Cauchy data from jumps and residues

A boundary value contains several normal derivatives. They form a weighted vector: taking another normal derivative costs one Sobolev order. The central construction below acts on this entire vector. Its leading symbol separates the modes entering the interior from the modes entering the exterior; its full operator is a projection up to a smooth kernel. Establishing that last assertion requires control of distributions at the boundary, beyond a calculation of eigenvalues.

This lesson treats smooth elliptic differential operators of arbitrary positive integer order on complex bundles of arbitrary common finite rank. It neither assumes a scalar factorization nor prescribes the number of boundary equations. The subsequent Fredholm theorem for complementing boundary problems is a separate unit.

The analytic prerequisites are Stable modes and the algebra of boundary data, Sections 1–3, for matrix ODE modes; Singularities along a submanifold and smooth boundary passage, Sections 10–13, for ordinary transmission and all layer traces; and Detecting regularity without choosing coordinates, Sections 5–6, 11 and 13, for properly supported bundle operators, two-sided parametrices and Sobolev bounds. We also use finite-dimensional linear algebra, smooth partitions of unity, Fourier inversion, distributional differentiation, smooth local flows and inverse functions, compactness, and the fundamental theorem of calculus. The projection is constructed directly from these tools; the later boundary Fredholm argument and all-real-order Poisson estimates are separate.

1. The weighted space of boundary jets

Let $X$ be a compact smooth manifold with boundary $Y$, let $E,F\to X$ be complex smooth bundles of common rank $N$, and let $$P:C^\infty(X;E)\longrightarrow C^\infty(X;F) \tag{CD1}$$ be differential, elliptic, and of order $m\geq1$. Ellipticity means that its principal symbol $p(x,\xi):E_x\to F_x$ is invertible for every nonzero real covector. Sections here are ordinary bundle sections. Choose a positive smooth density and Hermitian metrics for norms and distributional duality; these choices do not change the operator convention.

Fix an inward collar coordinate $t\geq0$ and an identification of the bundles in the collar with pullbacks from $Y$, constructed in Section 4. Write $D_t=-i\partial_t$. The Cauchy vector and its Sobolev space are $$\gamma u=(\gamma_0u,\ldots,\gamma_{m-1}u),\qquad \gamma_ku=(D_t^ku)|_{t=0+},\qquad \mathcal H_s(Y;E)=\bigoplus_{k=0}^{m-1}H^{s-k-1/2}(Y;E|_Y). \tag{CD2}$$ The space $\bar H^s(X^\circ;E)$ is the space of restrictions of $H^s$ sections on a neighborhood, with quotient norm. In particular $\bar H^0=L^2$. Equivalent finite chart constructions give the same spaces, by Section 11 of Detecting regularity without choosing coordinates.

For completeness the trace estimate needed here is a Fourier calculation. If $r>k+1/2$, Cauchy–Schwarz in the normal frequency $\tau$ gives $$\left|\int\tau^k\widehat w(\eta,\tau)\,d\tau\right|^2 \leq C_{r,k}\langle\eta\rangle^{2k+1-2r} \int\langle(\eta,\tau)\rangle^{2r}|\widehat w(\eta,\tau)|^2\,d\tau. \tag{CD3}$$ Indeed the integral of $|\tau|^{2k}\langle(\eta,\tau)\rangle^{-2r}$ is finite precisely in the indicated range. The substitution $\tau=\langle\eta\rangle z$ gives $$C_{r,k}=\int_{\mathbb R}|z|^{2k}(1+z^2)^{-r}\,dz, \qquad \int_{\mathbb R}|\tau|^{2k}\langle(\eta,\tau)\rangle^{-2r}\,d\tau =C_{r,k}\langle\eta\rangle^{2k+1-2r}.$$ Editorial restoration of the Fourier factors. In a collar chart of dimension $n=\dim X$, use the convention $\widehat w(\eta,\tau)=\int e^{-i(y\cdot\eta+t\tau)}w(y,t)\,dy\,dt$, the exact trace formula is $$\widehat{\gamma_kw}(\eta)=\frac1{2\pi}\int_{\mathbb R}\tau^k\widehat w(\eta,\tau)\,d\tau.$$ For the squared Sobolev norm in dimension $d$, retain the factor $(2\pi)^{-d}$. Multiplying (CD3) by $(2\pi)^{-(n-1)}(2\pi)^{-2}\langle\eta\rangle^{2r-2k-1}$ and integrating in $\eta$ gives $$\|\gamma_kw\|_{H^{r-k-1/2}(\mathbb R^{n-1})}^2 \leq\frac{C_{r,k}}{2\pi}\|w\|_{H^r(\mathbb R^n)}^2. \tag{CD3a}$$ Thus every Fourier factor is present in the trace estimate. For $n=1$, the tangential integral is over $\mathbb R^0$ with measure one, so the same calculation gives the point trace. The estimate also makes the trace independent of the extension: a smooth function vanishing on $t>0$ has zero trace there, and approximation in $H^r$ gives the same assertion for an extension difference. One can see the approximation directly by translations away from the boundary followed by convolution, first for compact charts; the Fourier translation factors converge to one and are uniformly bounded. Partitions and bundle frames finish the proof. Thus $\gamma:\bar H^s\to\mathcal H_s$ is continuous for $s>m-1/2$, in particular for every $s\geq m$.

The different orders in (CD2) are essential. An entry taking the $l$-th jet to the $k$-th jet must have order $k-l$ to preserve this scale. This statement defines the weight convention we will use; no general theorem about weighted systems is being invoked.

2. Division in the normal derivative

Let $B_j:C^\infty(X;E)\to C^\infty(Y;G_j)$, $1\leq j\leq J$, be boundary differential operators of total orders $m_j$; the ranks of the bundles $G_j$ may differ. There are boundary differential operators $B'_j,C_j$ such that $$B_j=B'_j+C_jP,\qquad \operatorname{ord}_t B'_j<m,\qquad \operatorname{ord}B'_j\leq m_j,\quad \operatorname{ord}C_j\leq m_j-m. \tag{CD4}$$ The convention is $C_j=0$ if $m_j<m$. Total order is not reduced to $m-1$.

Here is a division procedure that retains the order of matrix factors. Locally put $P=A_mD_t^m+\sum_{a<m}A_a(t,y,D_y)D_t^a$, with $A_a$ tangential of order at most $m-a$. The coefficient $A_m$ is invertible because $p(y,t,dt)=A_m$. If the current boundary operator contains $L_aD_t^a$ with largest normal exponent $a\geq m$, subtract the boundary restriction of $L_aD_t^{a-m}A_m^{-1}P$. The normal leading term is exactly $L_aD_t^a$. Every other term has lower normal order: either it differentiates a coefficient, or it uses $A_bD_t^b$ with $b<m$. All terms still have total order at most $m_j$; the multiplier of $P$ has order at most $m_j-m$. Iteration terminates. The inverse matrix occurs to the left of $P$, with the displayed intervening derivatives; it cannot be moved through the other coefficients. Multiply these local identities on the output by a partition of unity on $Y$ and sum. This proves (CD4) globally without choosing global frames.

