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

Hiroyuki Chuma; Kanji Otsuka; Yoichi Sato; Kodo Yamauchi

cs.NEOct 12, 2026

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

Authors: Hiroyuki Chuma, Kanji Otsuka, Yoichi Sato, Kodo Yamauchi

Organizations: Professor Emeritus, Hitotsubashi University · Professor Emeritus, Meisei University · Shuhari System

Abstract

We argue that tacit knowing is better modelled as an operating regime of a memory than as a kind of content. Our substrate is VaCoAl, an algebro-deterministic XOR-and-shift hyperdimensional memory over GF(2), realized as CASRAM and PyVaCoAl. In it, a DAG reads two ways. Repairing every collision (Rescue) makes the trace exactly auditable but ranks no path higher; tolerating them (Don't Care) yields a path confidence, CR $_2$ , silently favouring direct routes. This audit-valuation complementarity formalizes Polanyi's claim that tacit and explicit are not sharply divided. It separates contingent tacitness, which a new query brings out (AlphaGo's game tree), from regime-bound tacitness, statable only once it is gone (AlphaGo's hunches). (1) Polanyi's from-to structure reads as non-commutative binding. (2) Explicating the valuation destroys it, as with Polanyi's pianist, yet a closed-form law states it. Polanyi, Nonaka, Tsoukas and Collins each capture part of this, narrowly; even Nonaka and von Krogh's continuum cannot tell the classes apart. (3) Collins's three tacit knowledges become strata, one outside the substrate. (4) Our mentor-student DAG is a tacit-transmission graph. (5) Tao's remark that AI-solved problems spawn few follow-ups reveals rootless explicit knowledge: checkable results no one can trace or yet understands. 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. Art, craft and skill show the same decoupling of form from the maker's becoming. (6) On Mercier and Sperber's account an LLM is reason's arguing side set loose, a sophist; truth-seeking lies deeper, in Polanyi's conviviality, which we conjecture is the root of evolvability and, as the Buddhist idea that plants and lands attain buddhahood suggests, reaches beyond humans. Testable predictions follow.

Figures & tables

Position	What makes knowledge tacit	Can it become explicit?
Polanyi (1958, 1966, 1969)	Subsidiary particulars integrated toward a focal whole by the knower	Not sharply divided; explicit knowledge always rests on tacit understanding
Kuhn (1962/1970)	Competence carried by shared exemplars learned in practice	Only partly; exemplars do work that rules cannot
Nonaka & Takeuchi (1995); Nonaka & von Krogh (2009)	Personal, context-bound know-how and mental models; at the far end, embodied knowledge	Yes, by externalization through metaphor, analogy, concept, hypothesis, model; the embodied end cannot be represented (2009)
Tsoukas (2003)	An ineffable dimension of skilled knowing	No; it can be displayed and re-punctuated in dialogue, not converted
Collins (2010)	Three kinds: relational, somatic, collective	Relational: yes. Somatic: by machine reproduction. Collective: not in the foreseeable future

Table 1: Five positions on tacit knowledge.

Ref.	Paper	What this paper takes from it
2026a	Beyond LLMs, Sparse Distributed Memory, and Neuromorphics: VaCoAl (arXiv:2604.11665)	Architecture, Galois-field diffusion, Rescue vs Don’t Care, emergent STDP-like selection, mathematical auditability, HDC-AI as a complement to LLMs
2026b	Bridging Silicon and the Hippocampus (arXiv:2605.15652)	Two hippocampal regimes, $\mathrm{CR}_{2}$ as a model of multiplicative replay decay, Pearl’s rungs 2 and 3
2026c	How to Build Marcus’s Algebraic Mind: Substrate over Galois Fields (arXiv:2605.21379)	Operations over variables, recursive structure, individuals vs kinds; reversible binding and non-commutative bundling
2026d	Institutions and the Transmission of Upper-Tail Human Capital (arXiv:2605.31470)	Leibnizian hourglass, 46-fold rise in society membership, possession vs transmission, the Monastery Wall
2026e	How to Build Marcus’s Algebraic Mind: Thagard’s Brain–Mind Viewpoint (arXiv:2607.16573)	A reversible substrate for deeply nested bindings; demonstrated vs physically realized claims
2026f	How to Build Marcus’s Algebraic Mind: Minsky’s Emotion-Machine Viewpoint (arXiv:2607.16623)	Introspection as exact recovery of one’s own trace; valuation in Don’t Care, audit in Rescue

Table 2: The seven papers of the VaCoAl programme.

