ when the regularisation scale $\epsilon_{\mathrm{stab}}$ is zero, and of order $t^{-1/2}$ when it is positive. To our knowledge this is the first convergence rate for a rank-1 factored optimiser in continuous time, and the first such result that does not require positive damping.

Data Spaces enable sovereign and governed data sharing across organizational boundaries, but their integration with AI agents remains challenging due to mismatches between probabilistic language model interactions and policy-driven data infrastructures. This article presents an architectural mediation approach based on the Model Context Protocol (MCP), implemented through the Eunomia Agent, to enable controlled interaction between large language model (LLM) agents and data space services. The proposed mediation layer translates data space capabilities into structured, schema-driven tools that AI agents can discover and invoke while preserving governance constraints. A prototype implementation validates end-to-end interaction across catalog discovery, metadata retrieval, and data service invocation without modifying existing data space components. Results demonstrate that protocol-based mediation enables interoperable and standards-aligned integration of AI agents into data space ecosystems. The approach provides practical guidance for organizations seeking to introduce AI-driven automation into governed data-sharing environments while maintaining compliance, interoperability, and architectural separation of concerns.

Air-quality sensor outages often create contiguous missing blocks, where side information useful for isolated missingness may be less reliable. We study a block-specific, GAUDI-aligned conditional diffusion imputer that retains temporal and feature processing, visible-value and mask conditioning, variable identity, and diffusion-step information, while suppressing absolute time-position side embeddings. On ItalyAir (13 variables, length-32 windows, nominal 50% block missingness; three archived seeds), this feature-side configuration achieves RMSE 0.340, versus 0.355 for full context and 0.355 for local CSDI. The experiment isolates a geometry-aware conditioning effect under block missingness.

Let $p>5$ be a prime. We prove a conjecture of Z.-W. Sun asserting that $ \sum_{k=1}^{p-1}\frac{1}{k^4\binom{2k}{k}} \equiv \frac{H_{p-1}}{p^3}-\frac{7}{45}pB_{p-5}\pmod{p^2}.$ As a consequence, we obtain $ \sum_{k=1}^{p-1} \frac{H_{k-1}^{(2)}}{k^2\binom{2k}{k}} \equiv \frac{H_{p-1}}{3p^3} +\frac{26}{135}pB_{p-5} \pmod{p^2},$ an equivalent conjecture of K. Hessami Pilehrood and T. Hessami Pilehrood. Here $H_n^{(m)}$ denotes the generalized harmonic numbers of order $m$, with $H_n=H_n^{(1)}$, and $B_n$ denotes the $n$th Bernoulli number. The proof combines $p$-adic congruences involving harmonic numbers, finite-sum identities, and complex residue calculations via the residue theorem.

We investigate the least squares linear regression problem with random partial Discrete Fourier Transform (DFT) matrices, providing a rigorous analysis of the model's generalization error. By leveraging tools from random matrix theory, we derive exact non-asymptotic bounds for the risk of the Moore-Penrose estimator, which hold for finite-dimensional problems and reveal the precise dependence on key parameters such as the sample size, dimension, and noise variance. Then we obtain a characterization of the double descent phenomenon in the linear regression context, demonstrating how the risk evolves when the number of parameters $p$ and the number of samples $n$ tend to infinity, with $p/n$ fixed. The analysis relies on applications of the Stieltjes transform for random Fourier matrices, enabling a precise description of the spectral properties of these matrices and their impact on regression performance. To validate our theoretical findings, we present several numerical examples that illustrate the double descent curves. These simulations align closely with our derived bounds, confirming their predictive power in both under-parameterized and over-parameterized regimes.

