Geometric Signatures of Reasoning: A Spectral Perspective on Task Hardness

Chain-of-thought (CoT) reasoning enables large language models (LLMs) to solve complex problems by generating intermediate reasoning steps. While much attention has been paid to the length and content of these reasoning chains, far less is known about their internal geometry. We study the \emph{geometry} of CoT trajectories in the hidden state space of transformer models, formalizing each reasoning chain as a discrete curve in $\mathbb{R}^d$ and characterizing it through spectral, positional, and kinematic geometric functionals. We introduce the effective dimension $d_ρ$ as a measure of trajectory complexity and show theoretically that trajectories with flatter eigenvalue spectra correspond to harder tasks, as they explore more of the hidden dimensions. Lastly, we explore how kinematic features of the trajectory, mean position, positional dispersion, initial and current hidden states, mean velocity, mean speed, and speed dispersion, can be used to predict solution correctness before generation is complete, and may inform future early-stopping strategies. Experimentally, on mathematical reasoning problems from the MATH500 dataset, $d_ρ$ achieves $0.93$ AUC in distinguishing easy from hard problems, while kinematic features potentially can predict correctness from only the first $20\%$ of generated tokens. These correctness signatures transfer across questions of varying difficulty, establishing that the shape of a model's internal reasoning trajectory is a principled window into both task hardness and solution quality.

Phantoms and Disclosures: a Causal Framework for Auditing Synthetic Data

The rapid adoption of generative AI and Large Language Models (LLMs) has spurred interest in synthetic data as a privacy-preserving alternative to sensitive real-world datasets. However, generating high-utility synthetic data often carries the risk of memorizing and regurgitating private information from the training corpus. In this work, we present a customizable empirical auditing framework designed to detect and explain such data disclosures. Our framework introduces a mechanism to distinguish between "true disclosures"-where the system directly reproduces a user's information-and "phantom disclosures''-where the system incidentally generates a user's data. By partitioning input data into training and holdout sets and applying rigorous statistical hypothesis testing, we determine if observed disclosures are consistent with strict privacy baselines, such as zero-learning or specific Differential Privacy (DP) bounds. Crucially, this approach requires no model access, no canary insertion, and no reference model training -only the synthetic output and a held-out control set. We demonstrate that this framework effectively functions as a membership inference attack, providing empirical lower bounds on privacy leakage that are tighter than prior data-based auditing methods. Our approach is model-agnostic, applies to any synthetic data generation mechanism, and requires orders of magnitude fewer computational resources than shadow-model or canary-based alternatives.

InfoSFT: Learn More and Forget Less with Information-Aware Token Weighting

Supervised fine-tuning (SFT) provides the standard approach for teaching LLMs new behaviors from offline expert demonstrations. However, standard SFT uniformly fits all samples -- including those with low likelihood under the base model -- which can disproportionately drive training updates toward overfitting specific samples rather than learning the target behavior. Moreover, adapting to these unlikely samples induces substantial policy shifts that degrade prior capabilities. Existing methods mitigate this by filtering, regenerating, or down-weighting low-likelihood data. In doing so, they often suppress precisely the novel behaviors the base model has yet to learn. We propose InfoSFT, a principled weighting scheme for the SFT objective that concentrates learning signals on maximally informative, medium-confidence tokens -- those neither overly familiar to the base model nor too unlikely to cause instability. Requiring only a one-line modification to the standard token-wise loss, InfoSFT demonstrably improves generalization over vanilla SFT and likelihood-weighted baselines across math, code, and chain-of-thought tasks with diverse model families, while better preserving pre-existing capabilities.

Theoretical Perspectives on Data Quality and Synergistic Effects in Pre- and Post-Training Reasoning Models

Large Language Models (LLMs) are pretrained on massive datasets and later instruction-tuned via supervised fine-tuning (SFT) or reinforcement learning (RL). Best practices emphasize large, diverse pretraining data, whereas post-training operates differently: SFT relies on smaller, high-quality datasets, while RL benefits more from scale, with larger amounts of feedback often outweighing label quality. Yet it remains unclear why pretraining and RL require large datasets, why SFT excels on smaller ones, and what defines high-quality SFT data. In this work, we theoretically analyze transformers trained on an in-context weight prediction task for linear regression. Our analysis reveals several key findings: $(i)$ balanced pretraining data can induce latent capabilities later activated during post-training, and $(ii)$ SFT learns best from a small set of examples challenging for the pretrained model, while excessively large SFT datasets may dilute informative pretraining signals. In contrast, RL is most effective on large-scale data that is not overly difficult for the pretrained model. We validate these theoretical insights with experiments on large nonlinear transformer architectures.

Aletheia tackles FirstProof autonomously

We report the performance of Aletheia (Feng et al., 2026b), a mathematics research agent powered by Gemini 3 Deep Think, on the inaugural FirstProof challenge. Within the allowed timeframe of the challenge, Aletheia autonomously solved 6 problems (2, 5, 7, 8, 9, 10) out of 10 according to majority expert assessments; we note that experts were not unanimous on Problem 8 (only). For full transparency, we explain our interpretation of FirstProof and disclose details about our experiments as well as our evaluation. Raw prompts and outputs are available at https://github.com/google-deepmind/superhuman/tree/main/aletheia.

