Local Fault Repair of Perfect Resource Placements in Eisenstein–Jacobi Networks

Perfect resource placements in dense Eisenstein–Jacobi networks partition the network into hexagonal radius-t service cells. Thefault-free placement problem is already classified; this paper studies the complementary post-deployment problem of locally repairing such placements after resource failures. For the dense Eisenstein–Jacobi family generated by α = n + (n − 1)ω, with t = n − 1, we first prove failure-cell locality and candidate locality. For one failed resource, we prove that one nonfailed replacement cannot cover the failed hexagon and that two replacements always suffice; hence ρEJ(t) = 2 for every t ≥ 1. Among all minimum-size one-fault repairs, we prove the sharp minimum-overlap formula ΩEJ(t) = t2. The lower bound follows from the three-strip geometry of Eisenstein–Jacobi balls: every two-ball cover of a failed hexagonal cell contains a forced t × t axial interface region. We then extend the framework beyond one fault. For two failed resources, independent repair gives a universal four-replacement upper bound, but unlike the Gaussian case the EJ geometry is not always additive: we give explicit three-replacement constructions for two infinite neighboring displacement families and prove algebraic three-strip constructions for the infinite non-additive neighboring families. For additive neighboring pairs, we prove two algebraic lower-bound mechanisms: an endpoint-rigidity theorem for pairs with separated opposite axial endpoints, and a diagonal-corridor theorem for the remaining family D = ±(g1 + g2). Thus the closed-form two-fault section separates the non-additive neighboring families from the additive mechanisms without leaving the diagonal corridor as an open case. For q failed resources, independent canonical repair gives a universal 2q upper bound, and this bound is exact whenever failed cells are separated by more than 4t in Eisenstein–Jacobi distance. We also identify and prove infinite dense four-fault and six-fault cluster families: the four-fault cluster has exact repair number four rather than eight, and the six-fault cluster has exact repair number five rather than twelve, showing strong multi-fault subadditivity. For arbitrary multi-fault repairs, we prove an exact inclusion–exclusion identity for repeated coverage inside the failed region and its low-multiplicity specializations. A second exact optimization audit over 19,400 multi-fault instances for 2 ≤ t ≤ 12 and 3 ≤ q ≤ 6 confirms widespread subadditivity, including q = 6 instances with saving seven relative to independent repair. The results show that Eisenstein–Jacobi local repair is not a direct copy of the Gaussian case: the one-fault overlap is quadratic, neighboring two-fault repair can drop from four to three, and dense clustered repairs can reuse replacement balls across several failed hexagonal cells.