Papers

We propose PaRID (PaRallel Instruction Duplication), a software-directed soft error detection framework that requires only compile-time effort for multithreading parallel programs. PaRID addresses two key challenges: supporting parallel programs with mixed serial and parallel regions and minimizing performance overhead without relying on costly dynamic profiling. It combines parallel-aware code transformation with LLM-tuned performance modeling, guided by eight generalizable findings from an offline characterization study, to enable fast soft error detection in parallel applications. Evaluation on NPB benchmarks shows that PaRID reduces protection overhead from 162.79% to 59.84% on average and achieves up to 5x speedup while maintaining full error detection effectiveness.

Hiring is bilateral: employers assess fit, while candidates present and defend evidence of their qualifications. Yet résumé screening, the first gate, is commonly automated as a static, one-call judgment over a résumé-job pair. We study a two-agent alternative in which employer-side and candidate-side agents represent these roles, exchange evidence, and update their judgments before deciding who advances. We compare procedures on 600 constructed résumé-job pairs using GPT-5.5 and Claude Opus 4.7. Two-agent screening advances more applications (33.3% to 39.3% for GPT-5.5; 34.0% to 35.5% for Opus 4.7). Across three runs on the common 191-pair borderline pool, pass-instance rates rise from 4.5% to 26.2% and from 6.5% to 16.1%, respectively. This is not a uniform relaxation: two-agent screening rejects applications one-call advances, changing decisions in both directions. At similar pass volumes, the procedures advance different applications, and no one-call threshold recovers applications consistently selected by two-agent screening. Among discovery-selected cases re-executed in fresh runs, two-agent-only selections recur less often than shared selections, clearly under GPT-5.5 and less certainly under Opus 4.7, while a separate one-call follow-up shows no comparable decline. As hiring becomes agent-mediated on both sides, the screening procedure, not only the model behind it, shapes who reaches human review and how reliably that access recurs.

Suppose we observe the first $n$ points of a sequence of random variables having length $n+1$, and wish to estimate a functional of the unobserved final point and the empirical measure of the $n$ observed training points. Such next-token functionals include the probability that the next token is novel (also known as the surprise probability), the tail probability of the minimum distance between the next token and training points, and the test error of a classifier trained on the observed points. All of these quantities are classically estimated by the leave-one-out method, which is inconsistent under temporal dependence. We propose a leave-a-window-out estimator, which deletes a window of length $\tau$ after each index before forming the empirical measure and reduces to leave-one-out at $\tau = 1$. Under natural assumptions, we show that the error of our estimator decays at a parametric rate for any stationary $\beta$-mixing process that also admits a Marton coupling. Our results thus cover several natural functionals on a large class of stochastic processes. We complement these upper bounds with a sharp minimax lower bound for estimating the surprise probability on mixing Markov chains. Simulations on Markov chains, moving-average processes, and autoregressive processes show that our estimator succeeds in many scenarios where leave-one-out and add-constant baselines fail.

We consider the question of designing a universal family of sets $F \subset 2^{[n]}$ such that for any function $f:2^{[n]} \to R_{\geq 0}$ in a certain class, we have $$\max_{S \in F} f(S) \geq c(n) \cdot \max_{S \subset [n]} f(S).$$ We prove that there is a family of subpolynomial size such that for any nonnegative submodular function, $c(n) = \Omega(\frac{\log \log n}{\log n})$, and there is a family of logarithmic size such that $c(n) = \Omega(\frac{1}{\log n})$. We also prove that pairwise independence (which achieves a constant factor for graph cut functions), or even $k$-wise independence, does not imply a bound better than $O(\frac{1}{\sqrt{\log n}})$ for submodular functions. On the other hand, we prove that for any polynomially representable subclass of nonnegative submodular functions (such as the matroid connectivity functions for matroid representable over $F_q$), a constant-factor universal family of polynomial size always exists. For absolute XOS functions (a class that we introduce, in the form $f(S) = \max_i |\sum_{j \in S} w_{ij} + c_i|$ where $w_{ij}, c_i \in R$), we design a family of polynomial size such that $c(n) \geq \sqrt{\frac{\log n}{n}}$, and prove that there is no polynomial-size family achieving a factor better than $O(\sqrt{\frac{\log n}{n}})$.

