Characteristic classes, Bott normalization, fixed points, and external index theories

Editorial status, October 2026. Sections 1–10 retain the original formulations, including the positive receiving signs in (ISN12) and (ISN14). Those signs fail for the cited original cotangent orientation. Sections 11–15 prove the full global characteristic formula at the original scope, exhibit and correct its analytic receiving sign, and construct its exact defect. They also give complete independent proofs of the smooth fixed-point and adapted-family receivers and a stronger Fredholm compression theorem. The historical external-theorem statements below must be read with that explicit correction.

Written and dedicated to the public domain by Codex, September 2026 (CC0).

Several index formulas meet in this course, but they do not all have the same input. A Fredholm operator has an integer index. Its principal symbol has a stable homotopy class. A characteristic-class formula evaluates that class after a choice of orientation and normalization. A fixed-point formula evaluates a chain map. A Toeplitz index begins with a projection and a compressed operator. Treating these inputs as interchangeable would hide the maps that make each theorem meaningful.

This lesson gives those maps and records four exact scope boundaries. It imports one topological index theorem in a fixed primary-source convention. It then compares that theorem with the analytic symbol route already proved in the course, checks the Bott normalization that the course actually owns, proves that the fixed-point result has broader map scope than diffeomorphism pullback, and states the additional data required by broader hypoelliptic and Toeplitz theories.

Read Symbols, finite defects, and the index on a closed manifold for the analytic index and stable symbol homotopies; Relative Weyl projectors and the Chern cutoff form for the complete relative projector and its Chern form; The Bott operator, suspension, and reduction of the index to Euclidean space for the index-one Bott operator and suspension; Elliptic complexes, diagonal traces, and fixed points for the distributional Lefschetz formula; and Nonelliptic Fredholm operators on their exact adapted spaces for the proved anisotropic family.

1. The four carriers and the maps that are already proved

Let $X$ be a closed smooth manifold, let $E,F\to X$ be complex vector bundles, and let $$P:C^\infty(X,E\otimes\Omega_X^{1/2}) \longrightarrow C^\infty(X,F\otimes\Omega_X^{1/2}) \tag{ISN1}$$ be elliptic. Denote by $B^*X$ the radial compactification of $T^*X$, by $S^*X=\partial B^*X$ its cosphere boundary, and by $\pi$ the bundle projection. After multiplying the principal symbol by the positive order-changing factor used in the closed-manifold lesson, its boundary value is an isomorphism $$p:\pi^*E|_{S^*X}\longrightarrow \pi^*F|_{S^*X}. \tag{ISN2}$$ The exact relative carrier is $$\alpha(P) = [\,\pi^*E,\pi^*F,p\,] \in K^0(B^*X,S^*X) \cong K_c^0(T^*X). \tag{ISN3}$$ The last isomorphism extends a relative bundle triple by zero from the interior of $B^*X$, which is $T^*X$. Stable bundle addition, homotopy through boundary isomorphisms, and addition of a triple that extends as an isomorphism over $B^*X$ leave this class unchanged.

The analytic route proved earlier supplies $$\operatorname{a\!-\!ind}_X(\alpha(P)) =\operatorname{sind}(p) =\operatorname{ind}P. \tag{ISN4}$$ Here the middle term is defined by smooth approximation on one large cosphere and quantization; the proof of its independence uses a common exterior invertibility region. Thus (ISN4) does not presuppose a characteristic-class formula.

A square order-zero presentation has a second carrier. Suppose $$Q\in\Psi^0_{\mathrm{cl}}(X;E\otimes\Omega_X^{1/2}) \tag{ISN5}$$ is elliptic, and write $g=\sigma_0(Q)|_{S^*X}$. After stabilization, $g$ defines $$[g]\in K_1(C^\infty(S^*X)). \tag{ISN6}$$ Here is the actual stabilization when $E$ is not trivial. Choose a finite cover with smooth local frames and smooth real functions $\chi_a$ supported in those frame charts such that $\sum_a\chi_a^2=1$. Such functions follow from an ordinary smooth partition $\{\psi_a\}$ by replacing $\psi_a$ with $\psi_a/(\sum_b\psi_b^2)^{1/2}$; the denominator is everywhere positive. In local orthonormal frames, the map $i:E\to X\times\mathbb C^M$ whose components are $\chi_a v$ is a smooth fiberwise isometry. Extension by zero is smooth because each $\chi_a$ has support inside its frame chart. Keep the actual orthogonal projection $e=ii^*$ and complement $E^\perp=\ker i^*$, pulled back to the cosphere where needed. The original $g$ remains an automorphism of $\pi^*E$. Its global matrix representative and inverse are $$G=igi^*+(I-e),\qquad G^{-1}=ig^{-1}i^*+(I-e),\qquad [g]:=[G]. \tag{ISS1}$$ The products $i^*i=I_E$, $e(I-e)=(I-e)e=0$ prove both inverse orders exactly. The stabilized operator is $Q\oplus I_{E^\perp}$, conjugated by the smooth unitary bundle map $E\oplus E^\perp\to X\times\mathbb C^M$. Its symbol is $G$; its kernel and cokernel are those of $Q$, since the added identity is invertible. Thus its index is the original $\operatorname{ind}Q$.

The matrix class does not depend on the chosen embedding. For two isometries $i_1,i_2$, in the sum of their two trivial bundles put $$i_\theta v=(\cos\theta\,i_1v,\sin\theta\,i_2v),\qquad G_\theta=i_\theta g i_\theta^*+I-i_\theta i_\theta^*, \quad 0\leq\theta\leq\pi/2 . \tag{ISS2}$$ Since $i_\theta^*i_\theta=I_E$, the displayed inverse in (ISS1) with $i_\theta$ proves that the whole path is invertible. Its endpoints are $G_1\oplus I$ and $I\oplus G_2$. The whole comparison is also an actual conjugation. Keep $e_a=i_ai_a^*$ and define $$U_\theta= \begin{pmatrix} I+(\cos\theta-1)e_1&-\sin\theta\,i_1i_2^*\\ \sin\theta\,i_2i_1^*&I+(\cos\theta-1)e_2 \end{pmatrix}. \tag{ISS2a}$$ On the two embedded copies of $E$ this is exactly the two-plane rotation, and on both orthogonal complements it is the identity. Thus $U_\theta^*U_\theta=U_\theta U_\theta^*=I$, $U_\theta(i_1v,0)=i_\theta v$, and $G_\theta=U_\theta(G_1\oplus I)U_\theta^*$. Conjugation preserves the algebraic matrix $K_1$ class as well as the stable homotopy class. Block permutation does not change the stable class: each exchange of two complex coordinate vectors can be implemented by the continuous real two-plane rotation through $\pi/2$, followed by a complex phase path from $-1$ to $1$ on the one coordinate whose sign was changed. Conjugating by that path gives the required invertible homotopy. This proves the square-to-matrix comparison. In (ISN11) the ordinary differential is applied to a global matrix representative such as $G$, rather than to unrelated local matrices of an automorphism of a nontrivial bundle.

