Papers

The term "linear representation hypothesis" (LRH) has appeared across diverse subfields of artificial intelligence, neuroscience, and cognitive science. But previous works have not consistently treated the LRH as a falsifiable scientific hypothesis; we analyze these inconsistencies and examine their implications for how prior theoretical and methodological results should be interpreted. Based on this analysis, we argue that claims regarding linear representations become well-defined only through careful examination of the model, representation location, feature definition, and evaluation dataset. We therefore propose a more rigorous formalization of the LRH that makes these dependencies explicit and allows the hypothesis to be evaluated as a falsifiable scientific claim. Finally, we identify some non-trivial open problems that warrant further attention from the research community.

Public crash databases increasingly support automated safety analysis, but crash severity prediction remains difficult to translate into public-sector decision workflows when models are evaluated primarily as ordinary classifiers. This study reframes dementia-related crash severity modeling as a decision-aware triage problem in which a system must classify crashes into no-injury/property-damage-only (O), minor or moderate injury (BC), and fatal or severe injury (KA), while also controlling outcome leakage, reporting severe under-triage, calibrating confidence, and preserving every raw prediction for audit. Using 4,781 Texas crash records with structured fields and police narratives, we evaluate structured, narrative, fusion, calibrated fusion, BERT-family, and local large-language-model baselines under a stratified 70/15/15 split. In the reported split, leakage-controlled Gemma obtains the highest observed macro-F1 (0.545; 95% bootstrap CI [0.507, 0.583]). The best calibrated fusion model obtains macro-F1 of 0.522 and expected calibration error of 0.033. Selective deferral improves performance among cases retained for automatic classification. At 70% coverage, macro-F1 rises to 0.573 and severity cost falls to 0.577, while deferred cases are treated as candidates for a proposed human-review process and are not further evaluated in the present experiment. The study contributes a reproducible, leakage-controlled, and uncertainty-aware evaluation framework for crash AI systems, emphasizing auditability and selective deferral rather than accuracy alone.

For a dissection $D$ of the unit square into $n$ nondegenerate triangles, let $R(D)=\max_i a_i-\min_i a_i, \Delta(n)=\inf_D R(D).$ We prove that this infimum is attained for every $n\ge2$, and determine the exact minima for $n=5$ and $n=7$, allowing T-junctions. For five triangles, $\Delta(5)=\frac{5\sqrt5-11}{8};$ equality holds precisely when three areas equal $(3-\sqrt5)/4$ and two equal $(3\sqrt5-5)/8$. For seven triangles, $\Delta(7)=r_7$, where $r_7$ is the unique root in $(0,1/4900)$ of $864r^4+2160r^3-6060r^2+4972r-1.$ Every minimizer has four areas $(1+3r_7)/7$ and three areas $(1-4r_7)/7$, although its geometry need not be unique. The proofs combine finite combinatorial classification with exact symbolic and integer-interval certificates. For nine triangles, a tilted-strip construction gives the explicit algebraic upper bound $\Delta(9)\le 0.0001273496861283553341\ldots,$ which is the exact minimum within that topology. Conversely, every dissection in the complete single-cap two-rail zig-zag family, with arbitrary continuous areas, has range greater than $1/3500$; hence a global minimizer must lie outside that family. The exact value of $\Delta(9)$ remains open.

Phase matching is a fundamental problem in nonlinear optics that is normally constrained by material dispersion. The limited operation wavelengths within phase matching window limits the application, including the precise metrology using Kerr combs. Demanding applications like compact optical clock and astronomical spectroscopy requires atomic reference around 800 nm, where the natural phase matching is challenging. Here we revisit the concept of cavity phase matching (CPM), and fully reveal its advantage to engineer artificial phase matching beyond material dispersion. With the access of CPM condition in a monolithic high-Q fiber Fabry-Pérot resonator featuring a macroscopic cavity length, we achieve low noise Kerr comb generation around 800 nm, within the power budget of a single-mode laser diode. Inside a pump-integrated package of 19 cm3, low phase noise of -125 dBc/Hz at 100 kHz offset frequency is achieved for a 10.1 GHz repetition rate. Most importantly, the generated Kerr comb has been directly referenced to rubidium atomic transition for long-term stable operation. This result not only opens a new way for Kerr comb generation at arbitrary wavelengths, but also can be generalized to any other nonlinear optical frequency conversion application.

