The per-instance Jaccard score, or intersection over union (IoU), is standard in multi-label classification and binary segmentation. With s labels, its loss matrix has 2s outcomes and reports. Under the convention Jac(∅,∅)=1, we prove that the Jaccard score, shifted-loss, and ordinary loss matrices are nonsingular and that the loss columns have affine dimension 2s−1
Nearby in the stack
. The proof combines a finite MinHash Gram representation with Boolean Möbius inversion. For exact calibration, we prove
2s−1≤CCdim(LJac)≤2s−1
. The lower bound uses a factorially weighted distribution with
2s−1+1
supported outcomes and Bayes-optimal reports. Consequently, every exactly calibrated convex surrogate requires exponentially many prediction coordinates. We also give two polynomial-dimensional approximation guarantees with explicit regret transfers. A new
F1
-to-Jaccard transfer turns an existing
(s2+1)
-dimensional
F1
surrogate into a polynomial-time rule with asymptotic Jaccard regret at most
3−22
. For any
α>0
and
0<ρ<1
, a MinHash square-loss surrogate attains Jaccard-regret floor
α
uniformly over arbitrary conditional label distributions. With probability at least