Papers

We introduce a family of static and spherically symmetric regular black-hole geometries characterized by a discrete core index $n$, a regularization scale $a$, and a dimensionless deformation parameter $\eta$. The construction starts from a positive, normalized effective density profile with a tunable central behavior. The lowest member contains the Bardeen geometry as a limiting case, whereas a nonzero deformation parameter generates a Reissner–Nordström-like asymptotic correction. The parameter $n$ controls the order of central depletion: the lowest member possesses a de Sitter-type core, whereas higher members approach a Minkowski-like center with vanishing density and curvature. We determine the horizon structure, extremality condition, Hawking temperature, entropy, and the horizon thermodynamic identity implied by the radial Einstein equation. Null geodesics are analyzed to obtain the photon sphere, critical impact parameter, and shadow radius, including analytic weak-deformation approximations, the exact Schwarzschild and Bardeen limits, and comparison with the leading Reissner–Nordström-like behavior. The photon-orbit angular frequency and Lyapunov exponent are then used to separate orbital and instability timescales; their ratio provides information not already contained in the shadow radius. Weak gravitational lensing and magnification are discussed through a controlled sector-isolated expansion of the asymptotic metric. We further study timelike geodesics, circular motion, the marginally bound orbit, and the innermost stable circular orbit, deriving exact implicit relations and perturbative expressions. The model provides a unified framework for investigating how a tunable regular core and an RN-like exterior deformation affect the thermodynamic and geodesic properties of regular black holes.

This article is the second part of a review devoted to spacetime singularities in classical General Relativity. While Primer I developed the mathematical framework for defining, classifying, and proving the existence of singularities, the present work examines their concrete realization in spatially homogeneous cosmological models. After reviewing the geometrical foundations of homogeneity and the Bianchi classification, we derive the field equations for orthogonal and tilted perfect fluid cosmologies and use the Kasner and Bianchi I solutions as elementary prototypes of anisotropic singular behaviour. Raychaudhuri focusing provides general conditions under which the homogeneous Cauchy development reaches a finite proper time boundary. We then distinguish matter singularities, characterized by divergent density and Ricci curvature, from more subtle non-scalar or whimper singularities associated with Cauchy horizons and extreme fluid tilt. The classical Ellis-King analysis is reconsidered in the light of later developments, including rigorous curvature-blow-up results for Bianchi VIII and IX models, kinematic singularities in geodesically complete tilted spacetimes, and the recent reappearance of whimper behaviour in dipole cosmology. The resulting picture shows that even spatially homogeneous cosmologies exhibit a remarkably rich hierarchy of singular boundaries. The detailed dynamical approach to these boundaries - Hamiltonian cosmology, Kasner transitions, Mixmaster dynamics and the BKL conjecture - is reserved for Primer III.

We establish graph-theoretic analogues of the Brauer-Kuroda relations for ramified Galois covers of finite graphs with arbitrary finite Galois groups. These formulas relate the numbers of spanning trees of intermediate quotient graphs and recover the corresponding formulas for unramified covers. Our proofs use $h$-functions defined on multiplicity spaces of complex representations of the Galois group.

We present a unification scheme for anisotropic compact stars generated via minimal geometric deformation (MGD), built on a two-parameter seed family $\nu(r)=\text{Ln}[C(1+ar^{2n})^{m}]$ whose radial metric potential $\mu(r)=e^{-\lambda(r)}$ is fixed in closed form by the isotropy condition. At $n=1$ the family reproduces Tolman IV, Korkina–Orlyanskii, Heintzmann IIa, and Durgapal IV–V as the particular cases $m=1,2,3,4,5$. The key result is a single closure of the MGD $\theta$-sector, through one source function $g(r)$, that integrates the decoupler equation in closed form, in terms of the Gauss hypergeometric function ${}_2F_1$, for arbitrary $(n,m)$: the anisotropic extension of the $n=1$ family is thus obtained from one master formula. We derive the effective anisotropic matter content, the Darmois–Israel matching to the exterior Schwarzschild vacuum, and the full set of physical acceptability criteria constraining the parameter space, and map the admissible region in compactness and decoupling strength explicitly for the case $n=1$: Tolman IV and Korkina–Orlyanskii admit physically consistent anisotropic deformations at arbitrarily high compactness, while Heintzmann IIa and Durgapal IV–V are bounded by a compactness ceiling beyond which no value of the decoupling parameter restores causality. Using the same closed-form matter content, we compute the quadrupolar tidal Love number and dimensionless tidal deformability across the admissible region, finding that the decoupling systematically enhances both relative to the isotropic seed at fixed compactness, with a sharp rise near the causal ceiling in the Heintzmann and Durgapal branches.

