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Bridging Control, Inference, Transport, and Thermodynamics: From Theory to Applications in Learning
Review connects control theory, optimal transport, probabilistic inference, thermodynamics, and machine learning via free-energy optimization under constraints.
The review unifies five fields: control theory, optimal transport, probabilistic inference, non-equilibrium thermodynamics, and machine learning. The common conceptual thread is optimization of free-energy-like functionals under dynamical or statistical constraints. Selected applications are presented in reinforcement learning, variational inference, and generative modeling. The tutorial-style text assumes no prior familiarity and begins from physics principles.
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Unsolved Problem by Fields Medalist Breached by Two High School Students
Two high school students used Claude Opus 5 and GPT-5.6 Sol to help solve an open Lorentzian polynomials problem, posting a 75-page arXiv proof.
Aayush Bathija and Prince Rohatgi of Oak Park High School, mentored by UCLA postdoc Daniel Soskin, published the 75-page paper 'Bounded Ratios for Lorentzian Polynomials' (arXiv 2609.05341), solving an open problem in Fields Medalist June Huh's Lorentzian polynomial theory. The main structural theorem extends bounded coefficient-ratio characterization from quadratic to arbitrary-degree polynomials via discrete convexity conditions. The students used Claude Opus 5 and GPT-5.6 Sol for exploration and proof ideas but independently verified all arguments; the result follows an open letter from 25 Fields Medalists voicing concerns about AI's impact on mathematical rigor.
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When Intelligence Becomes Agency: A Theory of Governed, Proactive Agency for Symbiotic AI Systems
Researchers propose a formal framework for governed proactive AI agency, linking activation decisions to continuing authorization, accountability, and bounded delegation in symbiotic systems.
The paper defines the activation problem: persistent AI assistants must decide whether a situation warrants behavior, and whether to act, ask, monitor, defer, or deliberately refrain. It distinguishes autonomous from delegated agency and defines symbiotic agency as delegation under a standing, revocable mandate, coupled to the principal's situation with calibrated inference and bounded personalization. The framework links activation decisions to authorized perception, behavior selection, authority containment, traceable restraint, and constrained adaptation, and provides an agency classification method, evaluation framework, benchmark scenarios, and reference architecture for always-present assistants and embodied support systems.
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AI Doesn't Mean the End of Mathematics—at Least Not Yet
Schneier and Rafi argue frontier AI models produce notable mathematical results but cannot yet build genuinely new conceptual frameworks.
Bruce Schneier and Kasra Rafi, writing in The Guardian, argue current AI models are not yet as capable as experienced academic mathematicians despite striking results. They cite OpenAI's disproof of the unit distance conjecture, Anthropic's published cryptanalysis results, and Claude's attempt at the Riemann hypothesis as achievements in counterexample search and recombining known techniques. They contend AI has not yet developed substantial new conceptual frameworks, though they expect that capability sooner rather than later.
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