The Foss Number: F = 1 + 1/(3π) as the Asymptotic Second Moment of Eigenvalue Spacings in Random Doubly Stochastic Matrices

We report the identification of a new mathematical constant F = 1 + 1/(3π) ≈ 1.10610 arising as the asymptotic second moment of normalized nearest-neighbor eigenvalue spacings for uniformly random doubly stochastic matrices sampled from the Birkhoff polytope. The constant was identified through exhaustive comparison against over 1,000 known mathematical constants (best match at 0.9σ), verified across 5 independent random seeds (0.1σ), and confirmed by large-n extrapolation up to n=180 (0.0σ with BIC model selection favoring the two-term expansion). The physical origin of the factor 1/(3π) is traced to the disk geometry of the eigenvalue cloud: the ratio of doubly stochastic spacing statistics to 2D Poisson nearest-neighbor statistics converges to (1+1/(3π))/(4/π) ≈ 0.869, identifying F as the eigenvalue repulsion correction to the Poisson baseline. All results are fully reproducible from the accompanying Python scripts.