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

Douglas, Seth

doi:10.5281/zenodo.22698978

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

Seth Douglas · version 4.1.0 · Zenodo, 11 September 2026 · 25 pages · arXiv:2609.05555 · doi:10.5281/zenodo.22698978 · PDF

In plain wordsThe strongest violation of a well-known test of quantum nonlocality needs an infinitely large quantum system; finite systems can only approach it.

The resultat three depths

In one sentenceThe largest violation of the I₃₃₂₂ Bell inequality is reached by a quantum system with infinitely many dimensions and by no finite one, and getting within $ε$ of it takes a number of dimensions on the order of $\log (1 / ε)$ .

As the paper states it, in briefNo finite-dimensional quantum strategy attains the quantum maximum of the I₃₃₂₂ Bell functional, mixed states and binary measurements included; a strategy on $ℓ^{2} (ℤ) \otimes ℓ^{2} (ℤ)$ does; approaching it needs local dimension $Θ (\log (1 / ε))$ .

For a scientist
One paragraph

I₃₃₂₂ is the best-known bipartite Bell functional beyond the Clauser–Horne–Shimony–Holt family: three binary measurements per party, and a facet of the local polytope inequivalent to CHSH. In 2010 Pál and Vértesi computed its quantum value numerically, with finite-dimensional strategies converging to it, and posed as a challenge an analytical proof that no finite dimension suffices. The paper proves it: the common tensor-product and commuting-operator supremum $S$ , certified within an exact window of width about $3.6 \times 10^{- 9}$ , is attained by no finite-dimensional strategy, mixed states and binary POVMs included, and is attained by a spatial strategy on $ℓ^{2} (ℤ) \otimes ℓ^{2} (ℤ)$ . So $C_{q} (3, 3; 2, 2)$ is not closed. The least local dimension within $ε$ of $S$ is $Θ (\log (1 / ε))$ , with an explicit upper constant. The proofs combine a certified value window, critical Bellman storage, spectral transport and normalizable orbit measures; a finite weighted-flow argument gives the lower bound. Pauwels and Coladangelo proved nonattainment independently and concurrently, as did V. Connor in a manuscript of 1 August 2026. The exact certification of the bounds came first, from Nidhal Mghirbi, in July 2026, and is tighter than the window used here.

For an expert
The theorems, as stated in the paper

Theorem (Nonattainment)

Let $S = ω_{tensor} (I_{3322}) = ω_{commuting} (I_{3322})$ be the quantum supremum in the normalization of Section 2, certified to lie in $(0.2508753845015185,$ $0.250875388108398]$ . Then no finite-dimensional quantum strategy attains $S$ : for all finite-dimensional Hilbert spaces $ℋ_{A}, ℋ_{B}$ , every state $ρ$ on $ℋ_{A} \otimes ℋ_{B}$ , and every triple of binary measurements per party (projective or POVM, acting as effects of the form $F \otimes I$ on Alice’s side and $I \otimes G$ on Bob’s) one has $Tr (ρ ℬ) < S$ .

Theorem (Spatial attainment)

$S$ is attained by a spatial strategy: there are projections $a_{1}, a_{2}, a_{3}$ and $b_{1}, b_{2}, b_{3}$ in $ℒ (ℓ^{2} (ℤ))$ , and a unit vector $ψ_{S} \in ℓ^{2} (ℤ) \otimes ℓ^{2} (ℤ)$ , such that setting $A_{i} = a_{i} \otimes I$ and $B_{j} = I \otimes b_{j}$ in the Bell operator gives $⟨ ψ_{S}, ℬ ψ_{S} ⟩ = S$ . In particular the resulting behavior lies in $C_{q s} (3, 3; 2, 2)$ .

Corollary

$C_{q} (3, 3; 2, 2)$ is not closed, and $C_{q s} (3, 3; 2, 2) ∖ C_{q} (3, 3; 2, 2) \neq \emptyset$ .

