Prime Numbers and Quantum Physics: Signal Guarantees and Efficient State Preparation

This preprint presents a dyadic probability distribution for prime identification through Fourier coefficients of subsystem purity in an entanglement protocol. It proves a composite-signal lower bound over 2≤n≤2(d−1)2 ≤ n ≤ 2(d-1), gives an explicit state-preparation circuit, derives sampling-cost comparisons, and provides exact-arithmetic and numerical validation. The accompanying ZIP contains the LaTeX source, reproducibility scripts, QASM circuit, generated data, figures, and verification report. All included results are classical calculations or ideal circuit simulations.