Overview
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A finite arithmetic selector can be transported into a rational signal projector and a complex projective measurement by one explicitly normalized Fourier transformation. This chapter gives the full proof chain and follows a single exact twelve-dimensional example through every representation. The arithmetic input is the coprime residue set modulo twelve. Its Fourier synthesis is the integer-valued Ramanujan signal. The associated circulant matrix is an orthogonal projector of rank four. A normalized version of the signal is a quantum state, and an inverse Fourier transformation recovers the arithmetic support. An unsigned state has the same probabilities in the original basis yet zero probability on the selected support after transformation. The two states are orthogonal. A continuous relative-phase family makes that difference visible without changing the initial probability histogram.
The general finite Fourier, Möbius, divisor-projector, and polyphase identities are proved in the text. The formal companion covers the concrete modulus-twelve realization, including the sound interpretation of exact rational coordinates in the ordinary complex numbers, both unitarity identities, projector transport, state preparation and recovery, and the phase-sensitive probability conclusions. The formal companion contains ninety-six named theorems across five modules, including the twenty-two original bridge statements. The formal appendix identifies its exact coverage and verification evidence.
Ramanujan sums, Fourier orthogonality, rational cyclic projectors, and Ramanujan signal subspaces are classical. The contribution of this presentation is an exact, inspectable realization chain, its theorem-to-code concordance, and its explicit account of the information retained or discarded at each map. It makes no circuit-efficiency or quantum-advantage claim.