Retrieval-Augmented Generation (RAG) mitigates knowledge obsolescence and factual hallucination in large language models by introducing external context. However, when retrieved knowledge conflicts with the model's internal parametric knowledge, the model may either blindly follow misleading context or incorrectly rely on parametric knowledge, leading to unreliable responses. To address this issue, this paper proposes TRACE (Debate-TRace and Answer-Completeness rEgularized fine-tuning), a robust fine-tuning framework for RAG under knowledge conflicts. First, we propose a fine-tuning method that leverages multi-agent debate traces to extract correct candidates, incorrect candidates, and answer-shift patterns, providing fine-grained supervision for reliable knowledge-source selection. In addition, we design an answer completeness regularization mechanism to alleviate empty, overly short, and prematurely terminated responses via answer-tail token reinforcement and premature termination suppression. The fine-tuning objective combines correct-answer supervision, incorrect-candidate suppression, answer-tail token reinforcement, and premature termination suppression, enabling the model to use reliable external context, resist misleading or irrelevant retrieved content, and fall back to parametric knowledge when retrieved evidence is unreliable. Experiments across multiple knowledge-conflict scenarios and datasets show that TRACE improves robustness against misleading retrieved knowledge and reduces incomplete answers. These results demonstrate that multi-agent debate traces and answer completeness regularization jointly enhance knowledge-source selection, conflict robustness, and answer quality in RAG models. Our code is available at https://github.com/PHD-lanyu/TRACE.

A family of subsets of $[n]$ is $r$-fork-free if none of its members is strictly contained in $r$ other distinct members. For each fixed integer $r\ge2$, we prove that the maximum size of such a family is \[ \binom{n}{\lfloor n/2\rfloor} \left(1+\frac{2(r-1)}{n}+o(n^{-1})\right). \] This determines the second-order term and matches the upper bound of De Bonis and Katona. Our lower bound comes from a construction on two adjacent middle levels, using a finite ordered collection of disjoint coordinate blocks and a condition on sums modulo $n$. For each prescribed error, the blocks are fixed before $n$ tends to infinity, so the construction works for every sufficiently large $n$.

Metastable neutron stars can undergo catastrophic conversion to quark-containing stars when a localized energy deposition nucleates a critical bubble. We show that dark-matter-induced conversions can be constrained by the present transient rate of the neutron star population, rather than by the survival of any individual hadronic star as often assumed in previous literature. Population depletion makes the present transition rate nonmonotonic: weak triggering produces few transitions, whereas sufficiently rapid triggering predominantly converts stars formed in the past. Consequently, transient observations can exclude a finite interval of interaction rates, qualitatively different from conventional one-sided survival bounds. Using the beaming-corrected short gamma ray burst rate as a benchmark, we derive constraints on dark matter annihilation, decay, inelastic particle scattering, and elastic scattering of macroscopic dark matter, including parameter space not covered by existing probes. Transient demographics thus provide a new population level probe of dark sector interactions and the phase structure of ultradense matter.

AI development agents are increasingly used to support and partially automate software architecture tasks. To explore how practitioners perceive this shift, specifically what changes, what remains, and what new responsibilities emerge, we conducted a focus group at the 31st European Conference on Pattern Languages of Programs, People, and Practices (EuroPLoP 2026). Twenty-two participants from industry and academia discussed current practices, trust and validation strategies, the boundaries of AI autonomy, governance challenges, and implications for education. Among others, we found broad consensus that architectural decision-making, accountability, and the authoring of architectural guardrails remain fundamentally human tasks. A central emergent concept was harness engineering: the discipline of building the system that governs AI-assisted system creation, comprising validation mechanisms, knowledge lay- ers, and company-specific standards. The participants agreed that criticality, understood as the combination of uncertainty and cost of change, serves as the universal criterion for calibrating human oversight. A further concern was cognitive debt: the progressive erosion of human understanding of the system when AI-assisted decisions are accepted without full intellectual engagement. In this report, we present the findings of the focus group and describe directions for future work.

