Equally Damaged, Unequally Recoverable: Closure Complexity for Target-Relative Structural Repair of Failure-Degraded IoT Networks

Wireless sensor and Internet-of-Things (IoT) recovery is commonly treated as an optimization problem over coverage, connectivity, energy, reliability, or cost. This paper instead asks a different question: how difficult is a damaged network to repair? Building on the Closure Complexity (CC) framework of Akhtar (2026), we define CC as the minimum admissible structural cost required to move a degraded IoT network into a declared quality-of-service satisfaction region. We further introduce the Closure Optimality Gap (COG) for comparing recovery algorithms with exact CC. A fully reproducible benchmark evaluates 135 degraded networks and 810 method-instance evaluations across multiple network sizes, failure fractions, failure geometries, and fixed seeds. Exact CC is compared with Greedy, GA, PSO, NSGA-II, and Random Search. PSO reaches the exact optimum in 97.3% of paired feasible cases, while NSGA-II matches exact CC in all 110 feasible paired cases. More importantly, failure fraction shows no detectable monotonic association with CC (ρ = 0.050, p = 0.604), whereas degraded reliability is strongly inversely associated (ρ = −0.549, p < 10⁻⁹). Two nearly equally damaged networks nevertheless require 8.65 versus 5.25 repair-cost units. The results show that equal damage does not imply equal recoverability, positioning CC as a solver-independent measure of intrinsic repair burden.