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  pdftitle={A Translation-Invariant Fourier-Multiplier Branch Bridge with Constant-Coefficient and Nonlocal Benchmarks},
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\title{A Translation-Invariant Fourier-Multiplier Branch Bridge with Constant-Coefficient and Nonlocal Benchmarks}
\author{Duy Hoang Nguyen\\\small ORCID: \href{https://orcid.org/0009-0009-9228-7988}{0009-0009-9228-7988}}
\date{April 29, 2026}

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\small
\textbf{DOI:} \href{https://doi.org/10.5281/zenodo.19879889}{https://doi.org/10.5281/zenodo.19879889}\\
\textbf{ORCID:} \href{https://orcid.org/0009-0009-9228-7988}{0009-0009-9228-7988}\\
\textbf{License:} \href{https://creativecommons.org/licenses/by/4.0/}{CC BY 4.0}
\end{center}



\begin{abstract}
This paper does not introduce a new scalar Fourier-multiplier semigroup
theorem. Starting from standard branchwise $L^2$ Fourier-multiplier semigroup
ingredients, we prove a compatibility and normal-form realization theorem for
one fixed real branch algebra $A=\R[u]/(u^4-1)$ on $L^2(\R^d;A)$. The contribution
is to show that the intrinsic branch reduction of $A$, the three reduced
multiplier semigroups, the reconstructed $A$-valued flow, the relative
symbol-sum generator, the graph-core comparison with the componentwise strong
prototype, the coordinate-exact branch identities in the fixed branch-adapted Hilbert norm, and the
bounded-bulk observation pushforward all refer to the same operator-theoretic
object. Thus the result is a branch-algebraic compatibility theorem inside a
standard $L^2$ multiplier setting, not a replacement for scalar multiplier
semigroup theory or for general vector-valued multiplier theory.

The central operator-domain theorem is the maximal finite-a.e. symbol-sum
graph-core identity. The non-tautological point is not scalar multiplier
generation itself, and not a theorem about arbitrary unbounded sums, but the
comparison between the raw componentwise prototype and the closed relative
symbol-sum multiplier realization inside the prescribed \(L^2\) multiplier
model. The former may have a strictly smaller domain because it asks for
$a(D)$ and $b(D)$ separately, while the latter is governed by the reduced sums
\[
a+b,\qquad a-b,\qquad a+i b.
\]
The paper proves, before the reconstruction theorem is used, that the raw
prototype is nevertheless a graph core for the closed maximal finite-a.e. symbol-sum realization. The intrinsic plus, minus, and realified complex
branches reduce the formal multiplier
\[
\Gform=\IA\otimes a(D)+\Lu\otimes b(D)
\]
to the branch symbols $\lambda_+=a+b$, $\lambda_-=a-b$, and the
complex-coordinate representative $\lambda_c=a+i b$. Throughout the paper the
multiplier symbols are ordinary measurable finite-a.e. complex-valued functions;
extended-valued symbols or symbols singular on sets of positive measure are not
part of the theorem package. The last symbol is the transported scalar coordinate
of the real two-dimensional $A_c$-branch through $\mathcal I_c$; the precise normal-form convention is fixed once in
Proposition~\ref{prop:realified-complex-branch-normal-form}.

Under the reduced growth hypotheses, the three branch operators generate
$C_0$-semigroups by standard multiplier-semigroup arguments. These hypotheses are
used as a sufficient Hilbert-space multiplier package for this branch-adapted
$L^2$ theory, not as a minimal or classification-level condition for all possible
multiplier realizations. For the prescribed $L^2$ multiplier formula itself,
Lemma~\ref{lem:near-necessity-a2-prescribed-multiplier} records the matching
near-necessity: boundedness of $e^{t\lambda(D)}$ at one positive time is
equivalent to an upper essential bound on $\Re\lambda$. Their prescribed direct-sum assembly gives the
branch-coordinate $A$-valued flow. The same flow is also identified explicitly as
the $L^2$ Fourier multiplier with finite-dimensional matrix symbol
\[
\exp\bigl(t(a(\xi)\IAC+b(\xi)\Lu)\bigr),
\]
whose branch-coordinate blocks are $e^{t\lambda_+}$, $e^{t\lambda_-}$, and,
after the global real $c$-branch coordinate map, $e^{t\lambda_c}$. The reconstructed generator agrees with the relative
symbol-sum Fourier-multiplier realization. The reconstruction theorem itself is
stated without a global uniqueness claim; a separate corollary records only the
internal uniqueness inside the imposed branch-compatible class. All exact
operator-norm and graph-norm identities are stated with respect to the fixed
branch-adapted Hilbert norm; under an arbitrary equivalent coefficient norm, the
corresponding statements become two-sided estimates with equivalence constants
rather than literal isometries.

For the observation layer, we restrict throughout to bounded bulk maps
\[
\Ocal\in\mathcal L(L^2(\R^d;A),Z).
\]
This layer is deliberately modest: it records bounded-bulk pushforward,
masking, separated-channel bookkeeping, and inverse-on-range consequences of
sensor nondegeneracy estimates that are assumed or verified separately. It does
not prove an exact observability theorem, and it does not derive sensor lower
bounds from the multiplier dynamics alone. To keep this conditional hypothesis
from being merely a canonical projector in disguise, Section~\ref{sec:observation}
includes a Fourier-frame branch-sensor example: a finite family of multiplier
windows satisfying a pointwise frame lower bound gives a concrete bounded-below
separated branch sensor by Plancherel. All observation estimates involving
\(\omega_+\), \(\omega_-\), and \(\omega_c\) are upper-envelope estimates
transported through the chosen bounded sensor; they are not datum-specific
actual observed growth rates and they are not asserted to be sharp after
observation. Boundary observations, point sensors, trace maps, unbounded
admissible observations, time-dependent sensors, control-theoretic observability
estimates, variable coefficients, nonlinear perturbations, and inverse
identification are outside this paper.
\end{abstract}


\bigskip
\noindent\textbf{Keywords.} Branch algebra; Fourier multipliers; $C_0$-semigroups; realified complex branch; relative symbol-sum multiplier realization; graph core; bounded bulk observations; masking; sensor-projector intertwining; Fourier-frame branch sensors; conditional closed-range branch inverse.

\bigskip
\noindent\textbf{Preprint metadata.}
\begin{itemize}[leftmargin=2em]
    \item \textbf{Author:} Duy Hoang Nguyen.
    \item \textbf{Title:} \emph{A Translation-Invariant Fourier-Multiplier Branch Bridge with Constant-Coefficient and Nonlocal Benchmarks}.
    \item \textbf{Year:} 2026.
    \item \textbf{Date:} April 29, 2026.
    \item \textbf{Series position:} Paper 5 in the branch-calculus series, following Papers~\cite{nguyen-paper1,nguyen-paper2,nguyen-paper3,nguyen-paper4}.
    \item \textbf{Prior public series records:} Papers~\cite{nguyen-paper1,nguyen-paper2,nguyen-paper3,nguyen-paper4} are listed in the bibliography with their Zenodo DOI metadata.
    \item \textbf{Suggested citation:} Nguyen, D. H. (2026). \emph{A Translation-Invariant Fourier-Multiplier Branch Bridge with Constant-Coefficient and Nonlocal Benchmarks}. Preprint.
\end{itemize}


\section{Introduction}

\subsection{Role of Paper~5 in the series}

The present paper is the fifth step in a theorem-driven program built around a deliberately separated two-layer architecture: a deterministic branch-resolved backbone and a restrained observation layer applied only after deterministic propagation. The earlier papers established that this separation is not merely stylistic. It is the mechanism that allows exact formulas, exact reduction statements, and exact observation-level consequences to coexist without collapsing the deterministic and statistical aspects of the theory into a single undifferentiated model.

Paper~1~\cite{nguyen-paper1} introduced the basic branch calculus on the semisimple commutative algebra
\[
A=\R[u]/(u^4-1),
\]
and showed that the corresponding deterministic dynamics admit an exact branch decomposition together with an explicit scalar exponential calculus. It then attached a minimal Bernoulli readout layer at the level of observation, rather than at the level of dynamics, and derived exact expectation and variance formulas for the resulting observed scalar signals. In that setting the deterministic generator, the branch decomposition, and the observation protocol can all be written in closed form, so the first threshold discussion remains transparent and fully computable.

Paper~2~\cite{nguyen-paper2} kept the scalar deterministic backbone but turned from the forward problem to the inverse side of the theory. Once exact formulas for observed signals are available, one can ask which model parameters are recoverable from observation, how much information is lost under branch-selective measurement, and which recovery statements remain stable under perturbation of the data. In that sense Paper~2 converted the exact forward formulas of Paper~1 into an exact scalar information problem.

Paper~3~\cite{nguyen-paper3} enlarged the observation architecture rather than the deterministic coefficient class. By passing to tensorized finite-dimensional systems and to multichannel readout, it derived exact covariance formulas, interaction criteria, and observation-side envelope structure. Its main message was that one can substantially enrich the observable geometry while still keeping the theorem-bearing core exact, provided the deterministic coefficient layer remains finite-dimensional and sufficiently factorized.

Paper~4~\cite{nguyen-paper4} addressed the missing vertical extension. Instead of enriching the observation layer further, it lifted the deterministic coefficient class from scalars to operators and showed that the branchwise architecture survives on the tensorized state space
\[
\Xcal := A\otimesR X,
\]
with $X$ a real Banach or Hilbert space. The main achievement of Paper~4 was not a PDE theorem, but an abstract semigroup bridge: it identified the correct operator-valued state space, formulated the reduced branch generators, and proved an exact reconstruction principle at the semigroup level under an explicit reduced-generator package. That paper established the correct operator-theoretic architecture, but it stopped deliberately short of a concrete function-space realization in a translation-invariant Fourier-multiplier class.

For standalone preprint use, the earlier papers of the series are cited below with their public Zenodo metadata and DOI records. The citations are used here as internal series references: Paper~1 supplies the Bernoulli sector-readout protocol, Paper~2 supplies compatible model-jet and sampled-certificate selection, Paper~3 supplies tensor-sector observation identities and second-order bookkeeping, and Paper~4 supplies the branchwise operator-semigroup bridge on which the present Fourier-multiplier realization builds.

Paper~5 therefore has a sharply defined role. It is the stage of the present series at which the deterministic state is placed on an actual function space over physical space and the main theorem package is formulated directly for one concrete translation-invariant Fourier-multiplier class rather than for scalar models, tensorized finite-dimensional systems, or abstract operator pairs. Its task is not to build a general PDE theory for branch-valued dynamics. Its task is narrower and more structural: to identify one concrete function-space evolution class on which the branchwise semigroup program of Paper~4 actually lands, with local constant-coefficient differential-operator models appearing only as benchmark subclasses within that broader multiplier setting.

\subsection{The gap left by Paper~4}

Paper~4 provided the correct abstract bridge from scalar branch calculus to operator-valued semigroup dynamics, but it left open the fundamental question of realization. An abstract reconstruction theorem on $A\otimesR X$ does not by itself show that any concrete translation-invariant multiplier class satisfies the hypotheses in a mathematically natural way. In particular, once $X$ is replaced by a function space over $\R^d$, one must confront issues that do not appear at the purely abstract level: frequency localization, unbounded generators, domain questions, compatibility of branch projectors with function-space states, and the interaction between semigroup growth and observation geometry.

This is the point at which a concrete compatibility result in a Fourier-multiplier setting becomes useful. The slogan
\[
\partial_t Y = (\IA\otimes M_0 + \Lu\otimes M_1)Y
\]
is only a formal template until one specifies a concrete function space, a concrete generator class, and a concrete mechanism by which the branch decomposition is realized pointwise in frequency space or spectrally in the underlying spatial variables. Without that additional layer, the operator-valued theory remains structurally important but not yet realized on a concrete function space.

The main unresolved problem is therefore the following. Can one identify a concrete translation-invariant Fourier-multiplier class for which the branch algebra still acts cleanly, the reduced generators can still be written down explicitly, and the resulting full flow can still be constructed and characterized branchwise from reduced semigroups? The second unresolved problem is observational. Even if such a reconstruction exists, there is no a priori reason that branchwise growth or decay should remain visible after passing through bounded bulk observation maps. In a spatially extended setting, observation may suppress unstable frequencies, average out oscillatory branches, or detect only a restricted portion of the branch-resolved state.

The present paper addresses this restricted realization problem by identifying one theorem-bearing translation-invariant Fourier-multiplier class in which the architecture of Paper~4 is realized concretely. The intrinsic branch decomposition of the state space is transported by the Fourier transform into branch-coordinate multiplier form; the three reduced generators become explicit Fourier multipliers; the prescribed reduced semigroups assemble into a distributed multiplier flow on $L^2(\R^d;A)$; and the reconstructed generator is identified with the relative symbol-sum Fourier-multiplier realization by comparing their common branch-coordinate block forms. On the generator side, the paper proves, under the standing finite-a.e. multiplier-symbol convention, that the componentwise strong prototype $\Gcal$ is closable and is a graph core for that relative symbol-sum generator. The observation layer is correspondingly restrained: bounded bulk sensors yield post-composition pushforward, masking, separated-channel, and conditional lower-bound bookkeeping statements, with no universal threshold, inverse identification theorem, or intrinsic observability theorem inferred from the dynamics alone.


\subsection{Scope of the multiplier realization}

The theorem package in this paper is intentionally confined to one concrete setting:
linear, translation-invariant Fourier multipliers on the real Hilbert space
\[
L^2(\R^d;A).
\]
The formal prototype is
\[
\Gcal = \IA\otimes a(D) + \Lu\otimes b(D),
\]
where $a(D)$ and $b(D)$ are Fourier multiplier operators whose symbols satisfy the standing reality and reduced growth hypotheses. Within this setting the Fourier transform transports the intrinsic algebraic branch decomposition into a frequency-by-frequency branch-coordinate representation. The reduced symbols are
\[
a(\xi)+b(\xi),
\qquad
 a(\xi)-b(\xi),
\qquad
 a(\xi)+i\,b(\xi),
\]
and the resulting semigroup, generator, graph-core, and bounded-observation statements are proved only for this multiplier framework. The third symbol is always read through the global real-coordinate normal form of Proposition~\ref{prop:realified-complex-branch-normal-form}: it is the scalar representative of the real two-dimensional $A_c$-block only after applying the global real-linear map $\mathcal I_c$.

Local constant-coefficient differential-operator examples enter as benchmark subclasses, namely when the multiplier symbols are polynomial symbols coming from the chosen Fourier convention and satisfying the same standing assumptions. In particular, the polynomial symbols must satisfy the reality condition and the reduced $L^2$ semigroup growth bounds, equivalently the required upper bounds on $\operatorname*{ess\,sup}_{\xi}\Re\lambda_\bullet(\xi)$ for the reduced branch symbols. Non-polynomial multipliers, such as $|D|$-type symbols, may also be covered when those assumptions hold. Conversely, arbitrary polynomial symbols are not automatically included; polynomiality alone does not imply admissibility.

The neighboring directions deliberately excluded from this theorem package are collected once in Subsection~\ref{subsec:scope-and-nonclaims} and revisited as later directions in Section~8. The bounded-bulk observation formalism below follows the same convention: theorem-local statements keep only the hypotheses needed for the result being proved, while the global exclusions are not repeated at full length.

\begin{remark}[Benchmark status of differential-operator examples]\label{rem:differential-operator-benchmark-only}
The correct reading is
\[
\text{Fourier-multiplier theorem first; differential-operator benchmarks second.}
\]
Every local differential-operator model used later is a specialization of the multiplier theorem, not an enlargement of the theorem package beyond the stated multiplier hypotheses. Thus a constant-coefficient differential operator belongs to the benchmark subclass only after its Fourier polynomial symbols have been checked against the same assumptions imposed on general multipliers: the Hermitian reality condition and the reduced real-part upper bounds needed for $L^2$ semigroup generation. In particular, polynomiality alone is not an admissibility criterion.
\end{remark}

\subsection{Architecture-level contributions}

The paper has five architecture-level contributions. Each uses standard
Fourier-multiplier and $C_0$-semigroup tools at the branch level, but the
contribution lies in making the branch-algebraic assembly exact on
$L^2(\R^d;A)$ and in keeping the generator, graph, norm, and observation
statements attached to the same $A$-valued multiplier flow. They are stated here
in a compact form; the surrounding scope conventions in
Subsection~\ref{subsec:scope-and-nonclaims} and Section~8 delimit the theorem package.

\begin{enumerate}
    \item[\textbf{(I)}] \textbf{Branch splitting on $L^2(\R^d;A)$.} The distributed state space splits into the three intrinsic branches of $A$, and the Fourier transform carries this splitting into a frequencywise branch-coordinate representation. The reduced symbols are
    \[
    a(\xi)+b(\xi),\qquad a(\xi)-b(\xi),\qquad a(\xi)+i b(\xi),
    \]
    with the last expression understood only as the complex coordinate of the real two-dimensional $A_c$-branch.

    \item[\textbf{(II)}] \textbf{Reduced multiplier semigroups under sufficient growth hypotheses.} Under the stated reduced real-part bounds, used here as sufficient $L^2$ multiplier-growth hypotheses rather than as a minimal admissibility criterion, the plus, minus, and complex-coordinate branch generators produce the reduced semigroups $T_+$, $T_-$, and $T_c$ on their natural model spaces.

    \item[\textbf{(III)}] \textbf{Graph-core identity as the operator-domain spine.} Before using the reconstructed semigroup generator, the paper proves the main domain theorem
    \[
    \overline{\Gcal}^{\,\mathrm{graph}}=\Mss,
    \]
    where \(\Gcal\) is the raw componentwise strong prototype and \(\Mss\) is the closed relative symbol-sum Fourier-multiplier realization. The raw domain may be strictly smaller because it requires \(a(D)\) and \(b(D)\) separately, while \(\Mss\) only sees the combined symbol-sum action. The proof uses the explicit finite-symbol cutoff core
    \[
    \widehat{Y^{(n)}}=\mathbf 1_{\{|a|\le n,\ |b|\le n\}}\widehat Y
    \]
    modulo the symmetric null-set convention required by the real-state condition.

    \item[\textbf{(IV)}] \textbf{Branch-coordinate reconstruction and full matrix-symbol formula.} The reduced semigroups assemble into a prescribed distributed multiplier flow on $L^2(\R^d;A)$. The same flow is identified as the $A$-valued Fourier multiplier with matrix symbol $\exp(t(a\IAC+b\Lu))$, interpreted on the Fourier-side complexified coefficient fibre and restricted to the real-state symmetry. The reconstruction theorem makes no global uniqueness claim; internal uniqueness is stated separately only inside the imposed branch-compatible class. The reconstructed generator is then identified with the same relative symbol-sum realization \(\Mss\), so Theorem~\ref{prop:strong-prototype-symbol-sum-core} gives the graph-core link to the raw prototype.

    \item[\textbf{(V)}] \textbf{Bounded-bulk observation bookkeeping, masking, and checkable conditional inversion.} Bounded bulk sensors are used only after the deterministic multiplier flow has been reconstructed. The paper proves masking criteria, one-sided pushforward bounds, conditional observed-channel extraction under an imposed sensor-projector intertwining structure, and inverse-on-range consequences for separated branch sensors that are assumed or verified to be bounded below. No exact observability theorem is claimed. A Fourier-frame branch-sensor example shows that the bounded-below hypothesis can be checked by a concrete multiplier-frame condition rather than only by using the canonical branch projector.
\end{enumerate}

Taken together, these five contributions identify a concrete translation-invariant Fourier-multiplier class within the present series for which branch splitting, reduced semigroup generation, exact distributed multiplier-flow reconstruction, full matrix-symbol realization, graph-core identification, and a deliberately conditional bounded-bulk observation bookkeeping layer coexist in a single controlled framework.

The separation between standard components and the branch-algebraic assembly is essential. The
branchwise generation of $T_+$, $T_-$, and $T_c$ is an application of standard
$L^2$ Fourier-multiplier semigroup theory. Likewise, the use of bounded
observation maps is ordinary functional analysis. What is specific to this paper
is that these standard pieces are forced to pass through the same finite branch
algebra $A=\R[u]/(u^4-1)$, the same branch-adapted Hilbert norm, and the same
relative symbol-sum realization. The theorem package therefore identifies a
single coherent object: a concrete translation-invariant $A$-valued multiplier
flow whose reduced branch semigroups, reconstructed generator, graph core, exact
branch-norm identities, and bounded-bulk observation pushforward are mutually
compatible.

\subsection{What the direct-sum reduction does and does not prove}\label{subsec:nontrivial-assembly}

After applying the intrinsic branch decomposition, the reduced dynamics is
indeed a direct sum of three scalar multiplier problems. This paper does not
claim otherwise. The point is that this direct-sum representation is only one
coordinate description of a single real $A$-valued multiplier realization. Three
operator objects must be kept distinct:
\[
\Gform=\IA\otimes a(D)+\Lu\otimes b(D),
\qquad
\Gcal,
\qquad
\Acal.
\]
Here \(\Gform\) is the formal expression, \(\Gcal\) is the componentwise strong
prototype obtained by requiring the two pieces \(a(D)\) and \(b(D)\) to act
separately on the \(A\)-valued state, and \(\Acal\) is the branchwise
symbol-sum generator reconstructed from the reduced semigroups. The theorem
package identifies the reduced direct-sum semigroup with the generator obtained
from the symbol $a(\xi)I_{\AC}+b(\xi)L_u$, and then compares that closed
symbol-sum generator with the smaller componentwise strong prototype. The
direct-sum notation by itself does not identify these domains.

The reduced symbols are
\[
\lambda_+=a+b,
\qquad
\lambda_-=a-b,
\qquad
\lambda_c=a+i b.
\]
Thus the branchwise generator domain is governed by the integrability of
\(\lambda_+\widehat f_+\), \(\lambda_-\widehat f_-\), and
\(\lambda_c\widehat h_c\). The raw strong prototype instead requires the
separate integrability of \(a\widehat Y\) and \(b\widehat Y\). Cancellation in a
reduced symbol can therefore enlarge the reconstructed generator domain relative
to the componentwise strong domain. The graph-core identity
\[
\overline{\Gcal}^{\,\Acal\text{-graph}}=\Acal
\]
is the operator-domain statement that connects the raw formal prototype to the
actual branchwise generator. It is not a cosmetic restatement of a finite-dimensional
branch-coordinate reduction.

A one-dimensional cancellation test already exposes the issue. Take
\[
a(\xi)=-\xi^2+i\xi,
\qquad
b(\xi)=\xi^2+i\xi.
\]
Then
\[
\lambda_+(\xi)=a(\xi)+b(\xi)=2i\xi.
\]
The plus-branch component of the reconstructed generator has its natural domain
\(H^1(\R)\). The componentwise strong prototype, however, requires
\(a(\xi)\widehat f(\xi)\) and \(b(\xi)\widehat f(\xi)\) separately to lie in
\(L^2\), which forces \(f\in H^2(\R)\). Hence the raw componentwise domain can be
strictly smaller than the branchwise reconstructed domain even though it remains
a graph core. Section~7 returns to this example as a full diagnostic stress
test.

For clarity, the separation between standard ingredients and the paper's
operator-domain contribution is summarized as follows.
\begin{center}
\begin{tabular}{p{0.42\textwidth}p{0.48\textwidth}}
\hline
\textbf{Standard ingredient} & \textbf{Role in the present paper} \\
\hline
\(L^2\) Fourier-multiplier semigroup generation & Used branchwise under explicit
reduced growth hypotheses. \\
Finite-dimensional semisimple algebra splitting & Realized on
\(L^2(\R^d;A)\) together with the Fourier-side branch-coordinate model. \\
External direct sums of semigroups & Converted into one prescribed real
\(A\)-valued multiplier flow with a reconstructed generator on the original
state space. \\
Formal expression \(\IA\otimes a(D)+\Lu\otimes b(D)\) & Compared with the
branchwise symbol-sum generator through the graph-core identity
\(\overline{\Gcal}^{\,\Acal\text{-graph}}=\Acal\). \\
Bounded observation maps & Pushed through the same branch-resolved flow to
obtain signal splitting, masking, and conditional closed-range sensor inversion
only under explicit bounded-below sensor hypotheses. \\
\hline
\end{tabular}
\end{center}


\subsection{The graph-core identity as the operator-domain spine}
\label{subsec:graph-core-spine}

The graph-core statement is the operator-domain spine of the paper, but its
scope is deliberately narrow. It is the maximal finite-a.e. symbol-sum closure
statement for the \(L^2\) Fourier-multiplier realization fixed here, not a
general theorem about closures of arbitrary unbounded operator sums. The formal
expression
\[
\Gform=I_A\otimes a(D)+L_u\otimes b(D)
\]
first gives a componentwise strong prototype $\Gcal$, whose domain asks for the
two pieces $a(D)$ and $b(D)$ to exist separately. The reconstructed generator
$\Acal$, by contrast, is governed by the three reduced symbol sums
\[
\lambda_+=a+b,\qquad \lambda_-=a-b,\qquad \lambda_c=a+i b.
\]
Thus the correct comparison is not the raw equality $\Dom(\Gcal)=\Dom(\Acal)$,
which can fail, but the graph-closure identity
\[
\overline{\Gcal}^{\,\mathrm{graph}}=\Acal=\Mss.
\]
This identity is the precise sense in which the raw formal prototype and the
branchwise reconstructed generator describe the same multiplier realization. It
allows calculations on the smaller componentwise strong domain and then passes
to the maximal branchwise generator by graph-norm closure. The domain-gap stress
test in Section~7 is included to make this statement falsifiable rather than
rhetorical: it exhibits a case where the raw domains are genuinely different,
while the graph-core theorem still identifies the correct closure.

\subsection{Relation to existing literature and scope of novelty}

The mathematical background used in the present paper belongs to several established literatures. Fourier multipliers on $L^2$ and their frequency-side calculus are standard material in harmonic analysis; constant-coefficient evolution equations can be read through that multiplier perspective after Fourier transform; and the semigroup statements used here fit into the standard operator-theoretic framework for $C_0$-semigroups and multiplication-type generators; see, for example, \cite{grafakos,engel-nagel,pazy,hille-phillips,dunford-schwartz,haase}. Matrix- and operator-valued multiplier methods, especially in parabolic evolution problems, form a much broader theory than is needed here; the finite-dimensional branch block used below should be viewed as a deliberately elementary $L^2$ specialization rather than as a replacement for that literature; see, for example, \cite{denk-hieber-pruess,arendt-batty-hieber-neubrander}. On the observation side, the paper uses only bounded bulk observation operators together with one-sided norm estimates, masking consequences, and conditional closed-range sensor inverses. These statements are much weaker than the admissibility and exact observability questions studied in infinite-dimensional systems theory; for background see \cite{tucsnak-weiss,curtain-zwart}. None of those background ingredients is claimed here as new.

For clarity, Table~\ref{tab:literature-positioning} separates the background tools from the role they play in the present paper.
\begin{table}[htbp]
\centering
\begin{tabular}{p{0.34\textwidth}p{0.56\textwidth}}
\hline
Background literature & Role in the present paper \\
\hline
Scalar $L^2$ Fourier multipliers and constant-coefficient evolution & Provides the frequency-side calculus used to define the reduced branch symbols $\lambda_+$, $\lambda_-$, and $\lambda_c$. \\
$C_0$-semigroups, multiplication generators, and functional calculus & Provides the closed reduced generators and their graph domains. \\
Matrix-, vector-, and operator-valued multiplier theory & Supplies the broader context; the paper uses only a finite-dimensional branch-block specialization on $L^2$. \\
Real and complex Hilbert-space operator viewpoints & Clarifies why the $c$-branch is a real two-dimensional block transported to one complex scalar coordinate. \\
Observation and control of operator semigroups & Provides context for what is \emph{not} proved: no exact observability or admissibility theorem is derived here. \\
\hline
\end{tabular}
\caption{Literature positioning of the ingredients used in the paper.}
\label{tab:literature-positioning}
\end{table}

\begin{remark}[Novelty contract]\label{rem:novelty-contract}
No part of the paper claims novelty for the scalar $L^2$
Fourier-multiplier semigroup theorem, for finite-dimensional semisimple algebra
splitting, or for abstract direct sums of semigroups. These are background tools.
The contribution is the compatibility statement that, for the fixed algebra
$A=\R[u]/(u^4-1)$ and the stated multiplier hypotheses, the branch reduction,
reduced semigroups, reconstructed real $A$-valued flow, relative symbol-sum
generator, graph-core closure of the componentwise prototype, coordinate-exact
fixed-norm branch identities, and bounded-observation pushforward form one coherent
realization.
\end{remark}

Accordingly, the novelty claim of Paper~5 is deliberately narrower than a field-wide priority claim, but it is not merely an internal bookkeeping convention for the series. The contribution is the exact compatibility of these standard ingredients inside one fixed real branch algebra. In that setting an intrinsic branch decomposition is represented after Fourier transform in branch coordinates, reduced semigroups are generated on their natural model spaces, the full flow is reconstructed by a prescribed branch-coordinate direct-sum rule, the reconstructed generator is identified with the relative symbol-sum Fourier-multiplier realization through a common branch-coordinate description, and bounded bulk observation maps push that reconstructed flow into branch-induced signal decompositions that can, under an imposed sensor-projector intertwining structure, be analyzed and conditionally inverted only on the relevant closed ranges.

The paper should therefore be read as making a standalone compatibility statement inside a restricted multiplier class, with program-relative motivation. It identifies one concrete translation-invariant class in which the abstract bridge of Paper~4 can be realized concretely, and in which the distinction between the smaller raw domain of the componentwise strong prototype $\Gcal$ and the branchwise symbol-sum generator $\Acal$ becomes mathematically visible under the finite-a.e. multiplier-symbol convention: $\Gcal$ may be a proper sub-operator of $\Acal$, but it is closable and its operator closure is $\Acal$ in this multiplier setting. The norm identities used for this comparison are coordinate-exact identities for the fixed branch-adapted Hilbert norm; equivalent coefficient norms yield only two-sided estimates with constants.

Thus the word ``bridge'' in this paper means a precise compatibility theorem inside a fixed branch algebra: branch reduction, multiplier generation, semigroup reconstruction, graph realization, branch-adapted norm comparison, and observation pushforward commute with one another under the stated hypotheses. The global exclusions needed to keep this claim precise are collected once in Subsection~\ref{subsec:scope-and-nonclaims}.



\subsection{Scope and non-claims}\label{subsec:scope-and-nonclaims}

The sharpness of the present paper depends on a precise restriction of scope. Several neighboring problems are mathematically natural, but they are outside the theorem-bearing core. This subsection is the central non-claim registry for the paper: later sections retain only the local qualifications needed to state a theorem correctly, while the global exclusions are collected here to avoid scattering defensive caveats throughout the exposition.

The theorem package is limited to a translation-invariant linear Fourier-multiplier setting on $L^2(\R^d;A)$. It does not cover semilinear branch interaction, variable-coefficient or microlocal transport, boundary-value or trace observation, inverse identification of the multiplier symbols from data, a universal distributed threshold law, a classification of all closed realizations of the formal expression $I_A\otimes a(D)+L_u\otimes b(D)$, or minimal admissibility assumptions for every possible multiplier model. The complex branch is always read through the global real-coordinate normal form of Proposition~\ref{prop:realified-complex-branch-normal-form}; it is not read through a pointwise complex-fibre similarity.

These exclusions are part of the design. They isolate one theorem-bearing task and carry it out completely: a restricted translation-invariant Fourier-multiplier branch bridge whose constant-coefficient differential-operator and nonlocal examples are benchmarks rather than the defining scope of the theory. All later reminders of scope should be read as local pointers back to this registry, not as separate negative claims.

\subsection{Organization of the paper}

The remainder of the paper is organized as follows.

Section~2 recalls the branch algebra in the form needed for function-space lifting, defines the state space $L^2(\R^d;A)$, records the Fourier transform conventions for $A$-valued states, and introduces the prototype generator together with the standing admissibility assumptions on the multiplier symbols.

Section~3 establishes the intrinsic branch decomposition theorem together with its branch-coordinate Fourier representation. It derives the reduced branch symbols, proves that the state space decomposes into invariant branch subspaces, and separates the genuine one-dimensional plus/minus fiber reductions from the realified complex-branch representation obtained through $\mathcal I_c$.

Section~4 turns to the reduced branch operators on their natural model spaces---the real scalar space $\LtwoR$ for the plus and minus branches and the complex scalar space $\LtwoC$ for the complex branch---and proves their semigroup generation under explicit hypotheses. It then records the basic branchwise operator-norm envelopes that will later feed into the observation-bound analysis, and fixes the notation for the complex branch once and for all.

Section~5 first promotes the graph-core identity to the main operator-domain theorem: the componentwise strong prototype is a graph core for the closed relative symbol-sum multiplier. It then proves the branch-coordinate reconstruction theorem for the full distributed flow on $L^2(\R^d;A)$, writes the full evolution operator as a branchwise recombination of reduced semigroups, and identifies the same operator family as the finite-dimensional matrix-symbol Fourier multiplier
\[
\exp\bigl(t(a(\xi)\IAC+b(\xi)\Lu)\bigr).
\]
It formulates the corresponding mild solution concept and separates the reconstruction theorem from a later corollary recording internal uniqueness only inside the prescribed branch-coordinate reconstruction architecture.

Section~6 introduces the admissible class of bounded bulk observation maps and treats observation as a bounded-bulk bookkeeping layer rather than as an observability theory. It proves pushforward upper bounds and exact masking, formulates conditional observed-channel extraction under an imposed sensor-projector intertwining structure, records inverse-on-range consequences only after a separated branch sensor is bounded below, and supplies a Fourier-frame branch-sensor example showing one concrete way to verify that lower sensor hypothesis. A separate auxiliary corollary records a lower-bound transfer only when an external lower dynamical estimate is supplied on the same branch. The section also records a brief compatibility remark showing that the present distributed multiplier framework remains consistent with the observation-layer philosophy of Paper~1 without turning the deterministic flow into a stochastic evolution.

Section~7 presents three generator benchmarks, one observation benchmark, and a diagnostic cancellation stress test: a shifted heat-amplification benchmark, a mixed decay-oscillation model, a three-regime cancellation-coupled model, a frequency-window observation model, and a raw-domain cancellation stress test. These computations are included not as decoration but as benchmark calculations showing how branch geometry, observation geometry, and strict raw-domain mismatch interact in practice while preserving the graph-core identity $\overline{\Gcal}^{\,\mathrm{graph}}=\Acal$.

Section~8 concludes with the limitations of the present framework and the natural directions for continuation, including variable-coefficient extensions, semilinear perturbations, inverse multiplier-identification questions, boundary observations, and stronger spectral or microlocal theories. In particular, it explains why the natural next paper should study whether branchwise observation-bound, transported sensor-norm, masking, and conditional lower-bound structures persist or break under controlled nonlinear perturbations.


The aim of Paper~5 is therefore to identify a concrete translation-invariant multiplier class on which the branchwise semigroup architecture becomes exact, reconstructible, and observation-aware under a bounded bulk observation layer.

\section{Branch Algebra and \texorpdfstring{$A$}{A}-Valued \texorpdfstring{$L^2$}{L2} Spaces}

\paragraph{Standing convention on the spatial dimension.}
Throughout the paper we fix an integer \(d\ge1\). All scalar, complex,
and \(A\)-valued \(L^2\)-spaces are over \(\R^d\), equipped with Lebesgue
measure, unless a statement explicitly says otherwise. Fourier transforms
are taken on \(\R^d\) with the normalization fixed below, and all frequency
variables \(\xi\) belong to \(\R^d\). Any later benchmark that sets \(d=1\)
is a specialization of this standing \(d\)-dimensional framework, not a
change of the ambient theory.

\begin{remark}[Standing interpretation box: real states, Fourier fibres, and scope]
\label{rem:standing-interpretation-box-paper5}
All theorem statements below are to be read under the following fixed
interpretation, which is collected here to avoid repeating the same caveats in
full at every use.
\begin{enumerate}[label=\textup{(\roman*)}]
    \item The physical state space is the real Hilbert space
    \(L^2(\R^d;A)\). Its Fourier transform is represented inside
    \(L^2(\R^d;\AC)\) by the Hermitian real-state condition
    \[
    \widehat Y(-\xi)=\cA(\widehat Y(\xi))
    \qquad\text{for a.e. }\xi.
    \]
    Thus Fourier-side formulas may use the complexified coefficient fibre
    \(\AC\), but they are ultimately restricted to this closed real-linear
    Hermitian subspace.

    \item The complex branch coordinate
    \[
    \mathcal I_c(f\Pc+gJ)=f+ig
    \]
    is a global real-linear Hilbert-space identification. It is not a
    pointwise complex-linear scalarization of the complexified Fourier fibre
    \(\C\Pc\oplus\C J\). Consequently, the symbol \(\lambda_c=a+ib\) is a
    paired-frequency real-coordinate symbol, whereas full matrix-symbol
    statements keep the \(\Pc,J\)-block as a real two-dimensional block.

    \item Reduced branch symbols and reduced branch semigroups are
    symbol-sum multiplier objects on maximal finite-a.e. multiplier domains.
    They are not raw operator sums on \(\Dom(a(D))\cap\Dom(b(D))\) unless that
    equality is explicitly imposed or proved.

    \item The graph-core results are statements about the maximal finite-a.e.
    translation-invariant multiplier realization in the present Plancherel
    setting. They are not general closure theorems for arbitrary sums of
    unbounded operators.

    \item Observation results form a bounded-bulk pushforward and masking
    layer. They do not assert exact observability, unique continuation, or
    inverse recovery of the multiplier symbols unless those stronger hypotheses
    are explicitly added in the relevant statement.
\end{enumerate}
When a later theorem includes a short reminder such as ``on the real-state
subspace'', ``in branch coordinates'', or ``for the maximal symbol-sum
realization'', it refers back to this standing interpretation.
\end{remark}

\subsection{The branch algebra \texorpdfstring{$A$}{A}}

We begin by recalling only the portion of the algebraic structure from the
earlier papers that is needed for the distributed lifting. For the coefficient
algebra,
\[
A := \R[u]/(u^4-1)
\]
is viewed as a finite-dimensional commutative real algebra. Since
\[
u^4-1=(u-1)(u+1)(u^2+1),
\]
the algebra is semisimple and splits over $\R$ into two real one-dimensional branches and one irreducible real two-dimensional branch corresponding to the quadratic factor $u^2+1$. The present paper uses exactly this coarse branch structure and does not need the finer complex decomposition into the roots $i$ and $-i$ separately.

A convenient set of algebraic idempotents is given by
\[
\Pplus = \frac14(1+u+u^2+u^3),
\qquad
\Pminus = \frac14(1-u+u^2-u^3),
\qquad
\Pc = \frac12(1-u^2).
\]
A direct calculation shows that
\[
\Pplus^2=\Pplus,
\qquad
\Pminus^2=\Pminus,
\qquad
\Pc^2=\Pc,
\qquad
\Pplus\Pminus=\Pplus\Pc=\Pminus\Pc=0,
\]
and
\[
\Pplus+\Pminus+\Pc = 1.
\]
Thus these elements resolve the identity and isolate the plus, minus, and complex branches. The corresponding branch subspaces of $A$ are
\[
A_+ := \Pplus A,
\qquad
A_- := \Pminus A,
\qquad
A_c := \Pc A.
\]
The spaces $A_+$ and $A_-$ are one-dimensional over $\R$, while $A_c$ is two-dimensional over $\R$.

