The convex calibration dimension of the multi-label F₁ loss at two and three labels, and the F_β scale

rank F = rank(L - J) = rank L = s² - s + 2 and affdim L = s² - s + 1 for the multi-label F1 score/loss matrices, kernel-certified at s = 1, 2, 3, 4 (N ≤ 16, rank ≤ 14) from explicit integer certificates: a factorisation Λ·M = A·B through Fin r for the upper bound and a reduction X (Λ·M) Y = 1 + 1000003·W read modulo p for the lower bound, with Λ = lcm(1,…,2s) clearing every F1 denominator.

Perturbation control and the source's Remark 3.5: changing the single entry F(∅,∅) from 1 to 0 drops rank F from s² - s + 2 to s² - s + 1, certified at s = 2, 3, together with the inequality of the two ranks.

CCdim(L^(F₁)) ≥ 2 at s = 2: the explicit rational witness p = (9,7,2,3)/21, strictly positive on all four outcomes, whose Bayes-optimal reports are exactly ∅, 1, 1,2 (common risk 4/7, against 17/21 for 2) and whose active-constraint matrix has nullity μ = 1, so Ramaswamy–Agarwal (JMLR 17(14) 2016) Theorem 16 gives ‖p‖₀ - μ - 1 = 4 - 1 - 1 = 2.

No probability vector on the four s = 2 outcomes makes all four F1 reports tie: the unique solution of the tie system is (1/4, 3/4, 3/4, -3/4), which is not nonnegative. Hence the sufficient condition of RA 2016 Theorem 18 / Corollary 19 is unavailable at s = 2, and the RA Theorem-16 bound there cannot exceed 2.

CCdim(L^(F₁)) ≥ 3 at s = 3, one more than the source's own Theorem 4.1 at s = 3 (h·n = 1·2 = 2): the explicit rational witness p = (25,15,0,15,11,0,0,0)/66, supported on exactly ∅, 1, 1,2, 3, whose Bayes-optimal reports are exactly ∅, 1, 1,2, 1,2,3 (common risk 41/66) and whose active-constraint matrix (four nonnegativity rows, three tie rows, normalisation) is nonsingular, so μ = 0 and RA Theorem 16 gives 4 - 0 - 1 = 3.

Literature control: for the 4-class 0-1 loss the uniform distribution ties every report with full support and active-constraint nullity μ = 0, 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^(F₁)₂) = 2 and fsd(L^(F₁)₃) = 3 under the source's convention F(∅,∅) = 1; fsd ≤ 1 at s = 2 under F(∅,∅) = 0 and also when the witness support must avoid the empty outcome; fsd = 2 at s = 3 when the support avoids the empty outcome, and fsd ≤ 2 at s = 3 under F(∅,∅) = 0. So the source's Theorem 4.1 is tight at s = 3 inside its own witness class, the +1 of FC5 comes exactly from the empty-set convention, and the RA route is exhausted at s = 2 — CCdim(L^(F₁)₂) ∈ 2,3 cannot be decided by it.

Engineering: a reduction X M Y = 1 + p·W read modulo a small prime certifies a rank *lower* bound with certificate entries bounded by p and by r·N·(largest entry of M), where the adjugate route needs determinant-sized integers. Measured on this family: the determinants of the chosen r × r minors of Λ·L^(F₁) have 26, 38 and 69 decimal digits at s = 4, 5, 6, while the mod-p certificate at p = 1000003 has every entry of Y below 10⁶ and the largest entry of W equal to 5880, 35280 and 498960 at those same s (and at most 1.01 × 10⁷ at s = 7). The general lemma is rank_geₒfₘodCert and is loss-independent.

The F_β axis on this family's loss: F_β(A,B) = (1+β²)|A ∩ B| / (β²|A| + |B|) with the source's convention F_β(∅,∅) = 1, written in b = β². It restricts to the source's eq. (1) at β = 1 (fbScoreMat s 1 = scoreMat s), and satisfies the duality F_(1/β) = (F_β)ᵀ, so every rank statement is invariant under β ↦ 1/β and the two endpoints of the scale — precision at β → 0, recall at β → ∞ — are transposes of one another.

