Spinor and geometric Clifford bundles and connections are well defined

The logarithm series and each of its derivatives converge uniformly on smaller closed matrix balls, so the section $s(A)=\exp (b(\log A))$ in [F5] is smooth into the ambient Clifford algebra. The two covering sheets $\pm s$ give local charts on Spin; the projection is smooth by finite-dimensional Clifford multiplication and inversion, and translating the sheets gives compatible smooth inverse branches. In principal charts the equivariant map $\lambda :P\to P_{SO}(TM)$ is $(x,h)\mapsto (x,a(x)\rho (h))$ with continuous $a$, so it is a two-sheeted covering by [F5]. Every smooth local oriented orthonormal frame section therefore has continuous lifts after shrinking its domain. By [F0] and full AC choose countably many such domains covering $M$, with a smooth frame section and a continuous lift on each. Their continuous Spin transitions $h_{ij}$ lift the smooth frame transitions $g_{ij}$. At each point, translate the local logarithm/exponential section in [F5] to a smooth inverse branch of $\rho$ near $g_{ij}(x)$. After restricting to a connected neighborhood, the continuous lift $h_{ij}$ agrees with one of the two inverse branches, so $h_{ij}$ is smooth. The resulting principal charts have smooth transitions and define a smooth structure on the actual supplied lift. This structure is Hausdorff because the base is Hausdorff and each fiber is separated in a product chart. It is second-countable because it is covered by countably many charts $U_i\times \mathrm{Spin}(n)$ and the finite-dimensional Clifford group is second-countable. The right action is smooth in these charts. Realify the smooth complex representation $\sigma :\mathrm{Spin}(n)\to G{L}_{\mathbb{R}}(S_{0,\mathbb{R}})$. The associated-bundle theorem [F2] now applies to this smooth, second-countable principal bundle. Multiplication by $i$ commutes with $\sigma (h_{ij})$, so it descends to a smooth endomorphism $J$ with $J^2=-I$ and the associated charts are complex-linear; the Spin-invariant chirality operator and its eigenspaces descend as well.

Start with any positive Hermitian form on $S_0$ and average it over the finite group of signed monomials in the negative-square generators $c_0(e^j)$. Each generator becomes unitary; since its square is $-I$, its adjoint is its negative. Real linearity then makes every $c_0(v)$ skew-Hermitian, and unit vectors act unitarily. The positive generators are $c_{+}=-i{c}_0$, so their unit-vector products, and hence the full-even Spin action, are unitary too. Chirality is unitary and has square one, hence is selfadjoint. Thus this one form is both Clifford-compatible and Spin-invariant, and descends to a smooth associated Hermitian metric. Averaging over Spin alone would not ensure the vector adjoint relation. Let ${\omega }_i$ be the local $\mathfrak{s}\mathfrak{o}(n)$-valued Levi–Civita connection matrix and let ${\widehat{\omega }}_i=(d\rho {)}^{-1}{\omega }_i$ be its lift to $\mathfrak{s}\mathfrak{p}\mathfrak{i}\mathfrak{n}(n)$. The connection matrices $d\sigma ({\widehat{\omega }}_i)$ obey the frame-change law [F3], so they glue uniquely. In the positive-square cover, the infinitesimal lift of an orthogonal matrix $A$ is the negative Clifford bivector; substitution $c_{+}=-i{c}_0$ cancels that sign and gives the local spin action $\frac{1}{4}{\sum }_{a,b}⟨A{e}_a,e_b⟩c_0(e^a)c_0(e^b)$. Expanding the Clifford relation gives $[c(e^a)c(e^b),c(e^k)]=2({\delta }_{ak}c(e^b)-{\delta }_{bk}c(e^a))$, so the lifted connection intertwines the defining $\mathfrak{s}\mathfrak{o}(n)$ action and Clifford multiplication. Thus the glued spin connection is unitary, preserves chirality, and is Clifford-compatible.