Air-ground transportation research increasingly relies on co-simulation, yet constructing scenarios remains labor-intensive and difficult to validate. More importantly, a generated scenario may execute successfully while failing to realize the spatial, temporal, communication, or behavioral relationships requested by the user. This paper presents AURORA, a natural-language-driven agentic framework that treats air-ground scenario generation as a process of compilation with verification. Central to AURORA is the Air-Ground Scenario Graph (AGSG), a typed intermediate representation that explicitly connects agents, aerial missions, events, communication links, success conditions, and their cross-domain dependencies. This shared representation enables simulator-grounded parsing, joint road-airspace grounding, temporal planning, pre-execution feasibility checking, trace-based runtime verification, failure localization, and bounded repair within a unified workflow. We further introduce AURORA-Bench to evaluate not only whether generated scenarios execute, but whether they faithfully realize the requested interactions. Experiments across multiple language models show that structured execution substantially improves reliability, while runtime verification exposes silent failures that completion-based evaluation overlooks. Localized repair further resolves many violations without regenerating the entire scenario. The results show that reliable scenario generation requires verifying realized behavior, not merely executable code, and demonstrate the value of explicit intermediate representations for verifiable and repairable language-driven co-simulation.

Coding agents can recursively modify their own implementations, forming a loop of self-improvement. While prior work shows this can boost performance on coding benchmarks, existing approaches are costly and compute-intensive. We introduce a simple, sample-efficient self-improvement framework that significantly improves coding performance under strict budget constraints. We identify evaluation of candidate self-modifications as the main runtime bottleneck since prior approaches estimate their effectiveness by re-running a subset of benchmark tasks with the modified agent, which is time-consuming. We introduce Recursive Self Improvement via Fast Tree-search (SIFT), which augments these downstream task evaluations with an LLM-as-a-judge signal that performs pairwise comparisons between candidate patches, where the win-loss record is aggregated with a regularized Bradley-Terry model, and the resulting strength scores drive rank-based parent sampling inside a lightweight disaggregated tree search. Expensive downstream task evaluations are reserved only for the most promising nodes. Using a fully disaggregated tree search pipeline, the judge scores provide intermediate signal to guide exploration on promising candidate patches without being bottlenecked by slow evaluation runs. SIFT outperforms existing tree-search based self-evolution frameworks on the full Polyglot benchmark with significantly lower resource requirements in terms of CPU hours, wall clock time, and API cost.

Clinical B-mode images are widely available as potential data sources for quantitative ultrasound (QUS) analysis for tissue characterization. However, standard clinical ultrasound devices apply unknown log-compression to RF envelope data before display and storage. Previous work has demonstrated estimation of the underlying RF envelope statistics in the presence of an unknown compression law. Using Fisher information analysis, we show that finite-offset log compression causes severe information loss when estimating the Rayleigh scale $\sigma$, which controls diffuse speckle. For a single image window, unknown compression raises the minimum achievable variance for unbiased estimation of $\sigma$ by a compression-independent factor of approximately 34.98. When $M$ equal-sized windows share the same unknown compression settings, the excess variance decays as $1/M$; even in the most favorable regime, reducing the variance inflation factor below 1.1 requires 340 windows. Our analysis treats the contrast parameter $a$ as unknown and the boundary offset $b$ as known; estimating $b$ experimentally shows even larger variance. We validate this theory using synthetic estimation experiments and demonstrate RF-scale recovery on real RF-envelope windows from the OASBUD dataset. Together, these results clarify the limitations of using routine B-mode images for QUS.

Benchmark scores alone provide an incomplete basis for assessing the trustworthiness of modern artificial intelligence systems. Large language models (LLMs), agentic systems, and multimodal models (MLLMs) require different forms of assessment, yet their evaluation evidence must remain interpretable for development and oversight. We propose a unified framework that connects output-level, trajectory-level, and cross-modal assessment through eight trustworthiness dimensions: capability, robustness, safety, fairness, transparency, governance, oversight, and efficiency. The framework preserves system-specific metrics while mapping native measurements to common performance bands, accompanied by uncertainty estimates and traceable evidence. A meta-evaluation layer examines the validity, reliability, and reproducibility of the evaluation itself. Multidimensional profiles expose strengths and weaknesses, while safety-critical overrides prevent aggregate scores from masking critical failures. Mappings to governance frameworks, international standards, and European Union regulatory requirements connect technical assessment with oversight needs. The framework provides a structured basis for assessing both system performance and the credibility of the evidence supporting it, with empirical validation across deployment contexts remaining an essential next step.

