Papers

Topological Deep Learning extends graph-based learning to higher-order domains, such as hypergraphs, cellular, and simplicial complexes. These domains are typically constructed from patterns in an input graph through a process of graph lifting. Full-domain training constructs and stores the complete lifted representation before model execution. On large and dense datasets like Reddit (233k nodes and 57.3M edges), this global materialization becomes a severe computational bottleneck, often rendering training infeasible. To address this limitation, we introduce Cluster-TNN, a domain-agnostic framework that avoids this bottleneck by lifting locally instead. After partitioning the input graph during preprocessing, at runtime Cluster-TNN dynamically samples groups of node clusters, reconstructs their induced subgraphs to form mini-batches, and applies the chosen lifting within each mini-batch. Retaining all edges among sampled nodes preserves the connectivity needed to construct higher-order structures across clusters, producing topological mini-batches that existing Topological Neural Networks can process directly. Across 21 matched comparisons with full-graph execution, Cluster-TNN reduces peak GPU memory in every configuration, by 83.2% on average while maintaining competitive predictive performance. Notably, such a reduction enables, to our knowledge, the first training of multiple different higher-order Topological Neural Networks on large datasets such as Reddit and OGBN Products. These results establish Cluster-TNN as a general strategy for scaling Topological Deep Learning beyond the limitations of global domain construction.

Let $f$ be a postcritically finite endomorphism of the complex projective space $\mathbb{P}^N$ of degree at least two. We prove that every Fatou component of $f$ is preperiodic to the basin of a superattracting periodic orbit.

Social and service robots in public spaces need to anticipate which nearby person is about to approach and touch them, so that a response can be prepared before contact. It is largely unknown which cues support this anticipation when a model trained with one robot is used on another robot at a different site. We study this question with a fixed-reference pose residual (FRPR) model: a geometry predictor built from the person's bounding box and mask is trained and frozen, and a temporal network then learns from body pose an additive correction to its logit, so that every prediction splits exactly into a geometry term and a pose term. Between two public egocentric datasets recorded by different robots, HUI360 and SSUP-A, with every choice made on source data, the pose correction raised average precision (AP) from 0.277 to 0.321 from SSUP-A to HUI360 and gave no measurable gain in the opposite direction; the same asymmetry held over a stronger, source-selected geometry reference. Freezing gave no AP advantage over joint training, and simple geometric baselines and tree ensembles remained competitive or better, so the construction serves measurement rather than prediction accuracy. A head-orientation residual added small gains in both directions. Post hoc, whether a person faces the camera kept its discriminative direction across datasets, whereas head pitch reversed. With thresholds chosen on source data, the neural models that use geometry detected at most 17% of target interactions. Code and processed data are available at https://github.com/WeiZhou96/FRPR-interaction-anticipation.

Isolation Forest (IF) is a widely used baseline for unsupervised anomaly detection. Recent studies provide a closed-form expression for the infinite-forest limit for one-dimensional data. Inspired by the geometric interpretation of this formula, we introduce RIFT (Relative Isolation From Trees), a deterministic anomaly detection method that generates the minimum spanning tree and scores each point by the sum of the apparent sizes of tree edges as viewed from that point. For one-dimensional data, the RIFT score recovers the closed-form IF limit exactly. In higher dimensions, it provides a parameter-free generalization that is deterministic, robust to varying density and clustered anomalies and avoids the axis-parallel artifacts of IF. We further propose an ensemble variant for large datasets. Experiments on synthetic data and the ADBench benchmark demonstrate that the accuracy is comparable to IF, while the ensemble variant exhibits significantly lower variance across random seeds.

Dense retrieval usually ranks text chunks by their semantic similarity to a question. This ignores structure that many data systems already store, including section membership, session boundaries, and native order. We propose NativeScope, a scope-then-rank method for queries with a known anchor and relation. It represents a query as q -> (A, r, B). The anchor A and relation r select native units through belonging, before, or after operators, and the target term B ranks only chunks that overlap the selected scope. An internal variant, NS-FullQ, ranks the same candidates with the full question. We evaluate both methods on 200 controlled document and memory records derived from QASPER and LongMemEval under a 1,024-token budget. NativeScope attains native-unit recall of 89.28 percent for documents and 72.50 percent for memories, improving over instance-wide Dense RAG by 42.75 and 22.00 percentage points. NS-FullQ reaches 87.78 percent and 68.50 percent; its differences from NativeScope are inconclusive, locating the primary gain in relational scoping rather than the shorter ranking query. With automatic Top-1 anchors, memory recall falls to 35.50 percent. NativeScope is therefore effective when anchor coordinates and native relations are reliable, but hard scoping inherits errors from the localization interface.

