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\begin{document}

\section{Stable modes and the algebra of boundary
data}\label{stable-modes-and-the-algebra-of-boundary-data}

A frozen boundary equation asks which normal profiles remain bounded
inside the domain, and which boundary measurements determine those
profiles. For systems, the number of profiles counts algebraic
multiplicity. Eigenvectors alone can miss solutions. This lesson
constructs the bounded solution space from Cauchy data, then changes the
equation while preserving that space and its boundary measurements.

Section 13 of
\href{metric-foundation-bridges.pdf\#section*.74}{Metric
and topological foundations} proves the full original scalar and
finite-coordinate calculus. Its Section 13.12 gives the exact Gaussian,
Rayleigh, logarithmic-contour, matrix and half-line receiving maps,
retaining every original factor, endpoint and norm.

The complete original real-number, compactness, extrema and finite-norm
proofs are in Section 12 of
\href{metric-foundation-bridges.pdf\#section*.13}{Metric
and topological foundations}. Its Section 12.10 gives the exact
receiving minimum and spectral-contour maps; the original coordinates,
norms and constants are retained.

Throughout, vector spaces are finite-dimensional and complex,
coefficients are constant in the normal variable, and

\[
D=-i\frac{d}{dt},\qquad t\in\mathbb R.
\]

We call a solution \emph{stable} when it is bounded for \(t\geq0\).
Under the no-real-root hypotheses below, this is equivalent to
exponential decay of the solution and every derivative. No
diagonalizability, simplicity of roots, commutation of matrix
coefficients, or prescribed number of boundary equations is assumed.

\href{stable-prerequisite-bridges.pdf}{Polynomial and contour interfaces
for stable boundary models} proves scalar polynomial division over
\(\mathbb C\), Bézout, Cayley--Hamilton, complex triangularization, and
the contour moments for globally convergent power-series numerators with
specified spectral indices. That lesson now proves the complex-root
theorem and the full determinant, kernel and invariant-quotient maps in
Section 9. Sections 10.1--10.9 there supply the full finite bases and
direct sums, projection rank and trace, matrix algebra,
original-inner-product adjoints and operator norms. Compactness of the
original unit sphere retains its separate topological base. Section 16
below makes the original annihilator and Cauchy receiving maps explicit.
We also use its locally uniform exponential-series calculus, termwise
differentiation and integration under uniform convergence, and dominated
differentiation of exponentially decreasing tails. The spectral and ODE
consequences needed here are proved below.

\subsection{1. Spectral calculus without an
eigenbasis}\label{spectral-calculus-without-an-eigenbasis}

Let \(V\) be a finite-dimensional complex vector space and
\(A\in\operatorname{End}(V)\). Choose any norm. The series

\[
E_A(t)=\sum_{k=0}^{\infty}\frac{(itA)^k}{k!}
\]

converges locally uniformly with all derivatives, since its norms and
the norms of its differentiated terms are bounded by scalar exponential
series. Thus \(DE_A=AE_A\), \(E_A(0)=I\), and multiplication of
absolutely convergent series gives \(E_A(-t)E_A(t)=I\). If \(Du=Au\),
differentiating \(E_A(-t)u(t)\) shows that it is constant. Consequently

\[
u(t)=E_A(t)v                                                 \tag{S1}
\]

is the unique solution with \(u(0)=v\). In particular the solution space
has dimension \(\dim V\).

Here is the spectral information, including the nilpotent terms. Keep an
original least-degree nonzero annihilator \(m\) with leading coefficient
\(c_m\ne0\). Its named monic comparison has factorization
\(\prod_{\zeta}(z-\zeta)^{\nu_\zeta}\); the original polynomial is
\(m=c_m\prod_{\zeta}(z-\zeta)^{\nu_\zeta}\), with every unit and
multiplicity retained. Section 16.1 proves the full scalar-unit and
polynomial evaluation maps. The spectral label \(\lambda\) used there
denotes the same actual root as \(\zeta\) here, through the identity
\(\lambda=\zeta\). The factors are pairwise relatively prime. Repeated
Bézout identities give polynomials \(e_\zeta\) which are one modulo
\((z-\zeta)^{\nu_\zeta}\) and zero modulo every other factor. Their sum
is one modulo the named comparison, while the full original calculation
uses \(M_\zeta=m/(z-\zeta)^{\nu_\zeta}=c_mH_\zeta\) and
\(e_\zeta=(c_m^{-1}b_\zeta)M_\zeta\). Formula (SR2) retains both
original scalar factors. Evaluation at \(A\) therefore gives projections

\[
Q_\zeta=e_\zeta(A),\quad
Q_\zeta Q_\eta=0\ (\zeta\ne\eta),\quad
\sum_\zeta Q_\zeta=I.
\]

Their images are exactly

\[
V_\zeta=\ker(A-\zeta I)^{\nu_\zeta}.
\]

Indeed, the polynomial congruences show both inclusion in this kernel
and that \(Q_\zeta\) is the identity there. Thus
\(V=\bigoplus V_\zeta\). On \(V_\zeta\), write \(A=\zeta I+N_\zeta\),
where \(N_\zeta^{\nu_\zeta}=0\). The inverse and the evolution on that
summand are

\[
(zI-A)^{-1}|_{V_\zeta}
=\sum_{k=0}^{\nu_\zeta-1}\frac{N_\zeta^k}{(z-\zeta)^{k+1}},
\qquad
E_A(t)|_{V_\zeta}
=e^{it\zeta}\sum_{k=0}^{\nu_\zeta-1}\frac{(it)^kN_\zeta^k}{k!}. \tag{S2}
\]

The first identity follows by multiplying a finite geometric series; the
second follows from the exponential series for commuting \(\zeta I\) and
\(N_\zeta\). The entire-series contour moment formula from the preceding
lesson, applied to the first identity, yields

\[
E_A(t)=\frac{1}{2\pi i}\int_\Gamma
e^{itz}(zI-A)^{-1}\,dz,                                    \tag{S3}
\]

where \(\Gamma\) is a finite union of positively oriented closed
contours enclosing every eigenvalue once and no point of the spectrum on
the contours. Integrating the resolvent alone around a single eigenvalue
gives \(Q_\zeta\). These assertions also follow directly from (S2), so
multiple poles are included.

The full characteristic/pole comparison is proved in Sections
16.2--16.3, including the exact nonzero leading resolvent coefficient.
For later counting, \(\dim V_\zeta\) equals the algebraic multiplicity
of \(\zeta\) in \(\det(zI-A)\). The direct sum decomposes the
determinant into the determinants of the restrictions. A nilpotent map
is upper triangular with zero diagonal in a basis adapted to its
successive kernels; the restriction to \(V_\zeta\) therefore contributes
\((z-\zeta)^{\dim V_\zeta}\). The length of a nilpotent chain affects
polynomial factors in time, while the dimension of the whole generalized
eigenspace determines the number of solutions.

The exact original annihilator and scalar-unit calculation in Sections
16.1--16.4 proves these same operators and contour formulas with every
leading factor shown.

\subsection{2. A monic equation and its complete Cauchy
coordinates}\label{a-monic-equation-and-its-complete-cauchy-coordinates}

Let \(E\) have dimension \(N\), let \(m\geq1\), and put

\[
p(z)=z^m I+\sum_{j=0}^{m-1}p_jz^j,
\qquad p_j\in\operatorname{End}(E).                         \tag{S4}
\]

The scalar variable \(z\) commutes with every coefficient; the
coefficients need not commute with each other. The case \(E=0\) has only
zero solutions and determinant one; the following arguments can be read
with \(N>0\).

For \(x=(x_0,\ldots,x_{m-1})\in E^m\), define the companion map

\[
C_px=(x_1,\ldots,x_{m-1},-p_0x_0-\cdots-p_{m-1}x_{m-1}).
\]

If \(p(D)u=0\), its state \(J u=(u,Du,\ldots,D^{m-1}u)\) satisfies
\(D(Ju)=C_pJu\). Conversely, the first \(m-1\) rows of this first-order
equation imply that any state solution is the jet of its first
component, and the last row gives \(p(D)u=0\). Applying Section 1 proves
existence and uniqueness for arbitrary Cauchy data and gives
\(\dim\ker p(D)=mN\).

We will need the exact characteristic polynomial:

\[
\det(zI_{E^m}-C_p)=\det p(z).                              \tag{S5}
\]

For a direct determinant check, make the triangular change of variables
\(x_0=y_0\) and \(x_j=zx_{j-1}+y_j\) for \(1\leq j<m\); its determinant
is one. The first \(m-1\) rows of \((zI-C_p)x\) become
\(-y_1,\ldots,-y_{m-1}\), and the coefficient of \(y_0\) in the last row
is \(p(z)\). Move that last block row to the front and eliminate its
other entries using the identity block rows. The block-row permutation
has sign \((-1)^{N^2(m-1)}\), and the negative identities contribute
\((-1)^{N(m-1)}\). Their product is one. This proves (S5), as a
polynomial identity, without assuming that any \(p_j\) is invertible.

There is also a useful polynomial encoding of a jet. Set \(p_m=I\) and

\[
\mathcal U_x(z)=
\sum_{0\leq k\leq j<m}p_{j+1}x_{j-k}z^k.                 \tag{S6}
\]

This maps \(E^m\) bijectively onto the \(E\)-valued polynomials of
degree less than \(m\). To see this, read coefficients from the highest
power downwards: the coefficient of \(z^{m-1}\) is \(x_0\), the next is
\(x_1+p_{m-1}x_0\), and in general the next unknown \(x_r\) occurs with
coefficient \(I\), accompanied only by already recovered
\(x_0,\ldots,x_{r-1}\).

Let \(\pi_0:E^m\to E\) be the first coordinate. Solving \((zI-C_p)w=x\)
gives

\[
w_j=z^jw_0-\sum_{r=0}^{j-1}z^{j-1-r}x_r,
\qquad
p(z)w_0=\mathcal U_x(z).
\]

The latter equality follows by substituting the first formula into the
last block row; its coefficient of \(x_r\) is
\(\sum_{j=r}^{m-1}p_{j+1}z^{j-r}\). Consequently (S3), applied to
\(C_p\), gives the solution with jet \(x\):

\[
u(t)=\frac{1}{2\pi i}\int_\Gamma
p(z)^{-1}\mathcal U_x(z)e^{itz}\,dz.                       \tag{S7}
\]

The contours enclose all roots of \(\det p\). Formula (S7) is valid for
repeated roots and matrix poles of arbitrary order. It is the finite sum
of the pole moments computed from (S2). The triangular bijection above
proves uniqueness of the polynomial in this representation, as well as
existence. In particular, it does not require a factorization of \(p\)
into linear matrix factors.

Section 16.7 proves the corresponding original numerator, inverse
coordinates, resolvent and contour maps when the actual leading matrix
is any invertible \(p_m\). It retains its full determinant and both
block signs.

\subsection{3. Bounded solutions, derivatives, and
multiplicity}\label{bounded-solutions-derivatives-and-multiplicity}

First, boundedness of a solution of a monic constant-coefficient
equation controls all its derivatives. This fact is useful before any
spectral separation is imposed.

\textbf{Lemma.} If \(r\) is any monic matrix polynomial and \(r(D)u=0\),
then boundedness of \(u\) on \([0,\infty)\) implies boundedness there of
\(D^ku\) for every \(k\geq0\).

\textbf{Proof.} Write \(C=C_r\), with state space \(W\), and define

\[
h(v)=\int_0^1\|\pi_0E_C(s)v\|^2\,ds.
\]

If \(h(v)=0\), the continuous integrand vanishes on the interval. Its
derivatives at zero then vanish, so every Cauchy coordinate of \(v\) is
zero. The positive function \(h\) has a positive minimum \(c\) on the
unit sphere. Thus for every \(t\geq0\),

\[
c\|Ju(t)\|^2\leq
\int_0^1\|u(t+s)\|^2\,ds.
\]

All components of the state are bounded. Repeated differentiation of
\(D(Ju)=CJu\) expresses every higher derivative through a fixed power of
\(C\), proving the claim. The same argument with integration over
\([-1,0]\) proves the negative-half-line version. An equation with
invertible leading coefficient is first multiplied on the left by its
inverse. \(\square\)

Assume now that \(A\) has no real eigenvalues. Define

\[
q_A=\sum_{\operatorname{Im}\zeta>0}Q_\zeta
=\frac{1}{2\pi i}\int_{\Gamma_+}(zI-A)^{-1}\,dz,             \tag{S8}
\]

where \(\Gamma_+\) surrounds precisely the upper-half-plane spectrum.
This is a projection commuting with \(A\). On its image, (S2) shows that
\(E_A(t)\) and all its derivatives decay as \(t\to+\infty\): every
polynomial factor is dominated by the exponential
\(e^{-t\operatorname{Im}\zeta}\). On the image of \(I-q_A\), the
corresponding decay holds as \(t\to-\infty\).

Conversely, if \(E_A(t)v\) is bounded for \(t\geq0\), set
\(v_-=(I-q_A)v\). Then

\[
v_-=E_A(-t)(I-q_A)E_A(t)v.
\]

The norm of \(E_A(-t)\) on the lower spectral subspace tends to zero, by
(S2), while the last factor is bounded. Taking \(t\to\infty\) gives
\(v_-=0\). The negative-half-line assertion follows with signs reversed.
Hence initial evaluation identifies the two bounded solution spaces with
\(q_AV\) and \((I-q_A)V\), and every solution splits uniquely into one
of each kind.

For (S4), assume

\[
\det p(\xi)\ne0\qquad(\xi\in\mathbb R).                    \tag{S9}
\]

The companion characteristic identity and the lemma imply that its
stable solution space

\[
\mathcal M^+(p)=
\{u\in C^\infty(\mathbb R,E):p(D)u=0,
\ \sup_{t\geq0}\|u(t)\|<\infty\}
\]

corresponds under \(J\) exactly to \(q_{C_p}E^m\). All its derivatives
decay exponentially. The dimension is the sum of the algebraic
multiplicities of the roots of \(\det p\) in the upper half-plane. The
analogous lower space gives a direct-sum decomposition of all solutions.
The statement counts generalized modes, even if the scalar determinant
has high multiplicity and the space of ordinary eigenvectors is small.

