New Lower and Upper Bounds for the Grothendieck Constant

Adam Klivans; Alan Li; Anton Xue; Pravesh K Kothari; Raghu Meka; Rahul Saha; Swarat Chaudhuri

New Lower and Upper Bounds for the Grothendieck Constant

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves 6π/11 ≤ K_G ≤ π/(2 log(1+√2)) − 10⁻⁴, fixing the Grothendieck constant's tenths digit at 7 with a cubic–quintic rounding scheme and a universal barrier on Krivine schemes.

desk verdict Big claimed jump on both sides of K_G, but the lower bound rests on tight, unverified interval-arithmetic certificates; still deserves a serious referee. read the letter →

arxiv 2608.11158 v2 pith:XJ6VWX5Y submitted 2026-08-11 cs.CC cs.DS

Rahul Saha , Alan Li , Anton Xue , Swarat Chaudhuri , Adam Klivans , Pravesh K Kothari , Raghu Meka This is my paper · ORCID

classification cs.CCcs.DS MSC 46B0790C2268Q17

keywords GrothendieckconstantKrivineroundingschemelimitingaffinecoefficientinequalityinversemajorantsemidefiniterelaxationintervalarithmeticcertification

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes new interval bounds for the Grothendieck constant, the best universal factor by which sign arrangements approximate inner products: $\frac{6\pi}{11} \le K_G \le \frac{\pi}{2\log(1+\sqrt{2})} - 10^{-4}$, numerically $[1.7135\ldots, 1.7819\ldots]$. That fixes the previously unknown tenths digit of this decades-old constant at 7. The upper bound comes from limiting Krivine schemes, asymptotic high-dimensional rounding schemes, and a concrete cubic–quintic example whose certified inverse-majorant series beats Krivine's classical bound by roughly $3.5 \times 10^{-4}$. The lower bound comes from the opposite direction: instead of constructing hard instances, the paper proves a universal restriction, the affine coefficient inequality $b_3 \ge 2b_1 - 11/6$, on the Taylor coefficients of every Krivine correlation function, converts that restriction into a cap of $11/12$ on all admissible majorant parameters, and applies the known optimality of Krivine schemes to turn the cap into the lower bound. The computational steps are backed by reproducible interval-arithmetic certificates, and the paper reports the bounds were found by a long-running human–AI collaboration.

What carries the argument

The upper bound is carried by limiting Krivine schemes: pairs of odd sign functions $f,g : \mathbb{R}^k \to \{\pm 1\}$ whose coordinate correlations are arbitrary allowable maps $\rho_i(t) = \sum_{d\ \mathrm{odd}} c_{i,d}\,t^d$ with $\sum_d |c_{i,d}| \le 1$, rather than the fixed correlation $t$ of classical Krivine schemes; a central-limit-theorem argument shows such a scheme is a locally uniform limit of ordinary finite-dimensional Krivine schemes, so an inverse-majorant certificate for the limit is a genuine upper bound on $K_G$. The lower bound is carried by the affine coefficient inequality $b_3 \ge 2b_1 - 11/6$, proved by decomposing $f$ and $g$ into agreement and disagreement parts $h = (f+g)/2$ and $k = (f-g)/2$ and reducing the needed estimates to one-dimensional inequalities; being affine in $(b_1, b_3)$, the inequality survives mixtures and coefficientwise limits and forces every admissible inverse-majorant parameter to satisfy $\gamma \le 11/12$.

What would settle it

Two checks settle the claims. For the upper bound: re-run the published interval-arithmetic certificates (the paper says the suite finishes in under two minutes) and look for any violated enclosure, such as $b_1 \ge 0.881573822049$ or $\|D^3H\|_{L^2(\mathbb{T})} < 14.443$, or independently invert the cubic–quintic correlation series to high precision at $\gamma = 0.881545409$ and verify $M(\gamma) < 1$. For the lower bound: scan pairs of odd sign functions $f,g$ on $\mathbb{R}^2$ and $\mathbb{R}^3$ for any instance with $b_3 < 2b_1 - 11/6$, or verify that the limiting construction cited in Section 11 is really approximable by ordinary schemes with uniformly convergent kernels.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that the Grothendieck constant satisfies $\frac{6\pi}{11} \le K_G \le \frac{\pi}{2\log(1+\sqrt{2})} - 10^{-4}$. The upper bound is certified by the cubic–quintic limiting scheme: odd sign functions $f(w,x) = \mathrm{sgn}(w + \vartheta\,\mathrm{He}_3(x))$ and $g(w,x) = \mathrm{sgn}(w - \vartheta\,\mathrm{He}_3(x))$, whose coordinate correlations are the allowed power series $\rho(t) = (t - s_3^2 t^3 + s_5^2 t^5)/(1 + s_3^2 + s_5^2)$ in the first coordinate and $t$ in the second; its inverse majorant satisfies $M(\gamma) < 1$ at $\gamma = 0.881545409$, and a finite-dimensional approximation theorem transfers this certificate to ordinary rounding schemes, yielding $K_G \le 1.7818666069360661$. The lower bound is the assertion that no Krivine-style correlation function, mixed or in any dimension, can admit an inverse-majorant parameter above $11/12$; this follows from the affine coefficient inequality $b_3 \ge 2b_1 - 11/6$, which is preserved under mixtures and coefficientwise limits. Combined with the optimality of Krivine schemes, this barrier forces $K_G \ge 6\pi/11$.

