This manuscript provides a generalized, operator-theoretic extension of Mills' theorem for deterministic prime extraction. Starting from a short-interval prime-gap hypothesis, the paper derives the exact admissibility threshold for the exponent c, constructs the corresponding recursive prime chain, and proves the existence of a unique scalar anchor Ac that recursively generates the primes via the floor function of Ac raised to the power of cⁿ. Moving beyond classical number theory, the arithmetic sequence is lifted into an operator-theoretic domain by defining a diagonal spectral operator whose eigenvalues are the recursively selected primes. The manuscript proves that this operator generates a zero-dimensional spectral geometry. It establishes that the inverse operator is super-summable (lying in every Schatten class), derives the ultra-sparse counting law of order log (log (Lambda) ), and evaluates the exact logarithmic divergence of the heat-trace asymptotics. Furthermore, the paper analyzes the Dirichlet trace (the spectral zeta function) of this prime operator. Because the spectral sequence exhibits doubly-exponential lacunarity, the Ostrowski-Hadamard gap theorem is applied to rigorously prove that the imaginary axis serves as a strict natural boundary, mathematically forbidding any analytic continuation to the left half-plane. Finally, the framework sharply separates the arithmetic admissibility layer from the spectral renormalization layer, culminating in a Classification Theorem that generalizes these properties to any admissible discrete spectrum (Mills Systems) beyond the prime numbers.
Andrew Kim (Sat,) studied this question.