Section 16.5 also proves the exact bounded modes when real roots are
present; those extra modes are distinguished from the original
no-real-root decay assertion above.

\subsection{4. Parameters and fixed
projections}\label{parameters-and-fixed-projections}

Suppose \(A(y)\) is a \(C^r\) family on an open parameter set, where
\(0\leq r\leq\infty\), and every \(A(y)\) has no real eigenvalues. Near
a fixed parameter one can choose the same contours in (S8):
invertibility on a compact contour persists under small matrix
perturbations. The resolvent is \(C^r\), and, for one derivative,

\[
\partial_y(zI-A)^{-1}
=(zI-A)^{-1}(\partial_y A)(zI-A)^{-1}.                      \tag{S10}
\]

Repeated differentiation gives finite sums of ordered products of such
factors; their order cannot be changed when matrices do not commute.
Differentiating the contour integral proves that \(q_A(y)\) is \(C^r\).
Since the rank of a projection is its integer-valued trace, its rank is
locally constant. A basis of its image at one parameter remains a basis
after applying nearby projections, so the stable spaces form a \(C^r\)
vector bundle. A global eigenbasis or a splitting into individual
eigenvalue branches is unnecessary.

On a compact parameter set contained in this no-real-spectrum region,
there are constants \(\delta>0\) and \(C_k\) such that

\[
\|D_t^k E_{A(y)}(t)q_{A(y)}\|\leq C_k e^{-\delta t}
\qquad(t\geq0).                                           \tag{S11}
\]

To justify the common contour used for this estimate, matrix norms give
a common bound for all eigenvalues. If their distances from the real
axis had no positive lower bound, a convergent sequence of parameters
and a convergent subsequence of eigenvalues would give a real eigenvalue
at the limiting parameter, by continuity of the determinant. Choose a
rectangle enclosing all upper eigenvalues with its lower side strictly
above the real axis, and with its other sides beyond the common spectral
bound. Its distance from the spectra is positive. Formula (S3) with this
contour, and \(D_t^ke^{itz}=z^ke^{itz}\), proves (S11). In parameter
charts, derivatives through order \(r\) obey the corresponding estimate
by (S10). Constants are local on the parameter space; no global spectral
gap is asserted on a noncompact parameter set.

These results apply to \(C_{p(y)}\) when the coefficients of a
fixed-degree monic polynomial vary \(C^r\) and (S9) holds. Collisions
among roots in one half-plane do not damage the stable bundle or these
estimates.

Section 16.6 gives every ordered multiindex derivative with its full
multiplicity and the original contour length, spectral-gap and resolvent
constants.

\subsection{5. Collapsing rates within their
half-planes}\label{collapsing-rates-within-their-half-planes}

Let \(q=q_A\), let \(\lambda>0\), and let \(0\leq\tau\leq1\). Put

\[
A_\tau=(1-\tau)A+i\tau\lambda(2q-I).                      \tag{S12}
\]

\textbf{Theorem.} Each \(A_\tau\) has no real eigenvalues, and its upper
spectral projection is exactly \(q\). At \(\tau=1\), its solution with
initial value \(v\) is

\[
e^{-\lambda t}qv+e^{\lambda t}(I-q)v.                     \tag{S13}
\]

\textbf{Proof.} On an original primary summand \(V_\zeta\), the
restriction is

\[
\big((1-\tau)\zeta+i\tau\lambda\,
\operatorname{sgn}(\operatorname{Im}\zeta)\big)I
+(1-\tau)N_\zeta.
\]

Its only eigenvalue has the same strict imaginary-part sign as
\(\zeta\), including at both endpoints. The fixed direct sum of all
original upper summands is therefore the full upper primary subspace for
every \(\tau\), and the lower sum is its complementary lower subspace.
The spectral projection is uniquely the projection onto the first along
the second, hence equals \(q\). Formula (S13) follows by exponentiating
the restrictions \(i\lambda I\) and \(-i\lambda I\).

Distinct original eigenvalues can coalesce during this deformation, and
at the last endpoint nilpotent parts vanish. Individual generalized
eigenspaces labelled by \emph{distinct moving eigenvalues} therefore
need not stay individually identifiable. The two half-plane sums and
their projection are what remain fixed. \(\square\)

For parameter-dependent \(A\) and positive \(\lambda\), (S12) has the
same regularity as those data by Section 4. There is no additional
requirement that \(\lambda\) avoid the original eigenvalues.

\subsection{6. Polynomial coordinates adapted to two
half-planes}\label{polynomial-coordinates-adapted-to-two-half-planes}

Return to (S4), now with \(m>1\), and fix \(\lambda>0\). Write

\[
L(z)=z+i\lambda,\qquad R(z)=z-i\lambda.
\]

There are unique coefficients \(a_0,\ldots,a_m\in\operatorname{End}(E)\)
with

\[
p(z)=\sum_{j=0}^m a_j R(z)^jL(z)^{m-j}.                   \tag{S14}
\]

For completeness, set \(w=R(z)/L(z)\), so \(z=i\lambda(1+w)/(1-w)\) and
\(L(z)=2i\lambda/(1-w)\). Then the coefficients are those of the
polynomial

\[
\sum_{j=0}^m a_jw^j
=\sum_{k=0}^m\frac{i^{k-m}}{2^m}\lambda^{k-m}
 p_k(1+w)^k(1-w)^{m-k}.                                  \tag{S15}
\]

This gives an explicit formula

\[
a_j=\sum_{k=0}^m c_{jk}p_k\lambda^{k-m},\qquad
c_{jk}=\frac{i^{k-m}}{2^m}
\sum_{r=0}^{j}(-1)^{j-r}
\binom{k}{r}\binom{m-k}{j-r},                             \tag{S16}
\]

with binomial coefficients outside their ordinary range interpreted as
zero. The right side of (S15) has degree at most \(m\), proving
existence. If a combination in (S14) vanishes, divide by \(L(z)^m\) for
\(z\ne-i\lambda\); the polynomial \(\sum a_jw^j\) vanishes on infinitely
many \(w\), so each coefficient vanishes. This proves uniqueness and
invertibility of the change of basis. Comparing the coefficient of
\(z^m\) gives

\[
\sum_{j=0}^m a_j=I.                                      \tag{S17}
\]

Neither \(a_0\) nor \(a_m\), nor any intermediate coefficient, is
assumed invertible. The negative powers of \(\lambda\) in (S16) require
\(\lambda>0\); there is no claim of uniformity as
\(\lambda\downarrow0\).

\subsection{7. First-order coordinates that respect
decay}\label{first-order-coordinates-that-respect-decay}

For a solution of \(p(D)u=0\), define

\[
v_j=R(D)^jL(D)^{m-1-j}u,\qquad 0\leq j<m.                \tag{S18}
\]

The polynomials \(R^jL^{m-1-j}\) form a basis of the scalar polynomials
of degree at most \(m-1\), by the same argument as in Section 6. Thus
(S18) is an invertible, constant, scalar-block change from the Cauchy
state \((u,Du,\ldots,D^{m-1}u)\). There is no loss of a Cauchy
coordinate.

The transformed equations are

\[
\begin{aligned}
&\sum_{j=0}^{m-1}a_jL(D)v_j+a_mR(D)v_{m-1}=0,\\
&L(D)v_j-R(D)v_{j-1}=0\quad(1\leq j<m).
\end{aligned}                                             \tag{S19}
\]

The second equations follow by commuting scalar polynomials in \(D\);
the first is precisely (S14) applied to \(u\). We have only moved scalar
differential factors past constant matrix coefficients, never exchanged
two matrix coefficients.

These equations are exactly a first-order system, not merely
consequences that define a larger space. One way to check the converse
is to let \(T\) denote the invertible change in (S18). The leading
matrix of the left side of (S19) has row zero
\((a_0,a_1,\ldots,a_{m-2},a_{m-1}+a_m)\), and lower rows have \(-I,I\)
in adjacent columns. Its determinant is one by the factorization in the
next result at \(\tau=1\). Call that leading matrix \(C\), and the
constant matrix \(K\). Every jet initial value yields a transformed
solution, so \((CTC_p+KT)x=0\) for every initial vector \(x\). Hence
\(CD+K=CT(D-C_p)T^{-1}\). Invertibility of \(C,T\) proves the converse.

This change of state is an exact first-order reduction. The next
construction has a different purpose: it produces a homotopy of
fixed-degree polynomial systems beginning with the original equation
plus auxiliary equations. Choosing \(L(D)^m\) for those auxiliary
equations places their characteristic root at \(-i\lambda\), where they
have no stable solutions. Auxiliary equations \(D^m v=0\) would instead
introduce the real root zero.

\subsection{8. An explicit stabilization and its leading
coefficient}\label{an-explicit-stabilization-and-its-leading-coefficient}

For \(0\leq\tau\leq1\), define the degree-\(m\) polynomial
\(P_\tau(z)\in\operatorname{End}(E^m)[z]\) by its action on
\(U=(U_0,\ldots,U_{m-1})\). The first component is

\[
\begin{aligned}
(P_\tau(z)U)_0={}&
\big((1-\tau^m)p(z)+\tau^m a_0L(z)^m\big)U_0\\
&+\sum_{j=1}^{m-2}\tau^{m-j}a_jL(z)^mU_j\\
&+\tau\big(a_{m-1}L(z)^m+a_mR(z)L(z)^{m-1}\big)U_{m-1},
\end{aligned}                                             \tag{S20}
\]

and the remaining components are

\[
(P_\tau(z)U)_j=L(z)^mU_j-
\tau R(z)L(z)^{m-1}U_{j-1}\quad(1\leq j<m).               \tag{S21}
\]

For \(m=2\), the sum in the middle line of (S20) is empty; the last
block appears only once. At the endpoints,

\[
P_0(z)=\operatorname{diag}(p(z),L(z)^mI,\ldots,L(z)^mI),    \tag{S22}
\]

and every entry of \(P_1(z)\) contains \(L(z)^{m-1}\). After that scalar
factor is removed, its equations are (S19).

Let \(C_\tau\) be the coefficient of \(z^m\) in \(P_\tau\). Define
\(S:E^m\to E^m\) by \((SU)_0=0\), \((SU)_j=U_{j-1}\), and let \(H_\tau\)
have identity diagonal, zero off-diagonal entries except in its first
row, and

\[
(H_\tau)_{0j}=h_j
=\tau^{m-j}\sum_{k=j}^m a_k\quad(1\leq j<m).
\]

Then

\[
C_\tau=H_\tau(I-\tau S).                                 \tag{S23}
\]

To verify this, its first block is
\(I-\tau h_1=I-\tau^m\sum_{k=1}^ma_k=(1-\tau^m)I+\tau^ma_0\). Its
intermediate first-row blocks are \(h_j-\tau h_{j+1}=\tau^{m-j}a_j\),
and its last block is \(h_{m-1}=\tau(a_{m-1}+a_m)\). The lower rows are
exactly \(-\tau I,I\). These are the leading blocks of (S20)--(S21).

Both factors in (S23) are block triangular with identity diagonal.
Therefore

\[
\det C_\tau=1.                                            \tag{S24}
\]

Moreover, \(H_\tau=I+N_\tau\) with \(N_\tau^2=0\), while \(S^m=0\). Its
inverse is the explicit finite expression

\[
C_\tau^{-1}
=\left(\sum_{r=0}^{m-1}\tau^rS^r\right)(I-N_\tau).          \tag{S25}
\]

In block form, \((C_\tau^{-1})_{i0}=\tau^iI\), and for \(j\geq1\),

\[
(C_\tau^{-1})_{ij}
=\begin{cases}
\tau^{i-j}I-\tau^i h_j,&i\geq j,\\
-\tau^i h_j,&i<j,
\end{cases}
\qquad 0\leq i<m.                                        \tag{S26}
\]

Every entry has degree at most one in the \(a_k\); using (S17) to
replace \(I\) makes it a linear combination of them. No product of two
\(a_k\)'s occurs. The factorization uses only addition, composition with
scalar multiples of identity, and the relation (S17). Consequently the
inverse calculation also works for coefficients in any unital, possibly
noncommutative, algebra satisfying that relation. Only the determinant
statement is restricted here to finite-dimensional matrices.

\subsection{9. A determinant identity retaining every
root}\label{a-determinant-identity-retaining-every-root}

For every complex \(z\) and every \(0\leq\tau\leq1\),

\[
\det P_\tau(z)=\det p(z)\,L(z)^{m(m-1)N}.                 \tag{S27}
\]

\textbf{Proof.} First suppose \(L(z)\ne0\). The block of rows and
columns indexed \(1,\ldots,m-1\) is lower triangular with diagonal
\(L(z)^m I\), and therefore has determinant \(L(z)^{m(m-1)N}\).
Eliminating these rows expresses the auxiliary components as

\[
U_j=\left(\tau\frac{R(z)}{L(z)}\right)^jU_0.
\]

The Schur complement in the first row is

\[
(1-\tau^m)p(z)
+\tau^m\sum_{j=0}^ma_jR(z)^jL(z)^{m-j}=p(z).
\]

The block determinant formula proves (S27) whenever \(L(z)\ne0\). Both
sides are polynomials in \(z\), so equality on that set proves equality
also at \(z=-i\lambda\). The calculation never assumes invertibility of
\(p(z)\). \(\square\)

If (S9) holds, (S27) proves that \(P_\tau\) is invertible at every real
\(z\). It also records multiplicity: the upper-half-plane roots, with
algebraic multiplicity, are exactly those of \(\det p\); the auxiliary
factor adds only the lower root \(-i\lambda\), with multiplicity
\(m(m-1)N\). If this is already a root of \(\det p\), multiplicities
add. At \(\tau=1\), writing \(P_1=L^{m-1}F\), where \(F\) is the
first-order polynomial of (S19), gives \(\det F=\det p\). A zero of the
auxiliary factor is not discarded from the full equation.