Building autonomous agents that reproduce human behavior in realistic 3D environments has been a longstanding objective in AI. Cognitive neuroscience commonly believes that human behaviors are controlled by multiple control systems including goal-directed and habitual. However, existing human-like agent frameworks primarily focus on goal-directed system and habitual system is largely overlooked. In this paper, we address this gap by proposing GEMS, a Generative Embodied Multiple Behavior Control System that jointly models goal-directed and habitual behaviors for generating believable human-like behavior. GEMS employs a cognitive module to generate natural language behaviors and a motion generation module to construct them in 3D environment. The cognitive module comprises a Habitual Controller that retrieves habitual actions from habit memory in response to environmental stimuli, a Goal-directed Controller that proposes goal-directed actions for current goal, and an Arbiter that governs their influence and selects the final action, supporting diverse and naturalistic behavior patterns generation. To construct these behaviors, the motion generation module adopts a Text-to-Motion model to generate versatile and expressive 3D human-scene interactions without relying on predefined actions. Extensive evaluations, human and ablation studies demonstrate that human-likeness performance is substantially improved by GEMS. The efficacy of GEMS indicates the benefits of leveraging habitual behavior and multiple behavior control system coordination for believable embodied human-like agents design.

We develop an index policy for finite-horizon Bernoulli multi-armed bandits from minimax solutions to single-arm bandit (SAB) problems. Each SAB problem involves choosing between an unknown Bernoulli arm and a known reward. We show that minimizing worst-case regret of SAB problems over all non-anticipative policies admits an exact semi-infinite linear programming formulation. The resulting stopping policies offer a natural way to compare arms: the higher the known reward against which a policy continues sampling, the more promising the unknown arm. We turn this intuition into indices based on cumulative continuation probabilities, with a monotone adjustment and a reward-shortfall cap. By relating index errors to the regret of single-arm stopping policies, we establish a distribution-free regret bound of $4.45\sqrt{KT}+10.75K$ for $K$ arms and horizon $T$. This bound matches the minimax-optimal regret order established in the literature. The guarantee extends to rewards supported on $[0,1]$ through Bernoulli randomization. We also provide a finite-grid implementation with quantified approximation loss. In numerical experiments, the SAB-based index policy achieves lower worst-case regret than every tested benchmark policy across all evaluated numbers of arms and horizons, while closely matching the grid-based MAB minimax policy in the two-arm setting.

We study the global continuation of solution manifolds arising in dynamical systems. We present a rigorous continuation method based on a Chebyshev series expansion of the solution manifold. The branch is first approximated by a high-order Chebyshev interpolation polynomial, and an explicit error bound is then obtained by verifying the contraction of a quasi-Newton operator near this approximation. The contraction is formulated on a weighted $\ell^1$ space, giving a finer control than the typical $C^0$-error bound obtained from the uniform contraction theorem. In fact, the latter follows directly from our contraction operator. Furthermore, we discuss how our strategy applies naturally to pseudo-arclength continuation, where the continuation parameter fails to provide a valid local coordinate, and extends to multi-parameter continuation. Lastly, we detail two applications in which we compute a two-parameter family of steady-states for the Cahn–Hilliard equation, and a one-parameter family of steady-states undergoing saddle-node bifurcations for the Shigesada–Kawasaki–Teramoto system.