The odd differential-form class has its exact homotopy comparison too. For a smooth invertible matrix path $G_t$, let $\theta_t=G_t^{-1}dG_t$ and $\eta_t=G_t^{-1}\partial_tG_t$. Differentiating both actual inverse identities gives $d\theta_t=-\theta_t^2$ and $\partial_t\theta_t=d\eta_t+[\theta_t,\eta_t]$. The graded product rule gives $d(\theta_t^{2k})=0$. In the trace of the derivative of $\theta_t^{2k+1}$, each of its $2k+1$ terms rotates to the same written order with positive sign: moving a length-$j$ block past $2k+1-j$ one-forms has the even exponent $j(2k+1-j)$. The commutator term has trace zero. Consequently the full character in (ISN11) satisfies $$\partial_t\operatorname{ch}(G_t) =d\sum_{k\geq0}\frac{k!}{(2k)!} \operatorname{tr} \left(\frac{\eta_t\theta_t^{2k}}{(2\pi i)^{k+1}}\right). \tag{ISS2b}$$ Only the actual dimensions truncate this sum. The same graded cyclicity makes the trace of each positive even power zero, so each odd summand in (ISN11) is closed. Integration of (ISS2b) over the original parameter interval proves equality of the character classes at the endpoints of (ISS2), with every factorial and $2\pi i$ factor retained.

The pseudodifferential extension has an index map $$\operatorname{Ind}_{\mathcal E}: K_1(C^\infty(S^*X)) \longrightarrow K_0(\mathcal L^{-\infty}(X)) \cong\mathbb Z, \qquad \operatorname{Ind}_{\mathcal E}([g])=\operatorname{ind}Q. \tag{ISN7}$$ Equations (ISN4) and (ISN7) meet at the same operator: $$\operatorname{a\!-\!ind}_X(\alpha(Q)) =\operatorname{ind}Q =\operatorname{Ind}_{\mathcal E}([g]). \tag{ISN8}$$ This is the exact comparison available here. For a general relative triple with different bundles, the course does not replace the triple by an unproved canonical odd class. It first performs an actual stable square reduction, or it retains the relative carrier. The later closed-double boundary construction likewise lands first in (ISN3).

2. The characteristic-class formula in one fixed convention

Choose an affine torsion-free connection on $TX$, with curvature $$R\in\Omega^2(X;\operatorname{End}(TX)). \tag{ISN9}$$ In Denis Perrot’s convention the Todd form of the complexified tangent bundle is $$\operatorname{Td}\!\left(\frac{iR}{2\pi}\right) = \det\!\left( \frac{iR/2\pi}{\exp(iR/2\pi)-1} \right). \tag{ISN10}$$ Only degrees at most $\dim X$ contribute, so the power series in (ISN10) truncates as a differential form. For an invertible matrix-valued function $g$ on $S^*X$, the odd Chern character is $$\operatorname{ch}(g) = \sum_{k\geq0} \frac{k!}{(2k+1)!}\, \operatorname{tr}\!\left( \frac{(g^{-1}dg)^{2k+1}}{(2\pi i)^{k+1}} \right). \tag{ISN11}$$ Again only finitely many degrees can pair with the fundamental class.

External index theorem in the adopted convention. Let $X$ be closed; orient $S^*X$ by its canonical cosphere orientation. If $Q$ is an elliptic pseudodifferential operator of order at most zero on a complex vector bundle, with leading symbol class $[g]$, then $$\operatorname{ind}Q = \left\langle [S^*X], \pi^*\operatorname{Td}(T_{\mathbb C}X) \smile\operatorname{ch}([g]) \right\rangle. \tag{ISN12}$$ The leading degree-zero matrix is required to be invertible. A strictly negative-order operator has zero degree-zero symbol and is not thereby Fredholm on $L^2$; its actual order-changing Sobolev realization is treated in (ISN13). This theorem is an external input. The original-author source proves that the Radul cocycle is the pullback of the Poincaré dual of the Todd class and derives (ISN12) from the index map of the pseudodifferential extension. The formula is quoted with the source’s orientation, Todd form (ISN10), and odd Chern character (ISN11); no sign is imported from a different cotangent orientation.

For a square elliptic operator $P$ of order $m$, the exact order changer $J_E^{-m}$ from the closed-manifold lesson is invertible and has index zero. Hence $$Q=PJ_E^{-m}\in\Psi^0_{\mathrm{cl}}(X;E), \qquad \operatorname{ind}Q=\operatorname{ind}P, \qquad \sigma_0(Q)=p_m h^{-m}. \tag{ISN13}$$ This proves the order reduction needed before (ISN12). It does not alter the original principal symbol silently: the factor $h^{-m}$, the side on which it is composed, and the zero index of $J_E^{-m}$ are all displayed.

Combining (ISN8) and (ISN12) gives the course’s strongest exact characteristic statement for a square presentation: $$\operatorname{a\!-\!ind}_X(\alpha(Q)) = \operatorname{Ind}_{\mathcal E}([g]) = \left\langle [S^*X], \pi^*\operatorname{Td}(T_{\mathbb C}X) \smile\operatorname{ch}([g]) \right\rangle. \tag{ISN14}$$ A compact-support formula on $T^*X$ requires an additional, convention-sensitive comparison between its Thom or boundary orientation and the cosphere orientation. This lesson therefore uses (ISN14) as the fixed formula and does not replace it by an unsigned symbol $J(X)$.

3. What the Bott calculation normalizes

The Bott lesson proves three analytic facts without invoking full Bott periodicity:

1. the Euclidean Bott oscillator has index $1$;
2. its compactified symbol is homotopic, with the displayed orientation, to the Clifford symbol $p(x+i\xi)$;
3. suspension by the fibrewise Bott block preserves the symbol index: $$\operatorname{sind}(d)=\operatorname{sind}(q). \tag{ISN15}$$

In one complex phase coordinate, the boundary loop is $$b(e^{i\theta})=e^{i\theta}, \qquad b^{-1}db=i\,d\theta. \tag{ISN16}$$ The degree-one term of (ISN11) gives $$\int_{S^1}\operatorname{ch}(b) = \frac{1}{2\pi i}\int_0^{2\pi} i\,d\theta =1. \tag{ISN17}$$ This checks the coefficient and orientation against the analytic index-one calculation. Replacing $e^{i\theta}$ by $e^{-i\theta}$ changes the integral to $-1$, so the sign is not disposable.

