Supports of quasi-copulas

The paper proves, for the specific 4×4 quasi-transformation matrix Tr, that the fixed point QTr is a proper quasi-copula not associated with any doubly stochastic signed measure, and that its support is the self-similar attractor of the IFS consisting of the maps ωij with tij ≠ 0; moreover, dimH = dimB is the unique s ∈ (1,2) solving (1−r)^s + 9(r/3)^s = 1, with s(r) strictly increasing and bijective between (0,1) and (1,2) (Theorem 5 and Theorem 6; see the definition of T and the mixed-increment facts around Eq. (2) and Lemma 3) . The candidate solution establishes the same three claims via the same core steps: (i) first-level rectangle masses VQ(Rij) = tij, with a negative entry making QTr not 2-increasing; (ii) an increment factorization VQ(R) = tij VQ(ω−1ij(R)) implying S = ⋃tij≠0 ωij(S) and strong separation; and (iii) a similarity-ratio computation leading to the equation (1−r)^s + 9(r/3)^s = 1 and strict monotonicity of s(r). The only issues are minor: an index swap in naming the “1−r” map (ω14, not ω41) and an unnecessary metric rescaling trick (the nonzero maps are already isotropic contractions here). Substantively, the proofs agree.