Papers

We develop accelerated three-dimensional simulations of growing planet pairs that draw on a finite gas reservoir. An annulus compression (Scaling A) shortens the predicted clearing time and shrinks the computational domain at equal grid resolution while preserving key physical parameters. Planetary seeds start with Hill radii equal to the disk scale height, retaining three-dimensional accretion. We demonstrate the method in scaled PDS 70 and Solar-System configurations, assuming perfect Hill capture by growing planets on circular orbits. We compare Scaling A at radius-compression factor \(\Gamma=1.75\) with the unscaled PDS system at matched resolution, through a 1% reduction of the interplanetary gas inventory. The scaled run is nearly an order of magnitude faster, with similar accreted mass and a comparable partition of gas between the planets. For the Solar-System configuration at \(\Gamma=3\), the scaling forecasts a speedup of nearly two orders of magnitude at equal local resolution. The scaling is complementary to orbital advection. On eight CPU cores of a desktop computer, the scaled PDS 70 and Solar-System runs reduce the gas inventory between the orbiting planets by roughly half in less than a day. Accretion-driven orbital convergence narrows the shared feeding zone and, together with gas clearing, limits the planetary growth budget. In our scaled PDS 70 and Solar-System runs, the angular momentum carried by meridional inflow directly constrains delivery theory. These results establish a practical route to testing accretion theory against simulations at matched scaled parameters, before applying it to unscaled planetary systems.

We prove that reachability for constraint automata over finite binary words is undecidable. Eight registers suffice, with conjunctive guards using only equality, proper prefix, proper suffix, and infix incomparability, without word constants or concatenation. Finite-run nonemptiness, Buechi nonemptiness, and bounded Buechi nonemptiness are undecidable as consequences. For Boolean guards over equality, prefix, suffix, and prefix and suffix incomparability, Buechi nonemptiness is PSPACE-complete with arbitrarily many registers and word constants. The upper bound also allows positive infix atoms and yields PSPACE-completeness of prefix–suffix LTL satisfiability.

Quantum localization, the phenomenon by which wave diffusion is halted, here understood as the superposition of propagating waves, can have many different origins, such as onsite random potentials in Anderson localization. However, some systems exhibit complex geometries and could potentially reproduce similar results without external potentials. In this work, we study how localization of single particle states is affected by the underlying topology of a simple but highly predictive complex network model: the geometric soft configuration model $\mathbb S^1$. By tuning the model's temperature and heterogeneity parameters, we are able to generate different structural patterns and focus on the transition from a small-world to a non-small-world regime. Our study suggests that small-world networks tend to have extended states, which spread the whole system's size, in contrast to non-small-world networks which tend to localize, showing how complex connectivity can play the role assigned to random energetic disorder.

Currently, there is no common methodology or set of practices for evaluating memory-system performance in integrated simulation infrastructures. As a result, studies adopt different evaluation approaches, which can lead to conflicting conclusions even for the same simulation infrastructure. We propose a validation methodology that evaluates memory-system performance from three complementary views: the memory simulator, the CPU–memory interface and the simulated application. We use these views to detect, understand and correct memory-simulation inaccuracies arising from simulator integration and simplified modeling of microarchitectural features. The methodology reveals that the application's perception of memory performance can become effectively decoupled from the detailed memory simulation. Most importantly, we identify and correct a fundamental memory-model mismatch between delayed-response cycle-accurate memory simulators and ZSim, a representative parallel CPU simulator. We demonstrate the broader applicability of these enhancements by applying them to ZSim integrations with Ramulator 2, DRAMsim3 and DRAMSys 4, covering all major public cycle-accurate memory simulators. After the corrections and enhancements, all integrations reproduce the main bandwidth–latency trends and key characteristics of the actual system. Across the evaluated simulators, STREAM bandwidth differs from actual-hardware measurements by less than 5% in most experiments, while pointer-chase latency remains 15–18% lower than on actual hardware. All simulator enhancements are released as open source.

Open Radio Access Networks (O-RAN) empower operators with choice and customization through their disaggregated and interoperable architecture. However, ensuring interoperability in O-RAN presents challenges due to complex testing environments and evolving specifications. This paper addresses these challenges by proposing the Generative AI and Graph Neural Network-based Open RAN Testing and Automation (GNG-OTA) framework. GNG-OTA utilizes state-of-the-art AI techniques including generative AI, natural language processing, and graph neural networks to enable accurate, reliable, efficient, and automatic O-RAN interoperability and performance testing.