In order to represent the complex branch concretely inside the real algebra $A$, it is useful to introduce the element
\[
J := \frac12(u-u^3).
\]
Then $J\in A_c$, and one checks that
\[
J^2 = -\Pc,
\qquad
\Pc J = J \Pc = J.
\]
Hence the pair $(\Pc,J)$ plays the role of the standard basis $(1,i)$ on the complex branch. Every $a\in A$ can therefore be written uniquely in the form
\[
a = a_+\Pplus + a_-\Pminus + a_c^R\Pc + a_c^I J,
\qquad
 a_+,a_-,a_c^R,a_c^I\in\R.
\]
Equivalently,
\[
A \cong \R\Pplus \oplus \R\Pminus \oplus \mathrm{span}_{\R}\{\Pc,J\},
\]
where the last summand is canonically isomorphic to $\C$ as a real algebra through the identification
\[
\iota_c(a_c^R\Pc + a_c^I J) := a_c^R + i a_c^I.
\]
This realification viewpoint is the one used throughout the paper. It allows us to work on real state spaces while encoding the complex branch by a single transported reduced symbol of the form $a(\xi)+i b(\xi)$. The operator-level, global real-coordinate convention is fixed below in Proposition~\ref{prop:realified-complex-branch-normal-form}.

The action of left multiplication by $u$ is scalar on the two real branches and rotational on the complex branch:
\[
\Lu \Pplus = \Pplus,
\qquad
\Lu \Pminus = -\Pminus,
\qquad
\Lu \Pc = J,
\qquad
\Lu J = -\Pc.
\]
Accordingly, when $\Lu$ is restricted to $A_+$ and $A_-$ it acts by the scalars $+1$ and $-1$, whereas on $A_c$ it is represented by the real matrix
\[
\begin{pmatrix}
0 & -1\\
1 & 0
\end{pmatrix}
\]
in the ordered basis $(\Pc,J)$. Through the real-linear coordinate map $\iota_c$, this real rotation block is represented by multiplication by $i$. This is the algebraic origin of the complex reduced symbol used later.

For the Fourier-side formulation used later, we also fix the complexification
\[
\AC:=A\otimesR \C.
\]
All algebraic operators introduced above extend canonically by complex linearity to $\AC$; by abuse of notation we denote these extensions again by $\Pplus$, $\Pminus$, $\Pc$, $J$, and $\Lu$. We write
\[
\cA:\AC\to \AC,
\qquad
\cA(a\otimes z):=a\otimes \overline z,
\]
for the canonical conjugation. The physical state space of the paper remains the real Hilbert space $L^2(\R^d;A)$, while $\AC$ will be used only on the Fourier side, where complex scalar multipliers act naturally.

For later Plancherel and multiplier-domain statements, we fix the corresponding branch-adapted norm on the complexified coefficient algebra explicitly. Every $V\in\AC$ has a unique expansion
\[
V=z_+\Pplus+z_-\Pminus+z_c^R\Pc+z_c^I J,
\qquad
z_+,z_-,z_c^R,z_c^I\in\C.
\]
We equip $\AC$ with the Hilbertian norm
\[
\norm{V}_{\AC}^2
:=|z_+|^2+|z_-|^2+|z_c^R|^2+|z_c^I|^2.
\]
The induced $L^2(\R^d;\AC)$ norm is always the one used below. On the real subspace $A\subset\AC$ this norm restricts to the branch-adapted norm fixed on $A$, and the projectors $\Pplus$, $\Pminus$, and $\Pc$ are bounded orthogonal projections for this norm.

\subsection{The state space \texorpdfstring{$L^2(\R^d;A)$}{L2(Rd;A)}}

The state space of the paper is
\[
L^2(\R^d;A),
\]
consisting of equivalence classes of strongly measurable maps $Y:\R^d\to A$ such that $\norm{Y(\cdot)}_A\in \LtwoR$ for any fixed norm on the finite-dimensional space $A$. Since all norms on $A$ are equivalent, the specific algebra norm is immaterial for the topology. Nevertheless, for the later branchwise decomposition it is convenient to fix a branch-adapted Hilbertian norm from the start.

Using the decomposition of $A$ above, every state $Y\in L^2(\R^d;A)$ can be written uniquely as
\[
Y(x)=Y_+(x)\Pplus + Y_-(x)\Pminus + Y_c^R(x)\Pc + Y_c^I(x)J,
\]
with coefficient functions
\[
Y_+,Y_-,Y_c^R,Y_c^I \in L^2(\R^d;\R).
\]
We define
\[
\norm{Y}_{L^2(\R^d;A)}^2
:= \norm{Y_+}_{\LtwoR}^2
 + \norm{Y_-}_{\LtwoR}^2
 + \norm{Y_c^R}_{\LtwoR}^2
 + \norm{Y_c^I}_{\LtwoR}^2.
\]
With this choice, $L^2(\R^d;A)$ is a real Hilbert space canonically isomorphic to
\[
L^2(\R^d;\R)\oplus L^2(\R^d;\R)\oplus L^2(\R^d;\R^2).
\]

\begin{remark}[Branch-adapted norm as part of the theorem package]\label{rem:branch-adapted-norm-convention}
The fixed Hilbert norm above is not only a harmless topological choice. It is the
coordinate convention under which the branch projectors are orthogonal and under
which the later direct-sum norm formulae are literal equalities. If a different
but equivalent coefficient norm is placed on the finite-dimensional algebra
\(A\), the same topology is obtained, but the exact maximum and graph-norm
identities of Section~5 become two-sided estimates with constants depending on
the norm equivalence. Thus every occurrence of a coordinate-exact branch-norm identity
below is to be read in the branch-adapted Hilbert model fixed here. All exact
norm identities are stated relative to this fixed branch-adapted Hilbert norm;
equivalent Hilbert norms preserve generation, closedness, and graph-density
statements, but not the displayed constants or literal isometries.
\end{remark}

\begin{remark}[No intrinsic canonical norm claim]
\label{rem:no-intrinsic-canonical-norm-claim}
The algebra \(A=\R[u]/(u^4-1)\) does not by itself select the numerical
constants appearing in the exact norm formulae below. Those formulae are exact
because the paper has fixed the branch-adapted Hilbert coordinate norm in which
the three projectors are orthogonal and the \(c\)-branch coordinate map is
real-unitary. Replacing this norm by an arbitrary equivalent coefficient norm
leaves the same Hilbert topology and the same closed multiplier generators, but
it changes exact operator norms into equivalent estimates with comparison
constants. Consequently, no claim of an intrinsic canonical algebra norm is
being made.
\end{remark}
Via the identification $\iota_c$, the complex branch can also be represented by a single complex-valued coordinate
\[
h_c := Y_c^R + i Y_c^I \in L^2(\R^d;\C).
\]
In this paper the notation $Y_c$ is reserved for the actual $A_c$-valued branch element, while $h_c$ denotes its complex scalar coordinate. Thus the whole state space may be regarded equivalently as
\[
L^2(\R^d;A) \cong L^2(\R^d;\R) \oplus L^2(\R^d;\R) \oplus L^2(\R^d;\C),
\]
where the last summand is understood as a real Hilbert space. This hybrid notation is convenient later: the plus and minus branches are treated as real scalar semigroup components, whereas the complex branch is handled through a single complex reduced operator.

The decomposition above is not merely notational. It is the distributed analogue of the branch splitting from the finite-dimensional theory. In particular, it shows that the $A$-valued state space already carries an intrinsic branch geometry before any differential operator is imposed. The role of the multiplier generator will be to respect this geometry in a frequency-resolved way.

\begin{remark}
The use of $L^2(\R^d;A)$ rather than a Sobolev scale is deliberate. The purpose of Paper~5 is to establish an exact function-space multiplier realization of the branchwise semigroup architecture, not to optimize regularity. Once the branchwise reconstruction and observation pushforward mechanisms are proved at the $L^2$ level, stronger function spaces can be revisited in later work.
\end{remark}

\begin{remark}[Sobolev convention for the benchmark stress test]\label{rem:sobolev-convention}
Whenever Sobolev spaces appear in the benchmark and stress-test discussion, we use the Fourier convention
\[
H^s(\R^d)
:=
\{f\in L^2(\R^d):\ \ang{\xi}^{s}\widehat f(\xi)\in L^2(\R^d)\},
\qquad
\ang{\xi}:=(1+|\xi|^2)^{1/2}.
\]
This convention is used only to interpret the polynomial-symbol benchmark domains, for instance the diagnostic comparison between an \(H^1\) branch-generator domain and an \(H^2\) componentwise strong-prototype domain in Proposition~\ref{prop:stress-test-cancellation}.
\end{remark}

\begin{remark}[Real/complex Fourier convention]\label{rem:real-complex-convention}
This remark expands item~\textup{(i)} of the standing interpretation box,
Remark~\ref{rem:standing-interpretation-box-paper5}. The ambient physical
space $L^2(\R^d;A)$ is always treated as a \emph{real} Hilbert space. The plus and minus branches are therefore real $L^2$ spaces throughout. The complex branch is still represented through the real-linear isometric isomorphism
\[
\mathcal I_c:L^2_c(\R^d;A)\to L^2(\R^d;\C),
\]
and whenever $L^2(\R^d;\C)$ appears in the paper it is to be interpreted in one of two explicitly stated roles:
\begin{enumerate}[label=\textup{(\roman*)}]
    \item as a \emph{complex} Hilbert space when the reduced scalar semigroup $T_c(t)$ and its generator $G_c$ are being constructed;
    \item as a \emph{real} Hilbert space after transport back to $L^2_c(\R^d;A)$ through $\mathcal I_c$.
\end{enumerate}
There is, however, a necessary Fourier-side complexification: the Fourier transform of a real $A$-valued state takes values naturally in $\AC$, not in $A$ itself. Real physical states are recovered exactly by the reality symmetry
\[
\widehat Y(-\xi)=\cA(\widehat Y(\xi))
\quad\text{for a.e. }\xi\in\R^d.
\]
Thus the paper never complexifies the physical state space, but it does complexify the coefficient algebra on the Fourier side, where the multiplier calculus is carried out. Throughout the paper, every pointwise fiber formula involving the possibly complex-valued symbols $a(\xi)$ and $b(\xi)$ is therefore an operator formula on $\AC$. In such formulas the identity is denoted by $\IAC$, and $\Lu$ denotes the complex-linear extension of left multiplication by $u$ from $A$ to $\AC$.
\end{remark}

\begin{remark}[Real-linearity of the complex branch coordinate]\label{rem:complex-branch-real-linear-warning}
The map \(\mathcal I_c\) is always used as a real-linear isometric transport
from the physical \(A_c\)-branch to one complex scalar coordinate. Later
occurrences of \(G_c=\lambda_c(D)\), \(\lambda_c=a+i b\), or
\(\mathcal A_c=\mathcal I_c^{-1}G_c\mathcal I_c\) use the normal form of
Proposition~\ref{prop:realified-complex-branch-normal-form}.
\end{remark}

\subsection{Fourier transform of \texorpdfstring{$A$}{A}-valued states}

Because $A$ is finite-dimensional, the Fourier transform of an $A$-valued state is defined componentwise. For a Schwartz-class state
\[
Y = Y_+\Pplus + Y_-\Pminus + Y_c^R\Pc + Y_c^I J,
\]
we set
\[
\widehat{Y}(\xi)
:= \widehat{Y_+}(\xi)\Pplus
 + \widehat{Y_-}(\xi)\Pminus
 + \widehat{Y_c^R}(\xi)\Pc
 + \widehat{Y_c^I}(\xi)J,
\]
where for scalar components we use the normalization
\[
\widehat{f}(\xi)
:= (2\pi)^{-d/2}\int_{\R^d} e^{-i x\cdot \xi} f(x)\,dx.
\]
Since the scalar Fourier coefficients are generally complex, one has
\[
\widehat Y(\xi)\in \AC
\quad\text{for a.e. }\xi\in\R^d,
\]
not merely in $A$. Consequently, Fourier-side multiplier fibers are always read in $\AC$: expressions such as $a(\xi)\IAC+b(\xi)\Lu$ are complex-linear operators on $\AC$, whereas the physical formal prototype remains an operator expression on the real state space $L^2(\R^d;A)$. The Fourier transform therefore identifies $L^2(\R^d;A)$ with the closed real-linear reality-symmetric subspace
\[
\mathcal H_{\mathrm{sym}}
:=
\bigl\{\Phi\in L^2(\R^d;\AC): \Phi(-\xi)=\cA(\Phi(\xi))\text{ for a.e. }\xi\in\R^d\bigr\}.
\]
This set is closed and real-linear, but it is not a complex-linear subspace of $L^2(\R^d;\AC)$ in general: multiplication by $i$ reverses the Fourier-side reality condition. More precisely, by density the componentwise transform extends uniquely to a real-linear isometric isomorphism
\[
\F : L^2(\R^d;A) \to \mathcal H_{\mathrm{sym}} \subset L^2(\R^d;\AC),
\]
where $\mathcal H_{\mathrm{sym}}$ is equipped with the real Hilbert structure inherited from $L^2(\R^d;\AC)$.
With the branch-adapted norms on $A$ and $\AC$ fixed above, Plancherel's identity holds componentwise and hence in total:
\[
\norm{\widehat{Y}}_{L^2(\R^d;\AC)} = \norm{Y}_{L^2(\R^d;A)}.
\]
Conversely, the symmetry condition $\Phi(-\xi)=\cA(\Phi(\xi))$ is exactly the Fourier-side encoding of reality of the physical state.

The importance of the Fourier transform in the present paper is structural rather than cosmetic. Since the multiplier class under consideration is translation-invariant, each constant-coefficient operator becomes a multiplier after Fourier transform. The branch algebra then acts on the finite-dimensional complexified coefficient space $\AC$, while the spatial dependence is encoded frequency-by-frequency. This is precisely what makes the present multiplier framework tractable: it converts the distributed dynamics into a measurable family of finite-dimensional algebraic problems indexed by $\xi\in\R^d$.

In particular, if $T$ is a scalar Fourier multiplier with symbol $m(\xi)$, then for every $A$-valued state $Y$ one has
\[
\widehat{(I_A\otimes T)Y}(\xi)=m(\xi)\widehat{Y}(\xi),
\]
with the right-hand side interpreted in $\AC$. Since the branch projectors act only on the coefficient algebra, their complex-linear extensions commute with the Fourier transform in the obvious sense:
\[
\F(\Pplus Y)=\Pplus \widehat{Y},
\qquad
\F(\Pminus Y)=\Pminus \widehat{Y},
\qquad
\F(\Pc Y)=\Pc \widehat{Y}.
\]
This commutation is the basic mechanism behind the Fourier-side representation of the intrinsic branch decomposition. It implies that the distributed branch decomposition can be analyzed pointwise in frequency space without losing the real-space Hilbert structure, provided one remembers that the coefficient algebra has been complexified on the Fourier side.

\subsection{Branch projectors on distributed states}

The algebraic projectors $\Pplus$, $\Pminus$, and $\Pc$ lift to bounded operators on $L^2(\R^d;A)$ by pointwise multiplication:
\[
(\Pplus Y)(x):=\Pplus Y(x),
\qquad
(\Pminus Y)(x):=\Pminus Y(x),
\qquad
(\Pc Y)(x):=\Pc Y(x).
\]
No ambiguity should arise from reusing the same symbols for the algebra elements and the induced operators. Because the coefficient algebra is finite-dimensional, these lifted operators are bounded, idempotent, and pairwise orthogonal in the sense
\[
\Pplus^2=\Pplus,
\qquad
\Pminus^2=\Pminus,
\qquad
\Pc^2=\Pc,
\qquad
\Pplus\Pminus=\Pplus\Pc=\Pminus\Pc=0,
\]
and
\[
\Pplus+\Pminus+\Pc=I_{L^2(\R^d;A)}.
\]
Therefore
\[
L^2(\R^d;A) = \mathrm{Ran}(\Pplus) \oplus \mathrm{Ran}(\Pminus) \oplus \mathrm{Ran}(\Pc)
\]
as a topological direct sum of closed subspaces.

The ranges of these projectors are precisely the distributed branch spaces:
\[
L^2_+(\R^d;A):=\mathrm{Ran}(\Pplus),
\qquad
L^2_-(\R^d;A):=\mathrm{Ran}(\Pminus),
\qquad
L^2_c(\R^d;A):=\mathrm{Ran}(\Pc).
\]
The plus and minus branches are naturally isomorphic to scalar real $\LtwoR$, while the complex branch is naturally isomorphic to complex $\LtwoC$ viewed as a real Hilbert space. More precisely, the map
\[
\mathcal I_c : L^2_c(\R^d;A) \to L^2(\R^d;\C),
\qquad
\mathcal I_c\bigl(Y_c^R\Pc + Y_c^I J\bigr) := Y_c^R + i Y_c^I,
\]
is an isometric isomorphism of real Hilbert spaces.

Throughout the rest of the paper we distinguish the real and complex scalar spaces by writing
\[
\LtwoR:=L^2(\R^d;\R),
\qquad
\LtwoC:=L^2(\R^d;\C).
\]
The plus and minus branch model spaces are real Hilbert spaces modeled on $\LtwoR$, while Fourier-side scalar multiplier expressions such as $m\widehat f$ are measured in the complex space $\LtwoC$. To avoid ambiguity, formal domains and norm estimates below use the branch-specific notation $\LtwoR$ or $\LtwoC$ whenever the scalar field matters.

This point is essential for the later semigroup reduction. On the plus and minus branches, left multiplication by $u$ acts simply by the real scalars $+1$ and $-1$. On the complex branch, after transport by $\mathcal I_c$, left multiplication by $u$ becomes multiplication by $i$. Thus the three reduced branch models are not arbitrary choices: they are forced by the branch projector geometry itself.

The projectors are also compatible with the class of translation-invariant operators studied in the paper. If $T$ is any scalar Fourier multiplier, then
\[
\Pplus(I_A\otimes T)=(I_A\otimes T)\Pplus,
\qquad
\Pminus(I_A\otimes T)=(I_A\otimes T)\Pminus,
\qquad
\Pc(I_A\otimes T)=(I_A\otimes T)\Pc.
\]
Likewise, because the projectors belong to the commutative algebra generated by $u$, they commute with $\Lu$. Consequently they also commute with the prototype generator introduced below. This invariance is the functional-analytic shadow of the algebraic branch splitting and is the reason the reduced branch flows can later be reconstructed exactly.

\subsection{The prototype generator and admissible multipliers}

We now introduce the class of distributed generators studied throughout the paper. Let $a(\xi)$ and $b(\xi)$ be measurable complex-valued functions on $\R^d$, finite almost everywhere. Equivalently, the paper works with ordinary finite-a.e. Fourier multiplier symbols, not with extended-valued symbols. Formally, they define scalar Fourier multiplier operators $a(D)$ and $b(D)$ first on the complex scalar space $\LtwoC$ through
\[
\widehat{a(D)f}(\xi)=a(\xi)\widehat{f}(\xi),
\qquad
\widehat{b(D)f}(\xi)=b(\xi)\widehat{f}(\xi),
\]
with domains
\[
\Dom(a(D)):=\{f\in \LtwoC: a(\xi)\widehat{f}(\xi)\in \LtwoC\},
\]
\[
\Dom(b(D)):=\{f\in \LtwoC: b(\xi)\widehat{f}(\xi)\in \LtwoC\}.
\]
We begin with the formal prototype expression
\[
\Gform := I_A\otimes a(D) + \Lu\otimes b(D).
\]
To make it into a well-defined densely defined operator on $L^2(\R^d;A)$, we first introduce its \emph{componentwise strong realization}
\[
\Gcal Y := \Gform Y,
\]
on the domain
\[
\Dom(\Gcal):=
\Bigl\{Y\in L^2(\R^d;A):
 a(\cdot)\widehat{Y}(\cdot),\ b(\cdot)\widehat{Y}(\cdot)\in L^2(\R^d;\AC)\Bigr\}.
\]
Thus $\Gcal$ is the operator obtained by requiring the two scalar multiplier pieces to act separately on the full $A$-valued state after passing to the complexified Fourier side. On the Fourier side one has the fiberwise formula
\[
\widehat{\Gcal Y}(\xi)=\bigl(a(\xi)\IAC+b(\xi)\Lu\bigr)\widehat{Y}(\xi),
\qquad Y\in\Dom(\Gcal),
\]
where $\Lu$ now denotes the complex-linear extension of left multiplication by $u$ to $\AC$. For $Y\in\Dom(\Gcal)$, the Fourier-side expression is initially an $\AC$-valued object. The point of the reality assumption is that it is not merely $\AC$-valued: after writing
\[
Y=Y_+\Pplus+Y_-\Pminus+Y_c^R\Pc+Y_c^I J,
\qquad
Y_+,Y_-,Y_c^R,Y_c^I\in \LtwoR,
\]
the coefficient functions obtained from
\[
(a(\xi)\IAC+b(\xi)\Lu)\widehat Y(\xi)
\]
retain the Hermitian Fourier symmetry imposed by real-valued physical states. The coefficientwise lemma below records this coefficientwise check and shows that $\Gcal Y$ belongs to $L^2(\R^d;A)$, not merely to $L^2(\R^d;\AC)$. This makes clear why this class is a natural multiplier realization to study here: the spatial variable enters only through scalar multipliers, while the branch algebra acts in the finite-dimensional complexified coefficient space on the Fourier side.

The distinction between the formal prototype $\Gform$ and the strong realization $\Gcal$ is important. The genuine generator of the reconstructed full flow will be identified only later, after the reduced semigroup theory has been developed. In general that generator is a \emph{branchwise realization} associated with the reduced symbols $\lambda_\pm$ and $\lambda_c$, and the componentwise strong operator $\Gcal$ should be viewed a priori only as a canonical sub-operator of that later generator. Accordingly, throughout the paper the terms \emph{formal prototype} and \emph{strong prototype realization} refer to $\Gform$ and $\Gcal$, whereas the phrase \emph{full generator} refers to the branchwise realization $\Acal$ once Section~5 has been reached. The point of the present section is therefore to fix the formal architecture and the strong prototype domain, not yet to settle generator identification for the reconstructed full flow; the correct comparison later on is $\Gcal\subseteq \Acal$, and that inclusion may be strict when branchwise cancellation enlarges the reduced domains.

At the level of setup, the only admissibility package we need is the one that guarantees later semigroup generation for the reduced branches and compatibility with the real state space. To keep the paper narrow and concrete, we impose the following standing assumptions.

\begin{remark}[Standing finite-a.e. multiplier-symbol convention]\label{rem:finite-ae-symbol-convention}
Throughout the paper, the symbols
\[
a,b:\R^d\to\C
\]
are ordinary measurable symbols, finite almost everywhere. No extended-valued
symbol theory is used. In particular, the graph-core argument later relies on
the exhaustion of frequency space, modulo null sets, by finite-symbol level sets
of the form
\[
\{\xi: |a(\xi)|\le n,\ |b(\xi)|\le n\}.
\]
If a symbol were allowed to take infinite values on a set of positive measure,
that exhaustion would fail and the operator domain and graph-core statement
would require a different formulation. This is a scope restriction, not a claim
that finite-a.e. multiplier symbols are the only possible multiplier framework.
Other theories may treat extended-valued, form-defined, or singular symbols by
using different domains or realization procedures. They are excluded here in
order to keep the branch bridge at the level of finite measurable matrix symbols
and maximal \(L^2\) graph realizations. No such extension is claimed here.
\end{remark}

\begin{enumerate}
    \item[(A1)] \textbf{Reality symmetry.} The symbols satisfy
    \[
    a(-\xi)=\overline{a(\xi)},
    \qquad
    b(-\xi)=\overline{b(\xi)}
    \quad\text{for a.e. }\xi\in\R^d.
    \]
    This condition is used in the domain-sensitive sense appropriate for possibly unbounded multipliers: if $m\in\{a,b\}$, $f\in \LtwoR$, and $m(\cdot)\widehat f(\cdot)\in \LtwoC$, then $m(D)f$ is again real-valued. Equivalently, the natural complex multiplier domain of $m(D)$ is invariant under the real structure, and $m(D)$ maps its real-domain part into $\LtwoR$. The same domain-sensitive interpretation applies to the reduced real-branch symbols $\lambda_+=a+b$ and $\lambda_-=a-b$ whenever their unbounded multiplier realizations are used. All later uses of this symmetry are made after the harmless representative and symmetric-cutoff normalization recorded in Lemma~\ref{lem:hermitian-representatives-symmetric-cutoffs}.

    \item[(A2)] \textbf{Uniform upper bound on the reduced real parts.} For each branch $\bullet\in\{+,-,c\}$, define
    \[
    \lambda_+(\xi):=a(\xi)+b(\xi),
    \qquad
    \lambda_-(\xi):=a(\xi)-b(\xi),
    \qquad
    \lambda_c(\xi):=a(\xi)+i b(\xi).
    \]
    Since $a$ and $b$ were already fixed at the start of Section~2 as measurable finite-a.e. complex-valued functions on $\R^d$, the reduced symbols $\lambda_+$, $\lambda_-$, and $\lambda_c$ are automatically measurable and finite almost everywhere. The only additional branchwise growth input imposed here is that there exist numbers $\omega_\bullet\in\R$ such that
    \[
    \operatorname*{ess\,sup}_{\xi\in\R^d} \Re \lambda_\bullet(\xi)
    \le \omega_\bullet.
    \]
    These upper bounds are the only branchwise growth input needed for the $L^2$ semigroup generation theorem of Section~4; the bounded multiplier property of $e^{t\lambda_\bullet}$ is then recovered as a conclusion rather than imposed as an assumption. For the prescribed $L^2$ multiplier formula used in this paper, the finiteness of $\operatorname*{ess\,sup}\Re\lambda_\bullet$ is also the exact boundedness condition for $e^{t\lambda_\bullet(D)}$ at any positive time, as recorded in Lemma~\ref{lem:near-necessity-a2-prescribed-multiplier}. The word ``sufficient'' is therefore relative to the broader universe of possible realizations---weighted spaces, form methods, extrapolation spaces, or non-$L^2$ settings---and not a weakening of the condition inside the displayed branchwise $L^2$ multiplier model. The numbers $\omega_\bullet$ should be interpreted throughout as admissible branchwise operator-norm upper envelopes for the semigroups, not as datum-specific growth exponents. When $\omega_\bullet$ is chosen to be
    \[
    \omega_\bullet^{\mathrm{sh}}
    :=
    \operatorname*{ess\,sup}_{\xi\in\R^d}\Re\lambda_\bullet(\xi),
    \]
    it is the candidate sharp branchwise $L^2$ multiplier growth envelope in the fixed branch-adapted Hilbert norm of this paper. The sharpness for the complex branch follows from the standard complex $L^2$ multiplier norm formula, while the sharpness for the real plus and minus branches requires the Hermitian real-multiplier norm lemma proved below. After Corollary~\ref{cor:sharp-real-branch-envelopes}, this candidate envelope is indeed sharp for all three reduced branches in that fixed branch-adapted Hilbert norm; otherwise $\omega_\bullet$ is only a valid upper envelope.
\end{enumerate}

\begin{definition}[Standing multiplier-semigroup package]\label{def:standing-multiplier-semigroup-package}
We say that the multiplier pair \((a,b)\) satisfies the \emph{standing
multiplier-semigroup package} if the following requirements hold:
\begin{enumerate}[label=\textup{(H\arabic*)}]
    \item \(a,b:\R^d\to\C\) are measurable and finite almost everywhere, in
    the sense of the finite-a.e. multiplier-symbol convention of
    Remark~\ref{rem:finite-ae-symbol-convention}.
    \item The reality symmetry assumption \textup{(A1)} holds.
    \item The reduced real-part upper-bound assumption \textup{(A2)} holds for
    the reduced symbols
    \[
    \lambda_+=a+b,
    \qquad
    \lambda_-=a-b,
    \qquad
    \lambda_c=a+i b.
    \]
\end{enumerate}
This named package is only a bookkeeping device for the theorem statements
below. It makes explicit, whenever the reconstructed flow, its generator, graph
identifications, norm identities, or observation pushforwards are invoked, that
all three standing inputs---the finite-a.e. convention, \textup{(A1)}, and
\textup{(A2)}---are being used unless a statement explicitly says that a smaller
hypothesis set suffices.
\end{definition}

\begin{remark}[Exact role of the reduced-symbol growth package]\label{rem:scope-reduced-symbol-package}
Assumption \textup{(A2)} is used in this paper as an explicit growth package for
the present branch-adapted \(L^2\)-Fourier-multiplier construction. Inside the
prescribed multiplier formula
\[
T_\bullet(t)=\mathcal F^{-1}\bigl(e^{t\lambda_\bullet}\widehat{\cdot}\,\bigr),
\]
it is not merely decorative: finiteness of
\(\operatorname*{ess\,sup}\Re\lambda_\bullet\) is exactly the condition that this
multiplier be bounded on \(L^2\) at any positive time. Lemma~\ref{lem:near-necessity-a2-prescribed-multiplier}
records this one-symbol converse. The condition is called a sufficient
hypothesis only relative to questions outside the model fixed here: it is not a
classification of all possible multiplier generators, is not asserted as a
minimality theorem in an abstract generator class, and is not meant to rule out
alternative realizations in weighted, form-theoretic, extrapolation, or
non-Hilbert settings.

For the reduced symbols considered here, \textup{(A2)} gives the bound
\[
\|e^{t\lambda_\bullet(\cdot)}\|_{L^\infty(\R^d)}
\le e^{\omega_\bullet t},
\]
which is the input needed for the branchwise semigroup construction of
Section~4. If the chosen value is larger than
$\operatorname*{ess\,sup}\Re\lambda_\bullet$, it should be read only as an
admissible upper envelope. If it is chosen as
$\operatorname*{ess\,sup}\Re\lambda_\bullet$, the sharpness of the resulting
semigroup envelope is a statement in the fixed branch-adapted $L^2$ Hilbert
norm and is not left implicit: the real branches are justified by
Lemma~\ref{lem:sharp-real-hermitian-multiplier-norm}, and the three branch
identities are collected in Corollary~\ref{cor:sharp-real-branch-envelopes}.
\end{remark}

\begin{lemma}[Near-necessity of the reduced real-part bound for prescribed multipliers]
\label{lem:near-necessity-a2-prescribed-multiplier}
Let \(\lambda:\R^d\to\C\) be measurable and finite almost everywhere, and define
for \(t>0\) the prescribed Fourier multiplier
\[
T_\lambda(t)f:=\mathcal F^{-1}\bigl(e^{t\lambda}\widehat f\bigr)
\]
on complex \(L^2(\R^d)\) whenever this multiplier is bounded. Then the following
conditions are equivalent for any fixed \(t_0>0\):
\[
T_\lambda(t_0)\in\mathcal L(L^2(\R^d))
\qquad\Longleftrightarrow\qquad
\operatorname*{ess\,sup}_{\xi\in\R^d}\Re\lambda(\xi)<\infty.
\]
More precisely,
\[
\|T_\lambda(t_0)\|_{\mathcal L(L^2)}
=\|e^{t_0\lambda}\|_{L^\infty}
=\exp\Bigl(t_0\operatorname*{ess\,sup}_{\xi\in\R^d}\Re\lambda(\xi)\Bigr),
\]
with the convention that the last quantity is infinite when the essential
supremum is infinite. Consequently, for the prescribed branchwise multiplier
semigroups used in this paper, assumption \textup{(A2)} is not an arbitrary
regularity adornment: it is exactly the boundedness condition for the displayed
\(L^2\) multiplier formula at any positive time on each reduced branch. This is
a near-necessity statement for the prescribed \(L^2\) multiplier realization, not
a classification theorem for unrelated realizations in other spaces or domains.
\end{lemma}

\begin{proof}
The standard \(L^2\) multiplier norm formula gives
\[
\|T_\lambda(t_0)\|_{\mathcal L(L^2)}
=\|e^{t_0\lambda}\|_{L^\infty}.
\]
Since
\[
|e^{t_0\lambda(\xi)}|=e^{t_0\Re\lambda(\xi)},
\]
the multiplier is essentially bounded if and only if
\(\Re\lambda\) is essentially bounded above. Taking the essential supremum gives
the displayed norm identity. Applying this one-symbol statement to
\(\lambda_+\), \(\lambda_-\), and \(\lambda_c\) gives the branchwise conclusion.
For the real plus and minus branch spaces, the same conclusion is read through
the Hermitian real-multiplier norm convention used elsewhere in the paper.
\end{proof}


\begin{proposition}[Admissibility checklist for standard symbol sources]
\label{prop:admissibility-checklist}
The standing multiplier-semigroup package is verified in each of the following
standard situations. These are sufficient tests, not a classification of all
admissible pairs.
\begin{enumerate}[label=\textup{(\alph*)}]
    \item \textbf{Direct reduced-symbol test.} If $a,b$ are measurable,
    finite almost everywhere, satisfy the Hermitian symmetry \textup{(A1)}, and
    the three reduced real parts
    \[
    \Re(a+b),\qquad \Re(a-b),\qquad \Re(a+i b)
    \]
    are essentially bounded above, then $(a,b)$ satisfies the standing package.

    \item \textbf{Dissipative symbols plus skew-adjoint transport.} Let
    $q,\beta:\R^d\to\R$ be finite a.e. even measurable functions, and let
    $\tau:\R^d\to\R$ be finite a.e. odd and measurable. Set
    \[
    a=q+i\tau,
    \qquad
    b=\beta.
    \]
    If $q+\beta$, $q-\beta$, and $q$ are essentially bounded above, then the
    standing package holds. This includes local polynomial examples such as
    shifted heat operators with constant zero-order branch shifts, and allows
    skew-adjoint transport terms through the odd imaginary part $i\tau$.

    \item \textbf{Simple nonlocal branch coupling.} Let $\rho:\R^d\to[0,\infty)$
    be finite a.e., even, and measurable, let $\gamma\ge0$, and set
    \[
    a=-\rho^2,
    \qquad
    b=\gamma\rho.
    \]
    Then the standing package holds with
    \[
    \operatorname*{ess\,sup}\Re\lambda_+\le \frac{\gamma^2}{4},
    \qquad
    \operatorname*{ess\,sup}\Re\lambda_-\le 0,
    \qquad
    \operatorname*{ess\,sup}\Re\lambda_c\le 0.
    \]
    The choice $\rho(\xi)=|\xi|$ gives the $|D|$-type benchmark used later.

    \item \textbf{Bounded lower-order perturbations.} If $(a,b)$ satisfies the
    standing package and $h,k$ are bounded measurable Hermitian symbols, then
    $(a+h,b+k)$ also satisfies the standing package. The new upper envelopes are
    obtained by adding the corresponding $L^\infty$ bounds for
    \[\Re(h+k),\qquad \Re(h-k),\qquad \Re(h+i k).\]
\end{enumerate}
\end{proposition}

\begin{proof}
Part \textup{(a)} is exactly the definition of the standing package. For
\textup{(b)}, the parity assumptions give
\[
(q+i\tau)(-\xi)=q(\xi)-i\tau(\xi)=\overline{q(\xi)+i\tau(\xi)},
\qquad
\beta(-\xi)=\beta(\xi),
\]
so \textup{(A1)} holds. The reduced real parts are $q+\beta$, $q-\beta$, and
$q$, respectively, so the stated upper bounds give \textup{(A2)}. For
\textup{(c)}, the symbols are real and even. Moreover
\[
\Re\lambda_+=-\rho^2+\gamma\rho\le \gamma^2/4,
\qquad
\Re\lambda_-=-\rho^2-\gamma\rho\le0,
\qquad
\Re\lambda_c=-\rho^2\le0.
\]
For \textup{(d)}, bounded Hermitian perturbations preserve the finite-a.e. and
Hermitian symmetry conditions, while the three reduced real parts change by the
bounded quantities displayed in the statement. Thus their essential upper
bounds remain finite.
\end{proof}

\begin{remark}[Covered and excluded symbol sources]
\label{rem:covered-excluded-symbol-sources}
Proposition~\ref{prop:admissibility-checklist} is the intended practical
reading of the assumption package. The package is not an arbitrary list of
conditions designed only to make the theorems true; it is a short sufficient
checklist for the multiplier sources used in the paper.
\[
\begin{array}{p{0.43\textwidth}p{0.47\textwidth}}
\hline
\textbf{Covered after verification} & \textbf{Excluded from the theorem package}\\
\hline
Polynomial constant-coefficient symbols whose Hermitian parity and reduced
upper real-part bounds hold & Polynomial symbols whose reduced real parts are
unbounded above on some branch\\[0.3em]
Real dissipative symbols with bounded branch shifts & Symbols violating the
Fourier-side reality symmetry needed to preserve real-valued states\\[0.3em]
Skew-adjoint transport perturbations entering through odd imaginary symbols &
Symbols singular or infinite on sets of positive measure\\[0.3em]
Nonlocal even symbols such as $|D|$ when the reduced bounds hold & Boundary,
trace, point-sensor, variable-coefficient, or form-defined realizations not
represented by finite-a.e. translation-invariant multipliers\\[0.3em]
Bounded Hermitian lower-order perturbations of an already admissible pair &
Alternative semigroup realizations outside the fixed $L^2$ multiplier graph
framework of this paper\\
\hline
\end{array}
\]
The examples in Section~7 are therefore not decorative afterthoughts. They are
explicit witnesses that the standing package contains local polynomial,
nonlocal, boundedly perturbed, cancellation-coupled, and observation-sensitive
subclasses while still excluding settings where the branch bridge would require
new domain technology.
\end{remark}

\begin{lemma}[Hermitian representatives and symmetric cutoffs modulo null sets]
\label{lem:hermitian-representatives-symmetric-cutoffs}
Let \(m:\R^d\to\C\) be measurable, finite almost everywhere, and satisfy
\[
m(-\xi)=\overline{m(\xi)}
\quad\text{for a.e. }\xi\in\R^d.
\]
Then there exists a measurable representative \(\widetilde m\), equal to \(m\)
almost everywhere, and a symmetric null set \(N=-N\) such that
\[
\widetilde m(-\xi)=\overline{\widetilde m(\xi)}
\qquad\text{for every }\xi\in\R^d\setminus N.
\]
Replacing \(m\) by \(\widetilde m\) does not change the Fourier multiplier: its
\(L^2\)-domain and its action agree with those of \(m(D)\).

Moreover, if \(E\subset\R^d\) is measurable and symmetric up to null sets, then
there exists a measurable set \(E^{\mathrm{sym}}\), equal to \(E\) modulo null
sets, such that \(E^{\mathrm{sym}}=-E^{\mathrm{sym}}\) exactly. Consequently
multiplication by \(\mathbf 1_E\) and by \(\mathbf 1_{E^{\mathrm{sym}}}\)
define the same \(L^2\) Fourier multiplier.
\end{lemma}

\begin{proof}
Let
\[
N_0:=\{\xi\in\R^d:\ m(-\xi)\ne\overline{m(\xi)}\}.
\]
Then \(N_0\) is null. Set \(N:=N_0\cup(-N_0)\). Then \(N\) is symmetric and
null, and the Hermitian identity holds for every \(\xi\notin N\). Define
\(\widetilde m=m\) on \(\R^d\setminus N\), and define \(\widetilde m\)
arbitrarily on \(N\), for instance \(\widetilde m=0\). Then
\(\widetilde m=m\) almost everywhere and satisfies the stated symmetry outside
the symmetric null set \(N\). Since \(\widetilde m=m\) almost everywhere,
\[
m\widehat f\in \LtwoC
\quad\Longleftrightarrow\quad
\widetilde m\,\widehat f\in \LtwoC,
\]
and the two products agree almost everywhere whenever either belongs to
\(\LtwoC\). Hence the multiplier domains and actions coincide.

