Reliable deployment of language models requires two capabilities that appear distinct but share a common geometric foundation: predicting whether a model will accept targeted behavioral control, and detecting when its internal structure degrades. We show that geometric stability, the consistency of a representation's pairwise distance structure, addresses both. Supervised Shesha variants that measure task-aligned geometric stability predict linear steerability with near-perfect accuracy ($ρ= 0.89$-$0.97$) across 35-69 embedding models and three NLP tasks, capturing unique variance beyond class separability (partial $ρ= 0.62$-$0.76$). A critical dissociation emerges: unsupervised stability fails entirely for steering on real-world tasks ($ρ\approx 0.10$), revealing that task alignment is essential for controllability prediction. However, unsupervised stability excels at drift detection, measuring nearly $2\times$ greater geometric change than CKA during post-training alignment (up to $5.23\times$ in Llama) while providing earlier warning in 73\% of models and maintaining a $6\times$ lower false alarm rate than Procrustes. Together, supervised and unsupervised stability form complementary diagnostics for the LLM deployment lifecycle: one for pre-deployment controllability assessment, the other for post-deployment monitoring.

What you need to know as Elon Musk's lawsuit against Sam Altman begins It's sure to be cringe, and may end up costing OpenAI billions. OpenAI CEO Sam Altman speaks during the BlackRock Infrastructure Summit on March 11, 2026 in Washington, DC. In a few short days, jury selection will begin in the long-awaited case. At the end of that process, an Oakland federal court will task nine regular people with deciding if OpenAI defrauded Elon Musk when it announced, and recently completed, its reorganization to become a more traditional for-profit business . More than just being the venue where two billionaires will air their grievances against one another in public, the trial has the potential to reshape the AI industry.

Analysis of learned representations has a blind spot: it focuses on $similarity$, measuring how closely embeddings align with external references, but similarity reveals only what is represented, not whether that structure is robust. We introduce $geometric$ $stability$, a distinct dimension that quantifies how reliably representational geometry holds under perturbation, and present $Shesha$, a framework for measuring it. Across 2,463 configurations in seven domains, we show that stability and similarity are empirically uncorrelated ($ρ\approx 0.01$) and mechanistically distinct: similarity metrics collapse after removing the top principal components, while stability retains sensitivity to fine-grained manifold structure. This distinction yields actionable insights: for safety monitoring, stability acts as a functional geometric canary, detecting structural drift nearly 2$\times$ more sensitively than CKA while filtering out the non-functional noise that triggers false alarms in rigid distance metrics; for controllability, supervised stability predicts linear steerability ($ρ= 0.89$-$0.96$); for model selection, stability dissociates from transferability, revealing a geometric tax that transfer optimization incurs. Beyond machine learning, stability predicts CRISPR perturbation coherence and neural-behavioral coupling. By quantifying $how$ $reliably$ systems maintain structure, geometric stability provides a necessary complement to similarity for auditing representations across biological and computational systems.

Understanding the dynamic reorganization of brain networks is critical for predicting cognitive decline, neurological progression, and individual variability in clinical outcomes. This work proposes a multimodal graph neural network framework that integrates structural MRI, diffusion tensor imaging, and functional MRI to model spatiotemporal brain network reorganization. Brain regions are represented as nodes and structural and functional connectivity as edges, forming longitudinal brain graphs for each subject. Temporal evolution is captured via fractional stochastic differential operators embedded within graph-based recurrent networks, enabling the modeling of long-term dependencies and stochastic fluctuations in network dynamics. Attention mechanisms fuse multimodal information and generate interpretable biomarkers, including network energy entropy, graph curvature, fractional memory indices, and modality-specific attention scores. These biomarkers are combined into a composite prognostic index to quantify individual risk of network instability or cognitive decline. Experiments on longitudinal neuroimaging datasets demonstrate both predictive accuracy and interpretability. The results highlight the potential of mathematically rigorous, multimodal graph-based approaches for deriving clinically meaningful biomarkers from existing imaging data without requiring new data collection.

Phase transitions have been proposed as the origin of emergent abilities in large language models (LLMs), where new capabilities appear abruptly once models surpass critical thresholds of scale. Prior work, such as that of Wei et al., demonstrated these phenomena under model and data scaling, with transitions revealed after applying a log scale to training compute. In this work, we ask three complementary questions: (1) Are phase transitions unique to large models, or can they also be observed in small transformer-based language models? (2) Can such transitions be detected directly in linear training space, rather than only after log rescaling? and (3) Can these transitions emerge at early stages of training? To investigate, we train a small GPT-style transformer on a character-level corpus and analyze the evolution of vocabulary usage throughout training. We track the average word length, the number of correct versus incorrect words, and shifts in vocabulary diversity. Building on these measures, we apply Poisson and sub-Poisson statistics to quantify how words connect and reorganize. This combined analysis reveals a distinct transition point during training. Notably, these transitions are not apparent in standard loss or validation curves, but become visible through our vocabulary- and statistics-based probes. Our findings suggest that phase-transition reorganizations are a general feature of language model training, observable even in modest models, detectable directly in linear training space, and occurring surprisingly early as coherence emerges. This perspective provides new insight into the nonlinear dynamics of language model training and underscores the importance of tailored metrics for uncovering phase transition behaviors

There are thus no fair comparisons that can be made with such methods.

How can we detect when global events fundamentally reshape public discourse? This study introduces a topological framework for identifying structural change in media narratives using persistent homology. Drawing on international news articles surrounding major events - including the Russian invasion of Ukraine (Feb 2022), the murder of George Floyd (May 2020), the U.S. Capitol insurrection (Jan 2021), and the Hamas-led invasion of Israel (Oct 2023) - we construct daily co-occurrence graphs of noun phrases to trace evolving discourse. Each graph is embedded and transformed into a persistence diagram via a Vietoris-Rips filtration. We then compute Wasserstein distances and persistence entropies across homological dimensions to capture semantic disruption and narrative volatility over time. Our results show that major geopolitical and social events align with sharp spikes in both H0 (connected components) and H1 (loops), indicating sudden reorganization in narrative structure and coherence. Cross-correlation analyses reveal a typical lag pattern in which changes to component-level structure (H0) precede higher-order motif shifts (H1), suggesting a bottom-up cascade of semantic change. An exception occurs during the Russian invasion of Ukraine, where H1 entropy leads H0, possibly reflecting top-down narrative framing before local discourse adjusts. Persistence entropy further distinguishes tightly focused from diffuse narrative regimes. These findings demonstrate that persistent homology offers a mathematically principled, unsupervised method for detecting inflection points and directional shifts in public attention - without requiring prior knowledge of specific events. This topological approach advances computational social science by enabling real-time detection of semantic restructuring during crises, protests, and information shocks.