From Undecidability to Cryptographic Efficiency: A Spectral Bridge from the Nitescence Theorem to Arithmetic Structure and Sieve Acceleration

We establish a formal continuum linking logical decidability, spectral arithmetic, and sieve-based cryptanalysis. Building on the Nitescence Theorem and the Jinx's Theorem, we conjecture — and provide strong empirical evidence — that B-smooth integers, the core consumable of the Quadratic Sieve and GNFS, exhibit a measurable Fourier decay signature O(k^{-1/π(B)}), strictly separated from random baselines.

We introduce Spectral Sieve Pre-filtering (SSP), a lightweight screening layer based on Riemann zeros that eliminates 49–87% of GNFS candidate evaluations while retaining ≥ 95% of smooth relations, yielding smooth-relation density gains of up to ×7.4 on instances up to 18-digit. Empirically validated on a consumer-grade processor.

Implications for RSA key-strength assessment and post-quantum transition planning are discussed. Source code available (MIT licence). 

-------

Keywords: number theory, cryptanalysis, integer factorization, smooth numbers, GNFS, quadratic sieve, Fourier analysis, RSA, post-quantum cryptography, sieve algorithms, Riemann Zeta Function