Web agents often revisit the same sites, yet most evaluations discard the procedures learned in earlier successful interactions. We introduce EconSkills, a skill library and evaluation framework that distills verified EconWebArena trajectories into parameterized standard operating procedures for retrieving live economic data. Each skill records its scope, navigation procedure, site-specific guidance, verification checks, and recovery steps while replacing source-instance values with placeholders. EconSkills separates two questions: whether a known relevant procedure transfers to a held-out task, and whether an agent can retain that benefit when selecting from a library. In controlled transfer, matched skills improve success over no-skill prompting and require fewer steps on paired successes. Under the evaluated prompt formats, the parameterized skill prompt substantially outperforms the corresponding raw-trajectory prompt. With the full 50-skill library, retrieval is competitive with the no-skill baseline overall and performs best on tasks with a direct family match; approximate matches on other tasks offset these gains. Browser trajectories further identify when procedural guidance shortens portal-specific navigation and when semantic verification remains necessary. These results establish that reusable economic web procedures can transfer across task instances and provide a concrete design target for match-aware selection and context delivery.

We study the variation of the quasi-Assouad dimension of a set moving under a holomorphic motion. We show that the reciprocal of the quasi-Assouad dimension is inf-harmonic in the sense of Fuhrer–Ransford–Younsi. As a consequence, we obtain quasiconformal distortion bounds for quasi-Assouad dimension as well as an improved version of Smirnov's celebrated theorem on the dimension of quasicircles. Our approach is elementary in that it does not require optimal Sobolev regularity for quasiconformal mappings.

Recurrent-depth Transformers reuse one block across many steps, so information needed later must survive repeated rewriting of the hidden state. A natural remedy is to keep more history. We show that, in controlled cellular-automaton tasks, making history available is not the same as making it usable. Using Rule 30, where the correct state is known at every recurrent step, we test depth extrapolation and de- layed recall, the recovery of an earlier state after further computation. CoTFormer, which caches keys and values from every earlier step, extrapolates less far and recalls less accurately than a Block Universal Transformer (BUT) that keeps only its current state. Interventions show that its retained history can pull a corrected trajectory back toward failure, and that the cache block written at the requested step is neither necessary nor sufficient for recall. We introduce LSTM-UT, which adds a small, bounded, gated cell state to the shared block. Trained to depth 12, LSTM-UT keeps 99.7% exact-row accuracy at depth 60, where BUT gets no row fully correct, and one checkpoint stays above 99.95% at depth 1,000. It also improves delayed recall over both baselines, and the advantage largely persists at near-matched parameter counts. On these tasks, a small state under learned control proved more useful than a complete but unaddressed history. In OpenWebText2 language modelling, LSTM-UT outperforms BUT and, at equal width, reaches slightly lower perplexity than CoTFormer while CoTFormer needs up to 91% more training time per step; against a parameter-matched CoTFormer, LSTM-UT comes within 0.6 perplexity.

Let $d\in [0, n]$ and let $E\subset \mathbb{R}^n$ be a closed $d$-thick set. Given $p\in [1, \infty]$, $q\in (0, \infty]$, and $s\in (\frac{n-d}{p}, 1)$, we provide an intrinsic description of the trace-space of the Besov space $B^s_{p, q}(\mathbb{R}^n)$ to $E$. If, in addition, $p<\infty$, we give an intrinsic description of the trace-space of the Lizorkin–Triebel space $F^s_{p, q}(\mathbb{R}^n)$ to $E$.

