A nonlinear neural cognitive architecture via geometric regulation, hybrid dimensional control, projected interface coupling, and structural memory

This deposit presents Version 3 (R03) of CNIT (Coupled Neural and Interactive Topology), a geometric framework for nonlinear cognitive architectures in which computation is organised around local supports of variable dimension, a global supervisory field, and a hierarchy of depletion zones acting as relay interfaces. The system is regulated by a hybrid dynamics combining continuous geometric evolution with discrete dimensional transitions, all governed by a single Master Field operating at fixed dimension 2. ADVANCES R03 vs R02 Version 2 established the core geometric architecture but left six formal gaps. Version 3 R03 closes all six and and concludes the exploratory phase of the CNIT model validation, yielding highly promising results 1. Explicit activity index. The activity index, which was an undefined ghost variable in R02, is now constructed as the normalised L2-density of the local field over its support. Its dynamics between dimensional jumps and its update rule at jump times are both derived explicitly, making the dimensional control loop formally closed. 2. Anti-chattering guard automaton. R02 provided no protection against oscillation between adjacent dimensions. R03 introduces a guard automaton with separated promotion and demotion thresholds and a minimum residence time. A theorem proves that this eliminates chattering and bounds the total number of jumps on any finite interval, which is a prerequisite for the existence theory. 3. Fixed-dimension Master Field with dimensional relay. In R02 the Master Field was implicitly assumed to operate at the dimension of the highest cluster, creating a logical inconsistency as clusters were elevated. In R03 the Master Field operates at fixed dimension 2. Supervision of higher-dimensional clusters is delegated to depletion zones, which are elevated to match their associated cluster and carry a relay field with an explicit radial-angular decomposition. The effective authority transmitted to each cluster is given by a closed-form expression combining geometric attenuation, anomaly memory, and global field strength. 4. Anomaly memory in the depletion zones. Each depletion zone now carries a transmission coefficient governed by an explicit ordinary differential equation. The coefficient decays after detected anomalies and recovers exponentially between events, encoding a local history of instabilities without requiring global state storage. 5. Wisdom functional fully constructed. The wisdom functional, which appeared in the R02 Master Field equation as an undefined correction term, is now fully specified as a two-phase adaptive policy. Phase 1 is a multiplicative descent on a causal weight vector that identifies and progressively attenuates the components responsible for each past crisis. Phase 2 exploits the accumulated crisis history to generate preventive corrections when similar configurations recur. The residual correction to the Master Field is exactly the fraction of each past crisis not yet absorbed by the local weights. 6. Optimal ambient dimension. The minimal ambient dimension guaranteeing generic free separation of all support pairs is shown to equal twice the maximum local dimension plus one. The maximum local dimension is itself bounded explicitly by the ratio of the entropic threshold to the minimum pioneer jump cost. In the minimal configuration, the architecture operates in dimension 5. THEORETICAL RESULTS Seven theorems are stated and proved: finite-time extinction under subcritical coherence, interface load control under dynamic coupling, absence of chattering under the guard automaton, topological separation of free supports, stratified connectivity of supervision, existence of hybrid solutions in the Filippov-Caratheodory sense, and optimality of the ambient dimension. EXPLORATORY COMPUTATIONAL TESTS Six exploratory tests accompany the theoretical development. They cover topological segregation under heterogeneous forcing, sub-quadratic computational scaling under locality constraints, coherence recovery after perturbation, stability regulation near the critical boundary via dynamic coupling, bounded scar accumulation under chronic stress, and geometric FM demodulation of a signal embedded in high-amplitude noise. Notably ,in the last test, the CNIT output achieves a correlation of 0.9999 with the hidden signal without any external filter obviously , BUT WITHOUT ANY PRIOR TRAINING through geometric self-organisation alone. These tests do not constitute formal proofs but provide computational evidence that the theoretical constructions produce non-trivial and interpretable dynamical behaviour across a broad range of operating conditions. S.G Avril 2026