Radial gauge yields the Getzler oscillator as the leading operator

In radial gauge $x^i{A}_i(x)=0$. The curvature identity $F_{ji}={\mathrm{\partial }}_j{A}_i-{\mathrm{\partial }}_i{A}_j+[A_j,A_i]$ and ${\mathrm{\partial }}_i(x^j{A}_j)=0$ give $x^j{F}_{ji}=x^j{\mathrm{\partial }}_j{A}_i+A_i$. Therefore $\frac{d}{ds}(s{A}_i(sx))=s{x}^j{F}_{ji}(sx)$ and integration from $0$ to $1$ yields $A_i(x)={\int }_0^{1}s{x}^j{F}_{ji}(sx)ds=\frac{1}{2}{x}^j{F}_{ji}(0)+O(\mathrm{\mid }x{\mathrm{\mid }}^2)\mathrm{.}$ This derives the radial-gauge formula rather than assuming it.

In the local factorization $V\cong S\widehat{\otimes }W$ from [F3], spin curvature has Clifford degree two, while the multiplicity curvature commutes with Clifford multiplication. The leading radial connection term is $\frac{1}{4}{\mathcal{R}}_{ij}{x}^j$ with ${\mathcal{R}}_{ij}=2F_{ji}^S(0)$; multiplicity-connection terms are $O(\mathrm{\mid }x\mathrm{\mid })$ and have Getzler degree at most $0$. Thus ${\mathrm{\nabla }}_i$ has leading part ${\mathrm{\partial }}_i+\frac{1}{4}{\mathcal{R}}_{ij}{x}^j$ of degree $1$. The Lichnerowicz formula [F2] then shows $D^2$ has degree $2$ and the stated leading oscillator, since scalar curvature has degree $0$ and the relative curvature acts by exterior multiplication with degree $2$.

On a local factorization $V\cong S_0\widehat{\otimes }W$, the grading identity from [F3] is ${\mathrm{\Gamma }}_V={\mathrm{\Gamma }}_{S_0}\otimes {\mathrm{\Gamma }}_W$, with ${\mathrm{\Gamma }}_W=({\mathrm{\Gamma }}_{S_0}\otimes I){\mathrm{\Gamma }}_V$. For homogeneous $A$ and a Clifford monomial $c_I$, the ordinary trace factorization is ${\mathrm{Str}}_V(c_I\otimes A)={\mathrm{tr}}_{S_0}({\mathrm{\Gamma }}_{S_0}{c}_I){\mathrm{tr}}_W({\mathrm{\Gamma }}_{W}A)$; at top Clifford degree $n$, $A$ is even and the latter factor is ${\mathrm{str}}_W(A)$. This identifies the global graded trace with the spinor Clifford trace times the multiplicity supertrace; the overlap law for $W$ makes it independent of the chosen factorization. The radial transport recursion in [F1] has leading part $(x^i{\mathrm{\partial }}_i+j)u_j=-P_0{u}_{j-1}$; all metric corrections have positive coordinate order. By step 2.1, $P_0$ raises Getzler degree by at most $2$, so induction from $u_0(0)=I$ gives ${\mathrm{ord}}_G(u_j)\le 2j$. If $\mathrm{\mid }I\mathrm{\mid }<n$, then ${\mathrm{tr}}_{S_0}({\mathrm{\Gamma }}_{S_0}{c}_I)=0$: the product ${\mathrm{\Gamma }}_{S_0}{c}_I$ is a nonempty Clifford monomial, and conjugation by a generator inside it when its length is even, or outside it when its length is odd, changes its sign while preserving trace. For $I=(1,\dots ,n)$, ${\mathrm{tr}}_{S_0}({\mathrm{\Gamma }}_{S_0}{c}_1\cdots c_n)=(-2i{)}^m$. Hence $\mathrm{str}{u}_j(y,y)=0$ for $2j<n$, and the order-$n$ component is the unique solution of the leading oscillator transport recursion, with the multiplicity factor ${\mathrm{str}}_W$.