Language-model agents are increasingly asked to carry out work spanning days or weeks, such as an operations remediation or a research programme. Such a task outlives any context window, any process and any interval at which a person can attend. In this paper, we argue that a long-horizon agent must run continually without forgetting before it can learn continually. This ability lies in the harness around the model rather than in the model itself. We derive seven bottlenecks from the long-horizon setting and answer them with a hierarchical architecture of three parts: (i) levels indexed by time scale, each keeping a bounded file summarising the level below; (ii) a clocked tick as the unit of autonomous action; and (iii) cascaded intelligence, where work is escalated to a more capable model only after failing review. We report on a ten-day campaign in which an agent built on this architecture reproduced a published reinforcement-learning result with a human attending once a day, and show (1) the agent kept the thread across every context reset and session boundary of the campaign, (2) operating knowledge written early changed later behaviour with no change to model weights, and (3) where learned components would enter such a system. Overall, our experience suggests continual learning for these agents needs a substrate outliving every context and process, and the checks the harness already runs are where a learner belongs.

We present AMB3R-SLAM, a real-time monocular SLAM system capable of reconstructing kilometer-scale trajectories over 10k frames on a single consumer-grade GPU. Our model couples a lightweight front-end for low-latency online tracking with a hierarchical backend that progressively enforces local, mid-level, and global consistency. By avoiding bundle adjustment that relies on the static world assumption, our system naturally handles complex dynamic scenes out of the box. Furthermore, we demonstrate that our method can be extended to leverage stereo, RGB-D, and LiDAR as additional inputs. AMB3R-SLAM achieves strong camera tracking performance across 9 datasets, reducing the absolute trajectory error (ATE) of previous state-of-the-art methods on VBR and Oxford Spires by over 70%. With additional LiDAR input, our model further reduces ATE to sub-meter level on KITTI and VBR datasets.

Receiver Operating Characteristic (ROC) and Precision-Recall (PR) curves are widely used to assess the discrimination ability of forecasts for binary events, such as threshold exceedances or warnings of extreme events. In weather forecasting, forecasts are provided as spatial fields, yielding location-wise ROC and PR curves that are often aggregated to facilitate comparison. However, the effect of the aggregation strategy on performance assessment remains poorly understood. We investigate how different aggregation strategies for ROC and PR curves affect the assessment of discrimination ability. In particular, we identify conditions under which aggregation strategies satisfy two desirable properties for fair comparison: preservation of dominance between forecasts and preservation of concavity or achievability of the curves. We obtain sufficient conditions and propose two strategies satisfying them. They are compared with existing strategies from the literature, and we analyze their properties and highlight potential pitfalls that may lead to misleading interpretations. Based on these findings, we provide practical guidelines for the interpretation of aggregated ROC and PR curves. The proposed framework is illustrated with AI-based global weather forecasts, showing how different aggregation strategies can yield different rankings of competing forecasts.

Electrical resistivity is a fundamental physical property that provides powerful constraints on subsurface structure, lithology, fluid distribution, and mineralization. We present a continental-scale electrical resistivity model of Australia's shallow crust with quantified uncertainty, extending to depths of up to ~650 m. The model is derived from probabilistic inversion of more than 26.6 million airborne electromagnetic (AEM) soundings acquired along 352,600 line-kilometers from the AusAEM program. We develop a deep learning-based probabilistic inversion framework using invertible neural networks (INNs). To accommodate differences among AEM surveys, we trained 11 networks using a common set of synthetic resistivity models and the same training strategy, enabling consistent integration of inversion results across Australia. The entire inversion was completed in only 4.37 GPU hours, overcoming long-standing computational barriers to continental-scale probabilistic geophysical imaging. The resulting resistivity model exhibits clear spatial patterns that closely align with Australia's major geological provinces, sedimentary basins, aquifer and mineralizing systems. We further define the depth of investigation using posterior resistivity distributions to support uncertainty-informed geological interpretation. The resistivity model together with quantified uncertainty provides new physical constraints on regolith thickness, basin architecture, paleochannel and mineralized systems across Australia, serving as a foundation for interpreting continental-scale electrical structure and its geological, hydrological, and mineral system implications.