\subsection{10. The stable space is preserved, with an explicit
inverse}\label{the-stable-space-is-preserved-with-an-explicit-inverse}

Assume (S9), and write \(\mathcal M^+_\tau=\mathcal M^+(P_\tau)\), with
the definition extended to invertible leading coefficient by left
normalization.

\textbf{Theorem.} For every \(0\leq\tau\leq1\),

\[
\Pi_\tau:\mathcal M^+_\tau\longrightarrow\mathcal M^+(p),
\qquad U\longmapsto U_0                                  \tag{S28}
\]

is a linear isomorphism. On these spaces,

\[
L(D)U_j=\tau R(D)U_{j-1},\qquad
L(D)^jU_j=\tau^jR(D)^jU_0.                               \tag{S29}
\]

\textbf{Proof.} The leading coefficient is invertible by (S24), and the
determinant has no real zeros by (S27). Section 3 therefore applies to
\(P_\tau\): every stable solution and all its derivatives decay
exponentially.

Equation (S21) says \(L(D)^{m-1}(L(D)U_j-\tau R(D)U_{j-1})=0\). The
expression in parentheses is bounded. The kernel of \(L(D)^k\), for
\(k\geq1\), consists exactly of \(e^{\lambda t}\) times vector
polynomials of degree less than \(k\): substitute \(f=e^{\lambda t}g\)
and use \(L(D)f=-ie^{\lambda t}g'\). No nonzero such function is bounded
for \(t\geq0\). Cancellation is therefore valid on this bounded solution
space, giving the first recurrence in (S29). Iteration gives the second
one.

For \(j<m\), multiply the second identity by \(L(D)^{m-j}\). In the last
term of (S20), use \(R(D)L(D)^{m-1}U_{m-1}=\tau^{m-1}R(D)^mU_0\). The
first row of the differential equation becomes

\[
(1-\tau^m)p(D)U_0+
\tau^m\sum_{j=0}^ma_jR(D)^jL(D)^{m-j}U_0=p(D)U_0=0.
\]

Thus \(\Pi_\tau\) has the asserted codomain.

To construct the inverse, first observe that for a smooth exponentially
decreasing function \(g\), whose derivatives have the same property, the
unique bounded solution of \(L(D)f=g\) is

\[
(T_\lambda g)(t)
=-i\int_0^\infty e^{-\lambda s}g(t+s)\,ds.                \tag{S30}
\]

Differentiating the integral or integrating once by parts gives
\(f'-\lambda f=ig\), which is equivalent to \(L(D)f=g\). The difference
of any two bounded solutions is \(ce^{\lambda t}\), so is zero.
Differentiation under the integral is justified by the exponential
derivative bounds. If \(\|D^kg(t)\|\leq M_ke^{-\delta t}\), then

\[
\|D^kT_\lambda g(t)\|\leq
\frac{M_k}{\lambda+\delta}e^{-\delta t}
\quad(t\geq0).                                          \tag{S31}
\]

Every \(u\in\mathcal M^+(p)\) has these bounds by Section 3. Define

\[
U_0=u,\qquad U_j=\tau T_\lambda R(D)U_{j-1}\ (1\leq j<m). \tag{S32}
\]

These functions are defined on all real \(t\); for fixed \(t\), the
integration tail still lies in the decaying region. They and all their
derivatives decay on the positive half-line, satisfy (S29), and hence
satisfy every lower row of \(P_\tau(D)U=0\). Substituting the
recurrences into the first row gives \(p(D)u=0\), as above. This proves
surjectivity. If \(U_0=0\), each successive bounded solution of
\(L(D)U_j=\tau R(D)U_{j-1}\) is zero by uniqueness, proving injectivity.
\(\square\)

Since \(T_\lambda\) commutes with scalar constant-coefficient
differentiation on these functions, the lift can also be written

\[
U_j=\tau^j(T_\lambda R(D))^ju.
\]

At \(\tau=0\), this is exactly \((u,0,\ldots,0)\); no division by
\(\tau\) is needed. The same estimates show local \(C^r\) dependence of
the lift on \(C^r\) coefficients and on \(\lambda>0\). In Cauchy
coordinates this is an isomorphism of the stable bundles from Section 4.
On compact parameter sets, differentiation of (S30) in \(\lambda\)
introduces powers of \(s\), whose integrals remain finite because
\(\lambda\) has a positive lower bound and the stable decay estimates
are uniform.

\subsection{11. Boundary measurements, their orders, and the zero
endpoint}\label{boundary-measurements-their-orders-and-the-zero-endpoint}

For \(1\leq\ell\leq J\), let \(G_\ell\) be a finite-dimensional complex
vector space and

\[
B_\ell(z)=\sum_{r=0}^{d_\ell}b_{\ell r}z^r,
\qquad b_{\ell r}\in\operatorname{Hom}(E,G_\ell),
\qquad d_\ell\geq0.
\]

The boundary map is

\[
\mathcal B:\mathcal M^+(p)\longrightarrow
G:=\bigoplus_{\ell=1}^J G_\ell,
\qquad
u\longmapsto\big(B_\ell(D)u(0)\big)_\ell.                 \tag{S33}
\]

The empty family is allowed, with target zero. We call these
measurements \emph{complementing} precisely when (S33) is bijective. In
particular its domain and target dimensions must agree; injectivity
alone does not establish complementing data with arbitrary targets.

For any finite orders \(d_\ell\), composition with (S28) gives

\[
\mathcal B_\tau(U)=\big(B_\ell(D)U_0(0)\big)_\ell,
\qquad \mathcal B_\tau=\mathcal B\Pi_\tau.                 \tag{S34}
\]

Since \(\Pi_\tau\) is an isomorphism, \(\mathcal B_\tau\) is bijective
if and only if \(\mathcal B\) is. This statement includes both
\(\tau=0\) and \(\tau=1\), and does not require \(d_\ell<m\).

The preserved data form the commuting square

\[
\begin{array}{ccc}
\mathcal M^+_\tau&\xrightarrow{\mathcal B_\tau}&G\\
\Pi_\tau\downarrow&&\downarrow I_G\\
\mathcal M^+(p)&\xrightarrow{\mathcal B}&G.
\end{array}
\]

The left vertical arrow is invertible; the bottom measurement map is
invertible exactly when the top one is. This diagram concerns solution
spaces, so it does not extend an expression with singular coefficients
to arbitrary Cauchy vectors.

For the following alternative expression, impose the additional
hypothesis \(d_\ell<m\). There are unique coefficients
\(\beta_{\ell k}\in\operatorname{Hom}(E,G_\ell)\) such that

\[
B_\ell(z)=\sum_{k=0}^{m-1}
\beta_{\ell k}R(z)^kL(z)^{m-1-k}.                         \tag{S35}
\]

Repeating the change of variables with degree \(m-1\) gives

\[
\sum_{k=0}^{m-1}\beta_{\ell k}w^k
=\sum_{r=0}^{d_\ell}
\frac{i^{r+1-m}}{2^{m-1}}\lambda^{r+1-m}
b_{\ell r}(1+w)^r(1-w)^{m-1-r}.                          \tag{S36}
\]

Thus
\(\beta_{\ell k}=\sum_{r=0}^{d_\ell}c'_{kr}b_{\ell r}\lambda^{r+1-m}\),
where

\[
c'_{kr}=\frac{i^{r+1-m}}{2^{m-1}}
\sum_{a=0}^k(-1)^{k-a}
\binom{r}{a}\binom{m-1-r}{k-a}.
\]

The degrees and powers of \(\lambda\) follow from (S36), including when
the actual order is less than \(m-1\). From (S29),

\[
L(D)^{m-1}U_k
=\tau^kR(D)^kL(D)^{m-1-k}U_0.
\]

Consequently for \(0<\tau\leq1\) and \(U\in\mathcal M^+_\tau\),

\[
B_\ell(D)U_0
=\sum_{k=0}^{m-1}\tau^{-k}\beta_{\ell k}
L(D)^{m-1}U_k.                                          \tag{S37}
\]

This is equality of functions of \(t\), so it remains true at the
boundary after evaluation at zero. The factors \(\tau^{-k}\) make (S37)
unsuitable as a formula for an ambient boundary operator at \(\tau=0\).
The correct endpoint map is (S34). On the explicitly lifted stable
vectors (S32), the factors cancel with \(\tau^k\), but this cancellation
does not define \(\tau^{-k}\) on arbitrary vectors at zero.

Here the orders are orders in the single normal variable. In a boundary
PDE, tangential orders and their Sobolev shifts are additional data.
None of the finite-dimensional statements permits dropping them or
replacing an arbitrary boundary system by Dirichlet data.

\subsection{12. The reduction with its leading coefficient
retained}\label{the-reduction-with-its-leading-coefficient-retained}

Combine the preceding statements under exactly these assumptions: \(E\)
is finite-dimensional over \(\mathbb C\); \(m>1\); \(p\) has degree
\(m\) and leading coefficient \(I\); \(\det p(\xi)\ne0\) for every real
\(\xi\); \(\lambda>0\); and \(0\leq\tau\leq1\).

The family \(P_\tau\) in (S20)--(S21) begins with the stabilized
equation (S22), has no real characteristic roots, preserves the full
stable solution space by first-coordinate projection, and preserves
bijectivity of every finite collection of boundary measurements
transported by that projection. Its leading coefficient has determinant
one and inverse entries linear in \(a_0,\ldots,a_m\). The extra boundary
expression (S37) requires orders less than \(m\) and \(\tau>0\).

Define

\[
\widehat P_\tau(z)=C_\tau^{-1}P_\tau(z).                  \tag{S38}
\]

It is monic of degree \(m\). Left multiplication by the invertible
constant matrix changes neither the differential equation's solutions
nor any of their boundary measurements. Because \(\det C_\tau=1\), it
also leaves (S27) unchanged. At \(\tau=0\), (S23) gives \(C_0=I\), so
the starting polynomial is unchanged. At \(\tau=1\),

\[
\widehat P_1(z)=L(z)^{m-1}(zI-A_*)                        \tag{S39}
\]

for a matrix \(A_*\) with no real eigenvalues: the first-order factor is
monic and its determinant is \(\det p(z)\). Thus its stable equation can
be further deformed by Section 5. The scalar factor in (S39) has only
the lower root \(-i\lambda\); cancellation of that factor is valid for
bounded solutions by Section 10, and is not an equality of full solution
spaces.

The preceding construction begins with a monic polynomial because its
Cauchy coordinates use the leading identity. For an original polynomial
\[
p(z)=p_m z^m+\sum_{j=0}^{m-1}p_jz^j,\qquad p_m\in\operatorname{GL}(E),
\] retain \(p\) and every leading-coefficient factor. Set
\(r(z)=p_m^{-1}p(z)\) only to calculate the monic comparison, and let
\(P_\tau[r]\) and \(C_\tau[r]\) denote (S20)--(S23) formed from \(r\).
The expression \(p_m^{-1}p\) is used only for this comparison;
multiplication is on the left. The family attached to the original
equation is \[
\mathcal P_\tau(z)
=\operatorname{diag}(p_m,I,\ldots,I)P_\tau[r](z).
\] Its first row has the original left factor \(p_m\); the other rows
are unchanged. In particular, \[
\mathcal P_0(z)=\operatorname{diag}(p(z),L(z)^mI,\ldots,L(z)^mI),
\qquad
\det\mathcal P_\tau(z)=\det p(z)\,L(z)^{m(m-1)N}.
\] The determinant identity follows without cancellation:
\(\det r(z)=(\det p_m)^{-1}\det p(z)\), while left multiplication
contributes \(\det p_m\). The leading matrix is \[
\mathcal C_\tau
=\operatorname{diag}(p_m,I,\ldots,I)C_\tau[r],
\qquad
\det\mathcal C_\tau=\det p_m,\qquad
\mathcal C_\tau^{-1}
=C_\tau[r]^{-1}\operatorname{diag}(p_m^{-1},I,\ldots,I).
\] These orders follow from matrix multiplication; no commutation of
\(p_m\) with the other coefficients is used. Since the diagonal left
factor is invertible, \(\mathcal P_\tau(D)U=0\) if and only if
\(P_\tau[r](D)U=0\), and \(p(D)u=0\) if and only if \(r(D)u=0\). Thus
the first-coordinate stable isomorphism and every transported boundary
measurement remain exact for the original equation. The monic comparison
satisfies
\(\mathcal C_\tau^{-1}\mathcal P_\tau=C_\tau[r]^{-1}P_\tau[r]\); it does
not erase \(\det p_m\) from the original family. For singular \(p_m\),
this comparison is unavailable and the full \(mN\)-dimensional Cauchy
space can fail, as Problem 6 shows.

\subsection{13. Example: cubic time factors in a four-dimensional stable
space}\label{example-cubic-time-factors-in-a-four-dimensional-stable-space}

Let \(E=\mathbb C^2\), let \(\kappa\ne0\), and consider

\[
p(z)=
\begin{pmatrix}(z-i)^2&\kappa\\0&(z-i)^2\end{pmatrix}.
\]

It is monic of degree two, and its determinant is \((z-i)^4\), so every
solution is stable. To see all four coordinates directly, write

\[
u_2(t)=e^{-t}(a+bt),\qquad
u_1(t)=e^{-t}
\left(c+dt+\frac{\kappa a}{2}t^2+\frac{\kappa b}{6}t^3\right).
\]

Since \((D-i)(e^{-t}f)=-ie^{-t}f'\), substitution proves the equation.
The free constants \(a,b,c,d\) give all four solutions by Cauchy
uniqueness. Thus a second-order matrix equation can contain a cubic
polynomial times an exponential; the order of the scalar equation alone
does not bound the length of a generalized chain for the full state
matrix. In agreement, the upper-right entry of \(p(z)^{-1}\) is
\(-\kappa(z-i)^{-4}\).

