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

arXiv: https://arxiv.org/abs/2605.15652 · HTML: https://mlsanity.com/paper/2605.15652/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 61 references that are in the map.
- First posted on arXiv 15 May 2026.

## Builds on (most cited first, 3 of 3)

- [A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part I: Models and Data Transformations](https://mlsanity.com/paper/2111.06077/) (2021)
- [Linear Codes for Hyperdimensional Computing](https://mlsanity.com/paper/2403.03278/) (2024)
- [Beyond LLMs, Sparse Distributed Memory, and Neuromorphics <A Hyper-Dimensional SRAM-CAM "VaCoAl" for Ultra-High Speed, Ultra-Low Power, and Low Cost>](https://mlsanity.com/paper/2604.11665/) (2026)

## Built on by (most cited first, 2 of 2)

- [How to Build Marcus's Algebraic Mind: Algebro-Deterministic Substrate over Galois Fields](https://mlsanity.com/paper/2605.21379/) (2026)
- [Where Polanyi Meets Galois: Tacit Knowing as an Algebro-Deterministic Operation, Toward Re-rooting AI-Generated Science](https://mlsanity.com/paper/2610.16102/) (2026)

## Abstract (arXiv)

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.