发表机构
Max Planck Institute of Animal Behavior; University of Konstanz（马克斯·普朗克动物行为研究所; 康斯坦茨大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出规范框架，用环路和乐刻画集体中私有表征的共享条件，证明可共同持有内容由环路不变性决定，并推导未记录路径的最小均方误差。
AI 中文摘要
许多集体行为模型将方向、信念和意义置于一个由实验者定义的框架中。但生物体并不生活在该框架内：每个生物体在私有空间中表征世界，比较需要翻译。共识在成为动力学问题之前，首先是一个存在性问题。成对的双向翻译在环路周围不一定能一致地复合。这种返回变换（即和乐）能暴露出任何孤立的成对关系都无法承载的不匹配。我们发展了一个规范协变理论，其中物理预测在私有重标记下保持不变。对于互易的酉翻译，连接拉普拉斯算子的核与环路和乐的联合固定空间同构：一个集体能共同持有的，正是其所有环路保持不变的内容。盲子群判据识别不可见的缺陷。共同框架比较，如传统集体行为模型中的那样，占据平坦扇区。对于来自有限群$G$的独立均匀翻译，具有$N$个个体间$E$个关系的连通图以概率$|G|^{-(E-N+1)}$是平坦的。精确的环谱决定线性松弛下的持久性。其后果既是操作性的，也是结构性的。对于指定模型下未记录路径，且具有各向同性高斯源和读出噪声，我们推导了任何解码器可实现的最小均方误差。在固定信噪比下，共同保留的特征比例决定了其长路径极限。具有相同环路角度的网络，当保留轴不同时，可恢复内容可能不同。集体的关系几何决定了它能共享的状态和能恢复的内容。
英文摘要
Many models of collective behavior write headings, beliefs, and meanings in one experimenter-defined frame. Organisms do not live there: each represents the world in a private space, and comparison requires translation. Consensus becomes an existence problem before it becomes a dynamical one. Reciprocal pairwise translations need not compose consistently around a loop. This return transformation (the holonomy) can expose a mismatch no isolated reciprocal pair can carry. We develop a gauge-covariant theory in which physical predictions are invariant under private relabeling. For reciprocal unitary translations, the kernel of the connection Laplacian is isomorphic to the joint fixed space of loop holonomies: a collective can hold in common exactly what all its loops leave unchanged. A blind-subgroup criterion identifies invisible defects. Common-frame comparison, such as in traditional models of collective behaviour, occupies the flat sector. For independent uniform translations from a finite group $G$, a connected graph with $E$ relations among $N$ individuals is flat with probability $|G|^{-(E-N+1)}$. Exact cycle spectra determine persistence under linear relaxation. The consequence is operational as well as structural. For unrecorded routes under a specified model with isotropic Gaussian source and readout noise, we derive the smallest mean-squared error achievable by any decoder. At fixed signal-to-noise ratio, the jointly preserved feature fraction fixes its long-route limit. Networks with identical loop angles can differ in recoverable content when their preserved axes differ. The relational geometry of a collective sets both the states it can share and the content it can recover.
Comments19 pages; 7 figures
