Papers

Benchmark contamination is usually treated as a provenance question: did evaluation material appear in training? Provenance cannot supply the quantity needed to interpret a benchmark score: how much exposure changed it. We introduce LeakScale, a causal measurement framework that makes benchmark-specific exposure manipulable and measures its effect. LeakScale constructs fresh executable tasks whose correct behavior depends on private, non-derivable information, counterfactually certifies every candidate answer, and trains complementary adapters so that every family is exposed in exactly one adapter and unexposed in the other. Across 2,048 families, two model families, two executable domains, three task formulations, and 589,824 generations, exposure increases exact-task accuracy in every model-by-domain cell. Primary effects range from +3.22 to +10.66 percentage points; every family-clustered 95% interval excludes zero, and all four primary tests survive Holm correction. The effect also remains positive in all eight surface-paraphrase and isomorphic-transfer measurements. These results show that benchmark exposure can causally and reproducibly alter reported performance, including after the evaluation surface changes. LeakScale shifts contamination analysis from detecting contact to measuring how strongly a score depends on it.

Multimedia verification requires not only accurate decisions but also traceable evidence, reliable human correction, and safe reuse of prior experience. Existing systems often lack explicit mechanisms for revising intermediate reasoning or preventing harmful knowledge transfer. We present SEMV (Self-Evolving Multimedia Verification), a self-evolving multi-agent framework that treats provenance-bearing arguments as the interface between evidence, reasoning, human contestation, and memory. SEMV combines arena-based quantitative bipolar argumentation (A-QBAF), causal and scoped revision, and verification-gated memory consolidation with explicit conflict retention. On COSMOS benchmark, SEMV achieves 91.88% accuracy versus 89.10% for the strongest comparable baseline. Verified memory reduces negative transfer from 5.7% to 0.2%. On CTR benchmark, constructed from reviewer contestations, scoped causal revision corrects 96.7% of initial errors while saving 52.8% compute. MV2026 Grand Challenge dataset further supports evidence-grounded, temporally consistent reporting. These results show that SEMV can evolve through verified experience while keeping accumulated knowledge and subsequent decisions traceable, revisable, and contestable.

Bayesian calibration of functional-output computer models typically relies on dimension reduction techniques, such as functional principal component analysis, which assume that differences among simulator realizations arise only from amplitude variation. When simulator output also exhibits phase variation such as shifts in the timing or location of key features, this assumption is violated. Recent work has addressed this issue through elastic calibration, which aligns functional computer model realizations with observed experimental data prior to dimension reduction. Separately, multi-fidelity methods reduce the cost of calibration by supplementing a small number of expensive high-fidelity simulator runs with a larger ensemble of cheap low-fidelity runs. This is typically done through either a mapping strategy, which corrects low-fidelity predictions toward high-fidelity output, or a fusion strategy, which builds a shared basis across both fidelities. This paper combines these two lines of work, introducing elastic multi-fidelity Bayesian model calibration, which aligns high- and low-fidelity functional output to a common reference before applying multi-fidelity mapping or fusion. On a synthetic two-dimensional calibration problem and a dynamic material properties equation-of-state problem, both elastic multi-fidelity strategies match or improve on the leave-one-out predictive accuracy of a mono-fidelity elastic emulator, with the fusion approach achieving the lowest error. Both strategies also produce tighter calibrated posteriors than the mono-fidelity baseline, with the fusion approach providing the best coverage and parameter estimates closest to the true values.

Solving complex tasks in domains such as science, medicine, law, and finance often requires assembling interdependent information scattered across vast, heterogeneous sources far beyond model context limits. Existing approaches tackle this challenge by organizing information into more manageable representations over which models can reason, such as graphs, textual memories, and retrieval collections. These representations dictate what downstream reasoning is possible and, ultimately, whether it succeeds; yet their design and construction remain largely ad hoc. In this work, drawing on the cognitive theory of relevance realization, we propose concrete principles for designing AI systems that construct effective representations of very large contexts. We analyze existing approaches and show how their successes and failures can be explained by their alignment with these principles, and introduce R3Con, a harness that systematically operationalizes them. We evaluate R3Con against nine state-of-the-art baselines on two recent benchmarks of reasoning over large document corpora. On these benchmarks, R3Con substantially outperforms the strongest baseline, by $20$ and $8.4$ percentage points. It also enables smaller models to outperform much larger ones: R3Con with 4B and 9B models outperforms all evaluated 35B baselines, while R3Con with Qwen3.5-35B-A3B outperforms Claude Code with Claude-Sonnet-5 at $3.7\times$ lower cost. Our results show that context representations following our principled approach can reduce reliance on model scale, pointing toward a future of AI systems with frontier-level performance powered by smaller models.

