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.
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.
We develop an exact structural reduction for a characterized class of three-setting, binary-outcome Bell functionals, transforming the operator optimization into an alternating one-dimensional Jacobi variational problem with a Bellman storage dual. The framework includes square-scalar operator welds, a classification of sixteen universal weld branches, two-matching coupling graphs, and endpoint-padded path-completeness rigidity. As an application, it contains I3322 and the Gigena–Kaniewski family and gives the exact rational bound ω(I3322)≤0.250875384626, yielding a certified two-sided interval of width 1.1202346452×10−10 when combined with an independent exact lower certificate. The work is structural and self-contained; critical attainment and dimension-growth questions are treated separately in a companion paper.
We determine the quantum value of the three-setting, two-outcome Bell functional I3322. The value is the unique intersection q† 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 10−55, 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†. 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.
Let S be the common tensor-product and commuting-operator supremum of the 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 S, whereas a spatial strategy on ℓ2(Z)⊗ℓ2(Z) does. The finite-dimensional statement includes mixed states and binary POVMs. Consequently Cq(3,3;2,2) is not closed and Cqs(3,3;2,2)∖Cq(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)≤23.9010650log(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.
The quantum supremum of the I3322 Bell inequality is not attained in finite dimension
In 2010, Pál and Vértesi found a family of finite-dimensional strategies for the 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) 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.
The I3322 Bell inequality requires infinite dimensions
The I3322 inequality is one of the simplest bipartite Bell inequalities, with three possible questions and two possible answers per party. Yet, despite its simplicity, it has long been conjectured by Pál and Vértesi (Physical Review A, 2010) to possess the following quirk: no finite-dimensional quantum strategy can attain its maximal violation, but an infinite-dimensional strategy can. A Bell correlation, with five questions and three answers per party, that provably possesses the same property has since been discovered by Coladangelo and Stark (Nature Communications, 2020). However, proving that the same property holds for I3322, which lives in the simplest Bell scenario in which such a phenomenon could occur, has remained elusive. Here, we provide a proof of this conjecture.
An approximate proof, containing all the main ideas, was produced by GPT 5.5 Pro after various rounds of interaction. The presentation of this proof was revised substantially by the author for completeness, correctness, and clarity.
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.