\section{One bright channel is sufficient: microstructure universality}
\label{sec:one-bright-universality}

The exact inverse construction needs at least $k$ bright dimensions when the
geometric target potential has full rank.  That pointwise obstruction does
not survive approximation by rapidly varying geometry.  A single bright
dimension can synthesize the inlet response of any target covariant energy
in the class specified below.  Moreover the drainage projectors can converge uniformly to one
constant projector while retaining the prescribed nontrivial limiting
response.  The drainage strength and the microstructure frequency must be
taken to their limits in the appropriate order or scale.

Fix $I=[0,L]$.  Assume $w\in C^1(I)$ is positive, $g\in C(I)$ is
positive, $A\in C^1(I;\mathbb C^{k\times k})$ is skew-Hermitian, and
$S\in C^1(I;\mathbb C^{k\times k})$ is Hermitian positive semidefinite.
Set $V=S^2$.  In particular these conditions hold for a smooth uniformly
positive definite target potential $V$, with $S=V^{1/2}$.  The target
admittance $Y_{A,V}$ is the inlet map of
\[
 q_{A,V}[\xi]=\int_0^Lw
 \bigl(\|\xi'+A\xi\|^2+\xi^\dagger V\xi\bigr)dt,
 \qquad \xi(0)=z,\quad\xi(L)=0.
\]
Let $N_0=\binom{I_k}{0}\in\mathbb C^{(k+1)\times k}$, let
$b_0=e_{k+1}$, and let $M_0=b_0b_0^\dagger$.

For each integer $n\geq1$, define the one-row matrix
\begin{equation}
 r_n(t)=\left(e^{2\pi i nt/L},e^{4\pi i nt/L},
 \ldots,e^{2\pi i knt/L}\right),\qquad
 C_n(t)=r_n(t)S(t),
 \label{eq:one-bright-row}
\end{equation}
and integrate the $(k+1)\times(k+1)$ unitary frame equation
\begin{equation}
 U_n'=U_n\begin{pmatrix}A&-C_n^\dagger\\ C_n&0\end{pmatrix},
 \qquad U_n(0)=I_{k+1},\qquad U_n=[N_n\ b_n].
 \label{eq:one-bright-frame}
\end{equation}
The drainage projector is $M_n=b_nb_n^\dagger=I-N_nN_n^\dagger$.
It has rank one and spectrum $\{0^k,1\}$ at every $t$ and every $n$.
The finite-drainage energy and compressed admittance are
\begin{align}
 q_{\rho,n}[P]&=\int_0^L
 \bigl(w\|P'\|^2+\rho gP^\dagger M_nP\bigr)dt,\\
 z^\dagger Y_{\rho,n}z
 &=\min\{q_{\rho,n}[P]:P(0)=N_0z,\ P(L)=0\}.
 \label{eq:one-bright-admittance}
\end{align}
Only projector orientation changes; the local eigenvalues of the actual
drainage coefficient $\rho M_n(t)$ are always $\{0^k,\rho\}$.

\begin{theorem}[Rank-one universal response and two joint regimes]
\label{thm:one-bright}
For the explicitly constructed projectors above there are constants
$C_0,C_1,C_2$, depending on the fixed target data but not on $n$ or $\rho$,
such that
\begin{align}
 \|M_n-M_0\|_{L^\infty}&\leq C_0/n,
 \label{eq:one-bright-projector-rate}\\
 \|Y_{\rho,n}-Y_{A,V}\|&\leq C_1/n+C_2n^2/\rho.
 \label{eq:one-bright-target-rate}
\end{align}
There is also a constant $C_3$ such that
\begin{equation}
 0\preceq Y_{\rho,n}-Y_{\rm free}
 \preceq C_3\rho n^{-2}I_k,
 \qquad
 Y_{\rm free}=
 \left(\int_0^Lw(t)^{-1}dt\right)^{-1}I_k.
 \label{eq:one-bright-free-rate}
\end{equation}
Consequently:
\begin{enumerate}
\item If $n\to\infty$ and $\rho_n/n^2\to\infty$, then
$Y_{\rho_n,n}\to Y_{A,V}$.
\item If $n\to\infty$ and $\rho_n/n^2\to0$, then
$Y_{\rho_n,n}\to Y_{\rm free}$.
\item Choosing $n=\max(1,\lfloor\rho^{1/3}\rfloor)$ yields
$\|Y_{\rho,n}-Y_{A,V}\|=O(\rho^{-1/3})$, while simultaneously
$\|M_n-M_0\|_{L^\infty}=O(\rho^{-1/3})$.
\end{enumerate}
The third assertion is an achievable convergence rate from this
construction and bound; no optimality among all possible constructions or
all choices of $n$ is claimed.
\end{theorem}

