New Lower and Upper Bounds for the Grothendieck Constant

Abstract: We establish new bounds on the Grothendieck constant $K_G$: [ \frac{6π}{11} \le K_G \le \fracπ{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.

• The paper develops a third/fifth-Hermite-chaos Gaussian rounding scheme that nearly cancels the cubic and quintic inverse-correlation coefficients, targeting the bound K_G ≤ π/2 and improving the classical Krivine estimate by approximately 3.81714 × 10⁻⁴.
• The paper derives an explicit weighted Gaussian correlation formula and uses Wiener-algebra estimates, holomorphic inverse stability, and a Rouché-theorem transfer argument to connect the limiting model with finite-dimensional rounding schemes.
• The claimed numerical bound remains conditional because the draft does not supply the required interval-arithmetic certificate, while effective reservoir dimensions and reproducible computational details remain open practical issues.

A Gaussianized Chaos Krivine Scheme for the Grothendieck Constant

Problem setting and claimed contribution

The paper studies explicit upper bounds for the real Grothendieck constant $K_G$, equivalently the worst-case integrality gap of the canonical semidefinite relaxation for bipartite bilinear optimization. The classical Krivine construction yields

$$K_G\leq \frac{\pi}{2\log(1+\sqrt{2})}\approx 1.782213,$$

while the conjecture that this value is optimal was disproved by Braverman, Makarychev, Makarychev, and Naor (Braverman et al., 2011). The present work proposes a different mechanism for obtaining an explicit improvement: finite-dimensional Gaussian rounding functions constructed from normalized Hermite-chaos reservoirs.

The central claimed bound is

$K_G\leq \frac{\pi}{2}=1.7818322637\ldots,$

which improves the classical Krivine value by approximately

$3.81714\times 10^{-4}.$

The paper’s contribution is not merely a numerical choice of rounding functions. It develops an analytic framework connecting finite-dimensional pure-chaos constructions to a limiting weighted Gaussian correlation model, derives an explicit coefficient formula for that model, and establishes a transfer theorem showing that a strict inverse-majorant certificate for the limiting correlation implies the existence of a sufficiently high-dimensional finite scheme. However, as explicitly acknowledged in the manuscript, the principal numerical theorem remains conditional on an interval-arithmetic certificate that is stated but not supplied in the draft.

Hermite-chaos rounding construction

For an odd Hermite degree $d$, the paper defines a normalized reservoir

$P_{d,N}=N^{-1/2}\sum_{j=1}^N H_d(U_{d,j}),$

where the $U_{d,j}$ are independent standard Gaussian variables and $H_d$ is the orthonormal probabilists’ Hermite polynomial. The proposed third/fifth-chaos scheme uses

$$\eta=0.136419125,\qquad s_3=0.34101124,\qquad s_5=0.05276111.$$

The two odd threshold functions are

$$f_N=\operatorname{sgn}\!\left( Z+\eta H_3(X)+s_3P_{3,N}+s_5P_{5,N} \right),$$

and

$$K_G\leq \frac{\pi}{2\log(1+\sqrt{2})}\approx 1.782213,$$0

The degree-three component is anti-aligned across the two sides, whereas the degree-five component is aligned. This sign pattern follows the general rule

$$K_G\leq \frac{\pi}{2\log(1+\sqrt{2})}\approx 1.782213,$$1

so that degrees $$K_G\leq \frac{\pi}{2\log(1+\sqrt{2})}\approx 1.782213,$$2 receive signs $$K_G\leq \frac{\pi}{2\log(1+\sqrt{2})}\approx 1.782213,$$3. The construction is designed to alter the low-order terms of the correlation function in a direction favorable for inverse-series admissibility.