The endpoint collapse. At both ends of the F_β scale the rank falls from quadratic to linear: for every s, the precision score matrix (β → 0) and the recall score matrix (β → ∞) have rank exactly s + 1, and so do their loss matrices J - P, J - R (s ≥ 1); moreover affdim(J - P) = s, so Ramaswamy-Agarwal Theorem 12 gives CCdim(Lᵖʳᵉᶜₛ) ≤ s, a linear bound where every interior β has the quadratic one. Proved for all s at once, with no case split and no decide: a factorisation through ℝˢ⁺¹ for the upper bounds, an identity submatrix on ∅, 1, …, s for the score, and J - I with inverse J/s - I for the loss.

rank F_β = rank(L^(F_β) - J) = rank L^(F_β) = s² - s + 2 and affdim L^(F_β) = s² - s + 1 for every β ∈ (0,∞) at once, kernel-certified at s = 1, 2. The certificate is a single polynomial left inverse in b = β²: with the row scaling Λ_b(A) = ∏ₗ₌₁ˢ(b|A| + l), the matrix H(b) = diag(Λ_b)·M(b) is polynomial and X(b) = adj(H(b))·diag(Λ_b) satisfies X(b)·M(b) = det(H(b))·I, where every det H(b) produced has all coefficients of one sign (Descartes; exact Sturm count: no root in (0,∞)) — which is what makes one certificate serve every β.

CCdim(L^(F_β)₂) ≥ 2 for every β ∈ (0,∞): the explicit rational-function witness p(b) = (5+20b+18b²+4b³, 2+15b+16b²+4b³, 2+7b+2b², 3+8b+4b²)/(12+50b+40b²+8b³), b = β², is strictly positive on all four outcomes (every coefficient positive), has Bayes-optimal reports exactly ∅, 1, 1,2 and active-constraint nullity μ = 1, so Ramaswamy-Agarwal (JMLR 17(14) 2016) Theorem 16 gives ‖p‖₀ - μ - 1 = 4 - 1 - 1 = 2, against the upper bound affdim = 3 of FC11.

No probability vector on the four s = 2 outcomes ties all four F_β reports, for every β ∈ (0,∞): the three tie equations already force p_∅·(1+2β²) + p_(1,2) = 0, so a nonnegative solution vanishes identically (the unique solution of the full tie system is (1/4, (2+β²)/4, (2+β²)/4, -(1+2β²)/4)). Hence the Ramaswamy-Agarwal Theorem-16 route is exhausted at 2 at s = 2 for every β, their Theorem 18 / Corollary 19 sufficient condition is unavailable there for every β, and CCdim(L^(F_β)₂) ∈ 2,3 cannot be decided by that route for any β.

PROSE. rank F_β = rank(L^(F_β) - J) = s² - s + 2 for every s ≥ 1 and every β ∈ (0,∞). Proof (journal §4): a factorisation F_b = A·B(b) through ℝ^(s(s-1)+2) whose left factor A is independent of b and has full column rank (the subset-incidence ranks of the source's Lemma 4.1), plus full row rank of B(b), which reduces to nonsingularity of the Cauchy matrices (1/(bk+l))_(k,l). The source's β = 1 proof does not extend: its middle matrix C_(kk') = 2/(k+k') is symmetric and positive definite, whereas Cᵇ_(k,l) = (1+b)/(bk+l) is not symmetric and not diagonally symmetrisable (the required identity fₖ(bk+l) = fₗ(bl+k) fails already at b = 3, k = 2, l = 5), so the positive-middle rank identity is unavailable and a two-sided Cauchy argument replaces it.

PROSE. The Ramaswamy-Agarwal feasible-subspace bound for the F_β loss is β-dependent: fsd(L^(F_β)₃) = 4 for β² < 1/2 and 3 for β² ≥ 1/2, the threshold being exactly β = 1/√2. Below it the witness is supported on ∅, 1, 1,2, 3, 2,3 with five tied reports ∅, 1, 1,2, 1,3, 1,2,3 and μ = 0, its fifth coordinate 2(1-2β²)(1+β²) being positive exactly for β² < 1/2; above it the four-support witness of the β = 1 case takes over, its gap at the report 1,3 being (β²/8)(2β²-1)(β²+1)/(…) ≥ 0 exactly for β² ≥ 1/2. So CCdim(L^(F_β)₃) ≥ 4 for β² < 1/2 — twice the source's own Theorem 5.1 value h·n = 2 at s = 3, which is a β = 1 statement.