Locally the complex Clifford algebra in even dimension is a full matrix algebra by [F1]. Its matrix idempotents split $V$ into smooth subbundles and its matrix units identify those subbundles, giving a smooth evaluation isomorphism $V\cong S_0\widehat{\otimes }W$ with $W={\mathrm{Hom}}_{\mathrm{C}\mathrm{l}}(S_0,V)$. The induced multiplicity grading is ${\mathrm{\Gamma }}_W=({\mathrm{\Gamma }}_{S_0}\otimes I){\mathrm{\Gamma }}_V$, so ${\mathrm{\Gamma }}_V={\mathrm{\Gamma }}_{S_0}\otimes {\mathrm{\Gamma }}_W$. In a local factorization the difference ${\mathrm{\nabla }}^V-{\mathrm{\nabla }}^S\otimes I$ commutes with Clifford multiplication, because both connections are Clifford-compatible. The full matrix algebra has commutant $I\otimes \mathrm{End}(W)$, so this difference uniquely defines a smooth multiplicity connection ${\mathrm{\nabla }}^W$; it is unitary and preserves ${\mathrm{\Gamma }}_W$ because both original connections are unitary and preserve their gradings. On an overlap write the change of factorization as $T_{\beta }=T_{\alpha }(\sigma (h_{\alpha \beta })\otimes B_{\alpha \beta })$, where $B_{\alpha \beta }$ is the smooth invertible multiplicity transition. The connection transformation law [F3], after subtracting the spin-connection law, gives ${\omega }_{\beta }^W=B_{\alpha \beta }^{-1}{\omega }_{\alpha }^W{B}_{\alpha \beta }+B_{\alpha \beta }^{-1}d{B}_{\alpha \beta }$. Choose the overlap lifts $h_{\alpha \beta }$ on sufficiently small overlaps. On triple overlaps they can multiply to a locally constant sign $ϵ$; since the transitions of $V$ satisfy the cocycle identity and $\sigma (-1)=-I$, the corresponding $B$ transitions multiply to ${ϵ}^{-1}I$. Thus a global multiplicity bundle is not asserted without a global Spin lift. Conjugation by $B$, however, satisfies the cocycle identity, and the displayed local connection laws are consistent because $dϵ=0$. Their curvature satisfies $F^V=F^S\otimes I+I\otimes F^W$, with $F^S=c(R^{LC})$ by [F4], so $F^{V\mathrm{/}S}=I\otimes F^W$ and is genuinely a global commutant-valued curvature.

On overlaps, $F_{\beta }^W=B_{\alpha \beta }^{-1}{F}_{\alpha }^W{B}_{\alpha \beta }$ and ${\mathrm{\Gamma }}_W$ is preserved, so ${\mathrm{str}}_W\exp (-F^W\mathrm{/}(2\pi i))$ is invariant and patches to a global even form. The Bianchi identity follows directly from the local connection matrix: for $F=d\omega +\omega \wedge \omega$, expansion gives $dF+\omega \wedge F-F\wedge \omega =0$. Cyclicity of supertrace then shows that the relative character is closed. For a path of compatible connections with commutant-valued difference $a_t={\dot{\omega }}_t$, one has ${\dot{F}}_t=d^{{\mathrm{\nabla }}_t}{a}_t$; applying Bianchi and cyclicity gives ${\mathrm{\partial }}_t{\mathrm{str}}_W{e}^{\lambda F_t}=\lambda d{\mathrm{str}}_W(a_t{e}^{\lambda F_t})$, with $\lambda =-1\mathrm{/}(2\pi i)$. The transgression forms also conjugate by $B_{\alpha \beta }$ on overlaps, so they patch. This supplies the relative cohomological interpretation for a fixed geometric module, including when no global spinor bundle $S$ exists. For $V=S\widehat{\otimes }E$ with its product connection, the spin connection contributes $c(R^{LC})$ and the twist contributes $I\otimes F^E$, so $F^{V\mathrm{/}S}=F^E$ and the relative supertrace is the ordinary trace on the ungraded $E$.

In dimension zero, use $S_{0,x}=\mathbb{C}$ with ${\mathrm{\Gamma }}_{S_{0,x}}={\epsilon }_{x}I$ from The A-hat form and relative Chern character. The Clifford algebra is the scalar algebra, so evaluation identifies $S_{0,x}\widehat{\otimes }{\mathrm{Hom}}_{\mathbb{C}}(S_{0,x},V_x)$ with $V_x$. Its grading is ${\mathrm{\Gamma }}_{W_x}={\epsilon }_x{\mathrm{\Gamma }}_{V_x}$, making evaluation even. Every two-form is zero; thus both the spin and multiplicity curvatures are zero. On a discrete manifold these pointwise maps are smooth, the reference sign and the graded dimensions are locally constant, and the relative character ${\epsilon }_x{\mathrm{str}}_{V_x}I$ patches globally. For $V=S_0\widehat{\otimes }E$ with $E$ even, $W=E$ is even and the character is its rank. This establishes the relative statement at every signed point without a Spin lift; all connection overlap, Bianchi and transgression equations there reduce to zero identities.