Hat Heegaard Floer homology is a closed three-manifold invariant

By [F1] both full-module admissible endpoints are globally weak. This implication uses the zero-Chern sector whether or not it is represented by a generator; testing existing generators alone would be insufficient. Use the actual generic hat data of [F1], retaining any independently supplied generic endpoint count choices. Generator charts are auxiliary parts of the construction, and need not have been agreed in advance. The independent-generator operation in [F4] first compares arbitrary supplied data at its OWN fixed background and area form to the canonical choices used in the finite comparisons below. Transported charts on actual diagram isomorphisms still give literal operator identifications. All subsequent comparisons are the actual operations in [F4]. Every comparison below is between actual finite count complexes. No knot-specialized result is substituted for a closed-manifold operation.

First prove the based complete-disk comparison needed on ONE handlebody. Prepare two systems as neat auxiliary witnesses and make their intersections transverse. A handlebody is irreducible using exactly [F6], without any irreducibility assumption about Y: carry its finite graph neighborhood to the standard handlebody model in $S^3$. An interior sphere bounds the two Schoenflies balls. The connected boundary of the handlebody lies on one side, so the ball on the other side misses that boundary and lies inside the handlebody, since its collar at the sphere does. Remove closed disk intersections as follows. Take a circle innermost on the first family over the entire second family. Its cap misses that second family. Replace the second disk's interior cap by a parallel of the first cap and round. Their sphere bounds the just proved interior ball. No other proper disk lies inside that ball, since its boundary is on the exterior handlebody boundary. Thus the boundary-fixed cap transfer preserves the whole second system's topological ball cuts and reduces the circle count. Only auxiliary witnesses change, with the same boundary curves. If using the topological cap transfer, the NEW smooth cut spheres have their actual smooth parametrizations from [F6]; no ambient smooth tangency change or arbitrary TOP smoothing is asserted. Iterate finitely.

An outermost arc on a second-family disk now bounds a cap B missing the first family. Surgery of its first-family disk $D_i$ along B gives two disks pushed disjoint from that family. Cut the other first-family disks; the remaining manifold is a solid torus. Exactly one surgery disk is an essential meridian: their oriented relative classes add to the primitive old meridian, while an essential simple disk boundary is a primitive meridian and an inessential one bounds a ball by step 1.2. Two nonzero primitive classes could sum only to zero or twice a primitive class. Replacing $D_i$ by the essential disk $D_i^{\mathrm{\prime }}$ is therefore a complete system. In the original cut ball $D_i^{\mathrm{\prime }}$ separates the two faces of $D_i$; otherwise it would cut off a ball after regluing those faces and would be inessential. Choose its boundary cap not containing z. It contains exactly one $D_i$ face and finitely many other faces. Apply [F6]'s planar supplier successively with exactly two inner faces, rather than once to this whole cap. Choose a finite tree of disjoint joining bands in the cap, rooted at its unique $D_i$ face, and process it outward from the root. At each stage the frontier of a small neighborhood of the processed subtree is the moving rim. A thin neighborhood of that moving face, the next unused face and their joining band is a disk with two holes whose interior misses every other current face and z. Its two labels are the current moving disk and a retained $D_j$ with $j\mathrm{\ne }i$, so the supplier's distinct-disk hypothesis holds with $r=1$. Its completed slide preserves the cut ball and the retained disks. Use its finger-retraction identification, supported at that face and band, to carry the unprocessed cap and its remaining faces to the new cut-ball boundary; outside that neighborhood the unused faces are unchanged. Thus the next stage has the same description. If both faces of a retained disk occur, the second remains an unused face of that retained disk and is processed by a later two-face application; it is never paired with its other face in one application. A tiny auxiliary trivial arc near z, with second endpoint w, can be fixed before this construction and missed by all these cap neighborhoods, so the supplier's marked arc condition is retained. After finitely many stages the moving rim is the frontier of the whole tree neighborhood, and the remaining empty annulus in the cap gives the isotopy to $\mathrm{\partial }{D}_i^{\mathrm{\prime }}$. This is an actual based slide word, not just disk surgery or homology. The replacement uses one old subdisk and a parallel of B, so it strictly decreases the total remaining arc count. After finitely many exchanges the two families are disjoint. Their union cuts the first ball into balls: each proper disk has two rounded smooth sphere boundaries in the standard ball model, and [F6] gives both balls. Choose a tiny trivial arc near z missing this finite union, with a second endpoint w. The two original systems are two rooted-tree reductions of that common extended ball cut. [F6] compares them by slides fixing z,w and the arc. Forget the auxiliary w and arc. This proves complete-system comparison with no genus change, including empty genus-zero systems.

Prepare a finite same-system slide/isotopy word while its opposite system is fixed. Choose paired dual-loop annuli for that fixed complete system as in [F2], avoiding the initial region representatives. There is ONE winding number making every prescribed diagram and every isotopy interval in this finite word globally weak. At finitely many diagram vertices, test evaluations are rational linear functions of the opposite boundary coefficients, whose injectivity follows from the complete moving system. For each compact isotopy interval choose the test points to be regular values of its finite moving traces. Their finite preimage counts bound all partial traces uniformly. Across a slide there are only finitely many new endpoint relation matrices; take the maximum of these finitely many linear bounds. Thus the same $\pm N$ calculation in [F2], with N larger than that maximum, supplies strict opposite signs for EVERY nonzero periodic chain throughout the word. The initial winding path remains globally weak because its supports miss every initial region representative. On an isotopy interval apply the actual maps of [F2]. At a slide use its actual circular representative, which is globally weak by that theorem; it is isotopic as a whole family to the prescribed slid system. Both endpoints of this last isotopy are globally weak for the wound opposite family, so its actual isotopy equivalence aligns them. A joint own-family isotopy is factored into single-curve ones by a finite subdivision into graphs in disjoint annuli. This proves actual equivalences along the prepared word, without counting on an inadmissible slice.