For the set statement, assume \(E\triangle(-E)\) is null and set
\[
E^{\mathrm{sym}}:=E\cap(-E).
\]
Then \(E^{\mathrm{sym}}=-E^{\mathrm{sym}}\) exactly, and
\(E^{\mathrm{sym}}\triangle E\subset E\triangle(-E)\), so
\(E^{\mathrm{sym}}\) equals \(E\) modulo null sets. Therefore
\(\mathbf 1_E=\mathbf 1_{E^{\mathrm{sym}}}\) almost everywhere, and the
corresponding \(L^2\) multiplier projections are identical.
\end{proof}

\begin{lemma}[Domain-sensitive reality preservation for Fourier multipliers]
\label{lem:domain-sensitive-reality-preservation}
Let $m:\R^d\to\C$ be measurable, finite almost everywhere, and satisfy
\[
m(-\xi)=\overline{m(\xi)}
\quad\text{for a.e. }\xi\in\R^d.
\]
Let
\[
\Dom(m(D)):=\{f\in \LtwoC:m\widehat f\in \LtwoC\}.
\]
Then the real subspace is preserved in the domain-sensitive sense
\[
f\in \LtwoR\cap\Dom(m(D))
\quad\Longrightarrow\quad
m(D)f\in \LtwoR.
\]
Moreover, if $e^{tm(\cdot)}\in L^\infty(\R^d)$ for some $t\ge0$, then the bounded multiplier
$e^{tm(\cdot)}(D)$ preserves real-valued $L^2$ data.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:hermitian-representatives-symmetric-cutoffs}, we may replace
\(m\) by a Hermitian representative without changing the multiplier domain or
action. If \(f\in L^2(\R^d;\R)\), then
\[
\widehat f(-\xi)=\overline{\widehat f(\xi)}
\quad\text{for a.e. }\xi.
\]
Combining this identity with the Hermitian representative gives
\[
m(-\xi)\widehat f(-\xi)
=
\overline{m(\xi)}\,\overline{\widehat f(\xi)}
=
\overline{m(\xi)\widehat f(\xi)}
\quad\text{for a.e. }\xi.
\]
Thus \(m\widehat f\) satisfies the Fourier-side reality condition. If $m\widehat f\in L^2$, its
inverse Fourier transform is real-valued, which proves the unbounded multiplier statement.
The bounded semigroup-multiplier assertion follows from the same calculation because
\[
e^{tm(-\xi)}=e^{t\overline{m(\xi)}}=\overline{e^{tm(\xi)}}.
\]
\end{proof}


\begin{lemma}[Real Hermitian multiplier norm]
\label{lem:sharp-real-hermitian-multiplier-norm}
Let \(m\in L^\infty(\R^d)\) satisfy
\[
m(-\xi)=\overline{m(\xi)}
\quad\text{for a.e. }\xi\in\R^d.
\]
Let \(M_m^{\R}\) be the restriction of the Fourier multiplier \(m(D)\) to
\(\LtwoR\). Then \(M_m^{\R}\) is a bounded real-linear operator on
\(\LtwoR\), and
\[
\|M_m^{\R}\|_{\mathcal L(\LtwoR)}
=
\operatorname*{ess\,sup}_{\xi\in\R^d}|m(\xi)|.
\]
\end{lemma}

\begin{proof}
The Hermitian symmetry of \(m\) implies, by
Lemma~\ref{lem:domain-sensitive-reality-preservation}, that the bounded
multiplier \(m(D)\) preserves real-valued \(L^2\) data. Hence the restriction
\(M_m^{\R}\) is well-defined. Plancherel gives the upper bound
\[
\|M_m^{\R}f\|_{L^2}
=
\|m\widehat f\|_{L^2}
\le
\|m\|_{L^\infty}\|f\|_{L^2},
\qquad f\in\LtwoR.
\]
It remains to prove the reverse inequality. Set
\[
M:=\operatorname*{ess\,sup}_{\xi\in\R^d}|m(\xi)|.
\]
If \(M=0\), there is nothing to prove. Otherwise fix \(0<\varepsilon<M\) and
let
\[
E_\varepsilon:=\{\xi\in\R^d:\ |m(\xi)|>M-\varepsilon\}.
\]
The set \(E_\varepsilon\) has positive measure. Since
\(|m(-\xi)|=|m(\xi)|\) almost everywhere, \(E_\varepsilon\) is symmetric up
to a null set. By Lemma~\ref{lem:hermitian-representatives-symmetric-cutoffs},
we may replace it modulo null sets by an exactly symmetric set. Taking a
finite-measure positive subset and adjoining its reflection, we may choose a
symmetric measurable set \(F_\varepsilon\subset E_\varepsilon\) modulo null sets
with \(0<|F_\varepsilon|<\infty\). Define
\[
\widehat f_\varepsilon:=\mathbf 1_{F_\varepsilon}.
\]
Then \(\widehat f_\varepsilon(-\xi)=\overline{\widehat f_\varepsilon(\xi)}\),
so \(f_\varepsilon:=\mathcal F^{-1}\widehat f_\varepsilon\) belongs to
\(\LtwoR\). Moreover,
\[
\|M_m^{\R}f_\varepsilon\|_{L^2}^2
=
\int_{F_\varepsilon}|m(\xi)|^2\,d\xi
\ge
(M-\varepsilon)^2 |F_\varepsilon|
=
(M-\varepsilon)^2\|f_\varepsilon\|_{L^2}^2.
\]
Therefore
\[
\|M_m^{\R}\|_{\mathcal L(\LtwoR)}\ge M-\varepsilon.
\]
Letting \(\varepsilon\downarrow0\) proves the reverse inequality.
\end{proof}

\begin{lemma}[Real-structure preservation]
\label{lem:coefficientwise-real-structure-preservation}
Assume \textup{(A1)}. Let
\[
Y=Y_+\Pplus+Y_-\Pminus+Y_c^R\Pc+Y_c^I J,
\qquad
Y_+,Y_-,Y_c^R,Y_c^I\in \LtwoR,
\]
and suppose that
\[
a(\cdot)\widehat Y(\cdot),\ b(\cdot)\widehat Y(\cdot)
\in L^2(\R^d;\AC).
\]
Then
\[
\F^{-1}\Bigl((a(\cdot)\IAC+b(\cdot)\Lu)\widehat Y(\cdot)\Bigr)
\in L^2(\R^d;A).
\]
Equivalently, the componentwise strong prototype maps its natural real domain into
the real Hilbert space $L^2(\R^d;A)$, not merely into $L^2(\R^d;\AC)$.
\end{lemma}

\begin{proof}
Since the four coefficient functions are real-valued, their Fourier transforms satisfy
\[
\widehat{Y_\alpha}(-\xi)=\overline{\widehat{Y_\alpha}(\xi)},
\qquad
Y_\alpha\in\{Y_+,Y_-,Y_c^R,Y_c^I\}.
\]
By \textup{(A1)}, the scalar symbols satisfy
\[
a(-\xi)=\overline{a(\xi)},
\qquad
b(-\xi)=\overline{b(\xi)}.
\]
Hence every product $a\widehat{Y_\alpha}$ and $b\widehat{Y_\alpha}$ satisfies the same
Hermitian Fourier symmetry on its natural $L^2$ domain. This is precisely the
calculation in the preceding domain-sensitive lemma, applied
coefficient by coefficient.

Using
\[
\Lu\Pplus=\Pplus,
\qquad
\Lu\Pminus=-\Pminus,
\qquad
\Lu\Pc=J,
\qquad
\Lu J=-\Pc,
\]
we have
\[
(a\IAC+b\Lu)\widehat Y
=
(a+b)\widehat{Y_+}\Pplus
+
(a-b)\widehat{Y_-}\Pminus
\]
\[
\qquad
+
\bigl(a\widehat{Y_c^R}-b\widehat{Y_c^I}\bigr)\Pc
+
\bigl(b\widehat{Y_c^R}+a\widehat{Y_c^I}\bigr)J.
\]
Each scalar coefficient on the right is an $L^2$ function by the assumed
componentwise multiplier-domain condition, and each scalar coefficient has Hermitian
Fourier symmetry because it is a finite linear combination of such symmetric products.
Therefore the inverse Fourier transform of each scalar coefficient is real-valued.
The inverse Fourier transform of the full expression consequently has real coefficients
in the real basis
\[
\Pplus,\ \Pminus,\ \Pc,\ J,
\]
and hence belongs to $L^2(\R^d;A)$.
\end{proof}

Under Assumption \textup{(A1)}, the preceding domain-sensitive lemma
applies to $a$, $b$, and to the reduced real-branch symbols
\[
\lambda_+=a+b,
\qquad
\lambda_-=a-b.
\]
Thus the unbounded real-branch multipliers preserve real-valued data on their natural
domains, while their bounded semigroup multipliers preserve real-valued $L^2$ data whenever
they are defined by bounded symbols.

Assumptions (A1)--(A2) are intentionally modest, but they are not meant to
look artificial. Proposition~\ref{prop:admissibility-checklist} gives the
practical checklist: local polynomial dissipative examples, skew-adjoint
transport perturbations, simple nonlocal even multipliers, cancellation-coupled
pairs, and bounded lower-order perturbations are all admitted once the reduced
real-part bounds are checked. The point of Paper~5 is not to classify every
possible admissible multiplier pair, but to identify a concrete class large
enough to contain meaningful benchmark differential-operator and nonlocal
examples and small enough that the branchwise reconstruction program remains
exact.

\begin{proposition}[Basic setup facts for the strong prototype]\label{prop:strong-prototype-basic}
Assume \textup{(A1)}, and assume that $a,b:\R^d\to\C$ are measurable and finite almost everywhere. Then the componentwise strong prototype realization $\Gcal$ is a well-defined densely defined linear operator on $L^2(\R^d;A)$.
\end{proposition}

\begin{proof}
Linearity is immediate from the definition of $\Dom(\Gcal)$ and from the Fourier-side formula
\[
\widehat{\Gcal Y}(\xi)=\bigl(a(\xi)\IAC+b(\xi)\Lu\bigr)\widehat{Y}(\xi).
\]
It remains to check that this formula defines an operator on the real Hilbert space, rather
than only on the complexified Fourier-side coefficient space. Write
\[
Y=Y_+\Pplus+Y_-\Pminus+Y_c^R\Pc+Y_c^I J,
\qquad
Y_+,Y_-,Y_c^R,Y_c^I\in L^2(\R^d;\R).
\]
The domain condition gives
\[
a(\cdot)\widehat Y(\cdot),\ b(\cdot)\widehat Y(\cdot)\in L^2(\R^d;\AC).
\]
Therefore the coefficientwise real-structure lemma implies
\[
\F^{-1}\Bigl((a(\cdot)\IAC+b(\cdot)\Lu)\widehat Y(\cdot)\Bigr)
\in L^2(\R^d;A).
\]
Thus $\Gcal$ is well-defined as a real operator
\[
\Gcal:\Dom(\Gcal)\subset L^2(\R^d;A)\longrightarrow L^2(\R^d;A),
\]
not merely as an $\AC$-valued Fourier-side multiplier. It remains to prove density of
the domain. For each $n\in\mathbb N$, let
\[
E_n^0:=\{\xi\in\R^d: |a(\xi)|\le n\text{ and }|b(\xi)|\le n\}.
\]
Because \(a\) and \(b\) are assumed here to be measurable and finite almost
everywhere, the sets \(E_n^0\) increase to a full-measure subset of \(\R^d\).
By \textup{(A1)} and
Lemma~\ref{lem:hermitian-representatives-symmetric-cutoffs}, the harmless
Hermitian representatives of \(a\) and \(b\) make the raw truncation sets
symmetric up to null sets. Applying the symmetric-cutoff part of the same lemma,
replace the raw truncation set modulo null sets by the exactly symmetric cutoff
\[
E_n:=E_n^0\cap(-E_n^0).
\]
Then \(E_n=-E_n\) exactly, the Fourier multiplier \(\mathbf 1_{E_n}(D)\) is the
same \(L^2\)-operator as the cutoff associated with \(E_n^0\), and the sets
\(E_n\) still increase to a full-measure subset of \(\R^d\). Given any
\(Y\in L^2(\R^d;A)\), define \(Y^{(n)}\) by
\[
\widehat{Y^{(n)}}(\xi):=\mathbf 1_{E_n}(\xi)\widehat Y(\xi).
\]
Because $\mathbf 1_{E_n}(-\xi)=\mathbf 1_{E_n}(\xi)$ exactly, multiplication by $\mathbf 1_{E_n}$ preserves the Fourier-side Hermitian reality condition of each real coefficient of $Y$. Hence $Y^{(n)}$ is again an $A$-valued real state in $L^2(\R^d;A)$, not merely an $\AC$-valued Fourier-side truncation. Moreover $Y^{(n)}\to Y$ in $L^2(\R^d;A)$ by dominated convergence.
\[
\|a(\cdot)\widehat{Y^{(n)}}(\cdot)\|_{L^2(\R^d;\AC)}
\le n\|Y^{(n)}\|_{L^2(\R^d;A)},
\qquad
\|b(\cdot)\widehat{Y^{(n)}}(\cdot)\|_{L^2(\R^d;\AC)}
\le n\|Y^{(n)}\|_{L^2(\R^d;A)},
\]
so $Y^{(n)}\in\Dom(\Gcal)$ for every $n$. Hence $\Dom(\Gcal)$ is dense in $L^2(\R^d;A)$.
\end{proof}

Section~2 has now fixed all notation needed for the technical core of the paper. The algebraic branch projectors have been lifted to the function-space state space, the Fourier transform has been normalized so that the intrinsic branch decomposition can be expressed in branch coordinates on the Fourier side, and the admissible generator class has been pinned down at a level compatible with the later semigroup and bounded-bulk observation formalism. Section~3 turns to the first theorem-bearing step: the state-space branch decomposition of the prototype generator together with its branch-coordinate Fourier representation.


\section{State-Space Branch Decomposition and Branch-Coordinate Fourier Representation}

\subsection{Frequencywise form of the generator}

The transition from the algebraic theory to the distributed multiplier setting becomes transparent after Fourier transform. The branch geometry, however, is already present on $L^2(\R^d;A)$ through the constant algebraic projectors $\Pplus$, $\Pminus$, and $\Pc$ introduced in Section~2. The role of Fourier analysis in the present section is not to create that splitting, but to transport it frequencywise and to convert the translation-invariant prototype into a measurable family of finite-dimensional multiplier fibers. By construction, the prototype generator
\[
\Gcal = I_A\otimes a(D) + \Lu\otimes b(D)
\]
acts on the spatial variable only through scalar translation-invariant multipliers, while the branch algebra acts only on the finite-dimensional coefficient factor $A$. Consequently, for every $Y\in\Dom(\Gcal)$ and for almost every frequency $\xi\in\R^d$,
\[
\widehat{\Gcal Y}(\xi)
=
\bigl(a(\xi)\IAC+b(\xi)\Lu\bigr)\widehat{Y}(\xi).
\]
In other words, the full distributed problem is fibered over frequency space as a measurable family of finite-dimensional algebraic branch problems.

This observation is the basic reason the present multiplier class is tractable. At each fixed $\xi$, the spatially distributed generator collapses to the algebraic operator
\[
\widehat{\Gcal}(\xi):=a(\xi)\IAC+b(\xi)\Lu
\qquad\text{on }\AC.
\]
The intrinsic branch decomposition from Section~2 can therefore be transported pointwise in $\xi$, with no loss of $L^2$ structure, because the branch projectors commute both with the Fourier transform and with the multiplier part of the generator.

To make the branch reductions explicit, recall that left multiplication by $u$ acts by scalars on the two real one-dimensional branches and as the standard complex structure on the two-dimensional real branch. More precisely,
\[
\Lu\Pplus = \Pplus,
\qquad
\Lu\Pminus = -\Pminus,
\qquad
\Lu\Pc = J,
\qquad
\Lu J = -\Pc.
\]
Thus the action of $\widehat{\Gcal}(\xi)$ on the plus and minus branches is already scalar, whereas the complex branch becomes scalar only after passing to the real-complex identification introduced in Section~2.

The resulting picture is conceptually important. One is not performing an \emph{ad hoc} branch reduction of the multiplier generator after the fact, and one is not discovering a new splitting created by Fourier analysis itself. Rather, the semisimple branch algebra already forces a decomposition of the state space through the constant projectors $\Pplus$, $\Pminus$, and $\Pc$, while Fourier analysis merely transports that intrinsic decomposition frequencywise and makes the reduced symbols explicit.

\subsection{Reduced branch symbols}

Let $Y\in L^2(\R^d;A)$ and write its branch decomposition as
\[
Y = Y_+ + Y_- + Y_c,
\qquad
Y_+ := \Pplus Y,
\quad
Y_- := \Pminus Y,
\quad
Y_c := \Pc Y.
\]
Since the projectors commute with the Fourier transform, the corresponding decomposition of the Fourier transform is
\[
\widehat{Y}(\xi) = \widehat{Y_+}(\xi)+\widehat{Y_-}(\xi)+\widehat{Y_c}(\xi)
\qquad\text{for a.e. }\xi\in\R^d.
\]
The frequencywise action of the generator can then be read separately on each branch.

On the plus branch one has $\Lu=+1$, hence
\[
\widehat{\Gcal Y_+}(\xi)
=
\bigl(a(\xi)+b(\xi)\bigr)\widehat{Y_+}(\xi).
\]
On the minus branch one has $\Lu=-1$, hence
\[
\widehat{\Gcal Y_-}(\xi)
=
\bigl(a(\xi)-b(\xi)\bigr)\widehat{Y_-}(\xi).
\]
On the complex branch, write
\[
Y_c = f\Pc + gJ,
\qquad f,g\in L^2(\R^d;\R).
\]
Using $\Lu\Pc=J$ and $\Lu J=-\Pc$, one obtains
\[
\Gcal Y_c
=
\bigl(a(D)f-b(D)g\bigr)\Pc
+
\bigl(b(D)f+a(D)g\bigr)J.
\]
After applying the identification
\[
\mathcal I_c\bigl(f\Pc+gJ\bigr)=f+ig,
\]
this becomes the complex scalar multiplier model
\[
\mathcal I_c(\Gcal Y_c)
=
\lambda_c(D)\mathcal I_c(Y_c),
\]
where this identity is first read on the componentwise strong domain and the right-hand
side uses the symbol-sum multiplier convention for \(\lambda_c(D)\). Therefore,
after applying the global real-linear coordinate map $\mathcal I_c$,
\[
\widehat{\mathcal I_c(\Gcal Y_c)}(\xi)
=
\bigl(a(\xi)+i\,b(\xi)\bigr)\widehat{\mathcal I_c(Y_c)}(\xi).
\]
Thus the complex branch has the global real-coordinate scalar representative fixed by Proposition~\ref{prop:realified-complex-branch-normal-form}. The three reduced branch symbols are therefore
\[
\lambda_+(\xi):=a(\xi)+b(\xi),
\qquad
\lambda_-(\xi):=a(\xi)-b(\xi),
\qquad
\lambda_c(\xi):=a(\xi)+i\,b(\xi).
\]
They are not arbitrary auxiliary quantities; they are the explicit Fourier-side representatives of the intrinsic branch action of the full multiplier generator.

\begin{remark}[Reduced symbols are not a fibre diagonalization]
\label{rem:reduced-symbols-not-fibre-diagonalization}
The tuple
\[
(\lambda_+,\lambda_-,\lambda_c)
\]
is a reduced branch-coordinate representation, not a diagonalization of the
complexified coefficient fibre \(\AC\). The plus and minus components are
one-dimensional real idempotent branches. The \(c\)-component is a real
two-dimensional branch with basis \((\Pc,J)\), and its scalar notation
\[
\lambda_c=a+i b
\]
is used only after the global real-coordinate identification
\[
f\Pc+gJ\longleftrightarrow f+i g.
\]
Equivalently, in a Fourier-side matrix statement the \(c\)-branch is represented
by the realified block
\[
S_c(\xi)=
\begin{pmatrix}
a(\xi)&-b(\xi)\\
b(\xi)&a(\xi)
\end{pmatrix},
\]
not by a pointwise complex-linear scalarization of \(\C\Pc\oplus\C J\). The
word ``reduced'' is therefore used deliberately: it records the branchwise
coordinate model needed for semigroup reconstruction, while full matrix-symbol
statements keep the \(c\)-branch as a two-dimensional real block.
\end{remark}

It is useful to record the associated reduced multiplier operators already at this stage, even though their semigroup generation will be treated only in Section~4. On the model spaces corresponding to the three branches we set
\[
G_+ := \lambda_+(D),
\qquad
G_- := \lambda_-(D),
\qquad
G_c := \lambda_c(D).
\]
Here \(\lambda_\bullet(D)\) denotes the maximal Fourier multiplier with symbol
\(\lambda_\bullet\) on the corresponding reduced branch model space. Thus
\(G_+\), \(G_-\), and \(G_c\) are symbol-sum multiplier realizations. They are
not, in general, the operator sums \(a(D)+b(D)\), \(a(D)-b(D)\), or
\(a(D)+i b(D)\) on \(\Dom(a(D))\cap\Dom(b(D))\). On the componentwise strong
domain of \(\Gcal\), the separate terms \(a(D)\) and \(b(D)\) exist and the
symbolic strong-domain expressions agree with the corresponding reduced
multiplier actions; outside that strong domain, only the symbol-sum
multiplier interpretation is intended.

\begin{remark}[Symbol-sum convention for reduced operators]\label{rem:symbol-sum-convention}
Throughout the reduced branch theory, \(\lambda_\bullet(D)\) denotes the maximal
Fourier multiplier with symbol \(\lambda_\bullet\). Consequently the reduced
operators \(G_+\), \(G_-\), and \(G_c\) are not defined as operator sums on
\(\Dom(a(D))\cap\Dom(b(D))\). This convention is essential because cancellations
inside \(\lambda_+=a+b\), \(\lambda_-=a-b\), or \(\lambda_c=a+i b\) may strictly
enlarge the reduced domains relative to the componentwise strong domain.
\end{remark}

The content of the present section is that these reduced symbol-sum operators are not merely formal. On the componentwise strong domain they are exactly the operators induced by $\Gcal$ on the branch subspaces after the natural branch identifications.

\begin{definition}[Branch identification maps]\label{def:branch-identifications}
The natural branch identification maps are the real-linear isometric isomorphisms
\[
\mathcal I_+ : L^2_+(\R^d;A)\to \LtwoR,
\qquad
\mathcal I_- : L^2_-(\R^d;A)\to \LtwoR,
\qquad
\mathcal I_c : L^2_c(\R^d;A)\to L^2(\R^d;\C),
\]
obtained from the branch parametrizations $f\Pplus$, $g\Pminus$, and $f\Pc+gJ\leftrightarrow f+ig$ described in Section~2.
\end{definition}


\begin{remark}[Global real-coordinate normal form versus complex Fourier fibres]
\label{rem:global-real-c-branch-not-fibre-diagonalization}
This is the detailed version of item~\textup{(ii)} in
Remark~\ref{rem:standing-interpretation-box-paper5}. The map
\[
\mathcal I_c(f\Pc+gJ)=f+ig
\]
is a real-linear Hilbert-space isomorphism
\[
L^2(\R^d;\operatorname{span}_{\R}\{\Pc,J\})
\longrightarrow
L^2(\R^d;\C),
\]
where the coefficients \(f\) and \(g\) are real-valued physical-space
functions. It is not used as a pointwise complex-linear isomorphism
\[
\C\Pc\oplus \C J\longrightarrow \C
\]
on each complexified Fourier fibre. After Fourier transform, real physical
coefficients are represented by complex Fourier coefficients satisfying the
Hermitian symmetries
\[
\widehat f(-\xi)=\overline{\widehat f(\xi)},
\qquad
\widehat g(-\xi)=\overline{\widehat g(\xi)}.
\]
Thus the scalar notation \(h=f+ig\) for the \(c\)-branch is a global real
coordinate representation, equivalently a paired-frequency Fourier
representation. It is not a pointwise diagonalization of the complexified fibre
\(\C\Pc\oplus\C J\). In particular, expressions of the form
\(\mathcal I_cS(\xi)\mathcal I_c^{-1}\) are not used below as complex-fibre
similarities.
\end{remark}



\begin{proposition}[Global real-coordinate normal form for the complex branch]
\label{prop:realified-complex-branch-normal-form}
\label{lem:complex-branch-transport-convention}
Let
\[
A_c=\operatorname{span}_{\R}\{\Pc,J\},
\qquad
J=\Lu\Pc,
\qquad
J^2=-\Pc,
\]
and let
\[
\mathcal I_c:L^2_c(\R^d;A)\longrightarrow L^2(\R^d;\C),
\qquad
\mathcal I_c(f\Pc+gJ)=f+ig,
\]
be the real-linear branch-coordinate map. Then \(\mathcal I_c\) is an
isometric isomorphism of real Hilbert spaces and
\[
\mathcal I_c\Lu\mathcal I_c^{-1}=M_i,
\]
where \(M_i\) denotes multiplication by \(i\), read as a real-linear operator
on \(L^2(\R^d;\C)\).

Let
\[
\lambda_c(\xi)=a(\xi)+ib(\xi),
\qquad
G_c:=\lambda_c(D),
\qquad
\Dom(G_c)=\{h\in L^2(\R^d;\C):\lambda_c\widehat h\in L^2(\R^d;\C)\}.
\]
Define the closed real \(c\)-branch operator by
\[
\mathcal A_c
:=\mathcal I_c^{-1}G_c\mathcal I_c,
\qquad
\Dom(\mathcal A_c)=\mathcal I_c^{-1}\Dom(G_c).
\]
Then the following diagram commutes on \(\Dom(\mathcal A_c)\):
\[
\begin{array}{ccc}
L^2_c(\R^d;A)
& \xrightarrow{\ \mathcal I_c\ } &
L^2(\R^d;\C)
\\[0.45em]
\mathcal A_c \downarrow
&&
\downarrow G_c=\lambda_c(D)
\\[0.45em]
L^2_c(\R^d;A)
& \xrightarrow{\ \mathcal I_c\ } &
L^2(\R^d;\C).
\end{array}
\]
In particular, the transported complex branch has the maximal symbol-sum domain
\(\Dom(G_c)\). It does not impose the stronger componentwise requirement that
\(a\widehat h\) and \(b\widehat h\) belong separately to \(L^2\), nor does it
introduce an additional complex-linear eigenspace of the physical real fibre.

This is a global real Hilbert-space normal form. It is not a pointwise
complex-fibre diagonalization of \(\C\Pc\oplus\C J\). The notation
\(\lambda_c=a+i b\) records the scalar multiplier seen after the global real
coordinate identification \(f\Pc+gJ\leftrightarrow f+ig\), not a complex-linear
similarity on each Fourier fibre. All later occurrences of
\(G_c=\lambda_c(D)\), \(\lambda_c=a+i b\), and
\(\mathcal A_c=\mathcal I_c^{-1}G_c\mathcal I_c\) refer to this operator-level
normal form.
\end{proposition}

\begin{proof}
For \(f,g\in L^2(\R^d;\R)\), the algebraic identities
\[
\Lu\Pc=J,
\qquad
\Lu J=-\Pc
\]
give
\[
\Lu(f\Pc+gJ)=-g\Pc+fJ.
\]
Applying \(\mathcal I_c\) yields
\[
\mathcal I_c\Lu(f\Pc+gJ)
=
-g+i f
=
i(f+ig)
=
M_i\mathcal I_c(f\Pc+gJ).
\]
Hence \(\mathcal I_c\Lu\mathcal I_c^{-1}=M_i\) as a real-linear identity on
the global \(c\)-branch Hilbert space.

The multiplier \(G_c=\lambda_c(D)\) is then the maximal closed scalar
symbol-sum multiplier on \(L^2(\R^d;\C)\). Transporting this closed operator
through the real-linear isometry \(\mathcal I_c\) gives the closed operator
\[
\mathcal A_c=\mathcal I_c^{-1}G_c\mathcal I_c
\]
on the \(c\)-branch, with the displayed domain. The commutative diagram follows
immediately from this definition, and the domain statement is exactly the
maximal scalar symbol-sum definition of \(G_c\).

No pointwise inverse of \(\mathcal I_c\) on the complexified Fourier fibre
\(\C\Pc\oplus\C J\) is invoked. The scalar representation
\(G_c=\lambda_c(D)\) is a global real-coordinate representation of the
\(c\)-branch operator.
\end{proof}


\begin{remark}[Scope of the complex-branch normal form]
\label{rem:scope-complex-branch-normal-form}
Proposition~\ref{prop:realified-complex-branch-normal-form} is an operator-level
normal form on real Hilbert spaces. It should not be read as saying that the
complexified Fourier fibre \(\C\Pc\oplus\C J\) is reduced by
\(\mathcal I_c\) to a one-dimensional complex fibre. The scalar symbol
\(\lambda_c=a+i b\) is the symbol seen after the global real-coordinate
identification \(f\Pc+gJ\mapsto f+ig\). Fibrewise matrix-symbol statements on
\(\AC\) must still be written either with the full \(2\times 2\) realified
\(c\)-block or with the full \(\AC\)-matrix
\(a(\xi)\IAC+b(\xi)\Lu\).
\end{remark}

\begin{remark}[Physical coefficients versus Fourier-side real coordinates]
\label{rem:physical-vs-fourier-real-coordinates}
There are two different real two-component descriptions of the complex branch.
First, a physical \(A_c\)-state is written
\[
Y_c=f\Pc+gJ,
\qquad f,g\in L^2(\R^d;\R),
\]
and its Fourier coefficient pair is \((\widehat f,\widehat g)\). On this pair
the coefficient-level symbol action of \(a\IA+b\Lu\) is
\[
(\widehat f,\widehat g)
\longmapsto
(a\widehat f-b\widehat g,\;b\widehat f+a\widehat g).
\]
Second, after transport by \(\mathcal I_c\), one writes
\[
h_c=f+ig,
\qquad
\widehat h_c=X+iY.
\]
The realification matrix
\[
R_{\lambda_c}(\xi)=
\begin{pmatrix}
\operatorname{Re}\lambda_c(\xi)&-\operatorname{Im}\lambda_c(\xi)\\
\operatorname{Im}\lambda_c(\xi)&\operatorname{Re}\lambda_c(\xi)
\end{pmatrix}
\]
acts on the pointwise Fourier-side pair
\((X,Y)=(\operatorname{Re}\widehat h_c,\operatorname{Im}\widehat h_c)\), not on
the physical Fourier coefficient pair \((\widehat f,\widehat g)\). These two
pairs coincide only in special real-valued symbol situations. The standing
complex-valued Hermitian-symbol theory uses the transported scalar coordinate
\(h_c\) and its closed multiplier domain
\[
\Dom(G_c)=
\{h\in L^2(\R^d;\C):\lambda_c\widehat h\in L^2(\R^d;\C)\}.
\]
No stronger entrywise matrix-domain condition is imposed.
\end{remark}

\begin{definition}[Realification of complex scalar multiplication]
\label{def:realification-complex-scalar}
For a measurable complex-valued symbol
\[
\mu(\xi)=\mu_R(\xi)+i\mu_I(\xi),
\]
we write
\[
R_\mu(\xi):=
\begin{pmatrix}
\mu_R(\xi)&-\mu_I(\xi)\\
\mu_I(\xi)&\mu_R(\xi)
\end{pmatrix}.
\]
Thus scalar multiplication by \(\mu(\xi)\) on a complex scalar
\(z=x+iy\) is, after forgetting the complex structure, multiplication by the
real matrix \(R_\mu(\xi)\) on the pointwise real-imaginary coordinate vector
\((x,y)=(\Re z,\Im z)\). In particular, whenever the paper writes the
complex-branch reduced symbol
\[
\lambda_c(\xi)=a(\xi)+i b(\xi),
\]
the notation \(R_{\lambda_c}(\xi)\) refers only to the realification of scalar
multiplication by \(\lambda_c(\xi)\) on the complex value of the reduced
coordinate \(\widehat h_c(\xi)\). It is not, in general, the coefficient matrix
acting on the physical Fourier coefficient pair
\((\widehat f(\xi),\widehat g(\xi))\) appearing in
\[
Y_c=f\Pc+gJ,\qquad h_c=f+ig.
\]
On those physical branch coefficients the symbol-sum action is instead
\[
(\widehat f,\widehat g)
\longmapsto
(a\widehat f-b\widehat g,\; b\widehat f+a\widehat g).
\]
Only under additional real-valued symbol restrictions on \(a\) and \(b\) do
these two matrix descriptions collapse to the same pointwise real \(2\times2\)
matrix. The standing theory does not impose such real-valued restrictions; it
keeps the general complex-valued Hermitian symbol setting and uses the reduced
complex scalar coordinate for the \(c\)-branch. For the global taxonomy of the
three coordinate systems involved, see
Remark~\ref{rem:three-c-branch-coordinate-systems}.

When \(\mu\) is unbounded, the notation \(R_\mu(D)\), if used at the
operator level, is always understood in \emph{Fourier-side real coordinates},
not through physical-space real and imaginary parts. More precisely,
\[
\Dom(\mu(D))
:=
\{h\in L^2(\R^d;\C):\mu(\xi)\widehat h(\xi)\in L^2(\R^d;\C)\}.
\]
For \(h\in L^2(\R^d;\C)\), write
\[
\widehat h(\xi)=X(\xi)+iY(\xi),
\qquad
\mathscr R_\xi h:=(X,Y)=(\Re\widehat h,\Im\widehat h).
\]
The Fourier-side real-coordinate realization of the scalar multiplier is the
closed operator whose domain is
\[
\Dom_\xi(R_\mu(D))
:=
\Bigl\{(X,Y)\in L^2(\R^d;\R^2):
(\mu_RX-\mu_IY)+i(\mu_IX+\mu_RY)\in L^2(\R^d;\C)\Bigr\},
\]
and whose action is
\[
R_\mu(D)(X,Y)
=
(\mu_RX-\mu_IY,\,\mu_IX+\mu_RY)
\]
in Fourier multiplier coordinates. Equivalently,
\[
\mathscr R_\xi(\mu(D)h)=R_\mu(D)\mathscr R_\xi h,
\qquad h\in\Dom(\mu(D)).
\]
This notation is deliberately not the physical realification
\(h\mapsto(\Re h,\Im h)\). Indeed, in general
\(\widehat{\Re h}\ne\Re\widehat h\) and
\(\widehat{\Im h}\ne\Im\widehat h\). Thus the domain condition above is the
transported scalar multiplier condition in Fourier-side real coordinates, not
an independently imposed entrywise matrix-domain condition requiring the four
products
\[
\mu_RX,
\qquad
\mu_IX,
\qquad
\mu_RY,
\qquad
\mu_IY
\]
to belong separately to \(L^2\).
\end{definition}

\begin{remark}[No entrywise domain for the realified complex block]
\label{rem:no-entrywise-domain-realified-block}
The complex-branch multiplier domain is the scalar symbol-sum domain
\[
\Dom(G_c)
=
\{h\in L^2(\R^d;\C):\lambda_c\widehat h\in L^2(\R^d;\C)\}.
\]
When the same scalar multiplier is displayed in real coordinates, the notation
\(R_{\lambda_c}(D)\) denotes its Fourier-side real-coordinate representation:
\[
\mathscr R_\xi(G_ch)=R_{\lambda_c}(D)\mathscr R_\xi h,
\qquad h\in\Dom(G_c),
\]
where \(\widehat h=X+iY\) and
\(\mathscr R_\xi h=(X,Y)=(\Re\widehat h,\Im\widehat h)\). Its domain is
\[
\Dom_\xi(R_{\lambda_c}(D))
=
\Bigl\{(X,Y)\in L^2(\R^d;\R^2):
(\Re\lambda_c\,X-\Im\lambda_c\,Y)
+i(\Im\lambda_c\,X+\Re\lambda_c\,Y)
\in L^2(\R^d;\C)\Bigr\}.
\]
This Fourier-side representation should not be confused with the physical
realification \(h\mapsto(\Re h,\Im h)\). The latter has Fourier coordinates
\((\widehat{\Re h},\widehat{\Im h})\), whereas the displayed matrix acts on
\((\Re\widehat h,\Im\widehat h)\). Thus the condition is not the stronger
componentwise condition that all four products
\[
\Re\lambda_c\,X,
\qquad
\Im\lambda_c\,X,
\qquad
\Re\lambda_c\,Y,
\qquad
\Im\lambda_c\,Y
\]
belong separately to \(L^2\). This is the same symbol-sum convention used for
\(G_+\), \(G_-\), and \(G_c\), and it is essential when cancellations enlarge
the natural reduced domain.
\end{remark}


\begin{remark}[Three coordinate systems on the complex branch]
\label{rem:three-c-branch-coordinate-systems}
The \(c\)-branch is a real two-dimensional branch of the original algebra, not
a literal one-dimensional complex branch of the physical state space. The paper
uses three coordinate systems on this branch, and they must remain separate.

First, the physical \(A_c\)-branch coordinates are
\[
Y_c=f\Pc+gJ,
\qquad f,g\in L^2(\R^d;\R).
\]
On Fourier side these coordinates are the coefficient pair
\((\widehat f,\widehat g)\), and the coefficient-level symbol-sum action is
\[
(\widehat f,\widehat g)
\longmapsto
(a\widehat f-b\widehat g,\; b\widehat f+a\widehat g).
\]

Second, the reduced complex scalar coordinate is obtained by the real-linear
identification
\[
\mathcal I_c(Y_c)=h_c=f+ig.
\]
In this coordinate the same action is the scalar complex multiplier
\[
\widehat{G_ch_c}
=
(a+ib)\widehat h_c
=
\lambda_c\widehat h_c.
\]

Third, if the scalar complex Fourier value is decomposed pointwise as
\[
\widehat h_c(\xi)=X(\xi)+iY(\xi),
\]
then scalar multiplication by \(\lambda_c(\xi)\) is represented on the
pointwise real-imaginary pair
\((X,Y)=(\Re\widehat h_c,\Im\widehat h_c)\) by
\[
R_{\lambda_c}(\xi)
=
\begin{pmatrix}
\Re\lambda_c(\xi)&-\Im\lambda_c(\xi)\\
\Im\lambda_c(\xi)&\Re\lambda_c(\xi)
\end{pmatrix}.
\]

The first pair \((\widehat f,\widehat g)\) and the third pair
\((X,Y)=(\Re\widehat h_c,\Im\widehat h_c)\) are not the same coordinate system
in the general complex-valued Hermitian-symbol setting. Thus
\(R_{\lambda_c}\) acts on pointwise real-imaginary coordinates of the reduced
scalar complex value \(\widehat h_c\), not on the physical Fourier coefficient
pair \((\widehat f,\widehat g)\). The coefficient-level matrix
\[
\begin{pmatrix}a&-b\\ b&a\end{pmatrix}
\]
and the pointwise realification matrix \(R_{\lambda_c}\) coincide only under
additional real-valued symbol restrictions on \(a\) and \(b\). Those
restrictions are not standing assumptions of this paper.
\end{remark}

\begin{proposition}[Reduced complex-coordinate multiplier and its realification]
\label{prop:c-branch-real-block-not-complex-diagonalization}
Let \(G_c\) denote the reduced complex-branch Fourier multiplier with symbol
\(\lambda_c=a+i b\). Under the global real-linear identification
\[
\mathcal I_c:L^2_c(\R^d;A)\longrightarrow L^2(\R^d;\C),
\]
\(G_c\) is represented by scalar multiplication by \(\lambda_c\) on the reduced complex
coordinate \(h_c=\mathcal I_cY_c\). This is an operator-level coordinate representation, not a pointwise complex-fibre similarity.

