Sharp Power Loss Thresholds for Vector Min Uncut

For every fixed integer power q ≥ 2, we determine the exact rank-restriction gap of the convex elliptope relaxation of vector Min Uncut. In rank k > q the gap is C(k,q) = Γ((k−1+q)/2)Γ((k−q)/2)/(Γ((k−1)/2)Γ(k/2)); in rank k ≤ q it is unbounded. We give a positive-slack randomized polynomial-time algorithm with deterministically bounded random-bit use and exactly feasible algebraic outputs. Assuming the Unique Games Conjecture, the matching strict-factor promise distinctions are NP-hard. The rounding argument rests on an increasing-convex-order comparison for normalized planar ellipses, yielding the sharp global Gaussian chord-moment bound for every real power p ≥ 1 and integer k > p. Integer-power lower bounds use directional Sobolev energy, odd degree, and a dimension-uniform nonlocal endpoint. A rational power lift, including odd powers, and a geometric dictatorship test complete the integer-power classification. At critical rank k = q we prove the optimized dimension-uniform endpoint (R_q + o(1)) θ^q log(1/θ), where R_q = 2Γ(q−1/2)/(Γ(q/2)Γ((q−1)/2)). Its proof combines a sharp directional current lower bound, integral projective class retention, and smooth radial clipping. At this critical rank we also prove a finite-bit relaxation-sensitive rounding bound (R_q/q)t log(1/t) + O_q(t), with a matching finite rational integrality-gap family. A stronger full ball-kernel endpoint, an exact symmetric-covariance comparison, and independent-endpoint kernel smoothing yield the matching fixed-parameter UGC-conditional logarithmic promise coefficient. The optimized quadratic endpoint, sharp finite-gap consequence, and logarithmic promise hardness are inherited from Zheng; the critical extension concerns higher fixed integer powers, with a unified treatment recovering the quadratic case. Subcritical rates and fractional-power optimization lower bounds are not asserted.