The pair of measurements \(u(0)\in E\), \(Du(0)\in E\) is complementing
because every Cauchy jet is stable. Under the construction with
\(\lambda=2\), the stabilized system has determinant
\((z-i)^4(z+2i)^4\). Its stable dimension remains four, while its total
solution dimension is eight.

\subsection{14. Example: noncommuting coefficients and an unsuitable
trace}\label{example-noncommuting-coefficients-and-an-unsuitable-trace}

Consider

\[
p(z)=
\begin{pmatrix}(z-i)^2&1\\0&(z+2i)^2\end{pmatrix}.
\]

Its coefficients \(p_1=\operatorname{diag}(-2i,4i)\) and
\(p_0=\left(\begin{smallmatrix}-1&1\\0&-4\end{smallmatrix}\right)\) do
not commute: \(p_1p_0-p_0p_1\) has upper-right entry \(-6i\). The
determinant is \((z-i)^2(z+2i)^2\). A bounded solution must have
\(u_2=0\), since its second component solves \((D+2i)^2u_2=0\).
Therefore

\[
u(t)=\big(e^{-t}(a+bt),0\big),\qquad a,b\in\mathbb C.
\]

The stable dimension is two. The ordinary vector trace
\(u(0)\in\mathbb C^2\) has image \(\mathbb C\times\{0\}\) and a
one-dimensional kernel, despite equality of source and target
dimensions. In contrast, the two scalar measurements

\[
B_1(D)u(0)=u_1(0)=a,\qquad
B_2(D)u(0)=Du_1(0)=i(a-b)
\]

are bijective onto \(\mathbb C\oplus\mathbb C\). The entire construction
above applies without changing the coefficient order or imposing a
scalar boundary model.

\subsection{15. Problems with solutions}\label{problems-with-solutions}

\textbf{Problem 1: a real root at the boundary of the hypothesis.} Let
\(A_\varepsilon=\left(\begin{smallmatrix}i\varepsilon&1\\0&i\varepsilon\end{smallmatrix}\right)\),
for real \(\varepsilon\). Determine the positive-half-line bounded
initial values, including \(\varepsilon=0\). Explain the failure of a
continuous stable projection across zero.

\textbf{Solution.} The solution is
\(e^{-\varepsilon t}(v_1+itv_2,v_2)\). For \(\varepsilon>0\), every
initial vector is stable. For \(\varepsilon<0\), only zero is bounded.
At zero, precisely vectors with \(v_2=0\) are bounded, and their
solutions are constant. Thus the bounded dimensions are respectively
two, zero, and one. The upper spectral projection is \(I\) for positive
\(\varepsilon\) and zero for negative \(\varepsilon\), so cannot extend
continuously across zero. At zero, boundedness does not imply decay, and
the upper/lower splitting omits the real generalized eigenspace. This
identifies why the no-real-spectrum hypothesis is used.

\textbf{Problem 2: the smallest block homotopy.} For \(m=2\), write
\(P_\tau\), its leading coefficient, and an inverse of that coefficient.
Verify the determinant-one assertion without commutativity assumptions.

\textbf{Solution.} With \(a_0+a_1+a_2=I\),

\[
P_\tau=
\begin{pmatrix}
(1-\tau^2)p+\tau^2a_0L^2&\tau(a_1L^2+a_2RL)\\
-\tau RL&L^2I
\end{pmatrix}.
\]

Writing \(b=\tau(a_1+a_2)\), its leading coefficient and inverse are

\[
C_\tau=
\begin{pmatrix}I-\tau b&b\\-\tau I&I\end{pmatrix}
=\begin{pmatrix}I&b\\0&I\end{pmatrix}
\begin{pmatrix}I&0\\-\tau I&I\end{pmatrix},
\qquad
C_\tau^{-1}=
\begin{pmatrix}I&-b\\\tau I&I-\tau b\end{pmatrix}.
\]

Multiplying the two displayed factors or multiplying
\(C_\tau C_\tau^{-1}\) verifies the formulas; no two unrelated
coefficient matrices are interchanged. Each triangular factor has
determinant one. In addition, (S27) reads
\(\det P_\tau=\det p\,L^{2N}\).

\textbf{Problem 3: boundary orders beyond the coordinate basis.} Let
\(B(z)\in\operatorname{Hom}(E,G)[z]\) have arbitrary finite degree and
let \(p\) be monic of degree \(m\). Prove a unique division formula
\(B=Qp+R_B\) with \(\deg R_B<m\). Which factor order is required, and
what does this imply on solutions?

\textbf{Solution.} If \(B\) has leading term \(bz^d\) with \(d\geq m\),
subtract \(bz^{d-m}p(z)\). The leading term cancels because \(p_m=I\),
and the degree decreases. Iterating constructs
\(Q\in\operatorname{Hom}(E,G)[z]\) and \(R_B\) of degree below \(m\). If
two decompositions exist, a nonzero difference \(Q_1-Q_2\) of degree
\(s\) makes \((Q_1-Q_2)p\) have degree exactly \(s+m\), since its
leading coefficient is unchanged by multiplication by \(I\). It cannot
equal a polynomial of degree below \(m\); hence both differences vanish.
The required order is \(Qp\), which composes
\(E\xrightarrow{p}E\xrightarrow{Q}G\). On constant-coefficient
solutions, \(B(D)u=R_B(D)u\). This justifies normal-order reduction for
this algebraic model. A PDE with variable coefficients or tangential
orders needs its own division and mapping argument.

\textbf{Problem 4: a negative power of the homotopy parameter.} Take
\(m=2\), \(B(z)=z\), and any stable scalar solution \(u\). Find
\(\beta_0,\beta_1\), verify (S37), and identify its valid zero-parameter
interpretation.

\textbf{Solution.} Since
\(z=\tfrac12(z+i\lambda)+\tfrac12(z-i\lambda)\), one has
\(\beta_0=\beta_1=1/2\). For \(\tau>0\), the recurrence gives
\(LU_1=\tau RU_0\), so
\(\tfrac12 LU_0+\tfrac1{2\tau}LU_1=\tfrac12(L+R)U_0=DU_0\). At zero, the
stable lift has \(U_1=0\); inserting this into \(\tau^{-1}LU_1\) is
undefined. The correct boundary functional remains \(DU_0(0)\) from
(S34). Along a lifted family, its limiting value can be computed before
setting \(\tau=0\), using \(\tau^{-1}LU_1=RU_0\). It is not an ambient
formula at zero.

\textbf{Problem 5: nilpotent coalescence at the last endpoint.} Let
\(A\) be the direct sum of the blocks \(iI+N\) on \(\mathbb C^2\),
\(2i\) on \(\mathbb C\), and \(-3i\) on \(\mathbb C\), where
\(N=\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)\).
Describe (S12) when \(\lambda=1\).

\textbf{Solution.} The first block is \(iI+(1-\tau)N\), the next is
\(i(2-\tau)\), and the last is \(-i(3-2\tau)\). The upper projection is
always \(\operatorname{diag}(I_2,1,0)\). For \(\tau<1\), the first and
second upper blocks have distinct eigenvalues; at one, their eigenvalues
coalesce at \(i\) and the nilpotent part of the first vanishes. The full
\(i\)-eigenspace at one has dimension three. Thus individual primary
summands associated with distinct spectral values have merged, while the
upper spectral subspace has remained the same three-dimensional space
throughout.

\textbf{Problem 6: why the leading coefficient matters.} Treat
\(r(z)=\operatorname{diag}(z,1)\) as a degree-one matrix polynomial on
\(\mathbb C^2\). Determine the solutions and compare their dimension
with the formal count \(m\dim E\).

\textbf{Solution.} The equations are \(Du_1=0\) and \(u_2=0\), so the
solution space is one-dimensional, although \(m\dim E=2\). The leading
coefficient is \(\operatorname{diag}(1,0)\), which is singular. Initial
values are constrained, and multiplication by the inverse leading
coefficient is impossible. In contrast, a genuinely monic degree-one
polynomial supplies two free initial coordinates. This example concerns
the leading-coefficient hypothesis; its determinant also has a real
root, so it is not an example within the stable reduction theorem.

\subsection{16. Full original factors in the spectral and Cauchy
maps}\label{full-original-factors-in-the-spectral-and-cauchy-maps}

We make explicit every scalar and matrix leading factor used by Sections
1--5 and 12. The original characteristic multiplicities, nilpotent
orders, projection maps, Cauchy coordinates and contour conventions
remain in the calculation. The bounded-mode classification at real roots
and the complete original-leading-matrix numerator map are stated as
additional results; they do not silently change the no-real-root
assumptions of the earlier reduction.

The entry proof is \href{stable-prerequisite-bridges.pdf}{Polynomial and
contour interfaces for stable boundary models}, Sections 2--4 and 9:
original polynomial division and Bézout, full complex
roots/factorization, original scalar units, determinant signs, kernel
coordinates and both invariant quotient maps. We retain the finite basis
and scalar-calculus entry definitions used there. Each argument below
preserves the chosen original norm and actual coordinates; no Jordan
decomposition or selected smooth eigenvalue branch is assumed.

\subsubsection{16.1. The original annihilator and complete primary
projection
maps}\label{the-original-annihilator-and-complete-primary-projection-maps}

Let \(V\) be the original finite-dimensional complex vector space,
\(n=\dim V\), and \(A\in\operatorname{End}(V)\). Cayley--Hamilton in
Section 4 of \href{stable-prerequisite-bridges.pdf}{Polynomial and
contour interfaces for stable boundary models} supplies a nonzero
annihilator. Keep an original least-degree nonzero annihilator \(m\),
with original leading coefficient \(c_m\ne0\). That lesson proves that
\(m\) divides every annihilator. For \(n>0\), \(m\) is nonconstant,
because a nonzero constant acts as that nonzero scalar times the nonzero
identity. The full factorization proved there is \[
\begin{gathered}
m(z)=c_m\prod_{\lambda\in Z(m)}h_\lambda(z),\quad
 h_\lambda(z)=(z-\lambda)^{\nu_\lambda},\quad
 \nu_\lambda\geq1,\quad\\
 H_\lambda(z)=\prod_{\mu\ne\lambda}h_\mu(z),\quad
 M_\lambda(z)=m(z)/h_\lambda(z)=c_mH_\lambda(z).
\end{gathered} \tag{SR1}
\] Every original leading unit and root multiplicity is retained. A
least-degree choice is unique up to its nonzero original scalar unit, by
division in both directions. The named monic polynomial is exactly
\(c_m^{-1}m\), as a comparison; it does not replace \(m\) in this
calculation.

The coprime-factor proof in Section 3 of
\href{stable-prerequisite-bridges.pdf}{Polynomial and contour interfaces
for stable boundary models} gives
\(a_\lambda h_\lambda+b_\lambda H_\lambda=1\). In the original quotient
calculation set \(\beta_\lambda=c_m^{-1}b_\lambda\). The full identities
are \[
 a_\lambda h_\lambda+\beta_\lambda M_\lambda=1,\qquad
 e_\lambda=\beta_\lambda M_\lambda
              =c_m^{-1}b_\lambda\,c_mH_\lambda,\qquad
 Q_\lambda=(c_m^{-1}b_\lambda)(A)(c_mH_\lambda)(A).
                                                               \tag{SR2}
\] For the factor \(h_\lambda\), \(e_\lambda\) is one modulo that
factor; for every other factor it is zero. Thus each of
\(\sum_\lambda e_\lambda-1\), \(e_\lambda^2-e_\lambda\), and
\(e_\lambda e_\mu\) for \(\lambda\ne\mu\) is divisible by every factor.
The exact Bézout divisibility proof in Section 3 of
\href{stable-prerequisite-bridges.pdf}{Polynomial and contour interfaces
for stable boundary models} makes it divisible by their full product
\(H=c_m^{-1}m\). Its evaluation is zero because \(H(A)=c_m^{-1}m(A)=0\).
Consequently \[
 Q_\lambda^2=Q_\lambda,\quad Q_\lambda Q_\mu=0\ (\lambda\ne\mu),
 \quad\sum_\lambda Q_\lambda=I_V,\quad
 V=\bigoplus_\lambda W_\lambda,\quad
 W_\lambda=\operatorname{im}Q_\lambda
            =\ker(A-\lambda I)^{\nu_\lambda}.                  \tag{SR3}
\] Here the image equality is an exact map:
\(h_\lambda e_\lambda=\beta_\lambda m\) gives the first inclusion;
\(1-e_\lambda=a_\lambda h_\lambda\) gives \(v=Q_\lambda v\) on the
kernel. Applying \(Q_\mu\) to a zero sum proves directness. Each
projection commutes with \(A\), since it is an evaluated scalar
polynomial.

A different Bézout choice has the same factorwise remainders. Its
difference from \(e_\lambda\) is a multiple of \(H=c_m^{-1}m\), whose
full evaluation is zero. Thus it gives the same actual \(Q_\lambda\),
not just the same abstract summand. Under an original coordinate change
\(T\), polynomial evaluation gives \(f(T^{-1}AT)=T^{-1}f(A)T\): expand
every finite power and use the ordered adjacent products \(TT^{-1}=I\).
Applied separately to both factors in (SR2), this gives \[
 (c_m^{-1}b_\lambda)(T^{-1}AT)
 (c_mH_\lambda)(T^{-1}AT)=T^{-1}Q_\lambda T.                  \tag{SR4}
\] This is the exact coordinate map for the full original projection.

If \(V=0\), a least-degree annihilator can be the original nonzero
constant \(m=c_m\). Its root set is empty, and the empty projection sum
is zero, exactly the identity of the zero vector space. The empty
decomposition in (SR3) therefore includes that case without adding a
root or assigning a multiplicity.

\subsubsection{16.2. Actual nilpotent order and characteristic
multiplicity}\label{actual-nilpotent-order-and-characteristic-multiplicity}

On \(W_\lambda\) define the original restricted operator
\(N_\lambda=A|_{W_\lambda}-\lambda I_{W_\lambda}\). Equation (SR3) gives
\(N_\lambda^{\nu_\lambda}=0\). In fact every \(W_\lambda\) is nonzero
and this exponent is exact.