Parametric CAD modeling requires not only matching the target shape, but also preserving design intent under parameter edits. However, most existing methods focus on improving static shape generation accuracy, while the dependencies that govern later edits are often overlooked. To tackle this issue, we present DIP-CAD, a design-intent-preserving agent harness for generating parametric CAD models from images. DIP-CAD combines vision-language model reasoning with CAD kernel operations to establish dependencies that encode the design intent. In parametric CAD, these dependencies are mainly established through geometry references in modeling commands and constraints in parametric sketches. At the command level, ID-based explicit geometry reference binds modeling commands to selected faces, edges, or vertices. At the sketch level, a local-coordinate bridge ensures accurate sketch placement, providing the basis for correctly establishing sketch constraints. Constraints among sketch primitives and constraints to external projected geometry further preserve both internal sketch relationships and dependencies on existing geometry. To provide a more comprehensive data foundation for parametric CAD modeling tasks, we introduce the Geometry Explicit Reference Dataset (GERD), whose structured data format links stepwise modeling commands to intermediate CAD models through explicit geometry references. On GERD-EVL and a DeepCAD test subset, DIP-CAD improves mIoU for static model generation by 11.1% and 5.6%, respectively. In parameter-editing evaluations, the corresponding mIoU improvements are 7.0% and 4.2%. Case studies further demonstrate how parameter edits drive shape updates while preserving design intent.

We present PanoSeg3R, a feed-forward framework for 3D panoramic semantic segmentation. Unlike existing methods designed for perspective inputs, PanoSeg3R jointly predicts 3D geometry and multi-view semantic segmentation in one single forward pass. Built upon a pretrained reconstruction backbone that supports panoramic images, our approach extends feed-forward 3D reconstruction with a query-based mask decoder. Furthermore, we introduce an automatic panorama data curation pipeline that leverages the complementary strengths of off-the-shelf foundation models to generate reliable pseudo semantic annotations, substantially expanding the training data and improving zero-shot generalization. PanoSeg3R achieves state-of-the-art performance on panoramic 3D semantic segmentation, improving 3D mIoU by up to 16.02 on ScanNet++, while the curated training data further improves zero-shot performance by up to 4.26 and 43.28 mIoU on Stanford2D3D and ToF-360, respectively. Website: https://harryyoon777.github.io/PanoSeg3R/

In randomized clinical trials with longitudinal continuous outcomes, missing-not-at-random (MNAR) missingness often motivates conservative alternatives to mixed models for repeated measures (MMRM). Such caution is important for estimation, but estimation and testing need not require identical assumptions. Moreover, overly conservative primary analyses may reduce power, increase required sample size, and raise trial costs. We investigated the validity of MMRM-based testing under the global null of identical longitudinal outcome distributions across groups. Because valid testing minimally requires treatment-effect estimators to converge to the null under the null hypothesis, we investigated sufficient conditions for this property. We introduced a proportional observation condition requiring ratios of observation probabilities relative to a reference group, conditional on the full outcome vector, to be outcome-independent, and showed that, with arbitrary post-baseline visits and monotone missingness, this condition is sufficient for convergence to the null value. The condition allows observation to depend on unobserved outcomes and permits between-group differences in overall observation probabilities through outcome-independent dropout, making it clinically interpretable while accommodating outcome-dependent MNAR missingness. Synthetic and data-based bootstrap simulations showed negligible bias and empirical test sizes near 0.05, including nonmonotone missingness. Thus, MNAR missingness does not by itself imply that a more conservative primary testing procedure is required. This result does not justify treatment-effect estimation under alternatives, which still requires estimand-based interpretation and sensitivity analyses.

