Persistent memory is a central capability for AI agents, yet the mathematical foundations of memory retrieval, lifecycle management, and consistency remain unexplored. Current systems employ cosine similarity for retrieval, heuristic decay for salience, and provide no formal contradiction detection. We establish information-geometric foundations through three contributions. First, a retrieval metric derived from the Fisher information structure of diagonal Gaussian families, satisfying Riemannian metric axioms, invariant under sufficient statistics, and computable in O(d) time. Second, memory lifecycle formulated as Riemannian Langevin dynamics with proven existence and uniqueness of the stationary distribution via the Fokker-Planck equation, replacing hand-tuned decay with principled convergence guarantees. Third, a cellular sheaf model where non-trivial first cohomology classes correspond precisely to irreconcilable contradictions across memory contexts. On the LoCoMo benchmark, the mathematical layers yield +12.7 percentage points over engineering baselines across six conversations, reaching +19.9 pp on the most challenging dialogues. A four-channel retrieval architecture achieves 75% accuracy without cloud dependency. Cloud-augmented results reach 87.7%. A zero-LLM configuration satisfies EU AI Act data sovereignty requirements by architectural design. To our knowledge, this is the first work establishing information-geometric, sheaf-theoretic, and stochastic-dynamical foundations for AI agent memory systems.

翻译：持久记忆是AI智能体的核心能力，然而记忆检索、生命周期管理和一致性的数学基础仍未得到探索。现有系统采用余弦相似度进行检索，使用启发式衰减处理显著性，且未提供形式化的矛盾检测方法。我们通过三项贡献建立了信息几何基础。首先，提出一种从对角高斯族的费希尔信息结构导出的检索度量，该度量满足黎曼度量公理，在充分统计量下具有不变性，并可在O(d)时间内计算。其次，将记忆生命周期建模为黎曼朗之万动力学，通过福克-普朗克方程证明了稳态分布的存在性与唯一性，从而用具有理论保证的收敛性取代了手动调整的衰减机制。第三，构建了一个胞腔层模型，其中非平凡的第一上同调类精确对应记忆上下文间不可调和的矛盾。在LoCoMo基准测试中，数学层在六轮对话中相比工程基线实现了+12.7个百分点的提升，在最富挑战性的对话中达到+19.9个百分点。四通道检索架构在不依赖云端的情况下实现了75%的准确率。云端增强结果达到87.7%。零LLM配置通过架构设计满足了欧盟《人工智能法案》的数据主权要求。据我们所知，这是首个为AI智能体记忆系统建立信息几何、层理论和随机动力学基础的研究工作。