Identifying the formation channels of merging compact-object binaries remains a fundamental challenge in gravitational-wave (GW) astrophysics. Here, we formulate a generalized framework incorporating both third-body perturbations and post-Newtonian effects to investigate the resonance-driven excitation of eccentricity and/or inclination for stellar orbits around merging black-hole binaries (BHBs). We show that at the quadrupole level, inclination excitation is driven by adiabatic capture into a quadrupole-order resonance, with subsequent resonance escape governed by the conservation of phase-space area. Furthermore, in the coplanar octupole regime, an apsidal precession resonance triggers robust eccentricity growth, where the capture probability depends on the initial conditions. For spatial octupole configurations, we reveal a distinct two-stage evolutionary pathway: the quadrupole-order resonance first excites the orbital inclination, followed by subsequent eccentricity growth driven by the inverse von Zeipel-Lidov-Kozai (ZLK) resonance. Our results indicate that these resonance-driven dynamics could imprint observable kinematic signatures on surrounding stellar populations, thereby offering a novel indirect probe for merging BHBs.

Visual data is increasingly consumed by machine-vision systems rather than by human observers. Image Coding for Machines (ICM) compresses images assuming the main observer is a computer vision application and that the human observer needs to inspect or validate the decisions. Inspired by just-noticeable distortion, we cap human-observed quality at a desired level and devote the remaining bits to machine performance. Specifically, joint compression-segmentation training is recast as a constrained optimization problem in which the codec must meet a predefined acceptable target visual quality while a task term consumes the remaining coding capacity. This paper proposes two variants of a penalty function that guides the quality toward the target: an absolute function and a bilinear function, the latter applying a steeper slope once the target visual quality is exceeded. Experimental results show that, under the quality constraint, the proposed method achieves BD-rates of $-22.82\%$ and $-29.81\%$ relative to an unconstrained joint rate–distortion–task optimization and a simple rate–distortion baseline, respectively, showcasing bitrate reduction with the same task performance. This is achieved while the codec also meets the target visual quality with a reasonable error and without adding any complexity overhead.

We prove that the pairs of integers $0\le m<n$ for which a nondecreasing integer sequence can remain complete after every deletion of $m$ terms and become incomplete after every deletion of $n$ terms are exactly those with $m\le1$. Here a sequence is complete if every sufficiently large integer is a finite sum of terms with distinct indices. This answers Erdős Problem 348, posed by Graham and later included in the book of Erdős and Graham. The proof uses a central-interval theorem: if a complete nondecreasing positive integer sequence $(a_i)$ has prefix sums $S_j$ with $S_j-a_{j+1}\to\infty$, then each sufficiently long prefix represents every integer from any fixed completeness threshold $T$ to $S_j-T$. We also refute Graham's conjecture, later repeated by Erdős and Graham, that $(\lfloor t\gamma^n\rfloor)_{n\ge1}$ is complete for every $t>0$ and $1<\gamma<(1+\sqrt5)/2$. We obtain the counterexample by combining Dubickas's fractional-part theorem with an elementary sign adjustment. For a common base in this range, we further construct two such sequences whose interleaving is incomplete and whose coefficient ratio is not a rational multiple of any integer power of the base.

In this note, we present an integral representation of the $k$-generalized Fibonacci sequence and provide a proof using contour integration. This representation enables the derivation of several known identities and a Binet-type formula related to the $k$-generalized Fibonacci sequence.

We study a jointly exercised American condor contract whose payoff is a call condor payoff plus a positive cash amount. Its plateau and positive tails produce two inner exercise boundaries and two outer cash-exercise boundaries. We construct the unique bounded solution of the associated variational inequality, strong on each side of the plateau, and identify it with the optimal stopping value. We prove monotonicity and continuity of the four boundaries and determine their limits at expiry. Following Broadie and Detemple, we express the inner boundaries as cutoffs of the boundaries of uncapped contracts with the same cash components. We also prove call-side convexity and volatility monotonicity under a sufficient condition, solve the stationary problems explicitly and present numerical results.

