Papers

Deploying language-model agents onboard mobile robots requires task reasoning and responsive execution within limited compute and memory. We propose BT-Agent, an onboard architecture combining LLM-driven task decisions with predefined behavior-tree (BT) skills. These skills enforce execution conditions and return structured failure feedback. We demonstrate fully onboard operation with a 9B model on a quadruped across five task categories. Intervention tests show that local checks stop ongoing navigation without waiting for another model decision, while the agent remains active to retry or report constraints. To compare agent designs efficiently, we introduce a 161-task multi-fidelity benchmark, evaluated in text on all tasks and in Habitat on 39. Text supports 10.9$\times$ higher evaluation throughput, while simulation reveals how perception and motion failures change task outcomes. BT-Agent's largest success gains over the strongest Act/ReAct baseline occur at 4B and 9B: approximately 10 and 5 percentage points. Ablations identify structured tool calls and informative failure feedback as the most useful components.

Accessing and combining the spatial information carried by light at different wavelengths is important for linking structured-light encoding with spectrally distinct detection channels. Here, we introduce a correlation-enabled linear-optical framework for cross-wavelength imaging, bidirectional replication, and spatial information mixing. Shared random holographic modulation establishes correlations between nondegenerate optical fields, allowing spatial information to be retrieved from joint intensity fluctuations. In the demonstrated replicator, a spatial pattern encoded at one wavelength is read out in the detection coordinates of another through intensity correlations. Selecting diffraction-order pairs produces correlation kernels containing either the product or the conjugate product of two input field envelopes. These operations reproduce the spatial algebra underlying sum- and difference-frequency generation, while the optical fields retain their original carrier frequencies. The theory accounts for wavelength-dependent propagation and coordinate scaling, linking the input structures to observable correlation patterns. Together, these results establish a common framework for cross-wavelength spatial information processing through engineered optical correlations and joint detection.

Explicit 3D scene representations such as 3D Gaussian Splatting (3DGS) have recently enabled high-quality, real-time novel view synthesis. However, these methods assume static scenes, limiting realism in the presence of dynamic phenomena ubiquitous in real-world environments. While coherent motion (e.g., humans or vehicles) can be captured and reconstructed, high-frequency effects such as flowing water remain intractable under casual capture conditions, where dense multi-view, high-speed acquisition is not feasible. We present a method to animate water in static 3DGS scenes, enabling visually plausible motion without requiring dynamic capture or physical simulation. Our approach estimates 2D optical flow from reference images or model renderings and lifts it into a consistent 3D Eulerian motion field. We propose a cinemagraph-inspired algorithm that applies this motion field to deform Gaussian primitives, producing temporally seamless looping animations. The resulting representation retains real-time rendering performance, significantly enhances perceptual realism, and preserves free-viewpoint navigation within the scene. We evaluate our method on a newly captured dataset of in-the-wild waterfall scenes and demonstrate clear improvements over prior work in dynamic Gaussian splatting and 3D cinemagraph generation in visual quality and computational efficiency.

New physics emerges from new materials, which in turn rely on the development of new synthesis techniques. Identifying the enlargement of metastable synthesis window as enabling for realizing novel materials, we develop diffusion-activated epitaxy, a technique which combines two metastable synthesis methods of diffusion and epitaxy. We observe that within a narrow window of investigated synthesis space we can experimentally identify the stabilization of four distinct phases, three of which show X-ray diffraction and scanning transmission electron microscopy characterization data that are incompatible with the crystal structures registered in experimental and computational inorganic structure databases. These results suggest an enlarged metastable synthesis window is enabled by this synthesis approach, identifying doubly metastable synthesis as a promising method for exploration of quantum materials.