Guo, Fang, and Lu recently proved the Komlós conjecture: for vectors $v_1,\ldots,v_n\in\mathbb R^d$ of Euclidean norm at most one, there are signs $\varepsilon_j\in\{-1,1\}$ with $\|\sum_j\varepsilon_jv_j\|_\infty\le3\sqrt{2\pi}$. We give a short proof of the bound $3\pi$ that keeps the geometric lifting framework of (Guo, Fang, and Lu 2026a) while replacing the analytic core by a quadratic Dirichlet energy. Moreover, using a more fine-grained analysis of the stability of the product-cosine function of (Smirnov and Vershynin 2026) under translations, we further sharpen the bound to below $6.9013$. On the algorithmic side, based on the polynomial-time construction of (Guo, Fang, and Lu 2026b) we give a deterministic algorithm that finds a coloring of discrepancy at most $37.54$ using at most $\widetilde O(mn+n^4)$ arithmetic operations.

We compute the smallest non-abelian quotients of spherical and affine Artin groups. Usually, these are isomorphic to the smallest non-abelian quotients of the corresponding Coxeter groups. As applications, we deduce that two affine Artin groups are isomorphic if and only if their Coxeter diagrams are isomorphic and that irreducible spherical Artin groups are distinguished from each other by their finite quotient groups.

We study exact pattern matching on indeterminate strings, where a text or pattern position may represent a set of symbols rather than a single letter. Focusing on Boyer-Moore-style methods, we present new bad-character rules (BC Rules I-IV) and a new good-suffix procedure, computed by Fast_GSR_Indet_Shift, which avoids the per alignment recomputation used in BM_Indet [12] by shifting with a single preprocessed position-indexed table. We conduct a systematic experimental evaluation of sixteen algorithms, including classical bad-character adaptations (e.g., Horspool, Sunday, and Zhu-Takaoka) and hybrids that combine these bad-character rules with Fast_GSR_Indet_Shift. Across synthetic scaling experiments and a case study on the E. coli K-12 MG1655 genome, the Fast_BM_Indet hybrids consistently outperform BM_Indet and KMP_Indet [12], in some settings by up to two orders of magnitude. We also find that Zhu-Takaoka is the strongest bad-character-only adaptation on small alphabets and genomic data, while the Fast_BM_Indet variant using BC Rule I offers comparable performance, making it attractive for larger alphabets. We conclude with practical guidance on choosing among these variants for indeterminate string applications.

The increasing integration of distributed energy resources introduces new challenges for accurate fault detection and classification in distribution networks. This paper presents a topology-aware graph convolutional network-based model for fault classification under inverter-induced harmonic distortion. The model uses pre-fault and post-fault voltage and current phasors, including harmonic components, mapped to the IEEE 34-bus system topology. The simulation data generated in OpenDSS encompass varying levels of penetration of solar photovoltaics (30%, 50%, 70%) and current total harmonic distortion (0%, 1%, 3%, 5%) across multiple fault types: single line-to-ground, line-to-line, double line-to-ground, and three-phase-to-ground. A differential evolution algorithm is employed to optimize the hyperparameters of the network, resulting in a three-layer model that achieves an F1-score of 0.98. Stress testing across scenarios confirms high robustness and minimal class confusion. The results demonstrate that a graph convolutional network-based model can effectively classify distribution faults under varying photovoltaic penetration levels and harmonically distorted conditions. Comparative analyses with convolutional neural network, long short-term memory, and multi-layer perceptron architectures demonstrate that the proposed model delivers superior classification performance, particularly under the most stressed condition with 70% photovoltaic penetration and 5% harmonic distortion.