Robust Policy Optimization to Prevent Catastrophic Forgetting

Large language models are commonly trained through multi-stage post-training: first via RLHF, then fine-tuned for other downstream objectives. Yet even small downstream updates can compromise earlier learned behaviors (e.g., safety), exposing a brittleness known as catastrophic forgetting. This suggests standard RLHF objectives do not guarantee robustness to future adaptation. To address it, most prior work designs downstream-time methods to preserve previously learned behaviors. We argue that preventing this requires pre-finetuning robustness: the base policy should avoid brittle high-reward solutions whose reward drops sharply under standard fine-tuning. We propose Fine-tuning Robust Policy Optimization (FRPO), a robust RLHF framework that optimizes reward not only at the current policy, but across a KL-bounded neighborhood of policies reachable by downstream adaptation. The key idea is to ensure reward stability under policy shifts via a max-min formulation. By modifying GRPO, we develop an algorithm with no extra computation, and empirically show it substantially reduces safety degradation across multiple base models and downstream fine-tuning regimes (SFT and RL) while preserving downstream task performance. We further study a math-focused RL setting, demonstrating that FRPO preserves accuracy under subsequent fine-tuning.

Accelerating Scientific Research with Gemini: Case Studies and Common Techniques

Recent advances in large language models (LLMs) have opened new avenues for accelerating scientific research. While models are increasingly capable of assisting with routine tasks, their ability to contribute to novel, expert-level mathematical discovery is less understood. We present a collection of case studies demonstrating how researchers have successfully collaborated with advanced AI models, specifically Google's Gemini-based models (in particular Gemini Deep Think and its advanced variants), to solve open problems, refute conjectures, and generate new proofs across diverse areas in theoretical computer science, as well as other areas such as economics, optimization, and physics. Based on these experiences, we extract common techniques for effective human-AI collaboration in theoretical research, such as iterative refinement, problem decomposition, and cross-disciplinary knowledge transfer. While the majority of our results stem from this interactive, conversational methodology, we also highlight specific instances that push beyond standard chat interfaces. These include deploying the model as a rigorous adversarial reviewer to detect subtle flaws in existing proofs, and embedding it within a "neuro-symbolic" loop that autonomously writes and executes code to verify complex derivations. Together, these examples highlight the potential of AI not just as a tool for automation, but as a versatile, genuine partner in the creative process of scientific discovery.

Differentially Private Model-X Knockoffs via Johnson-Lindenstrauss Transform

We introduce a novel privatization framework for high-dimensional controlled variable selection. Our framework enables rigorous False Discovery Rate (FDR) control under differential privacy constraints. While the Model-X knockoff procedure provides FDR guarantees by constructing provably exchangeable ``negative control" features, existing privacy mechanisms like Laplace or Gaussian noise injection disrupt its core exchangeability conditions. Our key innovation lies in privatizing the data knockoff matrix through the Gaussian Johnson-Lindenstrauss Transformation (JLT), a dimension reduction technique that simultaneously preserves covariate relationships through approximate isometry for $(\epsilon,\delta)$-differential privacy. We theoretically characterize both FDR and the power of the proposed private variable selection procedure, in an asymptotic regime. Our theoretical analysis characterizes the role of different factors, such as the JLT's dimension reduction ratio, signal-to-noise ratio, differential privacy parameters, sample size and feature dimension, in shaping the privacy-power trade-off. Our analysis is based on a novel `debiasing technique' for high-dimensional private knockoff procedure. We further establish sufficient conditions under which the power of the proposed procedure converges to one. This work bridges two critical paradigms -- knockoff-based FDR control and private data release -- enabling reliable variable selection in sensitive domains. Our analysis demonstrates that structural privacy preservation through random projections outperforms the classical noise addition mechanism, maintaining statistical power even under strict privacy budgets.

Self-Boost via Optimal Retraining: An Analysis via Approximate Message Passing

Retraining a model using its own predictions together with the original, potentially noisy labels is a well-known strategy for improving the model performance. While prior works have demonstrated the benefits of specific heuristic retraining schemes, the question of how to optimally combine the model's predictions and the provided labels remains largely open. This paper addresses this fundamental question for binary classification tasks. We develop a principled framework based on approximate message passing (AMP) to analyze iterative retraining procedures for two ground truth settings: Gaussian mixture model (GMM) and generalized linear model (GLM). Our main contribution is the derivation of the Bayes optimal aggregator function to combine the current model's predictions and the given labels, which when used to retrain the same model, minimizes its prediction error. We also quantify the performance of this optimal retraining strategy over multiple rounds. We complement our theoretical results by proposing a practically usable version of the theoretically-optimal aggregator function for linear probing with the cross-entropy loss, and demonstrate its superiority over baseline methods in the high label noise regime.

DeepCrossAttention: Supercharging Transformer Residual Connections

Transformer networks have achieved remarkable success across diverse domains, leveraging a variety of architectural innovations, including residual connections. However, traditional residual connections, which simply sum the outputs of previous layers, can dilute crucial information. This work introduces DeepCrossAttention (DCA), an approach that enhances residual learning in transformers. DCA employs learnable, input-dependent weights to dynamically combine layer outputs, enabling the model to selectively focus on the most relevant information in any of the previous layers. Furthermore, DCA incorporates depth-wise cross-attention, allowing for richer interactions between layers at different depths. Our language modeling experiments show that DCA achieves improved perplexity for a given training time. Moreover, DCA obtains the same model quality up to 3x faster while adding a negligible number of parameters. Theoretical analysis confirms that DCA provides an improved trade-off between accuracy and model size when the ratio of collective layer ranks to the ambient dimension falls below a critical threshold.