The relative-projector lesson gives the even-form normalization used by the matrix index route. With the original phase orientation retained, its exact equality is $$(2\pi)^{-n}\frac{2}{i^n n!} \int (1-\psi)^n\operatorname{tr}\!\bigl[(db\wedge da)^n\bigr] = (2\pi)^{-n}\frac{1}{i^n n!} \int\operatorname{tr}\!\bigl[P(dP)^{2n}\bigr]. \tag{ISN18}$$ The full matrix pairing and the independent finite Weyl calculation now give the same analytic index for the actual original symbols. For the isotropic matrix class and exterior inverse of the matrix-index lesson, with its actual orientation $\Omega_{\mathrm{AN}}=dx_1\wedge d\xi_1\wedge\cdots \wedge dx_n\wedge d\xi_n$, they prove $$\begin{aligned} \operatorname{ind}a^w &=(2\pi)^{-n}\frac{2}{i^n n!} \int^{\Omega_{\mathrm{AN}}} (1-\psi)^n\operatorname{tr}[(db\wedge da)^n]\\ &=-(-2\pi i)^{-n}\frac{(n-1)!}{(2n-1)!} \int_{\partial B_R}^{\Omega_{\mathrm{AN}}} \operatorname{tr}[(a^{-1}da)^{2n-1}]. \end{aligned} \tag{ISS3}$$ The full central connection and compact curvature transgression are proved in the matrix-index lesson, (FM6)–(FM10) and (FC8). Its analytic finite-degree formula (AT1) proves that every integrated relative-projector coefficient below and above degree $n$ is zero, while the degree-$n$ coefficient gives the original analytic index. The independent direct derivation is From ordered Weyl errors to the exterior index form, equations (5)–(18), for every integer error power $N>n$. These are complete receiving proofs for (ISN18), retaining the original compact cutoff, both ordered errors and every coefficient.

Thus the course owns an analytic Bott index, a suspension identity, the odd one-dimensional check (ISN17), and the complete relative-projector normalization (ISN18). It does not prove that multiplication by the Bott class is an isomorphism on every compactly supported $K$-group. That full periodicity statement, and the full Thom-character compatibility needed to move between every compact-support and cosphere presentation, remain external.

4. The fixed-point theorem has broader map scope than diffeomorphism pullback

The fixed-point lesson starts with any smooth map $$\phi:X\longrightarrow X \tag{ISN19}$$ on a compact smooth manifold. It never assumes that $\phi$ is invertible or a local diffeomorphism. The exact transversality condition is $$\det(I-d\phi_x)\ne0 \quad\text{for every fixed point }x. \tag{ISN20}$$ Under this condition the graph conormal misses the diagonal conormal, the fixed set is finite, and the course proves $$L(\phi) = \sum_{\phi(x)=x} \operatorname{sgn}\det(I-d\phi_x). \tag{ISN21}$$

Every diffeomorphism is a smooth map, so (ISN21) includes diffeomorphism pullbacks whenever (ISN20) holds. The inclusion is strict. If $p\in X$ and $\phi(x)=p$ for all $x$, then $p$ is the only fixed point, $d\phi_p=0$, and $$\det(I-d\phi_p)=1. \tag{ISN22}$$ On a positive-dimensional $X$, this map is not a diffeomorphism, yet (ISN21) applies and gives $L(\phi)=1$.

An equivariant index theorem has different input. It begins with a group action, equivariant bundles and an equivariant operator, and usually returns a representation-valued or $K$-theoretic class. For a compact group $G$ and $g\in G$, character evaluation is the exact homomorphism $$\operatorname{ev}_g:R(G)\longrightarrow\mathbb C, \qquad [V]-[W]\longmapsto \operatorname{tr}(g|_V)-\operatorname{tr}(g|_W). \tag{ISN23}$$ The de Rham theorem supplies an exact representation-valued class as soon as a smooth action of a compact Lie group is given. Write $\phi_g$ for a left action, so $\phi_{gh}=\phi_g\circ\phi_h$, and retain the original harmonic spaces $K_j$ from the fixed-point lesson. Define $$\rho_j(g)=H_jM_{\mu^{1/2}}\phi_{g^{-1}}^* M_{\mu^{-1/2}}|_{K_j},\qquad V_{\mathrm{dR}}=\sum_j(-1)^j[K_j,\rho_j]\in R(G). \tag{ISR1}$$ The inverse in the pullback is essential to the representation order. For a closed form $f$, (EC14) proves that $f-H_jf$ is an exact form with its actual smooth primitive. Pullback preserves exact forms, and $H_jD_{j-1}=0$. Inserting $H_j$ between the two pullbacks therefore changes the result by zero after the outer $H_j$. Since $\phi_{g^{-1}}^*\phi_{h^{-1}}^*=\phi_{(gh)^{-1}}^*$, this proves $\rho_j(g)\rho_j(h)=\rho_j(gh)$, and $\rho_j(1)=I_{K_j}$. The matrix entries in any fixed harmonic basis are integrals of smooth pullbacks of that basis against fixed smooth forms. They depend smoothly on $g$, by differentiation under the integral on compact $X$. Thus (ISR1) is a finite-dimensional continuous representation in each degree, rather than an assumed equivariant class. The isomorphism $K_j\to H^j_{\mathrm{dR}}(X;\mathbb C)$ from (EC14) intertwines $\rho_j(g)$ with $\phi_{g^{-1}}^*$. Consequently the original evaluation map (ISN23) gives $$\operatorname{ev}_g(V_{\mathrm{dR}})=L(\phi_{g^{-1}}),\qquad \operatorname{ev}_{g^{-1}}(V_{\mathrm{dR}})=L(\phi_g). \tag{ISR2}$$ When $\phi_g$ has the nondegenerate fixed points of (ISN20), the already-proved kernel calculation gives the exact comparison $$\operatorname{ev}_{g^{-1}}(V_{\mathrm{dR}}) =\sum_{\phi_g(x)=x}\operatorname{sgn}\det(I-d\phi_{g,x}). \tag{ISR3}$$ Every determinant, group inverse and alternating degree is retained.