Polanyi’s term	What it names	VaCoAl counterpart
From–to relation	We attend from proximal particulars to a distal whole; the relation has a direction	Non-commutative binding $\mathrm{Bind}(R,F)=R\oplus\sigma(F)$ ; directed DAG edges
Subsidiary awareness	Particulars present and functioning but not attended	Fillers bundled into a composite vector that no query has unbound
Focal awareness	The object of attention	The filler returned by unbinding the composite against a chosen role
Tacit integration	Particulars fuse into a meaningful whole	Bundling and majority vote: the whole is the argmax of many partial agreements
The Identikit	A face communicated by assembling chosen features	Role–filler composition, $\mathrm{Face}=\sum_{i}\mathrm{Bind}(\mathrm{role}_{i},\mathrm{feature}_{i})$
Knowing how without telling how	Recognition without stateable rules	Coincidence of votes across blocks: $\mathrm{CR}_{1}$ reports how much agreement, not why

Table 3: Polanyi’s anatomy of tacit knowing and its counterparts in VaCoAl.

Class	Example from the programme	What exposes it	Under full rescue
Explicit	A single recorded assertion that one person trained another	Direct lookup	Unchanged
Contingently tacit	The Leibnizian hourglass: 5.3 paths per node upstream, 53.4 downstream	Exhaustive path counting over 25.5 million paths	Survives (2026d, Appendix J)
Contingently tacit	Possession versus transmission: Newton 91st, Leibniz 2nd on the Giant Score	A new composite of unbinding, similarity and path counts	Survives (2026d, Appendix J)
Regime-bound tacit	The preference for direct over circuitous lineages	No query returns it; a closed-form law describes it	Dissolves into lexicographic order
Outside the substrate	That the mainstream of mathematics ought to rank high	Judgement by historians and mathematicians	Not affected

Table 4: The classes of § 4.1 filled with findings of the programme.

Figure 1: One dial, two readouts. Full rescue yields an exact trace with no path valuation; Don’t Care yields valuation with approximate reads. The coarse historical structures do not move with the dial; the ranking inside the frontier does. Intermediate rescue rates have not been mapped.

Position	Class it describes (§ 4.1 )	What it gets right, in VaCoAl/PyVaCoAl terms	What it leaves out
Polanyi	Every class at once	Tacit and explicit are not separate stores but two readouts of one memory, set by one dial	What sets the boundary in a given substrate, and what moves it
Nonaka & Takeuchi; Nonaka & von Krogh	Contingently tacit; since 2009 also an embodied end that cannot be represented	Contingently tacit structure can be externalized by enlarging the family of operations; metaphor, analogy, hypothesis and model act as new queries	Why something stays unstated: one continuum of content cannot tell structure no query has yet reached from structure the stating regime destroys
Tsoukas	Regime-bound tacit	Regime-bound valuation cannot be converted at the object level; converting it destroys it	Contingently tacit structure, which a new operation does externalize (§ 4.6 )
Collins	Relational: contingently tacit; somatic: regime-bound; collective: outside the substrate	“Tacit” names several things, which sit at different strata of the substrate (§ 5 )	The dial that trades the somatic stratum against exact audit (§ 4.2 )

Table 5: The four positions of § 2.1 , read as descriptions of one substrate. Each is right about the class it describes and narrow in a characteristic way.

Collins’s kind	Collins’s criterion	Stratum in VaCoAl/PyVaCoAl	What reaches it	Status
Relational	Could be told with effort; tacit for contingent reasons	Bound structure: DAG edges and composite vectors	New read operations (unbinding, bundle matching, multi-hop traversal)	Explicable by enlarging $Q$ (§ 4.6 )
Somatic	The body does it and cannot say how; reproducible by machine	Regime-level valuation: $\mathrm{CR}_{2}$ decay, Occam-like selection, phase structure across $N$	A closed-form law about the regime, not a retrieved item	Explicable at the level of law (§ 4.4 )
Collective	Located in society; flexible social performance	Significance for a community: which lineage matters, which result deserves a seminar	Value binding, which the architecture lacks (2026f)	Outside the substrate; held by its users

Table 6: Collins’s three kinds of tacit knowledge as strata of the substrate.