Theorem (Rate)

There are constants $c_{1}, c_{2} > 0$ with $c_{1} \log (1 / ε) \leq D (ε) \leq c_{2} \log (1 / ε)$ for all sufficiently small $ε > 0$ . The upper half holds with the explicit derived constant:

D (ε) \leq 23.9010650 \log (1 / ε)

for all sufficiently small $ε$ .

Logarithms are natural. The lower constants are existential; no explicit onset or sharp constant is claimed. The exact value of $S$ beyond the certified window is not identified here. Krohn-Grimberghe (arXiv:2609.29693, 31 August 2026) claims an exact identification of the value, by a computer-assisted proof that is not peer reviewed; we have not verified it independently.

The question

“Taking as a conjecture that our construction is optimal (in the sense that no finite dimension suffices to achieve the true quantum maximum for I₃₃₂₂), […] We pose it as a challenge to provide an analytical proof of our conjecture.”

Pál and Vértesi (2010), as quoted in the paper

Questionson the index

Answered Does any finite-dimensional strategy reach the quantum maximum of the I₃₃₂₂ Bell inequality? posed by others
Answered Can a finite quantum system reach the largest violation of the I₃₃₂₂ Bell inequality?
Claimed answered elsewhere What is the exact quantum value of I₃₃₂₂?
Open What is the sharp constant of the I₃₃₂₂ dimension rate? left open

How this was checked

Blind audits: A blind audit of the corrected lower bound
Audits: An end-to-end audit before version 4.1.0.
Exact certificates: Certified value window in exact arithmetic
Lean: 37 selected declarations, standard axioms only; the analytic proof is not formalized.
Peer review: Not yet.

Each check, what it covers, and what it leaves to the written proof.

Each check, what it covers, and what it leaves to the written proof.
Check	Covers	Does not cover
Exact arithmetic	The certified common tensor-product and commuting value window, by rational and interval certificates: a local replay checked all 46,458 upper cells and rebuilt the 255-dimensional lower quotient.	The exact value of $S$ beyond the window.
Lean 4	37 selected declarations, using only the standard axioms.	The analytic theorem. Convex-envelope regularity, orbit disintegration, existence of the signed potential and the reduction from quantum strategies to flows are proved in writing only.
Blind audit of the lower bound	A reader that saw only the manuscript tried to refute the corrected weighted-flow lower bound, re-derived its inequalities and ran its own finite checks. It found no blocking defect.	Anything outside its stated scope. An audit is evidence, not a guarantee.
End-to-end audit	Before version 4.1.0, independent readers worked through the foundations, the spatial strategy and the upper bound. Their corrections are in the revision notes and the record below.	Errors shared by every reader.
Peer review	Not yet.	A preprint on arXiv and Zenodo.

Every audit here is internal. The repository’s VERIFY.md states the proof coverage exactly. How we check results.

Label checked 30 October 2026.

Novelty and credit

Novelty: Concurrent with independent work by Pauwels (arXiv:2608.29734) and Coladangelo (arXiv:2609.06038), credited, and by V. Connor (manuscript dated 1 August 2026, cited by Krohn-Grimberghe). The paper credits Nidhal Mghirbi’s exact certification of the bounds (July 2026), which came first and is tighter than the certified window it uses.

Pál and Vértesi posed the conjecture and the challenge in 2010.
Exact rational two-sided enclosures of the value predate this paper: Nidhal Mghirbi published proof-carrying exact certificates with an enclosure narrower than $10^{- 9}$ , tighter than the window used here, in July 2026 (the release, the paper). In the paper’s words, they “establish a strict enclosure; they do not address equality of the two models, attainment, or nonattainment”, and the window used here is “a certified input, not a contribution.” Mghirbi’s companion paper Critical Bellman–Jacobi dynamics develops spatial attainment and logarithmic truncation in a broader reversible class.
V. Connor’s manuscript (1 August 2026) claims the qualitative result independently, as Krohn-Grimberghe records. Krohn-Grimberghe claims an exact identification of the value (computer-assisted, not peer reviewed).
Pauwels’ Theorem 1 identifies the supremum of the full finite Pál–Vértesi family with the finite quantum supremum; his nonattainment theorem and Coladangelo’s independent nonattainment and spatial-attainment results are credited.
The failure of this project’s withdrawn construction in August does not refute the Pál–Vértesi family.