We study stationary discrete-time mean-field games with finite state and action spaces under the average-reward criterion, with an entropy-regularized objective and population-dependent dynamics and rewards. Under a minorization condition, the soft Q-operator is a span-seminorm contraction for the full transition kernel, so relative value iteration does not require the minorizing measure. When the Lipschitz constant of the kernel is below the minorization mass and the temperature exceeds an explicit threshold, the equilibrium fixed-point operator is a contraction: an equilibrium exists, its population distribution is unique, and a fixed-point iteration converges to it geometrically. We then learn the equilibrium model-free from a simulator, combining soft Q-learning at a fixed population distribution with an empirical transition kernel. We prove a finite-sample bound for soft Q-learning in the span seminorm, with temperature-independent constants, and derive convergence rates for the learned population distribution and policy. A congestion-aware routing game illustrates the results.

AI-text detectors are already applied to Kazakh writing, yet no study reports how often publicly available detectors flag Kazakh text whose human authorship is dated before ChatGPT. We present AINA-KK: 3,811 human passages from encyclopedic, news, web and educational pages that provably existed before ChatGPT, and 5,597 machine and gray-zone passages from 9 generators, including the Kazakh-built KazLLM and Sherkala, with script and length stress tests. Five English-trained detectors from the Hugging Face Hub flag up to 81.4% of human passages and 79.6% of published student essays; recalibrated, they catch at most 1.3% of machine text. Calibrated zero-shot detectors catch at most 50.3% of machine text from in-distribution generators. Fine-tuned encoders rank text almost perfectly, but mDeBERTa at its default threshold flags 24.4% of 50-word human excerpts, partly because machine outputs are shorter than human text; calibrated, it catches only 47.1% of KazLLM and 62.7% of Sherkala passages. Without script handling, machine-transliterated Latin-2021 text defeats every detector. Fixes that need no new data (calibration, script canonicalization, script-augmented and length-matched training, and an OR with a zero-shot scorer) form AINA-KK-Det. On test it flags 0.9% of human passages (95% CI 0.4-2.7; 1.5% on development data) and catches 85.2% of machine text from known generators, 92.9% from two unseen Gemini-family Gemma-4 models and 61.3% from the Kazakh-built ones; its configuration was chosen with test results in view. We release the data, code and detector at https://github.com/ozxmn/aina-kk.

Let $R$ be a commutative ring. A submodule $N$ of an $R$-module $M$ is semiprime if $m\in(N:m)M$ implies $m\in N$. It is known that every semiprime submodule of $R^n$ is an intersection of prime submodules, and Aryapoor extended this to finitely generated modules. We show that no finiteness assumption is needed: every semiprime submodule of an arbitrary $R$-module is the intersection of the prime submodules containing it. The proof is short and uses neither coordinates nor presentations; its key step is that a nonzero element $q$ of a semiprime module $Q$ satisfies $sq\notin\p Q$ for all $s\notin\p$, where $\p$ is any minimal prime over the annihilator of $q$. We define an $R$-module $M$ to be real if $\sum_i m_i\otimes m_i=0$ in $M\otimes_RM$ implies $m_i=0$ for all $i$, and a submodule $N\subseteq M$ real if the module $M/N$ is real. We prove the corresponding result for real submodules: every real submodule of an arbitrary module is the intersection of the real prime submodules containing it. On $R^n$ our notion of realness coincides with the coordinate definition used in earlier work on real Nullstellensätze for free modules.

We study the behavior of the $k$-th-order nonhomomorphicity of Boolean functions under restriction to affine subspaces. For odd $k\ge3$ and $0\le t<n$, we introduce the $t$-restriction $k$-th-order nonhomomorphicity $P_{k,t}(f)$, defined as the minimum of $P_k(f_{\mid A})$ over affine spaces of codimension at most $t$. For a direct sum of $m$ monomials of degrees at least two, we determine $P_{k,t}$ exactly. When $t<m$, a minimum is obtained by setting one variable equal to zero in each of the $t$ monomials of lowest degree. When $m\le t<n$, the minimum is zero. The proof combines multiplicativity of even Walsh moments with a contraction argument for affine restrictions. We establish a general formula for the Walsh transform of a Boolean function restricted to an affine subspace of arbitrary codimension, yielding an exact spectral characterization of the $t$-restriction normalized nonlinearity $\rnnl_t(f)$. As odd $k$ tends to infinity, $(1-2P_{k,t}(f))^{1/(k+1)}$ converges to $1-2\rnnl_t(f)$. We further prove that every non-affine function has a hyperplane restriction with strictly smaller odd-order nonhomomorphicity. Finally, we determine $\rnnl_1(T_{d,n})$ exactly for all threshold functions with $n\ge3$. This proves the two threshold-function conjectures left open in~\cite{SergeAnaPante2025}, including the previously unresolved odd-majority case, where a minimizing hyperplane is $x_1+x_2=0$.

