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Lower bound for the complex Grothendieck constant

What is the exact value of the complex Grothendieck constant KGCK_G^{\mathbb C}, the least KK such that ∣∑i,jaij⟨xi,yj⟩∣≤Kmax⁡∣εi∣=∣δj∣=1∣∑i,jaijεiδj∣\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 (aij)(a_{ij}) and all unit vectors xi,yjx_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 1.338071.33807 and Haagerup's 1987 upper bound of about 1.404911.40491; the value is open, and this entry records progress on the lower bound.

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Functional analysis
Posed by
Alexandre Grothendieck, Résumé (1953); the value of the complex constant an explicit open question at least since Haagerup's 1987 upper bound
Year posed
1953
Years open
73y
Solved
2026-09-07
Model
Odin Automatic AI Research Agent
Vendor
—
Collaborators
Shengtao Guo, Ethan X. Fang, Junwei Lu
Verification
Unreviewed
Publication
Preprint
Significance
30 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

The paper provesKGC>1.35584631827168. K_G^{\mathbb{C}}>1.35584631827168.
The previously recorded Davie lower bound is approximately 1.338071.33807, while Haagerup's upper bound is approximately 1.404911.40491. 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∞L_{\infty}-to-L1L_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 K∗\mathcal{K}_* denotes the supremum obtainable within that framework, it proves1.35584631827168<K∗<1.35584697425050. 1.35584631827168<\mathcal{K}_*<1.35584697425050. The upper endpoint here is a limitation of this specific criterion, not an upper bound on KGCK_G^{\mathbb{C}} itself.

What the AI did

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.

Verification

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∞L_\infty-to-L1L_1 estimate for the weighted Gaussian Hermite multipliers that turns the certified numbers into a bound on KGCK_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.

Sources

Submitted by VibeGene on

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