Programmable Emergence via Cognitive Engine: Longitudinal Evidence of AI Research Maturation

Current AI research systems---FunSearch, AlphaProof, The AI Scientist---optimize single-run outputs but cannot accumulate structured understanding across attempts. We present a cognitive architecture that enables longitudinal research maturation through seven mechanisms organized into three functional layers: state representation (knowledge graph, belief tracking, strategy register), attention allocation (cognitive gradient, dual plasticity, dormant traces), and self-regulation (context-aware interventions). The architecture is organized around two design principles: a Cognitive Genome of domain-agnostic primitives that develop domain-specific specialization through amplification of small initial biases, and principled linguistic-cognitive separation where six of seven mechanisms are pure computation with zero LLM calls. The architecture operates entirely as external scaffolding around a frozen language model (Claude Sonnet 4), requiring no weight updates. Over a 261-exploration campaign on the open Hadamard conjecture at order 668, the system exhibited four phases of cognitive maturation without external guidance: naive exploration, structural deepening, computational intensification, and definitive assessment. Controlled ablation across four mathematical problems showed +22. 5 percentage points in success rate (p = 0. 021, Cochran-Mantel-Haenszel) and +81\% in structural constraint extraction (p < 0. 001). Cross-domain transfer occurred across 24 unique domain pairs via knowledge graph bridge activation, without problem-specific instructions. A subsequent 47-exploration campaign on the open Williamson matrix conjecture at order 65 demonstrated cross-problem knowledge transfer: spectral analysis techniques accumulated during Hadamard research transferred to a structurally related problem, producing novel non-existence evidence and mathematical insights. The system did not solve either target problem, but matured in how it engaged with them---in a way that is quantifiably distinct from a stateless system encountering the same problems repeatedly. We term this phenomenon programmable emergence: qualitative behavioral transitions whose class is determined by architecture but whose timing and parameters emerge from accumulated experience.