For smooth data the equations $Pu=f,\ B_ju=g_j$ become exactly $$Pu=f,\qquad B'_ju=g_j-C_jf. \tag{CD5}$$ At low regularity the expression $C_jf$ may itself require traces not defined on $\bar H^{s-m}$. Thus (CD5) is first a smooth-data equivalence. The realization below is for the reduced boundary operators; it does not assert the continuity of an original operator of arbitrarily high normal order at every $s\geq m$.

Henceforth write $B_j$ for such a reduced operator. It has the unique collar form $$B_ju=\sum_{k=0}^{m-1}\mathcal B_{jk}(y,D_y)\gamma_ku, \qquad \operatorname{ord}\mathcal B_{jk}\leq m_j-k, \tag{CD6}$$ where a negative upper bound means that the differential operator is zero. Formula (CD3), followed by tangential differentiation, proves for every $s\geq m$ the bounded map $$u\longmapsto(Pu,B_1u,\ldots,B_Ju): \bar H^s(X^\circ;E)\longrightarrow \bar H^{s-m}(X^\circ;F)\oplus \bigoplus_jH^{s-m_j-1/2}(Y;G_j). \tag{CD7}$$ Large tangential orders are allowed; their targets can be negative Sobolev spaces.

3. The frozen test and its coordinate invariance

At $y\in Y$, choose an inward conormal $n$, and choose $\xi\in T_y^*X$ whose class modulo $\mathbb Rn$ is nonzero. Let $M^+(y,\xi,n)$ be the space of smooth $E_y$-valued functions on $\mathbb R$ satisfying $$p(y,\xi+D_tn)v=0 \tag{CD8}$$ and bounded for $t\geq0$. A reduced boundary system is complementing when $$v\longmapsto\big(b_j(y,\xi+D_tn)v|_{t=0}\big)_{j=1}^J: M^+(y,\xi,n)\longrightarrow\bigoplus_jG_{jy} \tag{CD9}$$ is a bijection. Here $b_j$ is the homogeneous principal symbol of total degree $m_j$, including zero terms when an actual order is smaller than the assigned order. Injectivity and surjectivity are separate requirements.

The matrix polynomial in (CD8) has invertible leading coefficient. Its determinant has no real zero: for real $\tau$, $\xi+\tau n\ne0$, and ellipticity applies. The companion equation and generalized spectral decomposition of Sections 1–3 of Stable modes and the algebra of boundary data therefore split its $mN$-dimensional solution space into $M^+$ and $M^-$. Each component is a finite sum of polynomial factors times $e^{it\lambda}$; the positive space uses $\operatorname{Im}\lambda>0$, and the negative space uses $\operatorname{Im}\lambda<0$. All derivatives of positive modes decay exponentially as $t\to+\infty$, and conversely a bounded mode has no negative component. This proof includes multiple roots and nontrivial Jordan blocks.

The definition does not depend on the representative $\xi$. If $\xi' =\xi+a n$, then $w(t)=e^{-iat}v(t)$ satisfies $(\xi'+D_tn)w=e^{-iat}(\xi+D_tn)v$. The boundary evaluations in (CD9) agree, and boundedness is unchanged. If $n'=c n$, $c>0$, use $w(t)=v(t/c)$; then $cD_tw$ is the pullback of $D_tv$, and again the maps agree. Tangential coordinate changes transport $\xi$ and all homogeneous symbols by the cotangent map. Frame changes multiply the equation and boundary values by invertible fiber maps. These operations preserve bijectivity. A different normal jet convention produces a triangular invertible transformation on the Cauchy vectors, with the corresponding tangential degrees; it therefore gives the same intrinsic test.

For a smoothly varying nonzero tangential covector, the companion matrix varies smoothly. Locally choose contours separating its upper and lower spectra; the resolvent integral gives smooth projections. These local projections agree by spectral uniqueness, so the stable spaces form bundles over the nonzero tangential cotangent space, homogeneous after the jet rescaling. The earlier formulation said that their ranks “need not be imposed as $mN/2$”. The exact qualification is dimension dependent: the reflection proof in Section9 forces half rank in dimension at least three. In dimension two the two rays can have different complementary ranks, as the first worked model shows. When $\dim X=1$, there is no nonzero tangential covector and this microlocal test is empty; the actual boundary Cauchy space is finite dimensional. We do not manufacture a half-rank conclusion in that case.

4. A neighborhood where the calculus is available

We justify the geometric extension rather than assume an invertible double. A smooth inward transverse vector field near $Y$ is obtained by combining inward coordinate vector fields with a partition of unity. Its local flow gives a map $Y\times[0,\epsilon)\to X$. Its differential at $t=0$ is invertible, so the inverse-function theorem gives local collars. A uniform $\epsilon>0$ exists by compactness. Injectivity after shrinking follows as well: otherwise two colliding flow points with times tending to zero have a common limiting boundary point, where local injectivity contradicts the collision. After rescaling we use a single collar coordinate $t$.

Choose a smooth connection on each bundle by patching the local trivial connections. Parallel transport along each collar segment solves a linear ODE with invertible fundamental solution; smooth dependence on its parameters gives an isomorphism between the bundle and the pullback of its boundary restriction. This constructs the normal derivative in (CD2). Attach $Y\times(-\epsilon,0)$ to obtain an open manifold $\widehat X$ containing $X$, and extend each collar bundle by this pullback.

Smooth coefficients extend across the boundary. One explicit local argument prescribes the boundary jets $a_k(y)=\partial_t^ka(y,0)$ on the negative side by $\sum_{k\geq0}a_k(y)t^k\chi(\lambda_kt)/k!$, with $\chi=1$ near zero. Choose $\lambda_k\to\infty$ so that the $k$-th summand, for $k>2r$, is at most $2^{-k}$ in the first $r$ derivatives on each of the first $r$ fixed compact sets. The factor $t^k$ permits this choice for every fixed derivative order below $k$. The resulting series has the prescribed jets and is smooth on the negative side through zero; use the original coefficient on the positive side. Finite partitions and frame transitions extend the operator, without requiring the coefficients of different frames to be extended inconsistently: extend each localized differential operator and sum it.

The extended principal symbol remains invertible after shrinking the added collar. Indeed its smallest singular value has a positive minimum on the unit cotangent sphere above the compact boundary; continuity preserves a positive lower bound nearby. We obtain an elliptic differential extension, still denoted $P$, on $\widehat X$.

