State exactly what each passage preserves
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The value of a bridge is the contract it supplies.
| Passage | Preserves | Additional requirement or loss |
|---|---|---|
| Selector \(D\) to \(P=FDF^\dagger\) | Rank, idempotence, orthogonality | A specified unitary Fourier basis |
| Vector \(x\) to \(F^\dagger x\) | Full vector, inner product, norm | Retain all complex coefficients |
| Vector \(x\) to \(Px\) | Selected component | Complementary component removed |
| Vector to all polyphase lanes | All sample values | Every lane and its address retained |
| Amplitudes to probabilities | Outcome probabilities in one basis | Relative phase discarded |
| Projective measurement | Born probabilities and conditional state | One outcome per run, conditional branch |
The two states \(\psi\) and \(\chi\) give a concrete obstruction: the map from coherent amplitudes to a single-basis probability histogram is not injective. Their common histogram cannot determine which arithmetic selection will appear after the Fourier transform. The guide uses an actual twelve-dimensional Hilbert space. A four-qubit realization would require a specified embedding and implementation. Neither a gate synthesis nor an efficiency bound is supplied here.