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arxiv: 2606.03991 · v2 · pith:CU4PLLDRnew · submitted 2026-06-02 · 💻 cs.DS

The Grothendieck Constant is Less Than frac{π}{2 log (1+ sqrt{2})} - 10⁻⁵

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

Pith reviewed 2026-06-28 07:49 UTC · model grok-4.3

classification 💻 cs.DS
keywords Grothendieck constantGrothendieck inequalityupper boundsemidefinite relaxationapproximation algorithms
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The pith

The Grothendieck constant satisfies K_G < π/(2 log(1 + √2)) - 10^{-5}.

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

The paper establishes a concrete numerical improvement to the upper bound on the Grothendieck constant. Earlier work had shown only that the constant lies strictly below π/(2 log(1 + √2)) by some positive but unspecified amount. The new argument refines the prior technique to guarantee an explicit gap of size at least 10^{-5}. A reader would care because the constant controls the worst-case ratio between certain quadratic forms and their semidefinite relaxations, so a sharper explicit bound improves quantitative guarantees in related approximation problems.

Core claim

We prove that the Grothendieck constant K_G is strictly less than π/(2 log(1 + √2)) by at least 10^{-5}, obtained by refining the 2011 analysis of Braverman, Makarychev, Makarychev, and Naor so that every step remains valid while an explicit positive gap appears.

What carries the argument

Refinement of the 2011 Braverman et al. proof technique that extracts an explicit gap of size 10^{-5}.

If this is right

  • The constant is separated from the known closed-form upper bound by a definite numerical margin.
  • Approximation algorithms whose analysis depends on the Grothendieck constant receive an explicit quantitative improvement.
  • The same refinement technique applies to other related constants that were previously known only up to an unspecified gap.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Further numerical or analytic tightening of the same proof skeleton could enlarge the explicit gap.
  • The result supplies a concrete target for independent lower-bound constructions that attempt to approach the constant from below.
  • It suggests that similar gap-extraction methods might be portable to other inequalities whose proofs currently stop at existential statements.

Load-bearing premise

The 2011 proof technique admits a refinement that produces an explicit positive gap of size at least 10^{-5} while preserving correctness of all steps.

What would settle it

An explicit construction or numerical computation that produces a Grothendieck constant instance whose value exceeds π/(2 log(1 + √2)) - 10^{-5}.

Figures

Figures reproduced from arXiv: 2606.03991 by Adam Klivans, Alan Li, Anton Xue, Pravesh K Kothari, Raghu Meka, Rahul Saha, Swarat Chaudhuri.

Figure 1
Figure 1. Figure 1: The degree 9 partitions. The resulting scheme is mixed with the hyperplane scheme to [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Iteratively applying the σ operator to a random initial partition f0. Converges to the two-cycle tiger partitions (f∞, g∞), where σ(f∞) = g∞ and σ(g∞) = f∞. Therefore, it is natural to consider whether maximizers of König’s bilinear form also translate to good improvements on upper bounds on KG. Below, we show numerical evidence against this. Following [BMMN11], define the best-response operator σ(f)(y) :=… view at source ↗
Figure 3
Figure 3. Figure 3: Normalized König’s bilinear form BK for affine radial rotating-hyperplane partitions as a function of the ambient dimension dimension d + 2. The dashed horizontal line is the hyperplane benchmark value of 2 π log(1 + √ 2) = 0.5610998523 . . .. Corollary 6.11 (Cosine-series form). For every fixed κ, β ∈ R, lim d→∞ Kd [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗
read the original abstract

We prove that the Grothendieck constant $K_G < \frac{\pi}{2 \log (1+ \sqrt{2})} - 10^{-5}$. This improves on the work of Braverman, Makarychev, Makarychev, and Naor (2011), who proved that $K_G < \frac{\pi}{2 \log (1+ \sqrt{2})} - \epsilon$ for an unspecified $\epsilon>0$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript proves that the Grothendieck constant satisfies K_G < π/(2 log(1 + √2)) - 10^{-5}. This refines the 2011 result of Braverman, Makarychev, Makarychev and Naor, which established the same strict inequality for some unspecified ε > 0, by extracting an explicit positive gap of size at least 10^{-5}.

Significance. An explicit numerical improvement on the upper bound for K_G supplies concrete error margins that can be propagated into approximation algorithms and SDP analyses relying on Grothendieck's inequality. The work demonstrates that the compactness argument of the 2011 paper can be made effective, which is a useful methodological step even if the numerical gap itself is modest.

major comments (1)
  1. [Main theorem and its proof] The central claim requires that every inequality in the refinement of the Braverman et al. argument (including approximations to f(t), integration bounds, and choice of test measures) be replaced by explicit, computable estimates whose error terms are rigorously controlled. The manuscript must demonstrate that these controls suffice to guarantee a gap of at least 10^{-5} rather than relying on floating-point optimization whose rounding could reduce or eliminate the claimed gap.
minor comments (1)
  1. [Abstract] The abstract states the result but does not indicate the length or structure of the proof; a one-sentence outline of the refinement technique would help readers assess the scope of the contribution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the significance of making the 2011 compactness argument effective with an explicit gap. We address the single major comment below.

read point-by-point responses

  1. Referee: [Main theorem and its proof] The central claim requires that every inequality in the refinement of the Braverman et al. argument (including approximations to f(t), integration bounds, and choice of test measures) be replaced by explicit, computable estimates whose error terms are rigorously controlled. The manuscript must demonstrate that these controls suffice to guarantee a gap of at least 10^{-5} rather than relying on floating-point optimization whose rounding could reduce or eliminate the claimed gap.

    Authors: We agree that the current manuscript's use of floating-point optimization for the test measures and numerical integration leaves the claimed gap of 10^{-5} vulnerable to rounding errors, and that explicit, computable bounds with controlled error terms are required for a fully rigorous proof. We will revise the manuscript to replace all such approximations by interval-arithmetic or other rigorous enclosures (e.g., for the function f(t), the integrals over the test measures, and the optimization step), and we will verify that the resulting lower bound on the gap remains at least 10^{-5}. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit numerical refinement of external 2011 result

full rationale

The paper refines the Braverman et al. (2011) existence proof for some ε>0 into an explicit lower bound of 10^{-5} on the gap. The cited prior work is by unrelated authors and the current derivation does not reduce any claimed prediction to a fitted parameter, self-definition, or load-bearing self-citation chain. No ansatz is smuggled via citation and no known result is merely renamed. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the full set of assumptions, lemmas, and background results invoked in the proof cannot be audited. The central claim rests on the correctness of an unspecified refinement of the 2011 argument.

axioms (1)
  • standard math Standard properties of the Grothendieck constant and the 2011 upper-bound construction as defined in prior literature.
    The claim builds directly on the definition and earlier bound from functional analysis and theoretical computer science.

pith-pipeline@v0.9.1-grok · 5636 in / 1231 out tokens · 25873 ms · 2026-06-28T07:49:35.685769+00:00 · methodology

discussion (0)

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Reference graph

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