Lirong Che, Peidong Wang, Yimeng Su, Jiong Wang, Yang Chen, Wenbo Fu, Zhenyu Huang, Tiancheng Xu 等 14 位作者
Embodied agents are usually organized around individual tasks, yet the physical world keeps changing between tasks and during execution. We introduce LIFE (Lifelong Intelligence from Experience), a framework in which a robot observes continuously, remembers selectively, and turns experience into new capabilities. RoboBeats, a family of lightweight streaming gates, tracks conditions set by the agent and discovers new events, deciding when an observation warrants a response or a memory write. Selected observations enter Plastic Memory, a flat store of images and structured records that keeps the source evidence, so new fields and indexes can be added when later questions need them. When existing tools fall short, Explore-Distill-Mount (EDM) turns solutions found during problem solving into validated, reusable tools and skills. Under matched protocols, RoboBeats substantially improves condition tracking over its base models and, at the same writing budget, retains more useful evidence for memory question answering than uniform sampling. A case study of limited scale, replaying 24 hours of continuous video recorded by a real robot, illustrates how the integrated system builds memory from a day of experience and uses it to answer everyday questions. LIFE takes a step toward robots that become more familiar with their surroundings and more capable through experience: learning not only to complete tasks, but to live in the world.
Mordell's problem asks for the largest product of squared pairwise distances among $n$ points in the complex plane with a prescribed sum of squared distances from the origin. When this sum equals $n$, the maximum for $2\le n\le5$ is known to be $n^n$, attained precisely by the $n$th roots of unity up to rotation and relabeling. For $n\ge6$, Chalk showed that a regular polygon together with its center gives a larger product than the regular $n$-gon. We prove that for $6\le n\le9$ the maximum is $(n-1)^{n-1}r_n^{n(n-1)}$, where $r_n=\sqrt{n/(n-1)}$. Up to the same symmetries, the unique maximizer consists of the origin and the vertices of a regular $(n-1)$-gon of radius $r_n$. For six points, algebraic elimination reduces the proof to an exact comparison of stationary values. For seven points, we compare all twelve classes in a published classification of equilibria. For eight and nine points, rigorous interval arithmetic establishes an inequality for every admissible pair. Summing over pairs gives the sharp global bound, and the equality conditions determine the configuration. An explicit ten-point comparison and a known instability result together exclude Chalk's configuration for every $n\ge10$. For each $n$ from two through eight, Mordell's problem and the minimum-diameter problem studied by Bateman and Erdős have a common optimal configuration.