Papers

Diffusion models have become a dominant paradigm for end-to-end autonomous driving, yet existing planners organize trajectory generation along the Gaussian-noise axis, treating driving utility only as an auxiliary guidance or selection signal. We instead formulate diffusion along a driving-quality axis. We propose QualiDrive, which constructs scene-specific quality ladders by ranking observed trajectories according to measured driving utility and learns to reverse this ordered degradation through ordered low-to-high-quality refinement. Each reverse step is conditioned on the scene, ego state, current trajectory, and target quality level, progressively refining trajectories toward higher driving quality. On NAVSIM, QualiDrive achieves 94.5 PDMS on v1 and a state-of-the-art 92.74 EPDMS on v2. QualiDrive improves Oracle EPDMS only modestly over endpoint-only quality supervision (98.52 to 98.57), but substantially increases the fractions of candidates above 0.95 and 0.98 EPDMS from 12.94%/9.71% to 21.53%/15.93%, raising selected EPDMS from 92.63 to 92.74. These results suggest that diffusion is not merely a process of denoising, but one of progressive quality-ordered refinement.

GPU acceleration has made motion planning one to two orders of magnitude faster, but it has not made the three paradigms—sampling, search, and optimization—comparable to one another: every GPU-accelerated planner is built around a single paradigm, so comparing across libraries also compares their collision checkers. All three, however, bottleneck on the same GPU primitive: batched collision detection and the batched forward kinematics beneath it. PrismPlan builds that primitive once, as a shared substrate of per-sphere collision–kinematics device code that every planner consumes. Because they all execute the same device code, the backend is held fixed by construction and the paradigms can be compared head-to-head. We run six planners over 1800 MotionBenchMaker problems on three robots (7–14 DoF, one a dual-arm whole-body composite) at an equal 2s budget. On the 7-DoF Panda, sampling (pRRTC) is the most reliable paradigm at 98%, while parallel search (wPA*SE) follows within four points at a 3-4 times$ lower median latency, and the local optimizers trail well behind when made to solve from scratch. On the 14-DoF dual-arm composite, sampling still solves about 90% of feasible problems, while no search or optimization planner clears a third of them. Our open-source code is available at https://github.com/XiaoyubotMotionTeam/PrismPlan.

Reinforcement learning with verifiable rewards improves mathematical reasoning when a final answer can be checked, but a correct answer does not certify the derivation behind it, and proof problems offer no answer to check. Replacing the check with a reasoning model as an in-loop verifier faces a fundamental obstacle: the policy must already produce occasional correct attempts for any reward to exist, which it may never do at olympiad level. We instead move verification and repair out of the training loop and into offline data generation with SPLICE, a workflow in which a second model, given only a reference solution, edits a reasoning model's traces as they are generated: it verifies progress, truncates a failed trace at its last verified step, and splices in an adaptively strengthened hint until progress resumes. The resulting dataset includes repaired examples whose sampled unaided trajectory failed, with failed continuations removed and progress repeatedly checked against the reference solution. A model finetuned on SPLICE traces outperforms the same model distilled from the reasoning model's unedited traces on AIME and, under a grading protocol validated against human grades, on olympiad proofs. The trained model's own traces show a corresponding behavioral signature: at difficult points it announces a branch in the style of the spliced hints and takes riskier steps, with predictive entropy rising there and just after, a pattern absent from the model trained on unedited traces.

Post-training large language models (LLMs) on two-player zero-sum games provides a verifiable reward, and the gains have been shown to transfer to general reasoning. Existing pipelines, however, rely on a handful of hand-written games and on advantage estimators tied to them, so both the supply of games and the advantage estimate on a large corpus limit scaling. We present GameSmith, which scales both. An LLM synthesises games inside templates whose reward, observation, and hidden-information code is fixed. Each game is verified by seed-paired replay and by an independent re-implementation of its specification, then expanded into encoding, rule, and size variants, giving 26 new games in 29 families and 336 verified variants from the three SPIRAL templates. The model is distilled from the self-play of a larger teacher (Qwen3-32B) on these games and then trained by self-play reinforcement learning with CRAB, an advantage estimator that credits each decision against a shrinkage estimate of the expected return of its position and is defined on every variant. On Qwen3-4B-Base, distillation raises the 13-benchmark mean pass@1 from 45.5 to 62.5, above the three SPIRAL games at the same budget, and self-play with CRAB reaches 64.9. The ordering holds across models, and transfer grows with the number of game families.

