Ozsváth–Szabó link-surgery filtered complex and exact triangle

The cube module and its increasing filtration are fixed by [F1]. The candidate total differential sums over ordered immediate-successor paths and their polygon-count maps, with diagonal disk differential. Each component from $I$ to $J\ge I$ lowers $n-\mathrm{\mid }I\mathrm{\mid }$ to $n-\mathrm{\mid }J\mathrm{\mid }$, so it preserves every increasing filtration subspace whenever the count is defined. Square-zero, count finiteness and identification of total homology still require proof. No filtered comparison between different presentations is a premise of this fixed-data assertion.

The square-zero substitution has a precise additional finite hypothesis. Denote polygon operations by ${\mu }_k$ and the chosen immediate top inputs by ${\theta }_{IJ}$. For $A<B$, form the finite top-input sum $M_{AB}={\sum }_{A=I_0<\cdots <I_r=B}{\mu }_r({\theta }_{I_0{I}_1},\dots ,{\theta }_{I_{r-1}{I}_r}),$ where every step changes one coordinate; ${\mu }_1$ is the differential on a single top input. Suppose the actual surgery data satisfy $M_{AB}=0$ for every such interval, in their actual auxiliary complexes and compatible labels. Expand the component $(D^2{)}_{IJ}$ as the sum over two consecutive input-containing polygon operations. The boundary identity of [F2], summed over all paths from $I$ to $J$, pairs these with exactly the terms obtained by inserting one $M_{AB}$ among the top inputs of an input-containing polygon. Every consecutive subpath is counted once in this insertion sum, including length-one differential terms. Thus $M_{AB}=0$ implies $D^2=0$. For $I=J$ this is the vertex differential identity. This is a finite substitution in the actual polygon identities; it does not calculate the top-input sums.

There are only finitely many vertex generators, coordinate paths and required arities. Once the actual admissible subordinate data and top inputs are supplied, [F2] gives finite counts for each term and hence for every differential coefficient. The filtration calculation in step 1.1 then applies literally. In the $n=1$ case, the matrix has diagonal vertex differentials and sole off-diagonal triangle map $f_{01}$; the resulting two-summand complex is its mapping cone with the usual homological shift. This is an actual algebraic identification whenever those data exist, not yet a comparison with the third filling. For $n=0$ it is exactly the given hat complex of $Y$.

Apply [F4] to the actual surgery data: its top cycles and vanishing $M_{AB}$ supply exactly the extra hypothesis of step 2.1. Together with the retained local count identities [F2], steps 1.1, 2.1, 3.1 give an actual finite differential preserving the stated filtration. In particular the branched-cover definition later uses a genuine complex, not just a candidate module.

For one coordinate, [F5] supplies the actual comparison of this cone with the third filling, so the cone's short exact sequence gives the surgery long exact sequence. The cyclic triad ordering is the one with slope labels $\lambda ,\mu +\lambda ,\mu$, as fixed in the Statement; the source's Section2, p.6, describes its cyclic symmetry. For zero coordinates the identity complex gives the asserted result.

For several coordinates use [F6], rather than composing unspecified vertexwise comparison maps. Filter the enlarged $\{0,1{\}}^{n-1}\times \{0,1,\mathrm{\infty }\}$ complex by the first $n-1$ coordinates. Its successive quotients are the acyclic single-coordinate three-term complexes from [F5], including the quadrilateral entry. A finite filtered complex with acyclic successive quotients is acyclic, by induction through the short exact sequences of filtration pieces. The subcomplex with last coordinate $\mathrm{\infty }$ and the quotient with last coordinate $0$ or $1$ therefore have isomorphic homology via the connecting map, with the usual cone shift. Induction identifies the latter cube's total homology with the all-$\mathrm{\infty }$ hat homology. The source provides the higher entries of this enlarged complex and their square-zero in [F4], [F5], [F6]; no new canonical comparison map is claimed.

Closed-manifold invariance [F3] identifies this final group with $\widehat{HF}(Y;{\mathbb{F}}_2)$. The construction and every count used here are the imported compatible surgery construction, combined with the retained AComega analytic package. Its geometric data and arities for the fixed link are finite; the cyclic Hamiltonian-copy sequence used in the single-coordinate argument is countable, so the existing countable-choice branch suffices. The source itself does not formulate a choice hierarchy; this accounting uses the retained analytic CC interface and the finite/countable auxiliary selections just specified. No full-AC classification, presentation-independent filtered quasi-isomorphism, or restriction to $S^3$ is introduced.