Topological Harmonic Cognition: Variational Deficit Resolution on Co-Evolving Neural Cellular Automata
abstract
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We propose a theory of cognition and reasoning based on a dynamically self-reconfiguring Neural Cellular Automaton (NCA) whose topology, transport geometry, and representational basis evolve in response to unresolved semantic tension. Unlike conventional neural architectures with fixed computational structure, the proposed system continuously adapts its internal orga- nization as different inputs induce different patterns of relational contradiction between nodes.
Each node maintains a local representation and exchanges information through transport op- erators defined on an evolving graph substrate. Training minimizes disagreement, or harmonic deficit, between neighboring nodes, causing the system to self-organize toward coherent global representations. Within this framework, questions correspond to persistent regions of geomet- ric and semantic tension distributed across the network. Reasoning is therefore interpreted as a dynamical process of tension reduction.
The system attempts to resolve contradictions by transporting representations across multiple perspectives through algebraic conjugation and symmetry-preserving transformations. When existing representational coordinates are sufficient, the tension collapses and a stable answer emerges. However, when contradictions cannot be re- solved within the current representational basis, the system undergoes structural bifurcation: new orthogonal basis directions are introduced, nodes divide into specialized substructures, and the topology rewires to accommodate previously unrepresentable relationships.
In this sense, insight corresponds to basis expansion driven by unresolved contradiction.We further propose that logical proof arises naturally from this geometric process. A conclusion is provable when it can be orthogonalized into the span generated by the transported premise representations. Reasoning therefore behaves analogously to Gram–Schmidt orthogonalization on a dynamically evolving semantic manifold: contradictions generate residual components, transport operators reinterpret these residuals from different perspectives, and orthogonal basis expansion introduces new conceptual dimensions whenever existing structures fail to eliminate the remaining deficit.
Complex reasoning chains emerge as iterative cycles of tension accumulation, conjugation, pro- jection, and bifurcation, eventually converging toward globally coherent representations.The framework unifies adaptive topology formation, abstraction, analogy, logical deduction, and conceptual discovery under a single variational principle: the minimization of harmonic deficit across a self-modifying geometric substrate. Intelligence is therefore modeled not as static computation on fixed architectures, but as continuous topological self-organization driven by the resolution of contradiction.