Rank Expansions of Cylindric-Partition Series and Modified Huang–Jiang–Oblomkov Sums

We give an exact expansion of normalized bounded cylindric-partition series in powers of qᴺ⁺¹, where N is the entry bound and, for balanced profiles, the module rank in the Huang–Lau–Ono identity. The expansion coefficients are linear combinations of character series with coefficients rational in q; the first correction is governed by fusion with the vector representation of affine 𝔰𝔩ₐ. An Euler product of profile translations gives all orders, including reflected images outside the nonnegative profile region. We specify profiles for a finite refinement of the linear-term conjecture of Huang–Jiang–Oblomkov and prove the resulting finite HJO identity for the two-row family and for (a, b) = (3, 4), (3, 5) at every rank. We obtain nonnegative Gaussian-coefficient formulas, and hence coefficientwise monotone convergence, for every three-row profile at levels 4 and 5. A fixed-boundary transformation proves the balanced cases of Warnaar's bivariate families at every three-row level b > 3 coprime to 3, and identifies specified wall cases with finite HJO identities. A dual-semimodule cycle model yields a determinant for every profile and reduces the rank-one finite HJO conjecture to restricted area–inversion identities. The general finite HJO identity remains conjectural. We also determine the least principal congruence level with trivial modular character for an explicit Jacobi realization of the character numerators.