Verifier-based selection improves LLM performance by generating multiple candidate solutions and using a verifier to select the most promising one. However, existing methods typically treat verification only as a ranking step and discard its feedback once a fixed candidate pool has been evaluated. In this paper, we ask whether verification can also improve the candidate set itself. To this end, we introduce LLM-as-an-Improver and propose Verify–Repair–Reselect (VRR), which uses verification feedback to generate and reselect improved candidates. VRR retains the initial winner while conditionally generating three complementary alternatives: repaired versions of the winner and runner-up, and a solution based on a new approach. It filters invalid and duplicate candidates using only inference-time information and then reselects the final answer under the original evaluation criteria. Across diverse models and code-generation and reasoning benchmarks, VRR improves over fixed-pool verifier-based selection in many settings and can recover correct solutions even when all candidates in the initial pool are incorrect. These results highlight a broader role for LLMs as improvers: verification feedback can not only select among existing solutions but also construct stronger candidates beyond the initial pool.

Local-to-global techniques for establishing spectral gaps have played a central role in the modern theory of Markov chain mixing times and the theory of high-dimensional expanders. One of the most striking results in this burgeoning literature is that a spectral gap for the global down-up walk on the facets of a pure simplicial complex can be reduced to sufficiently strong spectral expansion of just the codimension-2 links of the complex, a phenomenon colloquially referred to as "trickle-down". These types of theorems have had many important applications, including rapid mixing of the exchange walk on the bases of any matroid. In this primarily expository article, we give streamlined proofs of two such theorems in the literature, one by Oppenheim (2018) and one by Leake and Oveis Gharan (2025), via an integrated Bochner method. Moreover, in the latter setting, we quantitatively strengthen the dependence of the global spectral gap on the dimension of the complex and the spectral influence, resolving an open question of Leake and Oveis Gharan. Disclaimer: The proofs were developed through a couple of rounds of interaction with GPT-5.6 Sol Ultra. We later discovered that Guo and Zhang (2026) had independently proven the same strengthening of the trickle-down theorem of Leake and Oveis Gharan using an extremely similar argument, also found by GPT-5.6 Sol Ultra. The focus of their paper is the complexity of approximating the partition function of spin systems on planar graphs, not on the trickle-down phenomenon itself. In contrast, our motivation is primarily expository, and we hope to bring Bochner-type methods and their connections with the trickle-down phenomenon to the attention of a wider community of researchers.

High-quality pre-training data is a critical bottleneck for educational and STEM-specific language models targeting edge AI and on-device deployment where token budgets are tightly constrained. While major organizations train ever-larger models on private corpora, the open ecosystem lacks STEM-focused synthetic datasets that deliver high per-token learning value efficiently for small models. To address this gap, we introduce QVAC Genesis III, a 191.43B-token, STEM-focused multi-domain synthetic corpus covering 19 domains across several difficulty levels and different educational styles. QVAC Genesis III is built via a dual generation strategy that performs targeted teacher distillation using a weak edge-scale student model as signal: the student's failures are converted into corrective explanations, while its successes are expanded into contrastive option-level reasoning over all answer choices. We further introduce an LLM-as-a-parser evaluation protocol that extracts final answers from free-form outputs and tracks both accuracy and answer validity. To validate the effectiveness of our QVAC Genesis III data, we conduct controlled from-scratch ablations with 1.7B-parameter models, showing that models trained with QVAC Genesis III consistently outperform both models trained with the open-source synthetic corpus Cosmopedia-v2 and the publicly released Cosmo-1B model across ARC, GPQA Diamond, and MMLU STEM benchmarks, achieving up to +28.57% on ARC-E and +21.35% on ARC-C, while reaching a Valid Answer Rate of up to 99.45%.

Robots can recall prior failures without knowing whether recalled evidence remains valid, conflicts with current observations, or is sufficient to guide a decision. We present CoreSense, a robot-system integration architecture that combines traceable episodic evidence with a conflict-aware belief gate and bounded, auditable recommendations. The gate checks scope, provenance, time, contradiction, and support before it permits PROCEED, requests re-observation, abstains, or escalates. Evaluation follows three complementary layers without commanding a physical robot: offline public real-robot data, a frozen signal-level simulation, and a live cloud deployment path. On CableTrace-120 and BotFails-200, belief gating reduces protocol-defined unsafe proceeds from 20% and 40% to 0%. A disjointly calibrated raw-video policy also reaches 0% unsafe proceed, but overblocks every nominal episode. On public data, a ViFailback-BotFails visual detector reaches 0.778 AUROC yet remains all-blocking, whereas cycle-disjoint UR3 telemetry for protective stops yields 0% unsafe proceed, 36.1% overblocking, and 61.9% coverage; grip-loss transfer remains a negative result. Controlled physical corroboration yields 3.3%, 0%, and 42.0%, while conflict-aware fusion yields 4.7%, 0%, and 42.8%. Finally, 20/20 cloud recalls validate a CockroachDB Cloud-Amazon Bedrock deployment path. The evidence supports an auditable integration pattern, not autonomous recovery or certified safety.

