Bridging Silicon and the Hippocampus: Algebro-Deterministic Memory "VaCoAl" as a Substrate for Vector-HaSH and TEM

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.