Modern vision and language models process ever longer token sequences, and reducing them is the main lever on downstream cost. Token compression, however, faces a fundamental trade-off: selecting tokens discards information, while existing merging methods preserve it only through pairwise token-to-token matching or $MN$ token-to-output interactions. We introduce LapMerge, a differentiable merging operator that resolves this trade-off for ordered sequences: a learned mass distribution partitions the sequence into exactly $N$ equal-mass segments under a Laplace-kernel density, and prefix/suffix scans aggregate each segment, so both the forward and the backward pass run in $O(M+N)$ time. This combination (an exact output budget, end-to-end differentiability, and linear cost) is precisely what previous reducers trade off against one another. The budget can be changed without changing operator parameters, and the learned mass doubles as a token-level map of where the budget is allocated. Across vision and vision-language tasks, LapMerge matches or improves on strong token reducers while being up to two orders of magnitude faster at large sequence lengths; it is particularly effective when token order reflects spatial or temporal locality, and works both as a drop-in reduction module and as a trainable architectural component.

Natural disasters such as earthquakes, tropical storms, and floods cause extensive damage to buildings, and rapid, reliable damage maps are essential to guide emergency response. Automating this task, referred to as building damage assessment (BDA), from satellite imagery, has become an active research topic at the intersection of remote sensing, computer vision, and machine learning, particularly since supervised deep learning methods were shown to outperform earlier handcrafted approaches. Yet, despite a decade of intense activity and the release of several large-scale datasets, the field lacks a shared foundation. This chapter revisits the task of BDA from first principles, organized around three pillars: data, evaluation, and methods. We first clarify the candidate task for- mulations and argue for a sensor-agnostic, instance-based definition that tolerates missing pre-event imagery. We then review existing datasets along the axes of sensing modality, temporal configuration, and geographic coverage. On the evaluation side, we show that prevailing practices, largely based on a single aggregate score, are theoretically fragile, and we lay the foundations of a more informative, ranking-based, multi-score protocol. We further review methods ranging from task-specific deep networks to recent Earth observation foundation models, noticing that generalization to unseen events and regions remains an open difficulty. Finally, we discuss how these elements can inform the organization of a BDA challenge, including cross-validation, challenge, and private data-sharing setups designed to build trust between developers and users.

Action recognition is a crucial task in domains like robotics, patient monitoring, or surveillance. Recently, event-based action recognition has attracted much attention due to the event camera's ability to capture fast motions and its improved privacy over conventional vision cameras. While frame-based event representations work well for short event sequences, their discriminative ability declines sharply with longer and more complex event sequences. In this work, we introduce a multi-frame representation that greatly improves recognition performance over its single-frame counterpart. We adopt a masked auto-encoding framework to pre-train a Vision Transformer using a sparsity-aware masking algorithm and token selection mechanism to improve downstream performance and control computational complexity. We also introduce multiple mask tokens for the decoder with a position-aware attention mechanism, leading to further improved performance. Our proposed method shows impressive performance for challenging long-duration and complex event sequences, yielding state-of-the-art performance on THU-EACT-50 CHL and DailyDVS-200.

Many scientific models can be simulated but have intractable likelihoods, making frequentist uncertainty quantification difficult. We construct exact anytime-valid confidence sequences for the parameters of such models. At each observation, fresh simulations from each candidate null conformalize a positive predictable score into a conditional e-value; their product is a test martingale, which we invert over the parameter. Validity holds for any such score, whatever its accuracy. We show that the oracle likelihood-ratio score maximizes expected log growth, which motivates approximating it with an amortized neural ratio estimator. On synthetic benchmarks, an audit of model routing in an LLM API, and microlensing data, the resulting confidence sequences stay valid under continuous monitoring, where fixed-sample methods do not, and neural scores remain competitive with exact likelihood ratios and application-specific scores.

