Minimal-Energy Arc Constraints

Minimal-Energy Arc Constraints are the geometric limits that the minimal-energy (optimal) trajectory of a binary transition imposes, which the book predicts should appear as π-geometric modifications in vacuum fluctuation measurements. The book names them in its testable predictions for π as the geometric foundation of binary computation: Casimir measurements of vacuum fluctuations should show π-geometric modifications from Minimal-Energy Arc Constraints at nanometer precision, detectable by atomic force microscopy as systematic deviations from standard Casimir force calculations. The constraints follow from the requirement that binary transitions take Minimal-Energy Paths, tracing the Emergence Arc through binary state space; the same π geometry is predicted to appear in gravitational-wave strain data and fine-structure-constant variations, and the Planck-scale relation ①⥂⦜⌂ = 2 · ①⊕⌂ is tested alongside them. It connects to Minimal-Energy Paths, the Emergence Arc, Complex Arc Trajectory, Planck Scale Emergence, Fine Structure Relationship and the Intrinsic Geometric Constant.