We give exact rational certificates for two results in the bipartite Bell scenario with three two-outcome measurements per party. First, the maximum of the $I_{3322}$ (Froissart) functional over the almost-quantum set -- the $1{+}AB$ level of the Navascues-Pironio-Acin (NPA) hierarchy -- is certified to lie between $0.2514708838$ and $0.2514709016$ (width $<2\times10^{-8}$), in the normalization with classical bound $0$ and quantum maximum $\approx0.2508754$; to our knowledge no value for this landmark had been published. Second, we exhibit a single exact rational behavior $T^{*}$ carrying certificates in both directions: a rational moment matrix proving $T^{*}$ almost-quantum, and a rational dual object proving that every NPA level-2 completion of $T^{*}$ requires slack at least $q=4.5934\times10^{-4}$, so that $T^{*}$ is not quantum. The strict inclusion $Q\subsetneq\tilde{Q}$ in the minimal three-setting scenario is thereby established end-to-end with no reliance on floating-point computation. Anatomically, the witness is locally quantum: all nine of its $2\times2$ correlator subtables pass the Tsirelson-Landau-Masanes criterion, so its non-quantumness is irreducibly a three-setting phenomenon, concentrated in ordering-dependent pseudo-moments coupling each party's third setting to the other's first two; the level-2 certificate is equivalently an exact sum-of-squares identity that any quantum state values nonnegatively and the witness is forced to value at $-q$. The certificates follow the rational rounding-projection-lifting programme recently developed for noncommutative optimization; for NPA moment matrices in canonical form the required projection reduces to a per-moment uniform shift, making certification essentially free at these sizes, and we work the programme in both the feasibility and infeasibility directions on one witness. Machine-checkable certificates and verifiers requiring only the Python standard library accompany the paper. Files in this record: the paper (PDF); exact rational certificates for both theorems (i3322_1AB_certificate.json, i3322_level2_infeasibility_certificate.json); standalone verifiers requiring only the Python 3 standard library (verify_i3322_certificate.py, verify_level2_certificate.py); and the generating scripts (certify_landmark.py, certify_level2.py). See README_ancillary.txt for which artifact proves which theorem and how to run the verifiers. v2: corrects the mechanism attribution in Sec. 5 (anatomy) and adds the archive reference to Data Availability; theorems, certificates, and verifiers unchanged.
We give exact rational certificates for two results in the bipartite Bell scenario with three two-outcome measurements per party. First, the maximum of the I3322 (Froissart) functional over the almost-quantum set -- the 1+AB level of the Navascues-Pironio-Acin (NPA) hierarchy -- is certified to lie between 0.2514708838 and 0.2514709016 (width <2×10−8), in the normalization with classical bound 0 and quantum maximum ≈0.2508754; to our knowledge no value for this landmark had been published. Second, we exhibit a single exact rational behavior T∗ carrying certificates in both directions: a rational moment matrix proving T∗ almost-quantum, and a rational dual object proving that every NPA level-2 completion of T∗ requires slack at least q=4.5934×10−4, so that T∗ is not quantum. The strict inclusion Q⊊Q~ in the minimal three-setting scenario is thereby established end-to-end with no reliance on floating-point computation. Anatomically, the witness is locally quantum: all nine of its 2×2 correlator subtables pass the Tsirelson-Landau-Masanes criterion, so its non-quantumness is irreducibly a three-setting phenomenon, concentrated in ordering-dependent pseudo-moments coupling each party's third setting to the other's first two; the level-2 certificate is equivalently an exact sum-of-squares identity that any quantum state values nonnegatively and the witness is forced to value at −q. The certificates follow the rational rounding-projection-lifting programme recently developed for noncommutative optimization; for NPA moment matrices in canonical form the required projection reduces to a per-moment uniform shift, making certification essentially free at these sizes, and we work the programme in both the feasibility and infeasibility directions on one witness. Machine-checkable certificates and verifiers requiring only the Python standard library accompany the paper. Files in this record: the paper (PDF); exact rational certificates for both theorems (i3322_1AB_certificate.json, i3322_level2_infeasibility_certificate.json); standalone verifiers requiring only the Python 3 standard library (verify_i3322_certificate.py, verify_level2_certificate.py); and the generating scripts (certify_landmark.py, certify_level2.py). See README_ancillary.txt for which artifact proves which theorem and how to run the verifiers. v2: corrects the mechanism attribution in Sec. 5 (anatomy) and adds the archive reference to Data Availability; theorems, certificates, and verifiers unchanged.
