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Precise statement
What is the exact value of the complex Grothendieck constant $K_G^{\mathbb C}$, the least $K$ such that $\bigl|\sum_{i,j}a_{ij}\langle x_i,y_j\rangle\bigr|\le K\max_{|\varepsilon_i|=|\delta_j|=1}\bigl|\sum_{i,j}a_{ij}\varepsilon_i\delta_j\bigr|$ for every complex matrix $(a_{ij})$ and all unit vectors $x_i,y_j$ in any complex Hilbert space? Grothendieck proved it finite in 1953. Before this work the best bounds were Davie's lower bound of about $1.33807$ and Haagerup's 1987 upper bound of about $1.40491$; the value is open, and this entry records progress on the lower bound.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
Odin Automatic AI Research Agent
The paper's disclosure is one sentence, in the abstract and again under the heading "The role of AI in this proof": "Odin Automatic AI Research Agent was used to derive the lower bound and the proof." Taken at face value, as this site's classification rule requires, that is an AI-discovered claim, and it is the same disclosure this team gave for its Talagrand convolution entry. The paper says nothing about what Odin is, which models it runs on, or how it was steered, and no public description of the system was found, so the model maker is left empty rather than guessed at. The three named humans are Shengtao Guo, Ethan X. Fang and Junwei Lu.
Unreviewed arXiv preprint, v1 of 7 September 2026, with no independent endorsement and no peer review. The numerical part is certified by Arb ball arithmetic with outward rounding; the verification code and the exact rational inputs are supplied as arXiv ancillary files and in the linked repository, and were not run here. The analytic part - the dimension-independent $L_\infty$-to-$L_1$ estimate for the weighted Gaussian Hermite multipliers that turns the certified numbers into a bound on $K_G^{\mathbb C}$ - was not checked. Checked here: the arXiv record and abstract as submitted, that Davie's and Haagerup's bounds are as the paper states them, and that the catalog holds no entry on either Grothendieck constant, so the 2026 real-constant improvements the submitter mentions are not duplicates. Partial: a better lower bound, not the value.
The paper proves
$$
K_G^{\mathbb{C}}>1.35584631827168.
$$
The previously recorded Davie lower bound is approximately $1.33807$, while Haagerup's upper bound is approximately $1.40491$. Thus the result closes more than one quarter of the remaining lower-to-upper-bound gap, but it does not determine the exact value of the complex Grothendieck constant.
The construction uses finitely many additional complex Hermite projections together with a common radial weight to obtain a dimension-independent $L_{\infty}$-to-$L_1$ estimate. The numerical inequalities needed for the final bound are certified by interval arithmetic.
The paper also analyzes the limit of its particular weighted criterion. If $\mathcal{K}_*$ denotes the supremum obtainable within that framework, it proves
$$
1.35584631827168<\mathcal{K}_*<1.35584697425050.
$$
The upper endpoint here is a limitation of this specific criterion, not an upper bound on $K_G^{\mathbb{C}}$ itself.
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
VibeMath has not independently verified the mathematical claim.
AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.