Given a unital inclusion of $C^*$-algebras $\mathcal C \subset \mathcal A$ and a unital inclusion $\mathcal C \subset \mathcal B$ into a von Neumann algebra $\mathcal B$, we investigate the extreme points of unital completely positive maps from $\mathcal A$ to $\mathcal B$ that fix $\mathcal C$ denoted by $UCP_\mathcal C(\mathcal A, \mathcal B)$. This space is obviously convex and $C^*$-convex with respect to the $C^*$-algebra $\mathcal C' \cap \mathcal B$. In this article we show that the $\mathcal C' \cap \mathcal B$-extreme points are exactly the $*$-homomorphisms that fix $\mathcal C$ when $\mathcal A$ is commutative and $\mathcal B$ has a normal faithful center valued trace. This generalizes a result due to Farenick and Morenz where $\mathcal C = \mathbb C 1$ and $\mathcal B = M_n(\mathbb C)$.

Memory-dependent robotic manipulation often requires later actions to use information from earlier interactions. Existing vision-language-action (VLA) policies primarily rely on current observations, limiting historical information retention. Memory-augmented VLAs, such as MemoryVLA, address this limitation with external memory banks but require explicit storage and retrieval. To address these limitations, we propose StateMem, a single-state residual memory framework for VLA policies that uses prediction error to update a persistent memory token through low-rank residuals and to adaptively route cached prefixes. A training-free controller adjusts the routing threshold online, while fast correction compensates for stale prefix features during cache reuse. We evaluate StateMem on LIBERO, RoboMemArena, and real-world manipulation tasks. On LIBERO, StateMem achieves an average success rate of 97.6% and reduces the average VLM prefix refresh rate by 20.25% relative to full refresh. In the Occlusion category of RoboMemArena, StateMem achieves the best performance among single-VLA methods, reaching 21.8% Task Success Rate (TSR) and 44.3% Cumulative Success Rate (CSR). Across six real-world manipulation tasks, it achieves +21% in average success rate.

We construct explicit examples of skew braces $A$ for which the third term $A^3$ in the left series is not generated by the right-normed star products $a_1*(a_2*a_3)$ of length $3$ as a sub-skew brace.

Collective intelligence depends not only on what team members know, but also on how they organize their work. When the structure of a solution is unknown, useful roles and divisions of labor cannot be specified in advance; teams must learn from experience how to organize reasoning as it unfolds. Human teams routinely adapt this way, while existing AI agent teams rely on fixed protocols, explicit task decomposition, or routing. We introduce Self-Organizing Agent Teams (SAT), fixed teams of AI agents that learn reusable strategies from prior collaborations to organize roles, conversational phases, participation, and information flow. These strategies enable what we call collaborative computation: agents exchange, challenge, repair, and synthesize partial reasoning into solutions no member produced independently. In two independent settings, we learn teamwork strategies that transfer unchanged to unseen benchmarks, using only 15 mathematics and 25 graduate-level knowledge problems. Across five mathematics and physics benchmarks, self-organizing teams average 66.7% accuracy, versus 48.8% for their strongest member, 58.7% for compute-matched inference by that agent, and 59.0% for a perfect router over members' independent answers; on AIME 2026, they exceed this router by 13.4 points. Because gains vary across benchmarks, we ask when self-organizing collaboration helps. Across eight benchmarks, demonstrability (the organizational-psychology construct of whether a team can distinguish correct from incorrect reasoning) strongly tracks improvement over the strongest member (Spearman $\rho=0.90$, $p=0.005$): teams benefit most when correct reasoning can be recognized once it appears. More broadly, these results suggest that organization itself can become an agent capability: agent teams can learn how to reason together and produce solutions their members could not reach independently.

Dexterous manipulation in confined spaces requires local control of hand orientation. Without a wrist, a dexterous hand must obtain this local orientation through coordinated motion of the robot arm, often involving several joints and a more complex end-effector path. We present CRAFT-Wrist, a concentric 2-DoF wrist extension that mounts between a robot arm and the CRAFT Hand without modifying the hand. Two XC430-T240BB-T servos drive sideways and front-back rotation through short rigid transmissions. Our initial hardware prototype actuates both axes under load, achieving a demonstrated workspace of $\pm20^{\circ}$ in radial–ulnar deviation (left–right) and a front-back range of $+80^{\circ}$ in flexion and $-18^{\circ}$ in extension. We characterize the resulting finger-motor loading across several wrist postures and demonstrate how the wrist supplies local orientation in grasping, nail hammering, and blackboard wiping. Demonstrations and assembly instructions are available at https://craft-wrist.github.io/.