Large language model (LLM) routing distributes inference work across different models, advancing the cost-quality Pareto frontier of LLM serving. While coarse-grained routing at the session or query level has been widely adopted in production systems, recent algorithmic work shows that fine-grained token-level routing can yield substantial efficiency and quality gains. However, efficiently serving token-level routed inference poses significant challenges to existing systems. Built on single-LLM assumptions, current systems suffer from severe step desynchronization and frequent batch admission delays under token-level routing, and they also impose high implementation complexity on developers. To address these challenges, we design TokenRouter, an efficient and developer-friendly serving system for token-level routed LLM inference. TokenRouter follows the principle of request-centric programming, model-centric execution: developers describe routing logic from the perspective of a single request, while the runtime launches a subserver for each LLM and dispatches requests asynchronously. Each subserver employs a delayed-batching scheduler, whose optimal hyperparameters are derived from a mathematical throughput model of the system. Across diverse routing algorithms, workloads, and model pairs, TokenRouter achieves 2.01-64.15x higher decoding throughput than existing systems, substantially advancing the serving efficiency of token-level LLM routing. Our code is available at https://github.com/thu-nics/TokenRouter.

Microscopic robots require accurate task geometry despite changes in language, parts, and focus. We present a semantic-to-physical framework that maps instructions to constrained geometric operators, reuses frozen open-vocabulary perception, and integrates locally reliable focal-plane trajectories by confidence weighting and dynamic programming. Calibrated multi-view geometry connects 2-D paths to physical execution. Prompt, unseen-part, and geometry reconfiguration tests yield 6.30-6.59-pixel RMSE. Relative to part-specific U-Net training with 20-100 labels, the proposed zero-new-label configuration takes 15 rather than 72-165 min. Across nine part-illumination conditions, trajectory-space integration reduces RMSE from 14.41 to 6.28 pixels (56.4%) and P95 error from 20.07 to 8.13 pixels (59.5%) compared with image-first multi-focus fusion. An ablation isolates the roles of confidence and path-wise selection. In representative robot experiments, target-region coverage improves from 83.5% to 92.9%. Dispensing provides a measurable physical trace, not a task-specific limitation of the method.

Time-series foundation models (TSFMs) have achieved strong forecasting performance across domains. However, most adaptation methods remain static. Existing all-in-one methods learn a single set of dataset-level parameter updates and apply the same adapted model to every input. As a result, they cannot adapt the model parameters to the temporal patterns, seasonality and dynamics of each input time series. This limits their ability to produce forecasts that are tailored to heterogeneous inputs. To address this limitation, we propose AdaCast, a conditional parameter generation framework for time-series forecasting. AdaCast uses a generator to produce input-specific low-rank parameter updates for a frozen pretrained TSFM. These updates adapt the model to each input during both training and inference. Across six public benchmarks, AdaCast consistently outperforms static adaptation baseline in in-domain forecasting and improves zero-shot generalization to held-out datasets across domains. These results demonstrate that conditional parameter generation provides an effective approach for adaptive forecasting.