For the tilted family Jw=w(a1+a2−b1−b2)+∑ijsijEij containing the I3322 Bell functional at w=1, we map where and why the almost-quantum set (NPA level 1+AB) exceeds NPA level 2, using a subset-relaxation diagnostic: adjoin the twelve level-2 words to the 1+AB basis in subsets and measure each subset's share of the gap. The excess is sharply localized and structurally rigid. (i) At w=1, the four matched words A1A2,A2A1,B1B2,B2B1 on the marginal-anchored settings recover 92% of the gap, while either party's complete pair structure, all eight third-setting words, and every mismatched cross-party pairing recover exactly none; at the level of the certified separation witness T∗, infeasibility appears already under a single cross-party pair of anchored words, one per wing, while every single-party and every third-setting subset leaves T∗ feasible. (ii) The mechanism is algebraic: reversal-symmetrized ordering differences obey ⟨A1A2BjB3⟩−⟨A1A2B3Bj⟩=21⟨[A1,A2][Bj,B3]⟩, an expectation of commutator products across the wings, and sharp sequential measurement reads only anticommutators, so the excess lives in a sector unreadable in principle on either wing alone. (iii) The gap requires the functional's marginal anchor to oppose and co-locate with the aligned correlator block: moving either, aligning their signs, or deleting the marginals annihilates the gap. (iv) In the anchor weight the gap opens only on a finite window, w∈(0.90,0.95) to (2.25,2.50), with a two-lobed interior peaking near w=2 at 4.07×10−3 -- 7.7 times the separation of standard I3322, which sits just inside the lower edge. Experiments probing the almost-quantum boundary should therefore run the tilted functional near w=2, not the textbook one. The headline values are certified in exact rational arithmetic: the two-word infeasibility of T∗, the matched-four value, the peak, and one closure; remaining grid values are floating-point with two solver families agreeing to five digits. Files in this record: the paper (PDF); exact rational proof objects for the four headline certificates (window_paper_certificates.json); a standalone verifier requiring only the Python 3 standard library (verify_window_certificates.py); analysis scripts (why_level2.py, anchor_tracking.py); and certificate generators (certify_window.py, refreeze_window.py). See README_window.txt. Companion record with the certified separation and its certificates: doi:10.5281/zenodo.21534675.
The Navascués--Pironio--Acín (NPA) hierarchy gives the standard semidefinite outer approximations to quantum behaviors. Whether \emph{any} finite level can already equal the quantum set has remained open even in the bipartite scenario with two binary measurements per party. We demonstrate that \emph{no finite level} is exact. For the symmetric doubly tilted CHSH functional hα=A0B0+A0B1+A1B0−A1B1+α(A0+B0), set T=1−α. Its quantum maximum satisfies [ωQ(1−T)−(4−2T)]/T3→4/3, whereas every fixed NPA level satisfies [ωL(1−T)−ωQ(1−T)]/T3→+∞. Under the corresponding boundary rescaling, an explicit expectation of the positive operator ωQ(1−t2)I−Ht converges to the Motzkin polynomial. A bounded fixed-level error would therefore make the Motzkin polynomial plus a nonnegative constant a sum of squares, which is impossible. Consequently, every standard NPA relaxation based on a fixed finite list of words in the measurement projectors strictly contains the complete quantum set, and its nonquantum behaviors accumulate at a local deterministic behavior. Thus, the finite-level exactness of CHSH and all one-sided tilted CHSH maxima does not extend to an exact finite-level description of the complete quantum set in the minimal scenario.