This class is also the equivariant analytic index of the actual de Rham even-to-odd operator with invariant geometric data. To prove that identification, average the original Riemannian metric over normalized Haar measure. For completeness, on a compact Lie group such a measure can be obtained by choosing a positive density at the identity and extending it by left translations. The resulting smooth density has finite positive mass, so divide by that mass. A right translation pulls it back to another left-invariant density, hence to a positive constant multiple. The change-of-variables formula preserves its total mass one, so that constant is one. This proves the two invariances used in the average: $$\bar h_x(v,w) =\int_G(\phi_g^*h)_x(v,w)\,dg . \tag{ISR4}$$ Every integrand is positive on nonzero $v=w$, and compactness permits all coordinate derivatives under the integral. The result is a smooth positive metric. Haar measure is invariant on both sides for compact $G$, so a group change of variables proves $\phi_g^*\bar h=\bar h$. Keep the associated invariant density $\bar\mu$, induced form metrics and actual half-density differential $\bar D=M_{\bar\mu^{1/2}}dM_{\bar\mu^{-1/2}}$. The unitary pullbacks $\bar R_g=M_{\bar\mu^{1/2}}\phi_{g^{-1}}^* M_{\bar\mu^{-1/2}}$ commute with $\bar D$, hence with its adjoint, both full Laplacian summands and $\bar A_{\mathrm{ev}}=(\bar D+\bar D^*)_{\mathrm{ev}}$. The kernel and adjoint kernel are respectively the even and odd harmonic sums by (EC4)–(EC15). The equivariant Fredholm index is therefore their difference in $R(G)$. Each invariant harmonic space and each original $K_j$ map isomorphically to the same de Rham cohomology with the same $\phi_{g^{-1}}^*$ action. These isomorphisms prove $$\operatorname{ind}_G\bar A_{\mathrm{ev}} =[\ker\bar A_{\mathrm{ev}}]-[\ker\bar A_{\mathrm{ev}}^*] =V_{\mathrm{dR}}\quad\text{in }R(G). \tag{ISR5}$$ This constructs the representation class and its scalar receiving map for the de Rham operator. General equivariant characteristic formulas and fixed-submanifold evaluations when (ISN20) fails remain external subjects; neither is asserted by (ISR1)–(ISR5).

5. The exact hypoelliptic result and its boundary

The proved nonelliptic family is $$P_{r,m}=I+D_x^{2r}+D_y^{2m}, \qquad 1\leq r<m, \tag{ISN24}$$ on $\mathbb T^2$. Its full multiplier is $1+k^{2r}+\ell^{2m}$, and the exact adapted domain records that entire multiplier. Division by it proves an isometric bijection onto $H^\sigma$ for every real $\sigma$. Both Sobolev inclusions bounding this domain are strict for every real $\sigma$, by the full axis distributions (AH22)–(AH24). The same operator on the ordinary pair $H^{\sigma+2m}\to H^\sigma$ has dense, proper, nonclosed range. Equations (AH25)–(AH32) prove local hypoellipticity on every open subset of the original torus, using the full reciprocal and its exact off-origin kernel.

This result proves more than an isolated example because it covers every integer pair $1\leq r<m$, but it does not establish a general hypoelliptic index theory. Such a theory must specify its symbol class, anisotropic estimates, composition law, parametrix remainders, domain and range, and the topology in which the index is stable. Hypoellipticity says that output smooth on any open set forces input smooth on that same set; it does not imply Fredholmness on a chosen ordinary Sobolev pair. The ordinary realization of (ISN24) is the course’s exact counterexample to that implication.

6. The missing operation in a Toeplitz index problem

A Toeplitz problem begins with data absent from (ISN24): a Hilbert space $H$, an orthogonal projection $\Pi$, and an invertible or unitary operator $U$. The compressed operator is $$T_U=\Pi U\Pi:\Pi H\longrightarrow\Pi H. \tag{ISN25}$$ The basic Fredholm mechanism can be proved without an index formula. If $U$ is unitary and $[\Pi,U]$ is compact, put $S=\Pi U^*\Pi$ on $\Pi H$. Then $$\begin{aligned} ST-\Pi &=\Pi U^*(\Pi U-U\Pi)\Pi =\Pi U^*[\Pi,U]\Pi,\\ TS-\Pi &=\Pi U(\Pi U^*-U^*\Pi)\Pi =\Pi U[\Pi,U^*]\Pi. \end{aligned} \tag{ISN26}$$ Both errors are compact, so $T_U$ is Fredholm.

The unitary hypothesis can be removed from this Fredholm mechanism. Let $U$ be any bounded invertible operator on $H$, with $[\Pi,U]$ compact. Set $S=\Pi U^{-1}\Pi$ on the original $\Pi H$. Multiplication of the full commutator proves $$[\Pi,U^{-1}]=-U^{-1}[\Pi,U]U^{-1}, \tag{IST1}$$ because the right side expands to $-U^{-1}\Pi+ \Pi U^{-1}$. With both original compression orders unchanged, the exact errors are $$\begin{aligned} ST_U-\Pi&=\Pi U^{-1}[\Pi,U]\Pi,\\ T_US-\Pi&=\Pi U[\Pi,U^{-1}]\Pi . \end{aligned} \tag{IST2}$$ Both are compact by (IST1). The two-sided Fredholm criterion therefore applies to every such bounded invertible $U$. For unitary $U$, $U^{-1}=U^*$, so these are precisely the original (ISN26), with neither error deleted. Along an operator-norm-continuous path of invertible $U_t$ satisfying the same compact commutator condition, $\Pi U_t\Pi$ is norm-continuous and Fredholm at each point. The Fredholm stability lesson consequently proves that its index is constant on the path; no characteristic evaluation is assumed.

Equation (ISN26) identifies the operation prerequisites precisely. A Toeplitz index theorem must construct the relevant $\Pi$ and $U$, prove compactness of their commutator, identify the compressed symbol class, and evaluate the resulting Fredholm index. Neither the anisotropic multiplier lesson nor the radial-compression lesson supplies those data. Their index statements therefore cannot be relabeled as a Toeplitz theorem.

The top row keeps the original bundle automorphism and its actual global matrix representative (ISS1)–(ISS2). The middle row keeps the group inverse in the cohomology and character maps (ISR1)–(ISR5). The last row keeps both ordered compressed errors (IST1)–(IST2). The human source for the external characteristic theorem is Denis Perrot, Pseudodifferential extension and Todd class, arXiv:1112.1850v1, index corollary. All other displayed receiving maps have their complete proofs above. The reproducible source is index-scope-receiving-maps.py.

7. Four reader records and their exact dispositions

Each record points to a proved course equation or to the single marked external index theorem. The added exact constructions above preserve the original formulas and identify the previously omitted receiving maps.

8. Worked checks

The circle orientation. For $b(e^{i\theta})=e^{i\theta}$, (ISN17) is $+1$. Traversing the same geometric circle with parameter $e^{-i\theta}$ gives $$\frac{1}{2\pi i}\int_0^{2\pi}(-i)\,d\theta=-1.$$ The coefficient alone does not determine the sign; the orientation is part of the input.

A smooth map with singular derivative. For the constant map in (ISN22), the derivative has rank zero, while $I-d\phi_p=I$ is invertible. This separates the theorem’s actual condition from local invertibility of $\phi$.

A compression parametrix. Formula (ISN26) keeps the two products in their original order. Reversing $U$ and $\Pi$ without taking the adjoint commutator would not prove that both errors are compact.

A hypoelliptic non-Fredholm realization. For $P=I+D_x^2+D_y^4$, normalized Fourier modes on the $k$-axis have unit $H^{\sigma+4}$-norm while their images tend to zero in $H^\sigma$. Injectivity plus closed range would give a lower bound, so the ordinary range cannot be closed. On the adapted domain, the same multiplier gives an exact isometry.