More explicitly, if
\[
Y_c=f\Pc+gJ,\qquad h_c=f+ig,
\]
then the physical branch coefficient action on Fourier side is
\[
(\widehat f,\widehat g)
\longmapsto
(a\widehat f-b\widehat g,\; b\widehat f+a\widehat g),
\]
and this is equivalent to
\[
(a+i b)\widehat h_c
=
(a\widehat f-b\widehat g)+i(b\widehat f+a\widehat g).
\]
If one further writes the scalar complex multiplication
\(z\mapsto\lambda_c(\xi)z\) in pointwise real-imaginary coordinates
\(z=X+iY\), then it is represented by
\[
R_{\lambda_c}(\xi)=
\begin{pmatrix}
\Re\lambda_c(\xi)&-\Im\lambda_c(\xi)\\
\Im\lambda_c(\xi)&\Re\lambda_c(\xi)
\end{pmatrix}.
\]
This last matrix acts on \((X,Y)=(\Re z,\Im z)\), not on the physical Fourier
coefficient pair \((\widehat f,\widehat g)\) unless \(a\) and \(b\) are
real-valued in the relevant pointwise sense. At the unbounded-operator level,
this realification is taken in Fourier-side real coordinates:
\[
\mathscr R_\xi G_ch
=
R_{\lambda_c}(D)\mathscr R_\xi h,
\qquad h\in\Dom(G_c),
\]
where
\[
\Dom(G_c)
=
\{h\in L^2(\R^d;\C):\lambda_c\widehat h\in L^2(\R^d;\C)\}
\]
and
\[
\Dom_\xi(R_{\lambda_c}(D))
=
\Bigl\{(X,Y)\in L^2(\R^d;\R^2):
(\Re\lambda_c\,X-\Im\lambda_c\,Y)
+i(\Im\lambda_c\,X+\Re\lambda_c\,Y)
\in L^2(\R^d;\C)\Bigr\}.
\]
It is therefore not the entrywise domain obtained by requiring separately that
\(\Re\lambda_c\,X\), \(\Im\lambda_c\,X\), \(\Re\lambda_c\,Y\), and
\(\Im\lambda_c\,Y\) all lie in \(L^2\). This is precisely the separation
of coordinate systems recorded in
Remark~\ref{rem:three-c-branch-coordinate-systems}. Consequently, \(\lambda_c\) is reduced-coordinate notation for a real
two-dimensional branch operator, not a one-dimensional complex eigenblock of
the original real \(A\)-valued fiber, and no conjugate direction of a
complexified fiber is discarded.

\end{proposition}

\begin{proof}
The identity on physical branch coefficients follows directly from the
relations \(\Lu\Pc=J\) and \(\Lu J=-\Pc\):
\[
(a\IAC+b\Lu)(\widehat f\Pc+\widehat gJ)
=
(a\widehat f-b\widehat g)\Pc+(b\widehat f+a\widehat g)J.
\]
Under the complex branch identification
\(\mathcal I_c(f\Pc+gJ)=f+ig\), this becomes
\[
(a\widehat f-b\widehat g)+i(b\widehat f+a\widehat g)
=
(a+i b)(\widehat f+i\widehat g)
=
\lambda_c\widehat h_c.
\]
Finally, the displayed matrix \(R_{\lambda_c}\) is the standard pointwise
realification of scalar multiplication by \(\lambda_c\) on a complex scalar
\(z=X+iY\). At the multiplier level, however, the real-coordinate operator is
not obtained by passing to physical-space real and imaginary parts. It is the
Fourier-side representation of the closed scalar multiplier
\(G_c=\lambda_c(D)\): for \(\widehat h=X+iY\), its domain is precisely the set
of pairs for which
\[
(\Re\lambda_c\,X-\Im\lambda_c\,Y)
+i(\Im\lambda_c\,X+\Re\lambda_c\,Y)
\in L^2(\R^d;\C).
\]
No entrywise matrix-domain condition is imposed. Thus
\(R_{\lambda_c}(D)\), whenever this notation is used, is a Fourier-side
real-coordinate representation of the reduced scalar complex multiplier, not
an independent physical coefficient matrix on \((\widehat f,\widehat g)\) and
not an entrywise matrix multiplier built from the four separate coefficients
\(\Re\lambda_c\) and \(\Im\lambda_c\). The proof uses the physical pair only to
verify the algebraic symbol-sum identity and then uses the reduced complex
coordinate for the closed multiplier; it never identifies
\((\widehat f,\widehat g)\) with the pointwise real-imaginary pair
\((\Re\widehat h_c,\Im\widehat h_c)\).
\end{proof}




\begin{lemma}[Finite-dimensional form of the complex-branch transport]
\label{lem:complex-coordinate-real-ac-block}
On the finite-dimensional branch
\(A_c=\operatorname{span}_{\R}\{\Pc,J\}\), left multiplication by \(u\) has the
matrix
\[
[L_u]_{(\Pc,J)}=
\begin{pmatrix}
0&-1\\
1&0
\end{pmatrix},
\]
and the coordinate map \(\iota_c(x\Pc+yJ)=x+iy\) satisfies
\[
\iota_cL_u\iota_c^{-1}=M_i.
\]
The global real-coordinate convention in Proposition~\ref{prop:realified-complex-branch-normal-form}
is exactly the \(L^2\)-level, real-linear transport of this finite-dimensional identity; it is not a Fourier-fibre complex-linear similarity.
\end{lemma}


\begin{lemma}[Hermitian-pair collapse]\label{lem:hermitian-pair-collapse}
Let \(p,q:\R^d\to\C\) be measurable Fourier-side functions satisfying the
Hermitian symmetry conditions
\[
p(-\xi)=\overline{p(\xi)},
\qquad
q(-\xi)=\overline{q(\xi)}
\]
for almost every \(\xi\). Set
\[
r(\xi):=p(\xi)+i q(\xi).
\]
Then
\[
p,q\in L^2(\R^d;\C)
\quad\Longleftrightarrow\quad
r\in L^2(\R^d;\C).
\]
Moreover, whenever these conditions hold,
\[
\norm{r}_{L^2(\R^d;\C)}^2
=
\norm{p}_{L^2(\R^d;\C)}^2+
\norm{q}_{L^2(\R^d;\C)}^2.
\]
This is an integrated \(L^2\) identity. It is not a pointwise frequency identity:
for general complex values of \(p(\xi)\) and \(q(\xi)\), the pointwise expression
\(\abs{p(\xi)+i q(\xi)}^2\) contains a cross term. The cross term disappears
only after using the Hermitian symmetry and Plancherel. The Hermitian symmetry is
essential: without it, the implication \(r\in L^2\Rightarrow p,q\in L^2\) is
false in general.
Equivalently, if
\[
P:=\F^{-1}p,
\qquad
Q:=\F^{-1}q,
\]
then \(P,Q\in L^2(\R^d;\R)\) and
\[
\F^{-1}r=P+iQ.
\]
\end{lemma}

\begin{proof}
If \(p,q\in L^2(\R^d;\C)\), then \(r=p+iq\in L^2(\R^d;\C)\)
immediately. Conversely, assume \(r\in L^2(\R^d;\C)\). Since
\[
\overline{r(-\xi)}
=
\overline{p(-\xi)+iq(-\xi)}
=
p(\xi)-iq(\xi),
\]
we have
\[
p(\xi)=\frac12\bigl(r(\xi)+\overline{r(-\xi)}\bigr),
\qquad
q(\xi)=\frac1{2i}\bigl(r(\xi)-\overline{r(-\xi)}\bigr).
\]
Because \(r\in L^2(\R^d;\C)\) if and only if
\(\overline{r(-\cdot)}\in L^2(\R^d;\C)\), it follows that
\(p,q\in L^2(\R^d;\C)\).

For the norm identity, the pointwise cross term in
\(\abs{p+i q}^2\) is not asserted to vanish. Instead we pass to physical
space. Let
\[
P:=\F^{-1}p,
\qquad
Q:=\F^{-1}q.
\]
The Hermitian symmetries imply that \(P,Q\in L^2(\R^d;\R)\). Hence, by
Plancherel and by the real-valuedness of \(P,Q\),
\[
\norm{r}_{L^2(\R^d;\C)}^2
=
\norm{\F^{-1}r}_{L^2(\R^d;\C)}^2
=
\norm{P+iQ}_{L^2(\R^d;\C)}^2
=
\norm{P}_{L^2(\R^d;\R)}^2+
\norm{Q}_{L^2(\R^d;\R)}^2.
\]
Applying Plancherel again gives
\[
\norm{P}_{L^2(\R^d;\R)}^2=\norm{p}_{L^2(\R^d;\C)}^2,
\qquad
\norm{Q}_{L^2(\R^d;\R)}^2=\norm{q}_{L^2(\R^d;\C)}^2.
\]
Therefore
\[
\norm{r}_{L^2(\R^d;\C)}^2
=
\norm{p}_{L^2(\R^d;\C)}^2+
\norm{q}_{L^2(\R^d;\C)}^2.
\]
\end{proof}

\begin{lemma}[Complex-branch multiplier identity]
\label{lem:realified-complex-branch-multiplier}
Assume \textup{(A1)}. Let $Y_c\in L^2_c(\R^d;A)$ and write
\[
Y_c=f\Pc+gJ,
\qquad
f,g\in L^2(\R^d;\R),
\qquad
h_c:=\mathcal I_cY_c=f+ig.
\]
Define
\[
p(\xi):=a(\xi)\widehat f(\xi)-b(\xi)\widehat g(\xi),
\qquad
q(\xi):=b(\xi)\widehat f(\xi)+a(\xi)\widehat g(\xi).
\]
Then
\[
\bigl(a(\xi)\IAC+b(\xi)\Lu\bigr)\widehat{Y_c}(\xi)
=
p(\xi)\Pc+q(\xi)J
\]
for almost every $\xi$, and
\[
(a(\xi)+i b(\xi))\widehat h_c(\xi)=p(\xi)+iq(\xi).
\]
Consequently,
\[
\bigl(a(\cdot)\IAC+b(\cdot)\Lu\bigr)\widehat{Y_c}(\cdot)
\in L^2(\R^d;\AC)
\iff
(a+i b)\widehat h_c\in L^2(\R^d;\C).
\]
Moreover, whenever these equivalent conditions hold,
\[
\mathcal I_c\left[
\F^{-1}\Bigl(
\bigl(a(\cdot)\IAC+b(\cdot)\Lu\bigr)\widehat{Y_c}(\cdot)
\Bigr)
\right]
=
\F^{-1}\bigl((a+i b)\widehat h_c\bigr)
=
G_c\mathcal I_cY_c.
\]
\end{lemma}

\begin{proof}
Since $f$ and $g$ are real-valued,
\[
\widehat f(-\xi)=\overline{\widehat f(\xi)},
\qquad
\widehat g(-\xi)=\overline{\widehat g(\xi)}.
\]
By Assumption \textup{(A1)},
\[
a(-\xi)=\overline{a(\xi)},
\qquad
b(-\xi)=\overline{b(\xi)}.
\]
Therefore the Fourier-side complex-valued functions $p$ and $q$ satisfy the
Hermitian symmetries required in Lemma~\ref{lem:hermitian-pair-collapse}:
\[
p(-\xi)=\overline{p(\xi)},
\qquad
q(-\xi)=\overline{q(\xi)}.
\]

Using
\[
\Lu\Pc=J,
\qquad
\Lu J=-\Pc,
\]
we obtain
\[
\bigl(a\IAC+b\Lu\bigr)(\widehat f\Pc+\widehat gJ)
=
(a\widehat f-b\widehat g)\Pc+(b\widehat f+a\widehat g)J
=
p\Pc+qJ.
\]
On the other hand,
\[
(a+ib)\widehat h_c
=
(a+ib)(\widehat f+i\widehat g)
=
(a\widehat f-b\widehat g)+i(b\widehat f+a\widehat g)
=
p+iq.
\]

Since the norm on the finite-dimensional coefficient space $\AC$ is equivalent to the
coordinate norm on the branch block $\mathrm{span}_{\C}\{\Pc,J\}$, we have
\[
p\Pc+qJ\in L^2(\R^d;\AC)
\quad\Longleftrightarrow\quad
p,q\in L^2(\R^d;\C).
\]
By Lemma~\ref{lem:hermitian-pair-collapse},
\[
p,q\in L^2(\R^d;\C)
\quad\Longleftrightarrow\quad
p+iq\in L^2(\R^d;\C).
\]
Since $p+iq=(a+i b)\widehat h_c$, this proves the claimed domain equivalence.

Finally, when these conditions hold, set
\[
P:=\F^{-1}p,
\qquad
Q:=\F^{-1}q.
\]
The Hermitian symmetries imply that $P,Q\in L^2(\R^d;\R)$. Hence
\[
\F^{-1}(p\Pc+qJ)=P\Pc+QJ
\]
is an element of the real complex branch, and
\[
\mathcal I_c(P\Pc+QJ)
=
P+iQ
=
\F^{-1}(p+iq)
=
\F^{-1}\bigl((a+i b)\widehat h_c\bigr).
\]
This is precisely the stated operator identity.
\end{proof}

\begin{remark}[Complex coordinate versus Fourier-side complexification]
\label{rem:complex-coordinate-versus-complexification}
The scalar symbol \(a+i b\) is used only in the transported complex coordinate
fixed by Proposition~\ref{prop:realified-complex-branch-normal-form}. The
separate physical-versus-Fourier real-coordinate distinction is recorded in
Remark~\ref{rem:physical-vs-fourier-real-coordinates}.
\end{remark}

\subsection{Main theorem: intrinsic branch decomposition on \texorpdfstring{$L^2(\R^d;A)$}{L2(Rd;A)}}

We now state the initial theorem-bearing result of the paper. It is the distributed counterpart of the algebraic branch decomposition from the earlier papers and the concrete distributed precursor of the reconstruction theorem proved later. Its core content is an intrinsic decomposition of the state space; the Fourier transform enters only to display the corresponding branch-coordinate reduced representatives. For the plus and minus branches these representatives are genuine one-dimensional fiber reductions, while the complex-branch representative is obtained only through the global real-coordinate map $\mathcal I_c$.

For the complex branch, the scalar notation $G_c=\lambda_c(D)$ is governed only by the operator-level normal form of
Proposition~\ref{prop:realified-complex-branch-normal-form}. The coordinate
distinction between physical branch coefficients and Fourier-side realification
is recorded in Remark~\ref{rem:physical-vs-fourier-real-coordinates}.

\begin{theorem}[Branch decomposition and Fourier coordinates]\label{thm:distributed-branch-splitting}
Assume \textup{(A1)}. Let
\[
\Gform = I_A\otimes a(D)+\Lu\otimes b(D)
\]
be the formal prototype from Section~2, and let $\Gcal$ denote its componentwise
strong realization on the real state space $L^2(\R^d;A)$. The role of
\textup{(A1)} in this theorem is only to ensure that the componentwise multiplier
expression preserves the real $A$-valued state space and that the real-branch
reduced multipliers preserve the corresponding real branch-coordinate spaces;
no semigroup growth assumption from \textup{(A2)} is used here. Then the following
statements hold.
\begin{enumerate}[label=\textup{(\roman*)}]
    \item The subspaces
    \[
    L^2_+(\R^d;A):=\mathrm{Ran}(\Pplus),
    \qquad
    L^2_-(\R^d;A):=\mathrm{Ran}(\Pminus),
    \qquad
    L^2_c(\R^d;A):=\mathrm{Ran}(\Pc)
    \]
    are closed, pairwise complementary, and give the topological direct-sum decomposition
    \[
    L^2(\R^d;A)=L^2_+(\R^d;A)\oplus L^2_-(\R^d;A)\oplus L^2_c(\R^d;A).
    \]

    \item Each branch subspace is invariant under the componentwise strong realization $\Gcal$ in the sense that
    \[
    \Pplus\Dom(\Gcal)\subseteq \Dom(\Gcal),
    \qquad
    \Pminus\Dom(\Gcal)\subseteq \Dom(\Gcal),
    \qquad
    \Pc\Dom(\Gcal)\subseteq \Dom(\Gcal),
    \]
    and
    \[
    \Gcal\Pplus = \Pplus\Gcal,
    \qquad
    \Gcal\Pminus = \Pminus\Gcal,
    \qquad
    \Gcal\Pc = \Pc\Gcal
    \quad\text{on }\Dom(\Gcal).
    \]

    \item Under the branch identification maps from Definition~\ref{def:branch-identifications}, the strong realization intertwines with the reduced branch operators in the following domain-sensitive form: for each branch $\sigma\in\{+,-,c\}$,
    \[
    \mathcal I_\sigma\bigl(\Dom(\Gcal)\cap L^2_\sigma(\R^d;A)\bigr)
    \subseteq
    \Dom(G_\sigma),
    \]
    and for every $Y\in\Dom(\Gcal)\cap L^2_\sigma(\R^d;A)$ one has
    \[
    \mathcal I_\sigma(\Gcal Y)=G_\sigma\mathcal I_\sigma(Y).
    \]

    \item In branch coordinates, the Fourier-side representatives of the three reduced
    operators have symbols
    \[
    \lambda_+(\xi)=a(\xi)+b(\xi),
    \qquad
    \lambda_-(\xi)=a(\xi)-b(\xi),
    \qquad
    \lambda_c(\xi)=a(\xi)+i\,b(\xi).
    \]
    The plus and minus statements are genuine pointwise one-dimensional fiber reductions, since
    \[
    \Lu\Pplus=\Pplus,
    \qquad
    \Lu\Pminus=-\Pminus.
    \]
    The complex-branch statement is different: it is the scalar representative obtained
    after the global real-linear identification
    \[
    \mathcal I_c:L^2_c(\R^d;A)\to L^2(\R^d;\C),
    \qquad
    \mathcal I_c(f\Pc+gJ)=f+ig.
    \]
    More precisely, for every
    \(Y\in\Dom(\Gcal)\cap L^2_c(\R^d;A)\), if
    \(h:=\mathcal I_cY\), then
    \[
    h\in\Dom(G_c),
    \qquad
    \mathcal I_c(\Gcal Y)=G_ch,
    \qquad
    \widehat{G_ch}(\xi)=\lambda_c(\xi)\widehat h(\xi)
    \quad\text{for a.e. }\xi .
    \]
    Thus the complex-branch identity at this stage is only the domain-sensitive
    inclusion
    \[
    \left.
    \mathcal I_c\Gcal\mathcal I_c^{-1}
    \right|_{\mathcal I_c(\Dom(\Gcal)\cap L^2_c(\R^d;A))}
    \subseteq
    G_c.
    \]
    The genuine maximal conjugated complex-branch generator is introduced later as
    \[
    \Acal_c:=\mathcal I_c^{-1}G_c\mathcal I_c,
    \qquad
    \Dom(\Acal_c):=\mathcal I_c^{-1}\Dom(G_c).
    \]
    This is precisely the transported complex-branch realization fixed in
    Proposition~\ref{prop:realified-complex-branch-normal-form}.
\end{enumerate}
\end{theorem}

\begin{proof}
By \textup{(A1)} and Lemma~\ref{lem:coefficientwise-real-structure-preservation},
the componentwise expression defining $\Gcal$ maps its strong domain in
$L^2(\R^d;A)$ into $L^2(\R^d;A)$. Thus $\Gcal$ is a real operator on the stated
real Hilbert space. Moreover, the reduced real-branch symbols
\[
\lambda_+=a+b,
\qquad
\lambda_-=a-b
\]
satisfy the same Hermitian symmetry, so their multiplier realizations preserve
the real branch-coordinate spaces whenever the relevant domain condition holds.

Part \textup{(i)} is a direct consequence of the projector relations established in Section~2. Since $\Pplus$, $\Pminus$, and $\Pc$ are bounded idempotents with pairwise zero products and sum equal to the identity, their ranges are closed and yield a topological direct sum decomposition of $L^2(\R^d;A)$.

For part \textup{(ii)}, first note that the projectors commute with the Fourier transform because they act only on the coefficient algebra, now complexified on the Fourier side. They also commute with $I_A\otimes a(D)$ and $I_A\otimes b(D)$, since these are scalar multipliers on the spatial variable. Finally, they commute with $\Lu$ because each projector belongs to the commutative algebra generated by $u$. Therefore they commute with the componentwise strong realization $\Gcal$ on its domain. Domain invariance follows because, if $Y\in\Dom(\Gcal)$, then both $a(\xi)\widehat{Y}(\xi)$ and $b(\xi)\widehat{Y}(\xi)$ belong to $L^2(\R^d;\AC)$, and applying any bounded projector preserves this property.

For part \textup{(iii)}, consider first the plus branch. Every element of $L^2_+(\R^d;A)$ has the form $f\Pplus$ with $f\in L^2(\R^d;\R)$. Define
\[
\mathcal I_+(f\Pplus):=f.
\]
If in addition $f\Pplus\in\Dom(\Gcal)$, then
\[
\Gcal(f\Pplus)
=
(a(D)f)\Pplus + (b(D)f)\Lu\Pplus
=
\bigl(a(D)f+b(D)f\bigr)\Pplus,
\]
so
\[
\mathcal I_+\bigl(\Gcal(f\Pplus)\bigr)
=
a(D)f+b(D)f
=
\lambda_+(D)f
=
G_+\mathcal I_+(f\Pplus).
\]
Here the first equality is evaluated on the componentwise strong domain, where
\(a(D)f\) and \(b(D)f\) exist separately; the final equality uses the symbol-sum
multiplier convention \(G_+=\lambda_+(D)\).
The minus branch is identical, using $\Lu\Pminus=-\Pminus$. If $g\Pminus\in \Dom(\Gcal)\cap L^2_-(\R^d;A)$ and $\mathcal I_-(g\Pminus):=g$, then
\[
\mathcal I_-\bigl(\Gcal(g\Pminus)\bigr)
=
a(D)g-b(D)g
=
\lambda_-(D)g
=
G_-\mathcal I_-(g\Pminus).
\]

For the complex branch, every element of $L^2_c(\R^d;A)$ has the form $f\Pc+gJ$ with $f,g\in L^2(\R^d;\R)$. If $f\Pc+gJ\in\Dom(\Gcal)$, then by the relations $\Lu\Pc=J$ and $\Lu J=-\Pc$,
\[
\Gcal(f\Pc+gJ)
=
\bigl(a(D)f-b(D)g\bigr)\Pc
+
\bigl(b(D)f+a(D)g\bigr)J.
\]
Applying $\mathcal I_c$ yields
\[
\mathcal I_c\bigl(\Gcal(f\Pc+gJ)\bigr)
=
\bigl(a(D)f-b(D)g\bigr)+i\bigl(b(D)f+a(D)g\bigr)
=
\lambda_c(D)(f+ig)
=
G_c\mathcal I_c(f\Pc+gJ).
\]
The first equality is again a componentwise strong-domain identity, while
\(G_c=\lambda_c(D)\) is the full reduced symbol-sum multiplier. Hence the
transported complex-branch action of \(\Gcal\) is a restriction of \(G_c\).
This does not assert equality with \(G_c\) as an unbounded operator on its full
multiplier domain \(\Dom(G_c)\). Thus the complex-branch reduction is a realified
complex multiplier identity on the transported strong domain, not merely a formal
pointwise diagonalization.

Part \textup{(iv)} is the branch-coordinate Fourier reformulation of part \textup{(iii)}.
For the plus and minus branches, the identities
\[
\Lu\Pplus=\Pplus,
\qquad
\Lu\Pminus=-\Pminus
\]
give genuine pointwise one-dimensional fiber reductions with symbols
\[
\lambda_+=a+b,
\qquad
\lambda_-=a-b.
\]
For the complex branch, write
\[
Y_c=f\Pc+gJ,
\qquad
f,g\in L^2(\R^d;\R),
\qquad
h=\mathcal I_cY_c=f+ig.
\]
Using
\[
\Lu\Pc=J,
\qquad
\Lu J=-\Pc,
\]
one obtains, on the Fourier side,
\[
(a\IAC+b\Lu)(\widehat f\Pc+\widehat gJ)
=
(a\widehat f-b\widehat g)\Pc
+
(b\widehat f+a\widehat g)J.
\]
After applying the real-linear coordinate map $\mathcal I_c$, this becomes
\[
(a\widehat f-b\widehat g)+i(b\widehat f+a\widehat g)
=
(a+ib)(\widehat f+i\widehat g)
=
\lambda_c\widehat h.
\]
Thus the complex-branch symbol $\lambda_c=a+i b$ is the transported scalar
coordinate fixed by Proposition~\ref{prop:realified-complex-branch-normal-form};
Lemma~\ref{lem:realified-complex-branch-multiplier} supplies the corresponding
domain equivalence for real states.
\end{proof}

\begin{remark}[Necessity of the reality symmetry]\label{rem:A1-necessity-strong-realization}
The assumption \textup{(A1)} cannot be omitted from
Theorem~\ref{thm:distributed-branch-splitting} when the state space is the real
Hilbert space $L^2(\R^d;A)$. For example, if $a(\xi)=i$ and $b(\xi)=0$, then the
scalar multiplier $a(D)$ sends a nonzero real-valued function $f$ to $if$, which
is not real-valued. Hence the componentwise expression would be well-defined
only on the complexified state space, not as an operator
\[
\Dom(\Gcal)\subset L^2(\R^d;A)\to L^2(\R^d;A).
\]
Thus \textup{(A1)} is precisely the compatibility condition that lets the
Fourier-side complex multiplier formula descend to the real $A$-valued state
space.
\end{remark}

The theorem shows that the full multiplier architecture is branch-compatible already at the level of the strong prototype domain. What it does \emph{not} claim is equality of domains between the restricted strong realization and the reduced branch operators. That stronger identification belongs only to the reconstructed generator introduced in Section~5, whose domain is determined branchwise from the reduced semigroups rather than componentwise from the separate multiplier pieces.

Two remarks are worth making before moving on. First, the theorem does not yet assert well-posedness of the reduced flows for positive time; it identifies the correct reduced generators and the exact invariant splitting on which the semigroup theory should be run. The actual generation results belong to Section~4. Second, the theorem is stronger than a mere decomposition of initial data. Because the projectors commute with the generator, branch-supported initial states remain branch-supported under the full flow whenever the latter exists. Thus branch geometry is dynamically stable already at the linear function-space level.

\begin{remark}[What Fourier analysis does and does not do here]\label{rem:fourier-does-not-create-splitting}
The branch splitting proved in this section is not created by Fourier analysis itself. The splitting already exists on the physical state space because the semisimple coefficient algebra $A$ carries the fixed projectors $\Pplus$, $\Pminus$, and $\Pc$, which lift to bounded complementary projectors on $L^2(\R^d;A)$. The Fourier transform is used only to express the translation-invariant prototype in branch coordinates. In those coordinates $\lambda_+$ and $\lambda_-$ come from one-dimensional fiber reductions, whereas $\lambda_c$ is the scalar complex representative obtained after the real-linear map $\mathcal I_c$.
\end{remark}

\subsection{Interpretive remarks}

\paragraph*{What is proved in Section~3.}
The proved content of Theorem~\ref{thm:distributed-branch-splitting} is an exact intrinsic branch decomposition at the level of the function-space state space together with domain-sensitive intertwining for the componentwise strong prototype operator. In particular, the theorem identifies the correct reduced symbols and the correct invariant branch subspaces, but it does \emph{not} yet assert semigroup generation or full generator identification on the reconstructed flow. Those stronger statements belong to Sections~4 and~5.

\paragraph*{Interpretation.}
The significance of Theorem~\ref{thm:distributed-branch-splitting} is conceptual as much as technical. It shows how the branch program is realized in a distributed function-space setting rather than only in finite-dimensional algebraic models. The earlier papers established branch decomposition in scalar or abstract operator settings; the present theorem shows that, for the concrete translation-invariant multiplier class treated here, the same intrinsic decomposition acts on the whole Hilbert space of spatially extended states and is represented after Fourier transform in branch coordinates.

This should be understood as the correct multiplier-realization analogue of a reduced branch-coordinate representation, but with an important difference. The splitting is not driven by a spectral theorem for the full unbounded multiplier generator on $L^2(\R^d;A)$. Instead it is inherited from the fixed semisimple coefficient algebra $A$, and the Fourier transform simply distributes that algebraic splitting across frequencies. In that sense, the branch decomposition is more rigid than an accidental spectral decomposition: it is built into the architecture of the model.

The theorem also clarifies why the chosen multiplier class is a natural setting for the paper. If one were to start instead with variable coefficients, nontranslation-invariant operators, or nonlinear terms, the clean Fourier-side representation of the intrinsic branch decomposition would disappear immediately and would have to be replaced by a much subtler microlocal or perturbative analysis. Here, by contrast, the multiplier structure allows the distributed realization problem to reduce to a measurable family of exact finite-dimensional branch calculations. This is precisely the amount of structure needed for the restricted multiplier realization addressed here, rather than for several harder neighboring problems at once.

Finally, Theorem~\ref{thm:distributed-branch-splitting} prepares both halves of the remainder of the paper. On the deterministic side, it identifies the branchwise reduced generators $G_+$, $G_-$, and $G_c$ whose semigroup theory will be developed next. On the observational side, it already suggests the observation-dependent masking mechanism: different branches may grow, decay, or oscillate differently, and an observation map may enhance, suppress, or entirely mask one branch relative to another. Section~4 now turns to the analytic input required to turn this exact branch decomposition into a well-posed time-evolution theory.


\section{Reduced Multiplier Branches and Semigroup Generation}

\subsection{Reduced branch operators}

Section~3 identified the exact branch shadows of the strong prototype operator $\Gcal$. We now isolate them as autonomous evolution operators on the model spaces corresponding to the three branches. The point of the present section is analytic rather than algebraic: once the intrinsic branch decomposition is known exactly, one still has to prove that each reduced branch actually generates a strongly continuous semigroup on the underlying $L^2$ space.

\paragraph*{Standard multiplier proof convention.}
The generation, domain, density, closedness, and operator-norm estimates in this section are standard Fourier-multiplier and multiplication-operator arguments. To keep the main text focused on branch normal form, reduced-generator packaging, reconstruction, graph-core identity, and observation limitations, routine proofs are stated in compressed form and cross-referenced when useful to Appendix~\ref{app:routine-proof-ledger}. The novelty claimed by the paper is not a new scalar multiplier-generation theorem; the purpose of recording these facts is only to fix the maximal reduced multipliers used later.

\begin{definition}[Reduced branch multipliers]
\label{def:reduced-branch-multipliers}
For the plus and minus branches we work on the real Hilbert space $\LtwoR$ and define the maximal reduced Fourier multipliers
\[
G_+ := \lambda_+(D),
\qquad
G_- := \lambda_-(D),
\]
with domains
\[
\Dom(G_+) := \{f\in \LtwoR: \lambda_+(\xi)\widehat f(\xi)\in \LtwoC\},
\]
\[
\Dom(G_-) := \{f\in \LtwoR: \lambda_-(\xi)\widehat f(\xi)\in \LtwoC\}.
\]
For the complex branch we work first on the complex Hilbert space $\LtwoC$ and define the maximal reduced Fourier multiplier
\[
G_c := \lambda_c(D),
\]
with domain
\[
\Dom(G_c) := \{h\in \LtwoC: \lambda_c(\xi)\widehat h(\xi)\in \LtwoC\}.
\]
These are relative symbol-sum Fourier-multiplier realizations. Thus $G_+$, for example,
is $(a+b)(D)$, not the operator sum $a(D)+b(D)$ on
$\Dom(a(D))\cap\Dom(b(D))$. The same convention applies to $G_-$ and $G_c$.
This distinction is essential because cancellations in the reduced symbols may
strictly enlarge the reduced branch domains. In the plus and minus cases the
vectors $f$ are real-valued physical states, while the Fourier-side products
$\lambda_\pm\widehat f$ are complex-valued functions and are therefore tested in
$\LtwoC$.
\end{definition}


\begin{lemma}[Reduced multiplier domains]
\label{lem:reduced-multipliers-closed-dense}
For each branch \(\bullet\in\{+,-,c\}\), the reduced multiplier operator
\[
G_\bullet=\lambda_\bullet(D),
\qquad
\Dom(G_\bullet)=\{f:\lambda_\bullet\widehat f\in L^2\},
\]
is densely defined and closed on its natural branch Hilbert space: \(\LtwoR\) for
\(\bullet=+,-\), and \(\LtwoC\) for \(\bullet=c\). For the real branches the
domain, action, and density approximation are understood inside the Hermitian
Fourier-reality subspace corresponding to \(L^2_{\R}\).
\end{lemma}

\begin{proof}
This is the standard maximal multiplication-operator proof, with symmetric finite-symbol cutoffs used on the real branches to preserve the Fourier-reality condition. The detailed cutoff and closed-graph argument is recorded in Appendix~\ref{app:routine-proof-ledger}.
\end{proof}

It is useful to emphasize what has happened conceptually. The full branch-valued dynamics on $L^2(\R^d;A)$ reduce to three scalar multiplier problems:
\[
\partial_t f_+ = G_+ f_+,
\qquad
\partial_t f_- = G_- f_-,
\qquad
\partial_t h = G_c h.
\]
The plus and minus branches remain real scalar problems, while the complex branch becomes a complex scalar problem whose realification corresponds to the two-dimensional real branch $A_c$ of the coefficient algebra. Section~5 will reconstruct the full $A$-valued flow from these reduced branches, but that reconstruction makes sense only after the semigroup generation problem has been settled.

Lemma~\ref{lem:reduced-multipliers-closed-dense} records the basic closedness and density of these reduced domains before any semigroup theorem is invoked. Thus the generation result below is built on explicit closed multiplier operators, not on merely formal branch symbols.

\subsection{Admissibility assumptions}

The standing assumptions (A1)--(A2) from Section~2 were chosen so that semigroup generation for the reduced branches can be proved from explicit symbol conditions rather than assumed in advance. In particular, (A2) is used here as a sufficient $L^2$ Fourier-multiplier growth condition, not as a necessary condition for every possible semigroup realization of related symbols. We restate the operative part of that package here in a form tailored to the present section.

For each branch $\bullet\in\{+, -, c\}$, define the reduced symbol
\[
\lambda_\bullet(\xi)
\in
\{a(\xi)+b(\xi),\ a(\xi)-b(\xi),\ a(\xi)+i b(\xi)\}.
\]
No separate measurability assumption is needed here: each $\lambda_\bullet$ is measurable because it is built from the measurable symbols $a$ and $b$, while assumption (A2) provides the branchwise upper bound
\[
\Re \lambda_\bullet(\xi)\le \omega_\bullet
\quad\text{for a.e. }\xi\in\R^d.
\]
These estimates immediately imply that, for every $t\ge 0$,
\[
\abs{e^{t\lambda_\bullet(\xi)}}=e^{t\Re\lambda_\bullet(\xi)}\le e^{\omega_\bullet t}
\quad\text{for a.e. }\xi,
\]
so the Fourier multiplier family $e^{t\lambda_\bullet(\cdot)}$ is bounded on $\LtwoC$ with operator norm at most $e^{\omega_\bullet t}$, and on the real branches its Hermitian-symmetric restrictions act on $\LtwoR$. Thus the boundedness of the candidate branch semigroups is not an extra hypothesis: it is already forced by the reduced-symbol growth package.

The reality symmetry assumption from Section~2 also plays a nontrivial role. Because
\[
a(-\xi)=\overline{a(\xi)},
\qquad
b(-\xi)=\overline{b(\xi)},
\]
one has
\[
\lambda_\pm(-\xi)=\overline{\lambda_\pm(\xi)},
\qquad
e^{t\lambda_\pm(-\xi)}=\overline{e^{t\lambda_\pm(\xi)}}.
\]
By the preceding domain-sensitive lemma, the unbounded multipliers
$\lambda_\pm(D)$ preserve real-valued data on their natural domains, and the bounded
semigroup multipliers $e^{t\lambda_\pm(\cdot)}(D)$ preserve real-valued $L^2$ data for every
$t\ge0$. The complex branch carries no such real symmetry at the level of a scalar reduced
symbol; this is precisely why it is treated as a complex semigroup first and then
transferred back to the real branch space through the identification $\mathcal I_c$.
This does not mean that the pulled-back $A_c$-branch flow leaves the real state
space. It means only that the scalar coordinate
$h_c=\mathcal I_cY_c\in L^2(\R^d;\C)$ is treated as a genuinely complex
$L^2$ variable before being transported back to the real two-dimensional
branch fiber.

The present paper deliberately does not pursue the sharpest possible admissibility conditions. No attempt is made to characterize all multiplier symbols generating $C_0$-semigroups. The aim is narrower: to fix an explicit and checkable class large enough for the benchmark examples of Section~7 and small enough that the reconstruction program remains exact.

\begin{proposition}[Sufficient reduced-symbol growth criterion]\label{prop:reduced-symbol-criterion}
Assume that $a,b:\R^d\to\C$ are measurable and finite almost everywhere, and assume \textup{(A2)}. Then for each branch $\bullet\in\{+,-,c\}$ and each $t\ge 0$, the multiplier $e^{t\lambda_\bullet(\cdot)}$ belongs to $L^\infty(\R^d)$ and satisfies
\[
\|e^{t\lambda_\bullet(\cdot)}\|_{L^\infty(\R^d)}
\le e^{\omega_\bullet t}.
\]
Consequently, $e^{t\lambda_\bullet(\cdot)}(D)$ defines a bounded Fourier multiplier on $\LtwoC$.

If, in addition, \textup{(A1)} holds, then for the real branches $+$ and $-$, the multipliers
$e^{t\lambda_\pm(\cdot)}(D)$ restrict to bounded real-linear operators on $\LtwoR$, with the same norm bound.
\end{proposition}

\begin{proof}
The estimate is immediate from
\[
\abs{e^{t\lambda_\bullet(\xi)}}
=
e^{t\Re\lambda_\bullet(\xi)}
\le e^{\omega_\bullet t}
\]
for almost every $\xi\in\R^d$. This proves the $L^\infty$ bound and hence boundedness of the corresponding complex $L^2$ Fourier multiplier on $\LtwoC$.

Assume now also \textup{(A1)}. For $\bullet\in\{+,-\}$, the reduced real-branch symbols satisfy
\[
\lambda_\bullet(-\xi)=\overline{\lambda_\bullet(\xi)}
\quad\text{for a.e. }\xi\in\R^d.
\]
Therefore
\[
e^{t\lambda_\bullet(-\xi)}
=
\overline{e^{t\lambda_\bullet(\xi)}}
\quad\text{for a.e. }\xi\in\R^d.
\]
By Lemma~\ref{lem:domain-sensitive-reality-preservation}, the bounded multipliers
$e^{t\lambda_\bullet(\cdot)}(D)$ preserve the Fourier-side Hermitian symmetry of real-valued $L^2$ data. Hence their restrictions define bounded real-linear operators on $\LtwoR$. The restriction norm is bounded by the same $L^\infty$ multiplier bound obtained above.
\end{proof}

\begin{remark}[Exactness within the prescribed multiplier formula]\label{rem:reduced-symbol-criterion-sufficiency-only}
Proposition~\ref{prop:reduced-symbol-criterion} gives the forward estimate used
for the branch multipliers in this paper. Within the prescribed $L^2$ multiplier
formula, the corresponding boundedness condition is exact: by
Lemma~\ref{lem:near-necessity-a2-prescribed-multiplier}, boundedness of
$e^{t\lambda_\bullet(D)}$ at one positive time is equivalent to an upper
essential bound on \(\Re\lambda_\bullet\). The proposition is called a
sufficient criterion only because the paper does not attempt a classification
of all semigroup realizations of related symbols in other spaces, norms, or
domains.
\end{remark}

\subsection{Main theorem: reduced branch semigroup generation}

We now state the second main theorem of the paper. It is the analytic input that converts the intrinsic branch decomposition of Section~3 into well-posed time evolution.

\begin{theorem}[Reduced branch semigroup generation under prescribed real-part bounds]\label{thm:reduced-semigroup-generation}
Assume that \((a,b)\) satisfies the standing multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}. Thus the reduced
symbols $\lambda_+$, $\lambda_-$, and $\lambda_c$ are measurable and finite
almost everywhere and satisfy the branchwise real-part upper bounds of
\textup{(A2)}. Define the natural reduced branch model spaces
\[
X_+ := \LtwoR,
\qquad
X_- := \LtwoR,
\qquad
X_c := \LtwoC,
\]
where $X_+$ and $X_-$ are real Hilbert spaces and $X_c$ is a complex Hilbert space. Whenever these three spaces are assembled into one branch-coordinate model, the direct sum is understood as a real Hilbert direct sum, with the third component $X_c=\LtwoC$ regarded through its underlying real Hilbert structure.