We prove that, for any $f:S^2\to S^1$ and any $\varepsilon>0$, there exist orthogonal vectors $x,y\in S^2$ such that the length of the shortest arc between $f(x)$ and $f(y)$ is at most $\pi/2 +\varepsilon$. This proves a conjecture of Ghebleh from 2007 that the circular chromatic number of the real orthogonality graph is equal to four.

For infinite cardinals $\lambda \leq \kappa$, we calculate the calibers of topological spaces $X$ of cardinality $\kappa$ whose open sets are the subsets $A$ satisfying $|X \setminus A| < \lambda$. We also prove that the set of calibers of $X$ coincides with the set of calibers of the Vietoris hyperspace $\mathcal{H}(X)$, where the space $\mathcal{H}(X)$ has as its underlying set a collection of subsets of $X$ that includes all its finite subsets and is contained within the collection of its closed subsets.

Density-Driven Optimal Control (D2OC) provides an effective framework for steering multi-agent systems toward prescribed spatial distributions. However, existing D2OC formulations are primarily developed for Euclidean domains and do not directly account for intrinsic manifold geometry. This paper extends D2OC to second-order multi-agent systems evolving on Riemannian manifolds. The proposed Riemannian D2OC (R-D2OC) constructs a local distribution objective in the tangent space of each agent through logarithmic maps and uses its weighted center as the reference for a finite-horizon LQR. The resulting control is executed on the manifold through intrinsic second-order dynamics and parallel transport within a receding-horizon scheme. We establish local curvature-dependent bounds that quantify the approximation introduced by the tangent-space reduction and characterize the resulting target bias. Furthermore, we derive a conditional discrete-descent result showing that the closed-loop objective decreases when a local velocity-alignment condition is satisfied. Numerical simulations on a 3D ellipsoidal manifold demonstrate distribution-level control and empirically support the proposed approximation and descent results.

Lightweight and slim manipulators enable safe operation in human living environments. Proximal actuation using remote transmission mechanisms, such as wire-driven or Bowden cables, effectively reduces inertia and arm size by relocating motors near the base and transmitting torque to distal joints. Existing approaches either increase mass through additional components, such as pulleys for direction changes, or suffer from reduced transmission efficiency due to friction losses. Flexible shaft transmission avoids both mass increase and excessive friction losses, but faces increasing angular transmission error due to helical buckling caused by slack generated at joint bending. To address this problem, we propose SHAFT: a Slack-compensating, Helical-buckling-Attenuating Flexible- shaft Transmission mechanism. This mechanism compensates for slack through a proximal tensioner, improving the angular transmission error and efficiency of flexible shaft transmission without increasing the moving mass of the arm section. In a transmission path containing four 90-degree bends, the proposed mechanism demonstrated approximately 30% higher efficiency and approximately 65% lower angular transmission error compared to a flexible shaft transmission without a tensioner. Using this mechanism, we fabricated a 6-Degree-of-Freedom (DoF) arm with a 1-DoF gripper manipulator consisting of a rotary module housing motors with a tensioner, and a 280 g weight for the arm module. The proposed manipulator represents a novel remote actuation system for achieving lightweight construction with high efficiency, contributing to the acceleration of safe robot deployment in human environments.

