Why the phase sum is an integer
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A divisor identity certifies integrality without rounding trigonometric values. For \(q\geq1\), insert the Möbius coprimality identity into the finite phase sum:
\[
\mathbf1_{\gcd(a,q)=1}=\sum_{d\mid\gcd(a,q)}\mu(d).
\]
Write \(a=db\) after interchanging the finite sums. Character orthogonality evaluates the inner geometric series:
\[
\begin{aligned}
c_q(n)&=\sum_{d\mid q}\mu(d)\sum_{b=0}^{q/d-1}e^{2\pi ibn/(q/d)}\\
&=\sum_{\substack{d\mid q\\q/d\mid n}}\mu(d)\frac qd
=\sum_{e\mid\gcd(q,n)}e\mu(q/e)\in\mathbb Z.
\end{aligned}
\]
| Sample | Relevant divisor contribution | Exact result |
|---|---|---|
| \(n=0\) | \(2\mu(6)+4\mu(3)+6\mu(2)+12\mu(1)\) | \(4\) |
| \(n=2\) | \(\mu(12)+2\mu(6)\) | \(2\) |
| \(n=4\) | \(\mu(12)+2\mu(6)+4\mu(3)\) | \(-2\) |
| \(n=6\) | \(\mu(12)+2\mu(6)+3\mu(4)+6\mu(2)\) | \(-4\) |
The finite verifier also evaluates the twelve phase sums in the exact ring \(\mathbb Z[z]/(z^4-z^2+1)\). The plots use floating coordinates for rendering, while the sample identities use exact arithmetic.