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：Vector-HaSH与Tolman-Eichenbaum机器（TEM）提出，海马-内嗅皮层回路通过网格细胞支架实现记忆的因子化分解以进行组合性回放。与此同时，人类颅内脑电图显示尖波涟漪门控回忆过程，且多跳回放保真度呈乘性衰减。然而，这些领域目前缺乏统一的代数基础。我们提出VaCoAl——一种基于伽罗瓦域线性反馈移位寄存器的代-确定性超维记忆架构。其确定性伽罗瓦域扩散机制为Vector-HaSH的随机投影提供了基底级替代方案，在保持准正交性的同时确保位级精确可复现性。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。从生物学角度，VaCoAl的两种运行模式分别对应EC-CA3直接通路与EC-DG-CA3三突触通路，解释了其5.2亿年的进化保守性。独立细胞学研究证据支持DG-CA3通路实现了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·珀尔的因果阶梯理论建立联系。可逆的GF(2)绑定为do-算子（阶梯第二层）提供了手术级代数工具，而VaCoAl的双正交化架构则为反事实推理（阶梯第三层）提供了并行基底。最终，我们证明了这些形式对应关系并推导出可经颅内脑电图检验的预测，从而统一了计算神经科学、电生理学与超维计算领域。