# Where Polanyi Meets Galois: Tacit Knowing as an Algebro-Deterministic Operation, Toward Re-rooting AI-Generated Science

arXiv: https://arxiv.org/abs/2610.16102 · HTML: https://mlsanity.com/paper/2610.16102/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 41 references that are in the map.
- First posted on arXiv 12 Oct 2026.

## Builds on (most cited first, 10 of 10)

- [The Reversal Curse: LLMs trained on "A is B" fail to learn "B is A"](https://mlsanity.com/paper/2309.12288/) (2023)
- [Can LLMs Generate Novel Research Ideas? A Large-Scale Human Study with 100+ NLP Researchers](https://mlsanity.com/paper/2409.04109/) (2024)
- [What Do Large Language Models Know? Tacit Knowledge as a Potential Causal-Explanatory Structure](https://mlsanity.com/paper/2504.12187/) (2025)
- [Hyperdimensional Probe: Decoding LLM Representations via Vector Symbolic Architectures](https://mlsanity.com/paper/2509.25045/) (2025)
- [Bridging Silicon and the Hippocampus: Algebro-Deterministic Memory "VaCoAl" as a Substrate for Vector-HaSH and TEM](https://mlsanity.com/paper/2605.15652/) (2026)
- [How to Build Marcus's Algebraic Mind: Algebro-Deterministic Substrate over Galois Fields](https://mlsanity.com/paper/2605.21379/) (2026)
- [Institutions and the transmission of upper-tail human capital: scientific lineages across a millennium](https://mlsanity.com/paper/2605.31470/) (2026)
- [How to Build Marcus's Algebraic Mind: From Thagard's Brain--Mind Viewpoint](https://mlsanity.com/paper/2607.16573/) (2026)
- [How to Build Marcus's Algebraic Mind: From Minsky's Emotion-Machine Viewpoint](https://mlsanity.com/paper/2607.16623/) (2026)
- [Computational Prosopography across a Millennium: Mathematically Oriented Lineages Traced from the Fields Medalists](https://mlsanity.com/paper/2608.17915/) (2026)

## Abstract (arXiv)

We argue that tacit knowing is better modelled as an operating regime of one memory substrate than as a kind of content. The substrate is VaCoAl, an algebro-deterministic XOR-and-shift hyperdimensional memory over GF(2), realized as CASRAM (commercial SRAM-CAM) and PyVaCoAl (DRAM-CAM).In VaCoAl/PyVaCoAl a directed acyclic graph (DAG) reads two ways. Repairing every collision (Rescue mode) makes the trace exactly auditable but ranks no path higher; tolerating collisions (Don't Care mode) yields a path-integrated confidence, CR2, that silently favours direct routes. This audit-valuation complementarity is a formal counterpart of Polanyi's claim that tacit and explicit knowledge are not sharply divided. It separates contingent tacitness, which a new query brings out (AlphaGo's game tree), from regime-bound tacitness, statable only in a regime where it is gone (AlphaGo's hunches). (1) Polanyi's from-to structure reads as non-commutative binding, the Identikit as role-filler composition, recognition as majority-vote coincidence. (2) Explicating the valuation destroys it, as attending to his fingers can make Polanyi's pianist falter, yet a closed-form law states it. Polanyi, Nonaka, Tsoukas and Collins each capture part of this but narrowly; even Nonaka and von Krogh's continuum cannot tell contingent from regime-bound tacitness. (3) Collins's three tacit knowledges become strata, one outside the substrate. (4) Our mentor-student DAG is a tacit-transmission graph. (5) Terence Tao's remark that AI-solved problems spawn few follow-ups is diagnosed as rootless explicit knowledge: checkable results whose route no one can follow and that no one yet understands well enough to extend, teach or judge. To re-root them, LLMs generate, VaCoAl/PyVaCoAl keeps an auditable trace with its confidence, and people value, a role virtue epistemology shows cannot be offloaded. Testable predictions follow.