The linearity convention in this theorem is the following:
\[
T_+(t),T_-(t)\text{ are real-linear on }\LtwoR,
\qquad
T_c(t)\text{ is complex-linear on }\LtwoC.
\]
Whenever $T_c(t)$ is transported to $L^2_c(\R^d;A)$ or assembled with $T_+(t)$ and $T_-(t)$ in a branch-coordinate direct sum, it is viewed as a real-linear operator on the underlying real Hilbert space of $\LtwoC$.

Define
\[
\Dom(G_+) := \{f\in X_+ : \lambda_+(\xi)\widehat f(\xi)\in \LtwoC\},
\qquad
\widehat{G_+ f}(\xi):=\lambda_+(\xi)\widehat f(\xi),
\]
\[
\Dom(G_-) := \{f\in X_- : \lambda_-(\xi)\widehat f(\xi)\in \LtwoC\},
\qquad
\widehat{G_- f}(\xi):=\lambda_-(\xi)\widehat f(\xi),
\]
and
\[
\Dom(G_c) := \{h\in X_c : \lambda_c(\xi)\widehat h(\xi)\in \LtwoC\},
\qquad
\widehat{G_c h}(\xi):=\lambda_c(\xi)\widehat h(\xi).
\]
Then the following hold. This theorem proves only branchwise \(C_0\)-semigroup
generation and the exact reduced direct-sum Fourier-multiplier representation on the
reduced branch spaces. No assertion is made here about observation, inverse
recovery, sensor stability, or branch visibility.
\begin{enumerate}[label=\textup{(\roman*)}]
    \item $G_+$ and $G_-$ generate strongly continuous semigroups $\{T_+(t)\}_{t\ge 0}$ and $\{T_-(t)\}_{t\ge 0}$ on the real Hilbert spaces $X_+$ and $X_-$.

    \item $G_c$ generates a complex-linear strongly continuous semigroup $\{T_c(t)\}_{t\ge 0}$ on the complex Hilbert space $X_c$. In every full real branch reconstruction, this same family is regarded through its underlying real-linear action.

    \item The semigroups are given by the Fourier multiplier formulas
    \[
    \widehat{T_+(t)f}(\xi)=e^{t\lambda_+(\xi)}\widehat f(\xi),
    \qquad
    \widehat{T_-(t)f}(\xi)=e^{t\lambda_-(\xi)}\widehat f(\xi),
    \qquad f\in \LtwoR,
    \]
    and
    \[
    \widehat{T_c(t)h}(\xi)=e^{t\lambda_c(\xi)}\widehat h(\xi),
    \qquad h\in \LtwoC.
    \]

    \item The semigroup norms satisfy the branchwise bounds
    \[
    \norm{T_\pm(t)}_{\mathcal L(\LtwoR)}
    \le e^{\omega_\pm t},
    \qquad
    \norm{T_c(t)}_{\mathcal L(\LtwoC)}
    \le e^{\omega_c t}
    \qquad (t\ge 0).
    \]
\end{enumerate}
\end{theorem}

\begin{proof}
For each branch $\bullet\in\{+, -, c\}$ let $X_\bullet$ denote the corresponding model space defined above, and define
\[
T_\bullet(t)f := \F^{-1}\bigl(e^{t\lambda_\bullet(\cdot)}\widehat f\bigr),
\qquad t\ge 0,
\qquad f\in X_\bullet.
\]
By Proposition~\ref{prop:reduced-symbol-criterion}, the multiplier $e^{t\lambda_\bullet(\cdot)}$ belongs to $L^\infty(\R^d)$ and satisfies the standard multiplier estimate
\[
\norm{T_\bullet(t)}_{\mathcal L(X_\bullet)}
\le \|e^{t\lambda_\bullet(\cdot)}\|_{L^\infty(\R^d)}
\le e^{\omega_\bullet t}.
\]
For the real branches $X_+$ and $X_-$ this is only used as a restriction estimate obtained from the corresponding Hermitian-symmetric complex Fourier multiplier. No equality between the real operator norm and the essential supremum of the multiplier is used here.
This proves the operator-norm bound. For the dominated-convergence estimates below, fix
\[
C_\bullet:=e^{\max\{\omega_\bullet,0\}}.
\]
Then, whenever $0\le r\le 1$ and $\Re z\le\omega_\bullet$, one has
\[
|e^{rz}|\le C_\bullet.
\]

The semigroup property follows directly from the multiplicative identity
\[
e^{(t+s)\lambda_\bullet(\xi)}=e^{t\lambda_\bullet(\xi)}e^{s\lambda_\bullet(\xi)}
\qquad\text{for a.e. }\xi\in\R^d,
\]
whence
\[
T_\bullet(t+s)=T_\bullet(t)T_\bullet(s),
\qquad T_\bullet(0)=I.
\]

To prove strong continuity at $t=0$, let $f\in X_\bullet$. Then
\[
\widehat{T_\bullet(t)f-f}(\xi)=\bigl(e^{t\lambda_\bullet(\xi)}-1\bigr)\widehat f(\xi).
\]
For almost every $\xi$, one has $e^{t\lambda_\bullet(\xi)}\to 1$ as $t\downarrow 0$. Moreover, for $0\le t\le 1$,
\[
|e^{t\lambda_\bullet(\xi)}-1|^2\,|\widehat f(\xi)|^2
\le 2\bigl(C_\bullet^2+1\bigr)|\widehat f(\xi)|^2,
\]
and the right-hand side is integrable in the norm of $X_\bullet$. Dominated convergence therefore gives
\[
\lim_{t\downarrow 0}\norm{T_\bullet(t)f-f}_{X_\bullet}=0.
\]
Thus $\{T_\bullet(t)\}_{t\ge 0}$ is a $C_0$-semigroup on $X_\bullet$.

It remains to identify the generator. Let $A_\bullet$ denote the infinitesimal generator of $T_\bullet(t)$. Suppose first that $f\in\Dom(G_\bullet)$. By definition, the Fourier-side product $\lambda_\bullet\widehat f$ belongs to the relevant scalar $L^2$ space. For the plus and minus branches this means membership in $\LtwoC$ on the Fourier side, together with the Hermitian symmetry that makes the inverse Fourier transform a real element of $X_\bullet=\LtwoR$; for the complex branch it means ordinary membership in $X_c=\LtwoC$. Then
\[
\frac{T_\bullet(t)f-f}{t}
=
\F^{-1}\!\left(\frac{e^{t\lambda_\bullet(\xi)}-1}{t}\widehat f(\xi)\right).
\]
For almost every $\xi$,
\[
\frac{e^{t\lambda_\bullet(\xi)}-1}{t}\to \lambda_\bullet(\xi)
\qquad (t\downarrow 0),
\]
and the integral representation
\[
\frac{e^{tz}-1}{t}= z\int_0^1 e^{stz}\,ds
\]
shows that for $0<t\le 1$ and $z\in\C$ with $\Re z\le \omega_\bullet$,
\[
\left|\frac{e^{tz}-1}{t}\right|
\le |z|\int_0^1 |e^{stz}|\,ds
\le C_\bullet |z|.
\]
Hence
\[
\left|\frac{e^{t\lambda_\bullet(\xi)}-1}{t}\widehat f(\xi)\right|
\le C_\bullet|\lambda_\bullet(\xi)|\,|\widehat f(\xi)|,
\]
and the right-hand side belongs to the relevant scalar $L^2$ space because $f\in\Dom(G_\bullet)$. Dominated convergence yields
\[
\lim_{t\downarrow 0}\left\|\frac{T_\bullet(t)f-f}{t}-\F^{-1}(\lambda_\bullet\widehat f)\right\|_{L^2}=0.
\]
Hence $f\in\Dom(A_\bullet)$ and $A_\bullet f=G_\bullet f$, so $G_\bullet\subseteq A_\bullet$.

Conversely, if $f\in\Dom(A_\bullet)$, then there exists $g$ in the relevant scalar $L^2$ space such that
\[
\frac{T_\bullet(t)f-f}{t}\to g
\qquad\text{in the relevant scalar }L^2\text{ space}
\quad (t\downarrow 0).
\]
Passing to the Fourier side gives
\[
\frac{e^{t\lambda_\bullet(\xi)}-1}{t}\widehat f(\xi) \to \widehat g(\xi)
\qquad\text{in the relevant scalar }L^2\text{ space}.
\]
After extracting a subsequence, this convergence holds almost everywhere, and the pointwise limit must equal $\lambda_\bullet(\xi)\widehat f(\xi)$. Therefore $\lambda_\bullet\widehat f$ belongs to the relevant scalar $L^2$ space and $\widehat g=\lambda_\bullet\widehat f$, showing that $f\in\Dom(G_\bullet)$ and $A_\bullet f=G_\bullet f$. Thus $A_\bullet=G_\bullet$.

For the plus and minus branches, the preceding domain-sensitive lemma applied to the bounded multipliers $e^{t\lambda_\pm(\cdot)}$ shows that $T_\pm(t)$ preserve real-valued data, so they define $C_0$-semigroups on the real Hilbert space $\LtwoR$. For the complex branch, the same argument is carried out on the complex Hilbert space $\LtwoC$. This completes the proof.
\end{proof}

\begin{remark}[Reduced semigroup linearity convention]
The reduced theorem deliberately mixes two scalar settings. The plus and minus
branch semigroups act on real Hilbert spaces and are used only as real-linear
$C_0$-semigroups. The complex branch semigroup is complex-linear on
$\LtwoC$. When this complex branch is returned to the physical branch
$L^2_c(\R^d;A)$, or when the three branches are assembled into a full
$A$-valued state, $T_c(t)$ is viewed only through its underlying real-linear
operator. Thus no complex-linear structure is imposed on the full physical
space $L^2(\R^d;A)$.
\end{remark}

\begin{corollary}[Sharp branch semigroup envelopes in the fixed Hilbert norm]\label{cor:sharp-real-branch-envelopes}
Assume that \((a,b)\) satisfies the standing multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}, and let
$T_+(t)$, $T_-(t)$, and $T_c(t)$ be the reduced branch semigroups constructed in
Theorem~\ref{thm:reduced-semigroup-generation}. Set
\[
\alpha_\bullet
:=
\operatorname*{ess\,sup}_{\xi\in\R^d}\Re\lambda_\bullet(\xi),
\qquad
\bullet\in\{+,-,c\}.
\]
Then, for every \(t\ge0\),
\[
\|T_+(t)\|_{\mathcal L(\LtwoR)}=e^{\alpha_+t},
\qquad
\|T_-(t)\|_{\mathcal L(\LtwoR)}=e^{\alpha_-t},
\qquad
\|T_c(t)\|_{\mathcal L(\LtwoC)}=e^{\alpha_ct}.
\]
Equivalently, when \(\omega_\bullet=\alpha_\bullet\), the branchwise
operator-norm envelopes in Theorem~\ref{thm:reduced-semigroup-generation} are
sharp on their natural real or complex branch spaces equipped with the fixed
branch-adapted \(L^2\) Hilbert norms used throughout this paper.
\end{corollary}

\begin{proof}
For the plus and minus branches, apply
Lemma~\ref{lem:sharp-real-hermitian-multiplier-norm} to
\[
m_t^\pm(\xi):=e^{t\lambda_\pm(\xi)}.
\]
The reality symmetry assumption gives
\[
m_t^\pm(-\xi)=\overline{m_t^\pm(\xi)}
\quad\text{for a.e. }\xi,
\]
so
\[
\|T_\pm(t)\|_{\mathcal L(\LtwoR)}
=
\operatorname*{ess\,sup}_{\xi\in\R^d}|e^{t\lambda_\pm(\xi)}|
=
\operatorname*{ess\,sup}_{\xi\in\R^d}e^{t\Re\lambda_\pm(\xi)}
=
e^{\alpha_\pm t}.
\]
For the complex branch, the same identity follows from the standard complex
\(L^2\) Fourier-multiplier norm formula:
\[
\|T_c(t)\|_{\mathcal L(\LtwoC)}
=
\operatorname*{ess\,sup}_{\xi\in\R^d}|e^{t\lambda_c(\xi)}|
=
e^{\alpha_ct}.
\]
\end{proof}

\begin{remark}[Upper envelopes versus sharp Hilbert-norm envelopes]\label{rem:no-real-norm-equality-used}
The proof of Theorem~\ref{thm:reduced-semigroup-generation} needs only the upper
estimate
\[
\norm{T_\bullet(t)}_{\mathcal L(X_\bullet)}
\le
\|e^{t\lambda_\bullet(\cdot)}\|_{L^\infty(\R^d)}
\le e^{\omega_\bullet t}.
\]
Thus semigroup generation does not depend on any sharp norm identity. However,
when \(\omega_\bullet\) is chosen as
\[
\operatorname*{ess\,sup}_{\xi\in\R^d}\Re\lambda_\bullet(\xi),
\]
Corollary~\ref{cor:sharp-real-branch-envelopes} shows that the displayed
operator-norm envelope is sharp in the fixed branch-adapted \(L^2\) Hilbert
norm. For the real plus and minus branches, this sharpness uses the Hermitian
symmetry of the multiplier and is not merely imported from the complex
\(L^2\)-multiplier norm formula. No sharpness statement is made here for
arbitrary equivalent coefficient norms without equivalence constants.
\end{remark}

The theorem is exactly the analytic statement that was missing from Section~3. The intrinsic branch decomposition theorem identified the reduced generators and the invariant branch subspaces; the present result shows that those reduced operators are generated directly from explicit bounds on the reduced symbols rather than being assumed in advance as semigroup multipliers. In particular, there is now a well-defined time evolution on each branch separately, with explicit Fourier representation at every time.

\subsection{Basic branchwise operator-norm envelopes}

The growth estimate in Theorem~\ref{thm:reduced-semigroup-generation} is elementary at the multiplier level, but it deserves to be recorded in a form adapted to later use. The bounded-bulk observation estimates of Section~6 will not rely on exact formulas for arbitrary data. Instead they will use upper bounds on branchwise propagation, and those bounds are already available at the end of the present section.

\begin{proposition}[Branchwise growth envelopes]\label{prop:branchwise-growth}
Assume that \((a,b)\) satisfies the standing multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}, and let
$T_+(t)$, $T_-(t)$, and $T_c(t)$ be the reduced branch semigroups constructed in
Theorem~\ref{thm:reduced-semigroup-generation}. Then the reduced semigroups satisfy
\[
\norm{T_+(t)f}_{L^2}\le e^{\omega_+ t}\norm{f}_{L^2},
\qquad
\norm{T_-(t)f}_{L^2}\le e^{\omega_- t}\norm{f}_{L^2},
\]
for all $f\in \LtwoR$ and all $t\ge 0$, while
\[
\norm{T_c(t)h}_{L^2(\C)}\le e^{\omega_c t}\norm{h}_{L^2(\C)}
\]
for all $h\in L^2(\R^d;\C)$ and all $t\ge 0$.

Moreover, if the admissible exponents are chosen as
\[
\omega_\bullet
=
\operatorname*{ess\,sup}_{\xi\in\R^d}\Re\lambda_\bullet(\xi),
\qquad
\bullet\in\{+,-,c\},
\]
then the displayed operator-norm envelopes are sharp in the fixed
branch-adapted \(L^2\) Hilbert norms:
\[
\norm{T_+(t)}_{\mathcal L(\LtwoR)}=e^{\omega_+t},
\qquad
\norm{T_-(t)}_{\mathcal L(\LtwoR)}=e^{\omega_-t},
\qquad
\norm{T_c(t)}_{\mathcal L(\LtwoC)}=e^{\omega_ct}.
\]
\end{proposition}

\begin{proof}
The upper estimates are exactly the operator-norm bounds already obtained in the proof of Theorem~\ref{thm:reduced-semigroup-generation}. They are restated separately because later sections will use them as input for reconstruction and observation-bound arguments without repeatedly returning to the multiplier proof. The sharp identities in the essential-supremum case are precisely Corollary~\ref{cor:sharp-real-branch-envelopes}, and are to be read in the fixed branch-adapted \(L^2\) Hilbert norms.
\end{proof}

Two immediate consequences should be kept in mind. First, different branches may carry different admissible upper exponents $\omega_\bullet$. Thus a single full-state evolution may already exhibit branch-selective amplification or branch-selective decay before any observation is applied. Second, the estimates are upper bounds for all data but are not datum-specific growth laws. When $\omega_\bullet$ is chosen as $\operatorname*{ess\,sup}\Re\lambda_\bullet$, Proposition~\ref{prop:branchwise-growth} gives the sharp semigroup operator-norm envelope on the corresponding branch in the fixed branch-adapted $L^2$ Hilbert norm; a particular datum may still grow more slowly, depending on the distribution of its Fourier mass. If $\omega_\bullet$ is chosen larger than the essential supremum, it is only an admissible upper envelope. This is exactly the level of control required for the one-sided observation-bound results later on.

\subsection{Comments on the complex branch}

Proposition~\ref{prop:realified-complex-branch-normal-form} fixes the complex-branch convention used in this section: \(G_c=\lambda_c(D)\) is the maximal scalar multiplier with symbol \(\lambda_c=a+i b\) in the global real-coordinate model, and \(L^2_c(\R^d;A)\) is recovered as a real Hilbert space through the real-linear isometric isomorphism
\[
\mathcal I_c : L^2_c(\R^d;A) \to L^2(\R^d;\C),
\qquad
\mathcal I_c(f\Pc+gJ)=f+ig.
\]
Through this identification the complex semigroup $T_c(t)$ transfers back to a real semigroup on the complex branch of the original $A$-valued state space.

\begin{proposition}[Realified complex branch semigroup]\label{prop:realified-complex-branch}
Assume that \((a,b)\) satisfies the standing multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}, and let $T_c(t)$ and
$G_c$ be the complex reduced branch semigroup and generator constructed in
Theorem~\ref{thm:reduced-semigroup-generation}. Let
$S_c(t):=\mathcal I_c^{-1}T_c(t)\mathcal I_c$ on $L^2_c(\R^d;A)$. Then $\{S_c(t)\}_{t\ge 0}$ is a strongly continuous semigroup on the real Hilbert space $L^2_c(\R^d;A)$. Its generator is the realified branch generator
\[
\Acal_c:=\mathcal I_c^{-1}G_c\mathcal I_c,
\qquad
\Dom(\Acal_c):=\mathcal I_c^{-1}\Dom(G_c),
\]
and one has the domain-sensitive inclusion
\[
\left.\Gcal\right|_{\Dom(\Gcal)\cap L^2_c(\R^d;A)} \subseteq \Acal_c.
\]
Moreover,
\[
\norm{S_c(t)}_{\mathcal L(L^2_c)} = \norm{T_c(t)}_{\mathcal L(L^2(\C))}
\le e^{\omega_c t}.
\]
\end{proposition}

\begin{proof}
Since $\mathcal I_c$ is a real-linear isometric isomorphism and $T_c(t)$ is a $C_0$-semigroup on $L^2(\R^d;\C)$ with generator $G_c$, the conjugated family $S_c(t)=\mathcal I_c^{-1}T_c(t)\mathcal I_c$ is a $C_0$-semigroup on the real Hilbert space $L^2_c(\R^d;A)$ with generator $\Acal_c=\mathcal I_c^{-1}G_c\mathcal I_c$ and domain $\Dom(\Acal_c)=\mathcal I_c^{-1}\Dom(G_c)$. The norm identity follows from the isometry of $\mathcal I_c$.

For the inclusion, let $Y\in \Dom(\Gcal)\cap L^2_c(\R^d;A)$. By Theorem~\ref{thm:distributed-branch-splitting},
\[
\mathcal I_c(\Gcal Y)=G_c\mathcal I_c(Y).
\]
Hence $\mathcal I_c(Y)\in \Dom(G_c)$, so $Y\in \Dom(\Acal_c)$ and
\[
\Acal_c Y = \mathcal I_c^{-1}G_c\mathcal I_c(Y)=\Gcal Y.
\]
Therefore $\left.\Gcal\right|_{\Dom(\Gcal)\cap L^2_c(\R^d;A)} \subseteq \Acal_c$.
\end{proof}

The practical meaning is that one may compute with the scalar complex semigroup \(T_c(t)\), but the physical branch remains the real subspace \(L^2_c(\R^d;A)\). The global real-coordinate normal form of Proposition~\ref{prop:realified-complex-branch-normal-form} supplies the transport convention; the present proposition records the corresponding semigroup and generator after that transport.


\paragraph*{Notation discipline for the complex branch.}
From this point onward, statements about $T_c(t)$ and $G_c$ are statements on the \emph{complex} Hilbert space $L^2(\R^d;\C)$. Statements about $S_c(t)$ and $\Acal_c$ are statements on the \emph{real} branch space $L^2_c(\R^d;A)$ obtained by transport through $\mathcal I_c$. Whenever the paper returns to the full flow on $L^2(\R^d;A)$, the ambient space is again real. We use the symbol $Y_c$ only for the real $A_c$-valued branch component $Y_c=\Pc Y\in L^2_c(\R^d;A)$, and we write
\[
h_c:=\mathcal I_cY_c\in L^2(\R^d;\C)
\]
for its complex scalar coordinate. Expressions such as \(\lambda_c\widehat h_c\) always refer to the global real-coordinate model of Proposition~\ref{prop:realified-complex-branch-normal-form}. This convention will be used without further comment in Sections~5--8.

At this point the analytic ingredient needed for reconstruction is available. The intrinsic branch decomposition from Section~3 has been promoted to branchwise $C_0$-semigroup evolution: each reduced branch carries a well-posed $C_0$-semigroup, the corresponding operator-norm envelopes are explicit, and the special status of the complex branch has been fixed once and for all through a realification map. Section~5 can therefore turn from branchwise semigroup theory to the full question that motivated Paper~5 from the start: how to reconstruct the $A$-valued flow exactly from its reduced branches.



\section{Reconstruction of the Full Distributed Flow}

\subsection{Reconstruction problem}

Sections~3 and~4 provide the two ingredients from which the full distributed flow should be rebuilt: an exact branch decomposition of the state space and a well-posed reduced semigroup on each branch. The remaining question is whether these pieces recombine into a single strongly continuous semigroup on the original real Hilbert space $L^2(\R^d;A)$, and if so how the true generator of that semigroup is related to the prototype operator
\[
\Gcal = I_A\otimes a(D)+\Lu\otimes b(D)
\]
with its natural multiplier domain.

This is the point at which the paper must make the operator domains explicit. In the scalar and abstract operator settings of the previous papers, reconstruction could be stated formally once the branchwise objects had been identified. In the present multiplier-realization setting, however, one has to identify the closed symbol-sum realization, prove that the componentwise strong prototype is a graph core for it, check that the recombined operator family is well-defined on the function-space state space, and identify the true generator of the resulting $C_0$-semigroup. Only after these domain and reconstruction steps have been completed is it legitimate to say that the abstract bridge of Paper~4 has landed on a concrete translation-invariant multiplier class.

The reconstruction question is also the place where the special role of the complex branch becomes unavoidable. The plus and minus branches may be recombined directly as real scalar semigroups, but the complex branch is computed through the complex semigroup $T_c(t)$ and then pulled back to the real branch subspace through the realification map $\mathcal I_c$. The correct full flow is therefore not obtained by a naive sum of three real multipliers; it is obtained by summing the two real scalar branches together with the realified complex branch.

\paragraph*{Standard graph-closure proof convention.}
The graph-core theorem below uses ordinary $L^2$ multiplier cutoff approximations on finite-symbol level sets. It is written out before the reconstruction theorem because the branch-adapted domains are easy to confuse: the componentwise strong prototype domain, the relative symbol-sum maximal domain, and the reconstructed generator domain are not the same object before closure. The cutoff proof is standard; its role is to certify the exact domain comparison required by the branch bridge.

\subsection{Branchwise recombination formula}

Let $Y\in L^2(\R^d;A)$ and write its branch decomposition as
\[
Y = Y_+ + Y_- + Y_c,
\qquad
Y_+ := \Pplus Y,\qquad Y_- := \Pminus Y,\quad Y_c := \Pc Y,
\]
where the plus and minus components are selected by $\Pplus$ and $\Pminus$, respectively. More explicitly,
\[
Y = f_+\Pplus + f_-\Pminus + Y_c,
\qquad
f_+ = \mathcal I_+(Y_+),\quad f_- = \mathcal I_-(Y_-),
\]
and set
\[
h_c:=\mathcal I_c(Y_c)\in L^2(\R^d;\C)
\]
for the complex scalar coordinate of the real $A_c$-branch. Motivated by the reduced semigroup theory of Section~4, we define for each $t\ge 0$
\begin{equation}\label{eq:full-recombination-formula}
\mathcal T(t)Y
:=
\bigl(T_+(t)f_+\bigr)\Pplus
+
\bigl(T_-(t)f_-\bigr)\Pminus
+
\mathcal I_c^{-1}T_c(t)h_c.
\end{equation}
Equivalently, using the branch projectors and branch identifications,
\[
\mathcal T(t)
=
\mathcal I_+^{-1}T_+(t)\mathcal I_+\Pplus
+
\mathcal I_-^{-1}T_-(t)\mathcal I_-\Pminus
+
\mathcal I_c^{-1}T_c(t)\mathcal I_c\Pc.
\]
The point of this formula is that each summand acts only on its own branch and vanishes on the other two branches. Thus the three reduced semigroups do not compete or interfere at the reconstruction stage; they are assembled along mutually orthogonal branch channels. Our goal in this section is not to reconstruct the flow directly from the formal prototype operator alone, but to show that once the reduced branch semigroups are available, they determine a prescribed branch-coordinate full semigroup on the real function-space state space. The reconstruction theorem below is deliberately separated from the later internal-uniqueness corollary: reconstruction is the main assertion, while uniqueness is only a statement inside the already imposed branch-compatible class.

For later use, define the branch identification map
\[
\mathfrak I: L^2(\R^d;A) \longrightarrow \LtwoR\oplus \LtwoR\oplus \LtwoC,
\qquad
\mathfrak I Y := \bigl(\mathcal I_+(\Pplus Y),\, \mathcal I_-(\Pminus Y),\, \mathcal I_c(\Pc Y)\bigr).
\]
Here $\LtwoR\oplus\LtwoR\oplus\LtwoC$ is understood as a real Hilbert direct sum, with the third component $\LtwoC$ viewed through its underlying real Hilbert space structure. By the orthogonal branch splitting and the real-linear isometries $\mathcal I_+$, $\mathcal I_-$, and $\mathcal I_c$, this map is a real-linear isometric isomorphism onto that direct-sum branch model space.

\begin{definition}[Reduced real Hilbert direct-sum notation]
\label{def:reduced-direct-sum-notation}
On the branch-coordinate model
\[
\Xred:=\LtwoR\oplus\LtwoR\oplus (\LtwoC)_{\R},
\]
where \((\LtwoC)_{\R}\) denotes the complex Hilbert space \(\LtwoC\) regarded as a real Hilbert space, we write
\[
\Gred:=G_+\oplus G_-\oplus (G_c)_{\R}.
\]
Here \(G_+=\lambda_+(D)\) and \(G_-=\lambda_-(D)\) act on the real branch spaces, while \((G_c)_{\R}\) denotes the closed complex multiplier \(G_c=\lambda_c(D)\) viewed as a real-linear operator on \((\LtwoC)_{\R}\). Its domain is
\[
\Dom(\Gred)=\Dom(G_+)\oplus\Dom(G_-)\oplus\Dom((G_c)_{\R}),
\qquad
\Dom((G_c)_{\R})=\Dom(G_c)
\]
with the scalar complex domain regarded as a real linear subspace. Likewise, for \(t\ge0\),
\[
\Tred(t):=T_+(t)\oplus T_-(t)\oplus (T_c(t))_{\R}.
\]
Whenever a frequency-side reduced tuple display involving \(\lambda_+,\lambda_-,\lambda_c\) is used informally, it is only shorthand for this reduced direct-sum representation after applying the branch-coordinate isomorphism \(\mathfrak I\). It is not a diagonalization of the original real \(A\)-valued fiber.
\end{definition}

Because the norm on $L^2(\R^d;A)$ was chosen to be branch-adapted, the boundedness of $\mathcal T(t)$ is immediate from the boundedness of the reduced semigroups. Indeed, writing $Y=f_+\Pplus+f_-\Pminus+Y_c$ and $h_c=\mathcal I_cY_c$,
\[
\norm{\mathcal T(t)Y}_{L^2(\R^d;A)}^2
=
\norm{T_+(t)f_+}_{L^2}^2
+
\norm{T_-(t)f_-}_{L^2}^2
+
\norm{T_c(t)h_c}_{L^2(\C)}^2,
\]
so the branchwise growth bounds from Proposition~\ref{prop:branchwise-growth} already control the full reconstructed operator. The real work is therefore no longer boundedness but the semigroup property, strong continuity, and generator identification.

\subsection{Main theorem: prescribed branch-coordinate reconstruction}

We now state the third main theorem of the paper. This is a prescribed direct-sum reconstruction theorem: once the reduced branch semigroups have been constructed, they assemble into a concrete full semigroup on the original real state space. The theorem itself contains no global uniqueness assertion. The separate internal-uniqueness statement is recorded afterward as Corollary~\ref{cor:internal-uniqueness-prescribed-branch-class}, and the broader classification exclusions are those recorded in Subsection~\ref{subsec:scope-and-nonclaims}.

\begin{theorem}[Prescribed direct-sum reconstruction theorem]\label{thm:prescribed-multiplier-flow-reconstruction}
Assume that \((a,b)\) satisfies the standing multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}. Let
$G_+$, $G_-$, and $G_c$, and $T_+(t)$, $T_-(t)$, and $T_c(t)$, be the reduced
branch generators and semigroups constructed in
Theorem~\ref{thm:reduced-semigroup-generation}. Define $\{\mathcal T(t)\}_{t\ge 0}$ on $L^2(\R^d;A)$ by the branchwise recombination formula \eqref{eq:full-recombination-formula}.

This theorem concerns only the semigroup obtained from the prescribed branch-coordinate direct-sum reconstruction. It constructs the reconstructed flow and identifies its generator; it does not classify all closed full-side realizations of the formal expression.

Then the following statements are true.
\begin{enumerate}[label=\textup{(\roman*)}]
    \item $\{\mathcal T(t)\}_{t\ge 0}$ is a strongly continuous semigroup on the real Hilbert space $L^2(\R^d;A)$.

    \item Each branch subspace $L^2_+(\R^d;A)$, $L^2_-(\R^d;A)$, and $L^2_c(\R^d;A)$ is invariant under $\mathcal T(t)$ for every $t\ge 0$.

    \item The infinitesimal generator of $\{\mathcal T(t)\}_{t\ge 0}$ is the branchwise realization $\Acal$ defined by
    \[
    \Dom(\Acal)
    :=
    \Bigl\{Y\in L^2(\R^d;A):
    \mathcal I_+(\Pplus Y)\in\Dom(G_+),\ 
    \mathcal I_-(\Pminus Y)\in\Dom(G_-),\ 
    \mathcal I_c(\Pc Y)\in\Dom(G_c)
    \Bigr\},
    \]
    and, for $Y\in\Dom(\Acal)$,
    \[
    \Acal Y
    :=
    \mathcal I_+^{-1}\bigl(G_+\mathcal I_+(\Pplus Y)\bigr)
    +
    \mathcal I_-^{-1}\bigl(G_-\mathcal I_-(\Pminus Y)\bigr)
    +
    \mathcal I_c^{-1}\bigl(G_c\mathcal I_c(\Pc Y)\bigr).
    \]

    \item For every $t\ge 0$,
    \begin{equation}\label{eq:full-growth-bound}
    \norm{\mathcal T(t)}_{\mathcal L(L^2(\R^d;A))}
    \le
    \max\{e^{\omega_+ t},\ e^{\omega_- t},\ e^{\omega_c t}\}.
    \end{equation}
\end{enumerate}
\end{theorem}

\begin{proof}
We proceed in four steps.

\smallskip
\noindent\emph{Step 1: boundedness and semigroup property.}
Let $Y\in L^2(\R^d;A)$ and write
\[
Y=f_+\Pplus + f_-\Pminus + Y_c
\]
with $f_+=\mathcal I_+(\Pplus Y)$, $f_-=\mathcal I_-(\Pminus Y)$, and $Y_c=\Pc Y$. By the branch-adapted norm and Proposition~\ref{prop:branchwise-growth},
\[
\norm{\mathcal T(t)Y}_{L^2(\R^d;A)}^2
\le
e^{2\omega_+ t}\norm{f_+}_{L^2}^2
+
e^{2\omega_- t}\norm{f_-}_{L^2}^2
+
e^{2\omega_c t}\norm{\mathcal I_cY_c}_{L^2(\C)}^2.
\]
Since
\[
\norm{Y}_{L^2(\R^d;A)}^2
=
\norm{f_+}_{L^2}^2+\norm{f_-}_{L^2}^2+\norm{\mathcal I_cY_c}_{L^2(\C)}^2,
\]
we obtain the operator bound \eqref{eq:full-growth-bound}. In particular $\mathcal T(t)$ is bounded for every $t\ge 0$.

To prove the semigroup property, note that the three summands in \eqref{eq:full-recombination-formula} act independently on the three branch components. Since $T_+$, $T_-$, and $T_c$ are semigroups by Theorem~\ref{thm:reduced-semigroup-generation}, one immediately gets
\[
\mathcal T(t+s)Y = \mathcal T(t)\mathcal T(s)Y,
\qquad
\mathcal T(0)=I.
\]

\smallskip
\noindent\emph{Step 2: strong continuity.}
Let $Y=f_+\Pplus+f_-\Pminus+Y_c$ and set $h_c:=\mathcal I_cY_c$. Then
\[
\norm{\mathcal T(t)Y-Y}_{L^2(\R^d;A)}^2
=
\norm{T_+(t)f_+-f_+}_{L^2}^2
+
\norm{T_-(t)f_--f_-}_{L^2}^2
+
\norm{T_c(t)h_c-h_c}_{L^2(\C)}^2.
\]
Each term tends to zero as $t\downarrow 0$ because $T_+$, $T_-$, and $T_c$ are $C_0$-semigroups on their respective reduced spaces. Hence $\{\mathcal T(t)\}_{t\ge 0}$ is a $C_0$-semigroup on $L^2(\R^d;A)$.

\smallskip
\noindent\emph{Step 3: invariance of the branch subspaces.}
The recombination formula shows directly that the plus branch of $\mathcal T(t)Y$ depends only on the plus branch of $Y$, and similarly for the minus and complex branches. Therefore
\[
\mathcal T(t)L^2_+(\R^d;A)\subseteq L^2_+(\R^d;A),
\qquad
\mathcal T(t)L^2_-(\R^d;A)\subseteq L^2_-(\R^d;A),
\qquad
\mathcal T(t)L^2_c(\R^d;A)\subseteq L^2_c(\R^d;A).
\]
This proves part \textup{(ii)}.

\smallskip
\noindent\emph{Step 4: generator identification.}
Let $\widetilde{\Acal}$ denote the branchwise operator appearing in part \textup{(iii)}. We prove that $\widetilde{\Acal}$ is exactly the infinitesimal generator of $\{\mathcal T(t)\}_{t\ge 0}$.

First let $Y\in\Dom(\widetilde{\Acal})$. Write
\[
Y=Y_+ + Y_- + Y_c,
\qquad
Y_+:=\Pplus Y,\qquad Y_-:=\Pminus Y,
\quad
Y_c:=\Pc Y,
\]
so that $f_+:=\mathcal I_+(Y_+)\in\Dom(G_+)$, $f_-:=\mathcal I_-(Y_-)\in\Dom(G_-)$, and $h_c:=\mathcal I_c(Y_c)\in\Dom(G_c)$. By Theorem~\ref{thm:reduced-semigroup-generation},
\[
\lim_{t\downarrow 0}\left\|\frac{T_+(t)f_+ - f_+}{t} - G_+f_+\right\|_{L^2}=0,
\]
with analogous limits on the minus and complex branches. Pulling these limits back through the branch identifications and using the branchwise norm decomposition yields
\[
\lim_{t\downarrow 0}
\left\|\frac{\mathcal T(t)Y-Y}{t}-\widetilde{\Acal}Y\right\|_{L^2(\R^d;A)}=0.
\]
Hence $Y$ belongs to the generator domain of $\{\mathcal T(t)\}_{t\ge 0}$ and the generator acts as $\widetilde{\Acal}$ on $Y$.

Conversely, let $Y$ belong to the generator domain of $\{\mathcal T(t)\}_{t\ge 0}$. Then the difference quotient
\[
\frac{\mathcal T(t)Y-Y}{t}
\]
converges in $L^2(\R^d;A)$ as $t\downarrow 0$. Applying the bounded projectors $\Pplus$, $\Pminus$, $\Pc$ and then the bounded identifications $\mathcal I_+$, $\mathcal I_-$, $\mathcal I_c$, one sees that the reduced difference quotients converge in their respective branch spaces. Since on each branch the reduced evolution is exactly $T_+(t)$, $T_-(t)$, or $T_c(t)$, Theorem~\ref{thm:reduced-semigroup-generation} implies that
\[
\mathcal I_+(\Pplus Y)\in\Dom(G_+),
\qquad
\mathcal I_-(\Pminus Y)\in\Dom(G_-),
\qquad
\mathcal I_c(\Pc Y)\in\Dom(G_c).
\]
Thus $Y\in\Dom(\widetilde{\Acal})$, and the generator action coincides with the branchwise formula from part \textup{(iii)}. Therefore the generator of $\{\mathcal T(t)\}_{t\ge 0}$ is exactly $\widetilde{\Acal}$, which proves part \textup{(iii)}.