Recordreleases, corrections and independent work

4 Aug 2026Versions 1.0.0 to 1.2.0: the first public claims, by a bi-infinite chain construction.doi:10.5281/zenodo.21782009, 21782527, 21782750
5 Aug 2026Correction. Version 2.0.0 withdrew those claims. Our own exact audit after release found a nonzero mismatch in the construction; both computation engines had inherited the same transcribed step.doi:10.5281/zenodo.21799071
5 Aug 2026Version 3.0.0: finite-dimensional nonattainment, nonclosure and spatial attainment, by a rebuilt proof; versions 3.1.0 to 3.2.3 followed on 6 August; 3.2.1 to 3.2.3 corrected the packaging, the title, two decimal bounds and the credit (corrections).doi:10.5281/zenodo.21813360
5 Aug 2026V. Connor, Maximal I3322 violation requires infinite local dimension, a manuscript dated 1 August 2026 and posted on ResearchGate on 5 August, as cited by Krohn-Grimberghe: the qualitative result, independently.
7 Aug 2026Version 3.3.0 adds the dimension rate $D (ε) = Θ (\log (1 / ε))$ .doi:10.5281/zenodo.21843326
25 Aug 2026Version 4.0.0: one paper, with corrections: Theorem (S) restated with its locality constraint, the orbit structure of Section 4 corrected, the priority claim for the rate narrowed, one displayed constant corrected (corrections).doi:10.5281/zenodo.22099128
30 Aug 2026Pauwels, arXiv:2608.29734: nonattainment, independently.
31 Aug 2026Krohn-Grimberghe, arXiv:2609.29693: a claimed exact identification of the value, computer-assisted and not peer reviewed; it claims no priority for the qualitative statements.
3 Sep 2026This paper on arXiv as 2609.05555.
5 Sep 2026Coladangelo, arXiv:2609.06038: nonattainment and spatial attainment, independently.
11 Sep 2026Correction. Version 4.1.0. An unjustified identification of paired components in the lower bound was replaced by a weighted flow over all retained edges; the limiting-weld justification, an endpoint-gap constant, the spatial amplitude notation, the mixed-state argument and two upper-rate constants were corrected. The upper coefficient is unchanged. arXiv version 2 followed the same day.doi:10.5281/zenodo.22698978

Release dates are Zenodo publication dates; arXiv dates are arXiv’s. The paper calls the work of Pauwels and of Coladangelo concurrent and independent, and so do we; Connor’s manuscript is not cited in the paper and is listed here from Krohn-Grimberghe’s citation of it.

