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

Stop calculating, start listening. This work introduces The Jinx’s Theorem, a framework showing that arithmetic structures encode intrinsic and measurable information in the frequency domain. Rather than treating integers as symbols to be enumerated, this approach reveals them as spectral objects, whose internal structure emerges through harmonic interference patterns revealed by quantum Fourier analysis, validated on both NISQ quantum hardware and classical simulation. The theorem establishes that distinct arithmetic classes—such as perfect squares, prime numbers, and composite integers—exhibit stable and characteristic spectral fingerprints directly linked to their underlying structure. These predictions are confirmed using quantum Fourier circuits on IBM Quantum hardware (up to 20 qubits) and large-scale classical simulations analyzing 1,048,576 states and over 82,000 prime numbers in under 60 seconds on consumer hardware (Intel i3). The results demonstrate reproducible spectral signatures for: Perfect squares, exhibiting sparse spectra with a characteristic geometric decay predicted theoretically and confirmed empirically. Prime numbers, exhibiting stable alignment with the first 30 known non-trivial zeros of the Riemann zeta function. Composite integers, whose resonance patterns reflect internal arithmetic geometry and factor proximity. Beyond classification, the work introduces a resonance-based scanning and structural triangulation method enabling rapid diagnostics of composite integers without explicit factorization. While not constituting a general factorization algorithm, this spectral approach provides new tools for structural analysis of arithmetic objects and cryptographic parameter assessment. This release is provided as open science to ensure full transparency, reproducibility, and timestamped authorship. All theoretical developments, experimental results, figures, and implementation details are contained in the accompanying document. The Jinx’s Theorem proposes a change of perspective: arithmetic structure can be detected and explored through resonance, rather than brute-force computation. ------- Keywords: Arithmetical Harmonics • Spectral Signatures • Fourier Analysis • Quantum Fourier Transform (QFT) • Quantum Computing • Number Theory • Discrete Geometry • Structural Resonance • Pattern Detection • Signal vs Noise • Spectral Classification • Beyond Brute Force • IBM Quantum • Open Science.