4.1. Prescribing every original boundary jet

We supply the details of the extension used in Section 4 of the Cauchy-data lesson. Keep its original compact smooth manifold $X$, boundary $Y$, bundles $E,F$ of common rank $N$, differential operator $P$ of order $m\geq1$, inward time coordinate and convention $D_t=-i\partial_t$. No invertible operator on a closed double is assumed or constructed.

First work in a boundary chart with tangential domain $O\subset\mathbb R^{n-1}$. The following construction applies to any sequence $a_k\in C^\infty(O;\mathbb C^{q\times p})$, with its given matrix norm, including the actual sequence $$a_k(y)=\partial_t^ka(y,0),\qquad k=0,1,\ldots . \tag{CJ1}$$ Choose a fixed smooth compactly supported function $\chi$ on $\mathbb R$, equal to one on a neighborhood of zero, and retain a number $R>0$ with $\operatorname{supp}\chi\subset[-R,R]$. Such cutoffs are constructed with every endpoint constant in the finite-calculus lesson, Section 13.10. Choose compact sets $K_1\subset K_2\subset\cdots\subset O$ whose interiors cover $O$. They can be obtained from the compact exhaustion in (MG1)–(MG3); no bound for the jets outside these compact sets is imposed.

Set $K_0=K_1$. For $j\geq0$, a tangential multi-index $\beta$, and $k\geq0$, the full product derivative of an actual proposed summand is $$\begin{split} \partial_y^\beta\partial_t^j \left(\frac{a_k(y)t^k\chi(\lambda t)}{k!}\right) &= (\partial_y^\beta a_k)(y) \sum_{\ell=0}^{\min(j,k)} \binom j\ell\frac{t^{k-\ell}}{(k-\ell)!} \lambda^{j-\ell}\chi^{(j-\ell)}(\lambda t),\\ \sup_{K_r\times\mathbb R} \left\|\partial_y^\beta\partial_t^j \left(\frac{a_k(y)t^k\chi(\lambda t)}{k!}\right)\right\| &\leq \|\partial_y^\beta a_k\|_{K_r}\lambda^{j-k} \sum_{\ell=0}^{\min(j,k)} \binom j\ell\frac{R^{k-\ell}}{(k-\ell)!} \|\chi^{(j-\ell)}\|_\infty,\qquad\lambda\geq1 . \end{split} \tag{CJ2}$$ For the estimate, each displayed summand vanishes unless $|t|\leq R/\lambda$. No factorial, power of $R$, derivative of $\chi$, or binomial coefficient has been absorbed or omitted.

Let $D_{k,r}$ be the maximum, over all $|\beta|+j\leq r$, of the product of the coefficient bound and the entire finite sum on the last line of (CJ2), using $K_{\max(1,r)}$ for $r=0$. Each $D_{k,r}$ is finite. Set $\lambda_0=1$, and for $k\geq1$ choose the actual number $$\lambda_k= 1+\max\left\{1,k,\ \max_{0\leq r\leq\lfloor(k-1)/2\rfloor} (2^kD_{k,r})^{1/(k-r)}\right\}. \tag{CJ3}$$ This single choice works for every required compact set and derivative order: if $k>2r$ and $|\beta|+j\leq r$, then $$\sup_{K_{\max(1,r)}\times\mathbb R} \left\|\partial_y^\beta\partial_t^j \left(\frac{a_k(y)t^k\chi(\lambda_kt)}{k!}\right)\right\| \leq D_{k,r}\lambda_k^{r-k}\leq2^{-k}. \tag{CJ4}$$ Here $j-k\leq r-k<0$ and $\lambda_k\geq1$. The estimate includes $D_{k,r}=0$; the first two entries in the maximum still give a positive choice.

Define the entire series in the original variable $t$: $$B(y,t)=\sum_{k=0}^{\infty} \frac{a_k(y)t^k\chi(\lambda_kt)}{k!}. \tag{CJ5}$$ For each $r$, all derivatives of total order at most $r$ of the tail $k>2r$ converge uniformly on $K_{\max(1,r)}\times\mathbb R$, and their full bound is $\sum_{k=2r+1}^{\infty}2^{-k}=2^{-2r}$. For a larger fixed compact set, take a larger $r$ also exceeding the required derivative order; its exhaustion set contains that compact set. The remaining finitely many summands are smooth. The uniform differentiation theorem of the finite-calculus lesson, Section 13.7, therefore proves that $B$ is smooth and that all these series are its actual mixed derivatives. For each $t\ne0$, the series is even locally finite: $\lambda_k>k$ forces its $k$-th summand to vanish when $k>R/|t|$.

Because each $\chi(\lambda_kt)$ is identically one near zero, evaluating the termwise derivative there gives exactly $$\partial_y^\beta\partial_t^j B(y,0) =\partial_y^\beta a_j(y). \tag{CJ6}$$ For $k<j$ the polynomial derivative vanishes; for $k>j$ a positive power of $t$ remains; for $k=j$ its factorial cancels precisely and gives $a_j$. All positive derivatives of the cutoff at zero vanish. Thus (CJ5) prescribes an arbitrary smooth sequence of tangential boundary jets, not just a finite list.

For the actual coefficient $a(y,t)$ on $t\geq0$, keep it unchanged and define $$\widehat a(y,t)= \begin{cases} a(y,t),&t\geq0,\\ B(y,t),&t<0. \end{cases} \tag{CJ7}$$ Every mixed derivative on the two sides has the same continuous boundary value by (CJ1) and (CJ6). These values prove smoothness of the pasted function, not merely formal equality of jets. Indeed integrate its proposed continuous normal derivative along an interval crossing zero, separately on each side and add the two oriented integrals. The fundamental theorem of calculus gives the normal derivative at zero as well. Tangential differentiation on $t=0$ differentiates the common smooth function $a_0(y)$. Repeat this argument for each proposed mixed derivative; induction proves all orders and their continuity. The result equals the whole original coefficient on its positive domain.

If all $a_k$ have support in one compact tangential set $K\Subset O$, every summand and the entire $B$ have that same support. A fixed additional tangential cutoff equal to one near $K$ therefore introduces no missing boundary jet. Multiplying the negative-side extension by a fixed normal cutoff equal to one near zero also preserves every jet. These are the support facts used when extending localized differential operators below.

