Coupled Neural and Interactive Topology: A Nonlinear Cognitive Architecture via Multi-Linearity Braiding and Hierarchical Field Regulation

This paper introduces CNIT (Coupled Neural and Interactive Topology), a theoretical framework for nonlinear cognitive architectures departing from classical connectionist models on three fundamental points. First, the computational volume is structured by the decay of spherical potential fields rather than by weight adjustment — a process termed topological sculpting. Each active linearity is carried by a field sphere defined by dimensional parameters (action radius, boundary attenuation, fringe spacing), all controlled exclusively by a high-inertia Master Field. Learning in CNIT is not the addition of information but the progressive elimination of what the field no longer sustains. Second, the system supports the simultaneous coexistence of multiple linearities forming a braid — parallel logical flows maintained without destructive interference by confinement potentials. Where spheres interpenetrate, the two logical flows do not merge but coexist through a weighted field, each retaining its own identity. This architecture naturally supports multi-context reasoning, the simultaneous maintenance of contradictory hypotheses, and analogy-based inference across independent logical threads. Third, the Master Field operates as a self-referential regulator: it controls the geometry of all spheres, sets the maximum authorised entanglement between linearities, and decides its own propagation topology by dynamically assigning one of four roles to each node — transparent, modulator, nonlinear transformer, or blocking. This self-referential structure, in which the field shapes the network through which it propagates, opens perspectives toward architectures capable of genuine attentional focus, selective inhibition, and dynamic reconfiguration without external supervision. The system dynamics undergo a sharp bifurcation at a critical entropy threshold, separating a cruise regime of fine continuous adjustment from a crisis regime in which the braid is brutally simplified — the freeze. Crucially, each freeze is treated not as a failure but as structural experience: its analysis slowly modifies the long-term behaviour of the Master Field, producing what the paper terms structural wisdom — a form of meta-learning derived entirely from the system's own crisis history, without any external error signal. Three formal results are established concerning the extinction of incoherent spheres, the control of inter-sphere entanglement, and a logical consistency constraint showing that the Master Field cannot sever its own observability of the network. The mathematical framework draws on Morse theory, catastrophe theory, Filippov's discontinuous dynamical systems, Kuramoto synchronisation, and Tikhonov slow-fast decomposition. Beyond its theoretical contributions, CNIT suggests concrete directions for neuromorphic hardware implementation, for the design of cognitive systems capable of graceful degradation under overload, and for a new class of unsupervised architectures in which structural coherence — rather than prediction error — drives adaptation.