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Real-rootedness of the Poincar\'e polynomials of $\overline{\mathcal M}_{0,n}$: an AI-assisted proof

We prove real-rootedness for the Poincar\'e polynomial \[ P_n(t)=\sum_{i=0}^{n-3} \dim H^{2i}(\overline{\mathcal M}_{0,n};\mathbb{Q})t^i \] of the Deligne--Mumford moduli space $\overline{\mathcal M}_{0,n}$ of stable $n$-pointed rational curves, proving a conjecture of Aluffi--Chen--Marcolli. The proof starts from the Keel--Manin--Getzler recurrence, but its main new idea is a bivariate deformation $F_m(y,t)$ of the Poincar\'e polynomial. This deformation reveals a hidden interlacing structure not visible in the one-variable recurrence. For fixed $t<0$, the zero set of $F_m$ in the $y$-direction is controlled by a Sturm--Rolle argument on the interval $0<1-t$. The original polynomial is recovered on the slice $y=1$, and the ordered crossings of the moving roots through this slice give both real-rootedness and strict interlacing. Consequently, the Betti numbers of $\overline{\mathcal M}_{0,n}$ form an ultra-log-concave sequence. We further prove real-rootedness and ultra-log-concavity for the Poincar\'e polynomial of the Fulton--MacPherson space $\mathbb{P}^1[n]$ of $n$ ordered points in degenerations of the complex projective line. The proof for $\overline{\mathcal M}_{0,n}$ was obtained through an iterative AI-assisted workflow with Co-Mathematician, an agentic frontier-model system developed by Google DeepMind. The human role was to pose the problem, evaluate successive attempts, request repairs of gaps, compare the evolving argument with the literature, and assemble the final human-verifiable proof. Our additional human contribution was to observe that a similar residual deformation strategy applies to the Fulton--MacPherson spaces $\mathbb P^1[n]$, yielding the corresponding real-rootedness theorem.

Experiments in Agentic AI for Science

This paper details two novel frameworks for developing autonomous, agentic AI in scientific workflows. Both systems leverage a hybrid Local Body, Remote Brain architecture via Google Colab, utilizing Python-based local orchestrators to invoke large language model (LLM) cloud backends. The first agent, DeepTS/DeepCollector, automates the large-scale curation, extraction, and deduplication of time-series datasets. The second, DeepScribe, is an autonomous presentation analyzer that converts visually dense, mathematically complex physics lectures into structured scientific reports. Through practical systems engineering-such as granular attribute extraction (Cellular RAG), remote data inspection, and distributed concurrency controls-we demonstrate how agentic AI can overcome the context and reasoning limitations of current state-of-the-art systems to rigorously support scientific workflows. Finally, we outline a generalization of DeepTS to support deep knowledge graphs and discuss the application of this conceptual approach to high-energy physics (DeepQCD).

Authority Signals in Claude AI Health Citations: A Descriptive Analysis Using the Authority Signals Framework

This study seeks to determine the authority signals used by Anthropic's Claude AI in its presentation of sources when answering consumer health questions. While there exists a great deal of discourse around the quality of health citations that LLMs produce, there is limited information on the integrity of the sources the citations originate from, and to what extent the sources are, from what health professionals would consider, credible sources. This descriptive cross-sectional study used data from HealthSearchQA, which contains 3,172 consumer health questions curated by Google Research. After exclusions, a final dataset of 3,075 questions yielding 10,038 citations was analyzed. The Authority Signals Framework (Jacques et al., 2026) was applied to examine 10 authority signals across four domains for a disproportionate stratified sample of 542 sources. Established institutional sources accounted for 97.8% of all citations (n = 9,818). Medical Institutions were the most frequently cited organization type (36.5%), followed by Government Resources (31.6%) and Professional Associations (28.4%). Commercial Health Information comprised 2.2% (n = 220). The top 10 organizations accounted for 57.8% of all citations, with Mayo Clinic alone representing 24.7%. Among commercial sources in the focused sample, 86.4% displayed medical review statements, 82.5% used schema markup, and 71.8% had comprehensive content, while traditional institutional sources appeared in Claude's citations with or without these same markers. As Anthropic positions Claude for HIPAA-ready healthcare applications, these findings establish a baseline for Claude's citation behavior and demonstrate the utility of the Authority Signals Framework as a tool for ongoing, cross-platform evaluation of AI-mediated health information.

MadEvolve: Evolutionary Optimization of Trading Systems with Large Language Models

We explore the application of LLM-driven algorithm optimization to several common tasks in quantitative finance. MadEvolve, a general-purpose algorithm optimization framework inspired by DeepMind's Alpha-Evolve, was recently developed to optimize algorithms in computational cosmology. Here we demonstrate the utility of MadEvolve to optimize algorithmic trading strategies and alpha generation at the example of Bitcoin trading. On our simulation and backtesting setup, we achieve significant improvements on all tasks we considered, such as evolving feature sets for signal generation, optimizing separate components of the trading strategy, and jointly evolving the feature pipeline together with the execution strategy. Additionally, we compare our method to other agentic search approaches, specifically Claude Code, and carefully evaluate p-hacking probabilities on our simulation setup. Our findings strongly support the utility of AI-driven agentic and evolutionary algorithms for algorithmic trading and quantitative finance.

Dreaming Smoothly and Sample Efficiently with Gradient Penalized Latent Dynamics

Model-based reinforcement learning improves sample efficiency by learning a world model. However, existing latent world models such as DreamerV3 do not explicitly enforce local smoothness in their learned transition dynamics, leaving a useful inductive bias for transition dynamics learning unexploited. We propose GPLD, a gradient-penalized latent dynamics regularizer for DreamerV3 that applies a row-wise Jacobian penalty to the posterior latent distribution to encourage locally smooth transition learning. We show that this penalty can be interpreted as the continuous-latent analog of finite-difference smoothing of transition laws in discrete embedded-state MDPs, and estimate it efficiently using Hutchinson-style stochastic probes. Empirically, across DeepMind Control proprioceptive tasks, GPLD improves aggregate sample efficiency, with particularly strong gains on higher-complexity locomotion environments. On more challenging quadruped tasks, GPLD reaches high-return behavior earlier and exhibits more consistent late-stage learning over longer horizons. Explicit local smoothness regularization is a simple and effective way to improve latent world models for smooth continuous control environments. Code for GPLD is available at github.com/romils9/gpld-mbrl .

Reflex: Reinforcement Learning with Reflection Symmetry Exploitation in State-Based Continuous Control

Reinforcement learning has long struggled with poor sample efficiency. One promising approach to mitigate this problem is leveraging group-invariant Markov Decision Processes ($G$-invariant MDPs). Existing works in this direction have primarily focused on image-based RL and rotational symmetry such as $\mathrm{SO(2)}$, leaving state-based RL and reflection symmetry largely underexplored. In this work, we focus on state-based continuous control tasks and exploit reflection symmetry by introducing Reflex, a paradigm that seamlessly integrates with both on-policy and off-policy RL algorithms. We formalize two types of reflection-axial reflection and bilateral reflection, and characterize their corresponding transformations. Building on a theoretical analysis of symmetry-preserving optimal value functions and policies, Reflex integrates reflection symmetry into policy learning through principled symmetry regularization mechanisms. We integrate Reflex with PPO and SAC, and evaluate it on a suite of OpenAI Gym and DeepMind Control benchmarks, demonstrating superior performance over standard baselines while improving sample efficiency. Our code is available at https://github.com/TonyStark042/Reflex.