Four companion papers located, dissected, generalized, and laddered the excess of the almost-quantum set over NPA level 2 on tilted Imm22 functionals. This paper asks why the numbers are what they are, and answers for the bottom of the tilted I3322 window: the constants are algebraic all the way down. Three thresholds are proven exactly. The no-signaling-line visibility threshold of I3322 is the golden ratio conjugate, λ3∗=(5−1)/2 (explaining a numerical coincidence between two cost conventions, since 1/λ−1=λ precisely at φ; the same constant appears elsewhere in Bell physics as a critical detection efficiency, with no known derivation connecting the two). The quantum maximum of the I3322 correlator part is exactly 5, attained by a hexagonal configuration -- a non-facet functional whose value appears to be new. The I4422 threshold is λ4∗=210−6, the positive root of x4+12x2−4, by a symmetrization argument in which two feasibility intervals kiss exactly at the quartic root. Below the window, the flat branch is a theorem: an affine family of zero-slack rational duals proves the gap vanishes identically on [0,471/500], certified closure gives gap≤2.3×10−8 at w=2357/2500, and certified opening gives gap≥1.10×10−6 at w=1889/2000, so the lower edge lies in [0.942,0.9445]. Inside, ten dual-solver points spanning five orders of magnitude follow the onset law gap=21(w−22/3)2 to 1--2%, down to gaps of 5×10−10; the coefficient follows from the envelope theorem given the measured unit marginal susceptibility (1.00(2)), and the edge is a staggered pair of bifurcations, level 1+AB at we=22/3 and level 2 at weL2=0.955934(3). The edge constant is derived at first order: the flat branch's 1+AB completion obeys a Clifford--Wick pairing and carries an eight-dimensional relation ideal (the dual hexagon plus five operator sum rules); the marginal mode must break relations to turn on, and the cheapest tilt breaks exactly three while preserving five, at a price certified in both directions by exact arithmetic over Q(2) to be 82/3 per 4 units of anchor -- hence 22/3. The certificates' entries lie in 61Z[2] and 181Z[2], extending the series' exact-certificate method beyond the rationals for the first time. A failed derivation route -- a homogenized fractional SDP whose optimum sits at infinity, so that every finite iterate overestimates the edge by 1.7×10−4 -- is documented alongside the successes, in the company of the field's own precedent that at this boundary only exact objects adjudicate. Files in this record: the paper (The_Algebraic_Edge.pdf and .tex); the Q(2) certificate pair for the marginal-price theorem with its standalone verifier (q2_edge_certificates.json, verify_q2_edge.py); the lower-edge theorem's proof objects -- the zero-slack flat-dual family, the rational level-2 flat completion, and the closure/opening pair -- with verifier (edge_bracket_certificates.json, verify_edge_bracket.py); the onset-law data table (onset_table.json); generating scripts (gen_edge_bracket.py, gen_onset_table.py); and a README. Both verifiers require only the Python 3 standard library. Companion records: the certified separation (doi:10.5281/zenodo.21534675), the tilted I3322 window (doi:10.5281/zenodo.21603712), the genericity and necessity tests (doi:10.5281/zenodo.21616031), and the ladder anatomy (doi:10.5281/zenodo.21625241).
