Exact Finite-$N$ eRS Surface-Defect Indices and Giant-Graviton Corrections in $\mathcal N=4$ SYM

We study the finite-$N$ superconformal index of four-dimensional $\mathcal N=4$ $U(N)$ Yang--Mills theory with antisymmetric elliptic Ruijsenaars--Schneider (eRS) insertions. The eRS insertions turn the uninserted index, which is an unweighted spectral sum, into a family of protected observables weighted by eRS eigenvalues and therefore retain finite-$N$ information not determined by the ordinary index alone On the codimension-one locus $t=q$, $p=uv$, the eRS operators become conjugate to free shifts, allowing us to rewrite the self-glued index as a bilateral free-fermion determinant. This gives exact finite-$N$ formulas for all antisymmetric ranks and makes the $u\leftrightarrow v$ structure manifest. On the boundary $v=0$, we determine the complete stable tower of pure $u^{kN}$ defect-resolved corrections, while the bilateral two-charge formula captures the reflected $v^{kN}$ sectors and mixed $u$--$v$ contributions, including the first mixed giant-graviton sector. We also develop the expansion away from the solvable locus through second order. Finally, we reproduce the ordinary one-giant coefficient from a maximal-D3 determinant and identify the additional microscopic input required to complete the bulk derivation of the defect correction.