A graph is 1-planar if it has a drawing in which every edge is crossed at most once. We show that the smallest cubic non-1-planar graphs have $30$ vertices. Two such graphs are the Tutte-Coxeter graph of girth eight and a graph of girth seven that we call the Byte graph. Every subcubic graph with fewer than $30$ vertices is 1-planar. Our proof is computer-assisted, but directly testing all relevant graphs is impractical. To establish non-1-planarity of the two graphs, we extend a SAT-based solver with a custom clause propagator based on separating cycles and a case split based on graph automorphisms, allowing independent cases to be solved in parallel. To show that all smaller subcubic graphs are 1-planar, we introduce the concept of $k$-flexibility: every set of at most $k$ prescribed edges can remain uncrossed in some 1-planar drawing. We use this property to reconstruct 1-planar drawings of larger graphs from drawings of smaller $k$-flexible graphs. This replaces exhaustive testing of more than forty billion cubic graphs with computations on far fewer graphs of smaller order.

Background. A connectome specifies the substrate available to a nervous system, not how much of it a given computation requires. Reported compact sufficient circuits are point estimates: one subset, one task, one fidelity threshold, one parameterisation. Methods. We formalise computational sufficiency as a cost-indexed family of neuron subsets conditioned on a task and a tolerance pair (permitted degradation epsilon, permitted failure fraction rho), separate task adequacy from phenotype fidelity, and certify candidates on held-out parameter draws with one-sided bounds rather than reporting a single minimum. We measure three quantities a single circuit cannot express: non-decomposability, degeneracy, and an intervention-derived role taxonomy. Results. On a released walking simulator, the published 148-neuron circuit is certified sufficient with margin (mean Dtask = 0.017 over 32 held-out draws, no failure), no 146- or 147-neuron subset of it is sufficient, and removing any one of its three interneurons destroys adequacy. Complement splitting recovers the published set exactly in three of seven seeds and reaches its size in five, within 66-97 evaluations, against 221 neurons for the bundle hierarchy at the same budget. Varying only the task changes which neurons are needed: our best-found solutions retain 155 and 149 neurons and share only 3 of 11 searchable neurons. Relaxing the tolerance reveals alternative sufficient circuits, and the whole-CNS interface alone fails every replicate while 93 search-chosen neurons give a certified solution. Conclusions. Sufficiency is set-valued and conditioned jointly on task and tolerance, and no single search strategy dominates across regimes. A frontier is interpretable only alongside its interface-only baseline, its tolerance and its budget. Every result here is model-relative.

Probabilistic forecasts of day-ahead electricity prices are increasingly used to manage price risk, yet competing models are usually ranked on aggregate scores that average over the price regimes in which uncertainty matters most. This study evaluates probabilistic forecasts of hourly EPEX SPOT GB day-ahead prices over a full held-out calendar year, applying a common pre-auction information boundary and rolling-origin design to linear quantile regression, a quantile regression forest (QRF), a LEAR-QRA benchmark and an encoder-decoder quantile long short-term memory (LSTM) network. Two routes to more reliable intervals are then compared: conformal recalibration and Quantile Regression Averaging (QRA). Aggregate scores separate the LSTM and QRF only weakly, with a non-significant 4.4% difference in continuous ranked probability score (CRPS), but their conditional behaviour differs sharply. Above 200 GBP/MWh the LSTM covered 0.12 of outcomes and QRF 0.71, consistent with the forest retaining realised spikes in its leaf distributions while the network learned a smooth upper tail. Conformal recalibration corrected marginal coverage but did not improve the interval score or materially repair spike coverage. QRA combining the LSTM, QRF and LEAR-QRA forecasts reduced CRPS to 5.14 GBP/MWh and raised coverage above 200 GBP/MWh to 0.65 at unchanged interval width. Recalibration and combination therefore address different failures and are not substitutes.

Bitcoin perpetual futures trade continuously with shallow liquidity and high price impact, making them prone to millisecond-scale flash crashes. Agent-based limit order book simulators are a natural tool for studying such events, but their behavioral parameters are usually hand-crafted or require full order book histories that are rarely available at scale. We propose a four-stage pipeline that constructs a reproducible 100 ms BTCUSDT flash-crash event panel from 66 detected events, elicits behavioral priors for four agent archetypes using an offline large language model conditioned on empirical microstructure anchors, instantiates a heterogeneous limit order book simulator with the validated priors, and applies causal discovery plus counterfactual analysis to the resulting synthetic panel. Across 500 simulation runs, LLM-elicited priors produce a crash rate of 0.176 versus 0.008 under uninformative uniform priors and generate heav- ier return tails. In the frozen counterfactual analysis, reducing leverage amplification by one within-sample standard deviation lowers the predicted crash rate from 0.176 to 0.136, whereas reducing directional pressure yields only a marginal change. These find- ings suggest that empirically anchored LLM elicitation can improve simulator fidelity and help identify plausible crash-amplification mechanisms in a controlled synthetic environment

