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

• The introduction links AI-generated proofs to a follow-up gap: results may be checkable even when their human proponents cannot answer questions or give talks.The account also reports fewer seminars, workshops, collaborations, and newcomers, alongside open directions withheld for fear of being scooped.Source: §1 · p. 3
• Polanyi’s from–to structure becomes non-commutative binding, the Identikit becomes role–filler composition, and recognition becomes majority-vote coincidence (Polanyi 1958, 55–56; 1966, 4–10).The face example illustrates how features can be matched against a remembered whole (Polanyi 1966, 4–5).Source: §1 · p. 3
• VaCoAl/PyVaCoAl models tacit–explicit knowing as an audit–valuation trade-off: repairing collisions yields exact traces without path ranking, while tolerating them yields approximate, valued reads.The trade-off separates information made explicit by a new query from regime-bound tacitness that explication destroys.Source: §1 · pp. 3–4
• Collins’s relational, somatic, and collective tacit knowledges map to bound structure, substrate valuation, and value binding, with the last outside the substrate.The two hippocampal pathways provide a biological precedent for keeping both readouts.Source: §1 · p. 4
• The mentor–student DAG is recast as a tacit-transmission graph, with the Leibnizian hourglass, Kuhnian shift, and Monastery Wall treated as events in knowledge travel.The Newton–Leibniz case is presented as a historical prototype of the follow-up gap.Source: §1 · p. 4
• The proposed remedy for rootless explicit knowledge pairs LLM generation with auditable VaCoAl/PyVaCoAl traces and confidence, while leaving community valuation to people.The paper argues that intellectual virtues exercised in community valuation cannot be offloaded to technology (Pritchard 2023).Source: §1 · pp. 3–4

2.1 Five positions on tacit knowledge

• Polanyi’s from–to account and Kuhn’s shared exemplars emphasize knowing through participation and practice, while later positions dispute whether tacit knowledge can be converted or represented (Polanyi 1958; 1966; 1969; Kuhn 1962/1970; Tsoukas 2003; Nonaka & von Krogh 2009).Nonaka and Takeuchi model conversion between distinct contents; Collins distinguishes relational, somatic, and collective tacit knowledge (Nonaka & Takeuchi 1995; Collins 2010).Source: §2.1 · pp. 4–5
• The paper argues that the tacit-knowledge debate has stalled because each position answers a different question, and proposes VaCoAl/PyVaCoAl as a formal object for addressing them together.The programme comprises seven arXiv papers, with cited results divided between PyVaCoAl software demonstrations and CASRAM hardware claims.Source: §2.1 · p. 5, §2.2 · pp. 5–6

2.2 The VaCoAl programme

• VaCoAl replaces random projection with deterministic diffusion over GF(2), using block-wise retrieval, majority voting, XOR-and-shift binding, and multi-hop confidence.In Rescue mode collisions are repaired and candidates fall back to lexicographic order; in Don’t Care mode residual confidence accumulates in CR2 to rank candidates. Rescue, CR2-selection, and million-dimensional results are demonstrated in PyVaCoAl, while CASRAM is commercial but not yet benchmarked.Source: §2.2 · pp. 5–6

2.3 Tacit knowledge and machines

• Unlike learned-network debates about machine tacit knowledge, this paper asks where the boundary falls in an inspectable substrate and agrees that collective tacit knowledge resides in communities (Collins & Thorne 2026).It gives a counterpart to Budding’s Davies-style structure that does causal work without being stated (Budding 2025).Source: §2.3 · p. 7

3 Polanyi’s structure of tacit knowing, read algebraically

• The section presents its correspondences as a dictionary, with the argument about where the tacit–explicit boundary lies beginning in the following section.It describes the mapping as exact enough to tabulate.Source: §3 · p. 7

3.1 From–to as directed binding

• Polanyi’s asymmetric from–to relation corresponds to non-commutative binding, while mentor-to-student edges provide a directed transmission graph (Polanyi 1966, 10).The graph’s 369 bidirectional assertions among roughly 470,000 records were removed before traversal as a DAG.Source: §3.1 · p. 8

3.2 Subsidiary and focal as two reads of one vector

• A bundled vector supports holistic similarity and analytic unbinding as distinct reads; naming one filler does not recover the integrated similarity.The holistic read compares a whole composite to a reference, while the analytic read extracts one role’s filler.Source: §3.2 · p. 8

