Formal monodromy ($M_f=e^{2\pi i\Theta }$)

An unramified formal fundamental matrix $\hat{Y}=\hat{G}(\lambda ){\lambda }^{\Theta }{e}^{Q(\lambda )}$ acquires the right factor $e^{2\pi i\Theta }$ on a positive turn of the logarithm branch. This is formal monodromy. It can be nontrivial even when the true equation has an entire fundamental matrix and trivial actual monodromy, because Stokes matrices compensate for it.

When Stokes factors act on solution coefficient vectors, a full positive turn gives their ordered product equal to the inverse of formal monodromy for a polynomial system with no finite singularities. The true fundamental matrix is entire, so its actual monodromy is trivial, but its normalized formal branch changes by $M_f$. Indeed, write $Y_{j+1}=Y_j{S}_j^{-1}$; then $Y_{N+1}=Y_1(S_N\cdots S_1{)}^{-1}$, whereas normalized continuation after the positive turn gives $Y_{N+1}=Y_1{M}_f$. Cancelling the invertible fundamental matrix proves the identity. The uncorrected product is identity only when $M_f=I$. For a cubic-exponential two-by-two system with $U=1,V=-1,P=1,R=2,d=3/2$, the formal exponent is $\theta =1/2$, so $M_f=-I$ and the Stokes product is $-I$.