Domain	The explicit form	What is missing	Source
Mathematics	A counterexample to the Jacobian conjecture, verifiable by a brief calculation	A route a person can follow; the understanding needed to extend and teach it	Tao 2026b; Sanderson 2026
Go	Move 37	The general principles behind AI moves	Graepel 2026; Kim 2026
Art	Forms of a sophistication that once required the maker’s becoming	The maker’s becoming	Zoran 2026
Clinical skill	Adenomas detected with AI assistance	Unaided detection (28.4% to 22.4% without AI)	Budzyń et al. 2025
Programming	Working code in a new library	Conceptual understanding, code reading and debugging (about 17% lower)	Shen & Tamkin 2026

Table 7: Rootless results across domains: an explicit form arrives while its root is missing.

Figure 2: Three layers, one feedback loop. A result moves from generation to valuation through an exact trace; questions raised by the community return to the trace layer as new queries. The third layer is where people exercise the intellectual virtues that technology cannot take over (§ 8.3 ).

Zoran’s condition	Why it matters	Counterpart in the three-layer design
Design against convergence	Closed loops of generation lose rare variations first: models trained on generated content lose the tails of the original distribution (Shumailov et al. 2024)	Exact records are kept rather than overwritten by generated summaries; different configurations of the substrate yield different, historically defensible valuations (Leibniz- or Gauss-dominant lineages; 2026d, Appendix F), so plural readings can be shown instead of one converged ranking
Preserve friction and uncertainty where they serve exploration	Friction is where a person goes through the route; uncertainty marks where judgement is needed	$\mathrm{CR}_{1}$ at each step and $\mathrm{CR}_{2}$ for each path are displayed, not hidden; the difference between Rescue and Don’t Care is read as information (§ 8.1 )
Make provenance legible	A form whose route cannot be read cannot be extended or taught (condition 2 of § 7.1 )	Exactly reversible binding: every item arrives with its derivation (§ 4.6 ), and every ranking decision can be reconstructed (2026g)
Evaluate by these conditions, not by output quality alone	What counts is the becoming that a system makes possible	Predictions 4 and 7 measure follow-up capacity and unaided skill, not output quality (§ 10.1 )

Table 8: Zoran’s four conditions for generative systems and their counterparts in the three-layer design.

Associative Memory

Associative memory studies how systems store patterns or associations and retrieve them from partial, noisy, or context-dependent cues.

See all 120 papers

Apr 13, 2026

Beyond LLMs, Sparse Distributed Memory, and Neuromorphics <A Hyper-Dimensional SRAM-CAM "VaCoAl" for Ultra-High Speed, Ultra-Low Power, and Low Cost>

VaCoAl (Python: PyVaCoAl) attacks the Binding Problem algebraically rather than statistically: an SRAM/DRAM-CAM organised end-to-end around one primitive, XOR-and-shift over GF(2) via primitive-polynomial LFSRs. Rooted in Kanerva's Sparse Distributed Memory, it retrieves in a million-dimensional binary space by Galois-field diffusion; Binding and Unbinding are exactly reversible at O(L), giving compositional generalisation with post-hoc auditability. Unexpectedly, a path-dependent semantic selection mechanism emerges undesigned, functionally equivalent to Spike-Timing-Dependent Plasticity (STDP) and predictable a priori from a closed form matching measurement on tens of millions of records. It is the Don't Care (collision-tolerance) rule: not a defect to engineer away, but the source of its ability to rank paths by quality. Capability: reversible binding plus tolerated collisions yields a path-integral confidence measure (CR2) ranking candidates by accumulated reliability, absent from hash-based search. Necessity: repair the collisions fully and the ranking vanishes; a memory that never fails has no record of which paths were difficult. Position: we do not surpass large language models; the substrate is orthogonal, supplying the auditable path-ranking layer they lack. A mentor-student ontology of ~470,000 WIKIDATA scholars, traversed backwards from all 64 Fields Medalists to depth 57 and >25.5M paths, stress-tests multi-hop reasoning over a large DAG. Ablation shows Don't Care imposing a depth-dependent exponential penalty that prunes circuitous routes and preserves direct ones: an Occam's razor. Per-generation CR1 stays within 0.995-0.999, yet cumulative CR2 decays monotonically to ~0.905 by generation 56, matching the closed form 0.997^56 = 0.846 within observed variation. All measurements are from PyVaCoAl, a software DRAM-CAM; speed and power figures are projections.