We prove that four capacities of single-mode bosonic dephasing channels coincide under an average photon-number constraint: the unassisted quantum capacity, the unassisted private capacity, the two-way classically assisted quantum capacity, and the two-way public-classically assisted private capacity. The assistance may be interactive: the parties can exchange classical messages between channel uses and choose subsequent inputs based on earlier outcomes. For every finite value of the constraint, their common value is the coherent information optimized over Fock-diagonal input states satisfying the constraint. The proof uses a relative entropy comparison with the completely dephasing channel to derive chain rules for the Rains relative entropy and the relative entropy of entanglement. These chain rules give an upper bound on the rates achievable by adaptive protocols whose mean input photon number is constrained on average over all channel uses. We also show that both chain rules are attained by canonical purifications of finite-energy Fock-diagonal states.

Power grid cyber-physical security requires intrusion detection methods that account for the operational consequences of detection errors. However, the existing intrusion detection approaches primarily focus on the predictive performance of learning algorithms, while downstream operational costs are often optimized independently, which can lead to inferior operational performance. To help alleviate this problem, we propose a novel optimization-learning cyber-physical-risk-informed framework that communicates iteratively to minimize power grid operational costs. This framework translates machine learning prediction errors into power grid operational cyber-physical-risk costs. Experimental results on two power grid operation datasets, Industrial Control System (ICS) Cyber Attack and Sherlock, show that the proposed approach improves power grid operation costs compared with the traditional approach.

Using spectral and polynomial methods, we show several sharp spectral bounds for the clique and chromatic numbers of graph powers that only use spectral information from the original graph. In particular, we strengthen earlier spectral-radius estimates to obtain an asymptotically sharp colouring bound for general graph powers, and prove a sharp bound for powers of line graphs with a complete characterisation of equality. We also develop polynomial methods yielding upper and lower bounds for clique numbers of graph powers. Regarding the connection between the clique and coloring parameters of graph powers, we propose a spectral analogue of Wegner's conjecture, determine the extremal values for graph squares, and establish the conjecture for all powers when the spectral-radius bound is sufficiently small. Finally, we connect our spectral polynomial method with coding theory: we prove that for distance-regular graphs, optimisation over the adjacency algebra recovers Delsarte's linear programming bound for anticodes.

We establish finite-sample guarantees for conditional diffusion models learned from data and implemented with finitely many denoising steps. Our approach aligns the score-training objective with the sampling grid, allowing statistical estimation and numerical discretization errors to be analyzed together. Under a smooth conditional density model, we derive high-probability bounds on the generated distribution's error and quantify the required number of score evaluations. For every smoothness index $\beta>0$, a suitable grid yields averaged conditional Kullback–Leibler (KL) divergence $\widetilde O_p(n^{-2\beta/(d_x+d_y+2\beta)})$ relative to the original target with or without early stopping. Both the full and early-stopped samplers attain average target total-variation (TV) error $\widetilde O_p(n^{-\beta/(d_x+d_y+2\beta)})$. Here $n$ is the sample size, $d_x,d_y$ are the data and conditioning dimensions, and errors are averaged over the conditioning variable. The total-variation bound matches the rate established in existing conditional distribution-estimation theory, up to logarithmic factors.

We compare three heights attached to a curve of genus two: the weighted height of its Igusa–Clebsch invariants, the GIT height of a binary sextic defining it, and the Néron–Tate height on its Jacobian. For a morphism from an abelian variety to a weighted projective space we define a weighted Néron–Tate height and show that the weighted height differs from it by a linear form and a bounded function; the linear form vanishes when inversion is induced by an automorphism of the weighted projective space. For binary sextics we prove the explicit inequality $h_{\inv}\le h_{\text{GIT}} + \frac{1}{2} \log 302$, with nonnegative local differences at the finite places that vanish at places of good reduction. We then construct a family of semistable binary sextics over $E\times E$, where $E:y^2=x^3-2$, whose relative Jacobian carries a section of infinite order, and we show that the Jacobian over the base point is absolutely simple. Along every curve in the parameter space through the base point, the Néron–Tate height of the section is a positive multiple of the weighted moduli height up to an error of square-root size, and up to a bounded error when the curve is rational. As applications, we obtain explicit families of genus-two Jacobians of positive rank over $\mathbf{Q}$, an infinite congruence lattice of fibres of positive height, and canonical-limit formulas along elliptic curves in $E\times E$.

