One arithmetic selector, three realizations
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Four residues modulo twelve will become an integer signal, an exact rational filter, and a normalized quantum state. Every bridge will be written down. The guide follows one object through three coordinate systems. The starting condition is coprimality. The signal realization adds its selected waves. The quantum realization normalizes the amplitude vector and distinguishes coherent transformation from measurement.
We will build a projector that keeps exactly those modes, apply it to a signal with a complementary component, and account for all retained and discarded energy. Then we will compare two orthogonal states with identical initial measurement probabilities.
| Arithmetic | Signal | Quantum model |
|---|---|---|
| Coprime labels | Fourier support | Frequency-basis amplitudes |
| Rational cyclic projector | Exact convolution filter | Projective measurement |
| Cyclotomic phase sum | Integer samples | Signed coherent amplitudes |
These are exact finite mathematical constructions. The quantum interpretation uses the usual state-vector and Born-rule model. No hardware experiment, circuit-cost claim, or quantum speedup is implied.