Collision-resilient airframes allow quadrotors to survive impacts, but do not ensure autonomous post-collision recovery. Innovation gating in stock onboard estimators often rejects valid external velocity measurements after high-speed impacts, compromising recovery. We present REFLEX, a collision-recovery flight mode that runs entirely on the flight controller, requiring neither contact sensors nor a companion computer. Its proprioceptive recurrent policy stabilizes the vehicle without position, velocity, heading, or altitude estimates. The translational estimate is reinitialized while REFLEX is active, and a trust gate checks estimator consistency before returning control to the stock autopilot. The policy is trained in simulation using proximal policy optimization, with episodes initialized from post-impact states measured in 47 real collisions. In hardware experiments, REFLEX recovered 19 of 22 wall impacts at 5.03 $\pm$ 0.09 m/s, compared with 1 of 14 for the matched stock autopilot. These included 15 consecutive recoveries in a single flight. A separate exploratory flight demonstrated two additional recoveries at 6.02 and 6.05 m/s within the same flight, exceeding the previously reported velocity for autonomous wall-collision recovery by 64%.

We propose k-Hessian equations as auxiliary equations for degenera- tions of Kähler classes to nef classes of numerical dimension k. As proof-of-concept results, we provide a uniform mean-value inequality and uniform à priori estimates for Hessian equa- tions.

Over the past decade, Metaverse technology (AR/VR) has seen widespread adoption in numerous areas, including gaming, social networking, and online collaboration. Although recent studies have exposed vulnerabilities in VR systems, they often apply methods (e.g., pre-installed malicious apps or scripts) from conventional mobile platforms directly, demonstrating impractical attacks that fail to investigate unique privacy leakages specific to VR platforms. In this paper, we present an orthogonal attempt to analyze the attack surface of the power side channel on AR/VR's network connection by capturing the through-wall wireless power in the network transmission medium (e.g., Wi-Fi signals) non-intrusively in a comparatively long distance (e.g., 5.0m-8.0m) and further reconstructing multi-level privacy-aware user behaviors. We propose MetaSpy to validate the feasibility and practicality of our disclosed wireless power-based traffic analysis attack. It first captures through-wall wireless power transmitted from the Wi-Fi router of an adjacent room, identifies the device type, then infers (i) user interactions such as browsing websites, running AR/VR apps, and performing various in-app activities, (ii) unseen VR-specific activities in VR games (e.g., Population: ONE), and (iii) avatar-based user re-identification and virtual keystroke inference corresponding to head movements. We evaluated the effectiveness of MetaSpy in four commercial-off-the-shelf (COTS) VR headsets, and the empirical results demonstrate its competitive accuracy in unveiling multi-level user privacy within virtual scenes in through-wall settings at a longer distance (1.8x-80x) than most of the recent studies, and it shows resilience to a series of external interference and impact factors. In addition, we also propose and evaluate effective countermeasures to protect data confidentiality in AR/VR wireless transmission.

Large-scale personalization is shifting from embedding-table retrieval to LLM-based recommenders that generate the Semantic ID (SID) of the item to recommend, where each SID is a short sequence of discrete hierarchical content codes. A single instruction-tuned model can then unify retrieval, content understanding, and language, and recommend brand-new titles from their content alone using SID; then, how should the model generate the SID? Traditionally, models rely on a single language modeling head that generates tokens autoregressively, typically using constrained beam search. In contrast, an emerging body of work explores multi-token prediction (MTP), where dedicated heads predict all tokens in parallel, substantially reducing the decoding latency associated with beam search. We report an industry-scale, apples-to-apples comparison of these two generation paradigms: single-head autoregressive decoding vs. MTP. We introduce a new auxiliary ranking loss, defined over the joint SID score, to train the MTP objective. We evaluate the SID training objectives on both quality and training/inference cost, inside a multi-task LLM that jointly does personalized retrieval, content question-answering (QA), and instruction following over a large streaming-media catalog. Our experiments show single-head beam search consistently beats MTP on most product-critical metrics: ranking and content QA, and trains at roughly 2.6x lower dollar cost on Qwen3-8B scale; MTP wins only on raw inference throughput. Around this finding, we detail an efficient training and inference stack and report end-to-end gains over the previous non-SID baseline equipped with an auxiliary ranking head: +77% item cold-start ranking, +22% content-QA accuracy, +2.8x faster training, with language metric regression traced to SID vocabulary expansion. We address the language regression by reinforcement learning post training.