We identify explicit closed-form and variational formulae for interpolating the exact values and derivatives through order $s$ of a $C^{s,1}$ function $f:\mathbb{R}^d\to\mathbb{R}$ at $N$ distinct points in $[0,1]^d$. The reconstruction satisfies bounds on its global $C^{s,1}$ seminorm and its Lipschitz constant on $[0,1]^d$ that are independent of the sample size $N$, the latter being the sharp Whitney condition. For $s\ge2$, the reconstruction is real analytic away from the data points and definable in the o-minimal structure $\mathbb{R}_{\exp}$. Our nonlinear extension formula coincides with the formula of McShane (1934) for $s=0$ and with the formulae of Le Gruyer and Phan (2015) and Azagra, Le Gruyer, and Mudarra (2018) for $s=1$, and is new for $s\ge2$. For fixed $d$ and $s\ge2$, our closed-form formula is computable away from the data points by a circuit using only elementary unary and binary real operations, with $\mathcal{O}(N)$ gates and $\mathcal{O}(\log N)$ depth. This yields $\mathcal{O}(N)$ storage and $\mathcal{O}(\log N)$ parallel evaluation time. From the supplied coefficients and weights, the circuit can be compiled with $\mathcal{O}(N)$ one-time work in the exact-real word-RAM model. Thus, in this supplied-data setting, our nonlinear construction improves by a logarithmic factor on the $\mathcal{O}(N\log N)$ initialization bound of Fefferman and Klartag (2009), while retaining linear storage and attaining logarithmic query time through parallel evaluation.

We construct smooth closed embedded minimal surfaces with everywhere negative Gaussian curvature in the unit round four-sphere. More precisely, for every sufficiently large integer $m$, our construction gives a minimal surface $\Sigma_m$ of genus $m^2+1$ such that $\operatorname{Area}(\Sigma_m)\to 4\pi^2$ as $m\to+\infty$. This result yields a complete solution to Problem 101 in Yau's 1982 list of open problems.

We establish a sufficient condition for infinite log-concavity and apply it to five families of combinatorial sequences. We give a new proof of infinite log-concavity for the Boros–Moll coefficient sequences $(d_k(n))_{k=0}^n$ for fixed $n$. For the transposed sequences $(d_\ell(\ell+k))_{k\ge0}$, we prove Zhao's conjecture on infinite log-concavity for every fixed integer $\ell\ge3$. For the normalized sequences in Euler's difference table, we confirm the infinite log-concavity conjecture of Chen, Gu, Ma and Wang under essential iteration, in which both endpoints are discarded after each step. We also confirm a conjecture of Medina, Moll and Rowland on the infinite log-concavity of the coefficient sequences of polynomials arising from iterated primitives of $\log(1+x)$. Finally, we answer a question of Brändén and Chasse by showing that, for positive integers $d$, the sequence $(k^d)_{k\ge0}$ is infinitely log-concave if and only if $d\ne2$. For every real $d\ge3$, all its iterates are positive at every index $k\ge1$. The proofs combine holomorphic estimates with finite computer-assisted verification.

RNA function arises from the coupling of nucleotide sequence and three-dimensional structure, motivating their joint design. Coordinating global folding with nucleotide-level detail remains challenging under limited structural supervision. We introduce La-Ribo, a generative framework for RNA sequence-structure co-design via geometry-latent flow matching. La-Ribo retains a sparse phosphate-sugar–base scaffold and encodes nucleotide identity and local conformation in residue-wise latents. A shared flow network generates both jointly, and an RNA-specific decoder then reconstructs all heavy atoms. To expand supervision, we construct a quality-controlled corpus of 168,561 RNA structures, integrating experimental data with predictions from three folding models, including 10,631 MSA-supported structures generated in this work. La-Ribo improves designability and codesignability over the evaluated baselines across sampling budgets and two refolding models, and the same prior supports scaffold-conditioned inverse folding without additional training.

AI research agents automate the cycle of proposing, implementing, and evaluating experiments, opening a path toward recursive self-improvement. Yet their ability to propose experiments outpaces their capacity to execute them in real environments, making outcome prediction a key capability for sustained self-improvement under limited experimental budgets. We investigate language models as Research World Models (RWMs), which predict the outcomes of candidate interventions across research environments. Our evaluation draws on over 2,600 experimental records from nine research environments spanning pretraining, post-training, and inference, representing more than 171,000 H100 GPU-hours of experimentation. Research knowledge acquired from real experimental experience improves RWM predictions of unseen interventions within the same environment (Spearman +0.27), and can be reused across environments. For example, using only pretraining experience from OLMo3, Marin, and Nanochat, an RWM reduces selection regret in the Qwen3 environment by 78% compared with zero-experience setting. These benefits extend to multi-round Autoresearch under a fixed selection budget: RWMs with in-env and cross-env research knowledge increase the best gain achieved by 15.8% and 11.6%, respectively. Ablations across 13 language models used as RWMs show that adding research knowledge can improve intervention ranking more than changing models or increasing reasoning effort alone. These findings support language models as RWMs and motivate accumulating experimental data for future RWM training.