Load-bearing premise

The lower bound rests on an approximation step in Section 11 that converts an idealized limiting construction into ordinary rounding schemes; the paper cites its source but leaves the conversion at the level of a sketch, and if that conversion fails the lower bound does not follow.

Editorial extensions

If this is right

The tenths digit of the Grothendieck constant is 7; the constant is now known to lie in [1.7135…, 1.7819…].
The answer to a question posed in the 2011 paper that disproved Krivine's conjecture is yes: asymptotic families of higher-dimensional rounding schemes can improve on one- and two-dimensional schemes, opening a new search space.
A universal upper bound on Krivine-majorant parameters is a working route to lower bounds on K_G; any stronger affine constraint on higher Taylor coefficients would immediately tighten the 11/12 barrier.
The upper bound's numerical claims are tied to downloadable, reproducible interval-arithmetic certificates rather than to heuristic computation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The agreement–disagreement decomposition is not limited to the first two Taylor coefficients: analogous affine constraints on $b_5$, $b_7$, or mixed coefficients could tighten the $11/12$ barrier further, though the paper only hints at why this would be more involved.
The limiting-scheme framework separates the design of correlation maps from the choice of sign partitions; searching over additional Hermite directions (degrees 7, 9, …) or over allowable-map coefficients is a natural, testable route to a smaller upper bound.
The paper's numerical data suggest the fiber inequality holds with substantial headroom (an apparent true constant of $3 - \pi/2$ versus the proved value 1); proving that stronger inequality would give the lower-bound argument slack to absorb sharper optimality transfers.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the quick take on arXiv:2608.11158. If the certificates are right, this is the first real movement on both sides of K_G in decades: lower bound to 1.7135, upper bound to 1.7819, pinning the tenths digit to 7. The methods are genuinely new: limiting Krivine schemes for the upper bound, and an affine coefficient inequality for the lower bound. The analytic skeleton is coherent and the proof sketches are readable. They also ship deterministic interval-arithmetic certificates and scripts, which is more than most pure math papers do.

The main caveat, as you'd expect, is the computer-assisted part. The lower bound depends entirely on Theorem 9.3, the fiber inequality, and Appendix D reduces that to interval-arithmetic checks with some very tight margins: the splice margin is about 1.9e-6, and the worst medium-band margin is 2.3e-5. That is not the kind of slack you can ignore. The certificates are well-documented and the scripts exist, but I haven't run them, and a referee cannot verify the claims by reading the text alone. The proof of Lemma 11.2, the reworking of Naor-Regev, is only sketched, but the stress-test note convinced me that it is less load-bearing: the barrier argument could run off the original NR theorem.

The upper bound is also machine-assisted, but it has more analytic structure (the near-linearity proposition and the D^3H tail bound) and the numerical margins there are more comfortable. Still, it deserves careful checking.

Who's this for? Anyone working on Grothendieck-type inequalities or rounding schemes. The limiting-scheme framework alone is worth a read. I'd send it to referee, but I'd ask for the certificates to be independently run, and I'd want the authors to provide a human-verifiable proof of at least the core fiber inequality or a formalization of the interval arithmetic. The paper would be stronger if the lower bound were reorganized so that the numerical part is isolated and clearly stated as a lemma open to independent verification.

My recommendation: yes, send it to peer review. It's important, likely correct, and the shipping of code and certificates makes verification possible. But the referee needs to actually run the certificates, and the bar for acceptance should include a clean statement of what is proved by hand versus by machine.

