cs.NEOct 12, 2026
SaveWhere Polanyi Meets Galois: Tacit Knowing as an Algebro-Deterministic Operation, Toward Re-rooting AI-Generated Science
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, 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, 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 ; 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, |
| Knowing how without telling how | Recognition without stateable rules | Coincidence of votes across blocks: 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 (§ 4.6 ) |
| Somatic | The body does it and cannot say how; reproducible by machine | Regime-level valuation: decay, Occam-like selection, phase structure across | 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 | at each step and 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.
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