\begin{proof}
We give the geometric averaging, constrained-response estimate, and finite
drainage estimate separately.

\paragraph{Step 1: geometry converges to a constant projector.}
Let $Q'=QA$, $Q(0)=I_k$, and put
$\overline U=\operatorname{diag}(Q,1)$.  Write the frame coefficient in
\eqref{eq:one-bright-frame} as
$K_A+K_n$, where $K_A=\operatorname{diag}(A,0)$ and $K_n$ consists of
the two off-diagonal blocks $C_n,-C_n^\dagger$.
For $Z_n=U_n\overline U^\dagger$, direct differentiation gives
\[
 Z_n'=Z_nH_n,\qquad H_n=\overline U K_n\overline U^\dagger.
\]
All matrices $Z_n$ are unitary, and $H_n$ is uniformly bounded.  Its entries
are finite sums of nonzero-frequency exponentials multiplied by fixed
$C^1$ amplitudes.  Integration by parts therefore gives
\[
 R_n(t):=\int_0^tH_n(u)du,\qquad
 \|R_n\|_{L^\infty}\leq C/n.
\]
Integrating $Z_n'=Z_nR_n'$ by parts yields
\[
 Z_n(t)-I=Z_n(t)R_n(t)-\int_0^tZ_n'(u)R_n(u)du.
\]
Unitarity and the uniform bound on $H_n$ imply
$\|Z_n-I\|_{L^\infty}\leq C/n$.  Thus
$U_n-\overline U=O(n^{-1})$ uniformly.  The bright column of
$\overline U$ is exactly $b_0$, proving
\eqref{eq:one-bright-projector-rate}.

\paragraph{Step 2: the constrained energy synthesizes the matrix target.}
The exact frame identities give
\[
 N_n^\dagger N_n'=A,\qquad
 \Phi_n=C_n^\dagger C_n
 =S\,r_n^\dagger r_n\,S=:V_n.
\]
Although $V_n$ has rank at most one, the diagonal entries of
$r_n^\dagger r_n$ are one and each off-diagonal entry has nonzero frequency.
Hence $V_n$ converges weakly to $V=S^2$.  More quantitatively,
\[
 T_n(t):=\int_0^tw(u)(V_n(u)-V(u))du,
 \qquad\|T_n\|_{L^\infty}\leq C/n.
\]
For every $\xi\in H^1$ with $\xi(L)=0$, integration by parts gives
\[
 \left|\int_0^Lw\xi^\dagger(V_n-V)\xi\,dt\right|
 \leq 2\|T_n\|_{L^\infty}\|\xi'\|_{L^2}\|\xi\|_{L^2}
 \leq (C/n)\|\xi\|_{H^1}^2.
\]
There is no boundary contribution: $T_n(0)=0$ and $\xi(L)=0$.

The constrained target forms with potentials $V_n$ have minimizers whose
$H^1$ norms are bounded uniformly by $C\|z\|$.  To see this, a fixed
trace-preserving trial field gives a uniform upper energy bound because
$V_n$ is uniformly bounded.  Removing the bounded connection by the unitary
solution $Q$ and using the fixed traces controls the $H^1$ norm by this
energy and $\|z\|$.  Applying the preceding potential-energy estimate to
both the $V_n$ minimizer and the $V$ minimizer gives
\begin{equation}
 \|Y_{\infty,n}-Y_{A,V}\|\leq C/n,
 \label{eq:one-bright-constrained-rate}
\end{equation}
where $Y_{\infty,n}$ denotes the exactly constrained inlet admittance.