3.3 The Identikit as role–filler composition

• The Identikit’s role–filler composition gives a face record an exact form, while matching remains a similarity search by block-wise vote (Polanyi 1966).CASRAM performs matching on an embedding but the paper does not claim that the product binds faces as role–filler pairs; its encoding, rather than its matching, is the unstated part.Source: §3.3 · p. 8

3.4 Knowing how without telling how

• VaCoAl can report block agreement and confidence without stating why a winner won; multi-hop confidence is CR2 (2026a).This partial telling supplies trace and confidence that a human needs to re-root a result.Source: §3.4 · pp. 8–9

3.5 Indwelling and the fixed scaffold

• Fixed feedback polynomials shape every read and write but are never queried, making them VaCoAl’s analogue of a tool the user dwells in (Polanyi 1966, 17–18).The paper connects this fixed scaffold to grid-cell projections into hippocampal CA3 that precede later memory orthogonalization (2026a; 2026d, citing Chandra et al. 2025).Source: §3.5 · p. 9

4.1 A relational definition

• Information is structurally tacit when it affects outputs but no available query states it; a new query can expose it contingently, whereas regime-bound information vanishes in regimes that state it.The definitions are relative to the memory, its regime, and the family of read operations available.Source: §4.1 · pp. 9–10

4.2 The trade-off

• Full collision repair yields exact traces but uniform CR2=1, while tolerated collisions permit approximate reads whose CR2 ranks candidates within the frontier.The proposed decline in valuation dispersion as rescue rate rises remains unmeasured; coarse historical structures remain nearly bit-identical across modes.Source: §4.2 · p. 11, Figure 1 · p. 12

4.3 The pianist, exactly

• As with the pianist attending to his fingers, full rescue can remove traversal’s skill by eliminating the CR1 imperfection whose accumulation produced CR2 ordering.Source: §4.3 · p. 12

4.4 Articulation at the level of law

• At sufficient depth, CR1≈0.997 predicts CR2≈0.846 at generation 56, versus ≈0.905 measured across 25.5 million paths, with the residual consistent with survivor bias.The authors say designers can state the valuation as a law of the whole regime, while bounded search uses CR2 to rank candidates retained in the frontier.Source: §4.4 · p. 12

4.5 Four positions, one substrate

• Nonaka and Takeuchi capture contingent conversion, Tsoukas captures regime-bound loss, and Collins distinguishes relational, somatic and collective strata, with collective knowledge outside the substrate.Source: §4.5 · p. 13

4.6 Externalization with a trace

• New composite operations made graph structure explicit: Newton and Leibniz scores are 0.534 and 0.569, but path counts are 4 and 47, producing Giant Score ranks 91st and second.Unbinding FIELD also recovered astronomy signals of +0.0025 in 1250–1300 and +0.0065 in 1450–1500, while algebra stayed near the 0.5 baseline; reversible binding preserves derivations.Source: §4.6 · p. 14

4.7 A conjecture on ineffability

• The authors conjecture that ineffability may reflect read operations: similarity-completion integration resists articulation, whereas exact relational bindings do not.They note that semantic-pointer circular-convolution unbinding is only a quasi-inverse, with noise growing with the number and depth of bindings.Source: §4.7 · p. 14

5 Collins’s three tacit knowledges as strata of one substrate

• Collins’s relational, somatic and collective tacit knowledges map respectively to bound structure, substrate valuation and value binding, with the last outside the substrate.The two hippocampal pathways offer a biological precedent for keeping both readouts.Source: §5 · p. 14

5.1 Relational: tacit because no one could survey it

• Once a semantic operation could survey the 25.5-million-path graph, previously inaccessible structure became stated facts, including 47 of 64 lineages through Leibniz.The authors interpret this as relational tacitness caused by the cost of looking, not by resistance to telling.Source: §5.1 · p. 14

5.2 Somatic: tacit because of what the memory is

• With total capacity fixed at N · 2^m = 234, changing N shifts which lineage the memory values, and N = 512 alone reaches 57 generations.This valuation follows from finite depth, residual collisions and the multiplicative structure of CR2; the authors describe it as explainable in closed form.Source: §5.2 · p. 15

5.3 Collective: tacit because it lives in the community

• The architecture cannot determine which lineage matters; validating its rankings depends on collective knowledge held by historians and mathematicians.The community is therefore one of the system’s layers, not an optional add-on.Source: §5.3 · p. 15

5.4 A biological precedent for keeping both readouts

• The hippocampal analogy pairs similarity-based completion with holistic, hard-to-tell recognition and typed commitments with relations that can be stated.The authors present this as an interpretation, cautioning that biological noise makes bit-exact reversibility only approximate.Source: §5.4 · pp. 15–16

