Arbitrary metric Morse--Smale end counts form finite Morse chain complexes

The critical set is finite as in [F2]. The actual metric co-orientation construction of [F1] transports the unstable critical ray along the stable disk, and the ordered exact sequence $0\to T(W^u(p)\cap W^s(q))\to T{W}^u(p)\to TM\mathrm{/}T{W}^s(q)\to 0$ defines its smooth kernel ray. Its flow-first quotient is the stated orbit orientation. The index-drop-one orbit space is finite by [F2], so its signed or mod-two count defines every matrix entry on the finite critical basis. Reversing either endpoint ray reverses that entry, which gives exactly diagonal conjugation of the differential.

For an index-drop-two pair, repeat the autonomous height-path argument of [F2] with fixed critical endpoints. Its image is compact in the uniform metric and a limit splits into actual trajectories. Positive index drops allow at most one internal break. The unbroken orbit space has dimension one; the one-neck charts of [F3] make every broken point a boundary point with a half-interval neighbourhood and give inverse coverage. Thus the compactification is a compact one-manifold whose boundary is the finite union of products of two rigid orbit sets. No normalized-flow theorem is used beyond its local orientation convention.

Compute its boundary sign explicitly. At a broken pair $({\gamma }_1,{\gamma }_2)$ the two ordered intersection sequences give $o{r}_p=ϵ({\gamma }_1)[X_1,o{r}_r]$ and $o{r}_r=ϵ({\gamma }_2)[X_2,o{r}_q]$. Cancelling the intermediate ray gives the ordered two-dimensional trajectory ray $ϵ({\gamma }_1)ϵ({\gamma }_2)[X_1,X_2]$. The finite matching normal blocks of [F3] preserve this ray, because their determinants are positive. Choose a representative crossing on the first piece and let $t$ be the common positive translation coordinate. Increasing the neck time $T$ holds that first crossing representative fixed and moves the second representative to ${\gamma }_2(s-T+t)$, up to the finite smooth exterior crossing-time adjustments. On the two flow directions the columns are therefore $(1,1)$ for ${\mathrm{\partial }}_t$ and $(0,-1)$ for ${\mathrm{\partial }}_T$, of determinant $-1$. Exterior corrections and flat endpoint substitutions do not change its sign for sufficiently long necks. Removing the positive common flow direction first consequently orients ${\mathrm{\partial }}_T$ by $-ϵ({\gamma }_1)ϵ({\gamma }_2)$. The outward boundary ray for $\rho =1\mathrm{/}T$ is $-{\mathrm{\partial }}_{\rho }$, a positive multiple of ${\mathrm{\partial }}_T$. The boundary sign is thus $-ϵ({\gamma }_1)ϵ({\gamma }_2)$, independently of the intermediate index and hyperbolic rates.

By [F4] the sum of these signed boundary products is zero. Its negative is the coefficient of $q$ in ${\mathrm{\partial }}^{2}p$, so that coefficient vanishes over the integers. Modulo two, even boundary cardinality gives the same vanishing without rays. This holds for every index-drop-two pair; all other coefficients of the degree-minus-two composite are absent. Finite linear extension proves ${\mathrm{\partial }}^2=0$, giving the claimed chain complexes in the sense of Chain complex in an abelian category. When the end field is normalized, all definitions use the same finite basis, orbit counts and ordered rays, so they agree with the two normalized differential suppliers in the statement.