Semicircular Arc

The Semicircular Arc is the path through binary state space that minimizes computational energy while still completing a full binary state transition, obtained as the solution of the Euler-Lagrange equation. Applying the variational principle to state-space trajectories, the Euler-Lagrange equation d/dx(∂L/∂y') - ∂L/∂y = 0 yields the Semicircular Arc as the unique minimal-energy path, connecting to the energy minimization that governs Pre-Pulse Field evolution. Because the path is a semicircle, π emerges from optimization: together with the Emergence Arc and its arc length integral, the result shows geometry arising as an optimization principle and π as an intrinsic constant of binary transition. The book notes that even where the variational formulations show dimensional tension, the Euler-Lagrange solution still points to the semicircular arc, aligning the earliest transitions of the Pulse with the logic of variational mechanics, in which the simplest path through binary state space is also the most fundamental law of emergent geometry. It connects to Euler-Lagrange Equation, Variational Principle for Binary Transitions, Emergence Arc, Arc Length Calculation, Minimal-Energy Paths and Pre-Pulse Field.