Gigena et al. [npj Quantum Inf. 11, 82 (2025)] proved the exact quantum maximum of the doubly-tilted CHSH functional Bαβ=α⟨A0⟩+β⟨B0⟩+CHSH and observed that the NPA level required to reach it grows without evident bound toward the critical line α+β=2. We quantify the mechanism on the symmetric slice s=2−α−β: (i) the quantum value leaves the local bound cubically, cQ=4−s+s3/6−s4/36+O(s5); (ii) each NPA level overshoots quadratically, ck(s)=4−s+aks2+O(s3), with the almost-quantum coefficient computed exactly, a1+AB=3/64; (iii) the divergence of the required exact level is equivalent to positivity of the single sequence (ak) - proven for every k in the companion paper. We prove the supercritical side completely: for all α,β≥1 and every level the hierarchy is exact, via three explicit rational certificates realizing an affine identity. The hierarchy's exactness thus undergoes a phase transition at the critical line. On the subcritical side we certify the first four levels in exact arithmetic (rational pseudo-moments beating cQ, confirmed by Sturm's theorem). We identify the exact mechanism: rescaled to the critical corner, the limiting obstruction is the Motzkin polynomial, the classical nonnegative-but-not-sum-of-squares form, so the finite-level failure sits in the restricted-certificate regime. The phase boundary has a precise geometric reading via Nie's finite-convergence theorem and Marshall's boundary Hessian condition: a self-tested optimum is finitely NPA-certifiable whenever its boundary Hessian is nondegenerate (contact order two), which holds for the single tilt and fails exactly at the doubly-tilted cubic touch. Three verified errata in the published polynomial system of Gigena et al. are documented.
Gigena, Panwar, Scala, Araujo, Farkas and Chaturvedi [npj Quantum Inf. 11, 82 (2025)] determined the quantum maximum of the doubly-tilted CHSH functionals, observed that the Navascues-Pironio-Acin level needed for exactness grows without evident bound toward the critical tilt, and asked whether any finite level suffices. We answer this in the negative. For the symmetric critical family Bs=(1−s/2)(⟨A0⟩+⟨B0⟩)+CHSH we prove: for every NPA level k≥2 there are an explicit rational gk>0 and an sk∗>0 with ck(s)≥4−s+gks2 on (0,sk∗]; since the quantum value leaves the local bound only cubically, every finite level strictly overshoots on an interval: no finite level is exact on any neighbourhood of the critical point. Unconditionally a2>1/39, a3>1/188, a4>1/641. The proof is a primal construction: an exactly feasible moment curve at each level, built from level-uniform structural laws and one level-independent signed witness -- a closed-form class function y∗ with NkTΓ(y∗)Nk=ukukT at every level. The mechanism forces the sign: for k≥3 no quantum state and no smooth curve of quantum models can realize the gain direction, so the overshoot lives strictly in the non-quantum part of the NPA tangent cone. The proof is computer-assisted in the strict sense: finite exact-integer verifications with proven degree bounds are constituent parts of the argument; the chain has been re-verified against independent implementations, including a symbolic per-regime proof of the witness identity and a clean-room implementation written from the paper text alone. The one external input is the published quantum value of Gigena et al., cross-checked to twelve digits.
This record contains the manuscript and complete verification artifact for the Bell inequality J62_4422 of Pál and Vértesi.It certifies the exact rational bracket750075557472719639 / 10^18 ≤ β_Q ≤ 750091086083765047288227625 / 10^27,and exact attainment of 3/4 by the stated two-qubit strategy. The lower endpoint is an exact rational strategy. The upper endpoint is certified by a sum-of-squares chain for the retained 365-word relaxation level.The Krawczyk certificate localizes the underlying KKT zero and establishes a strict local maximum, including negativity of the border form in all 256 boundary strata; it does not establish global optimality of that localized point. Accordingly, this work does not claim the exact quantum value, qubit-restricted optimality, minimum optimizer dimension, or global optimality of the localized KKT point.The archive includes SHA-256 manifests, standard-library replay programs, independently written certificate checks, mutation controls, and pinned full Krawczyk replays. The acceptance programs are fail-closed under optimized Python. The manuscript and certificates are licensed CC BY 4.0; verification code is licensed MIT.Version 1.1 is a hygiene-only update: it replaces one machine-local default path in a helper with the portable repository root ".", updates the corresponding checker and certificate source bindings, and leaves the mathematical calculation, inputs, claims, and replay outputs unchanged.