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

Hiroyuki Chuma, Kanji Otsuka, Yoichi Sato

Two critiques of connectionist cognition converge on one missing capacity. In The Algebraic Mind, Marcus isolated three components any architecture must support -- operations over variables, structured representations, and individuals distinct from kinds -- showed perceptrons support none, and left neural implementation as conjecture. In Brain-Mind, Thagard made binding the single mechanism assembling perception, emotion, consciousness, and self, but rested it on circular convolution, a lossy algebra that degrades under the recursion his account of self and emotion demands. We argue one substrate answers both. VaCoAl is a hyperdimensional architecture built on one primitive, XOR-and-shift over GF(2), on primitive-polynomial linear-feedback shift registers; PyVaCoAl is its extended realization, adding a multi-stage rescue circuit, million-dimensional scale, and a Rescue-Rate phase transition (SRAM-CAM hardware and low-power claim remain future work). Binding, Bind(R,F) = R $\oplus$ shift(F), is exactly reversible and non-commutative: it fills Marcus's open register algebra (Pillar 1), supports compositional bundling at fixed dimension (Pillar 2), and separates individuals from kinds (Pillar 3), while removing the depth-degradation that afflicts convolution at Thagard's deepest recursions. We make three scoped claims. Capability: exact reversible binding at O(N) yields compositional generalization with post-hoc auditability no lossy or learned substrate offers. Necessity: two independent cognitive-architecture programs and a biological circuit (dentate gyrus-CA3) requiring the same reversible-compositional algebra is evidence, by convergent evolution not biomimicry, that it is substrate-independent. Position: we do not claim to surpass LLMs; the substrate is orthogonal, supplying the reversible, auditable, multi-hop relational reasoning statistical embeddings structurally lack.