Flow matching learns how to transport noise toward data, but its mean prediction does not directly report the endpoint uncertainty remaining at an intermediate state. This paper introduces Posterior Flow Matching (PFM), a general generative modeling framework that learns conditional velocity means and componentwise variances with a Gaussian posterior model. The same model evaluation supplies both a transport prediction and endpoint uncertainty, making posterior diagnostics available throughout generation. These learned moments support a mean-driven ordinary differential equation (ODE) and a posterior-driven stochastic differential equation (SDE), whose local noise scales adapt to the predicted uncertainty. Across image, point-cloud, molecular, and action generation, PFM improves generation quality and provides endpoint-posterior readouts throughout generation.

Therapeutic sutures can localize bioactive payloads to wounds, but burst release and insufficient coating durability can compromise regeneration and increase infection risk. In this work, we introduce a data-driven discovery strategy for bioresorbable polymer suture coatings that improve therapeutic retention and release kinetics. We develop GenPoly, an end-to-end multimodal generative framework that learns a shared polymer representation over paired polymer string sequences and molecular graphs via contrastive alignment, and couples this representation to property prediction, synthetic accessibility filtering, and motif-based candidate clustering to discover coating families beyond commonly used linear aliphatic polyesters. Starting from a clinically prevalent baseline, GenPoly identifies candidate coatings and selects representative polymers for experimental evaluation on nanoparticle-loaded sutures with recombinant human EGF to an in vitro human keratinocyte wound model. A discovered polymer coating of the LA-TMC family shows improved payload retention and sustained release, exhibiting a $3.3\times$ increase in rhEGF concentration, accompanied by more localized Ki-67-positive keratinocyte proliferation at the wound bed, in comparison with the current state-of-the-art candidates. These findings suggest new opportunities in AI-driven polymer discovery, and highlight the viability of GenPoly in engineering bioresorbable coatings for suture-based therapeutic delivery in targeted tissue regeneration.

Positional embeddings encode the spatial position of each image patch in a Vision Transformer (ViT), allowing the model to distinguish different spatial arrangements of visual content. However, we know little about how the choice of embedding shapes a vision transformer's internal representation. In this work, we show that Absolute Positional Embeddings (APE) and Rotary Positional Embeddings (RoPE) encode spatial position through different internal mechanisms: APE stores position in explicit features, while RoPE builds it inside attention and rebuilds it when it is withheld. Specifically, in APE models, sparse autoencoders recover causal row- and column-specific features, whereas interventions on RoPE reveal distinct rotary components that separately encode row and column information. Additionally, causal interventions selectively disabling RoPE's row or column rotations show that decodable row and column information can be independently removed and, once rotations are restored, partly reconstructed, in part by axis-specialized attention heads. Both mechanisms are causally important for classification: permuting APE's positional features across patches reduces top-1 accuracy by 24–31 points, and ablating the rotations of RoPE's primary rebuilding head reduces it more than ablating those of any other head. Our findings show that the choice of positional encoding fundamentally changes how spatial information is organized and processed inside Vision Transformers. More broadly, our work demonstrates how mechanistic analysis can reveal how models construct, reconstruct, and maintain structured representations, providing a basis for targeted interventions in vision models.

Self-evolving agents improve through repeated cycles of modification and evaluation. However, existing methods receive feedback on modifications only after costly evaluations, and aggregate scores may obscure distinctions between failures that require different repairs. We introduce predictive self-evolution through an editable Improvement Dynamics Model (IDM). Before evaluation, the IDM predicts task-level outcomes of a proposed modification and guides an optional revision. After evaluation, past modifications and their observed outcomes are used to improve the IDM, allowing predictive knowledge to accumulate across generations. The system can thus improve how it solves tasks, proposes modifications, and predicts their effects. Building on HyperAgents, we evaluate this approach across multilingual coding, paper review, robotics reward design, and Olympiad-level mathematical grading. Our approach outperforms HyperAgents across all four domains, with up to 46% higher average performance over generations. Gains also persist under matched token budgets that include both agent evolution and task evaluation, and extend to held-out tasks. Forecast accuracy improves as the IDM evolves, and the learned predictive knowledge transfers to improve evolution in a new domain.