To prove both assertions without a Jordan assumption, suppose that
\(N_\lambda^{\nu_\lambda-1}=0\), also including an empty \(W_\lambda\).
The original polynomial \[
 m_\lambda(z)=m(z)/(z-\lambda)
       =c_m(z-\lambda)^{\nu_\lambda-1}
                    \prod_{\mu\ne\lambda}(z-\mu)^{\nu_\mu}
                                                               \tag{SR5}
\] would annihilate \(W_\lambda\) by that supposed zero power. It
annihilates every other \(W_\mu\) through the retained full factor
\((A-\mu I)^{\nu_\mu}\). The direct sum (SR3) would then make it an
annihilator of \(V\), with its original nonzero leading coefficient
\(c_m\) and degree one less than \(m\). Minimality excludes this. Hence
\(N_\lambda^{\nu_\lambda-1}\ne0\); in particular \(W_\lambda\ne0\).

Choose an actual \(w\in W_\lambda\) for which
\(N_\lambda^{\nu_\lambda-1}w\ne0\). The original vectors
\(w,N_\lambda w,\ldots,N_\lambda^{\nu_\lambda-1}w\) are independent. If
their linear combination vanished, take its smallest index \(j\) with
nonzero coefficient and apply \(N_\lambda^{\nu_\lambda-1-j}\). The full
higher terms vanish by the known zero power, leaving that nonzero
coefficient times the unchanged nonzero vector
\(N_\lambda^{\nu_\lambda-1}w\), a contradiction. Thus \[
             1\leq\nu_\lambda\leq d_\lambda,
                 \qquad d_\lambda=\dim W_\lambda.             \tag{SR6}
\]

For the full determinant calculation choose a basis by successively
extending bases of \(\ker N_\lambda^j\) for \(j=0,\ldots,\nu_\lambda\).
The actual map sends each such space into the preceding one. In that
ordered basis its matrix is upper triangular with every diagonal entry
zero; no vector or matrix entry is rescaled. For the full original
change of basis \(T_\lambda\), the determinant identities proved there
give \[
 \det(zI_{W_\lambda}-A|_{W_\lambda})
   =\det(T_\lambda)(z-\lambda)^{d_\lambda}\det(T_\lambda^{-1}),
       \qquad \det(T_\lambda)\det(T_\lambda^{-1})=1.           \tag{SR7}
\] Choose bases of the original direct summands and concatenate them,
with full coordinate matrix \(T\). The original operator is block
diagonal in those coordinates. Its full characteristic polynomial is
therefore \[
 \chi_A(z)=\det(T)\prod_\lambda
       \bigl[\det(T_\lambda)(z-\lambda)^{d_\lambda}
                         \det(T_\lambda^{-1})\bigr]\det(T^{-1}).
                                                               \tag{SR8}
\] All determinant factors remain explicit and their inverse products
are proved; the characteristic leading coefficient one was computed from
the original permutation sums in the polynomial and contour lesson,
rather than imposed on the original annihilator. Each root's
characteristic multiplicity is exactly \(d_\lambda\), while the
annihilator exponent is exactly \(\nu_\lambda\). The opposite
determinant convention retains the full factor
\(\det(A-zI)=(-1)^n\chi_A(z)\).

An eigenvalue is a root of \(m\): if \(Av=\lambda v\), \(v\ne0\), then
the original finite polynomial gives \(m(A)v=m(\lambda)v=0\). Conversely
each \(W_\lambda\ne0\) has a nonzero vector in \(\ker N_\lambda\), for
example \(N_\lambda^{\nu_\lambda-1}w\) above. Thus \(Z(m)\) is precisely
the original spectrum. For \(n=0\), both spectra are empty and the
actual characteristic determinant is the empty product one.

\subsubsection{16.3. The original resolvent, pole orders and every unit
factor}\label{the-original-resolvent-pole-orders-and-every-unit-factor}

For \(z\notin Z(m)\), multiplication of the full finite geometric sum
gives on the original summand \[
 (zI-A)^{-1}|_{W_\lambda}
   =\sum_{k=0}^{\nu_\lambda-1}
                         \frac{N_\lambda^k}{(z-\lambda)^{k+1}}.
                                                               \tag{SR9}
\] Indeed multiplication by \((z-\lambda)I-N_\lambda\) produces the
identity minus \(N_\lambda^{\nu_\lambda}/(z-\lambda)^{\nu_\lambda}\),
and that full terminal term is zero by (SR3). The product in the
opposite order is the same since every factor is a power of the same
original restricted operator. Using (SR2)--(SR3), the full global
operator formula is \[
 (zI-A)^{-1}
 =\sum_\lambda\sum_{k=0}^{\nu_\lambda-1}
  \frac{(A-\lambda I)^k
        (c_m^{-1}b_\lambda)(A)(c_mH_\lambda)(A)}
       {(z-\lambda)^{k+1}}.                                  \tag{SR10}
\] Every actual scalar-unit and projection factor is present. Acting on
each original summand proves both inverse identities for this original
matrix; no leading coefficient has been dropped from the receiving map.

Multiplying (SR10) by the full power \((z-\lambda)^{\nu_\lambda}\) and
taking \(z\to\lambda\) gives \[
 \lim_{z\to\lambda}(z-\lambda)^{\nu_\lambda}(zI-A)^{-1}
       =(A-\lambda I)^{\nu_\lambda-1}
              (c_m^{-1}b_\lambda)(A)(c_mH_\lambda)(A)\ne0.     \tag{SR11}
\] All terms with a smaller pole power at \(\lambda\) tend to zero;
every term at a distinct root has a bounded nonzero denominator near
\(\lambda\) and also tends to zero after that multiplication. The
nonzero conclusion follows by applying the right side to the unchanged
\(w\) from Section 16.2. Hence the resolvent's pole order is exactly
\(\nu_\lambda\), which need not equal the characteristic multiplicity
\(d_\lambda\). This is an exact nonzero coefficient and limit, not
merely a distinction between two names.

For example, in the original three coordinates let \[
 A=\begin{pmatrix}\lambda&1&0\\0&\lambda&0\\0&0&\lambda\end{pmatrix},
 \quad
 (zI-A)^{-1}=
 \begin{pmatrix}
 (z-\lambda)^{-1}&(z-\lambda)^{-2}&0\\
 0&(z-\lambda)^{-1}&0\\0&0&(z-\lambda)^{-1}
 \end{pmatrix}.                                               \tag{SR12}
\] Direct multiplication gives both inverse identities and all displayed
zeros. The full determinant is \((z-\lambda)^3\). An original
least-degree annihilator is \(c_m(z-\lambda)^2\), for any original
\(c_m\ne0\): the actual nilpotent part has square zero and is nonzero. A
nonzero degree-zero annihilator is impossible; a degree-one annihilator
would have to vanish on all diagonal entries, leaving its nonzero
leading coefficient in the actual upper-right entry, also impossible.
Thus the characteristic multiplicity is three and the exact
annihilator/resolvent order is two.

\subsubsection{16.4. Original contour and exponential receivers at every
order}\label{original-contour-and-exponential-receivers-at-every-order}

Let \(\Gamma\) be a finite integer linear combination of the original
piecewise \(C^1\) closed paths, avoiding every root. Keep \[
 \operatorname{ind}_\Gamma(\lambda)
       =\frac{1}{2\pi i}\int_\Gamma\frac{dz}{z-\lambda}.        \tag{SR13}
\] For a numerator \(\phi\) having its asserted globally convergent
power series about each of these finitely many roots, the proved
entire-series moment formula of
\href{stable-prerequisite-bridges.pdf}{Polynomial and contour interfaces
for stable boundary models} gives the full receiving identity \[
 \frac{1}{2\pi i}\int_\Gamma\phi(z)(zI-A)^{-1}\,dz
 =\sum_\lambda\operatorname{ind}_\Gamma(\lambda)
    \sum_{k=0}^{\nu_\lambda-1}\frac{\phi^{(k)}(\lambda)}{k!}
        (A-\lambda I)^k
        (c_m^{-1}b_\lambda)(A)(c_mH_\lambda)(A).               \tag{SR14}
\] This follows by integrating the actual finite sum (SR10). No
sum/interchange beyond the proved scalar moments is needed. Every
factorial, root repetition, nilpotent power, contour index and original
projection factor remains. The stated entire-series hypothesis includes
constants, polynomials and exponentials. No formula for an arbitrary
holomorphic numerator on an arbitrary domain is inferred here.

For \(\phi=1\), all higher derivatives vanish exactly; a contour with
index one at precisely the selected roots and zero at the others
therefore gives the sum of their actual \(Q_\lambda\). For
\(\phi(z)=e^{itz}\), the original derivative is
\(\phi^{(k)}(\lambda)=(it)^k e^{it\lambda}\), so (SR14) retains all
powers of \(i,t\) and every factorial.

The corresponding exponential equality is obtained directly from the
original operator series, without a contour assumption: \[
 E_A(t)=\sum_{r=0}^{\infty}\frac{(itA)^r}{r!}
       =\sum_\lambda e^{it\lambda}
          \sum_{k=0}^{\nu_\lambda-1}\frac{(it)^k}{k!}
            (A-\lambda I)^k
            (c_m^{-1}b_\lambda)(A)(c_mH_\lambda)(A).          \tag{SR15}
\] On each original \(W_\lambda\), the two operators \(\lambda I\) and
\(N_\lambda\) commute. Expand every finite binomial power of their sum.
The scalar exponential majorants give absolute convergence on each
compact time interval, including the differentiated series, so
regrouping by the finitely many surviving powers \(k<\nu_\lambda\) is
valid. The full factorial relation \(\binom{r}{k}/r!=1/(k!(r-k)!)\)
gives the two series product. All higher nilpotent terms are the proved
zero operators, not omitted nonzero contributions. Acting on the full
direct sum proves (SR15).

Termwise differentiation of the actual series gives \(D_tE_A=AE_A\),
where \(D_t=-i\partial_t\), and repeated differentiation gives
\(D_t^jE_A=A^jE_A\). The original addition identity gives
\(E_A(-t)E_A(t)=I\). Thus every solution of \(D_tu=Au\) is
\(u(t)=E_A(t)v\), because differentiating the full product
\(E_A(-t)u(t)\) gives zero. Initial evaluation \(u\mapsto u(0)\) and
\(v\mapsto E_A(t)v\) are inverse linear maps; these are the actual
Cauchy maps used in the source.

\subsubsection{16.5. Complete bounded-mode classification, including the
real
roots}\label{complete-bounded-mode-classification-including-the-real-roots}

First every linear map is bounded for the original chosen norm. Choose
an actual basis \(v_1,\ldots,v_n\), retaining these vectors. If
\(v=\sum x_jv_j\), triangle inequality and Cauchy--Schwarz give
\(\|v\|\leq B|x|_2\), with the full constant
\(B=(\sum_j\|v_j\|^2)^{1/2}\). On the coordinate unit sphere the actual
function \(x\mapsto\|\sum_jx_jv_j\|\) is continuous by that inequality,
and is everywhere positive by independence. Coordinate compactness
supplies its actual minimum \(b>0\). Homogeneity then gives
\(b|x|_2\leq\|v\|\). For any original linear map \(T\), this proves \[
 \|Tv\|\leq
     \frac{(\sum_j\|Tv_j\|^2)^{1/2}}{b}\,\|v\|.               \tag{SR16}
\] Every original basis vector and both comparison constants are
retained. These are comparison maps, not replacements of the original
norm. In particular all projections and coordinate functionals used
below are bounded. The zero-dimensional case requires no positive sphere
minimum and every map is the zero map.

On \(W_\lambda\), (SR15) gives for an actual \(v\) \[
 \|E_A(t)v\|\leq e^{-t\,\operatorname{Im}\lambda}
        \sum_{k=0}^{\nu_\lambda-1}
                    \frac{|t|^k}{k!}\|N_\lambda^kv\|.         \tag{SR17}
\] For \(\operatorname{Im}\lambda>0\) every full term tends to zero as
\(t\to+\infty\). To prove the elementary exponential domination, choose
any \(0<a<\operatorname{Im}\lambda\). The scalar series gives
\(e^{at}\geq (at)^{k+1}/(k+1)!\) for \(t>0\), so
\(t^ke^{-at}\leq(k+1)!/(a^{k+1}t)\). The remaining factor
\(e^{-(\operatorname{Im}\lambda-a)t}\) is bounded and tends to zero.
Every derivative is covered by the same actual finite expansion with the
additional fixed factor \(A^j\).

Conversely take \(v\ne0\) and the largest \(j\) for which
\(N_\lambda^jv\ne0\). It exists and lies between zero and
\(\nu_\lambda-1\). Some actual coordinate functional \(\ell\) has
\(\ell(N_\lambda^jv)\ne0\); boundedness was proved in (SR16), without
changing \(v\) or \(\ell\). The complete scalar polynomial in \[
 \ell(E_A(t)v)=e^{it\lambda}
       \sum_{k=0}^j\frac{i^kt^k}{k!}\ell(N_\lambda^kv)         \tag{SR18}
\] has nonzero leading coefficient \(i^j\ell(N_\lambda^jv)/j!\). For
\(t\geq1\), the full reverse triangle inequality bounds its modulus
below by \[
 \frac{|\ell(N_\lambda^jv)|}{j!}t^j
  -\sum_{k=0}^{j-1}\frac{|\ell(N_\lambda^kv)|}{k!}t^k.
                                                               \tag{SR19}
\] If \(j\geq1\), the lower sum is at most \(t^{j-1}\) times its full
coefficient sum. Taking \(t\) at least twice that coefficient sum
divided by the actual nonzero leading modulus makes (SR19) at least half
the full leading term. If \(j=0\), the lower sum is empty and the
modulus is its exact nonzero constant.