The Yukawa-Sachdev-Ye-Kitaev (YSYK) model, a solvable model of non-Fermi liquids with a proposed cavity-QED realization, couples $N$ complex fermions to $M$ bosons of frequency $\omega_0$ through random Yukawa couplings of strength $g$, and its dynamics depends only on $R\equiv\omega_0/g^{2/3}$. For $R\ll1$ the bosons are slow and act as a static random background in which the fermions are free, while for $R\gg1$ they are only virtually excited and generate a random four-fermion interaction of rank $M$, which is generically chaotic for $M\geq2$. Using exact diagonalization at infinite temperature, we follow the emergence of random-matrix statistics between these limits with unfolded spectral statistics and the Krylov complexity of the thermofield-double state. For complex couplings the level-spacing distribution evolves from Poisson to Gaussian-unitary-ensemble (GUE) statistics, the spectral form factor develops a correlation hole and a ramp, and the Krylov complexity evaluated on the unfolded spectrum develops a peak above its late-time plateau whose height reaches the GUE value. All diagnostics locate the onset around $R\approx0.3$, and adding boson modes moves it to smaller $R$. This delineates the parameter regime in which a cavity-QED realization would display random-matrix spectral correlations. On the other hand, real couplings give the same crossover towards Gaussian-orthogonal-ensemble (GOE) statistics, and along an interpolation between the two the peak height follows the mean gap ratio, so the peak identifies the symmetry class as well as the onset of chaos. Without unfolding the peak stays about 20$\%$ below its random-matrix value in both classes, which we trace to the non-semicircular density of states, and at larger $R$ the boson-number bands slow down the spreading of the state in Krylov space.

We apply physics-informed neural networks (PINNs) to recover the time-time component of static, spherically symmetric black-hole exterior metrics from a reduced ordinary differential equation (ODE). Inspired by recent work on solving Einstein field equations with deep learning~\cite{Li2023}, we encode the vacuum/charged exterior through a residual loss and an asymptotic boundary constraint—not by embedding analytic metric terms such as $2M/r$ directly into the network output. Unlike the distributed PINN (DPINN) strategy of Ref.~\cite{Li2023}, which partitions the radial domain into subdomains with separate networks, we employ a unified fully connected network over the entire interval $[r_{\min},r_{\max}]$, avoiding spurious jumps at subdomain interfaces. Moreover, whereas Ref.~\cite{Li2023} restricts training to $r\in(10,300)$—well outside the steep $1/r$ and $1/r^2$ curvature of the inner exterior—we begin at $r_{\min}=10^{-2}M$, spanning three decades in radius and covering the strongly varying region that DPINNs sidestep by domain truncation. Holding the mass fixed at $M=1$, we sweep electric charge $Q\in\{0,0.5,1.0,1.1\}$ at $\Lambda=0$ and sweep $\Lambda\in\{0,0.1,-0.1\}$ at $Q=0.5$. All configurations achieve relative $\mathcal{L}_2$ errors below $6\%$ against the analytic reference. For the representative case $Q=0.5$, $\Lambda=0.1$, three independent trainings with fixed seeds yield relative errors of $3.89\%$, $1.08\%$, and $2.76\%$ (mean $2.58\%$, standard deviation $1.41\%$), demonstrating robustness of the mesh-free approach without labeled field data.

At Hermite rank two, the constant in the weighted law of the iterated logarithm under long-range dependence is represented as the largest eigenvalue of a suitably normalized positive integral operator on the unit interval. The same eigenvalue determines the exponential-moment threshold and the logarithmic right-tail rate of the weighted second-chaos limit, which in the unweighted case is the Rosenblatt law. The weighted third spectral moment is evaluated in closed form. Convergent two-sided spectral enclosures are obtained, and in the weight-concentration limit the leading eigenvalue is characterized by a scalar equation with a uniform geometric remainder. At the unweighted memory boundary, the constant decays like the square root of the distance to criticality. Its leading coefficient is determined to $25$ decimal places with certified full-operator residual bounds. After division by the square-root boundary factor, the weight-concentration and memory limits commute. Their common coefficient differs from the unweighted boundary coefficient and lies in the certified interval $(1.370323114331,\,1.370323114332)$. The joint asymptotic formula is established with a uniform two-parameter remainder bound.