Sparse attention visits few keys, yet finding them can still require scoring every key with every indexer head. We introduce DoSA (Dots Sparse Attention), a lossless method for making this computation conditional while preserving exact token selection. Its central observation is that Top-K selection requires sufficient evidence to reject a candidate, not necessarily its full score. A seed set supplies a lower bound on the winning threshold. Nonnegative head projections and joint residual bounds upper-bound positive contributions, while activation-informed aggregation bounds the magnitude subtracted by negative heads. The certificate directs full scoring toward candidates that remain eligible; exact-score filtering further reduces writeback and selection work. We establish the certificate and its exact-selection conditions, and analyze how geometry and selection margins govern its effectiveness. Our GPU implementation accelerates the complete Indexer by up to 2.74 times over SGLang and 1.69 times over LiteTopK across English, Chinese, and code workloads. End-to-end evaluation on GLM-5.2 and DeepSeek-V3.2 achieves up to 1.94 times and 1.50 times time-to-first-token speedups over the two baselines, respectively.

Evaluating recommender systems before deployment requires simulators that remain realistic over long, closed-loop interaction cycles. Current LLM-based agents generally function as surface imitators that generate realistic short-term responses but fail to capture the underlying user dynamics such as fatigue, affect, and shifting preferences. We propose PsyBer-Agent, a psychology-inspired simulator that models these dynamics through its Psy-Engine. This engine represents each user with three evolving, model-defined latent states. To calibrate simulated behavior against real interaction logs, we align log-conditioned reference trajectories with simulator-generated latent-state trajectories via Unbalanced Gromov–Wasserstein optimal transport. This approach provides structured supervision for a lightweight policy without the computational burden of fine-tuning the backbone LLM. We also introduce WebSim and the PsyBer Benchmark, which encompass multimodal scenarios and seven recommender backbones. Our framework evaluates behavioral fidelity, latent-state consistency, robustness, and responses to social-cue perturbations. On the tested horizons, PsyBer-Agent shows lower behavioral discrepancy than the prompt-only simulators while maintaining stable performance under large-scale concurrent simulation for up to 5M queued simulation tasks under bounded concurrency. Code is available in the supplementary material and at: PsyBer-Agent: https://anonymous.4open.science/r/CC66F/, WebSim Platform: https://anonymous.4open.science/r/D8EAX/.

Visual perception organizes incomplete sensory evidence into coherent structures, but where do the principles governing this organization come from? We investigate whether learning to predict missing visual information can give rise to structural priors resembling classical Gestalt principles. Masked autoencoders (MAEs) trained on natural images exhibit perceptual completion consistent with good continuation, symmetry, and global form. Critically, when local evidence surrounding a missing region is held constant, manipulating distant elements systematically changes its reconstruction, demonstrating that local predictions are constrained by the global organization of the visual input. We next manipulate visual experience by adapting MAEs to either natural images or texforms that preserve local image statistics while disrupting global spatial organization. This altered experience weakens globally supported completion and shifts perceptual inference toward local continuation when local and global constraints conflict. Together, these results suggest that perceptual organization reflects structural regularities of natural scenes acquired through visual prediction: predictions integrate structural priors across multiple spatial scales, and the balance between local and global organization adapts to the statistical structure of visual experience. More broadly, our findings suggest that classical Gestalt principles may emerge as adaptive perceptual priors learned from the statistical structure of the visual world.

In this note we prove $\mathrm{forb}(n,\mathbf{1}_{r-2}\times C_4)=\Theta(n^{r-1/2})$, thus providing a series of counterexamples of a 2005 conjecture of the first two authors. Here $\mathbf{1}_{r-2}\times C_4$ is the $r$-extension of the 4-cycle, that is the $r$-uniform hypergraph of four edges obtained by joining a set of $r-2$ vertices to each edge of $C_4$. $\mathrm{forb}(n,F)$ is the largest number of edges of a simple hypergraph $G$ on $n$ vertices without trace (induced subhypergraph) $F$.

We prove a conjecture proposed by Robert J. Lang over 20 years ago, stating that in a circle-packed origami design, any axial polygon with 5 or more sides can be reduced by properly adding a new circle. The proof is based on the structure of the medial axis of the region enclosed by the circles, and yields a concrete algorithm for finding the added circle. The result extends to configurations involving rivers.