It follows that when \(\operatorname{Im}\lambda<0\), every nonzero \(v\)
gives an unbounded solution on the positive half-line, since its scalar
lower bound contains \(e^{-t\operatorname{Im}\lambda}\). When
\(\operatorname{Im}\lambda=0\), it is bounded exactly when \(j=0\),
equivalently \(N_\lambda v=0\); then the solution is exactly
\(e^{it\lambda}v\). On a general \(v\), applying the bounded actual
projection \(Q_\lambda\) to a bounded solution isolates each such
component; components cannot conceal one another's growth. Thus the
complete positive-half-line bounded initial space is \[
 B_A^+=\bigoplus_{\operatorname{Im}\lambda>0}W_\lambda
       \ \oplus\
       \bigoplus_{\operatorname{Im}\lambda=0}
                         \ker(A-\lambda I).                  \tag{SR20}
\] With \(t=-s\), \(s\geq0\), the same full expansions and inequalities
reverse the imaginary-part signs, giving \[
 B_A^-=\bigoplus_{\operatorname{Im}\lambda<0}W_\lambda
       \ \oplus\
       \bigoplus_{\operatorname{Im}\lambda=0}
                         \ker(A-\lambda I).                  \tag{SR21}
\] These formulas prove the precise additional real-root modes rather
than simply discarding the no-real-root hypothesis. Under the original
no-real-spectrum assumption in Section 3 the extra sums are empty, so
its actual upper projection gives \(B_A^+=q_AV\), its lower projection
gives \(B_A^-=(I-q_A)V\), and every such solution and derivative decays
in its own half-line. The original stable dimension counts \(d_\lambda\)
in the upper half-plane, by (SR8), whereas time-polynomial degree is
bounded by \(\nu_\lambda-1\). Neither count can be substituted for the
other.

\subsubsection{16.6. Full ordered parameter derivatives and common
contour
bounds}\label{full-ordered-parameter-derivatives-and-common-contour-bounds}

For an original \(C^r\) family \(A(y)\), near a fixed parameter a
compact contour avoiding its spectrum stays disjoint from nearby
spectra. Indeed the original determinant is continuous on the compact
contour, has a strictly positive modulus minimum there, and small
changes of its full finitely many coefficients preserve half that
minimum. Inversion is continuous through its full cofactor matrix
divided by that nonzero determinant. Differentiating \((zI-A)R=I\) and
multiplying by the actual inverse gives \(\partial_jR=R(\partial_jA)R\),
with this exact product order.

For a nonzero multiindex \(\alpha\), \(|\alpha|\leq r\), the full
formula is \[
 \partial^\alpha R
 =\sum_{k=1}^{|\alpha|}
   \ \sum_{\substack{\alpha^1+\cdots+\alpha^k=\alpha\\
                     |\alpha^1|,\ldots,|\alpha^k|>0}}
    \frac{\alpha!}{\alpha^1!\cdots\alpha^k!}
    R(\partial^{\alpha^1}A)R\cdots
                      (\partial^{\alpha^k}A)R.               \tag{SR22}
\] Here every inner list is ordered; matrices are never commuted. One
direct proof retains the actual Neumann series
\(R(y+h)=\sum_{k\geq0}(R(y)(A(y+h)-A(y)))^kR(y)\) for the original
difference small enough that its operator product norm is less than one.
Its finite Taylor coefficient at \(h^\alpha\) is the sum over ordered
nonzero lists in (SR22); each coefficient is divided by
\(\alpha^1!\cdots\alpha^k!\), and differentiation multiplies it by the
full \(\alpha!\). Terms with more than \(|\alpha|\) differences have no
such coefficient. The remainder bounds in the finite Taylor theorem and
the geometric-series tail make the coefficient passage valid through
order \(r\). Equivalently, labeling each of the \(|\alpha|\) successive
derivative applications partitions the labels into ordered nonempty
batches; their multinomial counts are precisely the displayed
coefficients. Both constructions include every repeated derivative and
every original factor.

For a compact parameter set with no real spectrum, the original norm has
a common bound \(M\) for \(A(y)\), and every eigenvalue satisfies
\(|\lambda|\leq M\) by the original eigenvector inequality in Section
9.8 of \href{stable-prerequisite-bridges.pdf}{Polynomial and contour
interfaces for stable boundary models}. A hypothetical sequence of
spectral points whose imaginary parts tend to zero has a convergent
parameter subsequence and a bounded complex-point subsequence.
Continuity of the original full determinant would give a real eigenvalue
at the limit, contradicting the original hypothesis. Thus there is an
actual gap \(g>0\).

Choose an original positive rectangle enclosing all upper roots, with
lower height \(0<\gamma<g\), upper height greater than \(M\), and
left/right coordinates beyond \(-M,M\). Its orientation and all its
edges are retained; the proven rectangle index is one at every upper
root and zero at every lower root. Let \(L_\Gamma\) be its full length
and let \(C_R=\sup_{y,z\in K\times\Gamma}\|R(y,z)\|\). This is finite by
the original determinant/cofactor continuity and compactness. The
original contour formula therefore gives \[
 \|D_t^jE_{A(y)}(t)q_{A(y)}\|
 \leq \frac{L_\Gamma}{2\pi}
       \bigl(\sup_{z\in\Gamma}|z|^j\bigr) C_R e^{-\gamma t},
                     \qquad t\geq0.                          \tag{SR23}
\] The full \(1/(2\pi i)\) contour factor yields \(1/(2\pi)\) in the
norm bound, and \(D_t^j e^{itz}=z^j e^{itz}\) retains the original
differential convention.

For each parameter derivative in a fixed original chart, insert every
term of (SR22) into the same contour. Its original operator norm is
bounded by \[
 \frac{L_\Gamma}{2\pi}\sup_\Gamma|z|^j\, e^{-\gamma t}
 \sum_{k=1}^{|\alpha|}
 \ \sum_{\substack{\alpha^1+\cdots+\alpha^k=\alpha\\
                   |\alpha^\ell|>0}}
   \frac{\alpha!}{\alpha^1!\cdots\alpha^k!}
   C_R^{\,k+1}\prod_{\ell=1}^k
          \sup_{y\in K}\|\partial^{\alpha^\ell}A(y)\|.          \tag{SR24}
\] For \(\alpha=0\) the actual bound is (SR23). The finite derivative
bounds hold in each chart on the compact set under consideration. This
proves the full regularity and uniform decay statements in Section 4
even when roots collide in one half-plane; no selection of individual
smooth roots was used. It makes no global gap claim on a noncompact
parameter set.

\subsubsection{16.7. Original matrix-polynomial Cauchy and numerator
maps}\label{original-matrix-polynomial-cauchy-and-numerator-maps}

In this section \(m\) has the original meaning in Section 2 as
matrix-polynomial order. Keep \(p(z)=p_m z^m+\sum_{j=0}^{m-1}p_jz^j\),
with \(m\geq1\), \(p_m\in\operatorname{GL}(E)\), and all original matrix
coefficients in their original order. If \(E=0\), all state spaces and
solution spaces are zero and every determinant is the empty product one.
Otherwise let \(N=\dim E>0\). The actual original-equation companion and
jet maps are \[
 C(x_0,\ldots,x_{m-1})
  =(x_1,\ldots,x_{m-1},-\sum_{j=0}^{m-1}p_m^{-1}p_jx_j),
 \qquad Ju=(u,Du,\ldots,D^{m-1}u).                            \tag{SR25}
\] The original equation implies \(D(Ju)=CJu\) through its last row
after multiplying on the left by the displayed \(p_m^{-1}\). Conversely
the first \(m-1\) rows give every jet coordinate from the first, and
multiplying the full last row by the original \(p_m\) gives exactly
\(p_mD^mu+\sum_{j=0}^{m-1}p_jD^ju=0\). Both implications retain every
left factor and do not commute any coefficients. Therefore the actual
map \(u\mapsto Ju(0)\) is inverse to \(x\mapsto \pi_0 E_C(t)x\), by the
already proved first-order uniqueness; the original solution dimension
is \(mN\).

For a direct characteristic calculation make the original triangular
coordinate change \(x_0=y_0,\ x_j=zx_{j-1}+y_j\), \(1\leq j<m\), on the
state coordinates. Its original block diagonal is all identities, so its
determinant is one. The first \(m-1\) output rows of \((zI-C)x\) are
\(-y_1,\ldots,-y_{m-1}\). The coefficient of \(y_0\) in the last row is
exactly \(p_m^{-1}p(z)\), with this order. Move that last block row to
the front: its full sign is \((-1)^{N^2(m-1)}\). Each remaining negative
identity gives the total factor \((-1)^{N(m-1)}\). Eliminate every other
entry of the first row by those identity rows, retaining the original
determinant under each row addition. Thus \[
 \det(zI_{E^m}-C)
   =(-1)^{N^2(m-1)}(-1)^{N(m-1)}
           \det(p_m^{-1}p(z))
   =(-1)^{N^2(m-1)}(-1)^{N(m-1)}
           \det(p_m^{-1})\det p(z).                           \tag{SR26}
\] Both signs are retained and their product is one because their full
exponent is \(N(N+1)(m-1)\), an even integer. Also
\(\det(p_m^{-1})\det p_m=1\) by the original determinant multiplication
proof. Hence the nonzero original prefactor preserves every root and
multiplicity through an explicit scalar-unit map. It is not a
replacement of the original determinant. All formulas include \(m=1\),
where the negative block list and row permutation are empty.

The full original polynomial numerator of an arbitrary actual state is
\[
 \mathcal U_x^p(z)
     =\sum_{0\leq k\leq j<m}p_{j+1}x_{j-k}z^k,\qquad
 \mathcal U_x^p(z)=\sum_{k=0}^{m-1}u_kz^k.                    \tag{SR27}
\] Here \(p_m\) is the original invertible leading matrix, including at
the highest coefficient \(u_{m-1}=p_mx_0\). The inverse map from the
full numerator coefficients is the ordered triangular recursion \[
 x_r=p_m^{-1}
       \left(u_{m-1-r}
           -\sum_{q=0}^{r-1}p_{m-r+q}x_q\right),
                   \qquad r=0,\ldots,m-1.                    \tag{SR28}
\] For \(r=0\) the sum is empty. To verify each row, the coefficient of
\(z^{m-1-r}\) in (SR27) is \(\sum_{q=0}^r p_{m-r+q}x_q\); its last term
is precisely \(p_mx_r\). Multiplication on the left by the actual
inverse yields (SR28). Thus the original numerator map is a linear
bijection, with every matrix order and leading factor explicit. Its
monic comparison \(r(z)=p_m^{-1}p(z)\) has
\(\mathcal U_x^r=p_m^{-1}\mathcal U_x^p\); the comparison does not erase
\(p_m\) from either actual forward or inverse map.

To find the first coordinate of the actual resolvent solve
\((zI-C)w=x\). Its first \(m-1\) rows give \[
 w_j=z^jw_0-\sum_{q=0}^{j-1}z^{j-1-q}x_q,\quad
 p(z)w_0=\mathcal U_x^p(z),\quad
 \pi_0(zI-C)^{-1}x=p(z)^{-1}\mathcal U_x^p(z).                \tag{SR29}
\] For the last row, multiply by the original \(p_m\) and substitute all
preceding rows. Its right side becomes the full sum
\(p_mx_{m-1}+p_m\sum_{q=0}^{m-2}z^{m-1-q}x_q+
  \sum_{j=1}^{m-1}p_j\sum_{q=0}^{j-1}z^{j-1-q}x_q\). For each original
\(x_q\), its coefficient is exactly \(\sum_{j=q}^{m-1}p_{j+1}z^{j-q}\),
proving the second identity. All denominators are legitimate exactly
away from the original roots by (SR26) and the actual
determinant-inverse criterion; no linear factorization of a matrix
polynomial has been assumed. Applying the already proved exponential
contour receiver to the original \(C\) gives \[
 u(t)=\frac{1}{2\pi i}\int_\Gamma
                   p(z)^{-1}\mathcal U_x^p(z)e^{itz}\,dz,
                  \qquad Ju(0)=x.                            \tag{SR30}
\] Every root multiplicity, matrix pole order, original leading factor,
factorial in the earlier scalar pole moments and the original contour
prefactor is retained. This is the full original-equation receiving
extension of the monic formula (S7) in Section 2; its arbitrary
numerator is unique through (SR28).

Boundedness of the original first component gives boundedness of the
full original state without assuming spectral separation. With the
original state and component norms let
\(h(x)=\int_0^1\|\pi_0E_C(s)x\|^2\,ds\). If \(h(x)=0\), continuity makes
its integrand zero on the entire interval. The first-coordinate solution
is then identically zero there. Its derivatives at zero vanish, and its
Cauchy coordinates \(x_0,\ldots,x_{m-1}\) vanish by the first rows of
(SR25). Thus \(h\) is positive on the original nonzero state space. Its
continuity follows from the original finite matrix exponential and norm
bounds; the original unit sphere is compact by the retained coordinate
comparisons (SR16). Let \(c>0\) be its actual minimum on that sphere.
Homogeneity and the original flow addition rule then give \[
 c\|Ju(t)\|^2\leq
        \int_0^1\|u(t+s)\|^2\,ds,\qquad t\geq0.              \tag{SR31}
\] Hence a bounded \(u\) has a bounded original jet; conversely its
first-coordinate projection is bounded whenever its jet is bounded. The
exact derivative conventions are \(D_t^ku=\pi_0C^kJu(t)\) and
\(\partial_t^ku=i^k\pi_0C^kJu(t)\), retaining the full original factor
\(i^k\). The actual finite operator norm bounds prove boundedness for
every derivative. The negative-half-line proof uses the original
interval \([-1,0]\) with its own positive minimum and the same exact
Cauchy map. For the zero state space the implication is immediate and no
minimum on an empty sphere is introduced.

