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

TL;DR

The paper introduces VaCoAl, an algebraic hyperdimensional memory architecture based on Galois fields, providing a deterministic substrate for hippocampal memory modeling and linking it to electrophysiological observations and causal reasoning.

Contribution

It presents VaCoAl, a novel Galois-field based memory model that offers deterministic, reproducible memory representations and connects algebraic structures to biological and causal frameworks.

Findings

VaCoAl matches quasi-orthogonality of random projections with bit-exact reproducibility.

It explains multiplicative replay decay via the Confidence Ratio CR2.

Biological pathways align with VaCoAl's operating regimes, supported by cellular evidence.

Abstract

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.