What the construction establishes
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Coprimality, signal filtering, and coherent measurement share a specified exact operator. For the twelve-sample model, the guide has constructed every ingredient: the primitive-root selector, integer phase-sum signal, rational projector, complementary residual, complete order packets, normalized states, and alternative measurement probabilities. The retained arithmetic is not inferred from a suggestive resemblance. It follows from explicit finite Fourier identities.
The phase-erasure example is decisive. Two orthogonal normalized states have identical sample-basis probabilities. One inverse Fourier transform places all mass on the coprime labels for the signed state and none there for the unsigned state. Relative phase is therefore an operational part of the realization, not optional decoration.
The construction also states its losses. Projection removes a complementary subspace. Measurement returns one outcome and generally changes the state. Erasing phase destroys information. Keeping all complex Fourier coefficients or all polyphase lanes permits exact inversion.
Ramanujan sums, Fourier orthogonality, spectral projection, and the quantum state-vector rules are classical. The paper's purpose is their exact integration within Realization Calculus. Efficient preparation, gate synthesis, incomplete-data recovery, hardware robustness, and broad quantum advantage require separate results. Read together. The traditional paper supplies the full derivations and scope. The Lean guide supplies formal declaration contracts and certificate provenance. This guide supplies the exact worked example and its reproducible geometry.