This paper introduces fractal-symmetric sieve recursion into the SPR (Symmetric Prime Ratio) framework, establishing a multiscale analytical representation of prime number distribution. We rigorously prove the limiting correspondence between this recursive construction and the Bochner measure of the SPR framework and utilize fractal symmetry to express the remainder term of the canonical substitution formula in a recursively approximable form. Through harmonic analysis, we develop the theory of maximal prime gaps, deriving piecewise precise upper bounds and the asymptotic formula where the constant coincides with Granville’s corrected lower bound but originates from structural derivation within the SPR framework. Furthermore, we extend this unified framework to the seven Millennium Prize Problems and related mathematical theories, revealing their shared deep mathematical structure: discrete-continuous duality and spectral measure positivity. This paper also integrates the latest international mathematical advances from 2025–2026 in numerical computation of Landau–Siegel zeros, research on the de Bruijn–Newman constant, new proofs of the Li criterion, and fractal string spectral operator theory, demonstrating the cutting-edge nature and consistency of the SPR framework through mutual verification
Mei Shenglin梅生林 (Tue,) studied this question.