1.13 The central finite bridge theorem
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The arithmetic, signal, and state statements can now be gathered without leaving any undefined arrow.
Theorem QA18: Arithmetic, rational signal, and unitary realization
Let \(F\), \(D\), \(P\), \(c\), \(b\), \(x\), \(u\), \(\psi\), and \(\chi\) have the explicit modulus-twelve definitions above. Then:
\(F^\dagger F=FF^\dagger=I\).
\(P=FDF^\dagger\), \(P^2=P\), and \(P^\dagger=P\).
\(Px=c\) and \((I-P)x=b\), with \(\|x\|^2=48+102=150\).
\(Fu=\psi\) and \(F^\dagger\psi=u\).
\(u\), \(\psi\), and \(\chi\) are normalized, and \(\psi\) is orthogonal to \(\chi\).
\(\psi\) and \(\chi\) have identical original-basis probabilities.
Their inverse-Fourier coprime probabilities are respectively one and zero.
\(P\psi=\psi\), \(P\chi=0\), and \(P\) is not injective.
Proof
QA03 gives the unitarity identities. QA06 gives projector transport and the two projector identities. QA11 gives the filter and energy statements. QA12 gives preparation and recovery; the displayed normalizations and QA13 give statement five and statement six. QA14 gives statement seven. In Fourier coordinates \(\psi\) is supported on \(U_{12}\) while \(\chi\) is supported outside it, so \(P\psi=\psi\) and \(P\chi=0\). The normalized nonzero vector \(\chi\) is therefore in the kernel, proving noninjectivity. \(\square\)
The later formal publication theorem gathers the principal complex transport and probability identities into a single conjunction. Its supporting modules also supply the exact vector coefficients, filter recovery, residual reconstruction, and noninjectivity theorem. The general prose route above explains why the identities hold; the finite formal source certifies this literal example under its exact definitions.
One mathematical object has thus appeared in three realizations. Arithmetic selects coprime character labels. The signal realization is a rational circulant projector. The complex state realization uses a unitary basis change and a projective measurement. The complete vector information is preserved by the basis change, while selection and observation have separately stated losses.