Uniqueness of the stable Clifford–Hopf decomposition

Absorb the original common trivial shifts into the $E_i$ and common regular shifts into the modules. Their upper bundles $F_0=E_0\oplus {\mathrm{\Lambda }}^{+}$ and $F_1=E_1\oplus M^{+}$ are isomorphic by restricting the given isomorphism to the upper pole. Choose a regular module $R$ whose positive half contains both $E_i$, as in [F1], and put $A=M^{+}\oplus (R^{+}\ominus E_0)$ and $B={\mathrm{\Lambda }}^{+}\oplus (R^{+}\ominus E_1)$. Then $F_0\oplus A={\mathrm{\Lambda }}^{+}\oplus M^{+}\oplus R^{+}=F_1\oplus B$. Since $F_0\cong F_1$, adding a complement of $F_0$ shows $A$ and $B$ are ordinarily stably isomorphic.

Add trivial summands so the added $A,B$ can be identified, retaining the total bundle isomorphism. The zero parts become $E_0\oplus A=M^{+}\oplus R^{+}$ and $E_1\oplus B={\mathrm{\Lambda }}^{+}\oplus R^{+}$, which are invariant halves in the common ambient module $\mathrm{\Lambda }\oplus M\oplus R$. This invariance is essential; merely embedding $E_i$ into an arbitrary real complement would not give affine boundary data. Apply [F2], adding the common ambient upper bundle as a pulled-back summand. Since that is itself a module half, the boundary splittings remain invariant and the new upper gauge is $I$. If identifying $A,B$ required an additional common trivial bundle ${\epsilon }^t$, embed it in a regular positive half $S^{+}$ using [F1], and add the common pulled-back complement $S^{+}\ominus {\epsilon }^t$. The entire added zero part is then $S^{+}$, so the upper ambient and both zero parts again are invariant module halves.

We make the common ambient module $U$ from step 2.1 regular before applying [F1]. By [F3] choose a real bundle embedding $j:U\hookrightarrow X\times {\mathbb{R}}^a$. In an orthonormal frame of $V$, let $e_I$ run over all square-free monomials, including $e_{\mathrm{\varnothing }}=1$, and define $\iota (u)={\sum }_I{e}_I\otimes j(e_I^{-1}u)\in C(V)\otimes {\mathbb{R}}^a$. The coefficient of $1$ is $j(u)$, so [F4] gives injectivity. Under an orthogonal frame change the monomials in each degree transform by its exterior-power orthogonal matrix, and Clifford inversion on these monomials is the linear Clifford conjugation. Thus the sum is frame-independent. For each generator $e_i$, reindexing the signed permutation $e_I\mapsto e_i{e}_I$ proves $\iota (e_{i}u)=e_i\iota (u)$, so $\iota$ is module linear. Using the invariant metric on $U$, replace it by $\iota ({\iota }^{\ast }\iota {)}^{-1\mathrm{/}2}$; the positive factor commutes with the Clifford action, so this is an isometric module embedding preserving both given orthogonal end splittings. In the metric making the monomials orthonormal, left multiplication by each unit vector is orthogonal. Hence the orthogonal complement $Q$ of $\iota (U)$ is a module subbundle, and $U\oplus Q\cong C(V{)}^a$. Add ${\pi }^{\ast }{Q}^{+}$ with identity clutching to both triples and extend the upper-identity gauge by identity. Their affine end splittings now lie in this regular module. Absorb this and all common pulled-back additions into both $A,B$; their ordinary stable isomorphism is preserved.

Add the same regular Hopf bundles to raise winding, followed by the same pulled-back regular halves to raise upper rank, until the size conditions of [F1] hold with $d=\dim X+1$. These common additions do not alter the desired stable conclusion. Let ${\varphi }_{-}$ be the lower gauge over $D_{-}V$ and ${\varphi }_0$ its value at the zero section. Radial contraction gives a homotopy from ${\varphi }_0$ to ${\varphi }_{-}$. Glue the two lower bundles on $X\times [0,1\mathrm{/}2]$ and $X\times [1\mathrm{/}2,1]$ using ${\varphi }_0$. On the second interval use the homotopy followed by the first clutching map. Its final value is the second clutching map, because the upper gauge is $I$. This gives a global triple on $\widehat{V}\times I$, with fixed common upper module half and the two prescribed affine end triples.

Apply [F1] to $(X\times I,X\times \{0,1\})$, starting from the given end splittings and identity lower gauges. Its relative cells have dimension at most $\dim X+1$, exactly the connectivity arranged in step 3.2. We obtain an ungraded module subbundle over the cylinder restricting to $\mathrm{\Lambda }$ and $M$ with the same regular additions, and a zero-part real bundle restricting to the correspondingly enlarged $E_0\oplus A$ and $E_1\oplus B$.

For explicit endpoint identification, represent these subbundles by orthogonal projections in the fixed ambient bundle. Compactness gives a subdivision of $I$ on which adjacent projections $P,Q$ have $\mathrm{\parallel }P-Q\mathrm{\parallel }<1$ uniformly. Then $T=QP+(I-Q)(I-P)$ is invertible: on $\mathrm{im}P$ its $Q$ component is bounded below by $1-\mathrm{\parallel }P-Q\mathrm{\parallel }$, and similarly on the orthogonal complement. Its polar part $T(T^{\ast }T{)}^{-1\mathrm{/}2}$ intertwines $P$ and $Q$. For module projections all these operators commute with the Clifford action, so the polar part is module linear. Multiplying finitely many such isomorphisms identifies the endpoints. Thus $\mathrm{\Lambda },M$ are stably module-isomorphic and $E_0\oplus A,E_1\oplus B$ are ordinarily stably isomorphic. Since $A,B$ are stably isomorphic, complement cancellation gives $E_0{\sim }_s{E}_1$.