From Bernoulli Numbers to Selector Kernels: Fredholm Determinants, {\zeta}-Regularization, and the Bridge Between Discrete and Continuous Spectra

We construct a unified analytic framework connecting Bernoulli numbers, zeta-regularization, and Fredholm determinants associated with trigonometric selector kernels. Starting from the Bernoulli-Stirling algebra, Euler-Maclaurin corrections are reinterpreted as spectral traces of compact operators. This bridge transforms discrete combinatorial data into continuous spectral quantities, showing that their determinants interpolate between finite-rank projectors and the sine-kernel of random-matrix theory. In the continuum limit the Fredholm determinant becomes a Painleve-V~tau-function, revealing a hierarchy in which Bernoulli coefficients and zeta-constants jointly describe the local-global asymptotics of analytic regularization.