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

Abstract

In The Algebraic Mind (2001), Marcus held that any adequate cognitive architecture needs operations over variables, recursively structured representations, and an individual/kind distinction, and that multilayer perceptrons support none of them; he left a register-and-treelet implementation as a conjecture. Twenty-five years later a memory architecture built for unrelated reasons (speed, power and cost on commodity silicon) meets that specification operation for operation, through one mechanism rather than three. PyVaCoAl/VaCoAl is a hyperdimensional computing architecture built end-to-end on one primitive: XOR-and-shift over GF(2), realised by primitive-polynomial linear-feedback shift registers (LFSRs). It gives reversible binding Bind(R,F) = R xor shift(F), non-commutative bundling that distinguishes "dog bites man" from "man bites dog", and address-space individual/kind separation, at fixed dimension. Capability: exact reversible binding at O(L) cost supplies each pillar as an architectural primitive, not a product of training, with inspectability no lossy substrate offers. Necessity: weaken the primitive to an approximate inverse, as circular convolution does, and all three pillars degrade together -- what cannot be exactly decomposed was never composed, only mixed. Position: this is not the mind, the brain is not an LFSR, and we do not beat large language models; the substrate supplies the auditable symbolic layer they structurally lack. We develop the correspondence pillar by pillar, recast the treelet as a register set identified by a primitive generator polynomial, cite a companion Perspective on the dentate gyrus-CA3 circuit as a biological instance, and show that inflecting an unseen pseudoverb is a rung-3 query in Pearl's sense. Bit-exact reversibility holds in silicon and only approximately under biological noise; biological claims are structural only.