Analytic foundations for the singular instanton knot complex

By [F1] the prescribed singular data are realized by actual finite-group charts and determinant-one gauges, with no reducible stabilizers. By [F2] the critical points of the perturbed Chern–Simons functional form a finite nondegenerate set, and its genuine singular trajectory equation has the stated Fredholm Sobolev realization. All smooth finite-flow-energy trajectories have the critical asymptotics in those spaces by [F3]. Thus the zero sets used to define the complex are actual smooth finite-dimensional regular moduli, rather than only formal expected-dimensional spaces.

Let $o_{\alpha }$ be the rank-one integral orientation module of a critical point from [F4] and set $C={⨁}_{\alpha }{o}_{\alpha }$. This is finite free. For each ordered critical pair, sum the signed module maps over every relative class of index one. There are only finitely many such classes and finitely many trajectories in each translation quotient by [F3]. Hence this is a well-defined integral homomorphism $d:C\to C$. The index periods are divisible by four, so the relative grading modulo four from [F3] gives this differential degree one in the cochain convention; no unsupported absolute integer grading is claimed. Choosing bases of the modules identifies this complex with the displayed generator convention of the Definition.

For any two endpoint critical points, take all index-two relative classes. Their finite union of compactified translation quotients is a compact oriented one-manifold by [F3]–[F4]. Its boundary consists exactly of all ordered index-one broken pairs, without bubbles, missing collar sheets or constant nonregular limiting levels. The actual signed boundary comparison in [F4] identifies its algebraic boundary sum with one fixed convention sign times the corresponding coefficient of $d^2$. That boundary sum is zero. This holds for every endpoint pair, so $d^2=0$ over $\mathbb{Z}$, not only modulo two. No surgery or sutured excision theorem is used in this argument.

The continuation theorem [F4] constructs finite signed counts for the actual nonautonomous equation, its chain maps and inverse chain homotopies between any two regular metrics and allowed perturbations. Its finite-parameter energy estimate and low-dimensional compactness supply the composition and homotopy relations, including the constant-continuation identity. Therefore the homology of this finite free cochain complex is independent of those choices with the stipulated determinant-orientation sign convention.

An isotopy of the based knot and its earring data transports the actual branch charts, exterior bundle, holonomy perturbations and gauge-fixed operators. The pullback chain identification and the continuation equivalence of [F4] give the asserted isotopy isomorphism. The sign choices agree with the orientation-module convention in step 2.1. This proves precisely the analytic construction and invariance in the original Statement. Sutured-closure independence, the KHI rank detector, resolution-cube maps, instanton L-space surgery and a distinguished canonical class remain their own separate obligations; none has been inferred from this complex construction.