We establish that temporal averaging over multiple observations is the degenerate case of algebraic group action with the trivial group $G=\{e\}$. A General Replacement Theorem proves that a group-averaged estimator from one snapshot achieves equivalent subspace decomposition to multi-snapshot covariance estimation. The Trivial Group Embedding Theorem proves that the sample covariance is the accumulation of trivial-group estimates, with variance governed by a $(G,L)$ continuum as $1/(|G|\cdot L)$. The processing gain $10\log_{10}(M)$ dB equals the classical beamforming gain, establishing that this gain is a property of group order, not sensor count. The DFT, DCT, and KLT are unified as group-matched special cases. We conjecture a General Algebraic Averaging Theorem extending these results to arbitrary statistics, with variance governed by the effective group order $d_{\mathrm{eff}}$. Monte Carlo experiments on the first four sample moments across five group types confirm the conjecture to four-digit precision. The framework exploits the $structure$ of information (representation-theoretic symmetry of the data object) rather than the content, complementing Shannon's theory. Five applications are demonstrated: single-snapshot MUSIC, massive MIMO, single-pulse waveform classification, graph signal processing, and analysis of transformer LLMs. Techniques for blind group matching are described.

翻译：我们证明，多观测的时间平均是平凡群$G=\{e\}$的代数群作用的退化情形。通用替换定理表明，基于单次快照的群平均估计器可实现与多次快照协方差估计等效的子空间分解。平凡群嵌入定理证明，样本协方差是平凡群估计的累积，其方差由$(G,L)$连续谱控制，服从$1/(|G|\cdot L)$的衰减规律。处理增益$10\log_{10}(M)$ dB等于经典波束形成增益，表明该增益是群阶数的属性而非传感器数量的属性。DFT、DCT和KLT被统一为群匹配的特例。我们提出通用代数平均定理猜想，将上述结果推广至任意统计量，其方差由有效群阶数$d_{\mathrm{eff}}$控制。针对五种群类型的前四阶样本矩的蒙特卡洛实验以四位精度验证了这一猜想。该框架利用信息的结构（数据对象的表示论对称性）而非内容，是对香农理论的补充。文章展示了五个应用：单快照MUSIC、大规模MIMO、单脉冲波形分类、图信号处理及Transformer大语言模型分析。同时描述了盲群匹配技术。