We determine the quantum value of the three-setting, two-outcome Bell functional I3322. The value is the unique intersection \(q_\dagger\) 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 \(q_\dagger\). 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.
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16 pages; computer-assisted proof; complete verification artifact at DOI 10.5281/zenodo.22160086
We determine the quantum value of the three-setting, two-outcome Bell functional I3322. The value is the unique intersection (q_\dagger) 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_\dagger). 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.
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.
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.
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+1/100. On the other hand, all qubit-projective strategies are bounded by 22+5/250+2/32400, giving a fully analytic certified gap greater than 1/1000. 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.
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.
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.