We prove a Pólya–Szegö inequality for functions defined on an $n$-dimensional submanifold $\Sigma$ of a complete noncompact Riemannian manifold with nonnegative sectional curvature. The associated rearrangement is a Schwarz rearrangement on $\mathbb R^n$, and the constant depends on the $L^n$-norm of the mean curvature of $\Sigma$ and an isoperimetric quantity obtained by Brendle. As applications, we derive Sobolev, Log-Sobolev, Hardy, and Gagliardo–Nirenberg inequalities on submanifolds of arbitrary codimension under a small total mean curvature assumption. In the critical Sobolev case, we obtain Moser–Trudinger inequalities on finite-volume submanifolds and exact growth inequalities on submanifolds with infinite volume. Under suitable assumptions, the Pólya–Szegö constant equals one; in this case, the critical constants in the inequalities coincide with the sharp Euclidean ones.

We prove Villani's conjecture on entropy production for Kac's walk with uniform collision angles. For every $N\geq2$, with total collision rate $N$, the optimal entropy production constant is $2/(N-1)$. We show that the known lower bound is sharp by constructing two families of smooth strictly positive probability densities, invariant under coordinate permutations and sign changes. The first consists of normalized products of inverse powers; the second is obtained by conditioning products of Gaussian mixtures on the energy sphere. With $N$ fixed, we compute the leading terms of entropy and entropy production. Successive parameter limits yield two independent proofs of sharpness.

Joint Embedding Predictive Architectures (JEPAs) are becoming a core representation-learning primitive and a building block for latent world models across vision, video, audio, brain dynamics, and time series. Despite (potential of) wide deployment, current JEPA training pipelines are inefficient: each input is executed through multiple mask-specific branches, with redundant target-side work, and memory-bound token routing. These costs grow with the number of masks and limit GPU efficiency. We present M-JEPA, a mask-aware execution architecture that restructures JEPA training without changing the learning objective. M-JEPA separates mask-independent computation from mask-dependent routing, enabling shared context encoder execution, fused token routing and slicing with backward support, sparse target encoder execution over the union of target tokens, and masked patch embedding for sparse inputs. The resulting pipeline preserves training semantics while reducing computation, memory traffic, and synchronization overhead. We implement M-JEPA for five JEPA variants and evaluate it on NVIDIA A100 GPUs. Compared against the state-of-the-art baselines, M-JEPA achieves up to 1.7x end-to-end training speedup for 2-10 masks. Separately, with masked patch embedding, 4.75x patch-embedding speedup at high sparsity. These results show that execution restructuring, rather than changes to the JEPA objective, is a key lever for efficient JEPA training.

We investigate collective chirped stimulated Raman adiabatic passage (STIRAP) in a pair of identical three-level atoms using a symmetric six-state model. Numerical simulations reveal a pronounced oscillatory dependence of the population transfer to the doubly excited Rydberg state on the peak Rabi frequency, indicating dynamics beyond the conventional single-dark-state description of STIRAP. To identify the underlying mechanism, we develop a dressed-manifold description based on gauge-invariant projections onto nearly degenerate instantaneous dressed manifolds. The analysis demonstrates that the wavefunction remains predominantly confined to a two-dimensional dark manifold while becoming transiently coupled to a neighboring bright manifold during the pulse-overlap interval. We further show that in the dressed-manifold representation, the Rydberg-Rydberg interaction modifies the collective resonance structure while preserving the dark-bright manifold dynamics responsible for the oscillatory transfer; an appropriately chosen frequency chirp compensates the interaction-induced detuning and restores efficient population transfer. This dressed-manifold picture provides a consistent interpretation of oscillatory collective transfer and establishes a framework for controlling adiabatic dynamics in interacting multilevel systems by linearly chirped STIRAP, opening a route toward manifold-based control of Rydberg-mediated quantum gates, correlated-state preparation, and other coherent operations in interacting atomic systems.

