Scaling large models requires optimization strategies that ensure rapid convergence grounded in stability. Maximal Update Parametrization ($\boldsymbol{\mu}$P) provides a theoretical safeguard for width-invariant $\Theta(1)$ activation control, whereas emerging optimizers like Muon are only "half-aligned" with these constraints: they control updates but allow weights to drift. To address this limitation, we introduce the **Spectral Sphere Optimizer (SSO)**, which enforces strict module-wise spectral constraints on both weights and their updates. By deriving the steepest descent direction on the spectral sphere, SSO realizes a fully $\boldsymbol{\mu}$P-aligned optimization process. To enable large‑scale training, we implement SSO as an efficient parallel algorithm within Megatron. Through extensive pretraining on diverse architectures, including Dense 1.7B, MoE 8B-A1B, and 200-layer DeepNet models, SSO consistently outperforms AdamW and Muon. Furthermore, we observe significant practical stability benefits, including improved MoE router load balancing, suppressed outliers, and strictly bounded activations.

Each layer of a Transformer refines the hidden state toward a prediction, an iterative process resembling fixed-point iteration. Yet when should this iteration terminate? Existing early exit methods rely on output confidence as a proxy for internal convergence. We take a more direct approach by examining the geometry of the hidden state trajectory. We find that layer-wise updates exhibit a two-phase structure: large, volatile updates in early layers, followed by small, aligned updates as the model propagates an already-formed representation. The transition is remarkably sharp. This yields a simple criterion: exit when step size vanishes and direction stabilizes. We track the normalized update norm and cosine similarity between consecutive updates, exiting when both indicate convergence. The overhead is $O(d)$ per layer, independent of vocabulary size, requiring no learned components or architectural modifications. On LLaMA-2-7B and LLaMA-2-13B across question answering and commonsense reasoning tasks, this geometric criterion reduces FLOPs by 30--35\% while retaining over 98\% of full-depth accuracy.

Speculative Decodin promises to accelerate Large Language Model inference, yet its efficacy often degrades in production-grade scenarios. Existing evaluations typically overlook the compute-bound nature of high-concurrency regimes, where verification compute becomes the dominant bottleneck. Consequently, prior methods face a dilemma: static trees incur massive verification waste, while dynamic trees suffer from cumulative misjudgments and kernel incompatibility. To bridge this gap, we introduce ECHO, a high concurrency-oriented framework integrated into SGLang that reformulates speculative execution as a budgeted scheduling problem. Crucially, ECHO employs sparse confidence gating to manage the batch as a unified super-tree, elastically pivoting budget between depth and width to co-optimize the trade-off between reducing global verification steps and maximizing per-step efficiency. Extensive evaluations across diverse model scales—particularly the industrial-grade Qwen3-235B—demonstrate that ECHO consistently outperforms state-of-the-art baselines in both low-load and high-load scenarios, achieving up to 5.35$\times$ walltime speedup and delivering over 20\% relative speedup gain against the strongest baselines.

State space models (SSMs) have emerged as efficient linear-time alternatives to attention for long-sequence modeling. However, existing SSMs often suffer from instability and memory degradation over extended horizons due to poorly conditioned first-order updates and unbalanced update geometry. We introduce MuonSSM, a general framework that stabilizes SSM training by explicitly conditioning the geometry of memory updates rather than the recurrent transition matrix. MuonSSM augments SSMs with a momentum-based pathway and a lightweight Newton-Schulz transformation on low-rank input injections, yielding bounded and spectrally conditioned updates while preserving parallel scan complexity. Theory shows that MuonSSM improves gradient propagation, mitigates spectral amplification, and enriches memory representations over long horizons. Extensive experiments across language, vision, and time-series benchmarks show consistent gains in accuracy, robustness, and long-context performance when integrated into diverse SSM backbones. These results establish geometric conditioning of updates as a principled pathway to stable, scalable sequence modeling.