Regularity, compactness, and gluing for pointed Heegaard disks

Choose the actual background and path from [F1]. Its fixed endpoint neighborhoods are genuine positive time collars, and all marked local multiplicities of its actual strips are nonnegative. Each index-one solution is nonconstant, since a constant transverse generator has index zero. Its actual linearization is onto by [F1]. The globally finite list of actual rigid orbits in [F1] in particular makes every specified class's translation quotient finite, including the empty quotient. This uses the full-marked construction, not the zero-$z$ associated-graded branch. Every actual index-two strip is regular as well, so its translation quotient has the indicated one-dimensional interval charts.

Verify the original PER-WHITNEY-CLASS compactification. Fix an index-two class $\varphi$ and its endpoints. If any marked coefficient is negative, its actual moduli is empty by step1.1, and no positive-degree nodal configuration can have that class. Otherwise let $W,Z$ be its two marked vectors and take the actual compact one-manifold $M(x,y;W,Z)$ in [F2], initially across classes. The Whitney-class label is locally constant in its interior: a sufficiently small local-addition chart gives the straight relative homotopy, with the same ends and physical boundary. Extend the label over each of its finitely many added endpoints using its actual collar. That collar's connected open interval has a single class. For a broken pair [F3]'s scalar real neck cut-and-cap homotopy identifies it with the concatenation of the two classes. For a disk endpoint, use the ACTUAL attachment source plumbing: cut out its small physical half-disk, retain the fixed generator chart on the annulus, and cap its outer boundary by the constant generator. The inner component is the node-fixed disk, while the outer strip is constant. The preglue represents precisely this boundary-disk insertion. Its Newton correction is homotopic to that preglue by the local-addition straight section homotopy preserving the two torus boundaries and generator ends. Scalar contraction on the remaining small annular chart proves the same class for every sufficiently small collar member; no holomorphicity of this homotopy is claimed. Exhaustion in [F2,F3] makes this label locally constant at the endpoint too. Thus the fixed-class preimage is clopen in the compact one-manifold. It is itself compact and inherits exactly the interval/half-interval charts and collars with that class label. This proves the stated compactification for $\varphi$, rather than replacing it by the marked-vector union.

The boundary classification on this fixed-class part is EXACT: its endpoints are the matching broken index-one pairs and boundary-degeneration disks in [F2]. No other end can appear upon restricting a clopen subset. The actual energy/index argument in [F2] captures all limits before restricting: a nonconstant limiting strip has positive index by [F1], so total index two makes it index one or two and hence regular. Nonconstant boundary bubbles have index at least two, and spheres have index at least $2k$. This leaves the listed broken and unit-disk cases when $k\ge 2$; for $k=1$ the only further unit-sphere possibility passes through the generator and is excluded by the chosen actual sphere avoidance. The finite-domain uniform energy bound and energy-complete extraction in [F1,F2] leave no escaping periodic-domain contribution. Every listed matching broken end has the exhaustive collar in [F2,F3], and so does each actual disk end.

To verify cancellation in the ORIGINAL differential-square count, first keep all marked variables: assign a rigid strip weight $U^{n_w}{V}^{n_z}$. The matrix is finite by step1.1. Boundary parity for [F2]'s actual compact one-manifolds identifies the coefficient of each $U^W{V}^{Z}y$ in its square with the actual disk-end count, since its other endpoints are precisely the ordered rigid pairs with that monomial. If $k\ge 2$, an alpha unit disk in the component containing $w_i,z_i$ has count one and weight $U_i{V}_i$, and a beta unit disk in the component containing $w_i,z_{i+1}$ has count one and weight $U_i{V}_{i+1}$, by [F1]. Hence $D^2=({\sum }_i{U}_i(V_i+V_{i+1}))I$. If $k=1$, both actual unit counts are zero and the sphere alternative was excluded, so $D^2=0$. Now set EVERY $V_i=1$. This specialization is exactly the full filtered minus differential in the original definition, still counting every positive $z$ multiplicity. Its curvature becomes ${\sum }_i(U_i+U_i)=0$ in characteristic two, and hence $d^2=0$. The cancellation is in each polynomial coefficient of this differential-square count; it does not falsely assert that the alpha and beta disk insertions have the same Whitney class or the same full $z$ vector.

For an actual term from $x$ to $y$, [F4] gives $A(U^{n_w}y)=A(y)-{\sum }_i{n}_{w_i}=A(x)-{\sum }_i{n}_{z_i}\le A(x)\mathrm{.}$ The last inequality is the actual marked positivity in step1.1. Thus the finite full filtered minus differential preserves the Alexander filtration. An additive shift in the relative Alexander values has no effect on this inequality; no absolute normalization or invariance is needed.

When the target is a point there are no nonconstant strips and every asserted differential and compactification is empty, so the same conclusion holds directly. For every positive-dimensional balanced diagram the preceding chosen path supplies all the surjectivity, finite quotients, perclass compactifications, exhaustive broken collars, boundary cancellation, absence of additional sphere/periodic ends and filtration statements. The proof has retained all $w,z$ degrees and the full $m=h+k-1$ scope. It proves this chosen-data analytic lemma and no diagram-move, absolute-grading or combinatorial comparison theorem.