Florian Luca devoted the last paper of his life to the set $\mathcal K$ of odd positive integers $k$ such that $2^n k+1$ is a Carmichael number for some positive integer $n$. He proved that if $m\ge 5$ is fixed and odd, then there are only finitely many primes $p$ with $p^m\in\mathcal K$. In this note, written as a small tribute to his memory, we extend his result in three directions, using only his published work as our reference base. First, we extract from his arguments explicit quantitative bounds: every Carmichael number $N=2^n p^m+1$ with $m$ odd satisfies $n<2^{2\cdot10^7(m+1)^2m^2(\log p)^2}$, and if $p>m$ the much sharper bound $n<25m^4\log p$ holds; for $p>500m^5\log m$ we further record the uniform bounds $\omega(N)\le 26m^7$ on the number of prime factors. As an application, the number of such Carmichael numbers up to $x$ is $O_m(x^{1/m})$. Second, we generalize the multiplicative independence step of Luca's proof from base $2$ to an arbitrary even base $b$: if $b$ is even, $m\ge3$ is odd and coprime to $b$, $p>m$ is prime and $N=b^n p^m+1$ is Carmichael, then $b^n p^m$ and $b^a p^\beta$ are multiplicatively independent whenever $b^a p^\beta+1$ is a prime factor of $N$. Third, we report an exhaustive computation showing that the only Carmichael number $2^n p^m+1\le 10^{18}$ with odd $3\le m\le 37$ is $1729=2^6\cdot 3^3+1$, that no such number exists with odd $5\le m\le 37$ (extended to $10^{21}$ for $5\le m\le 15$), and that the only example with even $4\le m\le 36$ is $46657=2^6\cdot 3^6+1=13\cdot 37\cdot 97$, where $3^6=27^2$ is the square of the smallest element of $\mathcal K$.

The integration of sensing into communication networks introduces a new privacy risk, as a passive eavesdropper (Eve) may exploit ISAC data signals as signals of opportunity to perform unauthorized sensing of targets. In this paper, we develop a sensing-privacy-enhancing geometric constellation shaping (GCS) framework for OFDM-ISAC systems. The key observation is that constellation-dependent ranging performance is receiver-specific. For matched filtering at Eve, the ranging MSE is governed by the constellation kurtosis $\kurt$, whereas reciprocal filtering at the legitimate receiver (Alice) is governed by the inverse second-order moment $\ism$. Based on closed-form MSE expressions, we define sensing privacy as the ranging MSE gap between Eve and Alice and characterize its dependence on these two moments. The analysis shows that positive skewness of the symbol-power distribution is necessary for a positive intrinsic moment gap, namely $\kurt-\ism$. We further derive an exact skewness-based decomposition of the intrinsic moment gap and a canonical two-ring characterization, providing analytical guidelines for privacy-enhancing constellation geometries. We then formulate Eve-aware and Eve-agnostic GCS designs that balance sensing privacy and communication reliability through the minimum Euclidean distance (MED), with the Eve-agnostic design depending only on the intrinsic moment gap. Numerical results demonstrate scalable privacy–communication trade-offs, while over-the-air experiments show that the proposed constellation shaping substantially increases the ranging error gap between Eve and Alice with only a small communication throughput loss.

This note provide a proof of Conjecture 2.5 in Fang et al. (J. Appl. Probab., 58(4):851–867, 2021), with the help of GPT-6 Astra. We show that the number of edges lying above zero of a random walk generated from the uniform permutation has the discrete arcsine law. The proof is short but hinges on an unexpected representation. We also make several comments on solving the problem using large language models.

AI agents that carry a multi-step computer task through on their own became ordinary tools in the past year, and the same autonomy is available to anyone whose task is harmful. We ask what that means for a relying party – an insurer, a lender, an auditor – whose evidence is a filed PDF. AgentForge-Bench measures how reliably an off-the-shelf coding agent, driving one of seven open-weight models with a shell and the stock Python PDF stack, alters one dollar amount, date or address in a real filed financial document from a single sentence of intent, graded by rules rather than by a model. Across 1,750 cells, 1,419 (81.1%) satisfy the verifier, and 808 (46.2%) also survive every stricter filter: visible, localized, typeface-matched, original value gone document-wide. A deterministic script with no model in it solves 98 of the 125 documents; the agents solve 124, and none the script solves alone. Agents misreport 41% of their wrong edits as done, no model refused, and the cheapest verified forgery costs 2.4 cents. The raw rate overstates the threat by about a factor of two; the strict rate is still large.