Pooling earthquake-triggering parameters can reduce estimation variability on margins with few recorded sequences, but its forecast value must be distinguished from the stability of simply freezing a fitted law. We fit the ETAS model to the Mexican national catalog above magnitude 5.0, using the Slab2 coverage mask as the domain. On a 4.7-year holdout the model gains 1.45 bits per earthquake over a time-independent spatial Poisson reference (cluster-resampled interval [-0.03, +2.75] bits), with two clusters carrying 90% of the signed gain. Using the Global CMT catalog we fit eight disjoint subduction margins and a joint model on 3,210 earthquakes sharing six triggering parameters with a background rate per margin; BIC favors the joint model by 184 units, though a parametric bootstrap rejects its exact shared specification. Transferred into Mexico with the local background fixed, six of seven foreign triggering sets score within 0.07 nats per event of the home fit. In leave-one-margin-out evaluation, the excluded-pool law's event-weighted paired gain is +0.0334 [+0.0002, +0.0670] nats per event; its lower limit sits barely above zero and crosses zero under alternative cluster-linking rules, so we read it as neither conclusive superiority nor guaranteed noninferiority. Across 28 overlapping annual evaluations, local refitting minus a pooled law frozen before the first origin has mean gain -2.07 nats/year (moving-block interval [-6.92, +1.15]), while local productivity varies by a factor of 5.4. Coverage of these intervals is unmeasured, and a measured analogue fell short of nominal; the paired median contrast's significance depends on a Gutenberg-Richter slope that is conventional rather than measured. These results support pooling as a regularized alternative to local fitting, short of equivalence, a universal law, or an optimal update policy.

Large language models (LLMs) show strong zero-shot reasoning capabilities, yet their sensitivity to prompt context raises questions about reliability in deployment. This study investigates whether geographic context embedded in prompts influences LLM-based chronic kidney disease (CKD) screening, independent of any change in patient features. Using a community-based CKD dataset from Bangladesh (284 participants with 112 CKD and 172 non-CKD), we systematically varied the country attribute across 25 global contexts while holding all clinical inputs constant, and evaluated two open-weight LLMs (Qwen3 8B and Mistral 7B). Experiments were conducted on a common-correct subset of 156 participants for whom both models produced correct baseline predictions. Results show that non-CKD predictions were highly stable across geographic contexts, with both models maintaining consistent non-CKD predictions in nearly all settings. In contrast, CKD-positive predictions were considerably more sensitive. For Qwen3, CKD prediction dropped from a baseline of 65 to as low as 50 across country contexts, while Mistral showed smaller but non-negligible variation. Country contexts did not group according to known epidemiological or healthcare similarities; instead, groupings appear to reflect how country tokens are internally represented within each model. These findings demonstrate that geographic tokens can systematically shift clinical screening outcomes despite identical patient inputs, with implications for the deployment of LLM-based decision support in diverse healthcare settings.

Scaling exponents of neural dimensionality are read as properties of the code, and used to compare representations and benchmark models against cortex. Yet one recording of 11,311 mouse V1 neurons gives 0.25, 0.30 or 0.35, depending only on how it is subsampled. We show that such an exponent measures the population and the probe together. We split it exactly into a floor, which random subsets of the same sizes give, and a shift, which a declared axis of stimulus classes adds. The usual random-subsample exponent is almost all floor. Stimulus direction lifts the exponent in every grating recording by reading both symmetry sectors of the code: orientation, which repeats every half turn, and direction, which does not. The lift lives in trial-to-trial fluctuations that differ between directions, while the class means, which linear decoders read, pull it down. Two measured terms, how much the classes' fluctuations overlap and how the means sit against them, account for the shift. They give its sign where it changes: in Allen Neuropixels recordings, whose class means mostly raise the exponent, and in macaque V1. In an XY lattice whose classes stay on one orbit, temperature flips the sign as the two terms say, and in GPT-2 the shift ranks the indirect-object circuit's name-mover and backup heads. Reported with its path, its floor and its shift, an exponent says how much structure a probe finds and what carries it. Our open tool, rung, reports all three for any labelled neural recording or language-model layer: https://github.com/adilamin89/structure-in-the-zoom

