Homotopic continuation data give chain homotopic maps

In any sequence of augmented solutions, first extract a convergent parameter subsequence in $[0,1]$. The autonomous ends are fixed, so the square-root height modulus of [F2] is uniform; the compact-window field and all its finite-time evolution estimates are uniform over the compact parameter interval. The pointed-tail and unshifted-window extraction of [F2] therefore applies verbatim with the converging parameter included. The limit middle solves the limiting parameter equation. By [F1] its augmented dimension is nonnegative, so its vertical index difference is at least $-1$. Each nonconstant autonomous tail loses at least one index. Telescoping consequently bounds the total tail count by $d$, not by the vertical index difference. The triple of pointed height paths, unshifted middle window and parameter gives the same compact metrizable geometric topology as in [F2].

When $d=0$, no autonomous break is possible in step 1.1; moreover at parameter endpoints the fixed regular data have vertical index $-1$, so there are no endpoint solutions. Thus ${\mathcal{P}}_0(p,q)$ is a compact zero-dimensional manifold and is finite. Its signed count with the convention of [F3] defines $K$ on the finite critical basis, with degree $+1$; extending finitely and linearly gives the displayed homomorphism. This remains valid for isolated rogue solutions whose vertical derivative is not onto, because their full augmented derivative and orientation are supplied by [F1] and [F3].

When $d=1$, the extraction allows at most one autonomous break. The only added configurations are a negative rigid end trajectory followed by an augmented zero-dimensional middle, or an augmented zero-dimensional middle followed by a positive rigid end trajectory. Each broken stratum is locally a point, and the augmented fixed-anchor matching chart in [F2] gives its half-interval collar, including inverse coverage. At parameter endpoints [F1] gives the ordinary half-interval collar over each rigid endpoint-datum solution. No endpoint can coincide with an autonomous break, since the requisite augmented zero-dimensional middle has vertical index $-1$ and the parameter-end data are regular. The compactification is therefore a compact one-manifold with precisely these four boundary patterns. Its boundary products are finite by step 2.1 and the metric-end rigid finiteness proved in [F2].

Orient this one-manifold by the parameter-first endpoint sequence. By [F3], its parameter-end signed counts are ${\mathrm{\Phi }}^1-{\mathrm{\Phi }}^0$ and its two broken counts are respectively $K{\mathrm{\partial }}^{-}$ and ${\mathrm{\partial }}^{+}K$, both with positive product signs. By [F4] their sum is zero. Over $\mathbb{Z}\mathrm{/}2$ the same statement follows from even boundary cardinality. Comparing the coefficient at every degree-$k$ target $q$ of every degree-$k$ generator $p$, [F5] therefore gives ${\mathrm{\Phi }}_k^0-{\mathrm{\Phi }}_k^1={\mathrm{\partial }}_{k+1}^{+}{K}_k+K_{k-1}{\mathrm{\partial }}_k^{-}$. Finite linear extension gives this identity in every degree.

The identity of step 4.1 is exactly the chain-homotopy identity of [F5]; hence the maps induce the same homology map. Finally any two regular data with the same fixed ends admit an augmented-regular family relative to their parameter endpoints: the regularizing function and metric perturbations in the parameter-interior endpoint-transversality construction of [F1] preserve the endpoint data. Applying the preceding argument gives datum independence of the induced homology map and of the chain-homotopy class of the chain map.