The real Grothendieck constant

Upper bound, last-listed row

| $< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 10^{-4}$ | [SLXCKKM26] | Same team; via the first *asymptotic* Krivine-type rounding scheme (previous works only considered low-dimensional schemes). |value changed 2026-08-12

Lower bound, last-listed row

| $\dfrac{6\pi}{11}\approx 1.71360$ | [SLXCKKM26] | Major jump over the Davie–Reeds anchor $\approx 1.67696$; proved by establishing asymptotic limitations of Krivine schemes rather than by constructing explicit gap instances (as all prior lower bounds did). Together with the upper bound above, this determines the tenths digit of $K_G^{\mathbb R}$ to be $7$. |value changed 2026-08-12

Checked against material for its cited source

• The row shown above: bound row $< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 10^{-4}$ line 33 citing [SLXCKKM26] — the claim we checked: that the cited reference gives the upper bound this row lists, below π/(2 ln(1+√2)) − 10^-4, via the first asymptotic Krivine-type rounding scheme, as the row says. What we read supports it: the abstract gives the bound at this threshold with ≤, and Theorem 4.2 proves K_G ≤ 1.7818666069360661, below π/(2 ln(1+√2)) − 3.4737 × 10^-4 and so strictly below the row's threshold; the paper describes its scheme as the first asymptotic construction of rounding schemes. source link (read in arXiv version 2 (12 Aug 2026), the current version of the unversioned arXiv link the entry gives, at the abstract, Theorem 4.2 (p. 17) and Theorem 6.4 (p. 21), from page images; version 1's abstract and Theorem 4.2 were checked and read the same). Selected: chosen because something already looked wrong.
• Another row, not one shown above: bound row $< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 6.039\times 10^{-5}$ line 32 citing [LSXCKKM26] — the claim we checked: that the cited reference gives the upper bound this row lists, below π/(2 ln(1+√2)) − 6.039 times 10^-5, an improvement of 6.039 times 10^-5 over Krivine's bound, as the row says. What we read could not settle it: the value 6.039 × 10^-5, subtracted from Krivine's bound, matches, but every statement of it in the pages read uses ≤ where the row writes <; the paper's strict statements are at other thresholds: the title and abstract at 10^-5, and Theorem 1.2 at 9.7 × 10^-6. source link (read in arXiv version 3 (11 Aug 2026), the version whose author order matches the entry, at the abstract, Theorem 1.1 (p. 3) and Theorem 5.3 with its proof (pp. 16–17), from page images; version 1's Theorems 1.1 and 5.3 were checked and give the same bound, though Theorem 5.3's displayed construction parameters differ. Pages 6–13 and 18–28 and the appendices were not read). Selected: chosen because something already looked wrong.
• The row shown above: bound row $\dfrac{6\pi}{11}\approx 1.71360$ line 44 citing [SLXCKKM26] — the claim we checked: that the cited reference proves the lower bound this row lists, 6π/11 ≈ 1.71360, by establishing asymptotic limitations of Krivine schemes, as the row's comment says. What we read supports it: Theorem 7.1 states that the real Grothendieck constant is at least 6π/11, proved by bounding the admissible parameter of any Krivine scheme by 11/12 and using the asymptotic optimality of mixed Krivine schemes. source link (read in the arXiv preprint, version 2 (12 Aug 2026): the abstract, the summary on pp. 4–5, Theorem 7.1 on p. 22 and the end of its proof on pp. 35–36, from page images; Sections 8 to 11, where the barrier is proved, were not checked). Selected: drawn by its position in the list of cited rows, not because it looked wrong.
• Another row, not one shown above: Another reading of this same row is listed above. bound row $< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 6.039\times 10^{-5}$ line 32 citing [LSXCKKM26] — the claim we checked: that the cited reference proves the real Grothendieck constant is less than π / (2 ln(1+√2)) minus 6.039 × 10 raised to the power −5, with the inequality strict as the row writes it. What we read supports it, though not in one sentence: Theorem 1.1 says only "at most" at this threshold, while the paper's certified table (Table 2, degree 9) gives the constant at most 1.782153581, which is about 7.2 × 10 raised to the power −9 below the row's threshold, so the strict form follows by arithmetic we did. This verdict does not cover whether the paper's computer-assisted certification is correct. source link (read in the body, in the LaTeX source of arXiv v1, v2 and v3 (11 Aug 2026), with Theorem 1.1 and Table 2 re-read in the v3 HTML; the entry names no version). Selected: drawn by its position in the list of cited rows, not because it looked wrong.

3 bound row(s) here have been checked against material for their cited source (2 drawn by position, 1 chosen because something already looked wrong). Across this whole ledger 70 row(s) have been drawn by position, against 691 rows that name a source, counted on 2026-09-27: 64 were checked against material for their cited sources and for 6 the cited source could not be read at all. 5 more were chosen for suspicion and checked; they are counted apart, because a set selected for suspicion carries no rate. A row not listed here has NOT been checked.