Recently, a uniform $\frac{7}{8}$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and its transfer to Dirichlet $L$-functions was given by OpenAI in [Ope26a]. In its proof, there are several parameters $b=\frac{1}{8}, \ell=\frac{1}{6}$, and $\kappa$. In this paper, via varying $b$, $\ell$, and $\kappa$, following the same strategy, we slightly improve the bound $\frac{7}{8}=0.875$ to \[ B_{\mathrm{new}}=\frac{18\xi_{\mathrm{new}}+7}{18\xi_{\mathrm{new}}+9} =0.874957019420098946\ldots, \] where $\xi_{\mathrm{new}}$ is the unique root in $(0.38858,0.38859)$ of $1188\xi^3-2160\xi^2+171\xi+190=0$. Explicitly, with $\arccos\in[0,\pi]$, \[ \xi_{\mathrm{new}}=\frac{20}{33}+\frac{\sqrt{1391}}{33} \cos\!\left(\frac13\arccos\frac{28470}{1391^{3/2}} -\frac{2\pi}{3}\right). \] The limiting parameters are \[ \begin{gathered} b_{\mathrm{new}}=\frac{8(3\xi_{\mathrm{new}}-1)(12\xi_{\mathrm{new}}^2-40\xi_{\mathrm{new}}+15)} {9(2\xi_{\mathrm{new}}+1)(24\xi_{\mathrm{new}}^2-20\xi_{\mathrm{new}}+5)}, \ell_{\mathrm{new}}=\frac{5-6\xi_{\mathrm{new}}}{9+18\xi_{\mathrm{new}}},\qquad \kappa_{\mathrm{new}}=\frac{18\xi_{\mathrm{new}}+5}{18\xi_{\mathrm{new}}+9} =2B_{\mathrm{new}}-1. \end{gathered} \] We also prove its optimality within the explicitly specified exponent comparison system. This result has been formalized by Lean [Liu26].

Recursive self-improvement, the participation of AI systems in improving their own capabilities, is beginning to move from theoretical prospect to practice, posing both challenges and opportunities for safety alignment. Models evolve through frequent updates, and their safety alignment requires continual adaptation to each new checkpoint. Meanwhile, with evolving red-teaming methods exposing new vulnerabilities, safety improvement for each checkpoint needs to mitigate exposed vulnerabilities and generalize to risks not yet revealed. Following R$^2$AI, we term these goals resistance to known threats and resilience to unforeseen risks. Recursive self-improvement, in turn, inspires an approach to both goals: safety alignment could likewise advance through successive rounds of evaluation and update. We therefore introduce ReSI, a recursive safety improvement framework that implements this approach through automated research. In each round, ReSI applies diverse red-teaming methods to identify vulnerabilities in the current target model, develops training recipes, and promotes the update with the largest safety gain among those passing a Pareto gate on capability retention as the next target model. Across four dense and mixture-of-experts models, ReSI matches or exceeds evaluated frontier models on in-distribution and out-of-distribution safety benchmarks, and outperforms alignment baselines on nearly all safety evaluations while largely preserving general capabilities. In particular, ReSI reduces the mean X-Teaming attack success rate across the four models from 86.01% to 31.45%, well below GPT-5.6-Luna's leading frontier result of 56.69%, indicating stronger resilience to attacks unseen during training. These findings support recursive safety improvement as a practical path toward resistant and resilient AI.

