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

Abstract

Let $S$ be the common tensor-product and commuting-operator supremum of the $I_{3322}$ Bell functional in the Collins-Gisin normalization. Starting from a certified value window and Bellman/path equivalence, we prove that no finite-dimensional quantum strategy attains $S$, whereas a spatial strategy on $\ell^2(\mathbb{Z})\otimes\ell^2(\mathbb{Z})$ does. The finite-dimensional statement includes mixed states and binary POVMs. Consequently $C_q(3,3;2,2)$ is not closed and $C_{qs}(3,3;2,2)\setminus C_q(3,3;2,2)$ is nonempty. We also establish the dimension complexity $D(ε)=Θ(\log(1/ε))$: approaching $S$ requires and suffices local dimension logarithmic in inverse error. The constructive upper bound is $D(ε)\le 23.9010650\log(1/ε)$ for all sufficiently small $ε$, with natural logarithms; the lower constants are existential. The proofs use critical Bellman storage, spectral transport and normalizable orbit measures. A finite weighted-flow argument supplies the unrestricted quantitative lower bound without identifying distinct joint spectral components. This revision replaces an unsupported step in the earlier lower argument and records additional proof corrections. Prior numerical certification and independent concurrent attainment results are credited. Exact arithmetic and Lean 4 check specified numerical, scalar and finite accounting facts; the complete analytic proof is not formalized.