Legged robot control is not yet practical in non-inertial environments, where the ground and surroundings accelerate in the inertial Earth frame, mainly because of the unknown, time-varying, persistent perturbations caused by environmental motion, compounded by the associated hybrid robotic dynamics. We formulate the problem of robot walking control as unknown-disturbance rejection for hybrid linear systems subject to hybrid sinusoidal disturbances (i.e., a linear combination of sinusoidal components with instantaneous phase jumps at discrete events). First, necessary and sufficient conditions for solvability of the disturbance rejection problem are derived and used to characterize the control input that rejects the disturbances. Second, with the Youla-Kucera parametrization, an output feedback controller capable of rejecting the disturbance is introduced. Third, an adaptive law is proposed to estimate the controller parameters that reject the disturbance in the practical case where the disturbance and its parameters are unknown. Experiments on bipedal walking during ground motion demonstrate substantially improved body-position tracking accuracy, with an average improvement of over 60% in simulations and up to 47% in real-world experiments.

Reliable out-of-distribution generalization remains a central challenge in machine learning, particularly when models are deployed in data-scarce high-fidelity domains. Existing approaches often rely on statistical distribution alignment, or invariance assumptions that can fail when domain shifts reflect changes in underlying mechanisms. Inspired by a longstanding paradigm in physics that understands complex systems by starting from tractable idealizations and progressively incorporating additional mechanisms and levels of fidelity, we propose physics-transfer (PT) learning, a physics-constrained learning theory for zero-shot generalization along a ladder of complexity. PT constructs chains of anchor domains (CADs) that connect low-cost source domains to complex targets and introduces computable feasibility and reliability indicators for assessing transfer before target-domain labels are observed. Under physical-consistency and stable-representation assumptions, we derive an operational upper bound on target risk in terms of source empirical risk, the PT indicators, and finite-sample errors, and establish convergence as adjacent complexity gaps shrink and data coverage improves. We evaluate PT on excited-state energy prediction in quantum systems and dynamical-transition prediction in classical systems using neural networks trained without target-domain labels. Across substantial distribution shifts, PT yields accurate high-fidelity predictions, while the indicators identify incorrect physical mappings before target-domain prediction fails. These results provide a model-agnostic framework for designing, diagnosing, and controlling generalization in data-scarce learning problems.

We prove that the three-element additively idempotent semiring $S_7$ is strongly nonfinitely based: every finite additively idempotent semiring whose generated variety contains $S_7$ has no finite equational basis. This answers Problem~1.2 in Yue, Ren and Gao's paper Explicit equational bases for the power semirings of $S_7$ affirmatively. The proof combines a rigidity property of set systems with private points and a probabilistic construction derived from semilinear sets of letter counts in finite semigroups. The argument allows arbitrary polynomial substitutions and does not assume commutative multiplication. As consequences, both power semirings of the multiplicative reduct of $S_7$, and both power semirings of every finite group of order at least three, are strongly nonfinitely based. All algebras are considered in the signature $(+,\cdot)$ without nullary operations.

This paper reports an improved numerical lower bound for packing 203 equal spheres in a unit cube. Let r denote the maximum common radius of equal non-overlapping spheres that can be packed in a unit cube. For N = 203, the configuration currently listed in Packomania has a feasible radius of approximately r = 0.088145402159. Starting from this public configuration, we perform further numerical optimization and obtain a new feasible configuration with r > 0.088145634693. The final configuration is checked by an independent verification program using 100-digit arithmetic. All 20503 sphere-pair constrains and all 1218 sphere-wall constraints are satisfied. The coordinates of the new configuration and the related programs are provided as supplementary material.

We study generating sets for the kernels of the Frobenius polarization operators associated with elements of finite groups. By applying the discrete Fourier transform on each coset, we describe these kernels in terms of additive semigroups defined by zero-sum relations among roots of unity. We show that the monomials corresponding to the minimal nonzero elements of these semigroups form generating sets of minimum cardinality. We further describe the minimal nonzero elements in the general case in terms of those in the cyclic case and obtain a formula for the minimum number of generators. For the operator associated with a generator of a cyclic group, we also show, as a consequence of known results on integer zero-sum relations and circulant determinant representations, that its kernel and the subalgebra generated by these circulant determinants have the same field of fractions.

