RESTRICTED DYNAMIC GEOMETRIC COMPLEXITY: PATH-SPACE REDUCTION AND MÖBIUS–JACOBI RESPONSE

In structured preconditioning, researchers often focus on whether a specific metric family can reach a target condition number. However, this endpoint feasibility does not account for the geometric effort—the "path"—required to reach that state. This paper formulates this effort as a path-space value problem, defining Restricted Dynamic Geometric Complexity (RDGC) as the minimum affine-invariant length of a metric path that satisfies a Hessian-relative generalized-eigenvalue condition.

The author models RDGC as a dynamic programming problem on a Hadamard state space. By employing path elimination, the paper derives an exact min-plus semigroup and a Bellman principle, establishing that the geometric effort can be computed through a fixed-horizon kinetic energy gauge. To analyze the response of this complexity to interventions, the paper utilizes a global Hadamard Möbius–Jacobi theorem, which uses a Green operator to solve for bulk and terminal forcing, and a bordered Jacobi–KKT theorem to handle hard terminal inequality constraints.

The paper establishes that RDGC is a well-defined, affine-invariant distance to a spectral target. The global Hadamard Möbius–Jacobi theorem shows that, under geodesic convexity, a single uniformly coercive Green operator can generate exact force-to-curvature bounds and recursively determine finite-order responses. For hard condition targets, the bordered Jacobi–KKT theorem allows for the differentiation of the moving projection endpoint and multiplier, explaining why constrained interactions can exhibit different signs than unconstrained ones. The theory is validated through closed-form solutions for two-dimensional diagonal models and a three-dimensional protocol that demonstrates how specific update protocols can yield path metrics exceeding ambient projection distances.