Algorithmic Submersion Part III: From Binary Search to Least Squares

Why does human cognition rely on two seemingly disparate modes of inference---discrete classification (binary search) and continuous optimization (least squares)? Energy-Efficiency Theory (EET) reveals that these are not independent tools but two limiting regimes of a single energy-constrained cognitive architecture. Binary search is the atomic operation of Natural Causality, responsible for crossing topological barriers Eb^melt and restructuring models (structural phase transitions). Least squares, by minimizing squared prediction error, operates within a fixed model structure, sliding parameters toward local energy minima (parameter drift). This paper unifies both within the EET energy landscape: binary search performs global exploration in discrete hypothesis space, while least squares performs local gradient descent in continuous parameter space. The allocation between them is governed by the energy ratio = Ėₑ₄ₒ₏/Ė₌₀₈₍, and their consolidation into expert intuition is quantified by the Submersion Depth d = 1/ (1+). Under algorithmic submersion, binary search becomes expert pattern recognition, and least squares becomes intuitive calibration. This paper constitutes Part III of the Algorithmic Submersion series, extending the thermodynamic formalism (Part I) and architectural convergence (Part II) to the specific domain of cognitive algorithm selection. The framework yields three falsifiable Level VI predictions regarding cognitive load, expert intuition, and AI architecture efficiency. Keywords: Binary search; least squares; Energy-Efficiency Theory; algorithmic submersion; energy landscape; cognitive algorithms; proper pseudo-truth