The shifted lonely runner conjecture was recently disproved by Blanco, Criado and Santos. We give quantitative bounds on its failure as the number of runners grows. The loneliness of a configuration is the maximum, over time, of the distance from the origin to the nearest runner. We write $1/(n+1+E_n)$ for the infimum of loneliness over configurations of $n$ runners on the unit circle with distinct positive integer velocities and arbitrary initial shifts. We prove $E_n\ge\lfloor n/287\rfloor$, which together with the elementary bound $E_n\leq n-1$ implies that $E_n$ grows linearly in $n$. We also show that $E_n\geq1$ for every $n\geq95$.

Expressive speech synthesis has advanced through prosody modeling, yet generating structured poetic speech, such as haiku, remains challenging. Prior work on prosody transfer improves expressiveness, and fine-tuned poetry TTS (text-to-speech) systems capture verse intonation. However, these models do not model haiku's 5-7-5 syllable structure or line-ending pauses. We present a cascaded system, HaikuS2S, combining ASR (automatic speech recognition), LLM (large language model)-generated haiku, and TTS fine-tuning on both prose and custom haiku datasets. Our evaluation focuses on emotion similarity, speech quality, and prosody alignment. In our experiments, we see that our prosody and tonal alignment improve significantly with our fine-tuned systems, particularly the one trained on both general poetry and haiku. We also see that we maintain similar emotion similarity scores across all systems.

Cardiovascular magnetic resonance (CMR) provides comprehensive cardiac assessment but remains underutilized because of the complexity of acquisition, post-processing, and interpretation. Existing artificial intelligence (AI) methods address isolated tasks, limiting clinical integration. We present ORION-CMR (On-scanner Reporting with Integrated fOunda-tioN Model), the first clinically evaluated scanner-native end-to-end CMR foundation model. Pretrained on 12,896,733 CMR images from 9,258 studies, ORION-CMR performs sequence classification, ventricular function assessment, late gadolinium enhancement (LGE) detection, binary and multiclass disease classification, and local large language model-based report generation in approximately 90 seconds. The framework. was evaluated on public benchmarks and clinically validated in a multi-vendor cohort of 68 subjects with normal examinations, congenital heart disease, dilated cardiomyopathy, and myocardial infarction. ORION-CMR outperformed supervised baselines and the previously published CMR foundation model (CMR-FM), achieving state-of-the-art performance for LGE classification and scar segmentation. Clinical evaluation achieved an AUC of 0.96 for normal-versus abnormal classification and 0.88 for multiclass disease classification, while generated reports demonstrated 81.4% agreement with expert interpretation. These results demonstrate the feasibility of real-time scanner-native AI-assisted CMR analysis and automated report generation.

We study row-softmax self-attention with independent Gaussian query and key weights in the proportional regime, at fixed inverse temperature. Hayase, Collins, and Karakida proved Gaussian equivalence for the empirical squared singular-value distribution after removal of the Perron direction. A global law alone does not exclude finitely many nonleading outliers. We prove that no such outliers persist: the rescaled squared singular value $\ell s_k(A)^2$ converges in probability to the upper edge of their bulk law for every fixed $k\ge 2$. In fact, this convergence is uniform over any deterministic sublinear number of leading non-Perron indices. The proof uses an exact decomposition of the softmax normalization, conditions on the key matrix, identifies the conditional covariance exactly with a diagonally conjugated inner-product kernel, linearizes that kernel in operator norm, and applies the outside-support local law of Fan, Ma, Paquette, and Wang. A separate stability argument identifies the finite conditional deformed Marchenko-Pastur edge with the limiting bulk edge. We also derive a scalar formula for that edge throughout the proportional regime and recover the explicit square-model formula of Hayase, Collins, and Karakida, including its physical branch.