Test-time adaptation (TTA) methods for time-series forecasting update a deployed model, or a small adapter around it, from incoming ground truth. But the label of an $H$-step forecast exists only $H$ steps later, and real data pipelines add further delay. We build a leakage-free harness in which the label of forecast origin $s$ is released for updates only at step $s+d$ with $d \ge H$, and enforce this rule inside the released code of four recent TTA methods (TAFAS, COSA, PETSA and DynaTTA), run on their own backbones and checkpoints across five benchmarks (ETTm1, ETTh2, Weather, Electricity and Traffic). As references we add two closed-form correctors: a bank of recursive least squares (RLS) filters combined by a per-coordinate median, with no tunable hyperparameters and 56 microseconds per step on the 7-channel streams, and an ELF-style linear corrector. Under causal delayed labels the picture is asymmetric. On ETTm1 every audited method genuinely adapts, yet the RLS bank still beats three of the four at a fraction of their cost; only DynaTTA beats the bank, only at the minimum causal delay, and at roughly 2,500 times the per-update cost; the ELF-style corrector beats all four. On the other four datasets, the largest statistically significant improvement any published method achieves over its own frozen checkpoint is half a percent, on all four at least one published method is significantly worse than the frozen model at the minimum causal delay, and on drift-heavy ETTh2 longer label delays make every adapter that separates from the frozen model, ours included, significantly harmful. Leaky next-step updates inflate the apparent gains of simple adapters by up to 110%, and the backbone training recipe moves frozen online error by up to a factor of 25, more than any adaptation effect we measure. We release the harness, integration patches and all cached runs.

Learning from demonstration has enabled impressive robot behaviors. A common choice for policy learning is to use diffusion or flow matching (Flow-Policies), which often outperforms direct action regression trained with mean squared error (MSE-Policies). This gap is commonly attributed to multimodal demonstrations. We revisit this gap from the perspective of statistical modeling: how action-prediction residuals shape policy optimization. Our analysis of real-world robot demonstration data reveals substantial state-dependent variation in residual scales and heavier-than-Gaussian tails. While both MSE-Policies and Flow-Policies exhibit heavy-tailed action residuals, their training gradients behave differently: MSE allocates more gradient magnitude to observations with large action residuals, which hurts optimization. Motivated by these findings, we introduce heteroscedastic Student-t action regression (HT-Policies), which learns input-dependent residual scales and reduces the influence of heavy tails. HT-Policies predict action chunks with a single feed-forward pass and can reuse pretrained flow-matching-based policy networks as the backbone. Across four simulation benchmarks and real-robot evaluations, HT-Policies achieves success rates competitive with generative policy baselines, both when trained from scratch and from pretrained vision-language-action and world-action models, despite being faster in training and inference. Together, these findings shed light on the practical advantages of generative objectives in robot learning from demonstrations and offer an efficient direct-regression alternative for a range of architectures and tasks. Project page: https://projects.thelabone.org/regression-policy/

Text prompting is an indirect interface for poster generation, requiring users to encode inherently two-dimensional composition intent into a one-dimensional sequence of words. We introduce a Spatial Canvas Interface that enables users to directly compose generation intent in space through four complementary binding types: semantic, identity, text, and pixel, together with Text Specifications for individual elements and global appearance. Based on this interface, we develop Compo, a poster generation model adapted from a pretrained image editing model to understand Spatial Canvas inputs and Text Specifications. Compo supports both direct inference, where users explicitly construct the canvas, and agentic mode, where a high-level request is automatically translated into a planned Spatial Canvas. To train Compo, we develop a scalable pipeline that automatically constructs supervision data for different binding types and their combinations, enabling efficient adaptation without training a specialized poster generator from scratch. We further introduce a benchmark that evaluates adherence to individual binding types and their joint composition. Experiments show that Compo achieves stronger compositional controllability than both general-purpose image generation models and dedicated poster generation systems while maintaining high visual quality. By decoupling intent specification from visual generation, our work shifts poster generation from prompting toward composing.

In text-guided image editing, describing the desired change is often straightforward, but identifying the intended object or region can be cumbersome, especially when several objects look alike. We introduce a new image editing interface that lets users place spatial marks and optional short notes directly on the image. Together, these annotations form a canvas instruction that specifies where to edit and what to change. Our editor, VibeEdit, follows these instructions to perform object addition, removal, replacement, attribute modification, and movement without a separate text prompt. We construct 1.55 million source-target edit pairs with object masks and structured edit descriptions, from which we render canvas instructions during training. We adapt Qwen-Image-Edit with layer-decoupled conditioning that separately encodes source images and canvas instructions for image editing. We train the model with region-weighted supervised fine-tuning, followed by rubric-guided reinforcement learning to improve edit completion, local edit quality, and preservation of unedited regions. We evaluate VibeEdit on an independently constructed, human-curated benchmark of 419 cases emphasizing target selection among similar objects. VibeEdit achieves a VLM rubric score of 79.9 and an outside-region PSNR of 32.8 dB, compared with 67.4 and 24.0 dB for FireRed, the highest-scoring text-instructed baseline in our evaluation.