We perform direct numerical simulations of rotating turbulence to investigate the mechanisms responsible for the maintenance of large-scale geostrophic vortices. We confirm that the Reynolds stresses generated by inertial waves are consistent with the Reynolds-Averaged Navier-Stokes balance of the large-scale flow and demonstrate the predicted cyclone-anticyclone asymmetry, with cyclones exhibiting stronger azimuthal velocities and more compact cores. Wave-field analysis provides evidence that geostrophic vortices selectively reflect and absorb inertial waves according to the sign of their angular momentum. Wave absorption occurs either through the formation of a critical layer ahead of the reflection point or through enhanced viscous dissipation associated with increasing local wavenumber. As rotation increases, fewer inertial waves undergo resonant absorption, reducing angular-momentum transfer to the vortices and weakening the geostrophic flow. Consequently, the fraction of energy contained in geostrophic modes depends non-monotonically on the rotation rate, demonstrating that sufficiently rapid rotation can ultimately suppress, rather than enhance, turbulence two-dimensionalization.

Elastic buckling remains a critical concern in the design of deep spherical shell caps, given that experiments reveal a significant reduction of critical loads from those predicted by linearized eigen-buckling analysis. This discrepancy has been attributed to initial geometric imperfections and is often reconciled through a dimensionless parameter, namely the knockdown factor (KDF). However, conventional KDFs (NASA SP-8007; Eurocode 3) are highly conservative, as they represent the lower bound of an extensive body of experimental data. This study aims to narrow the conservatism in the KDFs for spherical shell caps through machine learning (ML) regression models trained on experimental data on spherical shell buckling. A hybrid model is proposed by combining Gaussian Process Regression (GPR) with Extreme Gradient Boosting (XGBoost), and its improved performance is demonstrated over conventional ML algorithms, i.e., the Artificial Neural Network (ANN), Support Vector Regressor (SVR), Random Forest Regressor (RFR), and Histogram-based Gradient Boosting (HGBoost). Each of these models uses six input features and a single output, i.e., the KDF. The most critical feature affecting the KDF is ascertained through a sensitivity analysis. The study concludes with a reliability-based design framework that relies on the predictive ML model to obtain safety-consistent, yet less conservative, KDFs.

Pulsed laser deposition (PLD) expands the compositional and structural possibilities of high-entropy oxide (HEO) thin films through far-from-equilibrium growth and kinetic arrest of metastable configurations. This review connects growth kinetics and epitaxial constraints to the roles of individual cations in shaping the resulting films. We focus primarily on rock salt MgCoNiCuZnO and its derivatives, with selected fluorite, spinel, and perovskite examples illustrating broader applicability. In MgCoNiCuZnO, Co valence and Cu coordination illustrate how chemistry and growth conditions jointly enable lattice tuning and coherent nanostructure formation, while aliovalent and size-misfit additions expand the accessible defect and compositional landscapes. These coupled degrees of freedom underpin anomalous crystallinity, the coexistence of extensive chemical disorder with exceptional crystalline quality and epitaxial order, including full coherence in selected systems, particularly rock salts grown on MgO. This combination opens opportunities to discover and engineer functional configurations in high-quality single-layer films and heterostructures.

We study the greedy sequence of positive integers introduced by Slater and Vélez. They proved that it is a permutation of the positive integers but left open whether its sequence of consecutive absolute differences is also a permutation. We prove that it is.