In the finite even exterior-form algebra, put $z=t\mathcal{R}\mathrm{/}2$, $B(t)=z\coth z$, $a(t)=(4\pi t{)}^{-m}{\det }^{1\mathrm{/}2}(z\mathrm{/}\sinh z)$, and define ${\kappa }_t(x,0)=a(t)\exp \text{\hspace{0.17em}⁣}(-\frac{x^{T}B(t)x}{4t})e^{-t{F}_y^{V\mathrm{/}S}}\mathrm{.}$ The formal series truncate in each exterior degree. The scalar identities $z\frac{d}{dz}(z\coth z)=z\coth z-z^2{\mathrm{csch}}^{2}z$ and $\frac{d}{dz}\log (z\mathrm{/}\sinh z)=1\mathrm{/}z-\coth z$ give $t{B}^{\mathrm{\prime }}=B-B^2+t^2{\mathcal{R}}^2\mathrm{/}4$ and $a^{\mathrm{\prime }}\mathrm{/}a=-\mathrm{tr}(B)\mathrm{/}(2t)$, using $n=2m$. For $L_i={\mathrm{\partial }}_i+(\mathcal{R}x{)}_i\mathrm{/}4$, symmetry of $B$, skew-symmetry of $\mathcal{R}$, and $\mathcal{R}B=B\mathcal{R}$ make the mixed term $x^T\mathcal{R}Bx$ vanish and give ${\sum }_i{L}_i^2{\kappa }_t=(-\frac{\mathrm{tr}B}{2t}+\frac{x^T{B}^{2}x}{4t^2}-\frac{x^T{\mathcal{R}}^{2}x}{16}){\kappa }_t\mathrm{.}$ These identities cancel the constant and quadratic terms in $({\mathrm{\partial }}_t-{\sum }_i{L}_i^2+F^{V\mathrm{/}S}){\kappa }_t$, so ${\kappa }_t$ solves the oscillator heat equation. Since $B(t)\to I$ and $a(t)\sim (4\pi t{)}^{-m}$ as $t\downarrow 0$, its normalized Gaussian has the required delta initial value. After factoring $(4\pi t{)}^{-m}{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(4t)}$, write ${\kappa }_t={\sum }_j{t}^j{v}_j(x)$. Substitution into the heat equation gives the same leading radial transport recursion and initial coefficient as the Getzler-leading $u_j$ in step 3.1. At each order that radial equation has a unique smooth solution by the heat-transport integral formula in [F1]; the singular homogeneous solution is excluded by smoothness at $x=0$. Thus the leading transport coefficients agree with this explicitly normalized oscillator solution, including the multiplicity supertrace.

The differentiated graph-norm remainder in [F1] ensures that this transport coefficient is the coefficient of the actual heat kernel: under $t\mapsto ϵt$, $x\mapsto \sqrt{ϵ}x$, and Getzler rescaling, the worst Clifford factor is ${ϵ}^{-n\mathrm{/}2}={ϵ}^{-m}$, canceled by the heat-kernel normalization ${ϵ}^m$. For any fixed finite collection of derivatives, choose the arbitrary remainder order $N$ in [F1] large enough; its rescaled error is then $O({ϵ}^N)$ and tends to zero. Thus no convergence of rescaled global kernels is assumed. The sign convention is fixed by $R(X,Y)=[{\mathrm{\nabla }}_X,{\mathrm{\nabla }}_Y]-{\mathrm{\nabla }}_{[X,Y]}$ and ${\sigma }_1(D)(\xi )=ic(\xi )$; reversing either changes both the radial curvature matrix and its oscillator coupling. All assertions are local in $y$ and patch over the diagonal.

If $m=0$, the coordinate space is a point and has no coordinate derivatives or curvature two-forms. The Dirac operator and $P_0$ are zero, and their heat operator is $I_V$ for every $t>0$. The normalized zero-dimensional Gaussian is $1$, so its coefficient is $u_0=I_V$ and all higher $u_j$ vanish. There is no condition $j<m$ to check. The grading in [F3] gives ${\mathrm{\Gamma }}_V={\epsilon }_y{\mathrm{\Gamma }}_W$, hence ${\mathrm{str}}_{V}I={\epsilon }_y{\mathrm{str}}_{W}I$. Thus the oscillator determines the stated coefficient at either orientation; no positive-dimensional volume-chirality formula is imposed at a negative point.