Rank-one regular zero-FI ADHM and ordinary ASD boundary

Stabilize a rank-one raw datum by $W^{\mathrm{\prime }}=\mathbb{C}\oplus \mathbb{C}$, $I^{\mathrm{\prime }}=(I,0)$ and $J^{\mathrm{\prime }}=(J,0{)}^t$, retaining $V,B_1,B_2$. Then $I^{\mathrm{\prime }}{J}^{\mathrm{\prime }}=IJ$, $I^{\mathrm{\prime }}{I}^{\mathrm{\prime }†}=I{I}^{†}$ and $J^{\mathrm{\prime }†}{J}^{\mathrm{\prime }}=J^{†}J$. The word span is unchanged and $\ker J^{\mathrm{\prime }}=\ker J$, so both moments and regularity are unchanged. On the finite stereographic chart the raw kernel matrix annihilates the added framing line, and its orthogonal projection is $P_1\oplus 1$. Thus [A1] produces a smooth rank-two ASD connection whose restriction to that chart is a line connection plus the flat added line and whose charge is $q_2=-k$. By [F2] its determinant curvature vanishes. On that chart this curvature is exactly the first line's curvature, so the whole rank-two curvature is zero there. Smoothness makes it zero also at infinity, hence $q_2=0$ and $k=0$. Thus no regular rank-one zero-FI datum exists for $k>0$, conditional on the exact retained rank-two input. By [F6], a regular complex rank-one orbit would have such a zero-real-moment representative, so the same emptiness conclusion applies to the complex regular quotient.

Independently, stabilize any framed unitary ASD line connection by the trivial framed flat line. The result is a smooth unitary ASD rank-two connection. Its determinant connection is the original line connection, since $\det (L\oplus \mathbb{C})=L$ and the added connection is flat. By [F2] the original connection is flat, so $q_1=0$. Its sector is therefore empty if $k>0$. If $k=0$, transport the unit vector fixed by its framing from infinity. Any two paths differ by a null-homotopic loop by [F3]; subdivide a compact null-homotopy into finitely many cells lying in parallel-frame coordinate balls. Boundary transports in each cell are identity and internal edges cancel, so the loop transport is identity. The resulting global unit parallel section gives the unique framed isomorphism to the trivial flat line. A parallel automorphism fixing its value at infinity is identity. On the raw side at $k=0$ every map is the unique zero-dimensional map, both regularity conditions hold, and $U(0)$ is trivial. This proves the point classification and absence of framed stabilizers.

To type first-order bundle families without presuming a global trivialization, first use step 1.2 to identify the special fiber with its framed parallel line. Choose local lifts of this frame in a locally free Hermitian $D$-line family. If a lift has Hermitian Gram scalar $1+\epsilon r$ with $r$ real, multiplication by $1-\epsilon r\mathrm{/}2$ makes it unitary, so unitary local lifts exist. Their transition functions have form $1+\epsilon f_{ij}$, with imaginary-valued smooth $f_{ij}$ and additive cocycle identity. For a subordinate partition $({\rho }_k)$, set $u_i={\sum }_k{\rho }_k{f}_{ki}$ on chart $i$; terms extend by zero where needed. On an overlap, $u_j-u_i={\sum }_k{\rho }_k(f_{kj}-f_{ki})=f_{ij}$. Changing local frames by $1-\epsilon u_i$ kills the transitions. A constant unitary infinitesimal scalar then matches the given framing at infinity. Hence every such family can be written globally on the fixed framed line as $d+\epsilon a$, with $a$ a smooth imaginary-valued one-form; the remaining gauge changes are $1+\epsilon \varphi$ with imaginary-valued $\varphi$ and $\varphi (\mathrm{\infty })=0$.

The coefficientwise ASD equation in this global frame is $(da{)}^{+}=0$. Write $a=i\alpha$ for a real one-form. Since $d\alpha$ is ASD, $\int \mathrm{\mid }d\alpha {\mathrm{\mid }}^2=-\int d\alpha \wedge d\alpha =-\int d(\alpha \wedge d\alpha )=0$ by [F4]. Continuity and nonnegativity imply $d\alpha =0$. Again [F4] gives $\alpha =df$; subtract the constant $f(\mathrm{\infty })$ and put $\varphi =-if$. The based gauge $1+\epsilon \varphi$ removes $a$. This gauge is unique: a residual imaginary scalar with zero derivative is constant and its value at infinity is zero. Ordinary framed stabilizers were already trivial in step 1.2. The first-order connection groupoid is therefore the trivial point, not merely a set with one known reduced point.

A regular raw first-order family with $k>0$ would reduce to a prohibited datum from step 1.1, so none exists; the connection family for $k>0$ has no allowable special fiber by step 1.2. At $k=0$ the rank-zero internal $D$-module and all raw maps are uniquely zero, with trivial internal gauge group. Its forward monad is the constant framed line and its forward connection is $d$, also over $D$. Step 3.1 supplies a unique framed gauge from every first-order connection object to this one. Thus this explicit construction is an equivalence of the two first-order groupoids. Reduction and involution-preserving scalar maps commute with it, because the zero datum and the constant line pull back to themselves and the framed trivializing isomorphism is unique. The same argument after complexifying the real tangent spaces gives zero complexified tangent spaces; it does not assert a correspondence for arbitrary complex connections without the Hermitian real condition.

Stabilization intertwines $T$ with the rank-two compact action having framing weights $(h,1)$. In the zero-charge classification, inverse rotations and framing changes send a flat framed object to a flat framed object; the unique framed parallel trivialization identifies it with the same point. Its uniqueness also makes the first-order identifications equivariant. They hold for every first-order Hermitian family defined above, hence also for the subset satisfying the exact locally-free/cohomology-base-change hypotheses retained in [A1]; no extra family statement is obtained from an orbit-set bijection. Positive-FI rank-one stable data and their non-locally-free sheaves and formal fixed-point series were not used in any step.