A disturbance in steady rectilinear translation along a capillary-gravity interface radiates waves only above a minimum speed $c_{min}$, where it experiences a wave resistance; below this threshold it generates no waves, and the response is merely a localised deformation of the interface around the source, free of wave resistance. Modulating the source amplitude at a frequency $\Omega$ shifts the forcing to the sideband pair $kV\pm\Omega$, where $k$ is the wavenumber and $V$ is the translation speed, opening resonance conditions absent from the steady problem; radiation then sets in below the steady threshold $c_{min}$. Here we derive the mean wave resistance $\langle R\rangle$ of a line source in closed form, and show it to be strictly positive at every nonzero speed $V$: the drag-free below-threshold regime disappears for any $\Omega\neq 0$. Crossing the radiation transitions—where the group velocity matches the source speed—produces integrable resonant peaks in $\langle R\rangle$, whose exponents we derive, including at the cusp where two transition branches merge. Below threshold, where a steadily moving source feels no drag, $\langle R\rangle$ rises from zero as $\Omega^5$ (at fixed $V$). Throughout, surface tension is essential to $\langle R\rangle$: because it controls the short, dominant capillary-scale waves, discarding it misrepresents $\langle R\rangle$ over almost the entire speed-frequency plane.

We initiate a systematic operadic approach to anti-pre-Lie algebras. We study the operad controlling anti-pre-Lie algebras, its relationship to the operad of Novikov algebras and its cyclic structure. We establish that the Lie bracket of an anti-pre-Lie algebra satisfies the standard identity of degree five. We also establish a quadratic maximality property of the anti-pre-Lie operad with respect to this identity. We then show that the anti-pre-Lie operad and its Koszul dual are not Koszul and give a full description of the latter. Finally, we identify the Lie elements in the Koszul dual operad with the operad of metabelian Lie algebras.

Belief propagation (BP) is attractive for quantum low-density parity-check (QLDPC) decoding because of its low complexity and low latency, but short cycles and degeneracy can limit its error-correction performance. Reinforcement-learned sequential scheduling (RL-S) substantially improves this performance by adapting the variable node (VN) update order, although Q-table storage and heap-based selection introduce additional overhead. We propose bucket RL-S (B-RL-S), which quantizes learned Q-values into integer priorities and schedules VN updates using dynamically maintained first-in, first-out buckets. Without retraining or changing the BP message updates, B-RL-S reduces policy storage and refreshes only priorities affected by local residual syndrome changes. A fixed number of priority levels enables constant-time VN selection and queue relocation. For the $[[288,12,18]]$ bivariate bicycle code over the depolarizing channel at $p=0.05$, B-RL-S with eight priority levels achieves approximately one fifth the block error rate (BLER) of flooding BP followed by order-10 ordered-statistics decoding. The respective caps are $1000$ sequential iterations for B-RL-S and $1000$ flooding iterations for the BP stage of the reference decoder. These results highlight strong error correction with inexpensive scheduling and no additional post-processing stage.

DiffusionNFT (NFT) fine-tunes diffusion models toward task-specific rewards, but optimizing these rewards does not guarantee fidelity to the full prompt. Although classifier-free guidance (CFG) strengthens prompt alignment, NFT is formulated as a CFG-free method, and applying CFG to the fine-tuned velocity field can reduce the optimized reward. This motivates optimizing rewards over the guided output distribution while seeking to retain its prompt alignment. However, applying standard NFT directly to guided samples can cause reward degradation or training collapse. We trace this failure to a mismatch between the reference velocity used in the NFT loss and the score of the guided sample distribution, which biases the improvement direction. We propose GuidanceNFT, which estimates the matching reference velocity through online flow matching on guided samples and uses it to correct both regression terms. With an exact reference, the corrected objective recovers NFT's intended update. In our experiments, the learned correction sustains reward improvement under both explicit and built-in guidance. On SD3.5-M, GuidanceNFT achieves higher optimized rewards than NFT with and without CFG across GenEval, text-rendering, and multi-reward objectives, together with better overall prompt alignment as measured by off-domain evaluation metrics. Experiments on models with built-in guidance, including Model Guidance and ELF, further demonstrate improved ImageNet generation, a better perplexity-entropy tradeoff on OpenWebText, and higher accuracy on GSM8K.

