[challenge]: An explicit, certified Tsirelson violation

Released by

Jie Wang (AMSS, Chinese Academy of Sciences) & Jin-Guo Liu (Hong Kong University of Science and Technology (Guangzhou))

Contact email

cacate0129@gmail.com

Method

Other

Challenge issue

Difficulty: ★★★★ (rated by Jie Wang)

Background

MIP* = RE (arXiv:2001.04383) refuted Connes' embedding conjecture non-constructively: no explicit correlation scenario with commuting-operator value strictly above every finite-dimensional value is known. Finding one is the most-cited open problem left by the refutation (arXiv:2109.12682, arXiv:2502.06697).

Research objective

Exhibit an explicit scenario (or tracial moment problem) with a certified strict gap between the commuting-operator value and the supremum over finite-dimensional models. Search over structured families — solution-group games (arXiv:1606.03140), compiled variants (arXiv:2507.17581) — with NPA-from-above converging to the commuting value and see-saw plus dimension-uniform outer relaxations capping the finite-dimensional value.

Verification plan

• Success gate: a rational NPA dual certificate proving val_co ≥ c1, together with a certified upper bound val_q ≤ c2 < c1 valid for all finite dimensions (via a convergent outer relaxation of the tensor model) — two exact numbers with a strict gap.
• Hope signal: candidates where the numerical gap persists across levels but the all-dimension upper bound resists certification — publishable evidence lists.
• Pivot signal: every certifiable all-dimension upper-bound technique provably converges to the commuting value too — that structural no-go is itself a significant theorem.

Why this may lead to research output

An explicit certified instance would be studied simultaneously by operator algebraists, complexity theorists, and physicists — the constructive counterpart to one of the decade's landmark results (arXiv:1008.1142, arXiv:1703.08618).

References

1. Ji, Natarajan, Vidick, Wright, Yuen, MIP* = RE, arXiv:2001.04383.
2. Goldbring, The Connes embedding problem: a guided tour, arXiv:2109.12682.
3. Slofstra, Tsirelson's problem and an embedding theorem for groups arising from non-local games, arXiv:1606.03140.
4. Cui, Falor, Natarajan, Zhang, A convergent sum-of-squares hierarchy for compiled nonlocal games, arXiv:2507.17581.

No repository files, tests, or entry points are named. Start by reading the cited MIP* = RE and solution-group-game references, then formalize a candidate scenario and its certification requirements. Done means producing exact c1 and c2 values with a certified strict gap, or documenting the stated no-go result.