\end{proof}

\begin{corollary}[Internal uniqueness inside the prescribed branch-compatible class]
\label{cor:internal-uniqueness-prescribed-branch-class}
Let \(\{S(t)\}_{t\ge0}\) be a strongly continuous semigroup on
\(L^2(\R^d;A)\). Assume that, for every \(t\ge0\) and every
\(\sigma\in\{+,-,c\}\),
\[
S(t)L^2_\sigma(\R^d;A)\subseteq L^2_\sigma(\R^d;A),
\]
and that the transported branch actions agree with the prescribed reduced
semigroups:
\[
\mathcal I_+S(t)\Pplus=T_+(t)\mathcal I_+\Pplus,\qquad
\mathcal I_-S(t)\Pminus=T_-(t)\mathcal I_-\Pminus,\qquad
\mathcal I_cS(t)\Pc=T_c(t)\mathcal I_c\Pc.
\]
Then
\[
S(t)=\mathcal T(t)\qquad\text{for every }t\ge0.
\]
\end{corollary}

\begin{proof}
Every \(Y\in L^2(\R^d;A)\) decomposes as
\[
Y=Y_++Y_-+Y_c,\qquad
Y_+:=\Pplus Y,\quad Y_-:=\Pminus Y,\quad Y_c:=\Pc Y.
\]
By branch invariance, each \(S(t)Y_\sigma\) remains in the corresponding
branch. The transported-action assumptions give
\[
\mathcal I_+S(t)Y_+=T_+(t)\mathcal I_+(Y_+),\qquad
\mathcal I_-S(t)Y_-=T_-(t)\mathcal I_-(Y_-),\qquad
\mathcal I_cS(t)Y_c=T_c(t)\mathcal I_c(Y_c).
\]
Pulling these identities back through the branch-coordinate maps gives
\[
S(t)Y_+=\mathcal I_+^{-1}T_+(t)\mathcal I_+(Y_+),\qquad
S(t)Y_-=\mathcal I_-^{-1}T_-(t)\mathcal I_-(Y_-),\qquad
S(t)Y_c=\mathcal I_c^{-1}T_c(t)\mathcal I_c(Y_c).
\]
After summing over the three branches, the right-hand side is exactly the
recombination formula defining \(\mathcal T(t)Y\). Hence
\(S(t)Y=\mathcal T(t)Y\) for every \(Y\), so \(S(t)=\mathcal T(t)\).
\end{proof}

\begin{remark}[No global realization uniqueness is asserted]
\label{rem:no-global-realization-uniqueness}
Corollary~\ref{cor:internal-uniqueness-prescribed-branch-class} is an
internal uniqueness statement only. It does not classify all closed
realizations of the formal expression
\[
\IA\otimes a(D)+\Lu\otimes b(D).
\]
The conclusion holds only after branch invariance and agreement with the three
prescribed reduced semigroups have been imposed. Thus the result is a
uniqueness statement inside a fixed branch-compatible reconstruction class, not
a global uniqueness theorem for arbitrary domains, arbitrary closed extensions,
or arbitrary full-side semigroup realizations of the formal operator.
\end{remark}


\begin{lemma}[The complex-branch block exponential]
\label{lem:complex-branch-block-exponential}
Let
\[
S(\xi)=a(\xi)\IAC+b(\xi)\Lu
\]
be the finite-dimensional coefficient symbol. On the ordered real branch basis
\((\Pc,J)\), the restriction of \(\Lu\) to the \(c\)-block is
\[
L_c=
\begin{pmatrix}
0&-1\\
1&0
\end{pmatrix},
\qquad L_c^2=-I.
\]
Consequently the restriction of \(S(\xi)\) to the complexified \(c\)-block is
\[
S_c(\xi)=a(\xi)I+b(\xi)L_c
=
\begin{pmatrix}
a(\xi)&-b(\xi)\\
b(\xi)&a(\xi)
\end{pmatrix},
\]
and its exponential is
\[
\exp(tS_c(\xi))
=
e^{t a(\xi)}
\begin{pmatrix}
\cos(t b(\xi))&-\sin(t b(\xi))\\
\sin(t b(\xi))&\cos(t b(\xi))
\end{pmatrix},
\]
where the scalar entire functions are evaluated at the measurable scalar
symbol \(b(\xi)\).

Equivalently, if \(Y_c=f\Pc+gJ\) with \(f,g\in L^2(\R^d;\R)\), and if
\(h_c=f+ig=\mathcal I_cY_c\), then the same matrix identity gives
\[
\widehat{\mathcal I_c\,\mathcal F^{-1}
\bigl(\exp(tS_c(\xi))\widehat{Y_c}(\xi)\bigr)}(\xi)
=
e^{t(a(\xi)+ib(\xi))}\widehat h_c(\xi).
\]
This is a global real-coordinate consequence of the \((\Pc,J)\)-block matrix
formula. It is not a pointwise complex-linear similarity
\(\C\Pc\oplus\C J\simeq\C\).
\end{lemma}

\begin{proof}
The identities \(\Lu\Pc=J\) and \(\Lu J=-\Pc\) give the displayed matrix
for \(L_c\). Since \(I\) and \(L_c\) commute and \(L_c^2=-I\), the power series
for the exponential splits into even and odd powers, giving
\[
\exp(t(aI+bL_c))=e^{ta}\bigl(\cos(tb)I+\sin(tb)L_c\bigr),
\]
which is exactly the displayed \(2\times2\) matrix.

For the coordinate statement, write the Fourier-side coefficient column of
\(Y_c\) in the basis \((\Pc,J)\) as \((\widehat f,\widehat g)^T\). The matrix
formula gives
\[
\begin{pmatrix}
\widehat f_t\\
\widehat g_t
\end{pmatrix}
=
e^{ta}
\begin{pmatrix}
\cos(tb)&-\sin(tb)\\
\sin(tb)&\cos(tb)
\end{pmatrix}
\begin{pmatrix}
\widehat f\\
\widehat g
\end{pmatrix}.
\]
Therefore
\[
\widehat f_t+i\widehat g_t
=e^{ta}\bigl(\cos(tb)+i\sin(tb)\bigr)(\widehat f+i\widehat g)
=e^{t(a+ib)}\widehat h_c.
\]
The calculation takes place in the global real branch coordinate
\(h_c=f+ig\); no inverse of \(\mathcal I_c\) on the complexified fibre
\(\C\Pc\oplus\C J\) is used.
\end{proof}

\begin{proposition}[Full matrix-symbol formula for the reconstructed flow]
\label{prop:full-matrix-symbol-semigroup-formula}
Assume that \((a,b)\) satisfies the standing multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}. Let
\(\{\mathcal T(t)\}_{t\ge0}\) be the reconstructed semigroup from
Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction}. For each
\(t\ge0\), define the finite-dimensional Fourier-side symbol
\[
S(\xi):=a(\xi)\IAC+b(\xi)\Lu
\]
on the complexified coefficient fibre \(\AC\), and set
\[
E_t(\xi):=\exp\bigl(tS(\xi)\bigr).
\]
Then, for every \(Y\in L^2(\R^d;A)\),
\begin{equation}\label{eq:full-matrix-symbol-semigroup-formula}
\widehat{\mathcal T(t)Y}(\xi)=E_t(\xi)\widehat Y(\xi)
\qquad\text{for a.e. }\xi\in\R^d.
\end{equation}
The identity is an equality for the matrix multiplier on the Fourier-side
complexified coefficient fibre \(\AC\), restricted to the Hermitian real-state
subspace in the sense fixed by
Remark~\ref{rem:standing-interpretation-box-paper5}. In particular, the scalar
notation \(e^{t\lambda_c}\) for the \(c\)-branch is used only after the global
real branch-coordinate identification.

Equivalently, under the branch-coordinate real Hilbert isomorphism
\(\mathfrak I\),
\[
\mathfrak I\,\mathcal T(t)\,\mathfrak I^{-1}
=
\Tred(t)
=
T_+(t)\oplus T_-(t)\oplus (T_c(t))_{\R},
\]
and the corresponding Fourier-side branch multipliers are
\[
e^{t\lambda_+(\xi)},\qquad
 e^{t\lambda_-(\xi)},\qquad
 e^{t\lambda_c(\xi)}.
\]
Thus the prescribed branchwise reconstruction is precisely the \(L^2\)
realization, on the real-state subspace, of the full finite-dimensional matrix
multiplier generated by the symbol \(a(\xi)\IAC+b(\xi)\Lu\).
\end{proposition}

\begin{proof}
The branch projectors \(\Pplus\), \(\Pminus\), and \(\Pc\) reduce the
coefficient operator \(\Lu\). On the plus and minus branches,
\[
\Lu\Pplus=\Pplus,
\qquad
\Lu\Pminus=-\Pminus,
\]
and hence
\[
S(\xi)\Pplus=(a(\xi)+b(\xi))\Pplus=\lambda_+(\xi)\Pplus,
\]
\[
S(\xi)\Pminus=(a(\xi)-b(\xi))\Pminus=\lambda_-(\xi)\Pminus.
\]
Therefore
\[
E_t(\xi)\Pplus=e^{t\lambda_+(\xi)}\Pplus,
\qquad
E_t(\xi)\Pminus=e^{t\lambda_-(\xi)}\Pminus.
\]

On the \(c\)-branch we do not use \(\mathcal I_c\) as a pointwise
complex-fibre similarity. Instead, Lemma~\ref{lem:complex-branch-block-exponential}
computes the restriction of \(S(\xi)\) directly on the ordered real basis
\((\Pc,J)\). Thus the \(c\)-block of the full matrix symbol is
\[
S_c(\xi)=
\begin{pmatrix}
a(\xi)&-b(\xi)\\
b(\xi)&a(\xi)
\end{pmatrix},
\]
and
\[
\exp(tS_c(\xi))
=
e^{t a(\xi)}
\begin{pmatrix}
\cos(t b(\xi))&-\sin(t b(\xi))\\
\sin(t b(\xi))&\cos(t b(\xi))
\end{pmatrix}.
\]
If \(Y_c=f\Pc+gJ\), \(f,g\in L^2(\R^d;\R)\), and
\(h_c=\mathcal I_cY_c=f+ig\), the same block calculation gives
\[
\widehat{\mathcal I_c\mathcal T(t)Y_c}(\xi)
=
e^{t(a(\xi)+ib(\xi))}\widehat h_c(\xi)
=
e^{t\lambda_c(\xi)}\widehat h_c(\xi).
\]
This last scalar formula is a consequence of the \(2\times2\) matrix-block
identity in global real branch coordinates. It is not a pointwise complex-linear
diagonalization of \(\C\Pc\oplus\C J\).

Together with the plus and minus branch computations, this shows that after
applying \(\mathfrak I\) the matrix multiplier \(E_t(\xi)\), restricted to the
Hermitian real-state subspace, agrees with the direct sum of the three reduced
multipliers
\[
e^{t\lambda_+(\xi)}
\oplus
 e^{t\lambda_-(\xi)}
\oplus
 e^{t\lambda_c(\xi)},
\]
where the third block is regarded as a real-linear operator on
\((\LtwoC)_{\R}\). These are precisely the Fourier multipliers defining
\(T_+(t)\), \(T_-(t)\), and \(T_c(t)\). Hence the operator obtained from the full
matrix-symbol formula coincides with the branchwise recombination formula
defining \(\mathcal T(t)\) in
Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction}. The branchwise
bounds from the standing package imply that the three scalar blocks are bounded
on their reduced \(L^2\) spaces, so the matrix-symbol multiplier is bounded on
the real-state subspace. This proves \eqref{eq:full-matrix-symbol-semigroup-formula}.
\end{proof}

\begin{remark}[No pointwise scalarization of the complexified \(c\)-fibre]
\label{rem:no-pointwise-c-fibre-scalarization}
The \(c\)-branch scalar multiplier \(e^{t\lambda_c}\) is a scalar
representation only in the global real branch coordinates
\(f\Pc+gJ\leftrightarrow f+ig\). It should not be read as a complex-linear
diagonalization of the fibre \(\C\Pc\oplus\C J\). Indeed, the formal map
\[
p\Pc+qJ\longmapsto p+iq
\]
is not injective on \(\C\Pc\oplus\C J\); for example,
\[
i\Pc-J\longmapsto i+i(-1)=0.
\]
The full Fourier-side statement is therefore the matrix identity of
Lemma~\ref{lem:complex-branch-block-exponential}, namely
\[
S_c(\xi)=
\begin{pmatrix}
a(\xi)&-b(\xi)\\
b(\xi)&a(\xi)
\end{pmatrix},
\]
restricted to the Hermitian real-state subspace, while the scalar symbol
\(\lambda_c=a+ib\) is only the corresponding paired-frequency real-coordinate
description.
\end{remark}

\begin{remark}[Scope of the prescribed reconstruction]
\label{rem:scope-branch-reconstruction-uniqueness}
Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction} constructs the
prescribed branch-coordinate semigroup and identifies its generator. The only
uniqueness assertion in this section is the separate
Corollary~\ref{cor:internal-uniqueness-prescribed-branch-class}; its scope is
limited by Remark~\ref{rem:no-global-realization-uniqueness}. Classification
questions for other closed realizations belong to the global scope registry in
Subsection~\ref{subsec:scope-and-nonclaims}.
\end{remark}

\begin{corollary}[Strong prototype sub-operator]\label{cor:strong-suboperator}
Assume that \((a,b)\) satisfies the standing multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}. Let $\Gcal$ be the
componentwise strong realization from Section~2 and let $\Acal$ be the
branchwise reconstructed generator from
Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction}. Then
\[
\Gcal\subseteq \Acal.
\]
In particular, for every $Y\in\Dom(\Gcal)$ one has $\Acal Y=\Gcal Y$.
\end{corollary}

\begin{proof}
Take $Y\in\Dom(\Gcal)$. By Theorem~\ref{thm:distributed-branch-splitting}, the branch components satisfy
\[
\mathcal I_+(\Pplus Y)\in\Dom(G_+),
\qquad
\mathcal I_-(\Pminus Y)\in\Dom(G_-),
\qquad
\mathcal I_c(\Pc Y)\in\Dom(G_c),
\]
and the action of $\Gcal$ agrees branchwise with the reduced operators on those components. Hence $Y\in\Dom(\Acal)$ and the branchwise formula for $\Acal Y$ collapses exactly to $\Gcal Y$.
\end{proof}


\begin{remark}[The three full operators and their domains]
\label{rem:three-full-operators-at-a-glance}
Three closely related operators are used in the sequel. They have the same
formal multiplier origin, but they are not the same object before the graph-core
identification is proved. The distinction is as follows.
\begin{center}
\small
\begin{tabular}{|p{0.16\linewidth}|p{0.37\linewidth}|p{0.37\linewidth}|}
\hline
\textbf{Operator} & \textbf{Domain condition} & \textbf{Meaning} \\
\hline
$\Gcal$ & Requires the two componentwise quantities
$a\widehat Y$ and $b\widehat Y$ to belong separately to $L^2$ after the
finite-dimensional coefficient operations are applied. & The raw componentwise
strong prototype. It is the literal operator obtained by applying the two
multiplier pieces separately. \\
\hline
$\Acal$ & Requires branchwise membership
$\lambda_+\widehat f_+\in L^2$, $\lambda_-\widehat f_-\in L^2$, and
$\lambda_c\widehat h_c\in L^2$ under the branch-coordinate map $\mathfrak I$. &
The generator reconstructed from the three reduced branch semigroups. \\
\hline
$\Mss$ & Requires only the combined symbol-sum expression
$(a\IAC+b\Lu)\widehat Y$ to belong to $L^2(\R^d;\AC)$. & The candidate relative symbol-sum Fourier-multiplier operator for the fixed pair $(a,b)$. Its real-state well-definedness and closedness are proved immediately below before it is used as a closed realization. \\
\hline
\end{tabular}
\end{center}
Thus $\Gcal$ is the smallest raw-domain object, $\Acal$ is the branchwise
reconstructed generator, and $\Mss$ is, at this stage, the candidate relative
symbol-sum multiplier operator. Only after the density and closedness result
below do we refer to $\Mss$ as a closed relative symbol-sum realization. The
later identities
\[
\overline{\Gcal}^{\,\mathrm{graph}}=\Acal=\Mss
\]
are therefore substantive domain statements, not notational redundancies.
\end{remark}

\begin{remark}[Notation ledger for formal, raw, closed, and flow objects]
\label{rem:notation-ledger-formal-raw-closed-flow}
The notation in the graph-core and reconstruction sections separates five
objects that should not be conflated.
\begin{center}
\small
\begin{tabular}{|p{0.16\linewidth}|p{0.30\linewidth}|p{0.44\linewidth}|}
\hline
\textbf{Symbol} & \textbf{Role} & \textbf{Status in the paper} \\
\hline
$\Gform$ & Formal expression & The algebraic differential or multiplier expression
$\IA\otimes a(D)+\Lu\otimes b(D)$ before a domain is fixed. \\
\hline
$\Gcal$ & Raw componentwise strong prototype & The operator obtained by requiring the separate componentwise multiplier pieces involving $a(D)$ and $b(D)$ to exist. This is the smallest domain object in the comparison. \\
\hline
$\Mss$ & Maximal finite-a.e. symbol-sum multiplier realization & The closed Fourier multiplier defined by the combined symbol $a\IAC+b\Lu$ on its maximal $L^2$ domain. The graph-core theorem identifies it as the closure target of $\Gcal$. \\
\hline
$\Acal$ & Reconstructed branchwise generator & The generator obtained from the plus, minus, and global real-coordinate complex branch semigroups. After the main identification theorem, $\Acal=\Mss$. \\
\hline
$\mathcal T(t)$ & Reconstructed semigroup & The branchwise assembled $C_0$-semigroup on $L^2(\R^d;A)$ whose generator is $\Acal$ and whose full Fourier-side representation is the matrix multiplier $\exp(t(a\IAC+b\Lu))$. \\
\hline
\end{tabular}
\end{center}
Thus the formal expression, the raw-domain prototype, the maximal symbol-sum
realization, the reconstructed generator, and the reconstructed flow are kept
separate until the relevant graph-core and generator-identification results have
proved the claimed equalities.
\end{remark}


\subsection{Relative symbol-sum realization and the main graph-core theorem}
\label{subsec:relative-symbol-sum-graph-core-main}

\begin{definition}[Candidate relative symbol-sum Fourier-multiplier operator]\label{def:relative-symbol-sum-realization}
Define the candidate operator $\Mss$ on $L^2(\R^d;A)$ by
\[
\Dom(\Mss)
:=
\Bigl\{Y\in L^2(\R^d;A):
\bigl(a(\cdot)\IAC+b(\cdot)\Lu\bigr)\widehat Y(\cdot)\in L^2(\R^d;\AC)\Bigr\},
\]
and, for $Y\in\Dom(\Mss)$,
\[
\widehat{\Mss Y}(\xi)
:=
\bigl(a(\xi)\IAC+b(\xi)\Lu\bigr)\widehat Y(\xi)
\qquad\text{for a.e. }\xi\in\R^d.
\]
This definition fixes only the Fourier-side symbol-sum operator determined by the multiplier pair $(a,b)$ and by the branch-coordinate architecture. Its domain is the set on which the combined multiplier
\[
(a\IAC+b\Lu)\widehat Y
\]
belongs to $L^2(\R^d;\AC)$. At this stage no closedness property is assumed, and no classification of arbitrary closed extensions of the componentwise strong prototype or arbitrary $C_0$-semigroup generators associated with the formal expression is asserted.

At the level of this definition, the displayed Fourier-side formula is read initially as an
$L^2(\R^d;\AC)$-valued symbol-sum formula. The next lemma verifies that, under
Assumption~\textup{(A1)}, this formula descends to the real state space
$L^2(\R^d;A)$. The following proposition then proves density and closedness;
only after that point do we call $\Mss$ a closed relative symbol-sum Fourier-multiplier realization.
\end{definition}

\begin{lemma}[Real-state well-definedness]
\label{lem:symbol-sum-real-state-well-defined}
Assume \textup{(A1)}. Let $\Mss$ be the candidate symbol-sum operator from
Definition~\ref{def:relative-symbol-sum-realization}. If $Y\in\Dom(\Mss)$, then
\[
\Phi_Y(\xi):=
\bigl(a(\xi)\IAC+b(\xi)\Lu\bigr)\widehat Y(\xi)
\]
satisfies the Fourier-side reality condition
\[
\Phi_Y(-\xi)=\cA\bigl(\Phi_Y(\xi)\bigr)
\qquad\text{for a.e. }\xi\in\R^d.
\]
Consequently,
\[
\F^{-1}\Phi_Y\in L^2(\R^d;A).
\]
Thus $\Mss$ is a well-defined real-linear operator
\[
\Mss:\Dom(\Mss)\subset L^2(\R^d;A)\longrightarrow L^2(\R^d;A).
\]
\end{lemma}

\begin{proof}
Write the intrinsic branch decomposition
\[
Y=Y_+ + Y_- + Y_c,
\qquad
Y_+=\Pplus Y,
\quad
Y_-=\Pminus Y,
\quad
Y_c=\Pc Y.
\]
The projectors $\Pplus$, $\Pminus$, and $\Pc$ commute with $\Lu$, and hence with the
Fourier-side symbol
\[
S(\xi):=a(\xi)\IAC+b(\xi)\Lu.
\]
Since $Y\in\Dom(\Mss)$, one has $S\widehat Y\in L^2(\R^d;\AC)$. Applying the bounded
finite-dimensional branch projections gives
\[
S\widehat{Y_+},
\qquad
S\widehat{Y_-},
\qquad
S\widehat{Y_c}
\in L^2(\R^d;\AC).
\]

On the plus branch, write $Y_+=f_+\Pplus$ with $f_+\in L^2(\R^d;\R)$. Then
\[
S(\xi)\widehat{Y_+}(\xi)
=
\lambda_+(\xi)\widehat f_+(\xi)\Pplus,
\qquad
\lambda_+:=a+b.
\]
Assumption~\textup{(A1)} implies
\[
\lambda_+(-\xi)=\overline{\lambda_+(\xi)}.
\]
Since $\lambda_+\widehat f_+\in L^2(\R^d;\C)$, Lemma~\ref{lem:domain-sensitive-reality-preservation}
implies that
\[
\F^{-1}(\lambda_+\widehat f_+)
\]
is real-valued. Equivalently, the plus-branch contribution to $\Phi_Y$ satisfies
Fourier-side reality symmetry and its inverse Fourier transform belongs to
$L^2(\R^d;A)$.

The minus branch is identical. Writing $Y_-=f_-\Pminus$, one has
\[
S(\xi)\widehat{Y_-}(\xi)
=
\lambda_-(\xi)\widehat f_-(\xi)\Pminus,
\qquad
\lambda_-:=a-b,
\]
and $\lambda_-(-\xi)=\overline{\lambda_-(\xi)}$. Again
Lemma~\ref{lem:domain-sensitive-reality-preservation} gives a real-valued inverse
Fourier transform for the scalar coefficient.

It remains to treat the complex branch. Write
\[
Y_c=f\Pc+gJ,
\qquad
f,g\in L^2(\R^d;\R),
\qquad
h_c:=\mathcal I_cY_c=f+ig.
\]
By Lemma~\ref{lem:realified-complex-branch-multiplier},
\[
S(\xi)\widehat{Y_c}(\xi)=p(\xi)\Pc+q(\xi)J,
\]
where
\[
p=a\widehat f-b\widehat g,
\qquad
q=b\widehat f+a\widehat g.
\]
The same lemma, using Assumption~\textup{(A1)}, gives the Hermitian symmetries
\[
p(-\xi)=\overline{p(\xi)},
\qquad
q(-\xi)=\overline{q(\xi)}.
\]
Because $S\widehat{Y_c}\in L^2(\R^d;\AC)$, the finite-dimensional equivalence of norms on
$\operatorname{span}_{\C}\{\Pc,J\}$ gives $p,q\in \LtwoC$. Therefore
\[
P:=\F^{-1}p,
\qquad
Q:=\F^{-1}q
\]
are real-valued, and
\[
\F^{-1}(p\Pc+qJ)=P\Pc+QJ\in L^2_c(\R^d;A).
\]

Combining the plus, minus, and complex branch contributions proves that
\[
\F^{-1}\Phi_Y\in L^2(\R^d;A).
\]
Equivalently, $\Phi_Y$ satisfies the Fourier-side reality condition
$\Phi_Y(-\xi)=\cA(\Phi_Y(\xi))$ for almost every $\xi$. Real-linearity follows from the
real-linearity of the domain condition and of the symbol-sum formula.
\end{proof}

\begin{lemma}[Symmetry of symbol-sum matrix cutoffs]
\label{lem:symbol-sum-matrix-cutoff-symmetry}
Assume \textup{(A1)}, and set
\[
S(\xi):=a(\xi)\IAC+b(\xi)\Lu
\]
as an operator on the complexified coefficient fibre \(\AC\). After replacing
\(a\) and \(b\) by the harmless Hermitian representatives supplied by
Lemma~\ref{lem:hermitian-representatives-symmetric-cutoffs}, there is a
symmetric null set \(N=-N\) such that, for every \(\xi\notin N\),
\[
S(-\xi)=\cA S(\xi)\cA^{-1},
\]
where \(\cA\) is the canonical conjugation on \(\AC\). Consequently, with
respect to the fixed branch-adapted fibre Hilbert norm,
\[
\|S(-\xi)\|_{\mathcal L(\AC)}
=
\|S(\xi)\|_{\mathcal L(\AC)}
\qquad (\xi\notin N).
\]
In particular, for every \(n\), the matrix-symbol level set
\[
E_n^0:=\{\xi\in\R^d:\ \|S(\xi)\|_{\mathcal L(\AC)}\le n\}
\]
is symmetric modulo null sets. The same conclusion remains true after imposing
an additional radial truncation such as \(|\xi|\le n\).
\end{lemma}

\begin{proof}
By \textup{(A1)} and
Lemma~\ref{lem:hermitian-representatives-symmetric-cutoffs}, replace \(a\) and
\(b\), without changing any multiplier domain or action, by representatives for
which
\[
a(-\xi)=\overline{a(\xi)},
\qquad
b(-\xi)=\overline{b(\xi)}
\]
holds outside a symmetric null set \(N\). The maps \(\IAC\) and \(\Lu\) are the
complex-linear extensions of real maps on \(A\). Hence they commute with the
coefficient conjugation in the sense
\[
\cA \IAC \cA^{-1}=\IAC,
\qquad
\cA \Lu \cA^{-1}=\Lu .
\]
Therefore, for \(\xi\notin N\),
\[
\begin{aligned}
S(-\xi)
&=a(-\xi)\IAC+b(-\xi)\Lu \\
&=\overline{a(\xi)}\IAC+\overline{b(\xi)}\Lu \\
&=\cA\bigl(a(\xi)\IAC+b(\xi)\Lu\bigr)\cA^{-1}.
\end{aligned}
\]
The conjugation \(\cA\) is antiunitary, and therefore isometric, for the fixed
branch-adapted Hilbert norm on \(\AC\). Conjugation by an isometry preserves the
operator norm, so
\[
\|S(-\xi)\|_{\mathcal L(\AC)}
=
\|S(\xi)\|_{\mathcal L(\AC)}
\]
outside \(N\). Thus \(E_n^0\triangle(-E_n^0)\subset N\cup(-N)\), so \(E_n^0\)
is symmetric modulo null sets. Since \(|-\xi|=|\xi|\), adding a radial cutoff
preserves this conclusion.
\end{proof}

\begin{proposition}[Density and closedness of the symbol-sum multiplier operator]
\label{prop:symbol-sum-density-closedness}
Assume \textup{(A1)}, and assume that the symbols are the measurable finite-a.e. complex-valued functions fixed in Section~2. Let $\Mss$ be the candidate relative symbol-sum Fourier-multiplier operator from Definition~\ref{def:relative-symbol-sum-realization}. Then $\Mss$ is densely defined and closed as a real-linear operator on $L^2(\R^d;A)$. Consequently, after this proposition, $\Mss$ may be referred to as the closed relative symbol-sum Fourier-multiplier realization associated with the fixed multiplier pair $(a,b)$ and the branch-coordinate architecture.
\end{proposition}

\begin{proof}
Set
\[
S(\xi):=a(\xi)\IAC+b(\xi)\Lu .
\]
The operator norm of $S(\xi)$ is finite for almost every $\xi$, because $a$ and $b$ are finite almost everywhere and $\IAC,\Lu$ act on the fixed finite-dimensional coefficient space $\AC$.

For density, define
\[
E_n^0:=\{\xi\in\R^d:\ \|S(\xi)\|_{\mathcal L(\AC)}\le n\}.
\]
By Lemma~\ref{lem:symbol-sum-matrix-cutoff-symmetry}, the matrix-symbol level
sets \(E_n^0\) are symmetric modulo null sets. Applying the symmetric-cutoff
part of Lemma~\ref{lem:hermitian-representatives-symmetric-cutoffs}, replace
\(E_n^0\) modulo null sets by exactly symmetric measurable representatives,
still denoted
\[
E_n.
\]
Then \(E_n=-E_n\) exactly, the characteristic multipliers associated with
\(E_n\) and \(E_n^0\) are the same \(L^2\)-operators, and, because
\(\|S(\xi)\|_{\mathcal L(\AC)}<\infty\) for almost every \(\xi\), the sets
\(E_n\) increase to a full-measure subset modulo null sets.

Given \(Y\in L^2(\R^d;A)\), define
\[
\widehat Y_n:=\mathbf 1_{E_n}\widehat Y.
\]
Because \(E_n=-E_n\), the cutoff preserves the Fourier-side real-state
condition: if
\[
\widehat Y(-\xi)=\cA\widehat Y(\xi)
\quad\text{for a.e. }\xi,
\]
then
\[
\widehat Y_n(-\xi)
=
\mathbf 1_{E_n}(-\xi)\widehat Y(-\xi)
=
\cA\bigl(\mathbf 1_{E_n}(\xi)\widehat Y(\xi)\bigr)
=
\cA\widehat Y_n(\xi).
\]
Thus \(Y_n:=\F^{-1}\widehat Y_n\) belongs to \(L^2(\R^d;A)\). Moreover,
\[
\|S\widehat Y_n\|_{L^2(\R^d;\AC)}
\le n\|\widehat Y\|_{L^2(\R^d;\AC)},
\]
so \(Y_n\in\Dom(\Mss)\). By dominated convergence, \(Y_n\to Y\) in
\(L^2(\R^d;A)\). Hence \(\Dom(\Mss)\) is dense.

For closedness, suppose
\[
Y_k\in\Dom(\Mss),\qquad
Y_k\to Y \quad\text{in }L^2(\R^d;A),\qquad
\Mss Y_k\to W \quad\text{in }L^2(\R^d;A).
\]
Taking Fourier transforms gives
\[
\widehat Y_k\to\widehat Y,
\qquad
\widehat{\Mss Y_k}\to\widehat W
\quad\text{in }L^2(\R^d;\AC).
\]
Passing to a subsequence if necessary, both convergences hold pointwise almost everywhere. Since
\[
\widehat{\Mss Y_k}(\xi)=S(\xi)\widehat Y_k(\xi)
\qquad\text{for a.e. }\xi,
\]
and $S(\xi)$ is a finite matrix for almost every $\xi$, pointwise passage to the limit gives
\[
\widehat W(\xi)=S(\xi)\widehat Y(\xi)
\qquad\text{for a.e. }\xi.
\]
Thus $S\widehat Y\in L^2(\R^d;\AC)$, so $Y\in\Dom(\Mss)$, and $\Mss Y=W$. Therefore $\Mss$ is closed.
\end{proof}

\begin{theorem}[Maximal finite-a.e. symbol-sum graph-core identity]
\label{prop:strong-prototype-symbol-sum-core}
Assume the finite-a.e. multiplier-symbol convention of
Remark~\ref{rem:finite-ae-symbol-convention} and the reality symmetry assumption
\textup{(A1)}. Let $\Mss$ be the closed relative symbol-sum Fourier-multiplier
realization from Definition~\ref{def:relative-symbol-sum-realization} and
Proposition~\ref{prop:symbol-sum-density-closedness}. Let $\Gcal$ denote the
componentwise strong prototype associated with the formal expression
\[
\Gform=\IA\otimes a(D)+\Lu\otimes b(D).
\]
Then $\Gcal$ is closable as an operator on $L^2(\R^d;A)$ and is a graph core
for the closed symbol-sum realization:
\[
\overline{\Gcal}^{\,\mathrm{graph}}=\Mss.
\]
Equivalently, for every $Y\in\Dom(\Mss)$ there exists a sequence
$Y^{(n)}\in\Dom(\Gcal)$ such that
\[
Y^{(n)}\to Y,
\qquad
\Gcal Y^{(n)}\to \Mss Y
\quad\text{in }L^2(\R^d;A).
\]
One admissible graph-core sequence is obtained by the finite-symbol cutoffs
\[
E_n^0:=\{\xi\in\R^d:\ |a(\xi)|\le n,\ |b(\xi)|\le n\},
\qquad
\widehat{Y^{(n)}}(\xi)=\mathbf 1_{E_n}(\xi)\widehat Y(\xi),
\]
where $E_n$ denotes an exactly symmetric representative of $E_n^0$ modulo null
sets. Thus the raw componentwise prototype can have a strictly smaller domain
than the symbol-sum realization, but it still determines that closed realization
by graph closure.

This graph-core identity depends on the finite-a.e. multiplier-symbol hypothesis.
It is a graph-closure assertion for the maximal \(L^2\) symbol-sum multiplier
realization, not an equality of raw domains and not a general closure theorem
for arbitrary unbounded sums. In general the
componentwise strong domain
\[
\Dom(\Gcal)\subseteq
\{Y: a\widehat Y,\ b\widehat Y\in L^2(\R^d;\AC)\}
\]
may be a proper subdomain of the symbol-sum domain
\[
\Dom(\Mss)=
\{Y:(a\IAC+b\Lu)\widehat Y\in L^2(\R^d;\AC)\},
\]
because cancellation may occur in the combined multiplier even when the two pieces
are not separately in $L^2$. The theorem asserts approximation in graph norm
after closure, not equality of the two domains before closure. This is a
multiplier-domain closure statement inside the prescribed finite-a.e. multiplier
framework; it does not use the reduced semigroup growth assumption
\textup{(A2)}. Outside this \(L^2\) multiplier setting, a different realization
procedure would require a different graph-core theorem.
\end{theorem}

\begin{proof}
First note that $\Gcal\subseteq\Mss$ at the level of the symbol-sum realization. Indeed, if $Z\in\Dom(\Gcal)$, then
\[
a(\cdot)\widehat Z(\cdot),\ b(\cdot)\widehat Z(\cdot)\in L^2(\R^d;\AC),
\]
and hence
\[
\bigl(a(\cdot)\IAC+b(\cdot)\Lu\bigr)\widehat Z(\cdot)\in L^2(\R^d;\AC).
\]
Thus $Z\in\Dom(\Mss)$ and $\Mss Z=\Gcal Z$.

By Proposition~\ref{prop:symbol-sum-density-closedness}, $\Mss$ is closed. Since $\Gcal\subseteq\Mss$, the graph closure of $\Gcal$ is contained in the graph of $\Mss$. Hence $\Gcal$ is closable. It remains to identify its operator closure by proving graph-density in $\Dom(\Mss)$. Let $Y\in\Dom(\Mss)$. For each $n\in\mathbb N$, first set
\[
E_n^0:=\{\xi\in\R^d:\ |a(\xi)|\le n,\ |b(\xi)|\le n\}.
\]
By Assumption~\textup{(A1)} and
Lemma~\ref{lem:hermitian-representatives-symmetric-cutoffs}, after replacing
\(a\) and \(b\) by harmless Hermitian representatives, the raw sets \(E_n^0\)
are symmetric under \(\xi\mapsto -\xi\) up to null sets. Since \(a\) and \(b\)
are finite almost everywhere, the sets \(E_n^0\) increase to a full-measure
subset of \(\R^d\); equivalently, \(\mathbf 1_{E_n^0}(\xi)\to1\) for almost
every \(\xi\). This is precisely where the finite-a.e. symbol hypothesis enters
the graph-core proof. Applying the symmetric-cutoff part of
Lemma~\ref{lem:hermitian-representatives-symmetric-cutoffs}, replace the raw
sets modulo null sets by exactly symmetric representatives, still denoted
\[
E_n.
\]
Then \(E_n=-E_n\) exactly, the Fourier multiplier \(\mathbf 1_{E_n}(D)\) is the
same \(L^2\)-operator as the cutoff associated with \(E_n^0\), and
\(\mathbf 1_{E_n}(\xi)\to1\) for almost every \(\xi\in\R^d\).

Define $Y^{(n)}$ by
\[
\widehat{Y^{(n)}}(\xi):=\mathbf 1_{E_n}(\xi)\widehat Y(\xi).
\]
Because $\mathbf 1_{E_n}(-\xi)=\mathbf 1_{E_n}(\xi)$ exactly, the truncation preserves the Fourier-side reality condition, so each $Y^{(n)}$ is again a real $A$-valued $L^2$ state. Dominated convergence gives
\[
Y^{(n)}\to Y
\qquad\text{in }L^2(\R^d;A).
\]

On $E_n$, both $a$ and $b$ are bounded by $n$. Hence
\[
a(\cdot)\widehat{Y^{(n)}}(\cdot)\in L^2(\R^d;\AC),
\qquad
b(\cdot)\widehat{Y^{(n)}}(\cdot)\in L^2(\R^d;\AC),
\]
so $Y^{(n)}\in\Dom(\Gcal)$. For these truncated states, the strong prototype and the symbol-sum realization agree:
\[
\Mss Y^{(n)}=\Gcal Y^{(n)}.
\]
Moreover,
\[
\widehat{\Mss Y^{(n)}}(\xi)
=
\mathbf 1_{E_n}(\xi)
\bigl(a(\xi)\IAC+b(\xi)\Lu\bigr)\widehat Y(\xi).
\]
Since $Y\in\Dom(\Mss)$, the function
\[
\bigl(a(\cdot)\IAC+b(\cdot)\Lu\bigr)\widehat Y(\cdot)
\]
belongs to $L^2(\R^d;\AC)$. Dominated convergence therefore yields
\[
\Mss Y^{(n)}\to \Mss Y
\qquad\text{in }L^2(\R^d;A).
\]
Consequently,
\[
\norm{Y^{(n)}-Y}_{\mathrm{gr}(\Mss)}\to 0.
\]
Thus $\Dom(\Gcal)$ is graph-dense in $\Dom(\Mss)$ for the graph norm of $\Mss$. Together with the already established inclusion $\Gcal\subseteq\Mss$ and the closability just proved, this gives the operator-closure identity
\[
\overline{\Gcal}^{\,\mathrm{graph}}=\Mss.
\]
\end{proof}

\begin{remark}[Scope of the maximal finite-a.e. graph-core theorem]
\label{rem:scope-maximal-finite-ae-graph-core}
This is the local graph-core expansion of item~\textup{(iv)} in
Remark~\ref{rem:standing-interpretation-box-paper5}. Theorem~\ref{prop:strong-prototype-symbol-sum-core} is a theorem about the
maximal finite-a.e. Fourier-multiplier realization
\(\Mss\) associated with the measurable matrix symbol
\(a(\xi)\IAC+b(\xi)\Lu\). It should not be read as saying that every
formal sum of two unbounded operators has the closure of its componentwise
strong restriction as its maximal realization. The proof uses three special
features of the present model: multiplication by a finite-a.e. measurable
symbol, symmetric finite-symbol cutoffs exhausting frequency space, and
Plancherel graph convergence. If any of those features is replaced by a
form-defined, variable-coefficient, boundary, singular-symbol, or abstract
operator-sum realization, the graph-core statement must be reformulated and
proved separately.
\end{remark}

\begin{remark}[Why the graph-core identity is not a domain tautology]
\label{rem:graph-core-not-domain-tautology}
The domains need not coincide. The componentwise prototype requires the two
separate products $a\widehat Y$ and $b\widehat Y$ to be square-integrable. The
symbol-sum domain only requires
\[
(a\IAC+b\Lu)\widehat Y\in L^2(\R^d;\AC).
\]
After branch reduction this may require only a reduced combination such as
$(a+b)\widehat f_+$, $(a-b)\widehat f_-$, or $(a+ib)\widehat h_c$ to belong to
$L^2$. Cancellations can therefore make
\[
\Dom(\Gcal)\subsetneq\Dom(\Mss).
\]
Theorem~\ref{prop:strong-prototype-symbol-sum-core} is a maximal finite-a.e.
symbol-sum closure theorem: it says that this smaller
componentwise prototype nevertheless determines the closed symbol-sum
realization by graph closure.
\end{remark}

\begin{remark}[No extended-valued graph-core statement]\label{rem:no-extended-valued-graph-core}
The proof of Theorem~\ref{prop:strong-prototype-symbol-sum-core} uses that
\[
E_n^0=\{\xi: |a(\xi)|\le n,\ |b(\xi)|\le n\}
\]
exhausts the frequency space modulo a null set. This follows from the standing
finite-a.e. convention on the multiplier symbols. If extended-valued symbols, or
symbols singular on a set of positive measure, were allowed, the same cutoff
argument would only exhaust the finite-symbol part of the frequency space. The
operator domain and the graph-core identity would then have to be reformulated.
The present paper makes no claim in that broader setting.
\end{remark}

\begin{proposition}[Branchwise symbol-sum domain splitting]
\label{prop:full-multiplier-domain-branch-splitting}
Let $\Mss$ be the closed relative symbol-sum Fourier-multiplier realization from
Definition~\ref{def:relative-symbol-sum-realization} and
Proposition~\ref{prop:symbol-sum-density-closedness}. For $Y\in L^2(\R^d;A)$ write
\[
(f_+,f_-,h_c):=\mathfrak IY.
\]
Then
\[
Y\in\Dom(\Mss)
\]
if and only if
\[
\lambda_+\widehat f_+\in \LtwoC,
\qquad
\lambda_-\widehat f_-\in \LtwoC,
\qquad
\lambda_c\widehat h_c\in L^2(\R^d;\C).
\]
Equivalently,
\[
\Dom(\Mss)
=
\mathfrak I^{-1}
\bigl(\Dom(G_+)\oplus\Dom(G_-)\oplus\Dom(G_c)\bigr).
\]
\end{proposition}

\begin{proof}
Write the intrinsic branch decomposition as
\[
Y=Y_+ + Y_- + Y_c,
\qquad
Y_+=\Pplus Y,
\quad
Y_-=\Pminus Y,
\quad
Y_c=\Pc Y.
\]
The projectors $\Pplus$, $\Pminus$, and $\Pc$ are finite-dimensional coefficient
projections and commute with $\Lu$. Hence the Fourier-side multiplier
\[
S(\xi):=a(\xi)\IAC+b(\xi)\Lu
\]
preserves the three branch blocks and satisfies
\[
S(\xi)\widehat Y(\xi)
=
S(\xi)\widehat{Y_+}(\xi)
+
S(\xi)\widehat{Y_-}(\xi)
+
S(\xi)\widehat{Y_c}(\xi)
\]
for almost every $\xi$. Since the coefficient space is finite-dimensional and the
branch projections give a direct sum decomposition, the branch coordinate norm is
equivalent to the ambient coefficient norm. Therefore
\[
S(\cdot)\widehat Y(\cdot)\in L^2(\R^d;\AC)
\]
if and only if each branch component
\[
S(\cdot)\widehat{Y_+}(\cdot),
\qquad
S(\cdot)\widehat{Y_-}(\cdot),
\qquad
S(\cdot)\widehat{Y_c}(\cdot)
\]
belongs to $L^2(\R^d;\AC)$.