6 The mentor–student DAG as a tacit-knowledge transmission graph

• The graph’s edges are apprenticeships, making its macro-structures events in the history of tacit-knowledge transmission.The authors distinguish this from a graph of explicit knowledge.Source: §6 · p. 16

6.1 What an edge carries

• Each edge records that one person trained another, while node vectors hold explicit attributes such as field, language, employer, society membership and dates.Unbinding the FIELD component recovers the twelfth-century translation movement, which the mentorship topology misses because knowledge travelled through books and translators.Source: §6.1 · p. 16

6.2 The hourglass as institutionalized socialization

• Leibniz’s lineage forms an hourglass of 5.3 paths per node upstream versus 53.4 downstream, alongside rising learned-society membership across the cohort.The authors read academies, correspondence and lineages as institutions that multiply socialization and connect them to follow-up activities such as talks and workshops.Source: §6.2 · p. 16

6.3 Newton, Thurston, and the follow-up gap

• Only four of 64 lineages pass through Newton, versus 47 through Leibniz, although the graph measures documented mentorship, not Newton’s influence through texts.The paper connects this thin apprenticeship channel to the follow-up gap, and compares it with Thurston’s account of a field emptied by difficulty learning key proofs and with AI results lacking human apprenticeship edges.Source: §6.3 · pp. 16–17

6.4 The Monastery Wall as a change of medium

• The Monastery Wall marks where personal academic mentorship began leaving structured records, not an absence of earlier transmission.The authors interpret it as a shift from communal to dyadic transmission.Source: §6.4 · p. 17

6.5 Is a second wall forming?

• The predicted AI wall would mark results with few or no human apprentice edges because no person learned the tacit root; this is a prediction, not an observation.The paper says its proposed re-rooting practices would prevent that hypothetical boundary.Source: §6.5 · p. 17

7.1 Definition

• A result is rootless when it is explicit and checkable, its route lacks a person-followable record, and no one yet understands it enough to extend, teach or judge it.The route condition varies by degree: Deep Blue’s route was long but readable, AlphaGo’s recorded but rested on network hunches, and an LLM’s neither recorded nor reliably reported.Source: §7.1 · p. 18

7.2 A worked case

• Tao’s Jacobian counterexample was briefly verifiable despite 1,329 equations against 360 degrees of freedom for a generic degree-seven map; its origin looked difficult to see.Tao’s digestion rebuilt the construction from low-degree polynomial multiplication but left one geometric explanation less than fully satisfactory; Weinreich reported tens of sequel papers within a week, many AI-assisted.Source: §7.2 · pp. 18–19

7.3 Why LLM results are rootless by default

• LLMs can produce hypotheses without walkable traces, explanations that may imitate rather than report their process, and reliable self-assessment of significance; rootlessness is a default, not a law.The paper attributes this to patchy, directional implicit relations, weak introspection and generation outrunning evaluation, while noting that a result ceases to be rootless once someone gains the understanding to extend, teach and judge it.Source: §7.3 · pp. 19–20

7.4 A SECI4 reading of the gap

• Autonomous AI problem-solving performs explicit-to-explicit combination rapidly but skips internalization and socialization, where people develop and transmit tacit understanding.The paper links expert evaluation to collective tacit knowledge formed through spoken discourse that LLMs cannot currently reach.Source: §7.4 · p. 20

7.5 Scooping, possession and transmission

• When generation is cheap, people may guard promising directions for fear of being scooped, starving transmission; credit for transmission rather than possession is proposed as a counterweight.The authors connect this concern to Tao’s account and Sanderson’s proposal to credit motivated explanations.Source: §7.5 · p. 20

8 Re-rooting: auditable abduction and a three-layer division of labour

• A result can be re-rooted by moving from a generator through a walkable trace with confidence to a community that values it.VaCoAl/PyVaCoAl is proposed to supply the currently missing middle layer.Source: §8 · p. 20

8.1 Three layers

• LLMs generate hypotheses and candidate connections; VaCoAl/PyVaCoAl stores reversible bound records and paths with CR1 retrieval confidence and CR2 path confidence.Community questions return to the trace layer as new queries, externalizing structure through new operations.Source: §8.1 · p. 21

8.2 Why the middle layer has to be exact

• Because LLMs compose facts less reliably when chains must be assembled internally, an exact multi-hop trace exposes intermediate nodes and stepwise confidence for people to inspect.Only the closed-form CR2 decay is approximate, so its error bars can accompany the trace.Source: §8.2 · pp. 21–22