This study develops an energy formulation for linear anisotropic Maxwell–Cole–Cole media with multiple fractional polarization channels, passive conductivity, prescribed currents, and finite-terminal Caputo memory with zero prehistory. A weighted solution class captures the integrable fractional initial layer generated by quiescent polarization. Fractional polarization power then separates into a nonnegative history storage and a nonnegative constitutive dissipation rate. Although the instantaneous rate may diverge at initialization, it remains integrable, while stored history energy and accumulated loss vanish initially. Combined with the Poynting theorem, this decomposition yields a global field-plus-memory identity that establishes electromagnetic passivity, closed-system energy decay, and conditional uniqueness under zero-supply boundary conditions. The framework also recovers the Debye, instantaneous-dielectric, vanishing-susceptibility, and vacuum limits under convergence assumptions. A positive diffusive representation provides a second decomposition whose storage never exceeds the history storage. Their difference is an exact nonnegative weighted variance that vanishes only for a zero polarization history. In agreement with these time-domain results, the finite-time harmonic response approaches the anisotropic multichannel Cole–Cole permittivity, whose Hermitian imaginary part ensures nonnegative cycle-averaged absorption. A reproducible single-channel study shows that fractional order broadens the dielectric-loss band, redistributes wave attenuation, strengthens the integrable initial power layer, and prolongs the step-response tail. An exact electric-step response verifies both energy representations. The formulation connects fractional initial layers, hereditary storage, passivity, diffusive relaxation, dispersion, absorption, and attenuation in a local constitutive model.

Sequence models with a fixed-size state avoid the quadratic cost of attention, but face a design choice. An input-dependent state transition gives content-dependent memory at the cost of FFT-based training, while a fixed transition keeps the O(T log T) convolutional path but reads memory the same way for every input. ResonatorLM, a recent model of the second kind, reads a bank of damped complex oscillators through a fixed learned phase per head. We make that readout phase a function of the current token, which adds input dependence to the read only, keeps the convolution exact, and costs 0.2 percent more parameters. An analysis of the untrained bank shows that phase does encode elapsed time, but an ordinary token falls below the background within 2-8 steps and no choice of damping extends linear reconstruction beyond about 16 tokens. In 6M-parameter character-level models on WikiText-2, the input-dependent phase matches the original at the training length and removes its failure beyond it: trained on 256 tokens and tested on 1024, perplexity falls from 12.81 to 3.76. We trace the failure to long-memory heads that accumulate more history than training exposed them to. The learned phase tracks the current character rather than distance and does not improve retrieval.

Small language models trained on a single local machine leave little room for architectural trial and error. Kairos combines bidirectional gated-DeltaNet linear attention and sliding-window attention over shared projections (LiZAttention2), block-wise Attention-Residual aggregation, a sparse Mixture-of-Experts (MoE) feed-forward layer and a multi-scale byte-level codec, trained with a uniform-noise denoising objective over one byte space shared by text, image, audio, video, lidar and control streams; this report asks whether the combination optimizes at all before asking whether it is useful. We give code-derived algorithms for each component, including a cross-session recurrent-state memory gate, then run convergence sanity checks: overfittinga single 440-byte sentence at 1–15M parameters, text only, with three seeds for codec and attention choices and single-run sweeps for seven further factors. The two attention variants tested at 3\,000 steps (LiZAttention2 and window attention alone) reach a minimum loss near $10^{-3}$. After 600 steps, the mean minimum loss ranges from 0.018 (patch codec) to 0.354 (depthwise convolutional codec) with an identical ranking on three seeds. An apparent advantage of shared over separate attention projections closes with more steps or more diverse data and reverses when the projections are frozen: it is a convergence-speed effect, not a capacity effect. These checks suggest that the implementation optimizes; they say nothing about generalization, multimodal training or comparison with existing models, which a planned pretraining run is meant to address. The Discussion sizes a 200M-total/25M-active target analytically; none of its throughput figures is measured. Code is available at https://github.com/fabienfrfr/Kairos . Python package at https://pypi.org/project/kairos-fm/ .

Small-data adaptation can improve speech detection while degrading speaker attribution. We study this discrepancy in a released streaming diarizer adapted on 7.5 h of two-party conversation and evaluated across six corpora. Adaptation substantially improves in-domain diarization performance and transfers to an independent corpus. However, this improvement is not consistent across evaluation scenarios as the additional confusion is mainly associated with impaired temporal identity consistency rather than speaker-count errors. A local-remapping diagnostic reveals different patterns of identity degradation across corpora, indicating that adaptation may alter how streaming models maintain speaker assignments over time. Rehearsal reduces the observed degradation but reduces the cross-domain transfer performance. These results highlight the need to jointly evaluate detection accuracy, identity consistency, and retention behavior when adapting streaming diarization systems.