Symmetric and unitary matrices are among the most important classes of structured matrices admitting efficient and stable eigenvalue algorithms. Notable examples include structure-preserving variants of the QR algorithm and their square-root-free counterparts. For unitary matrices, these algorithms were historically derived from recurrence relations for orthogonal polynomials on the unit circle. A matrix-based derivation using core chasing was later given in [3]; this family of algorithms is backward stable, but the underlying symmetries needed to cast out the square roots were not identified, and the question of structured backward stability was left open. In this paper we present a matrix-based derivation of the symmetric unitary core-chasing algorithm and use it to obtain a square-root-free variant. We give a matrix-based proof of backward stability and show that strict structured backward stability is impossible in general. Open-source Fortran code and numerical experiments demonstrate that the combination of exploiting the symmetry and removing the square roots reduces computation time without degrading accuracy.

Superconducting quantum circuits underpin large-scale quantum processors and are increasingly interfaced with optical systems, yet they are vulnerable to quasiparticle poisoning from high-energy radiation and optical fields. Gap engineering of Josephson junctions suppresses quasiparticle-induced relaxation, but quasiparticles near the junction can still shift the qubit frequency and cause phase errors. Here, we extend gap engineering beyond the junction using a flux-tunable transmon with a gold-on-niobium ground plane and island and strongly gap-engineered aluminum junctions designed to keep quasiparticles away from the junction leads. Compared with an all-aluminum transmon with nominally identical junctions, we demonstrate that the niobium device withstands more than an order of magnitude higher near-infrared optical power before excess relaxation appears. The Nb transmon shows no resolvable quasiparticle-induced frequency shift over the same power range. At the same time, we observe that dephasing and state-preparation-and-measurement infidelity increase with optical power even while relaxation remains protected. These results suggest that engineering the quasiparticle energy landscape beyond the junction can substantially increase qubit resilience, not only to radiation-induced error bursts, but also to nearby optical fields.

The paper considers a system of unidirectional first-order hyperbolic partial differential equations (PDEs) coupled with ordinary differential equations (ODEs), modeling two-phase advective transport processes subject to both boundary and distributed input delays. To address the system defined over two spatial domains, we design a set of three backstepping transformations involving both Volterra- and Fredholm-type integral operators, which introduce eight kernel functions: five associated with the PDE states and three with the ODE states. Since the distributed delayed variable spans the two spatial domains, its transformation must integrate over all state variables, leading to three-dimensional kernel functions with singular initial conditions involving Dirac delta functions that serve as sampling operators to establish mappings between variables of different dimensions. By employing the method of characteristics and successive approximations, we prove the well-posedness of the kernel equations. On this basis, we prove the invertibility of the Fredholm transformation and establish the exponential stability of the closed-loop system in the $L^{\infty}$ norm under the delay-compensated controller. Simulation results for a linear example and a nonlinear screw extrusion model are presented to demonstrate the effectiveness of the proposed method.

Non-Abelian gauge theories above four dimensions may admit an interacting ultraviolet fixed point through the competition between canonical scaling and gauge-field antiscreening. Using the four-loop $\overline{\mathrm{MS}}$ beta function continued to $D=4+2\epsilon$, we determine the largest dimension $D_{\mathrm{pt}}=4+2\epsilon_{\mathrm{pt}}$ below which successive loop corrections to the fixed point remain hierarchically ordered under a specified ratio criterion, and map $D_{\mathrm{pt}}$ over color and flavor numbers for fermions in the fundamental representation of $SU(N)$. At $D=5$, we construct rational anomalous dimensions with a fixed, pole-regularized denominator inspired by the functional renormalization group (fRG). Matching to the $\overline{\mathrm{MS}}$ series through one, two, and three loops produces progressively smaller shifts in the critical flavor number and coupling at the ultraviolet–infrared merger (the point where the infrared and ultraviolet fixed points coincide), yielding $N_f^{\mathrm{cr}}\simeq 2.09 \, N$ at large $N$ in the three-loop construction. This low-order stabilization, together with qualitative evidence from fRG flows including higher gauge operators, suggests a potential merger scenario that bounds the candidate asymptotically safe region in flavor number. The resulting color–flavor map provides guidance for five-dimensional model building by displaying both the baseline perturbative-control boundary and the candidate three-loop merger boundary. Four-loop matching, however, retains an interacting ultraviolet fixed point beyond the three-loop merger boundary, with no merger found for integer $2\le N\le15$ and $0\le N_f/N\le11/2$. The sensitivity of the fixed-point structure to higher-order terms leaves both scenarios open, highlighting fixed-point annihilation as a potential constraint on five-dimensional asymptotic safety.