The use of the reservoirs is structurally important. If two Gaussian inputs have correlation $$K_G\leq \frac{\pi}{2\log(1+\sqrt{2})}\approx 1.782213,$$4, then the corresponding degree-$$K_G\leq \frac{\pi}{2\log(1+\sqrt{2})}\approx 1.782213,$$5 Hermite components have correlation $$K_G\leq \frac{\pi}{2\log(1+\sqrt{2})}\approx 1.782213,$$6. Consequently, the third- and fifth-chaos reservoirs do not contribute ordinary correlation $$K_G\leq \frac{\pi}{2\log(1+\sqrt{2})}\approx 1.782213,$$7 in the limiting model; they contribute the weighted correlations $$K_G\leq \frac{\pi}{2\log(1+\sqrt{2})}\approx 1.782213,$$8 and $$K_G\leq \frac{\pi}{2\log(1+\sqrt{2})}\approx 1.782213,$$9. This gives the limiting construction a memory of the underlying chaos degrees while retaining a finite-dimensional realization for every reservoir length $K_G\leq \frac{\pi}{2}=1.7818322637\ldots,$0.

Limiting weighted Gaussian model

For fixed $K_G\leq \frac{\pi}{2}=1.7818322637\ldots,$1, the multivariate central limit theorem implies that the pairs of reservoir sums converge jointly to Gaussian pairs with correlations

$K_G\leq \frac{\pi}{2}=1.7818322637\ldots,$2

The limiting threshold pair is therefore a four-dimensional Gaussian model whose coordinates have different correlation scales. Its arcsine-normalized correlation is denoted by $K_G\leq \frac{\pi}{2}=1.7818322637\ldots,$3 and has an odd expansion

$K_G\leq \frac{\pi}{2}=1.7818322637\ldots,$4

This weighted correlation model is the principal conceptual object of the paper. It differs from a conventional finite-dimensional Gaussian threshold correlation because the limiting reservoir coordinates are assigned correlation powers determined by their chaos degrees. The finite schemes approximate this object coefficientwise and, more strongly, locally uniformly on the unit disk.

The paper emphasizes that this limiting model is not introduced as an informal Gaussian approximation. The finite reservoirs are genuine functions of finitely many independent Gaussian variables, and their limiting behavior is derived from the covariance identity

$K_G\leq \frac{\pi}{2}=1.7818322637\ldots,$5

The distinction matters because the eventual Grothendieck argument requires a finite-dimensional rounding scheme, not only an abstract limiting correlation function.

Explicit coefficient formula

A major technical component is the reduction of the limiting correlation coefficients to two-dimensional Gaussian integrals. Let

$K_G\leq \frac{\pi}{2}=1.7818322637\ldots,$6

Conditioning on the base variables $K_G\leq \frac{\pi}{2}=1.7818322637\ldots,$7 and $K_G\leq \frac{\pi}{2}=1.7818322637\ldots,$8, the reservoir contribution depends only on a one-dimensional Gaussian projection. The paper introduces the threshold coefficients

$K_G\leq \frac{\pi}{2}=1.7818322637\ldots,$9

with the closed form

$3.81714\times 10^{-4}.$0

The remaining continuous quantities are

$3.81714\times 10^{-4}.$1

For the third/fifth model, the coefficient $3.81714\times 10^{-4}.$2 is expressed as a finite weighted-degree sum over indices satisfying

$3.81714\times 10^{-4}.$3

The signs in this expression encode both the transformation $3.81714\times 10^{-4}.$4 and the anti-alignment of the third-chaos reservoir. This formula provides a concrete route to rigorous numerical certification: each coefficient is determined by finitely many explicitly specified two-dimensional Gaussian integrals and finite combinatorial factors.

The formula also clarifies why the construction can target selected low-order coefficients. The reservoir degrees determine which weighted degrees contribute to each $3.81714\times 10^{-4}.$5, while the relative signs determine whether the corresponding contributions enter positively or negatively.

Cancellation of inverse obstructions

The numerical parameters are chosen so that

$3.81714\times 10^{-4}.$6

while

$3.81714\times 10^{-4}.$7

Thus the cubic and quintic coefficients are numerically almost annihilated. This is the key mechanism behind the claimed improvement. If

$3.81714\times 10^{-4}.$8

and

$3.81714\times 10^{-4}.$9

then series reversion gives

$d$0

Consequently, suppressing $d$1 and $d$2 suppresses the first two nonlinear inverse coefficients. This is directly relevant because the Krivine preprocessing criterion controls the inverse Wiener norm

$d$3

The linear coefficient alone is not sufficient: the inverse-majorant condition is an infinite nonlinear constraint. The significance of the cancellation is that it creates substantial slack in the low-degree portion of this constraint, allowing the remaining higher-order terms to be controlled by a tail argument.

