  The regular languages with a neutral letter expressible in first-order logic
with one alternation are characterized. Specifically, it is shown that if an
arbitrary $\Sigma_2$ formula defines a regular language with a neutral letter,
then there is an equivalent $\Sigma_2$ formula that only uses the order
predicate. This shows that the so-called Central Conjecture of Straubing holds
for $\Sigma_2$ over languages with a neutral letter, the first progress on the
Conjecture in more than 20 years. To show the characterization, lower bounds
against polynomial-size depth-3 Boolean circuits with constant top fan-in are
developed. The heart of the combinatorial argument resides in studying how
positions within a language are determined from one another, a technique of
independent interest.