We present HorizonNav, a video diffusion-based framework for anticipatory robot navigation that transforms a single RGB image and a text prompt into a geometrically aligned representation of plausible surrounding 3D space. Given the initial observation and a textual description of the reconstruction objective, a vision-language model, guided by designed instructions, interprets the scene and constructs structured prompts for a video diffusion model. The resulting video sequence provides plausible future observations of the environment, which are subsequently processed using monocular ORB-SLAM3 for camera pose estimation and Video Depth Anything for dense depth prediction. Our central contribution is a geometry-guided depth alignment stage that combines the estimated camera poses with triangulated visual landmarks to constrain dense depth predictions across multiple views. This alignment improves the geometric consistency of the reconstructed 3D environment, addressing inconsistencies that temporal coherence of the generated video alone cannot resolve. The experimental evaluation of navigation in imagined 3D environment shows that across a 19-scene horizon study, planning-feasible scenes increase from 13/19 at P20 to 15/19 at P81, while successful RRT trials increase from 62/90 to 75/90. In a separate corridor study, 6 of 25 evaluable generated-camera trajectories violate the reconstructed collision constraints; RRT planner recovers feasible alternatives in 5 of those 6 cases. A larger archive contains 58 usable reconstructions from 67 attempted cases, of which 36 support a downstream plan and 22 do not, exposing reconstruction and free-space failure modes rather than hiding them without lidar. Across 15 corridor reconstructions, the mean reconstruction error is approximately 0.30 m at matched structural landmarks.

We construct a smooth embedded annulus in Euclidean 3-space with negative Gauss curvature in its interior whose boundary consists of strictly convex curves in planes tangent to the annulus, and which contains closed asymptotic curves. This gives a counterexample to a 1962 conjecture of Nirenberg. The construction is based on Lelieuvre's formula, which yields a closed curve that is then realized as an asymptotic curve on a ruled annulus. The second asymptotic curve is obtained from a periodic solution of a Riccati equation.

The network simplex method is a classical algorithm for minimum cost flow problems formulated as linear programs, that has attracted recent interest from the optimal transport community. Indeed, discrete optimal transport is a subclass of minimum cost flow for which network simplex techniques compute exact solutions while leveraging the inherent problem sparsity. However, extending these approaches to other, nonlinear optimal transport formulations has yet to be thoroughly investigated. In this paper, we adapt the network simplex algorithm to unbalanced optimal transport. Our proposed framework is applicable to various unbalanced formulations, and can be implemented explicitly in the most common cases, including semi-unbalanced optimal transport. The resulting algorithms are memory-efficient, and perform competitively with existing approaches, as shown by our experiments on a range of transport and barycenter calculations.

Given a bivariate copula $C$, the bivariate copula $\Psi(C)$ is defined as the Markov product of $C^T$ and $C$. In this paper we consider the diagonal section of $\Psi(C)$ under the additional assumption that its value at a single point is known. We determine the exact upper and lower bounds for the diagonal section of $\Psi(C)$, given that $\Psi(C)(a,a)=d \in [a^2,a]$, where $a \in [0,1]$ is a fixed point. Furthermore, we describe some additional properties of the transformation $\Psi$, such as its behavior under ordinal sums and the survival operation. We also study checkerboard copulas in the range of $\Psi$ and matrices associated to them.

