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arXiv:2609.05555

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

Seth Douglas

quant-phSubmitted Sep 3, 2026• Announced Sep 4, 2026• 0 citations• Revised Sep 11, 2026 (v2)
v1 · Sep 3, 2026v2 · Sep 11, 2026
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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.

Comments
25 pages. v2: Replaced the lower-bound argument with a weighted-flow proof; main theorem statements unchanged. Proof and verification archive: https://doi.org/10.5281/zenodo.22698978. Verification guide and scoped machine checks: https://github.com/Apsiape/i3322-exact-wall

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Let SSS be the common tensor-product and commuting-operator supremum of the I3322I_{3322}I3322​ 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 SSS, whereas a spatial strategy on ℓ2(Z)⊗ℓ2(Z)\ell^2(\mathbb{Z})\otimes\ell^2(\mathbb{Z})ℓ2(Z)⊗ℓ2(Z) does. The finite-dimensional statement includes mixed states and binary POVMs. Consequently Cq(3,3;2,2)C_q(3,3;2,2)Cq​(3,3;2,2) is not closed and Cqs(3,3;2,2)∖Cq(3,3;2,2)C_{qs}(3,3;2,2)\setminus C_q(3,3;2,2)Cqs​(3,3;2,2)∖Cq​(3,3;2,2) is nonempty. We also establish the dimension complexity D(ε)=Θ(log⁡(1/ε))D(ε)=Θ(\log(1/ε))D(ε)=Θ(log(1/ε)): approaching SSS requires and suffices local dimension logarithmic in inverse error. The constructive upper bound is D(ε)≤23.9010650log⁡(1/ε)D(ε)\le 23.9010650\log(1/ε)D(ε)≤23.9010650log(1/ε) for all sufficiently small εεε, 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.
We determine the quantum value of the three-setting, two-outcome Bell functional I3322. The value is the unique intersection q†q_\daggerq†​ of an analytic A-high stable curve of the exact Jacobi characteristic map with a central reverser curve. A computer-assisted proof isolates this intersection in a rational interval of width 10−5510^{-55}10−55, beginning 0.2508753845139765356173366109459512989013219739714764755. We prove that the supremum over finite-dimensional tensor-product strategies, the unrestricted spatial value and the commuting-operator value all equal q†q_\daggerq†​. No finite-dimensional strategy attains the value, whereas a normal infinite-dimensional spatial strategy and a commuting GNS strategy do. The proof combines an exact Bellman/path variational reduction, extraction of a positive bilateral calibrated Jacobi fibre from a maximizing weak-star state, a reflected-fibre central-incidence theorem and interval certificates for the unique stable/central intersection. A separate operator-positive three-remainder decomposition gives the commuting upper bound. The complete verification record is archived at DOI 10.5281/zenodo.22160086.
The quantum supremum of the I3322I_{3322}I3322​ Bell inequality is not attained in finite dimension
In 2010, Pál and Vértesi found a family of finite-dimensional strategies for the I3322I_{3322}I3322​ Bell inequality whose optimized values appeared to converge as the local Hilbert-space dimension grew. They conjectured that this limit is the supremum over all finite-dimensional quantum strategies, but that no finite-dimensional strategy attains it. We prove both claims. The proof uses the symmetry of the Bell functional to associate every strategy with a finite matrix of probabilities, one for each pair of spectral subspaces of Alice and Bob. This matrix gives an upper bound on the Bell value, and finite-dimensional strategies built from the repeating structure found by Pál and Vértesi approach it as the dimension grows. If the bound were attained exactly in finite dimension, the optimality conditions would then require a state that cannot be normalized. Consequently, the set of finite-dimensional quantum correlations is not closed in the (3,3,2,2)(3,3,2,2)(3,3,2,2) scenario, the smallest Bell scenario where this can happen. Moreover, approaching the supremum requires unbounded local dimension. The core of the proof was formalized in Lean~4.
This preprint constructs finite rational proof objects for both sides of the quantum value of I3322 Bell inequality. Let 𝑄t denote the supremum over finite-dimensional tensor-product binary measurements and𝑄c the commuting-operator value. On the complete word space of total length at most four, we construct a rational Gram certificate. The full signed setting-relabeling stabilizer is the dihedral group of order eight, reducing the certificate to five rational blocks. Ex-act affine reconstruction and a uniform diagonal shift of certified positive interior prove: 𝑄c (𝐼3322) ≤ 62718846253/250000000000 = 0.250875385012. Independently, a local-dimension-499 strategy specified by rational projectors and a nonzero unnormalized rational Schmidt vector has exact normalized value 𝐿499 = 0.25087538451397653548 . . .. Con-sequently, 𝐿499 ≤ 𝑄t(𝐼3322) ≤ 𝑄c (𝐼3322) ≤ 0.250875385012, with endpoint difference below 5 × 10−10. The lower certificate also proves 𝑄t(𝐼3322) > 1/4. Every theorem is checked using exact rational or integer arithmetic; floating-point computation is used only to discover candidates. Complete certificates, source code, tests, and reproducibility materials are archived at: doi:10.5281/zenodo.21462892.
Generalized measurements can be implemented projectively after enlarging the Hilbert space, but this dilation changes the available local dimension. We construct a Bell functional with rational coefficients that separates the two measurement models at local dimension two. An explicit three-outcome qubit positive-operator-valued measure with rational matrix entries attains 22+1/1002\sqrt2+1/10022​+1/100. On the other hand, all qubit-projective strategies are bounded by 22+5/250+2/324002\sqrt2+\sqrt5/250+\sqrt2/3240022​+5​/250+2​/32400, giving a fully analytic certified gap greater than 1/10001/10001/1000. To our knowledge, this is the first fully analytic Bell-functional separation between qubit POVMs and qubit projective measurements over arbitrary shared two-qubit states. Lean certificate for the separation theorem is provided for completeness. Separately, an exact level-3 noncommutative sum-of-squares certificate proves that the explicit qubit strategy attains the unrestricted finite-dimensional tensor-product quantum optimum.
We develop a critical Bellman–Jacobi theory explaining how a Bell supremum can be approached in finite dimension yet attained only by an infinite-dimensional spatial strategy. A strict wall gap and an exactly certifiable endpoint barrier yield finite-dimensional nonattainment, spatial attainment, and nonclosure for a parameterized class of three-setting binary Bell functionals; for the specific case of I3322, this resolves the Pál–Vértesi finite-dimensional nonattainment conjecture. The theory further derives algebraic tail equations, explicit resonance surfaces, weighted wall repulsion, and an exact logarithmic finite-section law, giving the unrestricted constructive bound D(ε)=O(log(1/ε)). Exact certificates and independent algebraic verification programs accompany the manuscript.
A post-release exact-arithmetic audit found a load-bearing global-amplitude gap in the Bellman upper-bound assembly of the earlier releases (v1.0.0-v1.2.0). Those releases are preserved as historical records; their exact-optimum, nonattainment, spatial-separation, and nonclosure claims are under correction and are not asserted by this version. What this version rigorously and unconditionally certifies: 0.2508753845015185 < omega_tensor <= omega_commuting <= 0.250875388108398 (window width 3.61e-9), with the lower endpoint an explicit 255-dimensional rational tensor-product strategy verified in exact arithmetic and independently reconstructed at 160-digit interval precision, and the upper endpoint an exact rational Bellman subsolution with an endpoint-clustered 25,601-knot witness, independently reconstructed cell by cell. The certificate stack, the audit that found the gap, a structural explanation (Bellman contact versus amplitude-normalization duality), and one recorded failed quantitative prediction are all included. Every load-bearing claim maps to replayable source and a frozen receipt.