We study how to detect and mitigate visual attention sinks in vision-language models (VLMs). Visual attention sinks are image tokens that absorb a large share of attention over the image regardless of the query or task, and they mislead methods that treat attention maps as a record of where the model “looked.” Prior work has detected sinks through massive activation dimensions and proposed redistributing their attention, but none have addressed what makes a token become a sink. Comparing six candidate detectors on ten VLMs from four families, we find that only the centered cosine similarity between an image token's key and the key of the language model's text-sink detects visual sinks. Thus, a visual sink is an image token whose key aligns with the text-sink's key. Moving a sink's key component along this direction to the median of the image keys sends its attention back to the text-sink (Are Borrowed) while leaving model answers unchanged~(Not Seen). Building on this, we propose a training-free, entropy-triggered re-look that first cleans the attention map, then crops the most-attended region of the cleaned map and inserts it into the reasoning trace. Cleaning adds negligible computation to the forward pass, and the re-look improves accuracy with a mean gain of $+4.3$ points on fine-detail VQA benchmarks across model families.

Grounded spatial reasoning about indoor scenes rests on a geometric reconstruction that the multimodal system does not itself produce. Feed-forward multi-view reconstruction models operate under a fixed view budget, making frame selection a critical upstream decision. We show that under this setting the conventional preference for sharp frames can be actively harmful. Across nine handheld room scans, selecting the sharpest 16 of 200 candidate frames is the worst strategy we test, reaching 48% room dimension error where uniform temporal sampling reaches 3%. We trace this failure to a coupling between image quality and camera motion: sharp frames tend to occur during slow motion and therefore concentrate the selection in a narrow angular region, leaving $192^\circ$ of the room unobserved in the case we illustrate and over $300^\circ$ in the worst cells. A $2\times2$ factorial experiment that independently manipulates viewpoint spread and per-frame sharpness shows that spread has roughly three times the main effect of quality, and that blurry but well spread selections outperform sharp but clustered selections in 8 of 9 captures. The advantage also diminishes as the frame budget increases and angular coverage saturates, consistent with this mechanism. Finally, we ask whether the failure is detectable at inference: the reconstruction model's multi-view consistency score increases with reconstruction error, while a vision-language model given the same frames reports confidence unrelated to its own spatial estimation error. Under small view budgets, viewpoint coverage is a more important constraint than per-frame image quality, and neither uncertainty signal available at inference reveals this upstream failure.

Split Gibbs sampling (SGS) is a versatile tool to sample from complex posterior distributions, such as those arising in imaging inverse problems. However, existing SGS instances are limited to quadratic variable couplings, restricting their applicability to Gaussian likelihoods. This letter genearalizes SGS to non-Euclidean geometry by introducing a coupling kernel defined through a Jensen-Bregman divergence. This framework not only enables posterior sampling in these settings but also unifies several ad hoc sampling schemes from the literature under a principled theoretical foundation.

Based on a semigroup approach, we prove quantitative uniform-in-time propagation of chaos for two systems of interacting particles with singular interaction kernels $K$ of type $L^k$, $k>d$, and a contractive Lipschitz confinement force field. First, we consider a system with moderate interaction, which means that the particles interact through the mollified kernel $K\ast V^N$, for some approximation of identity $V^N = N^{d\alpha} V(N^\alpha \cdot),\, \alpha>0$. Assuming that $\|K\|_{L^k}$ is sufficiently small, we obtain that for any $\varepsilon>0$ the uniform-in-time rate is of order $N^{-\min\left(\alpha,\frac12-\frac{d\alpha}{2}-\varepsilon\right)}$, plus an exponentially small in time contribution of the initial condition. Second, for the classical, non-mollified particle system we establish uniform-in-time propagation of chaos with rate $N^{-\min\left(\alpha,\frac12-\frac{d\alpha}{2}-\varepsilon\right)} + \|K-K\ast V^N\|_{L^{k}}$, plus an exponentially small in time contribution of the initial condition. The method relies on the regularity properties of the solution of the associated nonlinear Fokker-Planck equation combined with estimates of the Duhamel formulation of the mollified empirical measure, together with partial Girsanov transforms for the non-mollified system of particles.

In any noncommutative Carnot group, for every $0\le \alpha<1$, we show the existence of a $C^{1,\alpha}_\mathrm H$ surface that is unrectifiable with respect to the Euclidean distance. Precisely, the surface has Euclidean Hausdorff dimension higher than its topological dimension. We show that in step-2 Carnot groups the construction gives hypersurfaces of maximal Euclidean Hausdorff dimension, equal to $q-\frac{1+\alpha}{2}$, where $q$ is the topological dimension of the group. Here the range of unrectifiability is sharp, since for $\alpha=1$ the surface is rectifiable in the classical sense. In step-2 Carnot groups, it is also possible to prove that the $(q-\frac12)$-dimensional Euclidean Hausdorff measure of every $C^1_\mathrm H$ hypersurface is zero. These results extend the seminal work by Kirchheim and Serra Cassano in the first Heisenberg group.

