The unique all-tied distribution of the multi-label Jaccard loss, and the exhaustion of the feasible-subspace lower bound

rank S = rank(L - U) = rank L = 2ˢ and affdim L = 2ˢ - 1 for the multi-label Jaccard score/loss matrices, kernel-certified at s = 1,2,3,4 (N ≤ 16) from integer reductions M·Y = 1 + 1000003·W read modulo p, with Λ = lcm(1,…,s) clearing every Jaccard denominator; the shifted-loss half is proved for every s from the identity L - U = -S.

Perturbation control and the source's Remark 4.4: changing the single entry Jac(∅,∅) from 1 to 0 drops rank S from 2ˢ to 2ˢ - 1, certified at s = 2, 3, while rank L = 2ˢ and affdim L = 2ˢ - 1 are unchanged; together with the inequality of the two score ranks.

For every s, the distribution p⋆ uniform on the s + 1 outcomes ∅, 1, …, s makes all 2ˢ reports Bayes-optimal for the Jaccard loss, at common conditional risk s/(s+1); proved in Lean for general s from the source's own identity Σⱼ Jac(A,j) = 1.

p⋆ is the unique probability vector on the 2ˢ outcomes at which all 2ˢ Jaccard reports tie (proved for general s from three probe reports: ∅, the s singletons, and [s]). Hence for every s ≥ 2 there is no full-support all-tied distribution, the sufficient condition of Ramaswamy–Agarwal 2016 Theorem 18 / Corollary 19 is unavailable for the Jaccard loss at every s ≥ 2, and fsd(L^Jacₛ) ≤ 2ˢ - 2: their Theorem-16 route can never certify the source's upper bound 2ˢ - 1. In particular the uniform distribution on all 2ˢ outcomes does not tie the reports, unlike the 0-1 loss of RA Example 11.

The source's Theorem 5.2 lower-bound witness certified in the kernel at s = 2 and s = 3: the explicit rational distributions p = (3,2,0,2)/7 and p = (13,6,0,6,0,6,0,3)/34, whose support and Bayes-optimal set are both the (2ˢ⁻¹+1)-element family ∅ ∪ 1 ∪ D (common risks 4/7 and 21/34, every other report strictly worse), and whose active-constraint matrices are nonsingular, so μ = 0 and RA Theorem 16 gives ‖p‖₀ - μ - 1 = 2 and 4 = 2ˢ⁻¹.

Literature control: for the 4-class 0-1 loss the uniform distribution has full support, ties all four reports at risk 3/4, and the active-constraint matrix is nonsingular, so the same machinery returns 4 - 0 - 1 = 3 = n - 1, reproducing Ramaswamy–Agarwal 2016 Example 11 (CCdim(L⁰⁻¹ₙ) = n - 1).

Exhaustive values of the Ramaswamy–Agarwal feasible-subspace bound (the best bound their Theorem 16 can give): fsd(L^Jac₂) = 2, fsd(L^Jac₃) = 4, fsd(L^Jac₄) = 8 under the source's convention Jac(∅,∅) = 1; and fsd = 1 at s = 2, 3 at s = 3 both under Jac(∅,∅) = 0 and when the witness support must avoid the empty outcome. So the published lower bound 2ˢ⁻¹ is exactly the ceiling of the RA Theorem-16 route at s = 2,3,4, and 2ˢ⁻¹-1 is exactly its ceiling under the alternative convention — the remaining factor of two is a property of the method, not of the witness.

Engineering and measurement: at s = 3 and s = 4 every Jaccard tie set of size 2ˢ⁻¹+2 collapses the attainable support onto supp(p⋆), of size s+1 — all 28 six-element tie sets at s = 3 and all 8 008 ten-element tie sets at s = 4 give |Sₘax| = 4 and 5 respectively. This is what makes the s = 4 fsd upper bound affordable (1 001 s by tie-set enumeration, against 4.3 × 10⁹ (supp, opt) pairs for the direct scan), and if it holds for all s it gives fsd(L^Jacₛ) = 2ˢ⁻¹ for every s.