Equal-charge projection of the $\mathcal{N}=4$ index: exact large-$N$ formula and finite-rank $U(3)$ coefficients

The equal-charge branch of supersymmetric rotating AdS${}_5$ black holes has $Q_1=Q_2=Q_3$. The corresponding microcanonical sector of the $\mathcal{N}=4$ superconformal index is obtained by projecting to equal charges, or equivalently by extracting the constant term in the two charge-difference fugacities. We prove that for the large-$N$ multigraviton sector the projected index factorizes exactly as $${\mathcal{I}}_{\mathrm{\infty }}^{\mathrm{e}\mathrm{q}\mathrm{Q}}(x,p)=\prod _{k\ge 1}(1-p^k{x}^{3k})(1-p^{-k}{x}^{3k})\sum _{n\ge 0}\mathsf{p}(n{)}^3{x}^{6n},$$ where $\mathsf{p}(n)$ is the partition function. This factorization gives, for every spin sector, an explicit onset energy below which the large-$N$ coefficient is zero. Exact $U(3)$ computations show that finite-rank coefficients can nevertheless appear at energies where the large-$N$ coefficient vanishes, including beyond the classical $U(3)$ black-hole bound. We also determine the full line $j=6J_R+6$. In particular, with $j^{\ast }(J_R)$ denoting this large-$N$ onset energy, $$d_3^{\mathrm{e}\mathrm{q}\mathrm{Q}}(87,\frac{27}{2})=1,j^{\ast }(\frac{27}{2})-87=1554,$$ and the first giant-graviton sector already contributes one unit at this point. All coefficients are coefficients of the $(-1{)}^F$-graded index, not positive degeneracies. The main conclusion is that the high-spin tail survives the exact equal-charge projection.