Hiroyuki Chuma, Kanji Otsuka, Yoichi Sato

Professor Emeritus, Hitotsubashi University · Professor Emeritus, Meisei University · Shuhari System

Associative Memory Hyperdimensional Computing

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Jul 18, 2026

How to Build Marcus's Algebraic Mind: From Thagard's Brain--Mind Viewpoint

This paper reports a convergence neither program was looking for. Marcus's The Algebraic Mind named three things any architecture needs -- operations over variables, recursive structure, and individuals distinct from kinds -- showed multilayer perceptrons have none, and left a register-and-treelet implementation as conjecture. Thagard's Brain-Mind ran it from the other end, making binding the mechanism from which the whole of mind is assembled, and circular convolution load-bearing. Marcus leaves his register algebra open; Thagard's is lossy, degrading under the very recursion his own account demands. VaCoAl is a hyperdimensional computing architecture built end-to-end on XOR-and-shift over GF(2) via primitive-polynomial LFSRs; PyVaCoAl is its extended software realization (all results here); the silicon substrate exists as CASRAM. Bind(R,F) = R XOR Shift(F) is exactly reversible and non-commutative: it supplies all three pillars at fixed dimension and removes convolution's depth degradation. Capability: exact reversible binding at O(N) yields compositional generalization with post-hoc auditability, which no lossy or learned substrate offers. Necessity: two independent architecture programs and the dentate gyrus-CA3 circuit require the same algebra -- convergence, not biomimicry. New here: discrimination and failure-tolerance are one. Repair every collision (RR = 1) and the system is bit-identical to a hash dictionary: candidates become indistinguishable and the confidence path-integral collapses; a memory that never fails keeps no record of which routes were hard. Position: we do not surpass large language models but supply the auditable, multi-hop relational reasoning embeddings lack. No consciousness is implemented and no cognitive experiment reported; bit-exactness holds in silicon, approximately under biological noise; speed and power remain unbenchmarked.

Hiroyuki Chuma, Kanji Otsuka, Yoichi Sato

Institute of Innovation Research, Hitotsubashi University (Professor Emeritus) · Meisei University (Professor Emeritus) · Shuhari System

Computational Cognitive Modeling Hyperdimensional Computing Vector Symbolic Architectures Algorithmic Reasoning

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May 20, 2026

How to Build Marcus's Algebraic Mind: Algebro-Deterministic Substrate over Galois Fields

In The Algebraic Mind, Gary Marcus identified three components essential for any adequate cognitive architecture: operations over variables, recursively structured representations, and a distinction between mental representations of individuals and kinds. He argued that standard multilayer perceptrons supported none of these, acknowledging that a neural implementation using registers and treelets, constructed via developmental programs rather than gradient descent, remained a programmatic conjecture. Twenty-five years later, the required substrate is now available. Our newly developed PyVaCoAl/VaCoAl is a hyperdimensional computing architecture organized end-to-end around a single algebraic primitive: XOR-and-shift over GF(2), implemented by primitive-polynomial linear-feedback shift registers. The architecture supports reversible variable binding via Bind(R,F) = R XOR shift(F), non-commutative compositional bundling that distinguishes "the dog bites the man" from "the man bites the dog," and address-space individual/kind separation under the same algebra. A companion perspective argues that the dentate gyrus-CA3 circuit is a biological homologue of this same engine, with developmentally specified mossy-fiber targeting supplying the innate microcircuitry Marcus anticipated. In this paper, we map the correspondence between Marcus's three pillars and the operational commitments of PyVaCoAl/VaCoAl. We reinterpret the treelet as an algebraic register set indexed by a primitive generator polynomial, arguing that this architecture provides a functional neural substrate meeting Marcus's specifications far more closely than the tensor products, circular convolution, or temporal synchrony available in 2001. We also demonstrate how this substrate naturally extends to Pearl's rung-3 counterfactual reasoning, a capability the original treelet program did not directly target.

Hiroyuki Chuma, Kanji Otsuk, Yoichi Sato

Institute of Innovation Research, Hitotsubashi University, Kunitachi, Tokyo 186-8603, Japan (Professor Emeritus) · Meisei University, Hino, Tokyo, Japan (Professor Emeritus) · Shuhari System, Tokyo, Japan

Computational Cognitive Modeling Hyperdimensional Computing Vector Symbolic Architectures

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Abstract

Associative Memory

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