4.2. Constructing one globally injective inward collar

Choose finitely many boundary charts covering $Y$. Let $s_a\geq0$ be their original inward coordinate, and let $V_a$ be the corresponding inward coordinate vector field. A finite smooth partition $\phi_a$, with compact supports inside these charts and sum one on a neighborhood of $Y$, gives the actual field $$V=\sum_a\phi_a V_a . \tag{CJ8}$$ Such partitions are constructed by compactly supported chart cutoffs divided by their positive finite sum, with all denominator derivatives given by (BG1)–(BG2). At a boundary point any inward conormal $\ell$ satisfies $\ell(V_a)>0$ for every active chart. The positive direction of the one-dimensional normal quotient is unchanged by a smooth boundary chart change; its two inward defining-coordinate differentials differ there by a positive factor. Hence $\ell(V)=\sum_a\phi_a\ell(V_a)>0$. This proves transversality of the actual patched field.

To apply the open-domain flow theorem near a boundary point, extend the coordinate coefficients of this same field locally across $s_a=0$ by (CJ7). The extensions change no positive-side coefficient or boundary derivative. Shrink the coordinate neighborhood so that $\partial_Vs_a\geq c_a>0$, with its actual $c_a$, and so that a compact smaller boundary neighborhood lies inside it. The full original nonlinear flow theorem (NF1)–(NF21) gives a common small existence interval for this compact set. On it the fundamental theorem of calculus gives $$s_a(\Phi_t(y))=\int_0^t (\partial_Vs_a)(\Phi_\tau(y))\,d\tau \geq c_at,\qquad 0\leq t\leq\epsilon_a . \tag{CJ9}$$ The bounded-vector-field estimates in (NF1)–(NF3) keep these trajectories within the chosen chart on this time interval. Thus they stay on the original side and lie in the interior for $t>0$. Different local negative extensions produce the same positive trajectories: on that side they solve the same original equation and have the same initial value, so the uniqueness theorem identifies them on their common interval. A finite cover of compact $Y$ gives one $\epsilon_0>0$. Smooth parameter dependence proves the smooth map $\Phi(y,t)=\Phi_t(y)$ on $Y\times[0,\epsilon_0)$.

At the boundary its differential is the exact map $$d\Phi_{(y,0)}:T_yY\oplus\mathbb R\longrightarrow T_yX, \qquad (w,a)\longmapsto w+aV(y). \tag{CJ10}$$ It is bijective because $V(y)$ is transverse. Applying the original inverse theorem (IV1)–(IV10) to the local extended map proves a local diffeomorphism, and restricting to $t\geq0$ gives a local collar. In finitely many smaller boundary neighborhoods, continuity of the differential gives a common positive time interval on which it remains invertible. All restrictions concern the same original $\Phi$; no rescaling or replacement of its time variable or field is used.

Global injectivity after a further shrinking has a direct compactness proof. If it failed on every interval $[0,1/j)$ contained in the existing one, there would be distinct $(y_j,t_j)$ and $(z_j,u_j)$, with $0\leq t_j,u_j<1/j$, having equal images. Extract convergent subsequences of both boundary sequences, with limits $y,z\in Y$. Continuity and equality of the images give $y=z$. Both parameter points then lie, for all large $j$, in the same inverse-function neighborhood of $(y,0)$. Its injectivity contradicts their distinctness. Therefore some $0<\epsilon<\epsilon_0$ gives an injective local diffeomorphism on all of $Y\times[0,\epsilon)$. Its image is a relative open neighborhood of $Y$ in $X$; its local smooth inverse agrees on overlaps by injectivity. This is the required global collar with the exact original inward time.

4.3. Bundle transport and the actual ambient manifold

On each of $E$ and $F$, use a finite chart partition on $X$ to patch the local trivial connections: $$\nabla=\sum_a\psi_a\nabla^{(a)},\qquad \nabla(fs)=df\otimes s+f\nabla s,\qquad\sum_a\psi_a=1. \tag{CJ11}$$ The last equality verifies the Leibniz rule in the middle formula, so the sum is a connection on the original bundle. In a frame along a collar segment, let $A(y,t)$ be its connection matrix evaluated on the actual $V$. Parallel transport and its inverse satisfy $$Y'=-AY,\quad Y(y,0)=I_N,\qquad Z'=ZA,\quad Z(y,0)=I_N,\qquad ZY=YZ=I_N. \tag{CJ12}$$ The complete ordered integral, parameter derivative, uniqueness and inverse calculations (NF6)–(NF10) apply to these original coefficients. They prove smoothness in $y,t$ and every required finite collection of compact bounds. No scalar exponential or commutation of matrix factors is used.

To check the fiber map, if the old frame components equal $T$ times the new components, direct differentiation of that equality gives $A^T=T^{-1}AT+T^{-1}\partial_tT$. The transformed fundamental matrix is $$Y^T(y,t)=T(\Phi(y,t))^{-1}Y(y,t)T(y). \tag{CJ13}$$ The product rule proves its equation and its initial value; uniqueness proves that it is the constructed transport in the new frame. For a second frame change $R$, the full composite map is $$R(\Phi(y,t))^{-1}T(\Phi(y,t))^{-1}Y(y,t)T(y)R(y) =(T(\Phi(y,t))R(\Phi(y,t)))^{-1} Y(y,t)(T(y)R(y)). \tag{CJ13a}$$ This gives the actual ordered cocycle separately for $E$ and for $F$. These maps therefore glue and give smooth fiber isomorphisms $\mathcal T^E_{y,t}:E_y\to E_{\Phi(y,t)}$ and $\mathcal T^F_{y,t}:F_y\to F_{\Phi(y,t)}$. Their constructed inverses give the inverse bundle identifications. They are precisely the identifications with the two pullback bundles used to define the normal derivatives in (CD2).

Attach $Y\times(-\epsilon,0)$ by this collar. More explicitly, take the quotient of $X\sqcup(Y\times(-\epsilon,0])$ identifying $(y,0)$ with its original boundary point. Its charts are the original interior charts and product charts on $Y\times(-\epsilon,\epsilon)$, using $\Phi$ on the positive part. They agree smoothly because $\Phi$ is the original collar diffeomorphism. They make a manifold without boundary $\widehat X$ containing the original $X$.

The quotient is Hausdorff. Distinct points in the interior of either side have disjoint original neighborhoods; an interior point can also be separated from the other side by a neighborhood away from the seam. Two points on the seam are separated using disjoint boundary neighborhoods and their two-sided product neighborhoods. A seam point and an interior point are separated by shrinking a product neighborhood about the seam point. These neighborhoods form the quotient topology. Countable original atlases on $X$ and $Y$, together with countable interval bases in the product charts, give a countable base. Thus no separation or countability property is assumed for an unidentified gluing.

Use the pullbacks of $E|_Y,F|_Y$ on the negative product and the actual maps $\mathcal T^E,\mathcal T^F$ on the positive product. Their chart transitions and (CJ13) satisfy the complete smooth bundle cocycle. The quotient bundle construction (BG3)–(BG6) gives the two smooth extended bundles $\widehat E,\widehat F$, retaining the original ones over $X$. We continue to use their original names in the receiving lesson.