Referee Report

2 major / 4 minor

Summary. The paper claims new bounds on the Grothendieck constant: 6π/11 ≤ K_G ≤ π/(2 log(1+√2)) − 10^{-4}. The upper bound is obtained by introducing limiting Krivine rounding schemes, showing that they are limits of finite-dimensional classical schemes, and constructing a two-dimensional cubic–quintic scheme whose inverse majorant is certified numerically by interval arithmetic. The lower bound is obtained by proving an affine coefficient inequality b3 ≥ 2b1 − 11/6 for correlation functions of odd sign functions, transferring this to a universal barrier γ ≤ 11/12 for admissible inverse-majorant parameters, and then invoking a reworking of the Naor–Regev optimality construction to conclude K_G ≥ 6π/11. The lower-bound proof also relies on a computer-certified one-dimensional fiber inequality (Theorem 9.3, Appendix D).

Significance. If correct, the contribution is significant: it gives the first improvement over the Davie–Reeds lower bound by a non-infinitesimal amount, it constructs the first asymptotic family of rounding schemes for the upper bound, and together the bounds determine the tenths digit of K_G. The analytic skeleton is coherent: the limiting-scheme approximation theorem and the reduction of the upper-bound certificate to a small set of interval inequalities are clean, and the affine coefficient inequality is elegant. The manuscript is unusually transparent about its computational component, shipping deterministic Arb-based certificates with a one-command reproducibility driver. The lower bound, however, depends on two delicate points that need additional support before the main theorem can be regarded as fully established.

major comments (2)

[§11, Lemma 11.2] Lemma 11.2 is the only bridge from the affine coefficient inequality to the lower bound, but its proof is a sketch. The step that partitions the sphere into Borel sets and extends sampled signs as step functions is asserted to give ordinary mixed rounding schemes with uniformly convergent kernels, yet the manuscript does not show how the spherical construction of [NR14] yields Gaussian-input correlation functions of the form required by Definition 11.1, nor does it prove the claimed analyticity and coefficientwise convergence of H_k. Because Proposition 10.2 is applied to the limiting functions H_k, this gap is load-bearing: without a complete proof of Lemma 11.2, Theorem 7.1 does not follow from the affine coefficient inequality.
[§9.4 and Appendix D, Theorem 9.3] The fiber inequality V(u) ≤ 3νβ(u) − β(u)^2 is the main quantitative ingredient of the lower bound, but its proof is delegated entirely to interval-arithmetic certificates in Appendix D. The certified margins are very small in places: the splice inequality (S) has margin about 1.9×10^{-6}, and the worst medium-band envelope margin is about 2.3×10^{-5}. A coverage hole or an outward-rounding bug in those regions would invalidate Proposition 10.2 and hence the lower bound. The appendix is a detailed protocol, but it is not an independent verification. I would ask for either a human-verifiable analytic proof of the tightest certified regions, or an independent formal recheck of the certificates, before the lower-bound claim is accepted as fully established.

minor comments (4)

[§3.3, Theorems 3.7 and 3.9] Theorems 3.7 and 3.9 state the same finite-dimensional approximation result; one of them should be removed or the second should be explicitly labeled as a strengthened restatement.
[§3.3, Definition 3.6] In Definition 3.6, the phrase 'where the discontinuity sets have Gaussian measure zero' is grammatically attached to 'odd measurable functions'; it would be clearer to list it as a separate requirement on the pair (f,g).
[Abstract and Theorem 7.1] The numerical value of 6π/11 is quoted as 1.7135... in the abstract and 1.7136... in Theorem 7.1; these are truncations of the same number, but the two presentations should be made consistent.
[Appendix D, Table 2] For reproducibility, it would be helpful to state the exact version of python-flint/FLINT and the hash of the pinned environment together with the one-command driver, so that independent reruns can verify that the same ball arithmetic backend is used.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: both bounds rest on independent computational certificates and the external Naor–Regev optimality theorem; self-citations are contextual and not load-bearing.