A unit ball is surrounded with a ring of $n$ equal infinite circular cylinders. Each cylinder touches the ball and its two neighbors. Can one more cylinder of the same radius be added if the configuration may be rearranged? Can the $n$ cylinders instead be enlarged, and by how much? We prove that insertion is impossible for $3\le n\le7$ and possible for $n\ge9$. The case $n=8$ remains open. The case $n=6$ is Kuperberg's problem from 1990, which was resolved recently. We prove that the ring of four cylinders cannot be expanded. This settles a conjecture of Ogievetsky and Shlosman from 2018. For $n\ge5$ the largest possible expansion radius remains unknown.

Actor-partner interdependence models (APIMs) describe how one member's exposure is associated with that member's and the partner's outcomes, but actor and partner coefficients are causal only under additional design and identification assumptions. We formulate APIM-style questions as dyadic target trials under within-dyad interference and make three links between the associational and causal formulations explicit. First, under joint exchangeability, positivity, and a homogeneous correctly specified treatment-related conditional mean, APIM actor, partner, interaction, and joint coefficients equal the corresponding causal contrasts. Second, for binary dyadic treatment, multiplying the true member-specific propensities induces an exact one-parameter discrepancy delta(H) = Cov(A_1, A_2 | H): the four joint-cell errors are (+delta, -delta, -delta, +delta), total variation is 2|delta|, and IPW contrast bias has a closed-form representation. Third, the same discrepancy affects AIPW through its product with outcome-regression error. A 36-scenario simulation and a true-marginal factorization diagnostic show that large separate-propensity bias under residual dependence is primarily factorization bias, while flexible AIPW reduces nonlinear-model bias but not weak-support instability. In the strongest dependence condition, factorizing even the true marginals reversed the spillover contrast and inflated the interaction to nearly five times its true value. We apply the framework to 2,400 Health and Retirement Study couples studying retirement transitions and later depressive symptoms. Joint retirement was rare, residual treatment dependence was substantial, and estimates were sensitive to support and propensity flooring. The contribution is a design-and-diagnostic bridge from APIM questions to causal dyadic analyses, not a new general interference or AIPW theory.

Constructive classifiers add structure while they train: a level to a tree, a unit to a hidden layer, a split at a leaf. This paper asks what each of four such growth decisions buys, measured under one protocol on 24 datasets, and gives an exact diagnosis and fix for the one that buys nothing. The diagnosis concerns the natural way to deepen a soft decision tree: turn every leaf into a gate whose two children inherit the parent's class distribution, so the function is unchanged. I prove that this leaves the gradient of every new gate identically zero and with the gate at 1/2, gives the two children identical gradients, so under this construction the added level can never learn. That predicts where it costs: nothing on two-class problems, where two leaves already suffice and a great deal where more classes need more leaves. Measured, the cost is -0.2 points over 13 binary datasets and 40.2 points over 7 multi-class ones, and it tracks the number of classes (Spearman 0.64), not the number of features (0.03). The fix is any perturbation of the children. Neither its size nor its direction matters: a residual-directed initialisation changes accuracy by +0.20 points against noise. The other three decisions each buy one thing. Fitting a new hidden unit to the residual before installing it buys a smaller network but not a more accurate one. Splitting one leaf at a time buys sparsity; it lost accuracy until I found that the split started its two children identical and untrained, the same defect in another place; with the children inheriting the parent and the symmetry broken, per-leaf growth comes within 2.3 points of a complete tree using 23% of its splits. Requiring statistical significance before a node gets a more expressive split buys nothing. Every number comes from the measurement scripts.