On the plus branch, $\Lu\Pplus=\Pplus$, so
\[
S(\xi)\widehat{Y_+}(\xi)
=
\lambda_+(\xi)\widehat f_+(\xi)\Pplus.
\]
Thus
\[
S(\cdot)\widehat{Y_+}(\cdot)\in L^2(\R^d;\AC)
\iff
\lambda_+\widehat f_+\in \LtwoC.
\]
Similarly, on the minus branch, $\Lu\Pminus=-\Pminus$, hence
\[
S(\xi)\widehat{Y_-}(\xi)
=
\lambda_-(\xi)\widehat f_-(\xi)\Pminus,
\]
and therefore
\[
S(\cdot)\widehat{Y_-}(\cdot)\in L^2(\R^d;\AC)
\iff
\lambda_-\widehat f_-\in \LtwoC.
\]

It remains to treat the complex branch. Write
\[
Y_c=f\Pc+gJ,
\qquad
f,g\in L^2(\R^d;\R),
\qquad
h_c=\mathcal I_cY_c=f+ig.
\]
By Lemma~\ref{lem:realified-complex-branch-multiplier},
\[
S(\cdot)\widehat{Y_c}(\cdot)\in L^2(\R^d;\AC)
\iff
(a+ib)\widehat h_c\in L^2(\R^d;\C).
\]
Since $\lambda_c=a+ib$, this gives
\[
S(\cdot)\widehat{Y_c}(\cdot)\in L^2(\R^d;\AC)
\iff
\lambda_c\widehat h_c\in L^2(\R^d;\C).
\]
Combining the three branch equivalences proves the asserted domain splitting.
\end{proof}


\begin{remark}[Reduced branch-coordinate symbols and realified c-block]
\label{rem:reading-branch-block-matrices}
In the branch-coordinate symbol statements below, all direct sums are real
Hilbert direct sums, and the realification convention is the one fixed in
Proposition~\ref{prop:realified-complex-branch-normal-form},
Definition~\ref{def:realification-complex-scalar}, and
Definition~\ref{def:reduced-direct-sum-notation}. The third coordinate \(\LtwoC\) is used as the
complex-coordinate model of the real two-dimensional \(A_c\)-branch through
\(\mathcal I_c\). Thus the preferred operator-level reduced notation is
\[
\Gred=G_+\oplus G_-\oplus (G_c)_{\R},
\]
not an unqualified literal matrix \(\operatorname{diag}(\lambda_+,\lambda_-,\lambda_c)\) on the original real \(A\)-valued fiber. The symbol
\[
\lambda_c(\xi)=a(\xi)+i b(\xi)
\]
means scalar multiplication on the reduced complex coordinate \(\widehat h_c(\xi)\), viewed through the underlying real Hilbert structure when it is assembled with the two real branches.

The three coordinate systems on the \(c\)-branch are fixed once and for all in
Remark~\ref{rem:three-c-branch-coordinate-systems}. We recall the practical
consequence here because it is the main point needed in the reduced symbol
statements below. Any occurrence of \(\lambda_c\) in a reduced direct-sum display
refers to the reduced scalar complex coordinate \(h_c\), while any occurrence
of \(R_{\lambda_c}\) refers to the pointwise real-imaginary coordinates of
\(\widehat h_c\). Neither notation is a coefficient-level action on the
physical pair \((\widehat f,\widehat g)\). For that physical pair, the
coefficient-level formula remains
\[
(\widehat f,\widehat g)
\mapsto
(a\widehat f-b\widehat g,\; b\widehat f+a\widehat g).
\]
When \(a\) and \(b\) are complex-valued Hermitian symbols, this coefficient-level
formula is not the same object as the pointwise realification matrix
\(R_{\lambda_c}\). At the operator level, the notation \(R_{\lambda_c}(D)\)
therefore inherits the Fourier-side scalar multiplier domain \(\Dom_\xi(R_{\lambda_c}(D))\), as fixed in
Remark~\ref{rem:no-entrywise-domain-realified-block}; it never imposes an
entrywise matrix domain on the four separate products involving
\(\Re\lambda_c\) and \(\Im\lambda_c\). Consequently any displayed reduced branch-coordinate matrix is only a coordinate
mnemonic for the direct-sum operator \(\Gred\). It is not an assertion that the
full real \(A\)-valued Fourier multiplier is a literal complex-coordinate direct sum on the original
real state space, and no conjugate branch is being suppressed.
\end{remark}

\begin{proposition}[Reduced branch-coordinate symbol of the relative symbol-sum multiplier realization]\label{prop:symbol-sum-branch-block}
With the real-linear isometric isomorphism $\mathfrak I$ defined above, the relative symbol-sum Fourier-multiplier realization $\Mss$ has the following reduced branch-coordinate representation: writing
\[
(f_+,f_-,h_c):=\mathfrak I Y,
\]
one has
\[
Y\in\Dom(\Mss)
\iff
\lambda_+\widehat f_+\in \LtwoC,\quad
\lambda_-\widehat f_-\in \LtwoC,\quad
\lambda_c\widehat h_c\in L^2(\R^d;\C),
\]
and for such $Y$,
\[
\mathfrak I(\Mss Y)
=
\bigl(G_+f_+,\,G_-f_-,\,G_ch_c\bigr).
\]
Equivalently, on the reduced branch-coordinate model \(\Xred\),
\[
\mathfrak I\,\Mss\,\mathfrak I^{-1}
=
\Gred
=
G_+\oplus G_-\oplus (G_c)_{\R}.
\]
On Fourier side this direct-sum identity may be remembered by the reduced-coordinate mnemonic
\[
(\widehat f_+,\widehat f_-,\widehat h_c)
\longmapsto
(\lambda_+\widehat f_+,\lambda_-\widehat f_-,\lambda_c\widehat h_c).
\]
This mnemonic is not a literal complex-coordinate one-dimensional fibre matrix acting on the original real
\(A\)-valued fiber. Its third entry is scalar multiplication by \(\lambda_c\)
on the reduced complex coordinate \(\widehat h_c\), not on the physical
coefficient pair of the \(A_c\)-branch. This is the distinction fixed in
Remark~\ref{rem:three-c-branch-coordinate-systems}. The notation
\((G_c)_{\R}\) is used when the branch is assembled in the real direct sum.
\end{proposition}

\begin{proof}
The domain statement is exactly Proposition~\ref{prop:full-multiplier-domain-branch-splitting}.
It remains only to identify the action.

On the plus branch,
\[
(a\IAC+b\Lu)\widehat{Y_+}
=
\lambda_+\widehat f_+\Pplus.
\]
On the minus branch,
\[
(a\IAC+b\Lu)\widehat{Y_-}
=
\lambda_-\widehat f_-\Pminus.
\]
On the complex branch, Lemma~\ref{lem:realified-complex-branch-multiplier} gives
\[
\mathcal I_c\F^{-1}
\bigl(
(a\IAC+b\Lu)\widehat{Y_c}
\bigr)
=
G_ch_c.
\]
Therefore, after applying the branch identification map $\mathfrak I$,
\[
\mathfrak I(\Mss Y)
=
(G_+f_+,G_-f_-,G_ch_c).
\]
This proves the reduced direct-sum identity
\(\mathfrak I\Mss\mathfrak I^{-1}=\Gred\). The frequency-side tuple formula is only the corresponding reduced-coordinate mnemonic.
\end{proof}

\begin{proposition}[Reduced branch-coordinate direct-sum form of the reconstructed generator]\label{prop:full-symbol-branch-block}
Assume that \((a,b)\) satisfies the standing multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}, and let \(\Acal\) be
the reconstructed generator from
Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction}. With the real-linear isometric isomorphism \(\mathfrak I\) defined above, the reconstructed generator \(\Acal\) is the direct sum of the three reduced branch generators in branch coordinates: writing
\[
(f_+,f_-,h_c):=\mathfrak I Y,
\]
one has
\[
Y\in\Dom(\Acal)
\iff
\lambda_+\widehat f_+\in \LtwoC,\quad
\lambda_-\widehat f_-\in \LtwoC,\quad
\lambda_c\widehat h_c\in L^2(\R^d;\C),
\]
and for such \(Y\),
\[
\mathfrak I(\Acal Y)
=
\bigl(G_+f_+,\,G_-f_-,\,G_ch_c\bigr).
\]
Equivalently, on \(\Xred\),
\[
\mathfrak I\,\Acal\,\mathfrak I^{-1}
=
\Gred
=
G_+\oplus G_-\oplus (G_c)_{\R}
\]
on \(\Dom(\Gred)\). Here the \(c\)-branch component of \(\mathfrak I\) is
the complex scalar coordinate \(h_c=f+ig\). Thus the third direct-sum
component is \((G_c)_{\R}\), not a coefficient-pair matrix acting directly on
\((\widehat f,\widehat g)\). On Fourier side this means only the reduced-coordinate tuple formula
\[
(\widehat f_+,\widehat f_-,\widehat h_c)
\longmapsto
(\lambda_+\widehat f_+,\lambda_-\widehat f_-,\lambda_c\widehat h_c).
\]
It should not be read as a pointwise reduced-coordinate form of the full real \(A\)-valued fiber.
\end{proposition}

\begin{proof}
The domain statement is just the definition of $\Dom(\Acal)$ rewritten through the branch identification map $\mathfrak I$, using the definitions of $G_+$, $G_-$, and $G_c$ from Section~4. The action formula follows immediately from part \textup{(iii)} of Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction}. This is exactly the reduced direct-sum identity $\mathfrak I\Acal\mathfrak I^{-1}=\Gred$ fixed in Definition~\ref{def:reduced-direct-sum-notation}.
\end{proof}

\begin{corollary}[Identification with the relative symbol-sum multiplier realization]\label{cor:symbol-sum-realization-identification}
Assume that \((a,b)\) satisfies the standing multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}. Let $\Acal$ be the
reconstructed generator from
Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction}. By
Proposition~\ref{prop:symbol-sum-density-closedness}, the candidate operator
$\Mss$ from Definition~\ref{def:relative-symbol-sum-realization} is densely
defined and closed; in this corollary, $\Mss$ is used as the closed relative
symbol-sum Fourier-multiplier realization. Then
\[
\Acal = \Mss.
\]
Equivalently, the branchwise reconstructed generator agrees with the relative symbol-sum Fourier-multiplier realization of the formal prototype
\[
\Gform = \IA\otimes a(D)+\Lu\otimes b(D)
\]
because the two operators have the same branch-coordinate domain and action. This equality is an identification inside the prescribed Fourier-multiplier symbol-sum realization class fixed in Subsection~\ref{subsec:scope-and-nonclaims}.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:full-multiplier-domain-branch-splitting},
\[
\Dom(\Mss)
=
\mathfrak I^{-1}
\bigl(\Dom(G_+)\oplus\Dom(G_-)\oplus\Dom(G_c)\bigr).
\]
By the definition of the reconstructed generator $\Acal$ and
Proposition~\ref{prop:full-symbol-branch-block},
\[
\Dom(\Acal)
=
\mathfrak I^{-1}
\bigl(\Dom(G_+)\oplus\Dom(G_-)\oplus\Dom(G_c)\bigr).
\]
Thus
\[
\Dom(\Acal)=\Dom(\Mss).
\]
For every $Y$ in this common domain, write $(f_+,f_-,h_c)=\mathfrak IY$. Propositions
\ref{prop:symbol-sum-branch-block} and~\ref{prop:full-symbol-branch-block} give
\[
\mathfrak I(\Acal Y)
=
(G_+f_+,G_-f_-,G_ch_c)
=
\mathfrak I(\Mss Y).
\]
Since $\mathfrak I$ is injective, $\Acal Y=\Mss Y$ on the common domain. Hence
\[
\Acal=\Mss.
\]
\end{proof}




\begin{proposition}[Strong prototype graph core for the reconstructed generator]
\label{prop:strong-prototype-core}
Assume that \((a,b)\) satisfies the standing multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}. Then $\Gcal$ is closable and
\[
\overline{\Gcal}^{\,\mathrm{graph}}=\Acal=\Mss.
\]
Here the graph-theoretic equality \(\overline{\Gcal}^{\,\mathrm{graph}}=\Mss\) is exactly
Theorem~\ref{prop:strong-prototype-symbol-sum-core}; the additional
hypothesis \textup{(A2)} is used in this proposition only to identify the same
closed operator with the reconstructed semigroup generator \(\Acal\).
The statement should not be read as \(\Dom(\Gcal)=\Dom(\Acal)\). The raw componentwise domain may be strictly smaller than the reconstructed branchwise domain; the theorem says only that the former is a graph core for the latter. Equivalently, $\Dom(\Gcal)$ is dense in $\Dom(\Acal)$ with respect to the Hilbert graph norm of $\Acal$,
\[
\norm{Y}_{\mathrm{gr}(\Acal)}
:=
\Bigl(
\norm{Y}_{L^2(\R^d;A)}^2
+
\norm{\Acal Y}_{L^2(\R^d;A)}^2
\Bigr)^{1/2}.
\]
Thus $\Gcal$ is a graph core for the branchwise reconstructed generator. This is a closure statement: it allows computation on the smaller strong prototype domain and then passage to the reconstructed generator in graph norm, but it does not replace the branchwise generator domain by the raw componentwise domain. This final equality uses the semigroup-generator identification $\Acal=\Mss$ and therefore belongs to the full \textup{(A1)}--\textup{(A2)} reconstruction setting.
\end{proposition}

\begin{proof}
By Theorem~\ref{prop:strong-prototype-symbol-sum-core}, $\Gcal$ is closable and
\[
\overline{\Gcal}^{\,\mathrm{graph}}=\Mss
\]
as a multiplier-domain operator-closure statement. By Corollary~\ref{cor:symbol-sum-realization-identification},
\[
\Acal=\Mss.
\]
Therefore
\[
\overline{\Gcal}^{\,\mathrm{graph}}=\Acal=\Mss.
\]
The equivalence of the graph-density statement for $\Acal$ follows from the equality $\Acal=\Mss$, which identifies both the domains and the Hilbert graph norms.
\end{proof}


\begin{proposition}[Coordinate-exact reconstructed semigroup norm in the branch-adapted Hilbert norm]\label{prop:exact-full-semigroup-norm}
Assume that \((a,b)\) satisfies the standing multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}, and let
$\mathcal T(t)$ be the reconstructed semigroup from
Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction}. For every $t\ge 0$, the reconstructed semigroup is the direct sum of the reduced branch semigroups in branch coordinates:
\[
\mathfrak I\,\mathcal T(t)\,\mathfrak I^{-1}
=
\Tred(t)
=
T_+(t)\oplus T_-(t)\oplus (T_c(t))_{\R}
\]
on the reduced real Hilbert direct-sum model \(\Xred\). Here \((T_c(t))_{\R}\) denotes the complex semigroup \(T_c(t)\) regarded as a real-linear operator on \((\LtwoC)_{\R}\).
Consequently, with respect to the fixed branch-adapted Hilbert norm on
\(L^2(\R^d;A)\),
\[
\norm{\mathcal T(t)}_{\mathcal L(L^2(\R^d;A))}
=
\max\Bigl\{\norm{T_+(t)}_{\mathcal L(\LtwoR)},\ \norm{T_-(t)}_{\mathcal L(\LtwoR)},\ \norm{T_c(t)}_{\mathcal L(\LtwoC)}\Bigr\}.
\]
In particular,
\[
\norm{\mathcal T(t)}_{\mathcal L(L^2(\R^d;A))}
\le
\max\{e^{\omega_+ t},\ e^{\omega_- t},\ e^{\omega_c t}\}.
\]
This equality is part of the branch-adapted Hilbert model. It is not asserted as
a literal equality for arbitrary equivalent coefficient norms on \(A\); under
such norms it is replaced by equivalence estimates as in
Remark~\ref{rem:norm-dependence-exact-branch-identities}.
\end{proposition}

\begin{proof}
Let $(f_+,f_-,h_c)$ belong to the direct-sum branch model and write
\[
Y:=\mathfrak I^{-1}(f_+,f_-,h_c).
\]
By the reconstruction formula \eqref{eq:full-recombination-formula},
\[
\mathfrak I\bigl(\mathcal T(t)Y\bigr)
=
\bigl(T_+(t)f_+,\,T_-(t)f_-,\,T_c(t)h_c\bigr).
\]
This is exactly the reduced direct-sum identity
\[
\mathfrak I\,\mathcal T(t)\,\mathfrak I^{-1}=\Tred(t).
\]
Because $\mathfrak I$ is an isometric isomorphism, the operator norm of $\mathcal T(t)$ equals the operator norm of this direct-sum operator. For every $(f_+,f_-,h_c)$ one has
\[
\begin{aligned}
&\norm{(T_+(t)f_+,\,T_-(t)f_-,\,T_c(t)h_c)}^2 \\
&\qquad=
\norm{T_+(t)f_+}_{L^2}^2+\norm{T_-(t)f_-}_{L^2}^2+\norm{T_c(t)h_c}_{L^2(\C)}^2 \\
&\qquad\le
M_t^2\bigl(\norm{f_+}_{L^2}^2+\norm{f_-}_{L^2}^2+\norm{h_c}_{L^2(\C)}^2\bigr),
\end{aligned}
\]
where
\[
M_t:=\max\Bigl\{\norm{T_+(t)}_{\mathcal L(\LtwoR)},\ \norm{T_-(t)}_{\mathcal L(\LtwoR)},\ \norm{T_c(t)}_{\mathcal L(\LtwoC)}\Bigr\}.
\]
Hence
\[
\norm{\Tred(t)}\le M_t.
\]
The reverse inequality follows by testing the direct-sum operator on vectors supported in a single branch, namely $(f_+,0,0)$, $(0,f_-,0)$, and $(0,0,h_c)$. Therefore
\[
\norm{\Tred(t)}=M_t,
\]
and the coordinate-exact norm formula follows. The final displayed bound is immediate from Proposition~\ref{prop:branchwise-growth}.
\end{proof}

\begin{proposition}[Coordinate-exact graph-norm splitting in the branch-adapted Hilbert norm]\label{prop:graph-norm-splitting}
Assume that \((a,b)\) satisfies the standing multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}, and let $\Acal$ be
the generator of the reconstructed semigroup from
Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction}. Under the branch identification map $\mathfrak I$, the generator domain splits as
\[
\mathfrak I\bigl(\Dom(\Acal)\bigr)
=
\Dom(G_+)\oplus \Dom(G_-)\oplus \Dom(G_c).
\]
More precisely, with all state norms taken in the fixed branch-adapted Hilbert
norm, if $Y\in\Dom(\Acal)$ and
\[
(f_+,f_-,h_c):=\mathfrak I Y,
\]
then
\[
\mathfrak I(\Acal Y)=\bigl(G_+f_+,\,G_-f_-,\,G_ch_c\bigr).
\]
Consequently, in this fixed branch-adapted Hilbert coordinate system, the Hilbert graph norm of $\Acal$ is exactly the direct-sum Hilbert graph norm of the reduced generators:
\[
\begin{aligned}
\norm{Y}_{\mathrm{gr}(\Acal)}^2
&:=
\norm{Y}_{L^2(\R^d;A)}^2+\norm{\Acal Y}_{L^2(\R^d;A)}^2\\
&=
\norm{f_+}_{L^2}^2+\norm{f_-}_{L^2}^2+\norm{h_c}_{L^2(\R^d;\C)}^2\\
&\quad
+\norm{G_+f_+}_{L^2}^2+\norm{G_-f_-}_{L^2}^2+\norm{G_ch_c}_{L^2(\R^d;\C)}^2.
\end{aligned}
\]
Equivalently, $\mathfrak I$ is an isometric isomorphism from the graph space of $\Acal$ onto the real Hilbert direct sum of the graph spaces of $G_+$, $G_-$, and $G_c$, with the $G_c$ graph space again viewed through its underlying real Hilbert structure. This isometry is asserted for the fixed branch-adapted state norm and its induced Hilbert graph norm; arbitrary equivalent coefficient norms give equivalent graph norms with constants, not the displayed literal square identity.
\end{proposition}

\begin{proof}
The domain identity and the action formula are exactly the content of Proposition~\ref{prop:full-symbol-branch-block}. Since $\mathfrak I$ is an isometric isomorphism on the state space,
\[
\norm{Y}_{L^2(\R^d;A)}^2
=
\norm{f_+}_{L^2}^2+\norm{f_-}_{L^2}^2+\norm{h_c}_{L^2(\R^d;\C)}^2.
\]
Applying the same isometry to $\Acal Y$ and using the action formula gives
\[
\norm{\Acal Y}_{L^2(\R^d;A)}^2
=
\norm{G_+f_+}_{L^2}^2+\norm{G_-f_-}_{L^2}^2+\norm{G_ch_c}_{L^2(\R^d;\C)}^2.
\]
Adding the two identities yields the displayed graph-norm formula. The final sentence is just a restatement of this norm identity together with the coordinate-exact domain splitting.
\end{proof}

\begin{remark}[Norm dependence of coordinate-exact branch identities]\label{rem:norm-dependence-exact-branch-identities}
The coordinate-exact norm identity in Proposition~\ref{prop:exact-full-semigroup-norm} and
the coordinate-exact graph-norm splitting in Proposition~\ref{prop:graph-norm-splitting}
are identities for the fixed branch-adapted Hilbert norm chosen in
Section~2, especially Remark~\ref{rem:branch-adapted-norm-convention}, and for the induced Hilbert graph norm. If the coefficient space
$A$ is equipped with an arbitrary equivalent finite-dimensional norm
\(\|\cdot\|_*\), then there are constants \(0<m\le M<\infty\) such that
\[
 m\|Y\|_{\mathrm{ba}}
 \le
 \|Y\|_*
 \le
 M\|Y\|_{\mathrm{ba}},
\]
where \(\|\cdot\|_{\mathrm{ba}}\) denotes the branch-adapted norm. Consequently
one obtains, for example,
\[
\|\mathcal T(t)\|_{*,\mathcal L(L^2)}
\le
\frac{M}{m}
\max\{\|T_+(t)\|,\|T_-(t)\|,\|T_c(t)\|\},
\]
and analogous two-sided graph-norm equivalences. In general these are not
literal isometric equalities. Thus these coordinate-exact identities are norm-convention
statements in the branch-adapted Hilbert model, not intrinsic assertions for
every equivalent coefficient norm or for the abstract algebra independent of the
chosen Hilbert coordinate structure.
\end{remark}

\begin{remark}[Choice of graph norm]\label{rem:choice-of-graph-norm}
Throughout the statements involving exact graph-space identities or isometric
branch-coordinate graph-space splittings, the graph norm is the Hilbert graph
norm
\[
\norm{Y}_{\mathrm{gr}(\Acal)}
=
\Bigl(
\norm{Y}_{L^2(\R^d;A)}^2+\norm{\Acal Y}_{L^2(\R^d;A)}^2
\Bigr)^{1/2}.
\]
The additive graph norm
\[
\norm{Y}_{L^2(\R^d;A)}+\norm{\Acal Y}_{L^2(\R^d;A)}
\]
is equivalent to this Hilbert graph norm and therefore gives the same notions
of graph convergence, graph density, and graph core. It is not the norm used
for the exact isometric direct-sum graph-space splitting in
Proposition~\ref{prop:graph-norm-splitting}.
\end{remark}

\begin{remark}[Reduced-coordinate direct sums versus real branch realification]\label{rem:reduced-symbol-vs-real-block}
The preferred operator-level notation is
\[
\Gred=G_+\oplus G_-\oplus (G_c)_{\R},
\qquad
\Tred(t)=T_+(t)\oplus T_-(t)\oplus (T_c(t))_{\R}.
\]
This notation lives in the reduced branch coordinates supplied by \(\mathfrak I\). It should not be read as asserting that the original real \(A\)-valued multiplier is literally a complex-coordinate direct sum on \(L^2(\R^d;A)\). On the real state space the plus and minus branches are scalar, while the \(c\)-branch is represented through the reduced complex coordinate \(h_c=\mathcal I_cY_c\), with \(G_c\) and \(T_c(t)\) viewed as real-linear only after this coordinate transport. Any reduced frequency-side tuple display involving \(\lambda_c\) is therefore no more than a shorthand for the reduced tuple map
\[
(\widehat f_+,\widehat f_-,\widehat h_c)
\mapsto
(\lambda_+\widehat f_+,\lambda_-\widehat f_-,\lambda_c\widehat h_c).
\]
It is not an alternative real-state-space operator and not a matrix action on the physical Fourier pair \(\widehat f,\widehat g\) in \(Y_c=f\Pc+gJ\); see Remark~\ref{rem:three-c-branch-coordinate-systems} for the three coordinate systems involved.
\end{remark}

\begin{remark}[What is and is not identified here]\label{rem:what-is-identified}
Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction} identifies $\Acal$ as the infinitesimal generator of the canonical branchwise reconstructed semigroup determined by the reduced branch dynamics. Corollary~\ref{cor:symbol-sum-realization-identification} then identifies this reconstructed generator with the relative symbol-sum Fourier-multiplier realization of the full formal prototype
\[
I_A\otimes a(D)+\Lu\otimes b(D)
\]
through the common branch-coordinate model furnished by Propositions~\ref{prop:symbol-sum-branch-block} and~\ref{prop:full-symbol-branch-block}. This identification is internal to the symbol-sum Fourier-multiplier realization class described in Subsection~\ref{subsec:scope-and-nonclaims}. Proposition~\ref{prop:strong-prototype-core} then resolves the apparent closure question inside the present multiplier class:
\[
\overline{\Gcal}^{\,\mathrm{graph}}=\Acal=\Mss.
\]
Thus the operator-theoretic comparison proved here is not merely the raw-domain inclusion
\[
\Gcal\subseteq \Acal=\Mss,
\]
but the stronger statement that $\Gcal$ is a graph core for the relative symbol-sum reconstructed generator. The first inclusion may still be strict as a statement about raw domains, as the next remark shows.
\end{remark}

\begin{remark}[Cancellation can enlarge the reconstructed generator domain]\label{rem:domain-gap}
The inclusion in Corollary~\ref{cor:strong-suboperator} can be strict. Take $d=1$ and
\[
a(\xi)=-\xi^2+i\xi,
\qquad
b(\xi)=\xi^2+i\xi.
\]
Then the reduced plus symbol is
\[
\lambda_+(\xi)=a(\xi)+b(\xi)=2i\xi,
\]
so on the plus branch the reconstructed generator requires only $\xi\widehat f(\xi)\in L^2$, that is, $f\in H^1(\R)$. By contrast, the componentwise strong realization requires both $a(\xi)\widehat f(\xi)$ and $b(\xi)\widehat f(\xi)$ to lie in $L^2$, which forces quadratic control and hence $f\in H^2(\R)$ for plus-branch data $Y=f\Pplus$. This example shows a strict raw-domain gap, not a closure gap. The generator of the reconstructed full flow must be defined branchwise rather than through the componentwise prototype domain, but Proposition~\ref{prop:strong-prototype-core} shows that the smaller componentwise strong domain is nevertheless graph-dense in the symbol-sum branchwise domain.
\end{remark}

The theorem should be read as the main compatibility statement of the paper at the branchwise semigroup level. The reduced semigroups are not merely auxiliary computational devices; together with the fixed branch-coordinate prescription they determine the reconstructed strongly continuous evolution on the original real $A$-valued state space. The infinitesimal generator of that reconstructed evolution is the branchwise realization $\Acal$ associated with the same formal multiplier pair $(a,b)$ with which the paper started, while the componentwise strong realization $\Gcal$ remains canonically embedded in it by Corollary~\ref{cor:strong-suboperator}. The added operator-theoretic gain of Section~5 is then twofold: Corollary~\ref{cor:symbol-sum-realization-identification} records that the reconstructed generator and the relative symbol-sum Fourier-multiplier realization have the same branchwise domain and action, and Theorem~\ref{prop:strong-prototype-symbol-sum-core} is the main graph-core theorem for the symbol-sum realization, and Proposition~\ref{prop:strong-prototype-core} transfers it to the reconstructed generator after the identification \(\Acal=\Mss\).

\subsection{Mild solutions and invariant branch subspaces}

With Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction} in hand, the homogeneous Cauchy problem associated with the reconstructed generator
\begin{equation}\label{eq:homogeneous-cauchy}
\partial_t Y(t)=\Acal Y(t),
\qquad
Y(0)=Y_0\in L^2(\R^d;A),
\end{equation}
now has a canonical mild solution theory.

\begin{definition}
Let $Y_0\in L^2(\R^d;A)$. A mild solution of \eqref{eq:homogeneous-cauchy} with initial datum $Y_0$ is the orbit
\[
Y(t):=\mathcal T(t)Y_0,
\qquad t\ge 0,
\]
where $\{\mathcal T(t)\}_{t\ge 0}$ is the reconstructed semigroup from Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction}.
\end{definition}

This definition is standard from semigroup theory, but it takes on a special meaning in the present branch-valued context because the orbit can be written explicitly branch-by-branch. If
\[
Y_0 = f_{+,0}\Pplus + f_{-,0}\Pminus + Y_{c,0},
\]
then the mild solution is
\begin{equation}\label{eq:mild-branchwise-solution}
Y(t)
=
\bigl(T_+(t)f_{+,0}\bigr)\Pplus
+
\bigl(T_-(t)f_{-,0}\bigr)\Pminus
+
\mathcal I_c^{-1}T_c(t)\mathcal I_c(Y_{c,0}).
\end{equation}
Thus the full distributed evolution is recovered without any coupled matrix exponential on the whole coefficient algebra. One solves three reduced scalar problems and then recombines them.

\begin{corollary}[Branchwise invariance of mild solutions]
Let $Y_0\in L^2(\R^d;A)$.
\begin{enumerate}[label=\textup{(\roman*)}]
    \item If $Y_0\in L^2_+(\R^d;A)$, then $Y(t)=\mathcal T(t)Y_0\in L^2_+(\R^d;A)$ for all $t\ge 0$.
    \item If $Y_0\in L^2_-(\R^d;A)$, then $Y(t)=\mathcal T(t)Y_0\in L^2_-(\R^d;A)$ for all $t\ge 0$.
    \item If $Y_0\in L^2_c(\R^d;A)$, then $Y(t)=\mathcal T(t)Y_0\in L^2_c(\R^d;A)$ for all $t\ge 0$.
\end{enumerate}
More generally, if $Y_0$ contains contributions from several branches, each contribution evolves independently according to \eqref{eq:mild-branchwise-solution}.
\end{corollary}

\begin{proof}
This is an immediate consequence of the invariance statement in Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction} and the explicit formula \eqref{eq:mild-branchwise-solution}.
\end{proof}

For data in the generator domain, the mild solution is in fact a strong solution. We record this only briefly, since no additional difficulty is involved.

\begin{proposition}[Strong solutions for domain data]
If $Y_0\in\Dom(\Acal)$, then the mild solution $Y(t)=\mathcal T(t)Y_0$ belongs to
\[
C([0,\infty);\Dom(\Acal))\cap C^1((0,\infty);L^2(\R^d;A)),
\]
where $\Dom(\Acal)$ is equipped with the Hilbert graph norm
\[
\norm{Y}_{\mathrm{gr}(\Acal)}
:=
\Bigl(
\norm{Y}_{L^2(\R^d;A)}^2
+
\norm{\Acal Y}_{L^2(\R^d;A)}^2
\Bigr)^{1/2}.
\]
Moreover, $Y$ satisfies
\[
\partial_tY(t)=\Acal Y(t)
\qquad
t>0
\]
in the strong sense, and it has the right derivative at the initial time
\[
\partial_t^+Y(0)=\Acal Y_0.
\]
\end{proposition}

\begin{proof}
This is the standard differentiability and graph-norm continuity statement for
$C_0$-semigroups on generator-domain data, applied to the semigroup
$\{\mathcal T(t)\}_{t\ge 0}$ whose generator is $\Acal$. On the half-line
$[0,\infty)$, differentiability at the endpoint is understood as right
differentiability. Thus the ordinary strong derivative is asserted on
$(0,\infty)$, while at $t=0$ the generator identity gives the right derivative
$\partial_t^+Y(0)=\Acal Y_0$.
\end{proof}

The point of these statements is not merely formal completeness. They show that the branch decomposition is dynamically meaningful at the level of the distributed flow itself. A branch-supported initial condition remains branch-supported for all future times, and mixed initial data separate into branchwise propagating channels. This will be crucial in Section~6, where observation maps may enhance one channel, suppress another, or hide one branch almost completely while leaving the full state evolution perfectly well-defined.


\begin{proposition}[Concrete instantiation of the Paper~4 reduced-generator bridge]
\label{prop:paper4-instantiation-map}
Within the notation of the abstract branchwise semigroup bridge from Paper~4, the present multiplier realization is obtained by the following specialization:
\[
X_+=L^2_{\R}(\R^d),
\qquad
X_-=L^2_{\R}(\R^d),
\qquad
X_c=L^2_{\C}(\R^d),
\]
where the complex branch is read as a real Hilbert space after transport by \(\mathcal I_c\). The reduced branch generators are
\[
G_+=\lambda_+(D),
\qquad
G_-=\lambda_-(D),
\qquad
G_c=\lambda_c(D),
\]
with maximal multiplier domains as in Definition~\ref{def:reduced-branch-multipliers}. Under the branch-coordinate isometry \(\mathfrak I\), the reconstructed full generator is exactly
\[
\mathfrak I\,\Acal\,\mathfrak I^{-1}
=
G_+\oplus G_-\oplus (G_c)_{\R},
\]
and the reconstructed semigroup is
\[
\mathfrak I\,\mathcal T(t)\,\mathfrak I^{-1}
=
T_+(t)\oplus T_-(t)\oplus (T_c(t))_{\R}.
\]
Thus the abstract Paper~4 bridge is instantiated here by a concrete translation-invariant \(L^2\) Fourier-multiplier package, not merely by a narrative analogy.
\end{proposition}

\begin{proof}
The identifications of \(X_+\), \(X_-\), and \(X_c\) are precisely the branch-coordinate state spaces fixed in Section~3. Definition~\ref{def:reduced-branch-multipliers} gives the reduced generators, Theorem~\ref{thm:reduced-semigroup-generation} gives the reduced semigroups, and Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction} identifies the direct-sum generator and semigroup under \(\mathfrak I\). The complex branch is interpreted through Proposition~\ref{prop:realified-complex-branch-normal-form}. Hence the abstract reduced-generator architecture of Paper~4 is realized by the stated multiplier data.
\end{proof}

\subsection{Conceptual remarks}

\paragraph*{What is proved in Section~5.}
Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction} proves that the prescribed reduced branch semigroups reconstruct a branch-coordinate full $C_0$-semigroup on the real state space $L^2(\R^d;A)$ by a fixed branch-coordinate direct sum, and it identifies the generator of this full flow as the branchwise realization $\Acal$. The internal uniqueness of this prescribed assembly is recorded separately in Corollary~\ref{cor:internal-uniqueness-prescribed-branch-class}: any competing semigroup is compared only after the same branch invariance and the same prescribed reduced actions have been imposed. Proposition~\ref{prop:full-matrix-symbol-semigroup-formula} then identifies the same flow as the full finite-dimensional matrix-symbol Fourier multiplier with symbol $\exp(t(a\IAC+b\Lu))$ on the Fourier-side complexified coefficient fibre, restricted to the real-state symmetry. Proposition~\ref{prop:exact-full-semigroup-norm} identifies, with respect to the fixed branch-adapted Hilbert norm, the operator norm of the full flow as the coordinate-exact maximum of the reduced branch semigroup norms, while Proposition~\ref{prop:graph-norm-splitting} shows that the generator domain and Hilbert graph norm split coordinate-exactly under the branch identification map. For arbitrary equivalent coefficient norms these identities should be read as equivalent estimates with constants, not as intrinsic isometric facts of the algebra alone. The section does not classify arbitrary full-side spectral realizations or arbitrary closed semigroup realizations of the formal expression beyond this prescribed branch-coordinate and matrix-symbol multiplier realization.

\paragraph*{Interpretation.}
Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction} is the place where Paper~5 supplies the restricted realization left open by Paper~4. Paper~4 developed an abstract branchwise semigroup bridge on spaces of the form $A\otimes X$, but by design it stopped short of showing that a concrete unbounded multiplier generator on an actual function space satisfies all the hypotheses needed for a branch-coordinate reconstruction from reduced branch semigroups. The present section supplies that missing realization on the admissible class treated here: linear, translation-invariant Fourier-multiplier evolution on $L^2(\R^d;A)$, assembled from standard reduced multiplier semigroups but tied to a single $A$-valued symbol-sum realization, with constant-coefficient differential operators appearing only when the multiplier symbols are polynomial symbols satisfying the standing reality and reduced growth hypotheses.

Two aspects of the result are worth stressing. First, the reconstruction is exact rather than approximate. There is no perturbative argument, no asymptotic spectral reduction, and no appeal to an external spectral theorem for the full operator on the $A$-valued space. Once the reduced semigroups are available, the full flow is obtained by branchwise assembly because the coefficient algebra itself already contains the right splitting, and because the Fourier multiplier structure distributes that splitting across frequencies without distortion. Corollary~\ref{cor:internal-uniqueness-prescribed-branch-class} shows only that, within this prescribed branch-coordinate architecture, the assembly is determined by the fixed reduced dynamics. Second, the reconstruction is honest at the level of real state spaces. Even though the complex branch is computed using a complex semigroup, the final full flow lives on the original real Hilbert space $L^2(\R^d;A)$.