Consequently any two globally weak diagrams on the SAME embedded splitting have equivalent chosen complexes. Apply step 2.1 to their alpha systems. Wind the first beta system to prepare that finite alpha word by step 3.1, and perform it, ending with the target alpha and the wound beta, globally weak. Now apply step 2.1 to that beta and the target beta. Wind the fixed target alpha to prepare this beta word, and perform it. Its result is $({\alpha }^{\mathrm{\prime }},{\beta }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}})$, globally weak, where alpha-prime is isotopic to the target alpha. The target diagram itself is globally weak. The two-endpoint isotopy theorem [F2] therefore removes this last alpha winding while keeping the target beta fixed. The same construction covers mere changes of auxiliary disk witnesses: their boundary curves did not change, and all counts depend on those curves rather than the chosen interior preparation. Every map and its inverse homotopy in this comparison is the actual one in [F2] or [F4].

The location restriction in [F3] causes no gap. For a geometric stabilization elsewhere, choose a path on the old surface carrying its small disk to a disk in the z-region, avoiding z. A compact supported surface flow and its two-sided collar extension carry the stabilizing ball and its torus bubble along that path. This is an actual ambient transport of the stabilized splitting fixing z and carrying both handlebodies. It need NOT preserve their attaching systems. After transport, those systems and the standard z-region stabilized systems are complete systems in the SAME two handlebodies. Step 2.1 gives their based same-genus slide/isotopy comparison, and step 4.1 supplies its actual equivalences whenever globally weak chosen representatives are used. Hence [F3] is applied only at a z-region disk; all other location changes use actual same-splitting moves. In particular no assertion that $n_q=0$ at a point in an arbitrary old region, and no unproved target-degeneration gluing, has been inserted into the product stabilization proof.

Take the finite geometric word of [F5]. For every supplied intermediate splitting choose a complete globally weak diagram on that splitting by the finite winding theorem in [F1]; at the two ends use the given admissible diagrams. This preparation changes the disk systems, not the splitting or Y. A genus-preserving step or ambient diagram transport now compares diagrams on the same splitting after the actual transport, so step 4.1 applies. For a genus increase first stabilize its chosen lower diagram at a z-region disk by [F3]. Step 5.1 identifies its higher splitting with the prescribed higher splitting; the actual transported data are identified by [F4]. Step 4.1 then compares this higher global diagram with the chosen global diagram there. For a genus decrease use exactly the reverse equivalence, including the literal inverse P of [F3]. Thus there is an actual chain homotopy equivalence for each step of the word. Intermediate geometric diagrams themselves were never assumed admissible and received no count matrix.

Match the actual analytic choices, including independently supplied generator data and area forms. Fix one set of smooth separate-root generator charts for each of the finitely many geometric diagrams. Construct a canonical reference on its off-diagonal locus as follows. In each two-dimensional surface factor choose a smooth positive tensor with the required two real axes at its crossings. Relative to j take its Cayley matrix separately in each root factor, multiply by a symmetric configuration cutoff equal to one near the two tori and zero near the protected discriminant and point divisors, and retain the block diagonal matrix. It is permutation-equivariant, is the standard symmetric tensor near the protected set, and extends smoothly there because the cutoff vanishes on a neighborhood. Each two-dimensional block is positive; hence this reference is tame for the product form of EVERY positive surface area form. It need not be asserted to be a symmetric power of a nonholomorphic surface map. For the finite list of forms used by the actual small-triangle operations, choose generic perturbations sufficiently close to this reference to retain all their positive tameness margins and its prescribed generator and boundary tensors. First apply [F4]'s independent-generator continuation to EACH arbitrary endpoint at its OWN fixed j and area form, obtaining such a canonical complex. This step requires no original endpoint to be tame for another area form. Now canonical choices at a shared diagram are tame for one common selected form and have the common prescribed generator charts. Align their full torus values at fixed background by [F4] if needed, and apply its changing-background comparison with the common compatible boundary tensor, retained charts and ONE fixed form throughout that comparison. The raw-identity trivialization near the tori keeps the prescribed tensor there. Model operation data may likewise be chosen close to their block product-integrable references, and the same first continuation compares them to these canonical endpoint data at their own form. Thus the maps in step 6.1 compose on the actual endpoint complexes. This supplies the independent-chart and finite-area junctions explicitly, instead of extending a common-chart theorem by omission.

Compose these finitely many actual equivalences. Reverse their inverse maps in the reverse order; adding the transported finite chain homotopies gives inverse homotopies for the composites, by direct expansion of $gf+I=dK+Kd$. The resulting chain maps induce inverse isomorphisms on homology. This identifies the two hat groups for the given arbitrary closed oriented Y. Genus zero has the empty generator and zero differential by [F1]; the same stabilization and inverse maps include it. The conclusion is the asserted isomorphism class of the full group, with no diagram-independent filtered chain type, naturality or absolute grading inferred. The empty-generator loophole is excluded by the actual full-all-sector admissibility condition in step 1.1, not by assuming a generator must exist.