Hiroyuki Chuma

Hiroyuki Chuma, Kanji Otsuka, Yoichi Sato

What made useful knowledge cumulative was not discovery alone but the institutions that transmitted it. We provide the first exhaustive structural measurement of the network through which upper-tail human capital passed from master to student across a millennium. Using 470,000 mentor-student records from Wikidata (which integrates the Mathematics Genealogy Project and MacTutor Archive), and all 64 historical Fields Medalists as a fixed, ex ante tracer set, backward traversal yields a directed acyclic graph of 25.5 million paths reaching 57 generations. We document two institutional transitions. First, a 17th-century watershed concentrates lineage traffic on Leibniz: 47 of 64 lineages pass through him with a 10:1 downstream-to-upstream ratio, and seven independent attributes -- learned-society membership (a 46-fold rise per scholar), field, language, employer, institutional diversification, student production, and diffusion entropy -- re-organize coherently across the same window. This is the network signature of Mokyr's Republic of Letters, and it reframes the Newton-Leibniz priority dispute as a distinction between the possession and the transmission of upper-tail human capital: it is transmission that generates the spillovers on which growth depends. Second, 84% of lineages converge upstream on five 12th-13th-century Islamic and Byzantine scholars before terminating at an 11th-century boundary -- the ``Monastery Wall'' -- at which personal academic mentorship first becomes record-generating in Europe. Our claims are descriptive-structural, not causal. Because exhaustive traversal at this scale defeats standard tools, we also contribute a deterministic, algebraic graph-traversal instrument whose measurement bias we characterize in closed form, and report one emergent property of independent methodological interest.

Hiroyuki Chuma, Kanji Otsuka, Yoichi Sato

This paper reports an unexpected finding: in a deterministic hyperdimensional computing (HDC) architecture **that inverts the conventional role of Galois-field algebra -- employing it not for error correction toward a unique answer but as an engine for relative similarity and path-quality ranking -- **a path-dependent semantic selection mechanism emerges, equivalent to spike-timing-dependent plasticity (STDP), with magnitude predictable a priori from a closed-form expression matching measured values. Addressing catastrophic forgetting, learning stagnation, and the Binding Problem at an algebraic level, we propose VaCoAl (Vague Coincident Algorithm) and its Python implementation PyVaCoAl on ultra-high-dimensional SRAM/DRAM-CAM. Rooted in Sparse Distributed Memory, it resolves orthogonalisation and retrieval in high-dimensional binary spaces via Galois-field diffusion, enabling low-load deployment. Crucially, VaCoAl embeds a cognitive bound -- the Frontier Size -- into its architecture, ranking candidates by path-integral confidence (CR2) to achieve compositional generalisation; this bounded-rationality design produces STDP-like selection that error-correction paradigms structurally cannot attain. We evaluated multi-hop reasoning on about 470k mentor-student relations from Wikidata, tracing up to 57 generations (over 25.5M paths). HDC bundling and unbinding with CR-based denoising quantify concept propagation over DAGs. Results show a reinterpretation of the Newton-Leibniz dispute and a phase transition from sparse convergence to a post-Leibniz "superhighway", with structural indicators supporting a Kuhnian paradigm shift. VaCoAl thus defines a third paradigm, HDC-AI, complementing LLMs with reversible, auditable multi-hop reasoning.

Hiroyuki Chuma, Kanji Otsuk, Yoichi Sato

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 Otsuka, Yoichi Sato

Vector-HaSH and the Tolman-Eichenbaum Machine (TEM) propose that the hippocampal-entorhinal circuit factorizes content from a prestructured grid-cell scaffold and supports compositional memory via ripple-mediated replay. Human iEEG shows that hippocampal sharp-wave ripples (SWRs) gate episodic recall, ripple-locked cortical reactivation recapitulates encoding-time patterns, and multi-hop replay fidelity decays multiplicatively along sequence length. These literatures have advanced in parallel without a shared algebraic object. We show that VaCoAl, an algebro-deterministic hyperdimensional memory architecture built from Galois-field LFSRs, supplies that object. Specifically, deterministic Galois-field diffusion provides a substrate-level alternative to Vector-HaSH's random scaffold-to-hippocampus projection that satisfies the same quasi-orthogonality requirement, with matched second-moment statistics, stronger avalanche behavior, and bit-exact reproducibility. The path-integral Confidence Ratio $CR_2$, the product of per-step $CR_1$ values along an $n$-hop chain, is the natural functional form for multi-hop replay-fidelity decay under conditional independence of per-step reactivation, providing the first algebraically tractable model of reported multiplicative decay. STDP-like path selection in VaCoAl follows from architectural demands -- similarity preservation, compositional reversibility, and bounded-frontier search -- that also constrain hippocampal computation. We further argue that VaCoAl operating regimes share architectural commitments with the EC--CA3 and EC--DG--CA3 pathways, motivating an energy-capacity-plasticity reading of why both are conserved across $>$520 Myr of evolution and primate dentate-gyrus elaboration. We prove formal correspondences, derive testable iEEG predictions, and bridge computational neuroscience and hyperdimensional engineering.