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FrontierAnalysissignificance 35

Lower bounds for the complex Grothendieck constant

the complex Grothendieck constant KGCK_G^{\mathbb C}

Current best · higher is better
>1.338> 1.338
Alexander M. Davie (unpublished), 1984
steps
2
by AI
0
since
1953
19601970198019902000201020201.281.301.321.341.361953: ≥ 4/π = 1.27323… (Grothendieck's Gaussian argument, complex case)1984: > 1.338 (Alexander M. Davie (unpublished))2026-09-07: > 1.35584631827168 (Guo, Fang and Lu, with the Odin research agent) - candidate, under review

The line is the frontier over time, climbing as the bound goes up: higher is better here. Filled dots are steps that moved it; muted dots are results that did not. Orange dots are catalog entries, results with AI in the loop. Hollow dots are candidates under review and never move the line.Grey dots along the bottom edge are results from before the quantity had a number, placed at the worst end because they have no value on this axis. Dots that would overlap are nudged sideways a few pixels. Hover a dot for its value and attribution.

About this frontier

Grothendieck proved in 1953 that there is a universal constant KK with ∣∑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 and all unit vectors in a complex Hilbert space. The least such KK is the complex Grothendieck constant KGCK_G^{\mathbb C}, and its value is open: Haagerup's 1987 upper bound is 1.40491…1.40491\ldots, and he conjectured the truth is 1.4046…1.4046\ldots. This frontier tracks the lower bound, which the Gaussian argument put at 4/π4/\pi and Davie lifted to 1.3381.338 in 1984, where it stayed for forty-two years. The ceiling is Haagerup's number; nothing here claims to be at it.

Every step, newest first

DateValueWhoModelStatusSource
7 Sept 2026>1.35584631827168> 1.35584631827168
Finitely many complex Hermite projections under a common radial weight give a dimension-independent L-infinity-to-L1 estimate; the numerical inequalities are certified by Arb interval arithmetic. Closes more than a quarter of the Davie-to-Haagerup gap. Unreviewed arXiv v1, one week old, certificates not rerun here: a candidate.
Guo, Fang and Lu, with the Odin research agentOdin Automatic AI Research Agentcandidate
unreviewed
entry
1984>1.338> 1.338best
Never published under Davie's name. Reported in Haagerup's 1987 paper on the upper bound, which is the linked source, and in Pisier's survey: "Davie (unpublished) improved the lower bound to K_G^C > 1.338." Dated to Davie's Columbia lecture of 1984 on matrix norms related to the inequality (LNM 1166). This number stood for forty-two years.
Alexander M. Davie (unpublished)–historicalsource ↗
1953≥4/π=1.27323…\ge 4/\pi = 1.27323\ldots
The first lower bound, from the L1 norm of a standard complex Gaussian: Pisier, Section 4, "in the complex case ||g||_1 = (pi/4)^{1/2}, and hence K_G^C >= 4/pi". Grothendieck's Resume proves the real-case value pi/2 and the argument carries over word for word. Dated to the Resume; the source is Pisier's survey, where the complex case is written out.
Grothendieck's Gaussian argument, complex case–historicalsource ↗

Rows read in Pisier's 2012 survey (arXiv:1101.4195, Section 4), which writes out the Gaussian bound for the complex case and reports Davie's unpublished 1.338. The 1953 row is dated to Grothendieck's Resume, whose real-case argument Pisier says extends immediately; the Resume was not opened. Davie's bound never appeared under his name: Haagerup 1987 reports it and is the source, dated to Davie's 1984 Columbia lecture. The 2026 row is a candidate, an unreviewed arXiv v1 not rerun here, so it is hollow and the headline stays 1.338. Haagerup's 1.40491 is the ceiling every row is checked against.

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