When the original determinant has no real roots, (SR26), (SR20)--(SR21)
and (SR31) identify the original bounded solution space with the
companion's upper primary space and prove decay of every derivative. Its
dimension is the sum of the upper-root algebraic multiplicities of the
original \(\det p\), with the original \(\det(p_m^{-1})\) map displayed
in (SR26). The full leading-coefficient homotopy and transported
boundary measurements in Section 12 retain their separate original
formulas and proofs. Formulas (SR25)--(SR31) give the complete
original-equation Cauchy, numerator, resolvent and contour maps without
dropping its leading matrix.

\subsection{17. Bounded profiles with complex
parameters}\label{AN03-STB-BOUNDED-001}

This separate extension allows real characteristic roots and complex
stabilization parameters. Sections 3--12 keep their original stable,
no-real-root statements. Here bounded profiles may contain nondecaying
real-root modes. The complete classification and jet estimate in
Sections 16.5 and 16.7 provide the additional argument; no uniform
bundle claim through a real-root crossing is made.

\subsubsection{17.1. The original polynomial and its full stabilized
family}\label{the-original-polynomial-and-its-full-stabilized-family}

Keep the original finite-dimensional complex space, its chosen norm, and
\[
 p(z)=p_mz^m+\sum_{j=0}^{m-1}p_jz^j,\quad
 p_m\in\operatorname{GL}(E),\quad m>1,\quad N=\dim E,
 \quad D=-i\partial_t.
 \tag{SB1}
\] There is no restriction on the real roots of \(\det p\). Fix \[
 \tau\in\mathbb C,\qquad \lambda\in\mathbb C,
 \qquad \alpha=\operatorname{Re}\lambda>0,
 \qquad L(z)=z+i\lambda,\quad R(z)=z-i\lambda.
 \tag{SB2}
\] Use \(r(z)=p_m^{-1}p(z)\) only for the coefficient calculation
(S15)--(S17), retaining the original polynomial. Those identities use
\(\lambda\ne0\), scalar polynomial algebra and the actual ordered
coefficients; their proofs hold for complex \(\lambda\) without a
positivity assumption. Form \(P_\tau[r]\) by exactly (S20)--(S21), with
every displayed power of \(\tau\), including the empty middle sum when
\(m=2\). The original family and leading coefficient are \[
 \begin{split}
 \mathcal P_\tau(z)&=\operatorname{diag}(p_m,I,\ldots,I)P_\tau[r](z),\\
 \mathcal C_\tau&=\operatorname{diag}(p_m,I,\ldots,I)C_\tau[r],\\
 \mathcal C_\tau^{-1}&=C_\tau[r]^{-1}
                         \operatorname{diag}(p_m^{-1},I,\ldots,I),\\
 \det\mathcal C_\tau&=\det p_m.
 \end{split}
 \tag{SB3}
\] Indeed the factorization (S23) is still \(H_\tau(I-\tau S)\). Its
actual nilpotents satisfy \(N_\tau^2=0\), \(S^m=0\), so (S25)--(S26) is
the same finite inverse for every complex \(\tau\). Both triangular
determinants equal one. This proves (SB3), keeping the inverse product
in its required order.

The lower-block elimination in Section 9 also uses no real-parameter
restriction. For \(L(z)\ne0\) it has the same lower-block determinant
and the same Schur complement \(r(z)\). Thus, with both original
determinant factors visible, \[
 \begin{split}
 \det\mathcal P_\tau(z)
   &=\det p_m\,\det r(z)\,L(z)^{m(m-1)N}\\
   &=\det p_m\,\det(p_m^{-1})\det p(z)\,
                                  L(z)^{m(m-1)N}\\
   &=\det p(z)\,L(z)^{m(m-1)N}.
 \end{split}
 \tag{SB4}
\] Both sides are polynomials, so the identity holds also at \(L(z)=0\).
The auxiliary root is \(-i\lambda\), with imaginary part \(-\alpha<0\)
and multiplicity \(m(m-1)N\). If it is already an original root, the
multiplicities add. Every original real root remains in the determinant,
with its full algebraic multiplicity.

\subsubsection{17.2. The bounded inverse and all original derivative
bounds}\label{the-bounded-inverse-and-all-original-derivative-bounds}

Let \(\mathcal B^+(p)\) consist of smooth solutions of the original
equation on \(\mathbb R\) that are bounded for \(t\geq0\); define
\(\mathcal B^+(\mathcal P_\tau)\) in the same way. They include all
bounded real-root modes. By (SR31) and (SB3), boundedness of either
solution controls its complete original Cauchy jet and every derivative.
This implication requires neither spectral separation nor exponential
decay.

If \(g\) is smooth on \(\mathbb R\) and all its derivatives are bounded
on the positive half-line, set \[
 T_\lambda g(t)=-i\int_0^\infty e^{-\lambda s}g(t+s)\,ds.
 \tag{SB5}
\] For negative \(t\), split this integral at \(\max(0,-t)\). Its first
part is over a finite interval of smooth values; its tail has the same
exponential majorant as on the positive half-line. On a compact set of
\(t\)'s the finite interval can be chosen uniformly. This proves
convergence and justifies every differentiation in \(t\) by the
corresponding derivative majorant.

To verify the equation, integrate the derivative of
\(e^{-\lambda s}g(t+s)\). The upper endpoint is zero, the lower endpoint
is \(g(t)\), and consequently \(\int_0^\infty e^{-\lambda s}g'(t+s)\,ds
 =-g(t)+\lambda\int_0^\infty e^{-\lambda s}g(t+s)\,ds\). Including the
original factor \(-i\) gives \((T_\lambda g)'=\lambda T_\lambda g+ig\),
hence \(L(D)T_\lambda g=g\). Two bounded solutions differ by
\(ce^{\lambda t}\). Its norm is \(\|c\|e^{\alpha t}\), so boundedness
forces \(c=0\). Thus (SB5) is the unique bounded inverse. It commutes
with each \(D^k\), by the just-proved differentiation.

For \(d\geq0\), retain the actual constants
\(\|D^kg(t)\|\leq M_{k,d}e^{-dt}\), \(t\geq0\). Direct integration
yields \[
 \|D^kT_\lambda g(t)\|
       \leq\frac{M_{k,d}}{\alpha+d}e^{-dt}.
 \tag{SB6}
\] In particular \(d=0\) gives the full bounded, nondecaying case with
denominator \(\alpha\).

\textbf{Theorem.} For every parameter pair (SB2), first-coordinate
projection is a linear bijection \[
 \Pi:\mathcal B^+(\mathcal P_\tau)\longrightarrow\mathcal B^+(p),
 \qquad U\longmapsto U_0.
 \tag{SB7}
\] Its inverse is exactly \[
 U_0=u,\qquad U_j=\tau T_\lambda R(D)U_{j-1},\quad 1\leq j<m.
 \tag{SB8}
\] If \(\|D^ku(t)\|\leq M_{0,k,d}e^{-dt}\) for all required derivatives,
then the lifted components satisfy \[
 \|D^kU_j(t)\|\leq
 \frac{|\tau|^j}{(\alpha+d)^j}e^{-dt}
 \sum_{a=0}^j\binom ja |\lambda|^{j-a}M_{0,k+a,d}.
 \tag{SB9}
\] For \(j=0\), the single summand is the original bound.

\textbf{Proof.} For a bounded original family solution the lower row
says \(L(D)^{m-1}g_j=0\), where \(g_j=L(D)U_j-\tau R(D)U_{j-1}\). All
its derivatives, and thus \(g_j\), are bounded by (SR31). Substituting
\(g_j=e^{\lambda t}h_j\) gives
\(L(D)^{m-1}g_j=(-i)^{m-1}e^{\lambda t}h_j^{(m-1)}\). Thus \(h_j\) is a
vector polynomial of degree at most \(m-2\). If it is nonzero, an actual
bounded coordinate functional detects a nonzero leading coefficient. The
full polynomial lower bound (SR19), multiplied by \(e^{\alpha t}\), is
unbounded. Therefore \(g_j=0\). This proves every recurrence (S29) on
the present bounded space, including complex \(\tau\).

Substitute these recurrences in the original first row. Its complete
expression is \[
 p_m\left((1-\tau^m)r(D)U_0+
       \tau^m\sum_{j=0}^ma_jR(D)^jL(D)^{m-j}U_0\right)
                  =p_mr(D)U_0=p(D)U_0=0.
 \tag{SB10}
\] The last summand uses the additional \(R(D)\) in (S20), exactly as in
Section 10. This proves the codomain of (SB7).

Conversely, (SR31) bounds every derivative of a bounded original \(u\),
so each step (SB8) is defined by (SB5), has bounded derivatives by
(SB6), and satisfies the actual lower row. Equation (SB10) gives the
first row. This proves surjectivity. Uniqueness of the bounded inverse
successively shows that a lifted solution with \(U_0=0\) has every
component zero, proving injectivity. In particular \(\tau=0\) gives
\((u,0,\ldots,0)\) with no division by that parameter.

Finally \(R(D)=D-i\lambda\) and (SB6) give the recursion for the
derivative bounds \(M_{j,k,d}\leq |\tau|(\alpha+d)^{-1}
 (M_{j-1,k+1,d}+|\lambda|M_{j-1,k,d})\). Starting at \(j=0\), its two
finite sums combine by \(\binom{j-1}{a-1}+\binom{j-1}a=\binom ja\), with
each endpoint term retained. Induction proves exactly (SB9). \(\square\)

\subsubsection{17.3. Exact iterated kernels and parameter
derivatives}\label{exact-iterated-kernels-and-parameter-derivatives}

The lift has a full integral formula, rather than only a formal inverse
power. For \(j\geq1\), \[
 T_\lambda^jg(t)=\frac{(-i)^j}{(j-1)!}
        \int_0^\infty s^{j-1}e^{-\lambda s}g(t+s)\,ds.
 \tag{SB11}
\] Induction proves this formula. The step from \(j\) to \(j+1\) has an
absolutely integrable double integral, bounded on the positive half-line
by the complete product envelope \(M_{0,d}r^{j-1}e^{-(\alpha+d)(r+s)}\).
For negative \(t\), use the same finite-interval/tail split as above.
Fubini therefore applies. The substitution \(v=r+s\), with Jacobian one
and \(0\leq r\leq v\), has inner integral
\(\int_0^v r^{j-1}/(j-1)!\,dr=v^j/j!\). The factor becomes exactly
\((-i)^{j+1}\), proving (SB11) with its full factorial.

Only scalar constant-coefficient operators are commuted in the following
binomial calculation: \[
 U_j=\tau^j\sum_{a=0}^j\binom ja(-i\lambda)^{j-a}
                      T_\lambda^jD^au,\qquad j\geq1.
 \tag{SB12}
\] Indeed \(T_\lambda\) commutes with \(D\), and expansion of
\((D-i\lambda)^j\) supplies precisely these coefficients. No two
original matrix coefficients are commuted.

For each integer \(b\geq0\), differentiation in \(\lambda\) gives \[
 \partial_\lambda^bT_\lambda^jg(t)
  =\frac{(-i)^j(-1)^b}{(j-1)!}
       \int_0^\infty s^{j+b-1}e^{-\lambda s}g(t+s)\,ds,
 \tag{SB13}
\] and \[
 \|D^k\partial_\lambda^bT_\lambda^jg(t)\|
 \leq\frac{(j+b-1)!}{(j-1)!}
       \frac{M_{k,d}}{(\alpha+d)^{j+b}}e^{-dt}.
 \tag{SB14}
\] On a compact subset of \(\operatorname{Re}\lambda>0\), the displayed
integrals have common integrable derivative envelopes, which justifies
the differentiation. The scalar moment used here is exact:
\(\int_0^\infty s^he^{-cs}\,ds=h!c^{-h-1}\) for \(c>0\), \(h\geq0\). At
\(h=0\) it is direct integration. For \(h>0\), integration by parts has
zero upper endpoint and zero lower endpoint, and gives \(h/c\) times the
preceding moment. This proves every factorial in (SB14).

For \(c\geq0\) the full product rule for the original lift is \[
 \begin{split}
 \partial_\lambda^cU_j={}&\tau^j\sum_{a=0}^j\binom ja
   \sum_{\substack{0\leq b\leq c\\c-b\leq j-a}}
    \binom cb(-i)^{j-a}\frac{(j-a)!}{(j-a-c+b)!}\\
 &\hspace{15mm}\cdot\lambda^{j-a-c+b}
       \partial_\lambda^b(T_\lambda^jD^au),\qquad j\geq1.
 \end{split}
 \tag{SB15}
\] Its indices make every factorial and power defined; differentiated
polynomial terms outside this range are zero. Taking norms and using
(SB14) proves the explicit bound \[
 \begin{split}
 \|D^k\partial_\lambda^cU_j(t)\|
 \leq{}&|\tau|^je^{-dt}\sum_{a=0}^j\binom ja
   \sum_{\substack{0\leq b\leq c\\c-b\leq j-a}}
     \binom cb\frac{(j-a)!}{(j-a-c+b)!}
             |\lambda|^{j-a-c+b}\\
 &\hspace{8mm}\cdot\frac{(j+b-1)!}{(j-1)!}
             \frac{M_{0,k+a,d}}{(\alpha+d)^{j+b}}.
 \end{split}
 \tag{SB16}
\] Here \(\lambda\ne0\) by (SB2), and every power-zero factor equals
one. For the other parameter, the complete formula is \[
 \partial_\tau^qU_j=
 \begin{cases}
 \displaystyle\frac{j!}{(j-q)!}\tau^{j-q}(T_\lambda R(D))^ju,
                                      &0\leq q\leq j,\\
 0,&q>j.
 \end{cases}
 \tag{SB17}
\] At \(\tau=0\) the power-zero term is one. The coordinate \(U_0=u\) is
independent of both parameters; its positive-order parameter derivatives
vanish. Applying (SB17) to (SB15) gives all mixed derivatives, with
exactly the same finite product-rule coefficients.

