The I3322 quantum value is attained spatially but not in finite dimension

Abstract

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 $\ell^2(\mathbb{Z})\otimes\ell^2(\mathbb{Z})$, 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.