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Use the local coordinate $t=1/\lambda$. The differential system becomesFor the nonzero quadratic leading term, the coefficient of the local linear ordinary differential equation system has a pole of order four, so infinity is an irregular singular point of Poincaré rank three in this gauge. The matrix-valued function $A(\lambda )$ itself has a pole of order two; the extra factor $d\lambda /dt=-t^{-2}$ is essential when classifying the equation. If $A_0$ vanishes the degree and these orders decrease. The specified distinct eigenvalues $1,-1$ give the nondegenerate unramified case.

The leading exponential function factors have actions $q_{\pm }=\pm {\lambda }^3/3$, so their difference is $2{\lambda }^3/3$. Equal exponential magnitudes occur onThese are commonly called Stokes rays in the sectorial ODE convention. With the action-phase convention used by the wiki's Stokes line article they are anti-Stokes lines; the same-phase rays instead haveGiving both defining equations removes the nomenclature ambiguity; both sets contain six directions.

The standard construction with distinct leading eigenvalues uses six sectorial fundamental solutions per turn, with basic angular spacing $\pi /3$. They are fundamental matrices, each containing two independent column solutions. On the logarithm cover choose six consecutive overlapping sectors of width $\pi /3+2\delta$, with $\delta >\epsilon /2$ small. Their union has angular opening $2\pi +2\delta$, covering the requested $2\pi +\epsilon$. One often also writes a seventh index to compare the closing sector with the first after a full turn; that is a closing copy, not a seventh independent sector in the six-sector construction. The number is a generic sectorial fundamental solution normalization count, not a universal minimum for exceptional systems with vanishing Stokes matrices: the exactly diagonal system already has one global elementary fundamental matrix.

There is a missing qualification in the requested product identity. Polynomial coefficients give an entire invertible fundamental matrix, hence trivial actual monodromy about a large circle: the circle contracts through ordinary points. But the formal matrix generally has the form $\hat{G}{\lambda }^{\Theta }{e}^Q$, and a positive turn changes its logarithm branch by the formal monodromy $M_f=e^{2\pi i\Theta }$. This factor cannot be discarded.

To specify the printed product order, let $S_j$ act on column vectors of solution coefficients across the $j$th sector boundary, so adjacent normalized matrices satisfy $Y_{j+1}=Y_j{S}_j^{-1}$. Six transitions give $Y_7=Y_1(S_6\cdots S_1{)}^{-1}$. The normalized formal branch after the turn is ${\hat{Y}}_1{M}_f$, so sectorial uniqueness gives $Y_7=Y_1{M}_f$. ConsequentlyUsing Stokes matrices for right changes of fundamental matrices instead reverses/inverts the corresponding convention; the formal monodromy factor remains necessary. The formal monodromy correction to an entire-system Stokes product shows that the printed identity holds when $M_f=I$, including the corrected Lax specialization discussed below.

It does not follow from polynomiality alone. For example takeThe coefficient matching in question 2 gives $\Theta =\frac{1}{2}\mathrm{diag}(1,-1)$, hence $M_f=-I$ and the ordered Stokes matrix product is $-I$, not $I$, despite having no finite singularity. This is an explicit counterexample to the unqualified final clause.

Two sectorial fundamental solutions of the same linear system differ on their overlap by a constant invertible right factor, since $(Y_j^{-1}{Y}_{j+1}{)}^{\prime }=0$. With a common formal normalization these factors encode the exponentially small ambiguities invisible to the asymptotic expansion. They are called Stokes matrices. Equivalently their inverses act on column vectors of solution coefficients; the convention must be specified when ordering products.