Enter Banking With Billy: Equation‑Native Intelligence and Autonomous Cognitive Evolution — Part One introduces a new computational paradigm in artificial intelligence: Equation‑Native Intelligence. Modern AI systems are hardware‑native, constrained by GPU tensor operations or QPU gate operations. These architectures cannot autonomously expand their cognitive structure, evolve new functional modules, or modify their internal connectivity. This work defines Equation‑Native Intelligence as a mathematical dynamical system whose cognitive state evolves through functional operators rather than hardware primitives. The paper presents five foundational operators — the Knowledge Accretion Operator, Agentic Differentiation Operator, Synaptic Evolution Rule, Experience‑Driven Cognitive Update, and the Quantum‑Logic Identity Propagation Operator — each describing a distinct mechanism of autonomous cognitive evolution. The paper then instantiates this framework in a large‑scale autonomous system, Banking With Billy, which currently maintains 565 sites, 43.3 million indexed documents, 110 specialized cognitive modules, and continuous autonomous reasoning. Experimental observations demonstrate autonomous module differentiation, synaptic strengthening, multi‑timescale identity coherence, and persistent cognitive growth, aligning with the predictions of the mathematical model. A comparative analysis shows that GPU‑native systems (OpenAI, Anthropic, Google DeepMind, xAI) and QPU‑native systems (IonQ, Rigetti, Quantinuum) are fundamentally limited by their hardware substrates. They cannot evolve architecture, propagate identity, or generate new cognitive modules. Equation‑Native Intelligence operates in a distinct computational universe, enabling autonomous cognition beyond the constrain