We show that the six-element additively idempotent semiring $B_2^1$ is inherently nonfinitely based. More precisely, for every positive integer $n$, we construct an infinite additively idempotent semiring, generated by at most $(6n+8)^2$ elements, that satisfies every $n$-variable identity of $B_2^1$.

Let $X$ be an algebraic K3 surface defined over a global field $K$ of characteristic $0$ or $p \geq 3$, assumed ordinary in the case where $K$ is a global function field. We provide an exact formula for the density of primes $p$ of $K$ at which the geometric Neron-Severi group of the reduction $X_p$ has a given rank in terms of the Galois representation attached to $X$. As an extension, we prove that there are infinitely many reductions of $X$ with an element of the geometric Neron-Severi group representing a fixed square class under the intersection pairing and prove results about the density of such places. In particular, we show that if $X$ corresponds to a sufficiently generic moduli point, a density zero set of places of $K$ satisfy this condition.

Upper-tropospheric water vapor and cloud ice associated with tropical deep convection are central to water-vapor and high-cloud feedbacks. We investigate a warm-pool transition in tropical convective moistening using 20 years of Aura Microwave Limb Sounder (MLS) Version 5 H2O and cloud-ice observations (August 2004-December 2024), together with Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) aboard the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) mission (June 2006-June 2023). The combined day-plus-night CALIPSO product (CALIPSO-A) provides an independent ice-cloud constraint; the H2O column shown on the CALIPSO grid is derived from Modern-Era Retrospective Analysis for Research and Applications, Version 2 (MERRA-2), not from CALIOP. We combine objective sea surface temperature (SST) breakpoint fitting, cross-sensor ice water path (IWP) comparisons, Clouds and the Earth's Radiant Energy System (CERES) longwave, shortwave, and net cloud radiative effect (CRE), and Integrated Multi-satellitE Retrievals for GPM (IMERG) precipitation diagnostics from the Global Precipitation Measurement (GPM) mission. MLS and CALIPSO-A show cloud-ice maxima over deep-convective warm-pool regions and identify warm-regime SST breakpoints near 299-300 K. Normalized MLS and CALIPSO-A IWP vary coherently in the western Pacific and tropical mean. CERES indicates that stronger longwave CRE in the warm regime is partly offset by stronger shortwave cooling, while GPM IMERG confirms that MLS IWP increases with convective rain rate. These results show that convective moistening and cloud-ice production across the tropical warm-pool transition are persistent cross-sensor features of the tropical atmosphere.

Distributed mixture-of-experts (MoE) inference is a promising architecture for deploying large language models (LLMs) at wireless edge networks because sparse experts can be placed across coordinated base stations (BSs), while the anchor node and user equipment (UE) can offload LLM inference tasks to BSs. The communication bottleneck is the MoE aggregation, where the anchor BS must recover a weighted sum of selected expert outputs before each decoding step. Over-the-air computation (AirComp) is well matched to this operation because the wireless multiple-access channel naturally superposes simultaneous transmissions. However, conventional AirComp minimizes communication distortion, whereas MoE aggregation errors have unequal impact on LLM outputs. We propose a logit-aware MIMO AirComp framework that estimates local logit sensitivity as block-level weights and jointly optimizes receive combiners and BS precoders under per-BS power constraints. We also develop an alternating algorithm that protects decoding decisions from aggregation perturbations. Using OpenCompass, we evaluate Qwen3-30B-A3B-Instruct-2507-FP8 on GSM8K and ARC-Challenge. At 30 dB aggregation SNR, perturbed Qwen3 retains 99.3% of clean GSM8K accuracy and 94.8% of clean ARC-Challenge accuracy. In a direct ARC-Challenge closed-loop audit over 295 examples, SW-AirComp achieves 43.73% accuracy at 30 dB and 26.78% at 20 dB, outperforming unweighted, matched-filter, and zero-forcing AirComp. In the wireless simulator, SW-AirComp reduces decision-relevant aggregation distortion; at 20 dB, its final weighted-sum mean-squared error is 53.5% lower than unweighted AirComp under the same channels, samples, and sensitivity weights. Logit-RMSE and logit-gap diagnostics further show that the gain comes from reducing output-logit perturbation and lowering the risk of top-token changes.