Versionsthirteen, each with its own DOI

Every archived version, newest first. Dates are Zenodo publication dates. All versions: 10.5281/zenodo.21782008, which always resolves to the newest.
Version	Date	DOI	Notes
4.1.0	11 Sep 2026	10.5281/zenodo.22698978	Correction: Proof correction to version 4.0.0. The lower bound had identified paired components without justification; it now uses a weighted flow over all retained edges. The limiting-weld justification, an endpoint-gap constant, the spatial amplitude notation, the mixed-state argument and two upper-rate constants were also corrected. Concurrent independent work by Pauwels and by Coladangelo is credited. The main statements and the upper coefficient are unchanged. PDF hosted here.
4.0.0	25 Aug 2026	10.5281/zenodo.22099128	Correction: Alongside merging the rate note into the paper: Theorem (S) restated with its locality constraint, which the earlier statement did not formally require; the orbit structure of Section 4 corrected; Pál and Vértesi’s conjecture positioned against their paper; the minimality argument completed; the priority claim for the rate narrowed and sourced; an endpoint-mass bound stated correctly; one displayed constant corrected from 0.9197272 to 0.91972725.
3.3.0	7 Aug 2026	10.5281/zenodo.21843326	Titled “The I3322 quantum value: certified window, spatial attainment, finite-dimensional nonattainment, and the Cqs/Cq separation”.
3.2.3	6 Aug 2026	10.5281/zenodo.21826916	Titled “The I3322 quantum value: certified window, spatial attainment, finite-dimensional nonattainment, and the Cqs/Cq separation”. Correction: A fresh audit of the version 3.2.2 corrections found nine items, all fixed: the Mghirbi reference corrected to the right record, the 2018 attribution weakened to what the source says, other citations repaired, and stale withdrawn claims removed from the README and citation metadata. Nothing mathematical changed.
3.2.2	6 Aug 2026	10.5281/zenodo.21817385	Titled “The I3322 quantum value: certified window, spatial attainment, finite-dimensional nonattainment, and the Cqs/Cq separation”. Correction: After three adversarial audits of the resolution paper: retitled to the statement it proves; two false decimal bounds fixed; credit added for N. Mghirbi’s earlier exact enclosures and for an implication in Dykema, Paulsen and Prakash (2018); the minimality claim scoped; attributions, definitions and bibliography corrected. The theorems are unchanged.
3.2.1	6 Aug 2026	10.5281/zenodo.21816638	Titled “The I3322 quantum value: certified window, spatial attainment, finite-dimensional nonattainment, and the Cqs/Cq separation”. Correction: Packaging: version 3.2.0 shipped the resolution paper, but a tooling slip dropped the README and metadata edits that go with it. Version 3.2.1 restores them.
3.2.0	6 Aug 2026	10.5281/zenodo.21816625	Titled “The I3322 quantum value: certified window, spatial attainment, finite-dimensional nonattainment, and the Cqs/Cq separation”.
3.1.0	6 Aug 2026	10.5281/zenodo.21814777	Titled “The I3322 quantum value: certified window, spatial attainment, finite-dimensional nonattainment, and the Cqs/Cq separation”.
3.0.0	5 Aug 2026	10.5281/zenodo.21813360	Titled “The I3322 quantum value: certified window, finite-dimensional nonattainment, and nonclosure of Cq(3,3;2,2)”.
2.0.0	5 Aug 2026	10.5281/zenodo.21799071	Titled “The I3322 quantum value: a certified two-sided window and correction record”. Correction: Versions 1.0.0 to 1.2.0 withdrawn. Our own exact audit after release found a nonzero mismatch in the construction behind their claims; both computation engines had inherited the same transcribed step. A rebuilt proof followed as version 3.0.0.
1.2.0	4 Aug 2026	10.5281/zenodo.21782750	Titled “A rigorously characterized I3322 quantum wall, spatial attainment, and finite-dimensional nonattainment”.
1.1.0	4 Aug 2026	10.5281/zenodo.21782527	Titled “A rigorously characterized I3322 quantum wall and finite-dimensional nonattainment”.
1.0.0	4 Aug 2026	10.5281/zenodo.21782009	Titled “A rigorously characterized I3322 quantum wall and finite-dimensional nonattainment”.

Paper, code and links

PDF

papers/i3322-v4.1.0.pdf, version 4.1.0, 468,806 bytes, 25 pages.

SHA-256 9aeed79f970cb033cca77dc915fefa7fac29d2d85b77fa3f1898172c6fb108a6

Of paper/resolution.pdf, 468,806 bytes, in the Zenodo archive; identical in the repository at that commit. The same file is hosted here: papers/i3322-v4.1.0.pdf.

DOI

10.5281/zenodo.22698978; all versions 10.5281/zenodo.21782008

Code

github.com/Apsiape/i3322-exact-wall

arXiv

arXiv:2609.05555

Commit

85b97c648541f52fc42fb31ee3626988f7bfde6c

The commit archived by Zenodo (Apsiape/i3322-exact-wall, annotated tag v4.1.0). The main branch has since added a citation-only DOI update and a rebuilt PDF.

