1.7 The exact modulus-twelve arithmetic calculation
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We now fix \(q=12\). The coprime labels are
The signal is
The cosine formula comes from conjugate pairs. It exposes interference, while the integer divisor formula provides an independent arithmetic route to each value.
Theorem QA10: The exact phase and transform tables
Over one period,
and
Proof
For \(q=12\), QA04 becomes
Indeed, the only divisors \(d\) of twelve for which \(\mu(12/d)\ne0\) are \(2,4,6,12\), with coefficients \(2,-4,-6,12\). Evaluating the divisibility indicators at \(0,\ldots,11\) gives the sample table. QA05 gives value twelve at the four coprime labels and zero elsewhere. \(\square\)
At sample zero, all four phases agree and the value is four. At sample one, they cancel. At sample six, all four are minus one and the value is minus four. Those signs are the relative phase information that will later distinguish two states with the same initial probabilities.
1.7.1 A compact projector matrix
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Put \(P=P_{12}^{(12)}\). Its entries are
Equivalently, its first row is
and each subsequent row is the cyclic right shift of the preceding row. This specifies every entry of the twelve-by-twelve rational matrix. It agrees exactly with the literal matrix in the original formal module and with the indexed matrix in the later transport module.
With \(D=\operatorname{diag}(\mathbf1_{a\in U_{12}})\) and \(F=F_{12}\),
The six divisor shells of length twelve have orders \(1,2,3,4,6,12\) and ranks \(1,1,2,2,2,4\). Their ranks add to twelve, and their orthogonal sum is the identity by QA06.
1.7.2 The finite lanes of the same signal
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For \(L=3\) and \(M=4\), the coefficient-residue lanes are
Their four-point Fourier transforms are
For example, a full coefficient at \(k=1\) is reconstructed as
At \(k=9\), the same lane-frequency index \(1\) gives
The offsets distinguish two full Fourier labels with the same reduced frequency. These equations demonstrate the role of the phases in QA09. They do not identify a lane with a character-order shell.