The introduction of two-body XX interactions has been shown to improve the performance of quantum annealing in solving certain combinatorial optimisation problems and would enable the simulation of a broader class of many-body Hamiltonians. Due to experimental constraints, such interactions are typically not natively available on analogue quantum devices. To overcome this limitation we introduce the truss gadget. This is an exact construction that generates two-body XX interactions using only one-body X and Z fields and two-body ZZ interactions which are readily accessible on, for instance, superconducting quantum annealers. We experimentally validate the construction by reproducing the dynamics of a target XX Hamiltonian on a trapped-ion quantum processor. We then apply the gadget to an illustrative set of instances of the weighted maximum-independent-set problem on a superconducting quantum annealer. We empirically determine an approximate thousand-fold improvement in the probability of finding a state close to the ground state for these instances, over implementation without an XX interaction.

Generative molecular design typically relies on over-generation and post-hoc filtering. Inference-time steering methods like Feynman-Kac (FK) steering improve efficiency by biasing trajectories toward high-reward candidates, yet they optimize within a single fixed configuration. In de novo nanobody design, upfront commitment to conditioning choices like scaffold identity or target epitope creates a bottleneck. Since yield varies across these conditions, steering unpromising setups wastes compute. We recast generative condition selection as an inference-time resource-allocation problem. We propose two dynamic budget allocators: a joint Global Sequential Monte Carlo (SMC) formulation that steers and allocates compute across configurations concurrently, and a decoupled approach adapting Batched Successive Elimination (BaSE). Empirically, these dynamic allocators increase the yield of designable candidates per GPU-hour nearly 5$\times$ without sacrificing diversity across distinct therapeutic targets and generative configurations.

Scaling trapped-ion processors by adding ions remains challenging. A complementary route exploits the multilevel internal structure of each ion to increase the computational Hilbert space per ion. We introduce a hardware-native hybrid qudit-oscillator instruction set for a general d-level internal register coupled to bosonic modes and prove that three of its native operation families suffice for universality of one qudit coupled to several modes. Adding an entangling operation between ions extends universality to a connected register of several qudits. The architecture natively supports both phase-space and sideband instruction sets and is implemented in the open-source framework HybridLane. As a representative application, we reproduce the dissipative vibronic dynamics of a three-chromophore complex previously simulated on a hybrid qubit-oscillator superconducting platform. Moving from a one-qubit-per-state register to a single qudit stores the entire electronic manifold in one ion, so every electronic coupling becomes a local intra-ion transition and the lab-frame Hamiltonian compiles directly into the ISA. Under the stated sequential timing model from a 40Ca+ qudit processor, one Trotter step is projected to run about ten times faster on the qudit, with an average RMSE of approximately 1.1 percent against the published reference.

We find a robust scenario for the emergence of the Golden Ratio (GR) dynamical universality class, characterized by the dynamical exponent $z \equiv \varphi=\frac{1}{2}(1+\sqrt{5})$. Previously, its emergence required fine-tuning of system parameters. Here, the GR universality class appears generically in a wide class of Markov processes as a consequence of a global, circulant symmetry of the flux Jacobian. This symmetry leads to GR universality in systems where the total number of conservation laws is odd and larger than three. We introduce a multilane exclusion process with Bernoulli stationary measure as a minimal, parameter-free model, which features GR universality for all conserved modes, most of which are doubly degenerate. Our predictions, based on mode-coupling theory generalized to treat degenerate conserved modes, are corroborated by large-scale Monte Carlo simulations.

Coding agents often make useful progress before they fail, but retry-from-scratch discards the machine state created along the way. This makes failure recovery an interesting search problem: if intermediate environment states preserve meaningful progress, restoring and branching from them could outperform repeatedly restarting from the initial state. We implement Go-Explore-style search over full Daytona snapshots and find that higher and more efficient solve rates depend mainly on progress that sits outside the repository, such as databases, services, and other runtime state. Our results suggest that full-snapshot search is most effective when long-horizon coding work spans multiple services whose state cannot be reconstructed from a single diff.