4.4. Extending every differential coefficient with its support

Choose a finite smooth partition on $X$ subordinate to smaller bundle and coordinate charts, with each support compactly contained in its chart. Write the exact localized operator as an output multiplication: $$P=\sum_a\psi_a P,\qquad \psi_a P=\sum_{|\alpha|\leq m}A_{a,\alpha}(y,t)D^\alpha . \tag{CJ14}$$ Both bundles keep their original, possibly different, frames; $A_{a,\alpha}$ maps the local $E$ component fiber to the local $F$ component fiber. The convention $D^\alpha=(-i)^{|\alpha|}\partial^\alpha$ and every original coefficient are retained. Since the partition multiplies the output, no derivative of $\psi_a$ has been dropped from a different operator $P\psi_a$.

For a chart meeting the boundary, the normal jets of every coefficient in (CJ14) are compactly supported in one tangential set inside that chart. Extend each full matrix coefficient by (CJ1)–(CJ7), including every lower-order term. Multiply the negative-side extension by fixed chart cutoffs equal to one on a neighborhood of this support and of $t=0$. Their constant values there preserve every original jet. Extending by zero across the artificial chart edges is smooth, because these cutoffs and the compact tangential supports vanish on neighborhoods of those edges. This defines a differential operator on the extended bundles using its chosen local frames; outside that chart it is zero. An interior localized operator already vanishes near $Y$ and extends by zero on the negative collar.

Sum the finitely many extended localized operators. On $X$ this sum is exactly $\sum_a\psi_aP=P$, and along $Y$ every coefficient jet agrees with the original one. On overlaps, each localized term is an actual differential map between the extended bundles, so their sum is one intrinsic operator $\widehat P:C^\infty(\widehat X;\widehat E)\to C^\infty(\widehat X;\widehat F)$. We have not tried to extend the coefficient tables in two different frames independently and then declare them equal. The frame transitions act on these actual local operators with all their product-rule terms, while their positive restrictions are the original localized maps.

The same jet construction extends the original smooth density and the original Hermitian and cotangent metrics near $Y$. Their positivity persists after a shrinking, because on compact $Y$ their smallest eigenvalues have positive minima. This argument concerns the entire extended matrices; it does not discard their off-diagonal entries or replace the original positive-side metrics.

4.5. The full ellipticity bound and the receiving boundary distributions

In the original collar identifications, let $$\widehat p(y,t,\xi) =\sum_{|\alpha|=m}\widehat A_\alpha(y,t)\xi^\alpha, \qquad \widehat p(y,0,\xi)=p(y,0,\xi). \tag{CJ15}$$ These are the full ordered matrix principal symbols of the summed operators. Let the subscript $0$ on a norm denote the original norm at $t=0$, transported only for this comparison by the fixed collar identifications. The boundary unit cotangent sphere and unit $E$-fiber sphere are compact. Original ellipticity and continuity therefore give an actual $\mu>0$ such that $\|p(y,0,\xi)e\|_{F,0}\geq\mu\|\xi\|_{\xi,0}^m\|e\|_{E,0}$ for every boundary covector and fiber vector. For nonzero covectors and vectors this follows from their unit-sphere values by the full degree-$m$ homogeneity and linearity; at zero it is the displayed zero bound.

Over finitely many compact collar charts the fundamental theorem of calculus retains every coefficient difference: $$\begin{split} \widehat p(y,t,\xi)-p(y,0,\xi) &=\sum_{|\alpha|=m} \left(\int_0^t\partial_s\widehat A_\alpha(y,s)\,ds\right) \xi^\alpha,\\ \|(\widehat p(y,t,\xi)-p(y,0,\xi))e\|_{F,0} &\leq C|t|\|\xi\|_{\xi,0}^m\|e\|_{E,0}. \end{split} \tag{CJ16}$$ Here one may take $C$ to be the maximum over those charts of $\sum_{|\alpha|=m} \sup\|\partial_s\widehat A_\alpha\|_{E,0\to F,0} \sup_{\|\xi\|_{\xi,0}=1}|\xi^\alpha|$. Both suprema are taken over the actual compact chart families. The sum, all original monomials and the original matrix map norms remain. No assertion of ellipticity of each localized partition term is needed.

Choose $0<\delta<\epsilon$ such that $C\delta\leq\mu/2$; if $C=0$, any smaller positive $\delta$ works. The triangle inequality now gives $$\|\widehat p(y,t,\xi)e\|_{F,0} \geq(\mu-C|t|)\|\xi\|_{\xi,0}^m\|e\|_{E,0} \geq\frac\mu2\|\xi\|_{\xi,0}^m\|e\|_{E,0}, \qquad -\delta<t\leq0. \tag{CJ17}$$ This implies injectivity for every nonzero covector. The domain and codomain fibers have the same original rank $N$, so the full rank-nullity theorem gives bijectivity.

The actual extended norms also keep their comparison factors. On this compact family choose constants $\|f\|_{F,t}\geq c_F\|f\|_{F,0}$, $\|e\|_{E,t}\leq C_E\|e\|_{E,0}$, and $\|\xi\|_{\xi,t}\leq C_\xi\|\xi\|_{\xi,0}$, with $c_F>0,C_E>0,C_\xi>0$. Their existence follows by continuity and positivity on the corresponding compact original unit spheres. Equation (CJ17) consequently retains the bound in the actual extended metrics: $$\|\widehat p(y,t,\xi)e\|_{F,t} \geq\frac{c_F}{C_EC_\xi^m} (\mu-C|t|)\|\xi\|_{\xi,t}^{\,m}\|e\|_{E,t}. \tag{CJ18}$$ Thus the comparison norms used to prove persistence have not replaced the original working metrics. On the positive side the extended operator is still exactly the given elliptic $P$. Restrict the ambient manifold to $X^\circ\cup(Y\times(-\delta,\epsilon))$; it contains all of $X$ and the extended operator is elliptic everywhere there.

If $Y=\varnothing$, there is no collar work: take $\widehat X=X$ and the original bundles and operator. In dimension zero there are no nonzero covectors, so no unit cotangent minimum is asserted. If the common bundle rank is zero, the unique map of the zero-dimensional fibers is invertible and the extension is vacuous; no positive smallest singular value on an empty unit fiber sphere is invented. Otherwise the positive minima above apply exactly as stated.

The bundle parametrix construction Section 5 of Detecting regularity without choosing coordinates, Section 6 of Detecting regularity without choosing coordinates and Section 13 of Detecting regularity without choosing coordinates gives a properly supported classical operator $$T\in\Psi^{-m}(\widehat X;F,E),\qquad TP=I_E+R_E,\qquad PT=I_F+R_F, \tag{CD10}$$ with smooth properly supported remainder kernels. We may work in a relatively compact neighborhood of $X$; each support assertion below is localized there. Neither $T$ nor $P$ is assumed invertible. In particular we have not asserted that this differential extension exists elliptically on a closed double.

