Vector-HaSH and the Tolman-Eichenbaum Machine (TEM) propose the hippocampal-entorhinal circuit factorizes memory via a grid-cell scaffold for compositional replay. Concurrently, human iEEG shows sharp-wave ripples gate recall and multi-hop replay fidelity decays multiplicatively. Yet, these fields lack a shared algebraic foundation. We introduce VaCoAl, an algebro-deterministic hyperdimensional memory architecture built on Galois-field linear-feedback shift registers. Its deterministic Galois-field diffusion offers a substrate-level alternative to Vector-HaSH's random projections, matching quasi-orthogonality while ensuring bit-exact reproducibility. Furthermore, the path-integral Confidence Ratio CR2 provides an algebraically tractable model for the empirically observed multiplicative replay decay. Biologically, VaCoAl's two operating regimes align with the EC-CA3 direct and EC-DG-CA3 trisynaptic pathways, explaining their 520-Myr conservation. Independent cellular evidence supports that the DG-CA3 pathway implements a biophysical homologue of Galois-field arithmetic. We also link this framework to Judea Pearl's Ladder of Causation. Reversible GF(2) binding provides the surgical algebra for the do-operator (Rung 2), and VaCoAl's dual-orthogonalizer architecture supplies the parallel substrate required for counterfactual reasoning (Rung 3). Ultimately, we prove these formal correspondences and derive testable iEEG predictions, uniting computational neuroscience, electrophysiology, and hyperdimensional computing.

翻译：向量哈希（Vector-HaSH）与托尔曼-艾兴鲍姆机（Tolman-Eichenbaum Machine, TEM）提出海马-内嗅回路通过网格细胞支架实现记忆的因子化组合回放。与此同时，人类颅内脑电图（iEEG）显示尖波涟漪调控记忆提取，且多跳回放保真度呈乘性衰减。然而，这些领域缺乏共享的代数基础。我们提出VaCoAl——一种基于伽罗瓦域线性反馈移位寄存器的代数确定性超维记忆架构。其确定性伽罗瓦域扩散机制替代了向量哈希的随机投影方案，在保持准正交性的同时确保位精确可复现性。此外，路径积分置信比CR2为实验观测到的乘性回放衰减提供了代数可解模型。在生物学层面，VaCoAl的两种工作模式分别对应EC-CA3直接通路与EC-DG-CA3三突触通路，解释了其5.2亿年的保守性。独立细胞学证据支持DG-CA3通路实现伽罗瓦域算术的生物物理同构体。我们进一步将该框架与朱迪亚·珀尔（Judea Pearl）的因果阶梯理论关联：可逆GF(2)绑定提供do算子（第二层级）的手术式代数，而VaCoAl的双正交化架构则为反事实推理（第三层级）提供并行基底。最终，我们证明这些形式对应关系并推导出可检验的iEEG预测，统一了计算神经科学、电生理学与超维计算领域。