The reported inverse coefficients are consistent with this design. In particular, $d$4 and $d$5 are of order $d$6 and $d$7, respectively, while the first more substantial nonlinear inverse coefficients occur at higher degrees.

Krivine inverse-majorant criterion

The paper uses the standard analytic preprocessing interface for Krivine schemes. Given odd measurable functions $d$8 and their normalized Gaussian correlation $d$9, suppose the local inverse has expansion

$P_{d,N}=N^{-1/2}\sum_{j=1}^N H_d(U_{d,j}),$0

If the inverse is holomorphic on a neighborhood of the disk of radius $P_{d,N}=N^{-1/2}\sum_{j=1}^N H_d(U_{d,j}),$1 and satisfies

$P_{d,N}=N^{-1/2}\sum_{j=1}^N H_d(U_{d,j}),$2

then

$P_{d,N}=N^{-1/2}\sum_{j=1}^N H_d(U_{d,j}),$3

The proof embeds the original SDP vectors into a direct sum of tensor powers, with coefficients derived from the inverse series. The absolute-sum condition ensures that the resulting vectors have norm at most one. Gaussian projection through $P_{d,N}=N^{-1/2}\sum_{j=1}^N H_d(U_{d,j}),$4 and $P_{d,N}=N^{-1/2}\sum_{j=1}^N H_d(U_{d,j}),$5 then recovers the desired bilinear correlations.

For the hyperplane scheme, $P_{d,N}=N^{-1/2}\sum_{j=1}^N H_d(U_{d,j}),$6 and $P_{d,N}=N^{-1/2}\sum_{j=1}^N H_d(U_{d,j}),$7. The largest radius detected by the absolute-coefficient condition is $P_{d,N}=N^{-1/2}\sum_{j=1}^N H_d(U_{d,j}),$8, reproducing Krivine’s bound. The proposed construction instead targets the larger radius

$P_{d,N}=N^{-1/2}\sum_{j=1}^N H_d(U_{d,j}),$9

which yields

$U_{d,j}$0

The numerical proximity of $U_{d,j}$1 to the reciprocal linear coefficient is notable: the main burden is not establishing local invertibility, but proving that the complete inverse Wiener norm remains strictly below one.

Finite-dimensional transfer theorem

One of the strongest theoretical parts of the paper is the finite-dimensional transfer argument. It addresses a genuine difficulty: convergence of each fixed coefficient of $U_{d,j}$2 to the corresponding coefficient of $U_{d,j}$3 does not imply convergence of the absolute sum of the coefficients of $U_{d,j}$4.

The paper proves that the finite correlation functions $U_{d,j}$5 extend holomorphically to the unit disk, are uniformly bounded there, and converge locally uniformly to $U_{d,j}$6. The local uniform convergence follows from pointwise convergence on the real interval via the joint central limit theorem, followed by Vitali’s theorem.

The inverse functions are then handled using a contour argument. A holomorphic inverse branch $U_{d,j}$7 is defined on a disk of radius larger than the target radius. A contour surrounding the image of this disk is chosen so that $U_{d,j}$8 stays uniformly separated from the relevant target values. Local uniform convergence of $U_{d,j}$9 allows Rouche’s theorem to be applied on this contour, guaranteeing that $H_d$0 has a unique corresponding inverse branch $H_d$1.

Cauchy estimates provide a uniform geometric bound on the coefficients of $H_d$2. The proof separates the inverse series into a finite head and a uniformly controlled tail. Fixed-coefficient convergence controls the head, while the Cauchy estimate controls the tail. This establishes

$H_d$3

for all sufficiently large $H_d$4, provided the limiting inequality is strict.

This analytic transfer is essential. It avoids the invalid inference that finitely many matching coefficients automatically imply admissibility of the finite scheme. The theorem also establishes a modular division of labor: numerical work is performed on the limiting model, while complex analysis converts the limiting certificate into an actual finite-dimensional rounding scheme.