Comics are one of the great visual records of the twentieth century: hundreds of thousands of pages in which a mass readership saw its leaders, its wars and its heroes. Most of this record has never been studied as a whole; comics scholarship proceeds one book at a time. We argue that artificial intelligence, including generative models, can change this to the benefit of the medium itself. We propose PAGE, a framework that makes artificial intelligence an instrument for comics research. It parses every scan into panels, text and a bibliographic record; annotates each page with a vision-language model that follows a written codebook, as a human coder would; grounds samples of these readings in human annotation; and explains change across an archive, with estimates controlled for publisher and genre. We demonstrate the framework on 3,929 American comic books (1938–1956) and 87 volumes of Japanese manga, in three studies, each resting on a different capability. The first pairs text search with visual reading to recover how a mass medium portrayed political power. The second measures the page's visual form across an industry and recovers the hand of individual artists. The third follows who speaks across two comics traditions, checked against human annotation. Beyond research, the same instruments can make archives searchable by what is drawn, open comics to blind and low-vision readers, and help preserve fragile collections and credit uncredited artists. We outline a research agenda in which artificial intelligence strengthens the study, preservation and reach of the medium.

Disinformation on online social media has been on the rise in the past years. While there is extensive literature modelling social cascades, online disinformation cascades show distinctive characteristics: they provoke debunking reactions, they spread over networks with heterogeneous connectivity, and they compete for the limited attention of users — which recommendation algorithms may partly offset. We propose a model including these ingredients and study it through Monte–Carlo simulations and mean–field approximations, on random graphs and an empirical Twitter network. We find that debunking reactions are double–edged: they reduce the reach of appealing items but allow unappealing ones to reach audiences they would not otherwise reach. Degree heterogeneity makes cascades trigger at a much lower content appeal but limits their final reach: heterogeneous networks are easy to ignite yet hard to saturate. Limited attention cancels the effects from degree heterogeneity, although the mediation of recommendation algorithms may restore them.

Nearly every autonomous or semi-autonomous flipper controller assumes access to an elevation map of the terrain under and in front of the robot. The terrain beneath the chassis is occluded in the sensors, so the map is remembered through odometry integration, whose drift corrupts it. We present Stabilizing Anti-Stuck (SAS), a mapless controller that combines a stabilization term for body roll and pitch with an anti-stuck term that frees the robot when it gets stuck. As input, SAS uses the IMU, the flipper joint angles, and a motion cue that only tells whether the robot is moving. The cue is computed over a short window and never accumulated, so SAS does not inherit the odometry-integration drift that corrupts the map. We deploy SAS on a tracked robot with four flippers and evaluate it against manual teleoperation and prior flipper controllers in a large comparison, both in simulation and on a real obstacle. In simulation and on the real robot, SAS achieves traversal quality comparable to state-of-the-art map-based controllers. On a log hidden in tall grass, invisible to the LiDAR, the best map-based controller gets stuck while SAS traverses the hidden obstacle without a problem.

In the present work, we investigate the long-distance contributions to light hadrons and radiative decays of $\eta_c$, where the long-distance contribution is phenomenologically described by the meson loops. Our estimations indicate that the branching fractions of $\eta_c \to \mathcal{VV},\ \mathcal{V}=(\rho,\omega, \phi)$ can be roughly reproduced in the range $1.4<\alpha <1.6$. While the parameter range for $\eta_c \to \gamma \mathcal{V}$ is determined by the similarity of $\eta_c \to \gamma \mathcal{V}/\mathcal{VV}$ and $\chi_{c1} \to \gamma \mathcal{V}/\mathcal{VV}$. In this parameter range, our estimations indicate that the branching fractions for the processes $\eta_c \to \gamma \rho^0$ and $\eta_c \to \gamma \omega$ are of order $10^{-4}$, while the one for the process $\eta_c \to \gamma \phi$ is of order $10^{-5}$. The estimations in the present work are expected to be accessible by future experiments at BESIII and Belle II.

