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 with 64% throughput gain, single-pulse waveform classification at 90% accuracy, graph signal processing with non-abelian groups, and algebraic analysis of transformer LLMs.

翻译：我们证明，基于多次观测的时间平均是平凡群$G=\{e\}$作用下代数群作用的退化情形。一个一般替换定理表明，来自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价的子空间分解。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$(G,L)$连续统以$1/(|G|\cdot L)$的规律控制。处理增益$10\log_{10}(M)$ dB等于经典波束形成增益，从而确立该增益是群阶的性质而非传感器数量的函数。DFT、DCT和KLT被统一为群匹配的特殊情形。我们推测存在一个将上述结果推广至任意统计量的一般代数平均定理，其方差由有效群阶$d_{\mathrm{eff}}$控制。基于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。文中展示了五个应用：单快照MUSIC、吞吐量提升64%的大规模MIMO、90%准确率的单脉冲波形分类、非阿贝尔群的图信号处理，以及变压器大语言模型的代数分析。