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We introduce the Riemannian Lyapunov Framework (RLF), a unified framework that represents first order optimization as a closed loop controlled dynamical system on a Riemannian parameter manifold. Within this framework, optimizers are specified by a quadruple $(M, g, Φ, η)$ consisting of a Riemannian metric, an internal state transition map, a direction field generator, and a lifting parameter. Classical and modern methods including SGD, Adam, AdamW, Lion and Shampoo are recovered as specific instantiations of this quadruple. The central structural object of the framework is a time indexed family of target submanifolds in the extended state space of parameters, velocities, and internal memory, which we call the Normally Attracting Invariant Manifold (NAIM) family. These submanifolds organize training dynamics into two timescales: a fast contraction of the velocity state onto the current target submanifold, followed by a slow descent along it. Using Riemannian backstepping, we construct a strict Lyapunov function on the extended state space and prove Uniform Ultimate Boundedness (UUB) of the optimization trajectory under a descent alignment assumption and bounded stochastic disturbances. Under the additional Polyak-Lojasiewicz (PL) condition, the Lyapunov function converges linearly to a bounded neighborhood of the optimum, defined as the drift of the target direction field between consecutive iterates after parallel transport. To validate the framework's generativity, we derive three concrete instantiations of the quadruple, each exercising a distinct design axis, and show on geometric diagnostics, image classification benchmarks, and language modeling that their empirical behavior matches the theoretical predictions. The framework casts optimizer design as controller synthesis, complementing heuristic perspectives by providing a constructive template with provable guarantees.

We introduce Continuous-Action World Models (CoAWM), a framework for learning cross-embodiment spatiotemporal dynamics priors that support robot action generation across embodiments and execution frequencies. At its core is a shared latent function space that captures the spatial and temporal structure of robot actions. We learn this space through two stages. First, function pretraining brings trajectories from different embodiments and sampling frequencies into a common representation, learning shared dynamics priors through action reconstruction and consistency across temporal sampling views. By accommodating heterogeneous robot trajectories, this stage broadens the range of data sources available for pretraining and facilitates scaling to larger and more diverse datasets. Second, we connect the pretrained function space to a world-model backbone, enabling observations, language instructions, and the current robot state to condition the prediction of a latent action function. A single decoder takes any query time and returns the corresponding end-effector action. By separating shared dynamics priors from task-conditioned action selection and temporal sampling, our approach provides a unified foundation for transferring robot experience across embodiments and control frequencies.

Wearable devices generate continuous physiological and behavioral data, but converting these signals into clinically reviewable biomarker hypotheses remains labor-intensive. We introduce CoDaS, an AI co-data-scientist that integrates multi-agent hypothesis generation, deterministic statistical analysis, adversarial validation and literature-grounded interpretation under human oversight. Across three wearable cohorts comprising 9,279 participant-observations, CoDaS prioritized candidate associations for mental-health and metabolic endpoints after internal checks for replication, stability, robustness and leakage. The system identified related circadian-instability signals associated with depression, including sleep-duration variability in DWB ($ρ$ = 0.252, $p$ < 0.001) and sleep-onset variability in GLOBEM ($ρ$ = 0.126, $p$ < 0.001), and derived a wearable cardiovascular-fitness index associated with insulin resistance (steps/resting heart rate; $ρ$ = -0.374, $p$ < 0.001). Adding these features to demographic models produced modest gains ($ΔR^2$ = 0.040 for depression, 0.021 for insulin resistance). In a 12-clinician review totaling approximately 25 active hours, clinician validity judgments aligned with CoDaS confidence tiers ($ρ$ = 0.67, $p$ = 0.005), whereas added clinical value and confidence to act were rated lower. CoDaS supports traceable, hypothesis-generating prioritization of wearable candidate biomarkers.