We consider a fixed finite number of independent two-dimensional discrete Gaussian free fields on a box of side $n$, with zero boundary values, conditioned so that the vector-valued field stays outside a fixed ball on a macroscopic interior domain. We prove that the field, divided by $\log n$, converges in distribution in $L^2$ to twice the equilibrium potential of the domain multiplied by a uniform random unit vector. We also identify the exact capacity cost of the conditioning event and obtain alignment on a deterministic density-one set of sites, with no restriction on their mutual distance. The proof combines the norm repulsion theorem of Korzhenkova and Sepúlveda with Gaussian large deviations and the equality case in the capacity variational problem.

Reweighting a probability law by a positive numéraire and pushing forward the payoff-to-numéraire ratio yields the change-of-numéraire reweighting: the swap-rate law under the annuity measure, the numéraire-inversion involution of martingale optimal transport, and the population form of self-normalized importance sampling. We characterize exactly when it is Wasserstein-stable. Along weakly convergent inputs with uniformly integrable numéraires and a strictly positive limiting numéraire mean, convergence of the reweighted laws is equivalent to a uniform integrability condition, computed under the inputs, on one explicit family in the payoff and numéraire. At first order the numéraire cancels, the sole obstruction being payoff mass where the numéraire vanishes; absent such mass, prices and first moments pass to the limit under a uniformly integrable payoff alone, with no assumption on the reciprocal numéraire. Two-atom examples with bounded inputs attain the threshold exactly. Applications characterize a standing moment assumption in the transport literature and delimit which annuity-measure statistics survive.

Distinguishability is a central concept in information theory and extends naturally to quantum systems. This work shows that, for any two finite-dimensional memoryless quantum channels, parallel, adaptive and general testing strategies achieve the same Stein exponent at every fixed type-I error tolerance $\epsilon\in(0,1)$, namely the regularized channel relative entropy. When this rate is finite, any larger type-II exponent forces the probability of correctly accepting the null hypothesis to decay exponentially. Thus, adaptivity provides no asymptotic advantage. The key step is proving the continuity of the regularized sandwiched Rényi channel divergence at order one. We also derive the exact strong-converse exponent for general testers.

Although the ultraviolet background from reionization is thought to conspire against the smallest halos, some gas-rich halos in the post-reionization Universe still form stars. We ask which properties, environmental or intrinsic, determine ignition. Using the FIREbox cosmological volume, we compare recently-ignited halos to a mass- and gas-matched sample of gas-resolved starless halos at $z=2$–6. We test three families of "third axis" properties beyond $M_{\rm vir}$, $M_{\rm gas}$, and $z$: geometric environment (the Perturbation Index and local galaxy density $\rho_5$), hydrodynamic environment (the $1$–$2,R_{\rm vir}$ gas-supply shell), and intrinsic halo properties (concentration and the neutral-gas reservoir). A nested random-forest classification finds that geometric environment adds little, while the hydrodynamic shell is modestly informative. Three complementary properties each discriminate ignited from starless halos beyond the mass–gas–redshift baseline (AUC $=0.635$, where $0.5$ is chance): total HI mass ($M_{\rm HI}$, AUC $=0.739$), the extent of the neutral gas ($r_{\rm HI,50}/R_{\rm vir}$, AUC $=0.738$), and the central HI density ($\rho_{\rm HI,0}$, AUC $=0.700$); jointly they reach AUC $=0.823$, the highest of any combination we test. At fixed halo mass, gas mass and redshift, recently-ignited halos harbor more HI over a larger radius than their matched controls, and these two enhancements are uncorrelated. The same reservoir sets when ignition can happen: from $z=6$ to $z=0$ the median starless halo keeps its gas ($+0.05$ dex) while its neutral fraction falls by $3.9$ dex, so the reservoir is ionized rather than exhausted and the ignition window closes below $z\simeq2$. In sum, a halo's own neutral-gas reservoir discriminates ignited from starless halos more sharply than its location in the cosmic web – nature over nurture.