\paragraph{Step 3: quantify the finite-drainage error.}
The frame coefficients $K_A+K_n$ are uniformly bounded and their first
derivatives are $O(n)$.  Thus $N_n'=O(1)$ and $N_n''=O(n)$ uniformly.
The constrained minimizers $\xi_{n,z}$ have uniform $H^2$ bounds
$\|\xi_{n,z}\|_{H^2}\leq C\|z\|$.  Indeed, their Euler equation contains
$V_n$ itself, which is uniformly bounded, but not $V_n'$.  With $w\in C^1$
positive and $A\in C^1$, the equation first bounds
$(w(\xi_{n,z}'+A\xi_{n,z}))'$ in $L^2$, and then bounds
$\xi_{n,z}''$ in $L^2$.

It follows for the lifted minimizers $P_{n,z}=N_n\xi_{n,z}$ that
\[
 \|F_{n,z}\|_{L^2}
 =\|-(wP_{n,z}')'\|_{L^2}\leq Cn\|z\|.
\]
The sharp finite-drainage bound of
Theorem~\ref{thm:universal-sharp-correction} therefore gives
\[
 0\preceq Y_{\infty,n}-Y_{\rho,n}
 \preceq (Cn^2/\rho)I_k.
\]
Together with \eqref{eq:one-bright-constrained-rate}, this proves
\eqref{eq:one-bright-target-rate}.

\paragraph{Step 4: the opposite joint regime.}
For any ambient trial pressure with the prescribed traces,
weighted Cauchy--Schwarz gives
\[
 \int_0^Lw\|P'\|^2dt\geq
 \frac{\|z\|^2}{\int_0^Lw^{-1}dt}.
\]
Leakage is nonnegative, so $Y_{\rho,n}\succeq Y_{\rm free}$.
For the reverse bound use the free minimizer
\[
 P_{\rm free}(t)=N_0z\,
 \frac{\int_t^Lw(u)^{-1}du}{\int_0^Lw(u)^{-1}du}.
\]
Because $M_0N_0=0$ and $M_n-M_0=O(n^{-1})$,
\[
 \int_0^L\rho gP_{\rm free}^\dagger M_nP_{\rm free}dt
 =\rho\int_0^Lg\|M_nP_{\rm free}\|^2dt
 \leq C\rho n^{-2}\|z\|^2.
\]
This proves \eqref{eq:one-bright-free-rate}.  All joint-limit and
achievable-rate assertions follow from the two estimates.
\end{proof}

\begin{corollary}[Noncommuting limits]
\label{cor:one-bright-noncommuting}
For the same construction,
\[
 \lim_{n\to\infty}\lim_{\rho\to\infty}Y_{\rho,n}=Y_{A,V},
 \qquad
 \lim_{\rho\to\infty}\lim_{n\to\infty}Y_{\rho,n}=Y_{\rm free}.
\]
These limits differ whenever $Y_{A,V}\neq Y_{\rm free}$.
For example, with $w=g=1$, $A=0$ and $V=vI_k$, $v>0$,
\[
 Y_{A,V}=\sqrt v\,\coth(\sqrt vL)I_k,
 \qquad Y_{\rm free}=L^{-1}I_k,
\]
which are distinct.  Thus uniformly converging local projectors do not
determine the limiting boundary response when drainage strength is allowed
to grow singularly.
\end{corollary}

\paragraph{Real variant.}
For real symmetric targets and real skew-symmetric connections one may use
$r_{n,j}(t)=\sqrt2\cos(2\pi jnt/L)$ instead of complex exponentials.
The row has zero mean and its matrix second moment is $I_k$; the same
oscillatory-primitive estimates apply.  The frame is then orthogonal in
$\mathbb R^{k+1}$.  The one-bright construction is therefore not contingent
on complex transport amplitudes.

\paragraph{Scope.}
The exact geometric realization requires ambient dimension $2k$ in the
worst case, whereas the approximate microstructured realization requires
only $k+1$ dimensions.  These statements concern different notions of
realization and do not contradict one another.  These bounds alone do not identify the critical regime. Section~\ref{sec:critical-crossover} supplies its complete limit. The construction does not claim the
$O(\rho^{-1/3})$ rate is globally optimal, and does not assert physical
realizability or literature-first priority.