A Spectral Framework for Prime-Generating Operators and an Exploratory π-Polynomial Relation for the Fine Structure Constant

Euler’s formula ζ(2) = π2/6 reveals a deep connection between primes (via the Euler product)and the transcendental number π. This paper elevates the fact that “primes generate π”to a rigorous operator-spectral language: we define a prime-generating operator Agen,shuiwhose discrete spectrum is precisely the set of logarithms of all primes {log p}, and prove theregularized determinant identity det(1 − e−sAgen,shui ) = ζ(s)−1. This identity transforms thedistribution of primes from an arithmetic object into a spectral-geometric one, providing amore direct prime-spectral entry point for Weil’s explicit formula and Connes’ spectral triple.Within this framework, we report an exploratory numerical finding: the polynomial 4π3 +π2 + π = 137.0363038 differs from the CODATA 2018 inverse fine structure constant 1/α =137.0359991 by a relative error of 2.22 ppm. In a systematic scan of 901,500 equal-complexityπ-polynomials, this expression ranks 2nd, and is the only combination among the top 4 withall-positive coefficients, continuous powers, and no constant term. However, the coefficients(4, 1, 1) cannot be naturally derived from the analytic properties of the prime zeta functionP (s) = Pp p−s; hence this relation is provisionally classified as an exploratory numerical clue.Keywords: prime zeta function; operator spectrum; fine structure constant; regularized deter-minant; Riemann zeta function