9. Exercises with complete solutions

1. Order reduction. Let $P\in\Psi^m_{\mathrm{cl}}(X;E)$ be square and elliptic. Prove every assertion in (ISN13).

Solution. The closed-manifold lesson constructs $J_E^{-m}\in\Psi^{-m}_{\mathrm{cl}}(X;E)$ as an invertible map on every Sobolev scale, with principal symbol $h^{-m}I_E$ and index zero. Composition gives order $m+(-m)=0$ and principal symbol $p_mh^{-m}$, with this factor on the right. Fredholm index additivity gives $$\operatorname{ind}(PJ_E^{-m}) =\operatorname{ind}P+\operatorname{ind}J_E^{-m} =\operatorname{ind}P.$$

2. Reverse the Bott loop. Compute the odd Chern integral of $b_-(e^{i\theta})=e^{-i\theta}$.

Solution. Since $b_-^{-1}db_-=-i\,d\theta$, $$\int_{S^1}\operatorname{ch}(b_-) =\frac{1}{2\pi i}\int_0^{2\pi}(-i)\,d\theta=-1.$$ This is the exact opposite of (ISN17).

3. Locate the failure for the identity map. Let $\phi=I_X$ on a positive-dimensional compact manifold. Which part of the fixed-point proof fails?

Solution. Every point is fixed and $I-d\phi_x=0$. Thus (ISN20) fails, the graph conormal meets the diagonal conormal, the diagonal restriction used for isolated fixed points is unavailable, and the delta change-of-variables step has zero Jacobian. The cohomological Lefschetz number still exists; only formula (ISN21) is outside its hypotheses.

4. Verify both Toeplitz errors. Starting from $T=\Pi U\Pi$ and $S=\Pi U^*\Pi$, derive (ISN26) without commuting any factor.

Solution. On $\Pi H$, the identity is $\Pi$. Insert $U^*U=UU^*=I$: $$ST-\Pi =\Pi U^*\Pi U\Pi-\Pi U^*U\Pi =\Pi U^*(\Pi U-U\Pi)\Pi,$$ and $$TS-\Pi =\Pi U\Pi U^*\Pi-\Pi UU^*\Pi =\Pi U(\Pi U^*-U^*\Pi)\Pi.$$ The first middle factor is $[\Pi,U]$; the second is $[\Pi,U^*]$, which is compact because it equals $-U^*[\Pi,U]U^*$. Bounded multiplication on both sides preserves compactness.

5. Separate hypoellipticity from an ordinary Fredholm estimate. Why does smooth regularity for (ISN24) not repair its ordinary order-$2m$ realization?

Solution. Division by $1+k^{2r}+\ell^{2m}$ preserves rapid Fourier decay, which proves global smoothness. The full difference-kernel argument (AH25)–(AH32) proves local hypoellipticity. Along $(k,0)$, however, the multiplier grows only like $|k|^{2r}$, while the domain norm $H^{\sigma+2m}$ charges $|k|^{2m}$. Normalized modes therefore have images tending to zero. A bounded inverse on a closed range would forbid that sequence. Regularity of each solution and a uniform Fredholm estimate are different assertions.

10. Sources and exact scope

The characteristic formula is taken from Denis Perrot, Pseudodifferential extension and Todd class, arXiv:1112.1850v1. Its original-author TeX states the Todd form, odd Chern character and index theorem in the convention reproduced in (ISN10)–(ISN12), and proves the Radul-cocycle theorem from which the index formula follows.

The analytic Bott operator, its index one, and the exact suspension receiving maps are proved in The Bott operator, suspension, and reduction of the index to Euclidean space. That complete programme proof is the mathematical source used here. Full Bott periodicity is not a theorem asserted or imported in this lesson.

The fixed-point result used here is the course’s proved distributional chain-map theorem. Equations (ISR1)–(ISR5) construct the de Rham equivariant index class and prove its exact scalar fixed-point evaluation. General equivariant characteristic formulas, fixed-submanifold evaluations, broad hypoelliptic calculi and Toeplitz index evaluations remain external subjects. Equations (IST1)–(IST2) prove the compression mechanism for every bounded invertible $U$ with compact commutator.

11. Editorial supplement: the full global calculation and its receiving sign

This supplement retains (ISN1)–(ISN26), (ISS1)–(ISS3), (ISR1)–(ISR5), and (IST1)–(IST2) as the identifiable original text. It supplies the missing proof of the global characteristic calculation and corrects a sign in its analytic receiver. The correction has the same closed-manifold, bundle, operator-order, and invertibility scope as (ISN12). In particular it does not replace that statement by a Euclidean, trivial-bundle, differential-operator, or flat-connection theorem. The original Todd determinant (ISN10), positive odd character (ISN11), half-density realizations, and right order-changing factor (ISN13) all remain.

Write $n=\dim X>0$, $\omega=d\xi_i\wedge dx^i$, $L=\xi_i\partial_{\xi_i}$, and orient $S^*X$ as the outward boundary for $\omega^n/n!$. This is the exact orientation specified at lines 552–559 of Denis Perrot’s freely accessible original author source, arXiv:1112.1850v1. It is defined even when $X$ is not orientable: simultaneous changes in base and cotangent coordinates have positive total determinant. With that orientation, the proved formula is

$$\begin{split} \operatorname{ind}Q &=-\int_{S^*X}^{\iota(L)\omega^n/n!} \pi^*\det\!\left(\frac{iR/(2\pi)}{e^{iR/(2\pi)}-1}\right) \wedge \sum_{j\geq0}\frac{j!}{(2j+1)!} \operatorname{tr}\!\left( \frac{(G^{-1}dG)^{2j+1}}{(2\pi i)^{j+1}}\right),\\ \operatorname{a\!-\!ind}_X(\alpha(Q)) &=\operatorname{Ind}_E([g])=\operatorname{ind}Q. \end{split} \tag{IG1}$$

Here $G$ is precisely (ISS1), including the identity on the orthogonal complement of the actual embedded bundle. Only total degree $2n-1$ is integrated. The minus sign is not a change in either characteristic form or orientation. It is the correction to the receiving sign in (ISN12) and hence in (ISN14). Sections 11.1–11.8 prove (IG1) at this full scope.

11.1. An exact counterexample and its defect

On $X=\mathbb R/(2\pi\mathbb Z)$, keep $D_x=-i\partial_x$, the original half-density $|dx|^{1/2}$, and orthonormal vectors $(2\pi)^{-1/2}e^{ikx}|dx|^{1/2}$. Let $\Pi_+$ project onto $k\geq0$. Choose a smooth function $\beta(\xi)$, zero for $\xi\leq-3/4$, one for $\xi\geq-1/4$. Its integer values are exactly those of $\Pi_+$. Periodizing its inverse Fourier distribution gives the kernel of $\Pi_+$: all nonzero translations are smooth near the diagonal, since integration by parts in the compactly supported derivatives of $\beta$ bounds their differentiated kernels by every inverse power of the translation. Thus $\Pi_+$ is classical of order zero, with leading branches $1$ and $0$, without a holomorphic-boundary theorem.