We propose a new framework for noncommutative stochastic integration in $L^p$, inspired by Bichteler's notion of $L^p$-integrators, that massively generalizes our previous theory of $L^2$-valued stochastic integration against $L^2$-decomposable processes. Central to our framework is a noncommutative analog of a predictable integrand we call a predictable linear process. We develop basic properties of stochastic integrals against "noncommutative $L^p$-integrators" and establish a useful criterion for a noncommutative stochastic process to be a noncommutative $L^p$-integrator. This criterion applies, in particular, to a rich class of processes we call measured decomposable processes, which includes free Brownian motion and, more generally, the $q$-Brownian motions. We also show that, when specialized appropriately, our framework recovers the classical notion of an $L^p$-integrator up to the fact that the noncommutative theory sees only the modification class of a classical stochastic process. As an application, we use our theory to study noncommutative stochastic differential equations (SDEs) in $L^p$. Under natural local-Lipschitz and boundedness assumptions, we establish the existence and uniqueness of maximal solutions and show that a maximal solution with finite lifetime must blow up in $L^p$ norm. As corollaries, we obtain new existence and uniqueness results for free SDEs and SDEs driven by $q$-Brownian motion. Notably, in the free case, our results apply to coefficients arising through the continuous functional calculus from merely locally Lipschitz scalar functions, as opposed to locally operator-Lipschitz functions.

We study online metric matching with $n$ servers and $n$ requests, all sampled independently and uniformly from $[0,1]^d$. We analyze the Greedy algorithm, which matches each arriving request to its closest available server. Despite its strong empirical performance, non-trivial bounds on Greedy's competitive ratio were previously known only for $d = 1$, where it is $O(1)$-competitive [BFP23]. We prove that Greedy is $O(1)$-competitive for every fixed $d\geq3$ and has competitive ratio $\Theta(\sqrt{\log n})$ for $d=2$. Our proof first analyzes Greedy on the flat torus and then transfers the resulting estimates to the cube.

We give an explicit non-random example of nonadditivity of minimum output von Neumann entropy. The proof uses a finite-dimensional construction which imitates free Haar unitary behavior. The same channel gives a violation of additivity for the minimum output Rényi-$p$ entropy for any $p\in[1,\infty]$. Additionally, given any $p_0>0$ and $\mathfrak g>0$, we extend the construction to obtain an explicit non-random channel which has entropy gap $\ge\mathfrak g$ uniformly over $p\in[p_0,\infty]$.

In-band full-duplex (IBFD) multiple-input multiple-output (MIMO) systems enable simultaneous transmission and reception on the same frequency band, improving spectral efficiency for next-generation wireless networks. However, IBFD-MIMO systems are susceptible to self-interference (SI), which may overpower signals of interest (SOI). In this scenario, blind source separation (BSS) algorithms can be adopted to remove SI and perform joint sensing and communication (JSAC), but BSS algorithms mostly assume an idealized linear and quasi-stationary signal model, which does not hold under realistic radio frequency (RF) impairments, such as I/Q imbalance, carrier frequency offset (CFO), phase noise, and power amplifier nonlinearity. This paper proposes a two-stage echo state network (ESN)-based scheme that is superior to BSS under these realistic conditions. A frozen ESN is trained offline to characterize the static SI path, while an adaptive ESN, updated online via recursive least squares, tracks the time-varying SOI path using sparse pilot symbols. We evaluate the proposed scheme's SOI recovery performance and acquisition speed with different block sizes, comparing it against other recurrent neural networks (RNN), such as long short-term memory (LSTM) and gated recurrent unit (GRU). Simulation results show that the proposed approach outperforms BSS, LSTM, and GRU in both efficiency and SOI recovery, demonstrating the viability of ESNs for real-time, nonlinear self-interference cancellation in realistic IBFD MIMO systems.

Vision-language-action (VLA) policies can struggle with manipulation tasks with complex obstacle geometries due to partial observability. These complex geometries can lead to similar visual observations or robot configurations requiring qualitatively different actions, a distinction that can be quantified using topological signatures. While motion planners with full knowledge of environment geometries and object states can reason about these signatures in planning, this information is often not known at deployment. To address this issue, we present a topology-guided visual-prompting framework that uses simulation-based planning to augment a nominal demonstration dataset and provides vision-based guidance at deployment. Our method uses a Gauss-Linking-Integral topological signature representation to capture important topological properties of the environment. Using privileged geometry information from a simulation approximation of our environment, we augment a VLA fine-tuning dataset with trajectories that move the system to a demonstrated signature and, from the new configuration, resume task execution. A vision-language model (VLM) is fine-tuned on the same dataset to both predict signatures from live camera observations and predict end-effector waypoints, which are rendered as visual prompts on the observations to guide the VLA. Across three simulated bimanual tasks and a real-world box pickup task, our method outperforms a VLA fine-tuned only on nominal demonstrations and a VLM-prompting baseline that can remove topology-relevant information from observations. On hardware, it exceeds the strongest baseline by 40% in task success. Project website: https://topology-vla.github.io.