Learning from human preference data is the dominant route to aligning language models with human values. In linear social choice, where rewards are linear in a fixed feature representation of prompt-response pairs, Ge et al.[2024] show that fitting such a reward by minimizing any non-decreasing convex loss, including BTL, fails PO and PMC. Moreover, no rule that reads only the majority relation can satisfy PO once the output is required to be linearly induced. We ask what it costs to enforce these axioms anyway. To this end, we relax the linear model to allow per-candidate slack. We compute the relaxed linear reward with the smallest total slack that satisfies the axioms with a margin $\eta$, the minimum required difference between two reward values. Our solution satisfies the axioms under no assumptions about the voters or how comparisons were collected. We bound the optimal total slack by $O(1)$ when $\eta$ is at most $O(\frac{1}{m^2})$ for $m$ candidates. Furthermore, we exhibit an instance that forces this bound, concluding that the rate is tight up to constants. In practice, the no. of candidates far exceeds the feature dimension, and only a linear reward can be evaluated on unseen responses. We therefore introduce a new method that charges the linear part for each comparison it gets wrong. It simultaneously minimizes the total slack and the no. of violations, with a parameter $\lambda$ trading off between them. We show that the total slack is monotone but saturating in $\lambda$: raising it reduces the violations of the deployed linear reward and increases the slack, yet the slack stays below $(\eta+\Delta\sqrt{d})\lfloor m^2/4\rfloor$, where $\Delta$ and $d$ are the diameter and dimension of the features, respectively. Experiments on both synthetic and real-life preference data corroborate our theory and show that the linear reward output by our method beats linear BTL.

Atmospheric rivers (ARs) produce many of the world's most extreme precipitation events and hydrometeorological hazards. Although artificial intelligence weather prediction (AIWP) models have demonstrated skill comparable to or exceeding numerical weather prediction (NWP) systems for large-scale atmospheric variables, their ability to forecast AR-related precipitation remains insufficiently characterized globally. Here, we evaluate 24-hour precipitation forecasts from the Global Forecast System (GFS), Global Ensemble Forecast System (GEFS), GraphCast, and Artificial Intelligence Forecasting System (AIFS) from Day 1 through Day 10 globally and across North America, Europe, and Australia and New Zealand. Using the Extreme Weather Bench framework, forecasts are evaluated against Integrated Multi-satellitE Retrievals for GPM (IMERG) observations using measures of precipitation magnitude, spatial structure, and localization. GraphCast and AIFS exhibit greater spatial skill than GFS and GEFS, particularly for heavy precipitation and at longer lead times, and better preserve the spatial organization of AR-related precipitation through Day 10. However, this improved spatial skill does not translate into accurate precipitation magnitudes. AIWP models tend to overpredict moderate-to-heavy accumulations while underpredicting the heaviest precipitation at longer lead times, whereas NWP systems develop pronounced dry biases. These results reveal distinct strengths and limitations of AIWP for high-impact precipitation forecasting and provide a reproducible benchmark.

Accurate high-resolution wind field simulations are critical for resolving fine-scale atmospheric dynamics, yet the simulation of wind fields in Earth System Models (ESMs) remains limited by coarse spatial resolution and systematic biases. To address this, data-driven down scaling techniques have been widely used to enhance coarse-resolution ESM outputs. However, existing methods are typically tied to fixed discretizations, limiting generalization across models with different native resolutions. Here we formulate global near-surface wind downscaling as an operator-learning problem on continuous atmospheric state fields and develop a downscaling neural operator that maps coarse-scale fields to fine-scale counterparts across heterogeneous discretizations. The operator learning-based neural downscaling framework outperforms dominant baselines, recovers fine-scale physical structures, and generalizes to previously unseen ESMs and future climate scenarios without retraining, while preserving long-term wind projection trends. These findings establish a generalizable paradigm for high-resolution climate downscaling across diverse simulation outputs and future scenarios.