For every integer $w$, retain the full operator

$$Q_w=e^{iwx}\Pi_++(I-\Pi_+),\qquad g_w(x,+1)=e^{iwx},\quad g_w(x,-1)=1. \tag{IG2}$$

For $w\geq0$ its images are the disjoint Fourier sets $k<0$ and $k\geq w$; its kernel is zero and its cokernel is the span of modes $0,\ldots,w-1$. For $w=-d<0$ every mode occurs, and the only overlaps are $-d,\ldots,-1$; its kernel is exactly the span of $e_j-e_{j-d}$, $0\leq j<d$, and its cokernel is zero. Each range is closed and every defect vector is smooth. Consequently its index at every original Sobolev realization is $-w$. On the two components of $S^*X$, $\iota(L)(d\xi\wedge dx)$ gives orientations $+dx$ at $\xi=+1$, $-dx$ at $\xi=-1$. The full original forms give $\operatorname{Td}=1$ and

$$\int_{S^*X}\operatorname{ch}(g_w)=w, \qquad \operatorname{ind}Q_w-\int_{S^*X}\operatorname{ch}(g_w)=-2w. \tag{IG3}$$

In particular $w=1$ disproves the positive receiving sign, and $w=-1$ proves that reversing the loop reverses both integer values. This is also the literal operator and orientation comparison with (I27)–(I28) in Symbols, finite defects, and the index on a closed manifold.

For every closed $X$, define the defect homomorphism on the original stable square cosphere carrier by

$$\mathfrak d_X([g])= \operatorname{Ind}_E([g])- \int_{S^*X}\pi^*\operatorname{Td}(T_{\mathbb C}X) \wedge\operatorname{ch}([g]). \tag{IG4}$$

The proof below establishes both well-definedness and the exact relation $\mathfrak d_X=2\operatorname{Ind}_E=-2\langle\operatorname{Td},\operatorname{ch}\rangle$. Thus the locus on which the old positive formula happens to hold is exactly $\ker\operatorname{Ind}_E$; the quotient by that locus maps isomorphically onto the actual subgroup $\operatorname{im}\operatorname{Ind}_E\subset\mathbb Z$. Surjectivity onto all integers is proved for the circle by (IG2), rather than assumed for every manifold. These maps study the obstruction while preserving the original carrier.

11.2. Classical symbols, actual traces, and the regularized commutator

In a chart with the original half-density coordinates, use left quantization $(2\pi)^{-n}\int e^{i(x-y)\cdot\xi}a(x,\xi)u(y)\,dy\,d\xi$. Its full formal product is

$$a\star b=\sum_{\gamma\in\mathbb N^n} \frac{(-i)^{|\gamma|}}{\gamma!} \partial_\xi^\gamma a\,\partial_x^\gamma b. \tag{IG5}$$

Taylor’s formula in the intermediate variable, followed by integration by parts in its Fourier variable, proves this product with the exact order drop $|\gamma|$. Each finite remainder satisfies the corresponding symbol estimates: apply $(1-\Delta_y)^N$ to the oscillatory factor, divide by $\langle\eta\rangle^{2N}$, and take $2N$ greater than the dimension plus the finitely many differentiations in question. Coordinate changes and the original input/output half-density factors are those of Sections 4–6 and 13 of Detecting regularity without choosing coordinates; nothing here replaces them by functions on an oriented base. Their all-order remainder and asymptotic-summation proofs identify the quotient by smooth kernels with the complete formal symbol algebra $\mathcal A$. For a classical order-zero symbol its leading map $\lambda:\mathcal A^0\to C^\infty(S^*X)$ is an algebra map, because all $|\gamma|>0$ terms in (IG5) have negative order. For matrix symbols the word order is unchanged.

We need a holomorphic regularizer, and prove its construction. On the half-density line take the metric connection Laplacian $H=I+\nabla^*\nabla$, with principal symbol $h^2=|\xi|_x^2$. Its positive form is $q(u,v)=(u,v)+(\nabla u,\nabla v)$. Chart Fourier estimates and the positive metric on the finite covering make its form norm equivalent to the original $H^1$ norm. Here is the operator construction, without a form-representation theorem as an additional premise. On the Hilbert space $V=H^1$ with inner product $q$, the Hilbert representation proved in Lower bounds and spectral calculus, (PR37), gives a unique $Tf\in V$ satisfying $q(Tf,v)=(f,v)_{L^2}$ for every $v\in V$. Cauchy–Schwarz gives its boundedness. Let $Kf=Tf$ regarded in $L^2$. Then $K$ is bounded, positive and selfadjoint, because $(Kf,g)=q(Tf,Tg)=(f,Kg)$. If $Kf=0$, the defining equality and density of $V$ give $f=0$; selfadjointness then makes its range dense. The spectral calculus constructs its inverse $K^{-1}$ on the exact domain $\operatorname{ran}K$. Alternatively selfadjointness follows directly from $(K^{-1}\pm i)K=I\pm iK$, since $I\pm iK$ has the bounded spectral inverse. The weak defining equality identifies $K^{-1}$ with the distributional operator $I+\nabla^*\nabla$. The smoothing parametrix gives $Tf\in H^2$ for $f\in L^2$, and conversely any $u\in H^2$ satisfies $u=T(Hu)$. Thus its operator domain is exactly $H^2$. Its form domain is exactly $V$: on $H^2$, its square-root norm is $q(u,u)^{1/2}$ by spectral integration; smooth sections are dense in $V$ by chart mollification, and spectral truncation proves that the completion of $H^2$ in that square-root norm is $\operatorname{Dom}H^{1/2}$. Compactness of $H^2\hookrightarrow L^2$, proved by local Fourier truncation in (I2) of the closed-manifold lesson, makes its resolvent compact. Its spectrum lies in $[1,\infty)$, and its spectral calculus therefore defines $J=H^{1/2}$ and $J^{-s}=H^{-s/2}$.