Sparse autoencoder (SAE) features are typically discovered from how their activations co-vary with behaviour, yet they are often used through interventions such as steering or ablation. We take intervention attribution, the first-order effect of a specified intervention on the model output, as our estimand. For a broad class of response functions, we show that observational feature-discovery evidence generally does not identify this attribution, and derive an exact gradient-based decomposition of the gap between linear-probe predictions and intervention attribution. Across four model-SAE pairs, median probe-to-attribution magnitude ratios range from tens to hundreds, while attributions closely track small-step steering. For single-latent ablations with nonzero effects, gradient predictions match the realized sign in 91-97% of cases, compared with 17-35% for probe-implied changes. Attribution-selected sets also produce larger absolute mean ablation effects on held-out prompts than sets selected by correlation, rank tests or probe coefficients. Together, these results establish intervention attribution as a principled, gradient-estimable target for selecting SAE features for intervention.

The involution numbers $1,1,2,4,10,26,\ldots$ repeat modulo $3$, $5$ and $7$. We explain this, and many similar patterns, by one rule. Many sequences count permutations in which every cycle of length $m$ carries a weight $w_m$ (involutions, derangements, "sets of lists", and many others in the OEIS). Writing $A(n)$ for such a count, the rule is $A(n+k)\equiv A(n)A(k)$ modulo $\mathrm{lcm}(n,k)$. It generalizes our earlier proof of the case modulo $k$ for one such sequence, but now it holds for all weights at once, as a coefficientwise congruence between universal polynomials, and we find the largest modulus that works for the universal polynomials, $nk/\mathrm{rad}(\gcd(n,k))$, and show that nothing larger does. We then show that passing from $k$ to $pk$ behaves like raising to the $p$-th power modulo a high power of $p$, with a sharp exponent for the polynomials, which gives $A(p^r)\equiv(w_1+(p-1)w_p)^{p^{r-1}}$ modulo $p^r$ for odd $p$, and explains why the prime $2$ is special. A family of sequences defined by Lagrange inversion fits the same framework, which proves Bala's conjecture for the whole family (the single sequence A278070 had been proved earlier by an automated system) and the divisibility observed in the entries A251568 and A251569. As further applications we prove OEIS conjectures of Richman, Hanna, van Hoeij and Bala (A227937), and the theorems apply to 101 OEIS sequences, whose congruences are recorded in a supplementary catalogue. Higher-order versions of the content formula are treated in a separate paper.

The restricted active space coupled cluster singles and doubles (RAS-CCSD) model, and its pilot implementation is detailed. The approach offers a simple, pragmatic way to formulate a coupled cluster theory for open-shell and multireference systems. For ground state energies, errors (compared to FCI) are shown to be similar to those of unrestricted (spin-orbital) CCSD. For excited states, the errors of RAS-CCSD are generally larger, and adaptations to the methodology are needed to achieve an approach which is balanced for both ground and excited states. However, unlike unrestricted CCSD, the approach is spin-adapted and can be used to add dynamical correlation to a multideterminantal reference state.

Information-theoretic generalization bounds usually control dependence on the task sample, whereas selective privacy may constrain only a different, private coordinate. We study i.i.d. pairs $(X,Y)$, where a learner observes $(X^n,Y^n)$, leaks at most $\epsilon_n$ nats about $X^n$, and incurs loss on $Y$. A chain-rule transfer converts this mismatched constraint into the closed-form universal certificate $I(W;Y^n)\leq\min\{nH(Y),nH(Y\mid X)+\epsilon_n\}$. Combining the certificate with mutual-information generalization yields a bound governed by residual task uncertainty and normalized privacy leakage. We derive complementary task-compression and individual-sample refinements, identify a sharp obstruction showing why selective privacy alone need not give a vanishing certificate, and invert the bound to obtain the exact leakage allowance for a prescribed generalization tolerance. We prove exact information-level achievability for deterministic-task and independent-residual source families and show that the resulting $\sqrt{\epsilon_n/n}$ generalization rate is tight up to a universal constant. We also give a fixed-horizon adaptive application in which conditional leakage costs add.