We study mixed symplectic sums along connected unbranched cyclic multisections of degrees $p,q\geq 2$ in ruled surfaces. With \[ p(g-1)=q(h-1)=n, \] the resulting four-manifold has signature zero and Euler characteristic \[ e=4n\left(1-\frac1p-\frac1q\right). \] We classify the types having the rational cohomology ring of $\#_k(S^2\times S^2)$. They occur only for odd $k$ and, with $t=(k+1)/2$, $a=g-1$, and $b=h-1$, are exactly the solutions of \[ n=a+b+t,\qquad a\mid b+t,\qquad b\mid a+t. \] For each odd $k$ there are finitely many types; we give an explicit parametrization and tables for $k\leq 15$. Every type admits an equatorially product-framed symplectic realization and, possibly with different framings, a spin realization whose free integral intersection form is $kH$. We prove minimality, boundary incompressibility, an integral Mayer–Vietoris presentation, and a regular cyclic cover of degree $\operatorname{lcm}(p,q)$. The boundary subgroup $\pi_1(\Sigma_G)\times\mathbb Z$ embeds in the fundamental group; this gives a nontrivial Bass–Serre splitting and rules out word-hyperbolicity, cocompact real- or complex-hyperbolic lattices, and torsion-free cocompact irreducible bidisk lattices. There are five types for $k=3$ and seven for $k=5$; for $k=1$ we recover the degree pairs $(2,3)$, $(2,4)$, and $(3,3)$.

The magnitude of the generalized Euclidean cross product is a Gram volume whose degree under common scaling is fixed by the integer dimension of the spanning frame. We formulate a generalized Fractal Cross Product (FCP) as a nonlinear radial deformation with a prescribed positive degree $D$, which may be non-integer. The scalar construction applies in any ambient dimension $m\geq k$, while its canonically oriented vector form requires codimension one. It recovers the classical generalized cross product exactly at $D=k$ and retains orthogonality, alternation, rotation equivariance, and $D$-homogeneity, but is generally not multilinear. For exact self-similar frame systems, the construction also obeys a scale-balance law at the similarity dimension. We derive a differentiable, dimensionless image response and a non-circular empirical accumulation exponent obtained by regressing raw angular Gram responses across patch widths. Binary64 calculations recover the finite-frame identities to roundoff, while raster experiments recover the Sierpiński-triangle value 1.5849625 at three resolutions. In five-seed medical-imaging comparisons, FCP-centered fusion increased mean area under the receiver operating characteristic curve from 0.7264 to 0.8135 and from 0.5843 to 0.6765 on the random and hospital-separated Retinal Image Database for Optic Nerve Evaluation partitions, respectively, and from 0.7403 to 0.7475 on FracAtlas. On FracAtlas, balanced accuracy increased from 0.6186 to 0.6721 and the harmonic mean of precision and sensitivity from 0.3669 to 0.4457. These results support the utility of the complete fusion framework, but do not isolate the effect of FCP from that of its complementary descriptors and fusion head.

Gielis curves describe many natural shapes, and some of them have corners or cusps. We apply two recent ideas from the differential geometry of curves to Gielis curves: "regularizing" changes of metric and fractional curvatures based on the Caputo derivative. First, we show that the smoothness of a Gielis curve is controlled by the exponents $n_2$ and $n_3$, not by $n_1$. A change of coordinates, the Gielis chart, maps every Gielis curve that closes after one turn onto the unit circle. However, the corners and cusps of the curve reappear as a loss of regularity of the metric; they are transported, not removed. Second, for the fractional curvature defined by Caputo scaling, we distinguish two conventions that differ in the choice of parameter and derive their scaling laws under homotheties. We prove that the memory of the Caputo operator breaks the rotational symmetry: the fractional curvature takes different values at corresponding points of congruent arcs. For fractional order $\alpha<1$, the curves of constant fractional curvature are spirals, and the clothoid is a special case. We also introduce a fractional total turning for curves with corners, give closed forms for regular polygons, and show that fractional general and slant helices coincide with the classical ones. The Python library used in the computations is listed in the appendix.