We prove a dimension-free Markov–Bernstein-type inequality for polynomial mappings $(f_1,f_2)\colon\mathbb R^n\to\mathbb R^2$ of degree at most $d$. For product probability measures whose one-dimensional marginals have densities bounded by $1$, we obtain a lower bound for the integral of the Gram determinant of $\nabla f_1$ and $\nabla f_2$ in terms of the Gram determinant of the $d$-th derivative tensors $D^df_1$ and $D^df_2$, with a constant depending only on $d$. For pairs in a fixed Wiener chaos, the resulting Gaussian inequality answers a question of Nourdin, Nualart, and Poly concerning a lower bound for the expectation of the Malliavin determinant in terms of the determinant of the covariance matrix.

An ideal (language) model must continually update its knowledge. Existing work largely focuses on making these updates efficient after training. We investigate how the organization of time-varying knowledge during pre-training shapes the efficiency of subsequent updates. Models trained on chronologically ordered knowledge specialize in the most recently encountered knowledge; we call them specialists. In contrast, models trained on the same knowledge in randomly shuffled order perform well across all represented time periods; we call them generalists. Surprisingly, specialists require far fewer update steps than generalists to acquire new knowledge. To explain this advantage, we derive a law from first principles that relates update cost to the model's optimization dynamics and that closely predicts empirical results. Guided by these findings, we design a strategy for organizing pre-training data that preserves stable historical knowledge while enabling efficient updates to current knowledge. We also show that the benefits of temporal knowledge organization extend to real-world pre-trained models and existing knowledge update tools, and generalize to other knowledge orderings. Project page: https://horizon-lab.org/specialist/

Continuous diffusion language models (CDLMs) promise to bring the benefits of continuous flow modelling to language, including accelerated sampling and controllable generation. The field has grown rapidly, and the capabilities of CDLMs have begun to rival those of discrete diffusion. Yet, unlike for continuous modalities such as images or videos, our grasp of the levers that matter when training CDLMs remains limited. In this work, we focus on two of these levers, namely self-conditioning and embedding geometry, and shed light on some of the mechanisms behind them, which in turn suggest cheaper and more effective designs. We begin by showing that self-conditioning corrects a train/inference mismatch by exposing the model to inference-like states that are otherwise rare during training. These gains can be recovered without an extra forward pass by a two-level interpolant, coined $\mathrm{DoubleCat}$, which improves TinyGSM accuracy more than threefold. Self-conditioning also aids in recognising tokens dominated by large-norm Gaussians$\unicode{x2013}$an issue known as token identifiability. To address it directly, we introduce discrete reveals to $\mathrm{DoubleCat}$, yielding $\mathrm{SemiCat}$, which achieves discrete diffusion-level performance on TinyGSM, reaching $49.94\%$ with standard training hyperparameters and a tuned sampler. At the $1.7$B scale, $\mathrm{SemiCat}$ outperforms discrete diffusion on four of six likelihood-based evaluations, and surpasses one-hot flows on unconditional generation. On embedded spaces, we find that cross-entropy drives embeddings towards a geometry whose decoding error rate closely matches the one-hot one. We briefly explore a range of further options, such as hybrid corruption processes and noise schedules, including asynchronous ones. We believe this exploration can help the community better understand and advance the design space of CDLMs.

In 1981, Bourgain, Rosenthal, and Schechtman constructed uncountably many pairwise non-isomorphic complemented subspaces of $L_p$ for $1 < p < \infty$, $p \neq 2$. We extend their construction to rearrangement-invariant spaces and Hardy spaces. For any separable rearrangement-invariant space $X$ with non-trivial Boyd indices, we show that these subspaces are complemented, previously obtained by Ghawadrah via a different argument, and establish the following dichotomy: Either one can extract an uncountable family of pairwise non-isomorphic subspaces, or they are all isomorphic to $X$. To this end, we introduce an ordinal index that detects whether $X$ embeds complementably into an arbitrary separable Banach space. On the other hand, we give sufficient conditions which imply that in certain spaces close to $L_1$ and $H^1$, these subspaces fail to be complemented.