Eisenstein-Jacobi network fault repair yields quadratic overlap, non-additive benefits

In distributed systems built on Eisenstein-Jacobi (EJ) networks — essentially dense hexagonal tilings for resource placement — fault repair has unique structural properties. Bader Albader's paper proves that when a resource fails, one replacement can never cover the affected hexagon, but two always suffice for any radius t ≥ 1, yielding a repair number ρ_EJ(t)=2. A sharp minimum-overlap formula Ω_EJ(t)=t² emerges from the three-strip geometry of EJ balls, representing the redundant coverage required. These results are foundational for designing self-healing resource allocations in mesh-based networks.

For multiple failures, the picture becomes richer. Two failed resources can sometimes be repaired with just three replacements instead of four, defying simple additivity — a behavior absent in Gaussian networks. Independent canonical repair gives a universal 2q upper bound for q failures, exact when failed cells are more than 4t apart. Dense clusters exhibit subadditivity: infinite families with four and six faults need only four or five replacements, saving four and seven replacements respectively. Albader also derives an exact inclusion–exclusion identity for repeated coverage and validates findings with an audit of 19,400 instances. The work provides concrete bounds and formulas crucial for fault-tolerant distributed computing.