For fixed \(p\) and \(u\), these formulas prove holomorphic dependence
of the lift on \(\lambda\) in its right half-plane and polynomial
dependence on \(\tau\). To verify the analytic assertion directly,
expand \(e^{-hs}\) in (SB5) at a fixed \(\lambda\). For \(|h|<\alpha\),
the sum of integral norms in any bounded derivative seminorm is at most
\(M_{k,0}\alpha^{-1}\sum_{a\geq0}(|h|/\alpha)^a\). It converges, so
termwise integration gives the Taylor series in that seminorm. On a
compact negative \(t\) interval, the finite first part has uniformly
bounded smooth input values and an entire exponential series; the
remaining tail has the same convergent majorant, with its finite shift
factor. This proves the assertion in each local derivative seminorm as
well. Formula (SB11), finite products and (SB12) then give the asserted
dependence of every component. This concerns a fixed original solution
space; it does not assert smoothness of bounded spaces when an original
root crosses the real axis.

\subsubsection{17.4. Real modes, multiplicities and the exact Cauchy
maps}\label{real-modes-multiplicities-and-the-exact-cauchy-maps}

For the original companion \(C\) in (SR25), the bounded initial space is
exactly \[
 J\mathcal B^+(p)(0)=
  \bigoplus_{\operatorname{Im}\zeta>0}W_\zeta(C)
  \ \oplus\!\bigoplus_{\zeta\in\mathbb R}\ker(C-\zeta I).
 \tag{SB18}
\] Only actual roots contribute a summand. Formula (SR20) proves this
for the full state and (SR31) identifies first-coordinate boundedness
with state boundedness. The original Cauchy map and its inverse in
(SR25) identify it with the actual solution space.

The exact real-root kernel map, for each real \(\zeta\), is \[
 \ker p(\zeta)\longrightarrow\ker(C-\zeta I),\qquad
 v\longmapsto(v,\zeta v,\ldots,\zeta^{m-1}v),
 \qquad\text{inverse }\pi_0.
 \tag{SB19}
\] The first \(m-1\) rows of the eigenvector equation force precisely
these coordinates. The last row, multiplied on the left by the actual
\(p_m\), says \(p_m\zeta^mv+\sum_{j=0}^{m-1}p_j\zeta^jv=p(\zeta)v=0\).
Both directions follow, proving the map and inverse without dropping the
leading coefficient. Formula (SR26) retains its complete signs and
factor \(\det(p_m^{-1})\); hence every upper-root primary dimension
equals the algebraic multiplicity of that root in the original
\(\det p\). Consequently \[
 \dim\mathcal B^+(p)
 =\sum_{\operatorname{Im}\zeta>0}
                     \operatorname{mult}_\zeta(\det p)
       +\sum_{\zeta\in\mathbb R}\dim\ker p(\zeta).
 \tag{SB20}
\] Real bounded modes are counted by their geometric nullities, not by
substituting their entire algebraic multiplicities. Higher real Jordan
terms give the unbounded time polynomials already proved in
(SR18)--(SR20).

For real \(\zeta\), \(L(\zeta)\ne0\), since its imaginary part is
\(\alpha\). The original stabilized real-root kernel and its inverse are
\[
 v\longmapsto
       \left(\left(\tau\frac{R(\zeta)}{L(\zeta)}\right)^jv
                         \right)_{j=0}^{m-1},
 \qquad
 \ker p(\zeta)\simeq\ker\mathcal P_\tau(\zeta),
 \qquad\text{inverse }\pi_0.
 \tag{SB21}
\] The lower rows uniquely force these coordinates because
\(L(\zeta)\ne0\); the first row is the actual Schur complement
\(p(\zeta)v\) proved in (SB4). This proves the displayed bijection. For
\(u(t)=e^{it\zeta}v\), direct integration in (SB5) gives
\(T_\lambda u=-i(\lambda-i\zeta)^{-1}u\), and
\(L(\zeta)=i(\lambda-i\zeta)\). Thus (SB8) gives exactly the coefficient
in (SB21), preserving the original frequency and factor \(-i\). At
\(\tau=0\), its first coordinate is one times \(v\), and every
positive-index coordinate is zero. The auxiliary lower root supplies no
additional bounded mode by the classification (SR20).

\subsubsection{17.5. Every finite boundary measurement and its ordered
remainder}\label{every-finite-boundary-measurement-and-its-ordered-remainder}

For the original finite family (S33), with arbitrary finite normal
orders, put \[
 \mathfrak B u=(B_\ell(D)u(0))_\ell,
 \quad\mathfrak B_\tau U=(B_\ell(D)U_0(0))_\ell,
 \quad\mathfrak B_\tau=\mathfrak B\Pi.
 \tag{SB22}
\] The target remains the actual \(G=\bigoplus_\ell G_\ell\), including
the empty target. The bijection (SB7) proves
\(\ker\mathfrak B_\tau=\Pi^{-1}(\ker\mathfrak B)\) and
\(\operatorname{im}\mathfrak B_\tau=\operatorname{im}\mathfrak B\): the
first follows by evaluating (SB22), and the second follows in both
directions using the inverse lift. Thus bijectivity, kernel dimension
and image are preserved for every complex \(\tau\), including zero, and
every \(\lambda\) in (SB2).

When \(d_\ell<m\), equations (S35)--(S36) retain exactly their full
coefficients, including \(i^{r+1-m}\), \(2^{1-m}\), \(\lambda^{r+1-m}\),
and the entire signed binomial sum. The polynomial change of variables
needs only \(\lambda\ne0\), so applies here. The same recurrence proves
(S37) for \(\tau\ne0\) as an equality of functions on the lifted bounded
space. It remains unsuitable for an ambient operator at zero; (SB22) is
the actual endpoint map.

For any larger boundary degree one can also prove an exact polynomial
reduction without altering the original equation. If its current leading
coefficient is \(b\) and its degree is \(d\geq m\), subtract \[
 (b p_m^{-1})z^{d-m}p(z).
 \tag{SB23}
\] This cancels exactly the leading term \(bz^d\), with the original
composition order. Repeating the strict degree decrease gives
\(B(z)=Q(z)p(z)+B_{<m}(z)\), where \(\deg B_{<m}<m\). Uniqueness follows
because any nonzero difference \(Q\), with leading coefficient \(q\),
gives the nonzero leading coefficient \(q p_m\) in \(Qp\); invertibility
of \(p_m\) makes its degree at least \(m\). It cannot equal a polynomial
of degree less than \(m\). On actual solutions, constant-coefficient
differentiation and these ordered compositions give
\(B(D)u=Q(D)p(D)u+B_{<m}(D)u=B_{<m}(D)u\). Thus the actual
arbitrary-order measurements can be computed by this remainder and the
original coefficient formula (S36), with every subtracted term recorded.
This finite-dimensional normal-variable proof makes no reduction of
tangential PDE orders or their Sobolev shifts.

\subsubsection{17.6. The exact measurement map during rate
deformation}\label{the-exact-measurement-map-during-rate-deformation}

Return only for this paragraph to the original no-real-root branch (S9).
Let \(C\) be the original companion, \(q=q_C\) its full upper
projection, and \(\lambda_0>0\) a fixed real number. The deformation in
Section 5 is \[
 C_\theta=(1-\theta)C+i\theta\lambda_0(2q-I),
       \qquad0\leq\theta\leq1.
 \tag{SB24}
\] Its bounded initial space is the same \(qE^m\), by (S12). An original
profile is \(u(t)=\pi_0E_C(t)v\); the new first-order state profile is
\(w(t)=E_{C_\theta}(t)v\), for the same \(v\in qE^m\). Sending \(u\) to
\(w\) is a bijection: its forward map uses the unique original Cauchy
state, and its inverse is \(w\mapsto\pi_0E_C(t)w(0)\). First-order
uniqueness and (SR25) prove both compositions to be identities.

For the original boundary polynomial, keep every original coefficient
and derivative order in \[
 H_\ell=\sum_{r=0}^{d_\ell}b_{\ell r}\pi_0 C^r,
 \qquad\widetilde{\mathfrak B}_\theta(w)
                         =(H_\ell w(0))_\ell.
 \tag{SB25}
\] Since \(D^ru(0)=\pi_0C^rv\), this is precisely the transported
original measurement. Its commuting map is proved by substituting
\(w(0)=v\), so it preserves the original kernel, image and bijectivity.

If one instead applies the unchanged polynomial \(B_\ell(D)\) to the
first coordinate of the new state profile, the measurement is
\(\sum_r b_{\ell r}\pi_0C_\theta^rv\). The complete difference from
(SB25) is \[
 \begin{split}
 \Delta_{\ell,\theta}
  &=\sum_{r=1}^{d_\ell}b_{\ell r}\pi_0
       \sum_{a=0}^{r-1}C_\theta^{r-1-a}(C_\theta-C)C^a,\\
 C_\theta-C&=\theta\big(i\lambda_0(2q-I)-C\big).
 \end{split}
 \tag{SB26}
\] For each \(r\), expansion of the middle difference produces
\(C_\theta^{r-a}C^a-C_\theta^{r-1-a}C^{a+1}\). Successive terms
telescope, retaining their order, to \(C_\theta^r-C^r\). This proves
(SB26) without assuming commutation. The zero-order summand has zero
difference; for degree zero the displayed sum is empty.

A concrete original order-two example shows why this receiving map
matters. Take \[
 p(z)=(z-i)^2,\quad
 C=\begin{pmatrix}0&1\\1&2i\end{pmatrix},\quad
 q=I,\quad v=(1,i),\quad\lambda_0=2,\quad B(z)=z.
 \tag{SB27}
\] Here \(Cv=iv\), \(C_\theta v=i(1+\theta)v\), and \(v\) belongs to the
bounded upper space. The original measurement and its transported value
are both \(i\). Applying the unchanged derivative polynomial to the
deformed first coordinate instead gives \(i(1+\theta)\), with difference
\(i\theta\). Both profiles are exponentially decreasing. This proves
that preservation of the initial stable space does not itself preserve
unchanged derivative expressions; (SB25) supplies the exact
correspondence. The earlier stable-bundle statement is retained with its
original scope.

\subsubsection{17.7. A real double root and an exact complex
lift}\label{a-real-double-root-and-an-exact-complex-lift}

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio,alt={Original real and auxiliary roots, with the exact bounded lifted profile}]{../figures/bounded-boundary-original-modes.png}}
\caption{Original real and auxiliary roots, with the exact bounded
lifted profile}
\end{figure}

The figure uses the original polynomial \(p(z)=3z^2\), \(E=\mathbb C\),
\(m=2\), \(\lambda=1+i\), \(\tau=i\). The coefficient calculation gives
\(a_0=1/4,a_1=1/2,a_2=1/4\), since \(r(z)=z^2=(L(z)+R(z))^2/4\). The
full original determinant is \[
 \det\mathcal P_\tau(z)=3z^2(z+i\lambda)^2,
                  \qquad\det\mathcal C_\tau=3.
 \tag{SB28}
\] The original root zero has algebraic multiplicity two, but its
bounded original space consists exactly of constants and has dimension
one: \(D^2u=0\) gives \(u=a+bt\), bounded exactly when \(b=0\). The
auxiliary root \(-i\lambda=1-i\) also has multiplicity two and has no
bounded positive-half-line mode. Formula (SB21) gives the exact lifted
constant \(U=(1,-\tau)=(1,-i)\). The original first-row value at \(D=0\)
is \(3[-\tau^2\lambda^2/4+\tau^2\lambda^2/4]=0\), while the lower row is
\(-\lambda^2(-\tau)-\tau\lambda^2=0\). These are symbolic equalities
with all factors retained. The right panel shows
\(\operatorname{Re}U_0=1\), \(\operatorname{Im}U_1=-1\), and both zero
components. Its finite plotting interval is an illustration of this
exact profile, not a numerical proof of boundedness. The editable figure
source and parameters preserve these original coordinates. No novelty
claim is made for this extension.

If \(E=0\), both solution spaces and all kernels above are zero, every
determinant is the empty product one, and every measurement has the
zero-domain image. The proofs use no minimum on an empty sphere. All
statements about nonzero leading coordinates and roots then disappear,
while the zero-space bijections and arbitrary target convention remain
exact.

\subsection{Further questions and analytic
use}\label{further-questions-and-analytic-use}

The next analytic use of this algebra is a family over nonzero
tangential covectors. Section 4 supplies its stable vector bundle, and
(S33) supplies the finite-dimensional complementing map. A global
boundary parametrix also needs symbol calculus, coordinate invariance,
bundle gluing and adjoint theory; \href{calderon-cauchy-data.pdf}{Cauchy
data from jumps and residues} and
\href{general-boundary-fredholm.pdf}{Solving an elliptic system from
compatible boundary measurements} develop those analytic steps. A
fiberwise stable isomorphism alone does not establish their Sobolev
estimates or index statements.

For the analytic use of a homogeneous differential principal polynomial,
\href{calderon-cauchy-data.pdf\#section*.57}{Cauchy data from jumps and
residues, Section9} proves the exact reflection of the original
solutions and every Cauchy jet. Its positive mode bundle together with
the antipodal pullback of that bundle maps bijectively onto the entire
original jet bundle, with an explicit inverse. In dimension at least
three the original stable rank is necessarily half the total solution
rank; in dimension two the antipodal ranks are complementary and may
differ. These conclusions use the additional homogeneous differential
and ellipticity hypotheses proved there. They are not imposed on the
arbitrary matrix polynomials or bounded real-root constructions of this
lesson.

Two useful research questions are to identify when the stable bundle
admits a globally chosen system of boundary measurements, and to
determine which additional estimates survive when spectral roots
approach the real axis. The examples here already distinguish the local
rank condition from a global choice of boundary data, and show why
estimates uniform through a real crossing cannot be inferred from this
lesson.

\subsection{References}\label{references}

The finite-dimensional algebra, decay estimates, examples and solved
problems in this lesson are given above.

\href{stable-prerequisite-bridges.pdf}{Polynomial and contour interfaces
for stable boundary models}, Sections 2--9, proves the exact scalar
algebra and contour tools used here.

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