We classify reflexive four-polytopes for which the generic anticanonical hypersurfaces associated with the face fans of the polytope and its polar both have at most isolated singularities. There are exactly 590 such polytopes up to lattice equivalence in the Kreuzer-Skarke classification. Their two-dimensional faces belong to twelve lattice-isomorphism classes, all triangles, zonotopes, or reflexive polygons. The only nonsmoothable local models are the cyclic quotients $\frac13(1,1,1)$ and $\frac15(1,1,3)$ and the anticanonical cone over the first Hirzebruch surface; no singularity has a positive-dimensional reduced deformation space without a smoothing component. Within this classification, the Hirzebruch cone occurs in a unique mirror pair, associated with polytopes having 22 and 26 vertices. We exhibit this pair explicitly: one hypersurface is nonsmoothable, while all singularities of its mirror are locally smoothable. The crepant resolutions have Hodge numbers $(20,26)$ and $(26,20)$. At the opposite extreme, the self-dual 24-cell is the unique member of the classification for which both hypersurfaces are smooth before crepant resolution.

5We study homogeneous cosmological perturbations in the quadratic model $F(R,\mathcal G)=R+\alpha R^2+\beta\mathcal G^2$. Linearization of the modified Friedmann equation together with dust conservation yields a second-order equation for the homogeneous Hubble perturbation, or equivalently a third-order equation for the matter contrast. We then restrict the asymptotic analysis to backgrounds that satisfy the zeroth-order equations of the same quadratic theory. Besides the low-curvature dust branch, which approaches $H=2/(3t)$, the model admits a nontrivial vacuum de Sitter solution with $\beta H_{\rm dS}^6=1/96$. Its homogeneous curvature modes are stable only when $\alpha<-1/(6H_{\rm dS}^2)$. At high curvature the background equation also admits a vacuum big-rip asymptotic branch beginning with $H=7/(t_s-t)$; the first corrections are fixed by the quadratic couplings. The corresponding $\mathcal G^2$-dominated perturbation has one divergent homogeneous mode. The simultaneous Einstein limit is taken only as a consistency check and recovers the standard dust behavior.

Linear perturbations of a Schwarzschild-MOG black hole separate into scalar and gravito-vector sectors in the reduced strong-field theory at leading order in $\mu\bar G M\ll1$. The field redefinition $\widehat h_{\mu\nu}=h_{\mu\nu}-\psi\bar g_{\mu\nu}$ gives a decoupled Klein–Gordon equation for the scalar and the Reissner–Nordström equations for the gravito-vector sector. The scalar is stable under the stated spectral boundary conditions for $m_G^2\ge0$. A scalar-only excitation produces a conformal metric perturbation with null Hamiltonian contribution $H_1^{(s)}=-\psi H_0$. For fixed observer worldlines and conformally corresponding orthonormal frames, every linear scalar-to-critical-curve transfer coefficient vanishes, including monopole breathing, provided the conformal factor is smooth, positive, and preserves the capture boundary. The exceptional charged axial dipole instead induces a constrained metric perturbation. A first-order radial integral reconstructs this perturbation in a gauge regular at the future horizon and on the sphere, separating the radiative amplitude from stationary angular momentum without division by frequency. For a unit-incident canonical vector amplitude with zero angular momentum, matched asymptotic solutions give an explicit charge-dependent local metric response that vanishes as $\omega^2$ at fixed radius and fixed subextremal background. A separate infinitesimal Kerr–Newman benchmark yields the stationary impact-parameter midpoint shift and recovers $\delta C_b=-2a_{\rm rot}$ in the neutral limit. These results exclude linear scalar breathing of the specified angular critical curve and distinguish the low-frequency vector-induced metric response from a stationary rotation effect.