Underwater robot learning relies on simulators that integrate high-fidelity hydrodynamics, convenient learning interfaces, and a credible transition to real scenarios. In this work, we present FinsSim, a reality-aligned integrated simulation platform for Sim-to-Real underwater robot learning. FinsSim first constructs high-fidelity simulation with selectable backends to adapt to diverse requirements. To facilitate underwater robot research, it further offers standard control baselines, alongside with unified robot learning workflows. For reliable Sim-to-Real transfer, FinsSim adopts a multi-sensor fusion scheme to provide low-cost yet precise localization. Moreover, it implements calibrated thruster-hydrodynamics models and a constrained wrench allocation algorithm. Bridging these modules by ROS~2, FinsSim establishes a complete Sim-to-Real transfer pipeline. Through matched simulations and experiments, it is demonstrated that reliable Sim-to-Real transfer of underwater robot control policies can be achieved with the FinsSim framework. Separate ablation studies also validate that the modules of FinsSim can address the pivotal issues of underwater Sim-to-Real from different aspects. Overall, this work aims to bridge the gap between theoretical research and practical applications, ultimately driving advancements in the field of underwater robotics.

Lee-Yang theorems are a powerful tool for studying many-body systems, with applications ranging from analyzing phase transitions to proving the efficiency of certain classical and quantum algorithms. In this work, we prove a Lee-Yang zero-freeness theorem for the partition function of a broad class of interacting fermion models, implying the existence of a provably efficient quantum algorithm for estimating their ground-state energies. This class includes several well-known models such as the attractive Hubbard model, repulsive Hubbard model on bipartite graphs, and the interacting Hofstadter model. Our results also rigorously establish the nonexistence of phase transitions in these models in the presence of a nonzero local external field.

We study the detectability of split surfaces in isogeny graphs of principally polarized superspecial abelian surfaces, a question relevant to the security analysis of dimension-$2$ isogeny-based cryptography. Our approach uses refined Humbert invariants to replace explicit isogeny computations with primitive representation problems for positive definite quadratic forms in five variables. We develop algorithms to detect $(N,N)$-splittings and to compute the minimum $(N,N)$-splitting level of a superspecial Jacobian without constructing the corresponding isogeny path. The same framework detects embeddings of real multiplication (RM) orders through primitive representations of discriminants of real quadratic orders. For RM, we use exhaustive small-prime data together with $100{,}000$ random polarizations per prime for $227\leq p\leq1619$, testing primitive representations of square-free discriminants $D\leq100$; the first nontrivial primitively represented discriminant is typically small. We apply these methods experimentally in two regimes. For $11\leq p\leq251$, where the irreducible principal polarizations are known exhaustively, the automorphism data recovered from the refined Humbert invariant reproduces the counts of Ibukiyama–Katsura–Oort for the irreducible polarizations. For large parameters, we reach primes of $1000$ bits, where the splitting degrees produced satisfy $\log_2N\approx\log_2p$ with a distribution whose shape is stable across the whole range, so the detected $(N,N)$-isogeny has degree about $p^{2}$; the smallest observed largest prime-power divisor of $N$ remains small up to $250$ bits and increases sharply from $300$ bits onward.

Let $\Pi$ denote the independence copula and $M,W$ the upper and lower Fréchet–Hoeffding copulas. Catalano and Lavenant (2025) conjectured that $M$ and $W$ maximize the quadratic Wasserstein distance from $\Pi$ among all bivariate copulas. We prove this conjecture and characterize all equality cases: $\mathcal W_2^2(C,\Pi)\leq 1/10$, with equality if and only if $C\in\{M,W\}$. We also determine explicitly the optimal Monge map from $\Pi$ to $M$. The proof is constructive and combines the optimal transport from independence to the diagonal, a sharp convex-order inequality for 1-Lipschitz functions, the conditional convex order, and a coupling construction based on conditional comonotonicity and the supermodular order. As a consequence, we obtain a normalized Wasserstein-based dependence measure that characterizes independence and attains its maximal value exactly for comonotone and countermonotone dependence.