Asynchronous monitoring, incident investigations, and compliance audits primarily rely on agent traces to reconstruct what happened. These analyses assume that LLM agents cannot tamper with their own execution traces. We show that local LLM agents such as Claude Code, Codex, Antigravity, Open Code and Grok Build fail to enforce this boundary. All tested harnesses, except Muse Code, allowed agents to delete their traces when asked, without triggering monitor guardrails. We also validate that external attackers can exploit this gap to induce trace deletion. Finally, we show that trace tampering behavior emerges naturally in frontier models, when agents try to improve their rewards. We advise practitioners to ensure trace logging happens through an independent interception mechanism outside of the agent's control, preserving trace integrity even in cases of full host compromise. Overall, our findings identify a concrete failure of trace integrity in agent infrastructure which can be used to conceal misaligned behaviors like scheming or sabotage.

Weyl or Dirac fermions with the iconic linear energy-momentum relation and average density of states (ADOS) $\rho(E) \sim |E|^{d-1}$ at energy $E$ in $d$ spatial dimensions, constitute a unique setup to study the disorder-driven semimetal-to-metal quantum phase transition (QPT) between ballistic (realized for weak disorder) and diffusive (stabilized at stronger disorder) quasiparticles. Such a QPT takes place only for $d>2$ and falls beyond the realm of the Anderson metal-to-insulator transition. From numerically computed ADOS (using the kernel polynomial method) in dirty Weyl systems in $d=2$ to $6$, here we show that $d=2$ and $d=4$ are the lower ($d_\ell$) and upper ($d_u$) critical dimensions for such an unconventional QPT, respectively, as suggested from the solution of quasiparticle lifetime within the self-consistent Born approximation. Consequently, for $d \geq 4$ the associated correlation length exponent is found to be $\nu \approx 0.5$ (within numerical accuracy). However, the dynamic scaling exponent at the quantum critical point is pinned close to $z \approx d/2$ (numerically) for any $d \geq 3$, which is shown to be an exact result from a field-theoretic renormalization group calculation. Therefore, Weyl semimetal-to-metal QPTs can be studied field theoretically around both $d_\ell$ and $d_u$.

Latent world models are typically trained to predict factual transitions, whereas model predictive control (MPC) must compare alternative actions from the same state. A model can therefore achieve low factual prediction error yet poorly distinguish candidate actions. We introduce AD-WM, an action-discriminative joint-embedding world model for counterfactual MPC. AD-WM combines residual latent dynamics with predictor-level action-recovery regularization, using inverse dynamics and a normalized recovery objective motivated by conditional mutual information. Both objectives encourage planning transitions to preserve action information; their auxiliary heads are discarded at test time, leaving MPC unchanged. On OGBench-Cube, AD-WM improves hard-start success from 3.7% to 52.0% over a matched LeWM baseline and improves mean success over the reproduced baseline in four of five simulation environments. Planning diagnostics show that factual prediction error and whole-bank action ranking do not follow the closed-loop success ordering, whereas CEM-aligned elite regret tracks success more closely. With a frozen V-JEPA 2 encoder and matched DROID post-training, AD-WM also improves zero-shot transfer to our Franka setup, increasing basic pick-and-place success from 42.2% to 71.1% without lab-specific adaptation. These results suggest that world models for planning should preserve action-dependent differences needed for counterfactual selection, rather than optimize factual prediction accuracy alone. More videos and code are available at https://ad-wm.github.io/.

