The Jinx's Theorem: Spectral Fingerprinting of Arithmetic Structures Through Quantum Resonance

This work introduces the Jinx's Theorem, establishing that polynomial arithmetic sets of degree d exhibit characteristic Fourier spectral decay at rate k^(-1/d), providing a rigorous spectral separation from random sets of comparable density. The theorem is proved for d = 2 (perfect squares) via quadratic Weyl sum estimates, and a universal scaling conjecture is formulated and validated empirically for d = 2, 3, 4. Experimental validation is conducted at 2^20 resolution (1,048,576 states, over 82,000 prime numbers analysed) on consumer hardware (Intel i3-1005G1, 8 GB RAM). The results confirm: Perfect squares exhibit sparse spectra with geometric decay matching the predicted exponent. Prime numbers display stable spectral alignment with the first 30 non-trivial zeros of the Riemann zeta function. Composite integers produce structured resonance patterns reflecting factor proximity and internal arithmetic geometry. A resonance-based structural triangulation method is introduced, enabling sub-second discrimination between symmetric and asymmetric composite integers on consumer hardware without explicit factorisation. While not constituting a general factorisation algorithm, this spectral approach provides new tools for computational number theory and cryptographic parameter assessment. All source code, experimental datasets, and replication instructions are publicly available under MIT licence at: https://github.com/carylun/jinx-theorem