For F_(p,q)=diag(q repeated p times, -p repeated q times), this paper studies the least rank among factors minimizing one half of the squared Hilbert--Schmidt norm subject to CC^*-C^*C=2F. On q=2p+1, exact rational hive primal-dual certificates determine the complete finite cost-and-rank frontier for 4<=p<=32: the rank excess above q is 0 on p=4..7, 1 on p=8..14, 2 on p=15..26, 3 at p=27, and 4 on p=28..32. The resulting consecutive rank transitions at p=27 and p=28 are followed by a distinct cost-slope transition at p=29. Separately, exact certificates prove kappa(F_(53,107))=8847 with minimum attaining rank 115; its optimum is half-integral, while an earlier integer candidate of trace 8843 is refuted by an integral Farkas certificate. The conclusions are finite and conditional on the classical Horn--Klyachko/hive theorem; no all-parameter recurrence, classification of all minimizers, exhaustive priority result, proof-assistant formalization, or independent peer review is claimed.
For a traceless Hermitian matrix F in M_d(C), define kappa_d(F) = (1/2) min{||C||_HS^2 : CC^* - C^*C = 2F} and let r_*(F) be the least rank among its minimizers. Inertia gives the universal lower bound r_0(F) = max{n_+(F), n_-(F)}. We prove the exact low-dimensional classification r_*(F) = r_0(F) for d <= 7, including every singular spectrum. The bound fails sharply in dimension eight. For the entire explicit two-ray cone lambda(b,c) = (4c-3b,b,b,b,-c,-c,-c,-c), 0 <= b <= c, we determine the unrestricted and rank-four costs: their difference is min{b,c-b}, and every interior point has r_* = 5 > 4 = r_0. In particular, the integer spectrum (5,1,1,1,-2,-2,-2,-2) has unrestricted cost 13 and rank-four cost 14. Thus eight is the least dimension in which Hilbert--Schmidt norm optimality can force factor rank strictly above inertia. The proof is an exact computer-assisted Horn-polyhedral argument. Thirty-three sign--zero strata through dimension seven yield 813 exact face rays and 272 projected facets; every facet has a rational nonnegative Farkas certificate on the unrestricted cone. A separate exact replay verifies the dimension-eight phase and a two-valued dimension-nine witness. No floating point output is used as a proof object. Finally, for the target G_t with spectrum (5^(12t),-4^(15t)) in dimension 27t, we prove kappa_27t(G_t) = 87t and 17t+1 <= r_*(G_t) <= 18t. Since r_0(G_t) = 15t, the additive excess is at least 2t+1 and is therefore unbounded. The amplification is symbolic: a coarse-grid restriction of any fine hive transfers a frozen order-27 rank-17 lower certificate to every t by telescoping elementary rhombi.
