  In this article, we establish the logarithmic foundation for compactifying
the moduli stacks of the gauged linear sigma model using stable log maps of
Abramovich-Chen-Gross-Siebert. We then illustrate our method via the key
example of Witten's $r$-spin class to construct a proper moduli stack with a
reduced perfect obstruction theory whose virtual cycle recovers the $r$-spin
virtual cycle of Chang-Li-Li. Indeed, our construction of the reduced virtual
cycle is built upon the work of Chang-Li-Li by appropriately extending and
modifying the Kiem-Li cosection along certain logarithmic boundary. In the
subsequent article, we push the technique to a general situation.
  One motivation of our construction is to fit the gauged linear sigma model in
the broader setting of Gromov-Witten theory so that powerful tools such as
virtual localization can be applied. A project along this line is currently in
progress leading to applications including computing loci of holomorphic
differentials, and calculating higher genus Gromov-Witten
  invariants of quintic threefolds.