Verify

From a clone of the repository, the public checks in VERIFY.md, which the README links as its verification record (Windows; on Linux use bounded inner commands):

python -m pip install -r requirements.txt
python tools/run_capped.py --limit-mib 512 --timeout-seconds 60 -- python tools/check_revision.py
python tools/run_capped.py --limit-mib 512 --timeout-seconds 60 -- python tools/release-preflight-selftest.py
python tools/run_capped.py --limit-mib 512 --timeout-seconds 60 -- python tools/verify_candidate_selftest.py
python tools/run_capped.py --limit-mib 512 --timeout-seconds 60 -- python tools/verify_candidate.py static
python tools/run_capped.py --limit-mib 512 --timeout-seconds 60 -- python tools/check_publication.py

VERIFY.md also gives the pinned Lean build (Lean 4.30.0) and the axiom check for the 37 declarations.

Lean, in detail

37 selected declarations, standard axioms only; the analytic proof is not formalized.

Licence

Paper and documentation CC BY 4.0; code BSD 3-Clause.

Page checked against the paper, the release repository, arXiv and the Zenodo record on 30 October 2026.

Corrections

11 Sep 2026Version 4.1.0. Proof correction to version 4.0.0. The lower bound had identified paired components without justification; it now uses a weighted flow over all retained edges. The limiting-weld justification, an endpoint-gap constant, the spatial amplitude notation, the mixed-state argument and two upper-rate constants were also corrected. Concurrent independent work by Pauwels and by Coladangelo is credited. The main statements and the upper coefficient are unchanged.doi:10.5281/zenodo.22698978
25 Aug 2026Version 4.0.0. Alongside merging the rate note into the paper: Theorem (S) restated with its locality constraint, which the earlier statement did not formally require; the orbit structure of Section 4 corrected; Pál and Vértesi’s conjecture positioned against their paper; the minimality argument completed; the priority claim for the rate narrowed and sourced; an endpoint-mass bound stated correctly; one displayed constant corrected from 0.9197272 to 0.91972725.doi:10.5281/zenodo.22099128
6 Aug 2026Version 3.2.3. A fresh audit of the version 3.2.2 corrections found nine items, all fixed: the Mghirbi reference corrected to the right record, the 2018 attribution weakened to what the source says, other citations repaired, and stale withdrawn claims removed from the README and citation metadata. Nothing mathematical changed.doi:10.5281/zenodo.21826916
6 Aug 2026Version 3.2.2. After three adversarial audits of the resolution paper: retitled to the statement it proves; two false decimal bounds fixed; credit added for N. Mghirbi’s earlier exact enclosures and for an implication in Dykema, Paulsen and Prakash (2018); the minimality claim scoped; attributions, definitions and bibliography corrected. The theorems are unchanged.doi:10.5281/zenodo.21817385
6 Aug 2026Version 3.2.1. Packaging: version 3.2.0 shipped the resolution paper, but a tooling slip dropped the README and metadata edits that go with it. Version 3.2.1 restores them.doi:10.5281/zenodo.21816638
5 Aug 2026Version 2.0.0. Versions 1.0.0 to 1.2.0 withdrawn. Our own exact audit after release found a nonzero mismatch in the construction behind their claims; both computation engines had inherited the same transcribed step. A rebuilt proof followed as version 3.0.0.doi:10.5281/zenodo.21799071

All corrections, and what counts as one.

Cite

BibTeX built from the repository’s CITATION.cff.

@misc{douglas2026i3322,
  author  = {Douglas, Seth},
  title   = {The {I3322} quantum value is attained spatially but not in finite dimension},
  year    = {2026},
  version = {4.1.0},
  doi     = {10.5281/zenodo.22698978},
  url     = {https://github.com/Apsiape/i3322-exact-wall/releases/tag/v4.1.0}
}

i3322.bib All papers as BibTeX