We study normalized solutions to the 1-Laplace equation \[ -\mbox{div}\left(\frac{Du}{|Du|}\right)+\lambda \frac{u}{|u|}=f(u)+|u|^{1^*-2}u,\quad\mbox{in}~ \mathbb{R}^N \] subject to the constraint $\int_{\mathbb{R}^N}|u| \, \mbox{d}x=m$, where $N\geq2$, $1^*=\frac{N}{N-1}$ is the critical Sobolev exponent for the embedding from BV into $L^p$, $\lambda, m\in\mathbb{R}^+$ and $f:\mathbb{R}\to\mathbb{R}$ is a continuous nonlinear perturbation. We first establish a Pohozaev-type identity for the equation and use it to study the associated critical energy, and then prove the existence of normalized solutions via nonsmooth critical point theory. A key feature of our approach is the proof of the existence of a Palais-Smale sequence for a Lipschitz continuous functional under a Lipschitz continuous constraint. This is significantly different in methodology from previous problems with $C^2$ constraint.

A recent experiment [arXiv:2507.08795] has reported measurements of the interwire $X$-parity lifetime in a Majorana tetron device via the quantum capacitance of an auxiliary readout dot. Motivated by these measurements, we theoretically study the interwire parity lifetime in a minimal model of the coupled dot-tetron system subject to electrostatic noise. We show that coupling to the measurement dot can itself strongly modify the parity dynamics by generating an energy splitting that shifts noise-induced transitions to finite frequency and thereby stabilizes the measured states against low-frequency fluctuations. For representative noise strengths motivated by disordered nanowire devices, we find $X$-parity switching lifetimes of order microseconds in the measurement configuration even when the corresponding bare-tetron coherence time is only tens of nanoseconds long. We analyze both the dispersive regime, where the dot can be integrated out and the dynamics reduces to an effective two-level system, and the near-resonant regime, where all four low-energy states participate. In the latter case, rapid transitions organize the spectrum into two long-lived manifolds that approximately correspond to sectors of the combined dot-tetron $X$-loop parity, yielding effective two-state switching despite the underlying multilevel structure. Our results show that measurement-induced stabilization can substantially enhance the observed $X$-parity switching lifetime, such that the lifetime measured under continuous readout can greatly exceed the intrinsic idle $T_2^*$ coherence time of the isolated Majorana tetron qubit.

In 1991, Denef introduced an explicit formula for Igusa's local zeta functions in terms of embedded resolutions of singularities over $p$-adic fields with good reduction. In this paper, by slightly generalizing the $p$-adic McKay correspondence given by the second author in 2017, we extend Denef's result to the setting of $\mathbb{Q}$-embedded resolutions of singularities. Consequently, the strong monodromy conjecture with respect to Igusa's local zeta functions holds for weighted homogeneous polynomials with isolated singularities. Moreover, we use our formula to prove Igusa's conjecture for exponential sums in the case of non-degenerate weighted homogeneous polynomials with isolated singularities.

