The normalized filtered complex of a reverse mirror is dual

Put $t=\tau (C_K)$ and let $[x]$ be its nonzero total homology class in degree $n$. There is a cycle $x\in F_t{C}_{K,n}$ representing $[x]$. Since $H(F_{t-1}{C}_K)\to H(C_K)$ has zero image, every cycle in $F_{t-1}{C}_{K,n}$ is a boundary. Thus $x\mathrm{\notin }{B}_n+F_{t-1}{C}_{K,n}$, where $B_n$ is the subspace of boundaries. Extend a linear functional $\lambda$ that vanishes on $B_n+F_{t-1}{C}_{K,n}$ and satisfies $\lambda (x)=1$. It is a cocycle. For a finite-dimensional chain complex, $\ker ({\mathrm{\partial }}^{\vee })=B^{\perp }$ and $\mathrm{im}({\mathrm{\partial }}^{\vee })=Z^{\perp }$, so evaluation identifies $H(C_K^{\vee })=B^{\perp }\mathrm{/}{Z}^{\perp }$ with $H(C_K{)}^{\vee }$; hence $\lambda$ represents the nonzero dual class. It belongs to $F_{-t}{C}_K^{\vee }$. If $s<-t$, every element of $F_s{C}_K^{\vee }$ vanishes on $F_t{C}_K$, so its induced class pairs trivially with $[x]$ and maps to zero in total dual homology. Therefore $\tau (C_K^{\vee })=-t$.

Specify the corresponding reverse-mirror diagram as follows. Before applying $r$, exchange the handlebodies and attaching systems, keeping the oriented surface and the ordered marks: ${\mathcal{H}}^{\vee }=(\mathrm{\Sigma },\mathit{\beta },\mathit{\alpha },w,z)$. This diagram lies in $-S^3$. Its alpha bridge is the old beta bridge traversed from $w$ to $z$, and its beta bridge is the old alpha bridge traversed from $z$ to $w$, so its knot is $-K$. Applying $r:-S^3\to S^3$ therefore gives $K^{†}$; the compatible reflected Seifert surface has orientation $F^{†}=-r(F)$. This is the corresponding reflected diagram in the Statement, with the two handlebody labels exchanged, rather than surface reversal alone. Periodic domains and their $w$ coefficients are unchanged by attaching-system exchange, so admissibility is preserved. Work on ${\mathcal{H}}^{\vee }$ and push all data forward by $r$ at the end.

For an original strip $u(s,t)$ from $\mathbf{x}$ to $\mathbf{y}$, put $u^{\vee }(s,t)=u(-s,1-t)$ and $J_t^{\vee }=J_{1-t}$. The new boundary conditions are beta at $t=0$ and alpha at $t=1$, its ends are $\mathbf{y},\mathbf{x}$, and ${\mathrm{\partial }}_s{u}^{\vee }+J_t^{\vee }(u^{\vee }){\mathrm{\partial }}_t{u}^{\vee }=-({\mathrm{\partial }}_{s}u+J_t(u){\mathrm{\partial }}_{t}u)(-s,1-t)=0$. Precomposition is invertible and conjugates the linearized real operators, preserving their kernels, cokernels and indices. It reverses translation and hence gives a bijection on translation orbits, preserving regularity and mod-two counts. The domain map reverses both real coordinates, so it preserves orientation and every marked divisor degree: $n_w({\varphi }^{\vee })=n_w(\varphi )$ and $n_z({\varphi }^{\vee })=n_z(\varphi )$. In particular, it bijects the actual $n_w=0$ strips defining the total hat differentials, including those with $n_z>0$. Thus ${\mathrm{\partial }}^{\vee }{\mathbf{y}}^{\vee }$ contains ${\mathbf{x}}^{\vee }$ exactly when $\mathrm{\partial }\mathbf{x}$ contains $\mathbf{y}$, with the same coefficient. This proves the literal transpose identity for the transported regular data.

In the pointed Morse construction for ${\mathcal{H}}^{\vee }$, replace the compatible Morse function $f$ by $-f$ and negate the generator vector field, carrying its local trajectory modifications along with it. The ambient orientation and vector-field direction both reverse, so the oriented perpendicular plane, and hence its first Chern class, retain their physical orientations. Carry the prescribed zero-framed cap as well: its zero relative Euler number remains zero, and multiplying its boundary frame by a constant $-1$ changes no winding. The capped Seifert surface has the opposite orientation, so the absolute formula in [F1] gives $A_{F^{†}}({\mathbf{x}}^{\vee })=-A_F(\mathbf{x})$, with no additive shift. Pushforward by $r$ carries this pairing and the strip counts unchanged. Sending ${\mathbf{x}}^{\vee }$ to the dual basis functional is therefore a chain isomorphism by step 2.1. Its level is $-A_F(\mathbf{x})$, so its sublevel $s$ consists exactly of functionals annihilating the generators at levels at most $-s-1$, which is the displayed dual filtration. This proves the filtered chain isomorphism; step 1.1 gives its one-dimensional total homology and the tau sign formula. The literal isomorphism uses the corresponding transported diagram, whereas [F4] only states the homotopy-type comparison for general normalized models.