5. Transmission of the entire inverse symbol

The parametrix in (CD10), and its adjoint for the chosen smooth pairings, have ordinary transmission across $Y$. Here is the all-order verification. Let $p_m,p_{m-1},\ldots,p_0$ be the homogeneous polynomial pieces of the differential symbol. The local inverse starts with $t_{-m}=p_m^{-1}$. The recursion obtained by equating homogeneous terms in $p\#t=I$ expresses each $t_{-m-r}$ as a finite sum of products of $p_m^{-1}$, polynomial symbol pieces, and their base and frequency derivatives. In every product the order of the matrix factors is retained. It is rational in the frequency and homogeneous of integer degree $-m-r$.

A homogeneous rational function $h$ of integer degree $d$ has $h(-\xi)=(-1)^dh(\xi)$ wherever its denominator is nonzero, and differentiating gives $$\partial_x^\beta\partial_\xi^\alpha t_{-m-r}(y,0,-n) =(-1)^{-m-r-|\alpha|} \partial_x^\beta\partial_\xi^\alpha t_{-m-r}(y,0,n). \tag{CD11}$$ This includes every normal base jet and tangential frequency jet. The same parity also follows term by term in the inverse recursion, including derivatives of $p_m^{-1}$. Asymptotic summation leaves only smoothing ambiguity. Section 10 of Singularities along a submanifold and smooth boundary passage now proves ordinary transmission. The product and adjoint expansions preserve (CD11): a frequency derivative lowers both degree and parity by one, base derivatives preserve both, and degrees add under ordered multiplication. Transposition and conjugation do not alter the real signs. Therefore the adjoint $T^*$ also has ordinary transmission. The adjoint acts on the appropriate dual density bundles; identifying these through the chosen metrics introduces only smooth degree-zero factors.

Consequently $r^+Te^+$ and $r^+T^*e^+$ map smooth sections continuously to sections smooth up to the boundary. Here $e^+$ denotes zero extension and $r^+$ restriction from $\widehat X$ to $X$. The continuity is in all smooth seminorms on fixed compact sets: the transmission proof estimates each output seminorm with finitely many input derivatives and symbol seminorms. Section 13 of Singularities along a submanifold and smooth boundary passage also supplies all one-sided traces of boundary layers and all classical remainder orders, not merely their principal symbols.

6. A jump as a triangular distributional operator

For a smooth section $u$ in the collar, distributional differentiation gives $$D_t e^+u=e^+D_tu+i^{-1}(\gamma_0u)\delta(t). \tag{CD12}$$ For a test function this is integration by parts on $[0,\infty)$: the endpoint term in $\partial_te^+u$ is $+u(0)\delta$, and multiplication by $-i=i^{-1}$ fixes the sign. Induction, applying (CD12) once more to the regular term and differentiating the already supported terms, gives $$D_t^ae^+u=e^+D_t^au+i^{-1}\sum_{k=0}^{a-1}(\gamma_{a-1-k}u)D_t^k\delta, \qquad a\geq1. \tag{CD13}$$ Write $P=\sum_{a=0}^mP_a(t,y,D_y)D_t^a$. Define the boundary distribution operator $$\mathcal JU=i^{-1}\sum_{l=0}^{m-1}\sum_{j=0}^{m-1-l} P_{j+l+1}(t,y,D_y)\big(U_l(y)D_t^j\delta(t)\big). \tag{CD14}$$ The coefficient $P_{j+l+1}$ in this formula acts after multiplication by the derivative of $\delta$. Its normal-variable coefficients must not silently be replaced by boundary values: multiplication by $D_t^j\delta$ contains the corresponding lower normal derivatives of those coefficients. Formula (CD13) proves the exact identity $$Pe^+u=e^+(Pu)+\mathcal J\gamma u. \tag{CD15}$$ The map $\mathcal J:\mathcal D'(Y;E^m)\to\mathcal E'(\widehat X;F)$ is continuous. Indeed its pairing with a test section is a finite sum of pairings of $U_l$ with smooth tangential differential expressions in at most $m-1$ normal derivatives of that test section and of the coefficients. Its support is on the compact boundary. Formula (CD15) is independent of local charts because its left side and its regular term are intrinsic. It thereby identifies the patched operator (CD14) without a choice of density hidden in the normal delta.

7. A Green potential and its boundary operator

For smooth Cauchy data define $$KU=r^+T\mathcal JU,\qquad QU=\gamma KU. \tag{CD16}$$ The potential $KU$ is smooth up to $Y$, by the layer form of transmission in Section 13 of Singularities along a submanifold and smooth boundary passage. In the interior it satisfies $$PKU=r^+R_F\mathcal JU, \tag{CD17}$$ since the distribution $\mathcal JU$ itself is supported on $Y$. Applying $T$ to (CD15) gives the exact Green representation $$e^+u+R_Ee^+u=Te^+(Pu)+T\mathcal J\gamma u. \tag{CD18}$$ In particular, with $V=\gamma r^+Te^+$ and $S=\gamma r^+R_Ee^+$, $$\gamma u+Su=V(Pu)+Q\gamma u. \tag{CD19}$$ These identities initially concern smooth $u$. Restriction precedes every one-sided boundary trace. Unless the remainders vanish, $K$ is a solution operator modulo a smooth error and $Q$ is an approximate projection. Neither statement identifies the range of the full $Q$ exactly with the traces of all global null solutions.

Changing the proper two-sided parametrix changes $T$ by a smooth kernel near the compact working set: apply the comparison identity in Section 6 of Detecting regularity without choosing coordinates. Hence $K$ changes by a smooth kernel from $Y$ to $X$, and $Q$ changes by a smooth boundary kernel. Changing the extension outside a neighborhood of $X$ has the same conclusion near $Y$, by the local parametrix uniqueness. Such statements do not make the exact finite-dimensional global solution spaces independent of the extension.