Fourier coefficients of Eisenstein series induced from Speh representations are the kernels of the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan. Ginzburg and Soudry integrated these coefficients against cusp forms to obtain Eisenstein series induced from shorter Speh representations and arbitrary cuspidal data. We determine the corresponding local modules over a non-archimedean local field $F$ of characteristic zero. Let $q_F$ be the cardinality of its residue field. Let $J_s$ be the twisted Jacquet module of the representation induced from $\Delta(\tau,m+i)|\det|^s$. It carries an action of $G\times H$, where $G$ and $H$ are split symplectic or special orthogonal groups. Let $\sigma$ be an admissible representation of $G$ of finite length. We prove that $(J_s\otimes\sigma)_G$ is induced from $\Delta(\tau,i)|\det|^s$ and a fixed conjugate of $\sigma$, outside a finite set of values of $q_F^{-s}$ depending on $\tau$ and $\sigma$. No genericity assumption on $\sigma$ is needed. For irreducible $\sigma$, this determines all irreducible quotients of $J_s$ of the form $\sigma^\vee\boxtimes\pi$. The proof computes every orbit contribution, including those that vanish globally by cuspidality. It also gives an explicit finite set containing the exceptional parameters. This set is empty for irreducible supercuspidal $\sigma$ unless $G$ is the split group $\mathrm{SO}_2$. We prove the analogous result for the symplectic double cover. Over finite fields of large characteristic, we give the complete decomposition of the twisted Jacquet module when $\tau$ is cuspidal on $\mathrm{GL}_n$ and $n$ is larger than the rank of $G$.

An optimistic rollup is safe as long as at least one honest validator executes its state transition function (STF) and disputes any assertion that is inconsistent with the result of the execution. Allowing anyone to participate, however, does not give incentive to do so: as long as every assertion is correct, verifying assertion correctness goes unpaid. Attention mechanisms address this by paying nodes to execute the rollup's STF, regardless of whether they participate in the validation process. Since a single node operator can back many registered identities with a single execution, paying per identity does not buy execution diversity, that is, the number of operators that independently execute. In this work we provide a new modeling framework for this problem. We formalize the underlying cryptographic primitive as a new notion, arguments with registered provers, whose properties include a form of non-amortizability that, to our knowledge, has not been studied for modern succinct cryptographic proofs before. We design validation and attention reward mechanisms and analyze them as two coupled Tullock-like contests, in which an operator pays once for the required execution and then chooses how many identities to deploy in each role. We quantify the attainable range of execution diversities in equilibrium as a function of the marginal cost of deploying an extra attention identity. We use our framework as a lens through which we compare two attention mechanisms, TRACE (MARBLE 2026) and Proof of Diligence (AFT 2024). In particular, we show that as opposed to Proof of Diligence, in TRACE the marginal cost mentioned above can be adjusted to support a higher execution diversity. As a case study, we apply our results to Arbitrum One, a widely deployed optimistic rollup.

Let $A$ be a unital, simple, separable, and pure $C^*$-algebra. We prove that, for every $a\in A_+$, the set of positive elements in $A$ having the same Cuntz class as $a$ is path-connected. This may be viewed as a generalization to positive elements of the classical homotopy theory of projection classes and extends a line of results on the topology of constant Cuntz classes that previously required stronger regularity hypotheses to the class of pure $C^*$-algebras, which notably contains all selfless algebras.

The second part of Hilbert's 16th problem is one of the most fundamental open problems in the qualitative theory of ordinary differential equations and dynamical systems. A central unresolved question is whether the Hilbert number $H(d)$ is finite for arbitrary polynomial degree $d$. (A very recent manuscript from OpenAI asserts the finiteness of $H(d)$; see the remark at the end of this paper.) In the absence of a general upper bound theory, most existing work constructs explicit perturbed polynomial systems to obtain improved lower bounds for $H(d)$, while rigorous upper bound estimates remain largely unavailable. This paper focuses on improving the quantitative lower bound for polynomial systems through a systematic Hierarchical Melnikov Realization framework. We construct perturbed dynamics based on an anisotropic Chebyshev Hamiltonian, where hierarchically selected perturbation coefficients generate simple zeros of the first-order Melnikov function simultaneously across multiple disjoint period annuli. Two distinct 3-adic filtrations supply the necessary logarithmic correction factors, and a Borel–Gauss analysis establishes the required local rank condition under sufficiently strong anisotropy. As the main result of this work, we prove that $H(d)=\Omega\bigl(d^2\ln^2 d\bigr)$, which improves the previously known lower bound by a factor of $\ln d$. The full construction of limit cycles and the resulting dynamical bounds are formally certified using Lean 4 formal verification.