This letter develops closed-loop performance certificates for a transformer-based feedback policy. The policy is trained to imitate optimal Linear Quadratic Regulator (LQR) control across a family of heterogeneous Multiple-Input, Multiple-Output (MIMO) Linear Time-Invariant (LTI) systems. First, we establish a finite-sample excess-risk bound for the imitation loss minimized during training. Second, for each fixed problem instance, we derive regional closed-loop guarantees consisting of a forward-invariant operating region and a worst-case bound on deviation from the optimal rollout. Our main result is a probabilistic certificate for finite-horizon closed-loop near-optimality. Using an exact LQR cost identity, we express excess cost as a measurable per-rollout statistic and use independent calibration and validation rollouts to obtain a high-confidence bound on its violation probability. We evaluate the certificate on $28$ benchmark systems. This uses the base policy on seen systems and system-specific fine-tuned copies on unseen systems, with each rollout drawing the plant, cost, and initial condition from the corresponding certification distribution. All per-system certificates have violation probabilities below $3.1\%$, each at $95\%$ confidence; twenty systems certify suboptimality below $10\%$, with the tightest threshold equal to $4.8\times10^{-6}$.

Bowditch's theorem for hyperbolic groups establishes a fundamental correspondence between splittings over two-ended subgroups and the existence of local cut points in the Gromov boundary. While analogous results have been obtained for CAT(0) and relatively hyperbolic groups, no general theorem of this type exists for arbitrary finitely generated groups. The Morse boundary, introduced by Charney-Sultan and extended by Cordes, provides a quasi-isometry invariant boundary for any finitely generated group that naturally generalizes the Gromov boundary. In this paper, we prove that a splitting of a finitely generated group with connected Morse boundary over a two-ended Morse subgroup gives rise to a separating pair of points in the Morse boundary.

Underwater robot simulation requires diverse environments in which terrain, scene composition, tasks, and currents remain mutually consistent. We present AquaWorld, a world-generation framework that preserves these relationships through shared terrain structure. A language-conditioned plan generates 3D terrain and shared structural references that guide asset placement, task definition, and inflow specification under stochastic variation. The framework incorporates over 10,000 underwater-compatible assets, predicts reusable terrain-conditioned mean-flow fields through a CFD-supervised residual model, and supports conventional underwater vehicles and bio-inspired robotic fish. On 24 paired terrains, structure-consistent randomization produces substantially better cross-factor consistency than independent randomization. In a matched-budget policy-training comparison, structurally coherent randomization achieves a validation success rate 21% higher than independent randomization. In separate physical experiments, a simulation-trained visual navigation policy succeeds in 95% of physical tank trials without updating its perception or control modules. Overall, AquaWorld provides a practical way to generate varied underwater environments while retaining the structural relationships needed for flow simulation and robot learning.

We prove that the quantum code distance is NP-hard to approximate within an additive error of $c N$, for some constant $c >0$, where $N$ is the number of qubits. Our reductions are deterministic. This improves the previous square-root additive gap to $\Omega(N)$ and resolves the explicitly stated linear-gap question of Kapshikar and Kundu. Our result holds for CSS codes with identical $X$- and $Z$-check spaces, and with a constant rate and constant relative distance. For every fixed $\lambda>1$, there is a constant $c>0$ such that hardness still holds even when every nonidentity stabilizer has weight greater than $\lambda$ times the quantum distance. We also improve the hardness gap of graph state distance on $N$ vertices of Grigorescu, Jha, and Samperton from cube-root to $\Omega(N)$, resolving their explicitly stated open question. Both hardness results are asymptotically optimal since both distances are at most $N$. Our graph state distance hardness result holds for balanced bipartite graphs with a binary adjacency matrix that is its own inverse (mod 2). Our main technique for both hardness bounds above is classical: we show how to convert any code $C$ of length $m$ into a self-dual code $A(C)$ of length $N=\Theta(m)$ while exactly doubling the original coset metric. The conversion is deterministic and efficient. We call it the metric self-dual completion of $C$. It comes with a linear embedding $\tau: \mathbb F_2^m \hookrightarrow \mathbb F_2^N$. The embedding doubles all Hamming distances between vectors in $\mathbb F_2^m$ and all pairwise distances between corresponding cosets. The embedding also guarantees that all codewords of $A(C)$ of weight at most $2m$ are exactly $\tau(C)$.

