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This research definitively resolves the attainment question for the $I_{3322}$ Bell functional, proving that its quantum supremum is reached only in infinite-dimensional Hilbert spaces rather than any finite dimension. By combining rigorous mathematical certificates with machine-checked proofs in Lean 4, the study establishes that the set of quantum correlations is not closed in this specific bipartite scenario. These findings provide fundamental theoretical constraints on the dimensionality required to maximize nonlocality, distinguishing between tensor-product and commuting-operator frameworks.
The I3322 quantum value is attained spatially but not in finite dimension
arXiv:2609.05555v1 Announce Type: new 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.
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.
21 pages. Results first announced 5 August 2026 (Zenodo, DOI https://doi.org/10.5281/zenodo.21782008). Independent and concurrent with arXiv:2608.29734 (Pauwels) and A. Coladangelo, "The I3322 Bell inequality requires infinite dimensions" (to appear on arXiv, 3 September 2026). Verification guide and machine-checked certificates at this https URL
arXiv:2609.05555 [quant-ph]
(or arXiv:2609.05555v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2609.05555 arXiv-issued DOI via DataCite (pending registration)
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