= Comb tooth index ambiguity
One <beat> magnitude does not tell us the <integer> $n$. In the example, every proposed optical <frequency> $n(250\,\mathrm{MHz})+50\,\mathrm{MHz}$ produces the same signed <beat> with its corresponding tooth. A coarse independent optical measurement can select the <integer>: once the <beat> sign is known, an error bound below $f_r/2$ suffices if the candidate is selected by nearest distance.
There is also a useful two-spacing check. Keep the unknown <laser> and $f_0$ fixed, track the same tooth continuously, and increase $f_r$ by a stipulated $5\,\mathrm{Hz}$. Its <frequency> rises by $10\,\mathrm{MHz}$, so the signed <beat> falls from $30$ to $20\,\mathrm{MHz}$. Hence
$$
n=-\frac{\Delta b}{\Delta f_r}=\frac{10{,}000{,}000}{5}=2{,}000{,}000.
$$
This inference fails if the unknown drifts appreciably, the offset changes unnoticed, the <beat> sign is lost, or the tracker jumps to another tooth. The assumed tooth motion stays well below half the spacing in this example. If $f_0$ also changes by a measured amount, use $n=-(\Delta b+\Delta f_0)/\Delta f_r$.
For a fixed verified tooth number, the first-order <frequency> error is $\delta f_L=n\delta f_r+\delta f_0+\delta b$. Errors in these readings can be correlated because they share a reference. A small <beat> uncertainty therefore does not imply the same absolute optical accuracy: the reference's calibration and the factor $n$ remain in the error budget. A wrong tooth number is a separate discrete error, not a small correction. The beauty of the comb is that two controlled parameters organize millions of optical reference points, while the unknown's position is read electronically.
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