8.3 Why the third layer is human: a virtue-epistemic reading

• Intellectual humility, conscientiousness, and tenacity require human motivation and reflection, so technology cannot take over the work of understanding and valuing results.Seminars, refereeing, digestion, and teaching are settings where these virtues are exercised.Source: §8.3 · p. 22

8.4 Auditable abduction

• Auditable abduction would encode an LLM hypothesis in DAG records, compute consequences and edge-removal counterfactuals by unbind–rebind, and expose changed confidence along named paths.The proposal inherits the Reflective layer's limits: its reachability is argued rather than demonstrated, and sufficient conditions remain future work.Source: §8.4 · p. 22

8.5 Re-rooting already happens by hand

• In two 2026 episodes, people reconstructed walkable traces by hand before others could value the AI-assisted results.An exact-trace substrate could mechanize part of that reconstruction, leaving more human effort for understanding and teaching, but cannot replace understanding.Source: §8.5 · pp. 22–23

8.6 A complementary direction

• A complementary hyperdimensional-probe approach reads an LLM's residual stream, whereas the proposed three-layer route keeps the LLM computation opaque and supplies a trace beside it.The authors suggest reversible binding may sharpen such probes.Source: §8.6 · p. 23

9.1 Predictions

• The predictions test whether valuation falls as rescue rises, distinguish contingent from regime-bound tacitness, and assess whether self-read error tracks collision saturation.The proposed tests use CR2 dispersion, Rescue-mode reruns, and recovery error under controlled collision rates.Source: §9.1 · p. 23
• The proposed human study predicts that trace-supported researchers will outperform result-only and LLM-self-explanation groups on questions and extension, while an AI-heavy provenance graph may show thinner downstream traffic.The traffic prediction can fail if digestion practices close the gap.Source: §9.1 · pp. 23–24
• A further, explicitly speculative prediction is that relations learned through typed commitment will be easier to state than similarly complex relations learned through similarity completion.The experimental design is left open.Source: §9.1 · p. 24

9.2 Limitations

• The account uses tacit structurally, draws empirical anchors from one mentorship corpus and tracer set, and treats Polanyi, cyclist, and Identikit comparisons as structural correspondences rather than identities.The corpus over-represents Western, university-affiliated, doctoral-advisor transmission and misses some channels; only the two endpoint rescue settings are characterized.Source: §9.2 · p. 24
• CASRAM speed and power are unbenchmarked, auditable abduction remains a proposal, LLM evidence may change with model capability, and independent replication depends on a planned release.The paper says no argument depends on CASRAM performance; PyVaCoAl code is proprietary and the reference materials are to be deposited on acceptance.Source: §9.2 · p. 24

9.3 Open problems

• Open questions include formalizing audit–valuation complementarity, representing significance without making it another explicit item, and designing a practical interface linking proposal, trace, and seminar..Source: §9.3 · pp. 24–25

10 Conclusion

• Tacit knowing is presented as a memory's operating regime: one substrate can provide exact audit or path valuation, but not both at full strength.The account maps Polanyi, Tsoukas, Nonaka, and Collins onto different aspects or strata of that trade-off.Source: §10 · p. 25
• The mentor–student DAG, Leibnizian hourglass, Monastery Wall, and AI-solved problems without seminars are linked by possession without transmission..Source: §10 · p. 25
• The proposed VaCoAl/PyVaCoAl trace supplies confidence-bearing routes for AI results, but people retain the work of understanding and judging their worth.The proposal is paired with predictions that could fail.Source: §10 · p. 25

The VaCoAl programme

• The VaCoAl programme references works on SRAM-CAM, hippocampal memory, Thagard, Minsky, and scientific lineages..Source: The VaCoAl programme · p. 25

Tacit knowledge, machines and the history of science

• The cited literature on tacit knowledge and the history of science includes Polanyi, Collins, Nonaka, Tsoukas, Kuhn, and Pritchard..Source: Tacit knowledge, machines and the… · p. 26

Hyperdimensional computing, memory and causality

• References on hyperdimensional computing, memory, and causality include Kanerva, Eliasmith, Pearl, Thagard, and hippocampal work..Source: Hyperdimensional computing, memory and… · p. 26

Language models

• The language-model references include work on introspection, self-explanation, two-hop and reversal failures, and research-idea generation..Source: Language models · p. 27

Mathematics in the age of AI

• The mathematics-in-the-age-of-AI references include writing on AI-generated mathematics and Tao's discussion of follow-up activities and digestion..Source: Mathematics in the age of AI · p. 27