full rationale

The upper bound is obtained by constructing an explicit cubic–quintic limiting Krivine scheme and certifying the inverse-majorant inequality M(γ) < 1 at γ = 0.881545409. The proof chain is: Theorem 3.9 shows finite-dimensional approximation of limiting schemes; Proposition 6.1 reduces M(γ) < 1 to the analytic condition γ + ΔH < b1; Proposition 6.3 certifies b1, the finite coefficient sum, and the D^3H boundary norm by interval arithmetic in Appendix C. None of these steps defines K_G in terms of M(γ) or fits any parameter to the claimed value of K_G; the parameters s3, s5, ϑ and γ are chosen for the construction and then certified, not inferred from the target constant. The lower bound proceeds from the affine coefficient inequality b3 ≥ 2b1 − 11/6, proved via Gaussian Hermite expansions, the agreement–disagreement substitution, the rearrangement lemma, and the fiber inequality (Theorem 9.3). The fiber inequality is reduced in Appendix D to finite interval-arithmetic certificates (C1)–(C3) and the splice check (S). This is a computational proof with explicit, reproducible certificates; it is not circular unless the certificates themselves assumed the conclusion, which they do not. The key external input is Lemma 11.2, which reworks the construction in Naor–Regev [NR14] to show the inverse-majorant bound is asymptotically optimal. This is a citation to independent prior work by different authors, not a self-citation, and it is not used to assume the lower bound; rather, it supplies the sequence γ_k → π/(2K_G) to which the paper's universal barrier γ ≤ 11/12 is applied. The proof of Lemma 11.2 is sketched, and Appendix D's certificates are not independently re-verified here; these are correctness or completeness risks, not circularity. The paper's self-citations [LSX+26a] and [LSX+26b] are contextual (related work and AI methodology) and do not carry any load-bearing step of the mathematical derivation. Accordingly, no step reduces by definition or by self-citation to its own output, and the circularity score is minimal.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claims rest on four groups of free numerical parameters and on the three domain assumptions above. No new physical entities are introduced; 'limiting Krivine schemes' and 'mixed rounding schemes' are mathematical constructions, not new particles or forces.

free parameters (4)

cubic-quintic coupling eta and weights s3, s5 = eta=0.136419125, s3=0.34101124, s5=0.05276111
Chosen in Eq (7) to make the cubic and quintic coefficients of the correlation function nearly vanish; the certified upper bound is proven after this choice, so the bound is not parameter-free.
certification radius gamma = 0.881545409
The inverse-majorant radius at which M(gamma)<1 is certified in Theorem 6.4. It is a chosen test point, not fitted to K_G, but it is part of the numerical certificate.
fiber certificate constants c_M, r1, L = c_M=0.993405, r1=1.081596, L=1.55
Threshold constants that split the fiber inequality certificate into high-budget, medium, and tail regimes in Appendix D. They are hand-picked so the interval checks pass.
coefficient head cutoff N = N=251
Truncation degree separating computed coefficients from the analytically tail-bounded coefficients in the upper-bound proof.

assumptions (4)

domain assumption Odd measurable sign functions with Gaussian measure-zero discontinuity sets suffice for the Krivine rounding framework
Assumed throughout Sections 3-4 to define correlation functions and apply the finite-dimensional approximation theorem.
domain assumption The Naor-Regev construction [NR14] can be reworked to produce limiting mixed correlation functions H_k with admissible gamma_k converging to pi/(2K_G)
Lemma 11.2 states this as the bridge from the affine coefficient inequality to the lower bound. The proof is a sketch that refers to equations (4)-(6) of [NR14].
domain assumption Correctness of the interval arithmetic implementation (Arb/python-flint) used in all certified inequalities
Appendices C-D reduce the upper and lower bound proofs to finite checks in outward-rounded interval arithmetic. The paper ships scripts but does not formally verify the implementation.
standard math Standard analytic facts: Vitali's theorem, Rouché's theorem, Mehler's identity, bathtub principle, Hermite addition formula
Used throughout Sections 3, 9, and Appendices A-B without proof.

how reviews work

0 comments

Cite this review

Pith. "Pith review of New Lower and Upper Bounds for the Grothendieck Constant." pith.science (2026). https://pith.science/paper/XJ6VWX5Y

@misc{pith2026260811158,
  author       = {Pith},
  title        = {Pith review of: New Lower and Upper Bounds for the Grothendieck Constant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XJ6VWX5Y}},
  note         = {Machine review of arXiv:2608.11158}
}

abstract

We establish new bounds on the Grothendieck constant $K_G$: \[ \frac{6\pi}{11} \le K_G \le \frac{\pi}{2\log(1+\sqrt2)} - 10^{-4}. \] Methodologically, our lower bound approach differs from previous works by establishing limitations on the asymptotically optimal Krivine schemes, rather than giving explicit constructions of gap instances. Our upper bound is obtained by proposing and analyzing the first asymptotic construction of rounding schemes, whereas previous works only consider low-dimensional schemes. Together, these bounds determine the previously unknown tenths digit of $K_G$ to be $7$. The bounds were discovered by a long-running collaborative effort of humans and a long-horizon AI research system that we engineered.

Discussion (0). Continue with ORCID to comment.

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.