Appendix: Brachistodynamic Holonomy

Brachistodynamic Holonomy (BDH) is a conceptual framework describing the optimal flow state in collaborative systems—whether human-AI pairs or complex dynamical systems.

Etymologically derived from:

• Brachisto- (Greek brachistos, shortest/fastest): Evoking the Brachistochrone curve, representing the path of optimal efficiency or least action.
• -dynamic: Pertaining to the energetic evolution of the system over time.
• Holonomy: The geometric phase acquired by a system when it travels along a closed path, implying that the state has fundamentally evolved even if the coordinates appear unchanged.

In the context of Geometric Numerical Integration, BDH serves as a meta-principle: just as a symplectic integrator preserves the geometric structure of a Hamiltonian system to ensure long-time stability, a "brachistodynamic" workflow preserves the semantic and creative intent of the collaboration, ensuring that the resulting artifact is geometrically richer than the sum of its individual contributions.

Further exposition to be added.