Cognitive Engineering: A New Science of Mind The Unified Mathematical Framework for Biological and Artificial Cognition

The following summary provides an overview of Cognitive Engineering, a new scientific discipline established to unify mathematics, neuroscience, artificial intelligence, and psychiatry under a single framework.

Core Foundation

Cognitive Engineering addresses the 37-year-old challenge of catastrophic forgetting in neural networks by demonstrating that the phenomenon is a feature of the stability-plasticity trade-off rather than a structural flaw. The discipline is grounded in Arithmetic Spectral Theory, which posits that neural representation stability is governed by the same mathematical structures that dictate the distribution of prime numbers.

The Pure Kernel and Mathematical Unification

The framework identifies the first six prime numbers—2, 3, 5, 7, 11, and 13—as a "Pure Kernel." These primes define the L-EFM operator, which exhibits a unique "spectral trap" at the critical line of 0.5. This mathematical structure serves as a unified foundation for the Riemann Hypothesis, the Green-Tao Theorem, and the resolution of catastrophic forgetting in machine learning models.

The Cognitive Phase Diagram

Cognitive performance is mapped across five discrete cognitive states using a two-parameter control system:

• Stability (lr_embed): Manages memory preservation, analogous to hippocampal consolidation.

• Plasticity (lr_cls): Manages new learning, analogous to the balance of neurotransmitters like acetylcholine and cortisol.

The five identified states are:

1. Elder/Expert: Characterized by maximal stability and slow-wave sleep consolidation.

2. Healthy Adult: Represents optimal, balanced cognitive performance.

3. Average: An unoptimized baseline state.

4. Young/Student: Features high plasticity and rapid encoding.

5. Burnout/Overload: A pathological state involving stress-induced dysregulation and the collapse of oldest memories.

Topological Governor and Biological Isomorphism

To achieve stable, lifelong learning, the framework employs a "Topological Governor" within the architecture. This implementation functions through a biological isomorphism with the hippocampus:

• take_snapshot(): Functions like the CA1 region to handle pattern separation and memory trace consolidation.

• enforce_anchors(): Functions like the CA3 region to manage pattern completion and memory restoration.

Defining AGI

Cognitive Engineering introduces a formal, verifiable definition of Artificial General Intelligence. According to this standard, a system qualifies as AGI only if it satisfies three non-negotiable pillars:

• Architectural Invariance: Guaranteed by prime-anchored embedding rows.

• Continual Learning Integrity: Proven by the ability to achieve zero-loss or positive backward transfer.

• Numerical Determinism: Verified by the consistent application of the AST constant and the total absence of NaN/Inf training events.

The framework has been validated across five diverse production model architectures totaling 122 billion parameters, consistently demonstrating that catastrophic forgetting can be eliminated while improving backward transfer by an average of 1.55% with negligible memory overhead.