Here is the symbol proof, including complex parameters. Write the complete differential symbol of $H$ as $a_2+a_1+a_0$. On the separated contour rays, take $z$ outside a fixed cone about the positive real axis and $|z|\geq c>0$, and put

$$\begin{split} r_{-2}(x,\xi,z)&=(a_2(x,\xi)-z)^{-1},\\ r_{-2-j}&=-r_{-2} \sum_{\substack{k+|\gamma|+\ell=j\\0\leq k\leq2,\ 0\leq\ell<j}} \frac{(-i)^{|\gamma|}}{\gamma!} (\partial_\xi^\gamma a_{2-k}) (\partial_x^\gamma r_{-2-\ell}). \end{split} \tag{IG6}$$

Every original lower coefficient and multiindex remains in this recurrence. Differentiate it inductively. On each compact chart, with $\rho=(1+|\xi|^2+|z|)^{1/2}$, one obtains $|\partial_x^\alpha\partial_\xi^\beta\partial_z^q r_{-2-j}|\leq C_{\alpha\beta qj}\rho^{-2-j-|\beta|-2q}$; the initial inequality follows from the positive scalar $a_2$ and the angular separation of $z$, and each summand in the recurrence has exactly the displayed degree. Above zero frequency $r_{-2-j}(x,t\xi,t^2z)=t^{-2-j}r_{-2-j}(x,\xi,z)$. Use cutoffs at increasing $\rho$ to sum these symbols: choose the next cutoff radius to make each next differentiated tail smaller than $2^{-j}$ in the first $j$ seminorms. This proves all parameter remainders and preserves the finite chart supports. Multiplying the quantized sum by $H-z$ leaves a remainder decreasing in $\rho$ by any requested power. The identity $(H-z)^{-1}=B(z)+(H-z)^{-1}(I-(H-z)B(z))$, the spectral bound $\|(H-z)^{-1}\|\leq C/(1+|z|)$ on the rays, and the elliptic estimates at each Sobolev order show that the difference has a smooth kernel with every kernel derivative decreasing by any chosen inverse power of $|z|$. The input derivative assertion follows by the adjoint identity. Thus this is the actual resolvent, not only a formal inverse.

For $\Re u<0$, integrate $z^u(z-H)^{-1}/(2\pi i)$ around the positive spectrum, on two separated rays and a small circle of radius $c<1$, with positive contour orientation. On the compact joining arc, the ordinary elliptic recurrence applies for all sufficiently large $|\xi|$, uniformly in $z$; its finite low-frequency piece is smooth. This treats the arc crossing the excluded parameter cone without asserting the ray estimate there. For every $\lambda\geq1$ the scalar contour integral is exactly $\lambda^u$. The spectral integral in the proved calculus, Fubini, and the uniform resolvent bounds therefore identify the operator contour integral with $H^u$. This does not assume an eigenbasis. The elementary scalar contour identities and their actual logarithm branch are proved in the polynomial/contour and metric prerequisite lessons; the branch here is the one real on positive $\lambda$. In the homogeneous coefficients the contour can be moved to one surrounding $a_2$, by the elementary contour integral of each rational term in (IG6). Substitution $z=t^2v$ then gives homogeneous degrees $2u-j$, holomorphic in $u$; moving the small joining arc changes no pole enclosure at large frequency. All differentiated contour tails obey the preceding estimates. Extending by $H^{u+N}=H^NH^u$ supplies the entire holomorphic classical family. At $u=0$ it is the exact identity. Consequently $J^{-s}$ is a holomorphic family of order $-s$, with leading symbol $h^{-s}$, and $\log J=-\partial_sJ^{-s}|_{s=0}$ has symbol $\log h+\ell_0$, where $\ell_0$ is classical of order at most zero. In a Euclidean polar chart this is $\log|\xi|+q'_0$, retaining the degree-zero norm factor inside $q'_0$. The same construction on an auxiliary bundle can be made block diagonal for its degree-zero-form projection.

For a classical $A$ of order $m$, $\operatorname{Tr}(AJ^{-s})$ exists for $\Re s>m+n$. Indeed, an operator of order less than $-n/2$ is Hilbert–Schmidt: on each localized left-quantized piece Plancherel gives $(2\pi)^{-n}\int|a(x,\xi)|^2\,dx\,d\xi$, and the proper-kernel cutoff and smooth remaining pieces retain that property. An operator of order less than $-n$ is a product of two such operators by inserting powers of $J$. The trace-product theorem and trace of a smooth kernel are proved in Traces that survive passage to cohomology, (T6)–(T17). Finite Fourier kernel approximations prove the diagonal trace formula, with the same $(2\pi)^{-n}$, by trace-norm convergence.

Subtract finitely many homogeneous terms from the localized diagonal symbol. The remaining integral is holomorphic on successively larger half-planes. For each retained term the radial integral is

$$\int_1^\infty r^{m-s-j+n-1}\,dr= \frac1{s+j-m-n}. \tag{IG7}$$

Its angular and base integrals are holomorphic coefficients. This proves a meromorphic continuation with simple poles. At $s=0$, the residue is the globally defined functional

$$\operatorname{res}A=\frac1{(2\pi)^n} \int_{S^*X}^{\iota(L)\omega^n/n!} \operatorname{tr}(a_{-n})\,\iota(L)\frac{\omega^n}{n!}, \qquad \operatorname{TR}_J(A)=\operatorname{FP}_{s=0} \operatorname{Tr}(AJ^{-s}). \tag{IG8}$$

The orientation superscript identifies the integration orientation; the differential form under the integral is included explicitly, not counted a second time. The residue is independent of the chart: it is the residue of an actual global trace, and at $s=0$ the formal family equals $A$, so every local computation gives precisely its degree $-n$ coefficient. This argument also proves independence of the chosen positive order-one regularizer. For smoothing $A$, $\operatorname{TR}_J(A)=\operatorname{Tr}A$. An operator of order below $-n$ has zero residue.

Put $\delta A=[\log J,A]$. It is classical of order at most $m-1$: the scalar logarithmic leading term commutes at zeroth product order, and every other term of its commutator uses at least one frequency derivative. On the initial trace half-plane, cyclicity and the exact exponential commutator give

$$\begin{split} \operatorname{Tr}([A,B]J^{-s}) &=s\operatorname{Tr}\!\left( A\int_0^1J^{-(1-t)s}[\log J,B]J^{-ts}\,dt\right),\\ \operatorname{TR}_J([A,B])&=\operatorname{res}(A\delta B),\qquad \operatorname{res}[A,B]=0, \qquad \operatorname{res}\delta A=0. \end{split} \tag{IG9}$$

For the first equality, $[B,e^{-s\log J}]=s\int_0^1e^{-(1-t)s\log J}[\log J,B]e^{-ts\log J}\,dt$; differentiating that integral proves it with its positive sign. The integral symbol family at $s=0$ is $A\delta B$; (IG7) proves the finite-part and residue statements. The last identity follows from the exact zero trace of $[\log J,A]J^{-s}$, since $\log J$ commutes with $J^{-s}$. These trace manipulations are valid also for finite positive orders: insert enough powers of $J^{-1}$ to make the products Hilbert–Schmidt and apply the already proved trace-product theorem to those factors and their Sobolev conjugations. No trace of an unregularized non-trace-class commutator is taken. Graded matrix versions follow by the graded finite-dimensional trace.

