Bridging Silicon and the Hippocampus: Algebro-Deterministic Memory "VaCoAl" as a Substrate for Vector-HaSH and TEM
For computational neuroscientists and hyperdimensional computing engineers, this work bridges two previously parallel literatures with a shared algebraic object, though the novelty is incremental as it applies existing hyperdimensional methods to hippocampal modeling.
The paper introduces VaCoAl, an algebro-deterministic hyperdimensional memory architecture that provides a formal substrate for hippocampal models Vector-HaSH and TEM, offering bit-exact reproducibility and stronger avalanche behavior than random projections, and derives the first algebraically tractable model of multi-hop replay-fidelity decay as a product of per-step confidence ratios.
Vector-HaSH and the Tolman-Eichenbaum Machine (TEM) propose that the hippocampal-entorhinal circuit factorizes content from a prestructured grid-cell scaffold and supports compositional memory via ripple-mediated replay. Human iEEG shows that hippocampal sharp-wave ripples (SWRs) gate episodic recall, ripple-locked cortical reactivation recapitulates encoding-time patterns, and multi-hop replay fidelity decays multiplicatively along sequence length. These literatures have advanced in parallel without a shared algebraic object. We show that VaCoAl, an algebro-deterministic hyperdimensional memory architecture built from Galois-field LFSRs, supplies that object. Specifically, deterministic Galois-field diffusion provides a substrate-level alternative to Vector-HaSH's random scaffold-to-hippocampus projection that satisfies the same quasi-orthogonality requirement, with matched second-moment statistics, stronger avalanche behavior, and bit-exact reproducibility. The path-integral Confidence Ratio $CR_2$, the product of per-step $CR_1$ values along an $n$-hop chain, is the natural functional form for multi-hop replay-fidelity decay under conditional independence of per-step reactivation, providing the first algebraically tractable model of reported multiplicative decay. STDP-like path selection in VaCoAl follows from architectural demands -- similarity preservation, compositional reversibility, and bounded-frontier search -- that also constrain hippocampal computation. We further argue that VaCoAl operating regimes share architectural commitments with the EC--CA3 and EC--DG--CA3 pathways, motivating an energy-capacity-plasticity reading of why both are conserved across $>$520 Myr of evolution and primate dentate-gyrus elaboration. We prove formal correspondences, derive testable iEEG predictions, and bridge computational neuroscience and hyperdimensional engineering.