8. The complete weighted matrix and its residues

In collar coordinates the $(k,l)$-entry of $Q$ is a sum over $0\leq j\leq m-1-l$ of the one-sided traces $$Q_{kl}U_l=i^{-1}\sum_{j=0}^{m-1-l} \gamma_0D_t^kT P_{j+l+1}(U_lD_t^j\delta). \tag{CD20}$$ The full operator before the simple layer has classical order at most $k-m+(m-j-l-1)+j=k-l-1$. Derivatives hitting coefficients only lower the relevant homogeneous order. Transmission is preserved by composition with differential operators, either by (CD11) or by Section 11 of Singularities along a submanifold and smooth boundary passage. The layer trace theorem raises the degree by one through normal integration. Consequently $$Q_{kl}\in\Psi^{k-l}(Y;E|_Y,E|_Y),\qquad Q:\mathcal H_s\longrightarrow\mathcal H_s \quad\text{continuously for every }s\in\mathbb R. \tag{CD21}$$ This all-real-order assertion concerns a boundary pseudodifferential matrix; it is not an assertion about the trace of arbitrary interior distributions. It follows entry by entry from the closed-manifold Sobolev theorem Section 11 of Detecting regularity without choosing coordinates. The same argument gives the full classical expansions with the remainder in entry $(k,l)$ lowered by every prescribed integer. A finite matrix does not lose any summability or uniformity in combining these estimates.

Write the principal differential polynomial at the boundary as $p(y,\eta,\zeta)=\sum_{a=0}^mp_a(y,\eta)\zeta^a$, for $\eta\ne0$. Let $\Gamma_+$ be a positively oriented contour enclosing all its determinant roots in the upper half-plane and none in the lower half-plane. The leading entry is $$q_{kl}(y,\eta)=\frac1{2\pi i}\int_{\Gamma_+} \sum_{j=0}^{m-1-l}\zeta^{k+j} p(y,\eta,\zeta)^{-1}p_{j+l+1}(y,\eta)\,d\zeta. \tag{CD22}$$ It is the sum of upper-half-plane residues of this matrix-valued rational function. The factor comes from $i^{-1}$ in (CD14), the normal Fourier normalization $(2\pi)^{-1}$, and the contour factor $2\pi i$. In particular the inverse symbol acts on the left of the coefficient $p_{j+l+1}$; reversing their order generally changes the answer.

For high normal powers an ordinary improper real integral in (CD22) would not converge. The formula is instead the upper contour value of the analytically subtracted integral in Section 12 of Singularities along a submanifold and smooth boundary passage and Section 13 of Singularities along a submanifold and smooth boundary passage. Polynomial terms at normal-frequency infinity have inverse Fourier transform supported at $t=0$; their positive-side traces vanish and their closed contour integral vanishes. The rational proper part is evaluated by residues. This is exactly why (CD22) still applies to every $k,l$, without a false absolute-convergence assertion.

Positive rescaling $(\eta,\zeta)\mapsto(r\eta,r\zeta)$ in the contour gives $q_{kl}(y,r\eta)=r^{k-l}q_{kl}(y,\eta)$. The contours may be chosen uniformly in a small parameter neighborhood; differentiation under their integrals proves smoothness in $(y,\eta)$, including at collisions of roots within a half-plane. The invariant principal symbol is therefore a weighted endomorphism of the Cauchy bundle. In dimension one $Y$ is a finite set and every boundary operator has a smooth kernel; (CD22) has no nonzero tangential frequency to address.

9. Identifying the projection by a jump, without choosing roots

Fix $(y,\eta)$ with $\eta\ne0$. Suppress these parameters and take $U\in E_y^m$. The rational function $$a_U(\zeta)=i^{-1}p(\zeta)^{-1} \sum_{l=0}^{m-1}\sum_{j=0}^{m-1-l}p_{j+l+1}U_l\zeta^j \tag{CD23}$$ is smooth on the real axis and is $O(|\zeta|^{-1})$ at infinity. Let $v=\mathcal F^{-1}_{\zeta\to t}a_U$ in tempered distributions. Then $$p(D_t)v=i^{-1}\sum_{l+j<m}p_{j+l+1}U_lD_t^j\delta. \tag{CD24}$$ For $t>0$, contour closure in the upper half-plane expresses $v$ as a finite sum of exponential-polynomial terms associated with upper roots. For $t<0$, closure in the lower half-plane gives the lower-root terms. A rigorous closure first subtracts the rational partial fractions, so that each term is a power of $(\zeta-\lambda)^{-1}$; its inverse follows from the residue formula with $e^{it\zeta}$. The possible $\zeta^{-1}$ tail is thereby included without discarding its jump. Thus $v=v^+$ on the positive half-line and $v=v^-$ on the negative half-line, where $v^\pm\in M^\pm$. This also proves the existence of all one-sided derivatives.

Let $W_k=D_t^kv^+(0)-D_t^kv^-(0)$. Applying the same differentiation calculation as (CD13) to a two-sided piecewise smooth function shows that the singular part of $p(D_t)v$ is $i^{-1}\sum_{l+j<m}p_{j+l+1}W_lD_t^j\delta$. Compare it with (CD24). The coefficient of $D_t^{m-1}\delta$ gives $p_m(W_0-U_0)=0$. Next the coefficient of $D_t^{m-2}\delta$ gives $p_m(W_1-U_1)=0$, because the term involving $W_0-U_0$ has vanished. Continuing down this triangular system proves $$D_t^kv^+(0)-D_t^kv^-(0)=U_k,\qquad0\leq k<m. \tag{CD25}$$ No commutation or scalar factorization occurs in this argument.

For positive $t$, differentiate the residue expression for $v^+$ and let $t\downarrow0$. Formula (CD22) gives $\gamma v^+=qU$. Equation (CD25) then gives $\gamma v^-=(q-I)U$. Initial-data uniqueness for the companion equation identifies the solution space with $E_y^m$, and the spectral decomposition is direct. Hence $$E_y^m=\gamma M^+\oplus\gamma M^-,\qquad q|_{\gamma M^+}=I,\qquad q|_{\gamma M^-}=0. \tag{CD26}$$ For example, if $U=\gamma w^+$ with $w^+\in M^+$, (CD25) compares the direct-sum decomposition $U=\gamma v^+-\gamma v^-$ with $U=\gamma w^++0$, forcing $v^+=w^+$ and $v^-=0$. The negative case is the same direct-sum uniqueness with the opposite component. Thus $q^2=q$. It projects along the negative modes, and is generally not an orthogonal projection for an arbitrarily chosen Hermitian metric.

The proof also explains the complementing test: (CD9) is bijectivity of the principal boundary matrix restricted to $\operatorname{ran}q$, with the component weights of (CD2) and (CD6). No inverse for the full rectangular boundary matrix is requested.

Editorial strengthening: the exact antipodal correspondence. The original homogeneous differential polynomial relates the two opposite tangential covectors by reflection of the normal solution. The complete argument below gives the solution map, every jet, the weighted projection, its bundle inverse and all dimensional qualifications. The full Green operator retains its distinct lower-order and smoothing argument.

