Let S be the quantum supremum of the I_{3322} Bell functional in the Collins-Gisin normalization (classical bound 0; two-qubit maximum exactly 1/4). From a certified window S\in(0.2508753845015185,0.250875388108398] (independently and more tight
Let S be the quantum supremum of the I_{3322} Bell functional in the Collins-Gisin normalization (classical bound 0; two-qubit maximum exactly 1/4). From a certified window S\in(0.2508753845015185,0.250875388108398] (independently and more tightly enclosed by Mghirbi's prior certificates) and a certified equality of the tensor-product and commuting-operator suprema, we prove: (i) no finite-dimensional quantum strategy attains S – any finite local dimensions, pure or mixed states, projective or POVM measurements – proving the conjecture of Pal and Vertesi (2010); (ii) S is attained by a spatial strategy on \ell^2(Z)\otimes\ell^2(Z), the infinite-dimensional attainment those authors asserted, on an independent route. So C_q(3,3;2,2) is not closed – the smallest two-outcome bipartite scenario by input count where nonclosure is known – and C_{qs}(3,3;2,2)\setminus C_q(3,3;2,2) is nonempty, settling the attainment question raised by Dykema, Paulsen and Prakash. Nonattainment is proved via a concave critical Bellman storage, exact rational endpoint-exclusion certificates, reflection-gluing and a convex-envelope theorem: finiteness forces an exact maximizer's two equality transports to coincide, capping its value at 1/4<S. Attainment is proved by disintegrating a commuting maximizer's spectral measure over the orbits of its two response transports, yielding an \ell^2 Jacobi eigenvector with inherited normalizability. We also determine the dimension complexity: with S_d the optimum at local dimension \le d and D(\epsilon)=\min\{d:S-S_d\le\epsilon\}, D(\epsilon)=\Theta(\log(1/\epsilon)) – the upper half constructively, with D(\epsilon)\le 23.9010650\log(1/\epsilon); the lower half in a certificate chain in the accompanying repository. Cores of both halves are machine-checked in Lean 4.