Closure Instrument Theorems for Nonabelian Holonomy Systems: Unconditional Results for Compact Lie Groups

This note establishes general instrument-level theorems for nonabelian closure (holonomy) systems valued in compact Lie groups with bi-invariant metrics.

We prove three structural results that hold independently of geometry or dynamics:

(1) Exact delocalization: a fixed nontrivial coarse residue can be distributed across N local factors with total local curvature cost O(1/N).

(2) Unconditional flatness–matching tradeoff: suppressing local curvature on targeted subsets forces a quantitative mismatch energy, yielding an explicit mismatch-floor certificate.

(3) Order sensitivity: ordered products encode nonabelian (commutator) information that cannot be recovered from any abelian or purely local summary.

Together these results imply that purely local curvature or pointwise closure measurements are intrinsically incomplete instruments for detecting nonabelian structure. Reliable inference requires coarse holonomy or mismatch-based observables.

The theorems are intended as reusable methodological lemmas for closure-first modeling and inference frameworks. No physical interpretation or dynamical assumptions are required.

Scope: compact Lie groups with bi-invariant metrics; quadratic distance-squared costs.