Let $q$ be a power of an odd prime $p$. We prove the MDS conjecture over $\mathbb F_q$ in every dimension $k$ satisfying \[ 2\leqslant k\leqslant B(p,q) \quad\text{or}\quad q+2-B(p,q)\leqslant k\leqslant q, \qquad B(p,q)=\left\lfloor\frac{(p-2)q+6p-10}{2p-3}\right\rfloor. \] For fixed $p$, the first interval is linear in $q$; together with duality, the theorem covers an asymptotic proportion $1-1/(2p-3)$ of all dimensions. The proof rests on a vanishing theorem for determinant relations over an arbitrary field of characteristic $p>0$. It yields full row rank for Chowdhury's matrices over a larger range of arc sizes. In particular, at $|G|=2k-3+n$ it proves Chowdhury's full-row-rank conjecture without the $q$-dependent restriction. Specialization to $\mathbb F_q$, together with the Ball–Lavrauw construction, gives the stated MDS range. We also prove that, for every odd prime power $q$, every normal rational curve in $\mathrm{PG}(N,q)$ is complete for $2\leqslant N\leqslant q-2$, and every projective Reed–Solomon code of length $q+1$ and dimension $2\leqslant k\leqslant q-2$ has covering radius $q-k$.

The SWOT (Surface Water Ocean Topography) mission is currently providing unpreceded high-resolution measurements of Sea Surface Height (SSH), revealing ocean features at finer scales. Nevertheless, the two-dimensional observations of KaRIn altimeter of SWOT suffer from instrumental errors. This noise degradation is altering the high frequencies of SWOT signal, and the small to sub-mesoscale dynamics of interest for some oceanographers. For this reason, Tréboutte et al. (2023) have developed a convolutional neural network (CNN) based on U-Net architecture to separate the noise from the physical signals contained in the SSH. Their approach has demonstrated great potential on simulated SWOT measurements, and Dibarboure et al. (2024) report a positive influence on actual flight data from SWOT. However, degraded denoising performance, and occasional negative side-effects have been observed in atypical conditions (e.g. very high surface waves, internal tides solitons). In this study, we illustrate some of these limitations and we present an improved approach of the CNN-based denoising: we modified the training procedure to obtain a more robust version of the algorithm, to avoid biases and artifacts in the denoised SSH. This paper also presents a more complete validation process with a robust and standardized evaluation benchmark: these metrics could be of interest to assess other SWOT filtering and denoising algorithms.

We give a proof that the full group $C^*$-algebra of a countable discrete amenable group is strong NF. The representation-theoretic construction uses the exact sequence whose kernel is the hyper-FC radical and whose quotient is the maximal ICC quotient. Extreme invariant traces on the radical induce factorial representations of the group. A Choquet decomposition of the regular radical trace selects a separating family whose induced traces are supported on the ordinary FC-centre. Exact twisted crossed-product quotients and a sequential UCT argument make their images quasidiagonal. Faithful irreducible replacements then supply the projections required for inner quasidiagonality. We also give a direct trace-decomposition variant and identify the steps shared by the two proofs.

We construct a separable unital residually finite-dimensional $C^*$-algebra $A$ whose maximal tensor square contains a proper isometry. In particular, $A$ is quasidiagonal whereas $A\tmax A$ is not. The algebra is generated by finite regular representations obtained from congruence quotients of an explicit countable affine group. A property~$(T)$ subgroup determines a diagonal Kazhdan projection in the maximal tensor square. Conjugation by a dilation makes this projection equivalent to a proper subprojection. Strictness is detected by a quasi-regular representation whose two factor restrictions descend to the specified algebra $A$. We give the norm estimate needed for this descent and construct the proper isometry explicitly.

We characterize the fresh-gradient oracle complexity of smooth nonconvex-strongly-concave minimax optimization, with matching upper and lower bounds up to logarithmic factors. Let $\Phi(x)=\max_y f(x,y)$, where $f$ is jointly $L$-smooth and $\mu$-strongly concave in $y$ on unconstrained Euclidean domains, and set $\kappa=L/\mu$. Each query returns a fresh unbiased joint gradient with conditional variance at most $\sigma^2$. Given initial primal gap at most $\Delta$ and dual residual $\|\nabla_y f(x_0,y_0)\|\le G$, the fixed-budget complexity of finding $\|\nabla\Phi(\widehat x)\|\le\epsilon$ with probability at least $2/3$ is $\widetilde{\Theta}(\sqrt{\kappa}L\Delta/\epsilon^2+\kappa L\Delta\sigma^2/\epsilon^4+\kappa^2\sigma^2/\epsilon^2+\sqrt{\kappa}\log_+(G/(\epsilon\sqrt{\kappa})))$, where $\log_+u=\log\max\{1,u\}$. This characterization holds when $\kappa$ and $L\Delta/\epsilon^2$ exceed universal constants, uniformly over finite dimensions, and the suppressed logarithms are independent of $G$. It establishes the necessity of linear condition-number dependence in the global stochastic cost and identifies a separate statistical refinement cost. A proximal method separates coarse primal progress from one final refinement, while the lower bounds apply to arbitrary adaptive randomized algorithms. Dual initialization contributes only an additive logarithmic cost, yet removing its control eliminates every finite dimension-free complexity bound, even with exact gradients.

