Finite cyclic smooth bounding signature defects are independent

First take $X$ connected. The homomorphism $\psi$ gives an actual finite principal $G$-cover, even if its image is proper: form $(\tilde{X}\times G)\mathrm{/}{\pi }_1(X)$ using the action $\gamma \cdot (x,g)=(\gamma x,\psi (\gamma {)}^{-1}g)$, which is an action since $G$ is abelian, and retain right translation on the $G$ factor as the specified free fiber-transitive deck action. Over any simply connected coordinate ball choose one universal sheet; the quotient is that ball times the finite set $G$. These local identifications have locally constant sheet permutations, so define the smooth covering structure and orientation. At a chosen basepoint the lift of a loop $\gamma$ ends at $[\gamma \tilde{x},g]=[\tilde{x},\psi (\gamma )g]$, so this is the actual prescribed monodromy rather than its inverse. Its degree is exactly $d$, and its total space may be disconnected. It is compact by finitely many compact lift closures of smaller evenly covered charts. Pull back a Riemannian metric. By [F5] the base pair, and therefore its finite cover, have actual finite cellular complexes. Thus every Hilbert-module complex in [F2] is simply finite-dimensional in this application. A compact disconnected $X$ has finitely many components (its connected coordinate charts have a finite covering subfamily), so the subsequent construction applies on each and signatures add.

Apply the projections $e_r=d^{-1}{\sum }_{k=0}^{d-1}{\zeta }_d^{-rk}{t}^k$ to the finite regular complex of step1.1. Their orthogonality, sum one and traces $1\mathrm{/}d$ are the finite geometric-sum identities in [F2]; none uses surjectivity. Kernels, images and middle homology split literally by these projections, and every image is closed. Sesquilinearity makes different components orthogonal, so the positive and negative projections have trace $d^{-1}$ times the sum of their component ranks; the same statement holds after removing each component's radical. This proves the character-mean formula also for the actual possibly disconnected principal cover, without silently applying [F2]'s surjectivity hypothesis in a larger scope. Each character block is the one computed by [F1] on this actual finite cover, with a positive covering normalization. It has signature $\mathrm{sign}(X)$. If the cohomology pullback convention in [F1] reverses a character, all characters are still included and complex conjugation preserves each inertia count. The normalized sum of these $d$ identical integers is $\mathrm{sign}(X)$, proving the closed-index equality for every $\psi$. This step used only finite cyclic character lines; no approximation or general-group closed-index theorem is involved.

For the bounding data, use their supplied collars to form the closed smooth oriented manifold $X=W_0{\cup }_M(-W_1)$. The coefficient homomorphisms agree on the common based boundary group, so [F5] gives the actual glued homomorphism $\psi :{\pi }_1(X)\to G$. The intersection is the connected full boundary collar; no injectivity of its group maps is required. Apply [F3] in precisely its smooth entire-boundary scope. It says ${\mathrm{sign}}_G^{(2)}(X,\psi )={\mathrm{sign}}_G^{(2)}(W_0,{\psi }_0)-{\mathrm{sign}}_G^{(2)}(W_1,{\psi }_1)$ and $\mathrm{sign}(X)=\mathrm{sign}(W_0)-\mathrm{sign}(W_1)$. Step2.1 identifies the two left sides. Rearranging gives the asserted equality of defects, without any surjectivity restriction on the boundary map or extensions. Hence the value is well-defined among all the supplied smooth extensions whenever at least one exists.

For the knot data, [F4] supplies an actual smooth bounding extension with the exact oriented zero-surgery boundary and positive meridional coefficient homomorphism. Thus the smooth common value of step3.1 exists, and evaluation on this particular extension is [F4]'s displayed average. Its proof treats nontrivial roots directly even when the Alexander form is singular, and treats the trivial character by a separate ordinary metabolic calculation; no invalid specialization at $t=1$ is introduced here. The convention is ${\mathrm{sign}}^{(2)}-\mathrm{sign}$ throughout. The construction and equality now justify the indicated smooth finite-cyclic rho notation, but they do not compare it with a defect from a nonsmooth TOP bounding manifold or with an eta definition. All broader analytic and topological independence obligations remain separate.