1. The full polynomial and the actual solution map

Fix $y\in Y$, the original positive integer $m$, bundles $E,F$ of the same finite rank $N$, the original inward conormal $n$, and a representative $\xi$ whose class modulo $\mathbb R n$ is nonzero. Write the original principal polynomial, with every coefficient acting from $E_y$ to $F_y$, as $$p_y(\xi+\zeta n)=\sum_{a=0}^{m}p_a(y,\xi,n)\zeta^a. \tag{AP1}$$ Every original principal monomial has total degree $m$. Thus its contribution to the coefficient of $\zeta^a$ has degree $m-a$ in $\xi$. Termwise evaluation, keeping each matrix and each monomial, gives $$p_a(y,-\xi,n)=(-1)^{m-a}p_a(y,\xi,n),\qquad p_y(-\xi+\zeta n)=(-1)^m p_y(\xi-\zeta n). \tag{AP2}$$ These equations use real covector reversal, not complex conjugation. No coefficient or factor order is changed. The leading coefficient $p_m=p_y(n)$ is invertible. For real $\zeta$, the covector $\xi+\zeta n$ cannot vanish, so original ellipticity excludes real determinant roots. The original companion equation therefore has $mN$ independent initial coordinates and the direct sum of its upper and lower generalized modes, as proved in (CD8), (CD26) and the stable-mode provider.

For a full smooth solution on $\mathbb R$, put $(\mathscr Rv)(t)=v(-t)$. Direct differentiation at every order gives $$\begin{gathered} D_t^a\mathscr Rv=(-1)^a\mathscr R(D_t^av),\\ p_y(-\xi+D_tn)\mathscr Rv =(-1)^m\mathscr R\big[p_y(\xi+D_tn)v\big],\\ \mathscr R^2=I. \end{gathered} \tag{AP3}$$ In the middle equation the coefficient of each term is the unchanged ordered coefficient from AP1 multiplied by $(-1)^{m-a}(-1)^a$; their product is exactly $(-1)^m$. This proves a bijection of the full original solution spaces in both directions. A lower mode is a finite sum of polynomial factors times $e^{it\lambda}$, with $\operatorname{Im}\lambda<0$; replacing $t$ by $-t$ reflects every polynomial factor and replaces the exponential by $e^{it(-\lambda)}$. It now decays, with every derivative, as $t\to+\infty$. The inverse reflection proves surjectivity onto the positive modes. Upper modes transform into lower modes by the same calculation. Multiplicities and Jordan factors are all retained.

2. The exact Cauchy projection and a full bundle isomorphism

Keep every Cauchy coordinate $U_k=D_t^kv(0)$, $0\leq k<m$, and write $$\begin{gathered} \mathcal J_y=E_y^{\oplus m},\qquad S_y=\operatorname{diag}(I_{E_y},-I_{E_y},\ldots,(-1)^{m-1}I_{E_y}),\\ \gamma\mathscr Rv=S_y\gamma v,\qquad S_y^2=I. \end{gathered} \tag{AP4}$$ The Cauchy map is a bijection by the original companion initial-value theorem. Define $V_y^\pm(\xi,n)=\gamma M^\pm(y,\xi,n)$. AP3 and AP4 consequently give exact linear bijections $$S_y:V_y^-(\xi,n)\longrightarrow V_y^+(-\xi,n),\qquad S_y:V_y^+(\xi,n)\longrightarrow V_y^-(-\xi,n). \tag{AP5}$$ Let $q_y(\xi,n)$ be the original weighted residue projection (CD22). By (CD26), its image is $V_y^+(\xi,n)$ and its kernel is $V_y^-(\xi,n)$. The map $S_y(I-q_y(\xi,n))S_y$ is identity on $V_y^+(-\xi,n)$ and zero on $V_y^-(-\xi,n)$, using AP5 and $S_y^2=I$. Their direct sum is the whole original jet space; equality on the two summands proves $$\begin{gathered} q_y(-\xi,n)=S_y(I-q_y(\xi,n))S_y,\\ q_{kl}(y,-\xi,n)=(-1)^{k+l}\big(\delta_{kl}I_{E_y}-q_{kl}(y,\xi,n)\big). \end{gathered} \tag{AP6}$$ This argument uses no orthogonality, scalar factorization or simple-root assumption. It also preserves every weighted entry order $k-l$; the diagonal identity in the second formula has order zero only when $k=l$.

Choose the existing collar splitting to represent each nonzero tangential covector $\eta$, and let $Z=T^*Y\setminus0$, $\pi:Z\to Y$, $\mathfrak a(y,\eta)=(y,-\eta)$. The smooth mode bundles $V^\pm$ on $Z$ are supplied by the local contours in Section3. The fiber matrices $S_y$ are a smooth bundle map of $\pi^*(E|_Y)^{\oplus m}$. AP5 proves $\mathfrak a^*V^+\simeq V^-$, with map $w\mapsto S_yw$ and inverse $S_y$. More strongly, the entire original jet bundle is recovered by the following mutually inverse smooth maps: $$\begin{gathered} V^+\oplus\mathfrak a^*V^+\longrightarrow\pi^*(E|_Y)^{\oplus m},\\ (u,w)\longmapsto u+S_yw,\\ \pi^*(E|_Y)^{\oplus m}\longrightarrow V^+\oplus\mathfrak a^*V^+,\\ v\longmapsto\big(q_y(\eta)v,S_y(I-q_y(\eta))v\big). \end{gathered} \tag{AP7}$$ The second component lies in $V_y^+(-\eta)$ by AP5. Composing in the jet space gives $qv+(I-q)v=v$. Composing in the direct sum gives $q(u+Sw)=u$ and $S(I-q)(u+Sw)=w$, since $u\in V^+$ and $Sw\in V^-$. Smoothness follows from the actual smooth projection, with no choice of individual roots or frames. These maps give a bundle relation even when ranks on the two antipodal components differ.

3. Original representative, normal scale, frame and frequency comparisons

For $\xi'=\xi+a n$, where $a\in\mathbb R$, the original solution map is $G_av=e^{-iat}v$. The complete product rule gives the triangular jet matrix $$(W_aU)_k=\sum_{l=0}^k\binom{k}{l}(-a)^{k-l}U_l, \qquad \gamma G_av=W_a\gamma v,\qquad W_a^{-1}=W_{-a}. \tag{AP8}$$ The inverse identity follows from $G_aG_{-a}=I$ and bijectivity of the Cauchy map; every displayed coefficient is retained. The positive and negative spaces both transform by $W_a$, so $q(\xi+a n,n)=W_aq(\xi,n)W_{-a}$. Entrywise the full sign comparison is $$(-1)^k\binom{k}{l}(-a)^{k-l} =\binom{k}{l}a^{k-l}(-1)^l, \qquad S W_a=W_{-a}S. \tag{AP9}$$ At the negative covector the representative changes by $-a n$. Hence AP9 proves that AP5–AP7 are transported by the actual two original representative maps; they do not require replacing the original representative by a preferred one.