Wiener-algebra certification

The paper introduces a Wiener-algebra approach to control the inverse tail. Writing

$H_d$5

it defines

$H_d$6

The quantity $H_d$7 controls the Lipschitz constant of $H_d$8 on the unit ball of the Wiener algebra. A basic contraction argument proves inverse admissibility when

$H_d$9

Because the target radius is extremely close to $$\eta=0.136419125,\qquad s_3=0.34101124,\qquad s_5=0.05276111.$$0, this elementary criterion does not provide enough numerical margin. The paper therefore develops an a posteriori certificate based on a finite inverse head $$\eta=0.136419125,\qquad s_3=0.34101124,\qquad s_5=0.05276111.$$1 and a residual estimate. If $$\eta=0.136419125,\qquad s_3=0.34101124,\qquad s_5=0.05276111.$$2 denotes the Wiener norm of the computed inverse head, and $$\eta=0.136419125,\qquad s_3=0.34101124,\qquad s_5=0.05276111.$$3 is the residual after truncating the correlation series at degree $$\eta=0.136419125,\qquad s_3=0.34101124,\qquad s_5=0.05276111.$$4, then the tail bound is

$$\eta=0.136419125,\qquad s_3=0.34101124,\qquad s_5=0.05276111.$$5

where $$\eta=0.136419125,\qquad s_3=0.34101124,\qquad s_5=0.05276111.$$6 bounds the omitted correlation tail. The sufficient condition is

$$\eta=0.136419125,\qquad s_3=0.34101124,\qquad s_5=0.05276111.$$7

For $$\eta=0.136419125,\qquad s_3=0.34101124,\qquad s_5=0.05276111.$$8 and $$\eta=0.136419125,\qquad s_3=0.34101124,\qquad s_5=0.05276111.$$9, the manuscript states the numerical bounds

$$f_N=\operatorname{sgn}\!\left( Z+\eta H_3(X)+s_3P_{3,N}+s_5P_{5,N} \right),$$0

along with

$$f_N=\operatorname{sgn}\!\left( Z+\eta H_3(X)+s_3P_{3,N}+s_5P_{5,N} \right),$$1

These inequalities imply

$$f_N=\operatorname{sgn}\!\left( Z+\eta H_3(X)+s_3P_{3,N}+s_5P_{5,N} \right),$$2

and hence

$$f_N=\operatorname{sgn}\!\left( Z+\eta H_3(X)+s_3P_{3,N}+s_5P_{5,N} \right),$$3

The argument is conceptually well designed: it avoids attempting to estimate the inverse tail directly in a complex domain near the boundary and instead controls the nonlinear correction through a contraction in the Wiener algebra.

Conditional status of the numerical theorem

The paper is explicit that its main numerical conclusion is conditional. The “certificate module” states the inequalities required for the proof, but the draft does not provide a formal outward-rounded interval-arithmetic verification of those inequalities. The manuscript proposes using Arb or comparable certified arithmetic, together with rigorous Gaussian quadrature, Taylor models, Gaussian-tail estimates, or Hermite/Sobolev bounds.

This distinction is mathematically significant. The analytic statements—coefficient reduction, local uniform convergence, inverse-branch stability, and the Wiener-algebra implication—are presented as complete. The missing component is the rigorous validation of the numerical bounds on the coefficient head, coefficient tail, weighted nonlinear mass, inverse head, and residual. Therefore the result, in its supplied form, should be described as a conditional proof architecture rather than an unconditional established upper bound.

The numerical claim is nevertheless sharply specified. The required computations are finite except for the explicitly bounded coefficient tails, and the target inequalities have margins on the order of $$f_N=\operatorname{sgn}\!\left( Z+\eta H_3(X)+s_3P_{3,N}+s_5P_{5,N} \right),$$4 to $$f_N=\operatorname{sgn}\!\left( Z+\eta H_3(X)+s_3P_{3,N}+s_5P_{5,N} \right),$$5. Reproducibility would require publishing the exact interval enclosures, quadrature rules, precision, subdivision strategy, and code.