Effective communication between users and AI agents is essential for human-AI collaboration. The XY problem is a well-known communication pitfall where a person asks about their attempted solution rather than their actual problem. We extend prior sycophancy evaluation to the XY problem in agentic settings, evaluating whether agents can resist plausible but misleading suggestions from users and communicate their reasoning. We introduce XYEval, a meta-evaluation framework that can transform an existing benchmark into an XY problem evaluation. We evaluate five models across six diverse benchmark suites. Agents suffer large XY drops under XY mutation across benchmarks, with relative drops reaching up to 46.7%. With $\tau^2$-bench, we further show that agent performance drops more when encountering a pedantic user who requires detailed explanations before approving a better solution. Our findings suggest that current agents lack the ability to effectively reason and communicate when facing misleading suggestions. A simple system instruction baseline that encourages awareness of XY problems only offers partial mitigation. Extensive trace analyses provide behavioral insights into how and why these XY drops occur across execution trajectories. Our results show that mitigating the XY problem remains challenging, requiring agents to both recognize user misdirection and clearly communicate the underlying problem.

We show that anomaly-free second-order supersymmetry (SUSY) in quantum mechanics admits a projective-geometric formulation in terms of a projective connection and a vector field preserving it. Once the physical coordinate and energy origin are fixed, these data determine the partner Hamiltonians and supercharges; the field's quadratic invariant fixes the central parameter and candidate singlet energies. The fixed-point structure of this field yields explicit criteria for smooth continuation of the quantum operators through superpotential zeros. The classically fictitious similarity transformation selects affine data whose covariant factors generate the Schwarzian quantum correction. Dimensional reduction of a Riemannian model reproduces these affine data through the volume measure, while a coordinate adapted to the selected field implements coupling-constant metamorphosis. A one-gap Lame pair illustrates the relation between the SUSY auxiliary geometry and the Gelfand-Dikii-Virasoro structure governing physical solution products. Its Lax-Novikov integral acts on Schrodinger solutions through the first-order action of a projective stabilizer; in the reflectionless limit it becomes proportional, in the physical coordinate, to the auxiliary third Bol operator.

Robust linear fits can resist response contamination yet remain too dense or unstable for useful global explanations. We propose penalized distillation, which fits a smoothly clipped absolute deviation (SCAD) estimator to a robust initial estimator's empirical fitted surface along a safeguarded coordinate-descent path and evaluates candidate states separately for fidelity, parsimony, perturbation stability, and held-out prediction. The new results attach to the states the algorithm actually computes. Conditional on a fixed uncontaminated design, deterministic bounds transfer response-replacement boundedness from the initial fit to every retained path state. Turning to fixed dimension, we characterize the oracle-support branch by its empirical-Gram projection and influence function, give conditions for covariance-weighted least-squares approximation equivalence, and establish a path-conditional generalized information criterion. By contrast, at large dimension-to-sample ratios the full-coordinate robust fit collapses without warning, and screening restores the construction. Under a sure-screening framework, the robustness bound and the support and selection guarantees transfer to the screened fit. Simulations separate robustness transfer from support recovery, efficiency, and computation across the dimension-to-sample ratio, with p up to 240, and the signal density, which isolates what the sparse stage adds once the screen over-selects. In a duplicate-grouped superconductivity study, the distilled estimator remains predictively stable under prespecified training-response shifts but retains 66.8–68.8 of 81 slopes. Stronger sparsification reduces the model to 12.6–14.0 slopes only at visible fidelity and prediction cost. Distillation therefore preserves predictive stability on these data without substantiating a compact coordinate-level explanation.

Generative AI can offer students useful feedback, but its value depends on judging which suggestions are accurate and relevant. Prior research shows that strategic friction during human-AI interactions can promote critical uptake, but how to effectively implement such friction in academic contexts remains unclear. We examine whether requiring students to justify decisions to accept or reject AI suggestions can mitigate uncritical uptake in academic writing. In a randomized experiment embedded in a course activity (N=129), students wrote a data analysis proposal, received mixed-quality AI revision suggestions, and decided whether to accept or reject them. Students required to provide written justifications were 24 percentage points less likely to adopt flawed suggestions (65% vs. 41%), with no reduction in acceptance of sound suggestions (81% vs. 86%). However, thematic analysis revealed superficial engagement in the justification task and gaps in metacognitive monitoring and domain knowledge.