Borel's theorem presents the flag manifold cohomology as the coinvariant algebra $H^*(K/T; \mathbb{C}) \cong S(\mathfrak{h}^\vee)/I_W$, establishing the vanishing of characteristic forms $p_j(F_\mathfrak{t}) = \mathrm{tr}(F_\mathfrak{t}^j)$ non-constructively; projection from the Weil algebra $W(\mathfrak{k})$ yields closed primitives on $K$ failing to descend. We construct relative transgression primitives $\alpha_j \in (\Lambda^{2j-1}\mathfrak{m}^\vee)^T \cong {}^K\Omega^{2j-1}(K/T)$ satisfying $d_\mathfrak{m} \alpha_j = -p_j(F_\mathfrak{t})$ on $K/T$. Via connections $\theta_s = \theta_\mathfrak{t} + s\theta_\mathfrak{m}$, the curvature factors as $F_s = (1-s)(F_\mathfrak{t} - s\theta_\mathfrak{m}^2)$; expanding via non-commutative word sums and Euler's Beta integral yields a universal closed formula for all $j \ge 1$. These primitives exhibit a structural transition: $\alpha_2 = -\frac{1}{3}\mathrm{tr}(\theta_\mathfrak{m}^3)$ is curvature-free, $\alpha_3$ is linear in $F_\mathfrak{t}$, while at $j = 4$, $[F_\mathfrak{t}, \theta_\mathfrak{m}^2] \neq 0$ shatters cyclic trace symmetry into independent necklaces, producing the irreducible invariant $\mathrm{tr}(\theta_\mathfrak{m} \wedge F_\mathfrak{t} \wedge \theta_\mathfrak{m}^2 \wedge F_\mathfrak{t})$. On the $A_2$ root lattice ($\mathfrak{sl}_3(\mathbb{C})$), $T$-invariant cochains correspond to closed loops, $\alpha_2 = \Omega_- - \Omega_+$ is the oriented difference of triangles, and $d_\mathfrak{m}$ acts as an edge-fracturing operator witnessing the exactness of $c_2(E)$. While character theory encounters a multiplicity barrier at $k=4$ in Specht decompositions, a minimal Koszul DGA resolution $K_\bullet(p_2, \dots, p_k)$ resolves the coinvariant algebra, recovering the Poincaré polynomial $P_{K/T}(t) = \prod_{j=2}^k \frac{1-t^{2j}}{1-t^2}$ and Euler characteristic $\chi(K/T) = k!$ without matrix-rank ambiguities.

Papapetrou showed that the covariant derivative of a Killing vector field satisfies Maxwell's equations in vacuum. We extend Papapetrou's result to show that the covariant derivative of a Killing vector field satisfies Maxwell's equations in non-vacuum backgrounds as well, if we allow electromagnetic currents of purely geometric origin. We postulate that every Killing vector field gives rise to a physical electromagnetic field and, in a non-vacuum background, a physical electromagnetic current—hereafter called Killing electromagnetic field and Killing electromagnetic current, respectively. We show that the Killing electromagnetic field of the flat FLRW universe comprises a Killing magnetic field and a rotational Killing electric field; we derive an upper bound on the Killing magnetic field and find that the upper bound is consistent with the current upper bound on the cosmic magnetic field. We show that the time-like Killing vector of the Schwarzschild spacetime gives rise to a radial Killing electric field. We also show that in the weak field regime—and far from the matter distribution—the back reaction of the radial Killing electric field changes the Schwarzschild metric to Reissner-Nordstr$\ddot{\mbox{o}}$m metric; our calculation yields the converse of Wald's result. Next, drawing upon Rainich's work on Rainich-Riemann manifolds, we discuss the etiological question of how a physical electromagnetic field can arise out of geometry; we also argue that detection of the Killing electric field of flat FLRW spacetime may be within current experimental reach. Finally, we discuss the relevance of Killing electromagnetic currents and the aforementioned transmutation of Schwarzschild spacetime to Reissner-Nordstrom spacetime, to Misner and Wheeler's program of realizing “charge without charge”.

Let $G$ be a connected semisimple real Lie group with finite center and let $(\Gamma_N)$ be a tower of cocompact lattices. We prove that the weak injective radius property implies the spherical density hypothesis with the same parameter. Together with the converse theorem of Golubev and Kamber, this gives an equivalence in the cocompact Archimedean setting. The proof uses a uniform spherical-mass estimate near spectral walls and a positive pre-trace kernel, but no tempered-spectrum estimate or a priori spectral gap.

Policy gradient methods for the linear quadratic regulator optimize feedback gains using a cost over an infinite horizon. When this cost is evaluated at one fixed initial state, it need not diverge near every part of the stability boundary. We study whether the Euclidean gradient flow can reach this boundary in finite optimization time. For controllable planar systems with one input and positive definite quadratic weights, an exact representation of the cost yields a necessary and sufficient condition for the existence of such a collision. The condition reduces to a scalar root calculation and includes the critical case, where a double direction root attracts an open set of stabilizing gains. For a triangular family, the criterion gives an explicit algebraic parameter region, and every trajectory either converges to the Riccati gain or collides with the stability boundary. A sharp threshold for compact cost sublevels provides a convergence certificate. Along every colliding trajectory, the stability margin and the smallest eigenvalue of the accumulated state Gramian vanish linearly, although the Gramian remains positive definite before collision. Numerical experiments examine the dependence on initialization, slow passage near a critical direction, and the effect of adding excitation in a second state direction.