Density-Driven Optimal Control (D2OC) provides a principled approach to distributing multi-robot teams over non-uniform spatial distributions. Applying D2OC to nonholonomic robots, however, creates a gap between safety constraints imposed on a reference motion and the physical inputs that determine the actual robot motion. We address this issue by enforcing the safety constraint directly on the robot's physical inputs while preserving the density-driven coverage objective. The proposed framework combines D2OC with a control barrier function safety filter through a feedback-linearizing look-ahead point, allowing safety and actuator limits to be considered together during control. We further derive a safety margin that accounts for the look-ahead geometry, robot footprint, and motion during each control interval. Simulation results show that the proposed method maintains the required physical separation while achieving coverage performance comparable to a conventional reference-tracking approach, which can satisfy safety on the reference motion yet violate the corresponding physical clearance. Experiments on multiple nonholonomic robots in the Robotarium further demonstrate safe execution while driving the robots toward the desired spatial distribution. These results show that enforcing safety directly on the physical inputs can eliminate the mismatch between safety certification and physical robot motion in density-driven multi-robot coverage.

Let $M$ be the six-element commutative semigroup occurring as the common multiplicative reduct of the semirings $SR_6$ and $TR_6$. The closing paragraph of Shao, Ren, and Gao~\cite{ShaoRenGao2026} asks for the finite-basis and subvariety questions for the four remaining compatible additions on $M$. We answer these questions for the four isomorphism types $R_{01},R_{02},R_{11},R_{12}$. First, we classify all compatible additions on $M$: there are nine labelled additions and six isomorphism types, parametrized by $R_{ij}$ with $0\leq i\leq j\leq 2$. For each of the four new types we give a graph-theoretic criterion for every identity, an explicit infinite basis, and a proof of nonfinite basability. The generated varieties $\V(R_{01})$ and $\V(R_{02})$ have eleven subvarieties each, while $\V(R_{11})$ has sixty-six. The lattice $\Sub(\V(R_{12}))$ is countably infinite. Every identity in this variety reduces to a subset of twenty-five fixed identities together with two monotone path families $\gamma_n$ and $\gammaD_n$. This yields a canonical signature $(H,p,q)$, complete normal forms, explicit meet and join operations, and a formula for all covers. There are 153 fixed nodes, 43 one-parameter families, and 9 two-parameter families; exactly eighteen subvarieties are finitely based, and the unique limit subvariety is $\V(SR_6)$. The strong nonfinite-basis status of the four finite semirings remains open.

Atomistic studies of grain-boundary engineering, grain-size effects and dopant enrichment require reproducible models with prescribed microstructural features. We present GRAINSMITH, an open-source Python package for generating statistically controlled polycrystalline models with selectable grain-boundary morphologies for molecular dynamics and subsequent relaxation. Within supported feature combinations, a single configuration specifies grain-size and volume distributions, crystallographic texture, boundary-area-weighted disorientation-angle distributions, phase composition and grain-boundary dopant placement. Crystal construction supports 230 crystallographic space groups. Periodic Voronoi and volume-targeted Laguerre tessellations provide planar boundaries, while distinct geometry backends generate smoothly curved and band-limited self-affine boundaries. A registry of twenty-six checks assesses applicable inputs, construction properties and outputs. Each run exports LAMMPS data and Extended XYZ files together with structural and statistical descriptors and machine-readable provenance. For a fixed software version and computational environment, the configuration and random seed determine byte-reproducible atomic configurations and scientific data across supported worker counts. By combining statistical specification, boundary-morphology control and reproducible atomistic output, GRAINSMITH supports systematic studies of microstructural effects and quantitative comparisons across generated models.