Conifold boundary conditions constrain higher genus topological string free energies, but the cancellation of subleading poles is not implied by their leading singular behavior. We prove a regularity theorem for connected elliptic completed cycle traces and use it to establish the Nekrasov-Shatashvili conifold gap for the maximal contact relative Gromov-Witten theory of the projective plane with a smooth cubic boundary and a top Hodge class insertion. The higher genus local/relative correspondence expresses the ordinary potential in terms of the relative potential and differentiated lower genus contributions. We identify its conifold polarized correction with an elliptic trace. A central difference identity resums each non-vacuum partition weight into a content product multiplied by a regular exponential. For arbitrary regular parts of the input potentials and any nome with a simple zero, the normalized non-vacuum connected contribution is divisible by the conifold coordinate and by the square of the coupling. The entire polar part and finite part are consequently determined by the vacuum constants. A triangular induction transfers the known ordinary local projective plane gap to the relative theory. We also obtain finite part identities, finite formulas for regular Taylor coefficients, and reconstruction of the holomorphic ambiguities. The argument is formal in the coupling and gives holomorphic remainders at every fixed genus.

Traditional dynamical system models, including Koopman operator representations, are fundamentally equality-based, whereas many analysis and control tools rely on inequalities. This mismatch motivates representations that are intrinsically aligned with certification tasks involved in the analysis and control synthesis problems. In this paper, we propose a data-to-certificates (D2C) paradigm that bypasses explicit model construction and directly learns certificates from data. We introduce supereigenfunctions of the Koopman operator as an inequality-based generalization of eigenfunctions that define exponential growth envelopes encoding stability, safety, and uncertainty propagation, thereby serving as certificates for a range of control objectives. We establish their theoretical foundations and show that the associated rates recover intrinsic dynamical quantities such as Lyapunov exponents. Two complementary constructions are developed: a geometric approach based on the multiplicative ergodic theorem (MET), and a resolvent/Gramian formulation that enables computation directly from trajectory data. The resulting framework yields certificates that can be used for stability and contraction analysis, as well as stabilizing and safety-critical control synthesis via convex quadratic programming-based optimization program. Numerical examples demonstrate the effectiveness of the proposed data-driven certification approach for stabilization, contraction, and safe control design.

In the Steiner Point Removal problem, we are given a graph $G=(V,E)$ with an edge-length function $\ell_G: E\rightarrow \mathbb{R}_+$ and a subset $T\subseteq V$ of terminals. The goal is to find a minor $H=(T, E_H)$ of $G$ on vertex set $T$ such that the shortest path metric derived from $G$ on the edges of $H$ preserves the distance between every pair of terminals within a small multiplicative stretch. Filtser proved that a stretch of $O(\log |T|)$ can be achieved (in polynomial time), while Chen and Tan more recently proved a lower bound of $\Omega\left(\sqrt{\frac{\log |T|}{\log\log |T|}}\right)$ on the achievable stretch. Their lower bound is via a simple construction involving low-degree high-girth graphs. The existence of such graphs is guaranteed through the existence of low-degree high-girth expanders. In this work, we improve the lower bound to $\Omega(\sqrt{\log |T|})$ using the same simple construction of Chen and Tan but with a more careful analysis that exploits the expansion property.

In infectious disease transmission, regression and variable selection for time-to-event outcomes are complicated by transmission-induced dependence that violates standard independence assumptions. Pairwise survival analysis addresses this dependence by modeling contact intervals in ordered pairs, defined as the time from the onset of infectiousness in the pair's first individual to their first infectious contact with the second individual. We propose a penalized pairwise accelerated failure time model for contact intervals that accommodates simultaneous covariate effects on infectiousness and susceptibility while jointly representing within-population transmission and infection from external sources. For variable selection with many candidate covariates, we develop adaptive LASSO penalized maximum likelihood estimation with a coordinate descent algorithm, treating distributional baseline parameters as unpenalized nuisance parameters. We establish selection consistency and asymptotic normality of the adaptive LASSO estimator under profile-likelihood regularity conditions in a local asymptotic regime in which information accumulates over all ordered pairs that are ever at risk of transmission during follow-up. Simulation studies show accurate variable selection and improved estimation accuracy compared with the unpenalized estimator and the LASSO. We illustrate the method using 2009 household influenza A (H1N1) surveillance data from Los Angeles County, where the adaptive LASSO produces a sparse, interpretable model and improves the precision of selected covariate effect estimates, including the effect of antiviral prophylaxis on susceptibility.