SBCM Note #6: The Thermodynamic Limit of AI-Driven Growth (v2.0) — Digital Sprawl and the Conservation of Management Cost —

Target: Critique of Aghion, Jones, and Jones (2019) & GPT-5.2 Pro's Proof Recent demonstrations by GPT-5.2 Pro have validated the mathematical correctness of the Aghion-Jones growth model, which predicts a "Finite-Time Singularity" driven by AI self-improvement. However, this proof relies on a fatal assumption: The dematerialization of cost. This note presents a counter-proof based on the SBCM Field Theory: Information is Physical. Just as urban expansion led to administrative inefficiency, unmanaged AI growth triggers "Digital Sprawl," causing management costs to explode exponentially, thereby canceling out the Singularity. Key Concepts: The "Vacuum" Fallacy: Critique of models assuming constant depreciation in computing. Law of Digital Sprawl: Intelligence is Compute, and Compute requires Energy/Land. Complexity Penalty (γ): The thermodynamic cost of maintaining "Order" (Intelligence) against "Chaos". Thermal Wall: The inevitable stalling of growth due to heat/cost constraints. SBCM Solution: Application of "Mesh Refinement" and "Standard Block" partitioning to localize entropy and prevent collapse. Conclusion: Without the "Standard Block" architecture to manage the cost of complexity, the Singularity will consume itself before it saves us. Code is Law, but Physics is the Absolute Judge.