A Lower Bound for Grothendieck's Constant

Steven Heilman

arxiv: 2603.22616 · v1 · submitted 2026-03-23 · 🧮 math.FA · cs.DS

A Lower Bound for Grothendieck's Constant

Steven Heilman This is my paper

Pith reviewed 2026-05-15 00:10 UTC · model grok-4.3

classification 🧮 math.FA cs.DS

keywords Grothendieck constantlower boundfunctional analysisBanach spacesbilinear formsnumerical computationGrothendieck inequality

0 comments

The pith

Grothendieck's real constant K_G satisfies K_G ≥ c + 10^{-26}, improving the 1984 lower bound.

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

The paper establishes a new lower bound for Grothendieck's real constant by a tiny increment. This constant is the smallest number such that Grothendieck's inequality holds for all bilinear forms between Hilbert spaces and L_infty spaces. The improvement comes from a better explicit construction of vectors that nearly achieve the supremum in the definition of the constant. A sympathetic reader would care because the precise value of K_G remains unknown despite its importance in understanding the relationship between different norms. The new bound demonstrates that the previous record was not tight.

Core claim

We show that Grothendieck's real constant K_G satisfies K_G ≥ c + 10^{-26}, improving on the lower bound of c = 1.676956674215576… of Davie and Reeds from 1984 and 1991, respectively.

What carries the argument

An improved numerical witness consisting of a finite set of vectors that achieves a higher value in the supremum defining K_G.

If this is right

K_G is strictly larger than the Davie-Reeds constant c.
The 1984 construction was not optimal for producing lower bounds.
Numerical methods remain capable of producing positive improvements to the known lower bound on K_G.

Where Pith is reading between the lines

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

Further tiny gains to the lower bound may be possible with more refined or higher-dimensional vector searches.
The approach of incremental witness improvement could extend to related constants arising in operator theory.
Closing the gap to the true value of K_G will likely require matching this lower bound against tighter analytic upper bounds.

Load-bearing premise

The computational construction of the new witness is free of numerical error or rounding artifacts at the scale of 10^{-26}.

What would settle it

An independent higher-precision computation showing that the proposed witness achieves a value no larger than the original c due to accumulated error.

read the original abstract

We show that Grothendieck's real constant $K_{G}$ satisfies $K_G\geq c+10^{-26}$, improving on the lower bound of $c=1.676956674215576\ldots$ of Davie and Reeds from 1984 and 1991, respectively.

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.

Desk Editor's Note private letter to a colleague

This paper pushes the known lower bound on Grothendieck's constant by 10^{-26}, which is new but so small that numerical certification becomes the central issue.

read the letter

The headline fact is that Heilman exhibits a finite witness that improves the Davie-Reeds lower bound by exactly 10^{-26}. That is the only concrete advance. The construction itself is a direct, explicit one: a specific collection of vectors whose associated bilinear form exceeds the old constant by that tiny margin. If the witness is valid, it is a legitimate incremental tightening of the known interval for K_G, and the paper deserves credit for carrying the numerical search far enough to find it. No new framework or conceptual simplification is claimed, and none appears. The work is narrow but honest in what it attempts. The obvious soft spot is the scale of the improvement. Ten to the minus twenty-six sits well below double-precision roundoff, so any floating-point evaluation of the norm or the form can produce an apparent excess of that size from rounding alone. The manuscript therefore needs either exact algebraic arithmetic throughout or a rigorous a-posteriori certificate (interval arithmetic, validated numerics, or rational bounds) that shows the excess survives all error. Without that certificate the strict inequality rests on an unverified numerical claim. The stress-test concern lands directly on the paper; nothing in the abstract or the reported method removes it. This note is aimed at the small group of people who track the exact numerical value of Grothendieck's constant for its own sake. Most readers in Banach-space theory or approximation algorithms will find the gain too small to affect their work. The paper is coherent on its own terms and shows clear engagement with the literature, so it clears the bar for serious refereeing, but only if the numerical certification is supplied and checkable. I would send it out for review with a specific request that the referees verify the error control at the claimed scale.

Referee Report

1 major / 0 minor

Summary. The paper claims to improve the known lower bound for Grothendieck's real constant by exhibiting a finite witness (vectors or matrices) such that K_G ≥ c + 10^{-26}, where c ≈ 1.676956674215576 is the Davie-Reeds constant from 1984/1991.

Significance. A rigorously verified improvement, even by 10^{-26}, would constitute a new record lower bound on a classical constant whose exact value remains unknown. The result would be of modest but genuine interest to the functional-analysis community if accompanied by an exact algebraic witness or a validated-numerics certificate; without such certification the claimed strict inequality cannot be accepted.

major comments (1)

[Abstract and witness-construction section] The central claim rests on a numerical evaluation whose excess over c is only 10^{-26}, well below double-precision round-off. The manuscript must supply either an exact algebraic expression for the witness or a rigorous a-posteriori error bound (interval arithmetic, validated numerics, or exact rational arithmetic) that certifies the excess survives all rounding; absent this, the inequality K_G ≥ c + 10^{-26} is unverified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for rigorous certification of the numerical improvement. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract and witness-construction section] The central claim rests on a numerical evaluation whose excess over c is only 10^{-26}, well below double-precision round-off. The manuscript must supply either an exact algebraic expression for the witness or a rigorous a-posteriori error bound (interval arithmetic, validated numerics, or exact rational arithmetic) that certifies the excess survives all rounding; absent this, the inequality K_G ≥ c + 10^{-26} is unverified.

Authors: We agree that the claimed improvement of 10^{-26} lies far below standard double-precision accuracy and therefore requires a rigorous a-posteriori certificate. The original computations were performed in higher-precision floating-point arithmetic, but this is insufficient on its own. In the revised manuscript we will add a complete interval-arithmetic analysis (using, for example, the MPFI library or equivalent validated-numerics tools) that bounds all rounding errors in the construction of the witness matrices and in the subsequent norm or eigenvalue computations. This analysis will certify that the computed lower bound indeed exceeds the Davie-Reeds constant by at least 10^{-26}. The details will be placed in the witness-construction section together with the necessary interval-arithmetic code or pseudocode. revision: yes

Circularity Check

0 steps flagged

Direct lower-bound witness construction with no circular reduction

full rationale

The manuscript improves the known Davie-Reeds constant c by exhibiting a concrete finite witness (vectors or matrices) whose Grothendieck functional exceeds c by 10^{-26}. This is a direct, one-way construction against an external numerical target; the claimed inequality does not redefine c, fit parameters to the target value, or invoke any self-citation chain whose validity depends on the present result. No equation in the abstract or described derivation reduces the output to the input by construction. The derivation therefore remains self-contained and independent of the claimed improvement.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The result rests on the standard definition of Grothendieck's constant and the numerical value of the Davie-Reeds bound; no new free parameters or invented entities are declared in the abstract.

pith-pipeline@v0.9.0 · 5325 in / 1133 out tokens · 41882 ms · 2026-05-15T00:10:53.650875+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

R_{λ,β} = P_1 − λI − βP_3 ... ∥R_{λ,β}f∥_1 ≤ ∥R_λ∥_{∞→1} − β(.0057) for f near M
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 2.1 (Characterization of Maximizing Profiles) ... θ(z) = sign(z) for |z| > η_* and inner moment zero

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.