Vector-HaSH and the Tolman-Eichenbaum Machine propose the hippocampal-entorhinal circuit factorizes content from a grid-cell scaffold, supporting compositional memory via ripple-mediated replay. Human electrophysiology shows multi-hop replay fidelity decays multiplicatively. We show VaCoAl, an algebro-deterministic hyperdimensional memory built from Galois-field linear-feedback shift registers, supplies a shared algebraic object.Specifically: (i) deterministic Galois-field diffusion offers a substrate-level alternative to random projections, ensuring quasi-orthogonality and bit-exact reproducibility; (ii) the path-integral Confidence Ratio provides a tractable model of multiplicative decay in multi-hop replay; (iii) VaCoAl's STDP-like path selection follows from architectural demands - similarity preservation and bounded search - constraining hippocampal computation.We map two distinct VaCoAl regimes to the EC-CA3 direct and EC-DG-CA3 trisynaptic pathways. Cellular evidence, including mossy-fiber detonator transmission and granule-cell sparse coding, supports a reading where the DG-CA3 pathway implements biophysical homologues of Galois-field arithmetic with approximate reversibility.Crucially, we connect this to Pearl's Ladder of Causation. Reversible GF(2) binding supplies the surgical-modification algebra required by the do-operator (rung 2). The dual architecture (Regime A anchoring the factual world, Regime B minting counterfactual worlds) supplies the parallel non-interfering substrate counterfactual reasoning provably requires (rung 3), yielding a profound Pearl-based evolutionary rationale.The framing proceeds in two tiers: VaCoAl is offered first as architectural correspondence, then as biophysical realization with approximate reversibility. We prove formal correspondences and derive testable iEEG predictions, bridging computational neuroscience and hyperdimensional computing.

翻译：Vector-HaSH与Tolman-Eichenbaum机器提出，海马-内嗅皮质回路将内容从网格细胞支架中分解出来，通过波纹介导的重放支持组合记忆。人类电生理学表明，多跳重放保真度呈乘性衰减。我们证明，由伽罗瓦域线性反馈移位寄存器构建的代数学确定性超维度记忆结构VaCoAl，提供了一个共享的代数对象。具体而言：(i) 确定性伽罗瓦域扩散为随机投影提供了基质级替代方案，确保准正交性与位精确可重复性；(ii) 路径积分置信度比为多跳重放中的乘性衰减提供了可处理模型；(iii) VaCoAl基于STDP的路径选择源于架构需求——相似性保持与有界搜索——从而约束了海马计算。我们将VaCoAl的两种不同模式映射到内嗅皮质-CA3直接通路与内嗅皮质-齿状回-CA3三突触通路。包括苔藓纤维引爆式传递和颗粒细胞稀疏编码在内的细胞学证据支持如下解读：齿状回-CA3通路以其近似可逆性实现了伽罗瓦域算术的生物物理学同源性。关键之处在于，我们将此与珀尔的因果阶梯联系起来。可逆伽罗瓦域GF(2)绑定提供了do算子（第二层级）所需的手术式修改代数。双重架构（模式A锚定现实世界，模式B铸造反事实世界）提供了反事实推理可证明必需的非并行干扰基质（第三层级），从而形成了深刻的基于珀尔的演化论依据。该框架分为两个层次：首先将VaCoAl作为架构对应关系提出，随后作为具有近似可逆性的生物物理学实现。我们证明了形式对应关系并推导出可检验的颅内脑电图预测，从而架起了计算神经科学与超维度计算的桥梁。