A Non-Circular Theory of Correct Intermediate Concepts: Recognition, Stability, Certification, and Structured Assembly

This theory develops a non-circular foundation for treating intermediate concepts as scientifically real objects rather than as post hoc artifacts of a particular architecture. Its central claim is that correct intermediate modules should be defined at the level of structural classes, not surface realizations, and should be judged by whether they are recognizable across near-optimal systems, stable under perturbation and recombination, reusable through an admissible assembly domain, and resistant to cheap substitution by flatter or fragmented surrogates. The framework therefore introduces a typed ontology of problem families, real-use contexts, concept libraries, realizations, interfaces, and admissible assembly, together with a multi-axis carrying-cost regime that prices representation, calling, assembly, leakage, and over-refinement.

On this foundation, the theory builds a theorem-bearing program centered on three flagship results: recognition, stability, and weak cheap-substitution impossibility. It then extends this core with intrinsic cohesion and assembly geometry, an obstruction-sensitive branch, an endogenous certification regime, a learning-theoretic path for recovering approximately correct intermediate modules, and an illustrative architecture in which retrieval, call, assembly, and audit become first-class operations. The overall aim is not merely to improve interpretability, but to establish a new mathematical language for describing, learning, certifying, and evaluating intelligence through the structure of its intermediate concepts.