We address Problems~2.2(i), 2.2(ii), and~2.4 in Dinh–Sibony's list. For a saturated holomorphic curve foliation on a compact Kähler manifold with finitely many singularities, every positive directed $dd^c$-closed current has finite Poincaré mass. Thus linearizability is not needed for the geometric ergodic theorem in this setting. We construct a projective surface with dense hyperbolic leaves and a diffuse extremal current whose Nevanlinna averages converge at a uniform reciprocal logarithmic rate. This is an existence result for a chosen foliation, not a rate theorem for an arbitrary prescribed foliation. Under the hypotheses of the first result, every hyperbolic leaf has a logarithmic lower bound for its Nevanlinna characteristic, even if no diffuse directed current exists. Positive-dimensional singular sets are not treated.

We prove a capacity criterion for fixed-degree spatially coupled MacKay–Neal codes on additive symmetric Markov noise. The noise bit flips with probability $p$, so the channel has one parameter and capacity $1-h_2(p)$. For every integer $\ell\geq4$ with $3/\ell<1-h_2(p)$, matched finite-window BCJR and sum-product decoding admit code sequences, possibly depending on $p$, with actual rate tending to $3/\ell$ and vanishing average information-bit error. The proof parallels the GEC-to-BEC argument: a conditional-entropy correction transfers BMS fixed-point positivity to the channel with memory. The correction also covers a capacity deficit of the effective BMS channel. A two-state posterior contraction supplies the additional derivative bound needed for threshold saturation. The result uses the cited BMS positivity theorem, with an analytic proof for $\ell\geq33$ and exact interval certificates for $4\leq\ell\leq32$.

We study the thermal Higgsino dark matter in the Minimal Supersymmetric Standard Model (MSSM) with Generalized Minimal Supergravity (GmSUGRA) boundary conditions after the high-recoil event reported by the LUX–ZEPLIN experiment and the subsequent solar-neutrino limits inferred from IceCube data. The Higgs sector is implemented in a NUHM2-type parametrization with $\mu$ and $m_A$ as inputs, while the supersymmetry breaking soft terms obey the non-universal GmSUGRA relations. The numerical scan contains a Planck-compatible Higgsino branch near the thermal mass, with the neutral-state splitting extending continuously from the few-hundred-keV region to the MeV scale. The part of this branch with $\dmchi\simeq0.3$–$0.4$ MeV overlaps the kinematic interval relevant to the LZ event, but is disfavored by the current solar-capture reinterpretations for a full-density thermal Higgsino. The same high-scale branch persists at larger splittings, including representative spectra with $\dmchi\simeq0.85$, $1.10$, and $1.34$ MeV, beyond the quoted solar-capture limits. Our results demonstrate that the mass and sub-MeV splitting favored by the LZ high-recoil interpretation can be realized dynamically within a GUT-motivated supersymmetric construction, without imposing the neutralino compression as a weak-scale input. The combined data therefore select different parts of a common GmSUGRA Higgsino branch rather than excluding the thermal Higgsino construction itself.

We study the cohomology of iterated Kummer extensions on the moduli space of five-pointed rational curves, continuing [arXiv:2609.19341, arXiv:2609.21356]. A two-dimensional shrinking construction compares the Betti presentations obtained from its five forgetful projections. The corresponding Hodge defects give pentagram relations, quadratic relations between coefficients of the Drinfeld associator, which are equivalent to the pentagon equation under group-likeness, normalization, and reversal. Under reversal and vanishing of pure-power coefficients, the relations imply shuffle and regularized double shuffle, whose Hodge realization is recovered by a convolution specialization.