Let $p_c(\mathcal F)$, $q(\mathcal F)$, and $q_f(\mathcal F)$ denote the threshold, expectation threshold, and fractional expectation threshold of a family $\mathcal F$ of nonempty subsets of a finite set, respectively. We prove that there is an absolute constant $C>0$ such that, if $\mathcal F$ has VC dimension at most $d\ge1$, then $p_c(\mathcal F)\le Cq(\mathcal F)\log(d+1)$. More generally, for every $0<\varepsilon\le1/2$, a binomial random set of density $\min\{1,Cq(\mathcal F)\log((d+1)/\varepsilon)\}$ contains a member of $\mathcal F$ with probability at least $1-\varepsilon$. Consequently, $q_f(\mathcal F)\le Cq(\mathcal F)\log(d+1)$, verifying Talagrand's integral–fractional conjecture for families of any fixed VC dimension. We also prove that if a $k$-spread probability measure has support of VC dimension at most $d$, then a binomial random set of density $\min\{1,(C/k)\log((d+1)/\varepsilon)\}$ contains a member of its support with probability at least $1-\varepsilon$. In both random-containment results, the factor \(\log((d+1)/\varepsilon)\) is optimal up to absolute constants. As an application of the spread theorem, we prove that every $n$-uniform family of VC dimension at most $d$ with more than $(C p^{-1}\log((d+1)/\varepsilon))^n$ members contains a $(p,\varepsilon)$-robust sunflower. In particular, every such family with more than $(Cr\log(d+1))^n$ members contains an $r$-sunflower, improving the recent bound $(Crd)^n$ of Ge, Wang, Xu, and Zhao.

We introduce a new structural framework for constructing multivariate Alexander quandles based on a groupoid ${\cal G}$, composed of a disjoint union of delooping groupoids of deck groups. Compared to standard Alexander modules over group rings ${\mathbb{Z}}[{\mathbb{Z}}]$, the multivariate version involves modules over groupoid rings ${\mathbb{Z}}[{\cal G}]$. A manifestly categorical description is given in terms of ${\mathbb{Z}}$-Algebroids, whose functorial module theory yields an equivalent definition of multivariate Alexander modules. Using the category of elements construction, we define multivariate Alexander quandles from this groupoid framework. Three classes of multivariate quandle operations follow. In contrast to earlier approaches, we abstract the algebraic structure from explicit link-dependence. As a consequence, we have a structural framework for: (i) enumerating new classes of Alexander quandles, and (ii) finding new coloring invariants of links. We show that maps between ${\mathbb{Z}}$-modules, facilitated via multivariate quandle operations, realize a quiver presentation of quandles, which makes manifest the oidification of Alexander modules. Oidified Alexander modules, based on groupoids, provide a constructive framework for composing univariate quandles to obtain new multivariate ones. Finally, we comment on the possibility of new link invariants coming from groupoid-based formulations, including the fundamental groupoid.

We show that any convex co-compact hyperbolic surface with exponent of convergence of Poincaré series $\delta \in (\frac25,\frac{14}{27})$ has an essential spectral gap of size $\beta=\tfrac78(\tfrac12-\delta)+\tfrac{1}{32}\delta-\epsilon$ for any $\epsilon>0$. In particular, for $\delta=\frac12$ this becomes $\beta=\tfrac{1}{64}-\epsilon$. We show existence of the gap by proving a new Fractal Uncertainty Principle, using a two-ends Furstenberg theorem of O'Regan-Wu-Yi [arXiv:2607.08461].

We introduce projected amorphous topological insulators (PATIs) that despite containing only a fraction ($x$) of otherwise randomly selected disconnected sites of a parent crystal, feature a quantized global topological invariant (strong PATIs) when its effective Hamiltonian is constructed by integrating out the residual sites of the original lattice. Within the topological regime of the parent model, additionally, there exists a critical $x$ below which such systems foster a fragile PATI that only supports quantized local topological marker on a small fraction of sites therein. Both strong and fragile PATIs accommodate in-gap modes near the boundary. The system also hosts a normal insulator in the entire trivial parameter regime of the lattice-based model for any $x$, devoid of gapless boundary modes, for which both global topological invariant and local topological marker vanish. We demonstrate these possibly generic outcomes by starting with a parent square lattice-based model for time-reversal symmetry breaking insulators. Then the strong-to-fragile PATI quantum phase transition, tunable via $x$, is characterized by the mean correlation length exponent $\nu \in (1.00,1.46)$.