Reducible ends and compactification of the one-instanton moduli space

Prepare precisely the actual reducibles. Apply [F2] to their finite set and choose disjoint bounded invariant slice neighborhoods. Its constructed smoothing section ${\sigma }_r$ is supported in their union and vanishes at the central reducible points. On smaller neighborhoods its zeros are equivariant smooth graphs over ${\mathbb{C}}^3$; the punctured quotients are smooth oriented five-manifolds, and cutting their graph-radius gives projective links. There are no additional reducible solutions inside these slice neighborhoods: the stabilizer of a nearby orbit is a subgroup of the center's circle by the local slice construction; a reducible orbit must therefore be fixed by that circle. On its fixed diagonal subspace the equivariant off-diagonal perturbation is zero. The diagonal equation is the abelian self-dual equation, whose only local slice solution is the center because its $H^1$ is zero. Outside these neighborhoods the equation remains unperturbed, so [F2] lists all remaining reducibles. The construction is bounded and smooth into $H^3$ in these finitely many slices.

Compactness for this prepared equation and a genuine middle region. The supports in step 1.1 are in bounded $H^3$ slice balls. For a zero in such a ball the equation $d_{A_i}^{-}a=-F_{A_i}^{-}-(a\wedge a{)}^{-}-{\sigma }_r(a)$ has bounded $H^3$ right side, since $H^3$ is an algebra and the section is uniformly $H^3$ bounded. Together with $d_{A_i}^{\ast }a=0$, the Green estimate [F5] bounds $a$ in $H^4$. Thus zeros there have compact $H^3$ closure, including along the edges of the support. Any sequence of zeros outside these balls is unperturbed, so [F4] supplies its smooth or ideal subsequence. The bounded balls cannot approach an ideal point: their curvature has a uniform $L^4$ bound from the $H^3$ bound, so its energy on a ball of volume tending to zero tends uniformly to zero by Holder. Therefore a sufficiently small whole ideal collar from [F4] misses all the supports. The prepared equation is literally unperturbed there, retaining its entire regular end. Delete a smaller portion of that collar and smaller reducible cone neighborhoods. The remaining zero set is compact and irreducible, by the two sequence arguments just given. Cover this compact set by finitely many bounded irreducible Coulomb-slice collars, avoiding the protected smaller end and cone neighborhoods, and choose larger/smaller radii as in [F3].

Relative middle transversality with all ends preserved. Apply the finite-dimensional cokernel-direction and Sard proof of [F3] to $\mathrm{\Phi }+{\sigma }_r$ on these middle collars, multiplying its new directions by bumps supported there. The fixed ${\sigma }_r$ is allowed in those charts: transition gauges on bounded chart sets have uniform $H^4$ bounds, and its original finite smooth target vectors and coordinate cutoffs consequently have uniformly bounded $H^3$ representatives. The derivative is a compact perturbation of the original Fredholm slice derivative, so its index remains five. The actual compact zero-container in [F3] and its positive-minimum argument exclude all new middle zeros outside the chosen smaller covering neighborhoods when the finite parameters are small. Around the protected cone and ideal collars the equation and its regularity remain exactly as in steps 1.1–2.1; on the compact transition overlaps, openness of surjectivity preserves their regularity. Thus a sufficiently small regular finite parameter makes every irreducible zero transverse. The same proof gives openness and density for all further allowed small middle sections. Both perturbations are zero outside the finite original and middle slice neighborhoods. For every resulting zero sequence in their union the same bounded-section $H^4$ estimate gives a converging $H^3$ subsequence; outside their union [F4] applies unchanged. This proves compactness of the full ideal completion and excludes any other ends, without applying an unperturbed compactness theorem to a nonzero perturbation.

The actual smooth cut bordism. The determinant line orients the transverse irreducible zero set by [F3], and its compact-derivative homotopy includes the fixed reducible preparation. At the ideal stratum, [F4]'s actual center-scale diffeomorphism and homeomorphic ideal extension provide a smooth collar $X\times [0,{\lambda }_0)$. At each reducible, [F2]'s graph-radius coordinate gives the smooth punctured cone collar $(0,\delta )\times {\mathbb{C}\mathbb{P}}^2$. Delete its part of radius less than a smaller positive number. These finitely many collars are disjoint from each other and from the ideal collar. Step 3.1 makes the remaining interior compact up to the displayed collars, so the cut completion is a compact smooth five-manifold with exactly those boundary components. The induced boundary orientation on each projective link is its complex orientation or its negative. On the unique component meeting the connected ideal boundary choose the determinant orientation's global sign so this boundary orientation agrees with the given orientation of $X$; arbitrary other components cause no additional boundary strata. The number of links is exactly the number of actual reducibles, bounded by [F2]. This proves the compatible perturbation, compactified-strata and smooth cut-bordism assertions.