Local Fault Repair of Perfect Resource Placements in Dense Gaussian Networks

Perfect resource placement in dense Gaussian networks partitions the network into Lee balls centered at resource nodes. The fault-free placement problem is already classified; this paper studies the complementary post-deployment problem of repairing such placements after resource faults. The paper gives exact local repair theorems for the dense Gaussian placement generated by $t+(t+1)i$; by conjugation and rotation symmetry, the same results hold for the companion generator $(t+1)+ti$. For one failed resource, we prove failure-cell locality, derive the exact replacement number ${\rho }_G(1)=3$ and ${\rho }_G(t)=2$ for all $t\ge 2$, and prove the sharp minimum-overlap formula ${\mathrm{\Omega }}_G(t)=t+1$ among minimum-size repairs. The overlap lower bound is proved from the corner structure of equal-size Lee balls in the rotated coordinates $u=x+y$ and $v=x-y$, where Gaussian Lee balls become parity-constrained squares. For two failed resources, we prove exact additivity: every pair of failed resource cells requires exactly four local replacements for $t\ge 2$, and four always suffice. The two-fault lower bound reduces all relevant resource displacements to two canonical neighboring cases and exhibits four mutually incompatible failed-cell corners in each case. For multi-failure repairs, we prove a general inclusion–exclusion identity for overlap inside the failed region; hence the formula remains exact for arbitrary higher-order dense cores. When a canonical repair instance is certified to have maximum multiplicity three, the identity reduces to the compact correction ${\mathrm{\Omega }}_{\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{r}\mathrm{a}}=P_2-A-C_3$. A ground-truth audit over 7,494 Gaussian cases recomputes coverage from lattice geometry, verifies all exact formulas, and records reproducible multiplicity witnesses.