Relation to earlier and potential subsequent work

The construction follows the general direction initiated by the refutation of Krivine’s conjecture (Braverman et al., 2011), but it differs from the earlier perturbative mixing method. The earlier approach mixes rounding strategies and proves that a first-order perturbation improves the Krivine value. The present scheme instead engineers a single high-dimensional threshold pair whose Hermite-chaos structure directly cancels low-order inverse obstructions.

The paper also mentions a degree-$$f_N=\operatorname{sgn}\!\left( Z+\eta H_3(X)+s_3P_{3,N}+s_5P_{5,N} \right),$$6 extension with parameters

$$f_N=\operatorname{sgn}\!\left( Z+\eta H_3(X)+s_3P_{3,N}+s_5P_{5,N} \right),$$7

and reports an estimated radius

$$f_N=\operatorname{sgn}\!\left( Z+\eta H_3(X)+s_3P_{3,N}+s_5P_{5,N} \right),$$8

corresponding to an estimated bound

$$f_N=\operatorname{sgn}\!\left( Z+\eta H_3(X)+s_3P_{3,N}+s_5P_{5,N} \right),$$9

This extension is not part of the main theorem and is presented only as preliminary numerical evidence. It suggests that successive odd-chaos reservoirs may offer additional degrees of freedom for suppressing higher inverse coefficients. The possibility is consistent with the earlier observation that degree-three and degree-five reservoirs target the first two nonlinear inverse obstructions.

The manuscript also identifies effective dimension as an unresolved issue. The transfer theorem proves existence of a sufficiently large reservoir length but does not quantify it. A collision expansion for the normalized reservoir sums is proposed as a route toward explicit rates. Such bounds would be necessary if the construction were to be used as a practical approximation algorithm rather than solely as an existence proof for $$K_G\leq \frac{\pi}{2\log(1+\sqrt{2})}\approx 1.782213,$$00.

Implications for approximation algorithms and theory

If the certificate is completed, the work would provide an explicit finite-dimensional rounding scheme improving the classical Krivine guarantee by a directly specified amount. Since Grothendieck’s inequality controls the integrality gap of a broad class of bipartite SDPs, the result would have consequences for approximation algorithms involving matrix cut norms and related optimization problems. The practical benefit of the numerical improvement is modest, and the current lack of an effective dimension bound limits immediate implementation. Nevertheless, the construction is algorithmically concrete in principle: it consists of evaluating polynomial threshold functions of Gaussian projections and choosing random signs.

Theoretically, the paper contributes a flexible interface between Gaussian analysis, Hermite expansions, analytic combinatorics of inverse functions, and Banach-space rounding schemes. The weighted-degree formula provides a systematic way to search over higher-chaos reservoirs. The inverse-majorant transfer theorem is potentially reusable in other settings where a limiting analytic correlation model is easier to certify than finite-dimensional realizations.

The main conceptual implication is that optimizing a Krivine scheme can be formulated as an inverse-series design problem. Rather than optimizing only the leading correlation coefficient, one can manipulate the Hermite structure of the rounding functions to cancel selected nonlinear terms of the inverse. This viewpoint may support automated parameter search using symbolic Hermite calculations, certified optimization, and semidefinite or interval constraints.

Conclusion

The paper develops a Gaussianized third/fifth-Hermite-chaos rounding scheme for improving the classical Krivine upper bound on the real Grothendieck constant. Its central mechanism is the cancellation of the cubic and quintic coefficients of the limiting normalized correlation, which suppresses the first nonlinear coefficients of the inverse series. The paper combines an explicit weighted-chaos coefficient formula, a Wiener-algebra inverse certificate, and a Rouche-theoretic finite-dimensional transfer theorem.

The claimed bound

$$K_G\leq \frac{\pi}{2\log(1+\sqrt{2})}\approx 1.782213,$$01

would improve the classical Krivine value by approximately $$K_G\leq \frac{\pi}{2\log(1+\sqrt{2})}\approx 1.782213,$$02. In the supplied draft, however, this conclusion remains conditional on a missing interval-arithmetic certificate. Completing and independently reproducing that certificate, together with deriving effective reservoir dimensions, are the principal steps required to turn the analytic construction into a fully unconditional and practically implementable result.