For a formal inverse $P$ to the full symbol of $Q$, choose any actual smoothing-error parametrix, in the original order and on the original half-density bundle. Its analytic index is

$$\begin{split} \operatorname{ind}Q &=\operatorname{Tr}(I-PQ)-\operatorname{Tr}(I-QP) =\operatorname{Tr}(QP-PQ)\\ &=\operatorname{res}(Q\delta P) =-\operatorname{res}(P\delta Q). \end{split} \tag{IG10}$$

To prove the first equality for any such parametrix, start with the bounded inverse of $Q$ from its kernel complement onto its closed range and set it zero on the cokernel. The elliptic alternative and all-order parametrix identity in (I4)–(I7) make that inverse a pseudodifferential parametrix with smooth finite-rank errors exactly the two defect projections. Their traces are their dimensions. The difference between any two smoothing-error parametrices is smoothing, since $P-P_0=P(I-QP_0)+(PQ-I)P_0$. Changing the two error traces by this difference gives zero by trace cyclicity. This proves the claimed equality without an excision theorem. The final minus sign follows from $0=\operatorname{res}\delta(PQ)=\operatorname{res}((\delta P)Q+P\delta Q)$, because $PQ=1$ in the formal algebra. This is the exact sign missing in the original author’s line 1353 and in (ISN12).

Define $c(a_0,a_1)=\operatorname{res}(a_0\delta a_1)$. The derivation identity and (IG9) give $c(a_0,a_1)=-c(a_1,a_0)$ and $c(a_0a_1,a_2)-c(a_0,a_1a_2)+c(a_2a_0,a_1)=0$. If the regularizer changes, the two logarithms differ by a classical order-zero symbol $s$; the cocycles differ by the explicit Hochschild coboundary $-b\eta$, $\eta(a)=\operatorname{res}(sa)$. Thus neither a residue uniqueness theorem nor a cyclic-cohomology excision theorem is a premise of this construction.

11.3. The formal bimodule and its finite trace

The following construction is algebraic. Its parameter $\varepsilon$ is an indeterminate, not the heat time or a change in the original Weyl parameter. Let $V=\Lambda T^*_{\mathbb C}X$, and let $\mathcal A(V)$ be the full classical formal symbol algebra acting on forms. Its local generators are $x^i,\xi_i,\psi^i=dx^i\wedge,\bar\psi_i=\iota(\partial_{x^i})$, with

$$[x^i,\xi_j]=i\delta^i_j,\qquad \psi^i\psi^j+\psi^j\psi^i=0,\qquad \bar\psi_i\bar\psi_j+\bar\psi_j\bar\psi_i=0,\qquad \psi^i\bar\psi_j+\bar\psi_j\psi^i=\delta^i_j. \tag{IG11}$$

On a symbol $u$, define $a_Lu=a\star u$ and $b_Ru=(-1)^{|b||u|}u\star b$, with $b$ a differential symbol. Left and right actions graded commute; right multiplication represents the graded opposite algebra. In particular $\partial_{\xi_i}=(x^i_L-x^i_R)/i$, $\partial_{x^i}=i(\xi_{iL}-\xi_{iR})$, and the right anticommutator is $\psi_R^i\bar\psi_{jR}+\bar\psi_{jR}\psi_R^i=-\delta^i_j$. These signs are retained in every subsequent contraction. Let $\mathcal L$ be the image of the left/right actions, and $\mathcal S=\mathcal L[[\varepsilon]]$. In a chart an element of the filtered space $\mathcal D^m_k$ has no powers below $\varepsilon^k$, and its coefficient at $\varepsilon^r$ is a sum

$$\sum_{|\alpha|\leq r,\ \beta,\eta,\theta} (s_{r\alpha\beta\eta\theta})_L (\psi^\eta\bar\psi^\theta)_R \partial_x^\alpha\partial_\xi^\beta, \qquad \operatorname{ord}s_{r\alpha\beta\eta\theta} \leq m+\frac{r+|\beta|-3|\alpha|}{2}. \tag{IG12}$$

Here the odd indices each have entries zero or one, and the coefficients arise from the actual image $\mathcal L$; an arbitrary series is not declared to be a left/right operator. The full Leibniz expansion proves $\mathcal D^m_k\mathcal D^{m'}_{k'}\subset\mathcal D^{m+m'}_{k+k'}$. More explicitly, if $\gamma\leq\alpha$, $\zeta\leq\beta$ derivatives survive on the right, the difference between the resulting coefficient order and the permitted order for $\partial_x^{\gamma+\alpha'}\partial_\xi^{\zeta+\beta'}$ is at most $-3(|\alpha|-|\gamma|)/2-(|\beta|-|\zeta|)/2$. Thus each summand satisfies (IG12), and, for fixed surviving indices, the remaining infinite $\beta$-tail converges in decreasing symbol orders. This proves multiplication in the complete formal algebra, not a rearrangement of a numerically convergent infinite operator sum.

A generalized Laplacian is an even $\Delta\in\mathcal D^{1/2}_1$ whose local difference from $\Delta_0=i\varepsilon\partial_{x^i}\partial_{\xi_i}$ is in $\mathcal D^0_1$. Under $y=y(x)$, the exact generator transformations are

$$\psi^i\mapsto\frac{\partial y^i}{\partial x^j}\psi^j, \quad\bar\psi_i\mapsto\frac{\partial x^j}{\partial y^i}\bar\psi_j, \quad\xi_i\mapsto \frac{\partial x^j}{\partial y^i}\xi_j -i\partial_{x^k}\!\left(\frac{\partial x^j}{\partial y^i}\right) \psi^k\bar\psi_j. \tag{IG13}$$

The last term follows by applying the Lie derivative to a form, including its changing basis. Expand every scalar right multiplication by (IG5). It gives $\partial_{\xi_i}\mapsto(\partial y^i/\partial x^j)_L\partial_{\xi_j}\pmod{\mathcal D^{-1}_0}$ and $i\varepsilon\partial_{x^i}\mapsto(\partial x^j/\partial y^i)_L i\varepsilon\partial_{x^j}\pmod{\mathcal D^{1/2}_1}$. Multiplying the two comparisons proves $\Delta_0\mapsto\Delta_0\pmod{\mathcal D^0_1}$. A finite subordinate partition therefore constructs a global $\Delta$.

Direct commutation with $\Delta_0$ gives $[\Delta,s]=i\varepsilon((\partial_xs)\partial_\xi+(\partial_\xi s)\partial_x+\partial_x\partial_\xi s)+[\Delta-\Delta_0,s]$. By (IG12), $[\Delta,\mathcal D^m_k]\subset\mathcal D^m_{k+1}$; consequently the series $\sigma_t(s)=\sum_{r\geq0}t^r\operatorname{ad}_\Delta^r(s)/r!$ preserves $\mathcal D$ and is its inverse for $-t$. The formal exponential $e^{t\Delta}$ exists in $\mathcal S$ because $\Delta$ has a factor $\varepsilon$. For $u\in\mathcal D^0_1$, the identity