We give a computer-assisted proof that the real Grothendieck constant is strictly smaller than $1.77987164$. The proof provides a complete interval-arithmetic realization of Ren's five-variable construction. A finite Fourier calculation encloses every scalar coefficient through degree $403$; a fourth-derivative estimate on the entire unit circle bounds the remaining coefficient sum by $10^{-4}$. Exact rational arithmetic then certifies the final inequality. We include the transfer to all finite real matrices and arbitrary vector dimensions, the analytic quadrature errors, the rational parameters, and source code for checking and regenerating the certificate.
We give a computer-assisted proof that the universal real Grothendieck constant satisfies $K_G^{\mathbb R}\le 1.7813$. The construction combines an explicit odd Hermite threshold of degree $11$ with a signed correlation polynomial of degree $51$. A sufficient inverse-majorant inequality is certified by enclosing the sc...
We determine the hundredths digit of the real Grothendieck constant by proving $1.773 \leq K_G \leq 1.7799$. Furthermore, numerical heuristics suggest the estimate $K_G \approx 1.779$. We start by simplifying the Gaussian duality theory of $K_G$, based on Hermite projection games and Krivine rounding schemes. The bound...
Steven Heilman, Chris Jones, Giulio Malavolta· 1 citation
We establish the coefficient sign patterns in the Third Borwein conjecture and the modulus-three Cubic Borwein conjecture. The analytic arguments apply for $n\ge1750$ and $n\ge500$, respectively. They combine exact dissections and positive coefficient identities near the boundary with saddle point estimates that preser...
We give an elementary proof of the absence of Landau--Siegel zeros with the explicit constant $c=1/100$. For every primitive quadratic Dirichlet character $\chi$ of conductor $q\ge3$, each real zero $\beta\in(0,1)$ of $L(s,\chi)$ satisfies $(1-\beta)\log q>c$. The proof combines integer coefficient counting, polynomial...
We prove $K_G^{\mathbb{C}}>1.35584631827168$ for the classical complex Grothendieck constant, closing more than one quarter of the gap between Davie's lower bound and Haagerup's upper bound. The numerical part of the proof is rigorously verified by interval arithmetic. The lower bound and the proof were discovered by t...
We show that $1.4<K_G^{\mathbb{C}}<1.404898554746$, where $K_G^{\mathbb{C}}$ is the complex Grothendieck constant. The upper bound improves on Haagerup's bound of $1.40490913\ldots$ from 1987, and the lower bound improves on Davie's bound of $1.33807$ from 1984. The upper bound combines ideas from the real and noncommu...
Steven Heilman, Chris Jones, Giulio Malavolta· arXiv.org· 0 citations
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