This is precisely why Paper~5 should be regarded as a concrete multiplier compatibility result within the present program rather than as a mere reformulation of the earlier algebra. The branchwise semigroup architecture is now tied to a concrete translation-invariant multiplier class, with a reconstructed generator, an actual function-space state space, and a well-posed time-evolution theorem. The present paper also gives a relative symbol-sum operator-theoretic identification of the full formal prototype at the level appropriate to the multiplier class: the reconstructed generator equals $\Mss$, and the componentwise strong prototype $\Gcal$ is a graph core for it. The remaining sections therefore no longer need to ask whether the distributed flow exists or how it is reconstructed. They can instead focus on what the reconstructed flow looks like under observation, and on how branch geometry and observation geometry interact in specific distributed examples.


\section{Bounded-Bulk Observation Bookkeeping, Masking, and Conditional Range Inversion}
\label{sec:observation}

\subsection{Observation scope: pushforward, not observability}

The passage from scalar or finite-dimensional branch dynamics to spatially distributed multiplier evolution changes the role of observation in a decisive way. In the earlier papers the observed quantity was either already scalar or was obtained from a finite list of branch-sensitive components by a controlled readout rule. Once the state lives in $L^2(\R^d;A)$, however, the observation mechanism is no longer merely a bookkeeping device. It becomes an additional geometric layer that can amplify, suppress, average, or entirely annihilate different portions of the branch-resolved distributed state.

\paragraph{Scope of the observation results.}
All observation results in this section are confined to bounded bulk maps
\[
\Ocal\in\mathcal L(L^2(\R^d;A),Z).
\]
They concern only the pushforward of the algebraic branch splitting through such maps: observation-space projectors, masking, one-sided norm estimates, transported separated sensor norm comparisons, and conditional closed-range branch inverse under explicitly stated or independently verified bounded-below hypotheses. The section is not an observability theory; sensor lower bounds enter as external hypotheses or through the Fourier-frame verification below. The full list of excluded observation models is fixed in Subsection~\ref{subsec:scope-and-nonclaims}; it is not repeated theorem by theorem.

\paragraph{Terminological convention.}
Throughout this section, the phrases \emph{conditional closed-range branch inverse}, \emph{closed-range sensor recovery}, and \emph{bounded-below branch sensor} have the restricted meaning fixed in the scope registry: recovery is from the projected observed channel, on the actual closed range of the separated branch sensor, after the corresponding bounded-below estimate has been assumed as an input.

This additional layer appears even before randomness enters the picture. The present paper remains fully deterministic at the level of the distributed flow: for each initial datum $Y_0\in L^2(\R^d;A)$, the reconstructed semigroup of Section~5 produces a unique orbit $Y(t)=\mathcal T(t)Y_0$. Observation enters only after this deterministic evolution has been formed. The correct question is therefore not whether the distributed evolution itself is stable or unstable in some abstract sense, but rather how the already reconstructed branchwise flow appears after passing through a distributed observation operator.

Two new effects must be kept separate. The first is \emph{branch selectivity}. An observation map may be sensitive to one branch and insensitive to another, even when the full state contains nontrivial contributions from both. The second is \emph{spatial selectivity}. Since the state is distributed over frequency and physical space, an observation can average away oscillations, suppress unstable low frequencies, or inspect only part of the spatial pattern. These two effects combine to produce masking phenomena that have no analogue in the one-dimensional scalar framework.

For this reason Section~6 does not seek a universal observable threshold. The deterministic work has already been done by the multiplier reconstruction and graph-core results. The observation layer records, and only records, what survives after post-composing that deterministic flow with bounded bulk sensors. It is a bounded-operator pushforward layer, not a second theorem package of comparable strength to the multiplier reconstruction and graph-core results. First, every bounded bulk observation map yields a branch-induced additive splitting of the reconstructed flow at the level of observed signals. Second, the branchwise semigroup bounds from Section~4 push forward to one-sided observation estimates, while exact invisibility occurs whenever the observation annihilates a given branch subspace. Third, under an additional branch-compatible separating-projector hypothesis on the observation space, the full observed signal can itself be decomposed back into branch-labelled observed channels. Fourth, transported separated sensor norm comparisons and closed-range sensor inversion are obtained only after the corresponding separated sensor has already been assumed or independently verified to be bounded below on the relevant branch. The Fourier-frame sensor construction in Proposition~\ref{prop:fourier-frame-separated-branch-sensors} supplies one such verification within the bounded multiplier class. Thus Section~6 is a conditional pushforward-and-range-inverse bookkeeping formalism within the bounded-bulk class fixed in the scope registry.


\paragraph*{Observation-tier convention.}
Section~\ref{sec:observation} uses four strictly different levels of observation statements.
\begin{enumerate}[label=\textup{(\roman*)}]
    \item \emph{Pushforward} means that bounded sensors transport the already constructed branchwise semigroup bounds to one-sided observed upper envelopes.
    \item \emph{Observed-channel extraction} means that, after imposing sensor-projector intertwining, the observed signal can be projected onto branch-labelled observed channels.
    \item \emph{Lower-bound transfer} means that an externally supplied observation-side nondegeneracy estimate and an externally supplied branchwise dynamical lower estimate are combined.
    \item \emph{Closed-range sensor inversion} means inversion only on the actual closed range of a separated branch sensor that has already been assumed bounded below.
\end{enumerate}
These four levels are the full observation scope used below; the excluded observation problems are those listed in Subsection~\ref{subsec:scope-and-nonclaims}.

\begin{remark}[Observation layer as bounded pushforward bookkeeping]
\label{rem:observation-layer-bookkeeping-not-observability}
This is the local observation-layer expansion of item~\textup{(v)} in
Remark~\ref{rem:standing-interpretation-box-paper5}. The observation layer has lower mathematical strength than the deterministic
semigroup and graph-core construction. Its statements are consequences of
post-composing the already reconstructed flow with bounded operators, applying
the branch projectors, and using bounded-below sensor estimates only when they
are explicitly assumed or independently verified. Thus Section~\ref{sec:observation}
should be read as a bounded-bulk pushforward, masking, and conditional
range-inverse bookkeeping layer. It is not an exact observability theorem, not a
unique-continuation statement, and not an inverse theory for recovering the
multiplier symbols from observed data.
\end{remark}

\subsection{Bounded bulk observation class}

Throughout this section, $Z$ denotes a real Banach space that serves as the observation space. An observation map in this paper always means a bounded bulk operator
\begin{equation}\label{eq:observation-operator}
\Ocal : L^2(\R^d;A)\to Z.
\end{equation}
The word \emph{bulk} is part of the hypothesis: the map acts boundedly on the whole ambient $L^2$ state space. This is the sensing class used for the paper's branchwise pushforward and masking statements; the excluded trace, pointwise, time-integrated, and unbounded sensing mechanisms are listed in Subsection~\ref{subsec:scope-and-nonclaims}.

This restriction is mathematically natural for two reasons. First, bounded linear maps are stable under composition with the branch projectors and therefore interact cleanly with the reconstructed semigroup. Second, boundedness is exactly what is needed to transfer branchwise $L^2$ semigroup bounds to branchwise bounds for the observed signal without introducing additional regularity, trace, or admissibility theory.

\paragraph*{Standard bounded-operator proof convention.}
The observation arguments below use only elementary bounded-operator estimates, projector identities, compactness obstruction, and the closed-range consequence of a bounded-below sensor estimate. They are deliberately kept as bookkeeping statements rather than elevated into new observability results. Routine proofs are compressed in the main text and summarized in Appendix~\ref{app:routine-proof-ledger}; the main mathematical work remains the branch normal form, multiplier realization, graph-core identity, and explicit observation limitations.

\begin{definition}[Bounded bulk observation map]\label{def:bounded-bulk-observation}
Let $Z$ be a real Banach space. A \emph{bounded bulk observation map} is a bounded linear operator
\[
\Ocal:L^2(\R^d;A)\to Z.
\]
For each branch $\sigma\in\{+, -, c\}$ we write
\[
\Ocal_\sigma := \Ocal\vert_{L^2_\sigma(\R^d;A)}
\]
for the restriction of $\Ocal$ to the corresponding invariant branch subspace.
\end{definition}

Since the branch projectors $\Pplus$, $\Pminus$, and $\Pc$ extend to bounded projectors on $L^2(\R^d;A)$, one may equivalently regard the branch restrictions as the bounded maps
\[
\Ocal\Pplus,
\qquad
\Ocal\Pminus,
\qquad
\Ocal\Pc.
\]
Thus branch-selective sensitivity is encoded entirely in the way $\Ocal$ acts on these three projected components.

\begin{remark}[Bounded bulk sensors only]\label{rem:bounded-bulk-only}
The observation class of Section~6 consists only of bounded \emph{bulk} maps on the ambient space $L^2(\R^d;A)$. It does not include boundary traces, point evaluations, local flux measurements, time-dependent observation families, or unbounded admissible observation operators in the sense of control theory. In particular, no unbounded operator
\[
C:D(C)\subset L^2(\R^d;A)\to Z
\]
is considered, and no time-integrated admissibility or exact-observability estimate such as
\[
\int_0^T\|C\mathcal T(t)Y_0\|_Z^2\,dt\le C_T\|Y_0\|_{L^2}^2
\]
or its reverse is asserted. Accordingly, Section~6 should be read as a bounded-bulk observation pushforward and conditional range-inverse formalism for the reconstructed multiplier flow, not as a general PDE observation framework for broader models.
\end{remark}

\subsection{Branchwise observed signals}

Let $Y_0\in L^2(\R^d;A)$ and let
\[
Y(t)=\mathcal T(t)Y_0,
\qquad t\ge 0,
\]
be the reconstructed mild solution from Section~5. The observed signal associated with $\Ocal$ is defined by
\begin{equation}\label{eq:observed-signal}
 z(t):=\Ocal Y(t)=\Ocal\mathcal T(t)Y_0,
 \qquad t\ge 0.
\end{equation}
To make the branch structure visible at the observed level, we decompose the initial datum as
\[
Y_0=Y_{0,+}+Y_{0,-}+Y_{0,c},
\qquad
Y_{0,+}:=\Pplus Y_0,
\quad
Y_{0,-}:=\Pminus Y_0,
\quad
Y_{0,c}:=\Pc Y_0.
\]
Since the full semigroup preserves each branch subspace by Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction}, it is natural to define the branchwise observed contributions by
\begin{equation}\label{eq:branchwise-observed-contributions}
 z_\sigma(t):=\Ocal\mathcal T(t)Y_{0,\sigma},
 \qquad \sigma\in\{+, -, c\},
 \qquad t\ge 0.
\end{equation}
The full observed signal is then the sum of these three branchwise components,
\begin{equation}\label{eq:observed-signal-sum}
 z(t)=z_+(t)+z_-(t)+z_c(t).
\end{equation}
Formula \eqref{eq:observed-signal-sum} is elementary, but it should not be dismissed as a tautology. The point is not that the observation space itself carries an intrinsic branch decomposition. Rather, the branch splitting of the state induces an additive splitting of the observed signal without requiring any further spectral decomposition at the observation level. The observation acts on a flow that has already been reconstructed branchwise, and therefore each observed component can be estimated using the semigroup bound of its own branch.

To make this explicit, recall from Section~5 that if
\[
Y_{0,+}=f_{+,0}\Pplus,
\qquad
Y_{0,-}=f_{-,0}\Pminus,
\qquad
Y_{0,c}\in L^2_c(\R^d;A),
\]
then
\[
\mathcal T(t)Y_{0,+} = \bigl(T_+(t)f_{+,0}\bigr)\Pplus,
\qquad
\mathcal T(t)Y_{0,-} = \bigl(T_-(t)f_{-,0}\bigr)\Pminus,
\]
and
\[
\mathcal T(t)Y_{0,c}=\mathcal I_c^{-1}T_c(t)\mathcal I_c(Y_{0,c}).
\]
The branchwise observed contributions are therefore determined by the reduced semigroups together with the way $\Ocal$ sees each branch subspace.

\subsection{One-sided bounded-bulk observation pushforward and masking}

We can now formulate the foundational observation theorem of the paper. It collects the structural identities and one-sided upper bounds that are needed before any sharper visibility statement can be attempted: a branch-induced additive splitting of the observed signal, pushforward of the branchwise semigroup upper bounds through a bounded bulk observation map, and exact masking when the observation annihilates a branch subspace. The theorem is intentionally one-sided; the stronger observation-space separation, conditional norm-comparison, and conditional lower-bound statements are proved only afterward under additional structure on the sensor.

\begin{theorem}[One-sided bounded-bulk observation pushforward and masking]\label{thm:observation-pushforward}
Assume that \((a,b)\) satisfies the standing multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}, and let
$\mathcal T(t)$ be the reconstructed semigroup from
Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction}. Let $Z$ be a real Banach space, let $\Ocal:L^2(\R^d;A)\to Z$ be a bounded linear observation map, and let $Y_0\in L^2(\R^d;A)$ with branch decomposition
\[
Y_0=Y_{0,+}+Y_{0,-}+Y_{0,c}.
\]
Set $Y(t)=\mathcal T(t)Y_0$, and define $z(t)$ and $z_\sigma(t)$ by \eqref{eq:observed-signal}--\eqref{eq:branchwise-observed-contributions}. Then the following hold.
\begin{enumerate}[label=\textup{(\roman*)}]
    \item \textbf{Branch-induced additive splitting of the observed signal.} For every $t\ge 0$,
    \[
    z(t)=z_+(t)+z_-(t)+z_c(t).
    \]
    More explicitly,
    \[
    z_+(t)=\Ocal\bigl(T_+(t)f_{+,0}\,\Pplus\bigr),
    \qquad
    z_-(t)=\Ocal\bigl(T_-(t)f_{-,0}\,\Pminus\bigr),
    \]
    and
    \[
    z_c(t)=\Ocal\mathcal I_c^{-1}T_c(t)\mathcal I_c(Y_{0,c}).
    \]

    \item \textbf{One-sided observation bounds.} If the reduced semigroups satisfy the branchwise growth bounds of Proposition~\ref{prop:branchwise-growth}, then for every $t\ge 0$,
    \begin{align*}
    \norm{z_+(t)}_Z &\le \norm{\Ocal\Pplus}\,e^{\omega_+ t}\norm{f_{+,0}}_{\LtwoR},\\
    \norm{z_-(t)}_Z &\le \norm{\Ocal\Pminus}\,e^{\omega_- t}\norm{f_{-,0}}_{\LtwoR},\\
    \norm{z_c(t)}_Z &\le \norm{\Ocal\Pc}\,e^{\omega_c t}\norm{\mathcal I_c(Y_{0,c})}_{L^2(\R^d;\C)}.
    \end{align*}
    Consequently,
    \[
    \norm{z(t)}_Z
    \le
    \sum_{\sigma\in\{+, -, c\}} \norm{z_\sigma(t)}_Z,
    \]
    so the observed signal inherits an observation-dependent upper operator-norm envelope from the branchwise semigroup estimates. This bound does not identify the actual growth rate of a specific datum and does not assert sharpness of the observed signal envelope for the chosen sensor.

    \item \textbf{Exact masking criterion.} If $\Ocal\vert_{L^2_\sigma(\R^d;A)}=0$ for some branch $\sigma\in\{+, -, c\}$, then
    \[
    z_\sigma(t)=0
    \qquad \text{for all } t\ge 0
    \]
    and for every initial datum $Y_0$. In particular, growth or decay occurring entirely inside that branch is completely invisible to the observation map.
\end{enumerate}
\end{theorem}

\begin{remark}[State-induced, not observation-intrinsic, splitting]\label{rem:observation-not-intrinsic-splitting}
The identity \eqref{eq:observed-signal-sum} should be read primarily as a branch-induced additive splitting of the observed signal, not as an intrinsic decomposition of the observation space $Z$. No such observation-space structure is needed for Theorem~\ref{thm:observation-pushforward}. In particular, the theorem does not assume a priori the existence of bounded projectors $Q_+$, $Q_-$, and $Q_c$ on $Z$, does not assume a splitting $Z=Z_+\oplus Z_-\oplus Z_c$, and does not assume any intertwining relation of the form $\Ocal P_\sigma = Q_\sigma \Ocal$, where $P_+=\Pplus$, $P_-=\Pminus$, and $P_c=\Pc$. All branch information used up to this point enters from the state-side decomposition of $L^2(\R^d;A)$ and from the branch invariance of the reconstructed semigroup. A later subsection introduces these extra observation-space projectors as an additional hypothesis and shows that, when they do exist, the observed branch channels can be extracted directly from the full observed signal. This extraction is still an observed-channel statement, not a state-reconstruction statement.
\end{remark}

\begin{proof}
This is the elementary bounded-observation pushforward argument: branch invariance gives the additive decomposition, boundedness of \(\Ocal P_\sigma\) and the branch semigroup envelopes give the one-sided estimates, and annihilation of a branch gives exact masking. The routine details are collected in Appendix~\ref{app:routine-proof-ledger}.
\end{proof}

\begin{remark}[Observation envelopes are not observed growth laws]\label{rem:observation-envelope-not-growth-law}
The constants appearing in Theorem~\ref{thm:observation-pushforward} have two
separate origins: the branchwise semigroup envelopes
\(e^{\omega_\sigma t}\) and the sensor norms \(\|\Ocal P_\sigma\|\). Even when
\(\omega_\sigma=\operatorname*{ess\,sup}\Re\lambda_\sigma\) gives the sharp
operator-norm envelope on the unobserved branch in the fixed branch-adapted
\(L^2\) Hilbert norm, the observed channel
\(z_\sigma(t)=\Ocal\mathcal T(t)Y_{0,\sigma}\) need not realize that envelope.
The initial datum may have Fourier mass away from near-maximal growth regions,
the sensor may suppress those regions, the sensor may vanish on the branch, or
several observed branch contributions may cancel in \(Z\). Therefore Section~6
uses \(\omega_\sigma\) only as a one-sided propagation envelope. Any lower bound,
sharp observed rate, or reconstruction statement requires the additional
sensor-projector and bounded-below hypotheses imposed later in this section.
\end{remark}

\begin{proposition}[Algebraic aggregate cancellation obstruction]\label{prop:aggregate-cancellation-obstruction}
Let $Z$ be a nonzero real normed space, and fix arbitrary thresholds
\[
a_\sigma>0\qquad (\sigma\in\{+,-,c\}).
\]
Branchwise lower bounds in $Z$, by themselves, do not imply a lower bound for
an aggregate sum. More precisely, there is no strictly positive lower bound,
depending only on the three separate lower thresholds $a_+$, $a_-$, and $a_c$,
for the aggregate norm
\[
\norm{w_+ + w_- + w_c}_Z
\]
under the purely algebraic hypotheses
\[
\norm{w_\sigma}_Z\ge a_\sigma
\qquad
(\sigma\in\{+,-,c\}).
\]
Equivalently, for every such triple of positive thresholds there exist
vectors $w_+,w_-,w_c\in Z$ satisfying all three lower bounds but
\[
w_+ + w_- + w_c=0.
\]
Consequently, any theorem that infers a nontrivial lower bound for an
aggregate observed signal from branchwise lower bounds must include an
additional separation, positivity, orthogonality, sign, phase, coercivity, or
non-cancellation hypothesis.
\end{proposition}

\begin{proof}
Choose $e\in Z$ with $\norm{e}_Z=1$.  Let
\[
R\ge \max\{a_+,a_-,a_c/2\}
\]
and define
\[
w_+:=Re,\qquad w_-:=Re,\qquad w_c:=-2Re.
\]
Then
\[
\norm{w_+}_Z=R\ge a_+,
\qquad
\norm{w_-}_Z=R\ge a_-,
\qquad
\norm{w_c}_Z=2R\ge a_c.
\]
However,
\[
w_+ + w_- + w_c=Re+Re-2Re=0.
\]
Thus the aggregate norm can vanish while all three algebraic summands satisfy
the prescribed positive lower thresholds. Therefore no strictly positive lower
estimate for the aggregate norm can be inferred from those separate lower
thresholds alone. Such a lower estimate would require extra structure, for
instance bounded projectors separating the channels, an orthogonality or
coercivity mechanism, a phase/sign restriction, or an unconditional lower
decomposition constant on the particular observation subspace under
consideration.
\end{proof}

\begin{remark}[No dynamical realizability is asserted]\label{rem:no-dynamical-realizability-in-algebraic-cancellation}
Proposition~\ref{prop:aggregate-cancellation-obstruction} is only an algebraic
obstruction. It does not assert that the branch trajectories generated by the
present semigroups can realize an arbitrary cancelling triple. In the actual
observation problem the summands have the constrained form
\[
z_\sigma(t)=\Ocal\,\mathcal T(t)Y_{0,\sigma},
\qquad \sigma\in\{+,-,c\},
\]
or the corresponding transported branch-coordinate form. These vectors are
therefore tied to the same time parameter, the same bulk observation map, the
same branch-coordinate architecture, and the branchwise semigroup dynamics.
The proposition records only that branchwise lower bounds, without an
additional structural non-cancellation mechanism, are insufficient to imply an
aggregate lower bound.
\end{remark}

\begin{remark}[Lower thresholds versus exact norm data]\label{rem:thresholds-vs-exact-norm-data}
The obstruction concerns lower thresholds on the separated channels, not the
trivial reverse-triangle information available from exact norm data. If the
three exact values $r_\sigma=\norm{w_\sigma}_Z$ are known, then the reverse
triangle inequality gives only
\[
\norm{w_+ + w_- + w_c}_Z
\ge
\max_{\sigma\in\{+,-,c\}}
\left(r_\sigma-\sum_{\tau\ne\sigma}r_\tau\right)_+ .
\]
This bound is positive only in a dominance regime. It does not produce a
positive lower visibility estimate from branchwise lower thresholds alone.
\end{remark}

\begin{remark}[Separated channels versus total observed signal]\label{rem:separated-vs-total-cancellation}
Theorem~\ref{thm:observation-pushforward} gives only the triangle upper bound
\[
\norm{z(t)}_Z\le \norm{z_+(t)}_Z+\norm{z_-(t)}_Z+\norm{z_c(t)}_Z.
\]
Proposition~\ref{prop:aggregate-cancellation-obstruction} explains, at the algebraic level, why there is no converse lower bound from separated lower thresholds alone. As emphasized in Remark~\ref{rem:no-dynamical-realizability-in-algebraic-cancellation}, this is not a dynamical realizability statement for arbitrary branch trajectories. The later projectors $Q_+$, $Q_-$, and $Q_c$ are therefore not cosmetic: they are the additional observation-space structure that prevents separated branch information from being lost in the aggregate signal.
\end{remark}

\subsection{Sensor-projector intertwining and transported branch-sensor norm equivalences}


Theorem~\ref{thm:observation-pushforward} required no structure on the observation space beyond boundedness of $\Ocal$. To go further, we now impose an additional sensor-projector intertwining structure on the observation space. This extra structure does not arise from the branch dynamics alone; rather, it encodes a priori compatibility between the sensor and the state-side branch projectors. Under this imposed compatibility, the observed branch channels can be extracted directly from the full observed signal itself, not merely from the state-side decomposition used to define $z_+$, $z_-$, and $z_c$. This is a conditional observed-channel extraction statement inside an externally supplied sensor-projector architecture; no such projectors are asserted for an arbitrary bounded bulk sensor, and no branch state is reconstructed at this stage.

\begin{definition}[Sensor-projector intertwining structure]\label{def:separating-observation-projectors}
Let $Z$ be a real Banach space and let $\Ocal:L^2(\R^d;A)\to Z$ be a bounded linear observation map. For branch-indexed formulas below we write
\[
P_+:=\Pplus,
\qquad
P_-:=\Pminus,
\qquad
P_c:=\Pc.
\]
A \emph{sensor-projector intertwining structure on $Z$ for $\Ocal$} is a triple of bounded projectors
\[
Q_+,Q_-,Q_c\in\mathcal L(Z)
\]
such that
\[
Q_\sigma Q_\tau = \delta_{\sigma\tau}Q_\sigma,
\qquad
Q_+ + Q_- + Q_c = I_Z,
\qquad
Q_\sigma\Ocal = \Ocal P_\sigma
\]
for all $\sigma,\tau\in\{+, -, c\}$. Each range $Q_\sigma Z$ is equipped throughout with the inherited Banach norm
\[
\norm{w}_{Q_\sigma Z}:=\norm{w}_Z,
\qquad w\in Q_\sigma Z.
\]
Because $Q_\sigma$ is bounded and idempotent, $Q_\sigma Z$ is a closed Banach subspace of $Z$ with this norm.
\end{definition}

\begin{proposition}[Conditional extraction of separated observed channels]\label{prop:observation-space-branch-separation}
Assume the hypotheses of Theorem~\ref{thm:observation-pushforward}; in particular,
assume that \((a,b)\) satisfies the standing multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}. Suppose in addition
that $\Ocal$ admits a sensor-projector intertwining structure $(Q_+,Q_-,Q_c)$ in
the sense of Definition~\ref{def:separating-observation-projectors}. Then for every initial datum $Y_0\in L^2(\R^d;A)$ and every $t\ge 0$,
\[
z(t)=Q_+z(t)+Q_-z(t)+Q_cz(t),
\]
and the separated observed channels coincide exactly with the branchwise observed contributions:
\[
Q_\sigma z(t)=z_\sigma(t)=\Ocal\mathcal T(t)Y_{0,\sigma},
\qquad \sigma\in\{+, -, c\}.
\]
In particular, once such observation-space projectors are imposed, the plus, minus, and complex observed channels can be extracted directly from the single full observed signal $z(t)$. This is extraction of observed channels, not reconstruction of the underlying branch states. No inverse map from observed channels to branch states is constructed in this proposition.
\end{proposition}

\begin{proof}
Since $Q_+ + Q_- + Q_c = I_Z$, one has
\[
z(t)=I_Zz(t)=Q_+z(t)+Q_-z(t)+Q_cz(t).
\]
For a fixed branch $\sigma\in\{+, -, c\}$, the intertwining relation $Q_\sigma\Ocal=\Ocal P_\sigma$ and the commutation $P_\sigma\mathcal T(t)=\mathcal T(t)P_\sigma$ from Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction} give
\[
Q_\sigma z(t)
=
Q_\sigma\Ocal\mathcal T(t)Y_0
=
\Ocal P_\sigma\mathcal T(t)Y_0
=
\Ocal\mathcal T(t)P_\sigma Y_0
=
\Ocal\mathcal T(t)Y_{0,\sigma}
=
z_\sigma(t),
\]
which is the claimed identity.
\end{proof}

\begin{remark}[Interpretive status of conditional observation-space branch separation]
Proposition~\ref{prop:observation-space-branch-separation} is an intertwining statement, not an intrinsic branch-separation theorem for arbitrary sensors. The branch geometry on the observation space is assumed from the outset through the existence of projectors $Q_+$, $Q_-$, and $Q_c$ satisfying $Q_\sigma\Ocal=\Ocal P_\sigma$. Thus the proposition shows that an already branch-compatible sensor architecture faithfully transports the state-side branch splitting to the observation space at the level of observed channels; it does not show that such an architecture must exist for a general bounded bulk sensor, and it does not by itself reconstruct the underlying branch states.
\end{remark}

\begin{definition}[Assumed bounded-below separated branch sensor]\label{def:boundedly-visible-branch-sensor}
Assume that $\Ocal$ admits a sensor-projector intertwining structure in the sense of Definition~\ref{def:separating-observation-projectors}. For a branch $\sigma\in\{+, -, c\}$, set
\[
\Ocal_\sigma := \Ocal\vert_{L^2_\sigma(\R^d;A)}:L^2_\sigma(\R^d;A)\to Q_\sigma Z.
\]
The separated branch sensor $\Ocal_\sigma$ is said to satisfy an \emph{assumed bounded-below branch-sensor estimate} on the full branch space $L^2_\sigma(\R^d;A)$ if there exist constants
\[
0<m_\sigma\le M_\sigma<\infty
\]
such that
\[
m_\sigma \norm{Y}_{L^2(\R^d;A)}
\le
\norm{\Ocal_\sigma Y}_{Q_\sigma Z}
\le
M_\sigma \norm{Y}_{L^2(\R^d;A)}
\qquad
\text{for all } Y\in L^2_\sigma(\R^d;A).
\]
Equivalently, $\Ocal_\sigma$ is assumed to be a bounded embedding of the full branch space into $Q_\sigma Z$ with closed range. In infinite dimensions this assumption is closer to choosing an equivalent branch norm in observation space than to ordinary partial sensing. The resulting inverse is the inverse on the actual closed range of $\Ocal_\sigma$.
\end{definition}

\begin{remark}[Strength of bounded-below branch sensor]
\label{rem:global-bounded-visibility-is-strong}
The lower bound in Definition~\ref{def:boundedly-visible-branch-sensor} is an input assumption giving a global lower bound on the entire infinite-dimensional branch space $L^2_\sigma(\R^d;A)$. It should therefore be read as a closed-range embedding assumption, equivalently as the choice of an observation norm that remains comparable to the full branch norm, not as a generic feature of bounded sensors. Finite-rank, compact, smoothing, local-averaging, or narrow frequency-window sensors do not satisfy this estimate on the full branch space in the usual infinite-dimensional setting.

Consequently, the transported norm-equivalence and closed-range inverse statements below are conditional statements for sensors that have already been assumed bounded below on the relevant branch. Smoothing or frequency-selective sensors usually require a restricted state class, such as a finite-dimensional truncation, a band-limited subspace, or another closed subspace on which the restricted sensor is bounded below.
\end{remark}

\begin{example}[Canonical separated branch sensor]
\label{ex:canonical-separated-branch-sensor}
Let
\[
Z:=L^2_+(\R^d;A)\oplus L^2_-(\R^d;A)\oplus L^2_c(\R^d;A)
\]
with its Hilbert direct-sum norm, and define
\[
\Ocal Y:=(\Pplus Y,\Pminus Y,\Pc Y).
\]
Let $Q_+$, $Q_-$, and $Q_c$ be the coordinate projectors on $Z$. Then
\[
Q_\sigma\Ocal=\Ocal P_\sigma,
\qquad \sigma\in\{+, -, c\},
\]
so $\Ocal$ admits a sensor-projector intertwining structure. Moreover, on each branch,
\[
\norm{\Ocal_\sigma Y}_{Q_\sigma Z}=\norm{Y}_{L^2(\R^d;A)},
\qquad Y\in L^2_\sigma(\R^d;A).
\]
Hence every branch sensor $\Ocal_\sigma$ satisfies the assumed bounded-below branch-sensor estimate with $m_\sigma=M_\sigma=1$. This example shows that the hypothesis is nonempty. It also illustrates its meaning: the sensor must retain the full branch norm, up to equivalence of norms, rather than merely sample, smooth, or partially window the branch.
\end{example}

\begin{proposition}[Fourier-frame separated branch sensors]
\label{prop:fourier-frame-separated-branch-sensors}
Fix a branch $\sigma\in\{+,-,c\}$ and identify $L^2_\sigma(\R^d;A)$ with its scalar branch-coordinate model space $X_\sigma$, namely $\LtwoR$ for $\sigma=+$ and $\sigma=-$, and $\LtwoC$, viewed as a real Hilbert space, for $\sigma=c$. Let
\[
m_1,\ldots,m_N\in L^\infty(\R^d)
\]
be scalar multiplier symbols satisfying the pointwise frame bounds
\[
0<m^2\le \sum_{j=1}^N |m_j(\xi)|^2\le M^2<\infty
\qquad\text{for a.e. }\xi\in\R^d.
\]
Define
\[
\mathcal B_\sigma f:=\bigl(m_1(D)f,\ldots,m_N(D)f\bigr).
\]
For every branch, regard this as a map into $L^2(\R^d;\C^N)$, with the target read as a real Hilbert space. Then $\mathcal B_\sigma$ is bounded and bounded below:
\[
m\norm{f}_{X_\sigma}
\le
\norm{\mathcal B_\sigma f}_{L^2(\R^d;\C^N)}
\le
M\norm{f}_{X_\sigma}.
\]
If real-valued plus or minus branch outputs are desired, one may additionally impose the usual real-state symmetry on the symbols $m_j$; the bounded-below estimate itself only requires the real Hilbert-space target $L^2(\R^d;\C^N)_\mathbb R$. After transporting this estimate back through $\mathcal I_+$, $\mathcal I_-$, or $\mathcal I_c$, it gives a bounded-below separated branch sensor in the sense of Definition~\ref{def:boundedly-visible-branch-sensor}.
\end{proposition}

\begin{proof}
By Plancherel,
\[
\norm{\mathcal B_\sigma f}^2
=
\sum_{j=1}^N\norm{m_j(D)f}_{L^2}^2
=
\int_{\R^d}\left(\sum_{j=1}^N |m_j(\xi)|^2\right)|\widehat f(\xi)|^2\,d\xi.
\]
The frame bounds give
\[
m^2\norm{f}_{L^2}^2
\le
\norm{\mathcal B_\sigma f}^2
\le
M^2\norm{f}_{L^2}^2.
\]
Taking square roots gives the desired two-sided estimate. The real plus and minus branch cases are immediate. The complex branch is the same calculation after transport through the real-linear isometry $\mathcal I_c$, with $\LtwoC$ viewed as a real Hilbert space. Pulling the estimate back through the appropriate branch-coordinate isometry gives the asserted separated branch sensor estimate.
\end{proof}

\begin{remark}[Complementary frequency-window frame sensors]
\label{rem:complementary-frequency-window-frame-sensors}
A simple non-canonical example is obtained by taking a measurable set $E\subset\R^d$ and
\[
m_1=\mathbf 1_E,
\qquad
m_2=\mathbf 1_{\R^d\setminus E}.
\]
Then $|m_1(\xi)|^2+|m_2(\xi)|^2=1$ for a.e. $\xi$, so the two-channel sensor
\[
f\longmapsto \bigl(\mathbf 1_E(D)f,\mathbf 1_{\R^d\setminus E}(D)f\bigr)
\]
is isometric on the scalar branch-coordinate space. This is not the canonical branch projector: it is a frequency splitting sensor inside a fixed branch. More general overlapping frequency windows give bounded-below branch sensors whenever the multiplier-frame lower bound stays positive.
\end{remark}

\begin{lemma}[Bookkeeping transport of an assumed bounded-below separated sensor estimate]\label{thm:conditional-branch-norm-transfer}
Assume the hypotheses of Proposition~\ref{prop:observation-space-branch-separation};
in particular, assume that \((a,b)\) satisfies the standing multiplier-semigroup
package of Definition~\ref{def:standing-multiplier-semigroup-package}. Fix a branch $\sigma\in\{+, -, c\}$ and let
\[
\Ocal_\sigma := \Ocal\vert_{L^2_\sigma(\R^d;A)}:L^2_\sigma(\R^d;A)\to Q_\sigma Z.
\]
Suppose that $\Ocal_\sigma$ satisfies the assumed bounded-below branch-sensor estimate on the full branch space $L^2_\sigma(\R^d;A)$ in the sense of Definition~\ref{def:boundedly-visible-branch-sensor}, with constants $m_\sigma$ and $M_\sigma$. The lower estimate below is not derived from the semigroup; it is exactly the previously assumed bounded-below estimate applied to the branch orbit. Then for every initial datum $Y_0\in L^2(\R^d;A)$ and every $t\ge 0$,
\[
m_\sigma\norm{\mathcal T(t)Y_{0,\sigma}}_{L^2(\R^d;A)}
\le
\norm{Q_\sigma z(t)}_{Q_\sigma Z}
=
\norm{Q_\sigma z(t)}_{Z}
=
\norm{z_\sigma(t)}_{Z}
\le
M_\sigma\norm{\mathcal T(t)Y_{0,\sigma}}_{L^2(\R^d;A)}.
\]
This conclusion is only the transport of the assumed two-sided sensor estimate to
vectors lying on the branch orbit. It does not assert observability for arbitrary
bounded sensors and does not reconstruct the branch state; conditional closed-range branch inverse is
reserved for Corollary~\ref{cor:branch-reconstruction-observation}, where the inverse is
restricted to the actual closed range of \(\Ocal_\sigma\).
\end{lemma}

\begin{proof}
By Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction}, the branch component $\mathcal T(t)Y_{0,\sigma}$ remains in $L^2_\sigma(\R^d;A)$ for every $t\ge 0$. Applying the assumed bounded-below branch-sensor estimate from Definition~\ref{def:boundedly-visible-branch-sensor} to the single vector $\mathcal T(t)Y_{0,\sigma}$ gives
\[
m_\sigma\norm{\mathcal T(t)Y_{0,\sigma}}_{L^2(\R^d;A)}
\le
\norm{\Ocal_\sigma\mathcal T(t)Y_{0,\sigma}}_{Q_\sigma Z}
\le
M_\sigma\norm{\mathcal T(t)Y_{0,\sigma}}_{L^2(\R^d;A)}.
\]
Because $\mathcal T(t)Y_{0,\sigma}\in L^2_\sigma(\R^d;A)$ and Proposition~\ref{prop:observation-space-branch-separation} identifies the extracted observed channel with the restricted sensor output, one has
\[
\Ocal_\sigma\mathcal T(t)Y_{0,\sigma}
=
\Ocal\mathcal T(t)Y_{0,\sigma}
=
z_\sigma(t)
=
Q_\sigma z(t).
\]
Substituting this identity into the previous display and using the inherited norm convention
$\norm{w}_{Q_\sigma Z}=\norm{w}_Z$ for $w\in Q_\sigma Z$ yields the claimed transported sensor norm comparison.
\end{proof}

\begin{remark}[Scope of the transported sensor-norm statement]\label{rem:scope-conditional-branch-norm-transfer}
The lower estimate
\[
\norm{\Ocal_\sigma Y}_{Q_\sigma Z}
\ge
m_\sigma\norm{Y}_{L^2(\R^d;A)}
\]
is an input assumption on the separated branch sensor. The lemma only says that once this closed-range embedding estimate is available, it is transported along the branch orbit by the reconstructed semigroup.

Thus the result applies directly to embedding-like or closed-range sensors, and to sensors restricted to admissible subclasses on which a lower bound has been proved separately. It does not imply that compact, smoothing, local-averaging, finite-rank, or narrow frequency-window observations are globally visible on the full infinite-dimensional branch space.
\end{remark}

\begin{proposition}[Compact sensors are not bounded below on infinite-dimensional branch spaces]\label{prop:compact-sensors-not-globally-visible}
Let $X_\sigma=L^2_\sigma(\R^d;A)$ be infinite-dimensional and let $Z_\sigma$ be a Banach space. If
\[
\mathcal B:X_\sigma\to Z_\sigma
\]
is compact, then $\mathcal B$ cannot be bounded below on $X_\sigma$. Equivalently, there is no constant $m_\sigma>0$ such that
\[
\norm{\mathcal B Y}_{Z_\sigma}
\ge
m_\sigma\norm{Y}_{X_\sigma}
\qquad
\text{for all }Y\in X_\sigma.
\]
\end{proposition}

\begin{proof}
This is the standard compactness obstruction: a bounded-below compact operator would have closed range and a bounded inverse on that range, so the identity would factor through a compact map and would itself be compact on an infinite-dimensional Banach space. Details are recorded in Appendix~\ref{app:routine-proof-ledger}.
\end{proof}

\begin{corollary}[Conditional separated lower-bound transfer]\label{cor:conditional-lower-bound-separated}
Assume the hypotheses of Lemma~\ref{thm:conditional-branch-norm-transfer}; in
particular, assume that \((a,b)\) satisfies the standing multiplier-semigroup
package of Definition~\ref{def:standing-multiplier-semigroup-package}. Suppose in addition that there exist a subset $\mathfrak X_\sigma\subseteq L^2_\sigma(\R^d;A)$ and constants $c_\sigma>0$ and $\gamma_\sigma\in\R$ such that
\[
\norm{\mathcal T(t)Y}_{L^2(\R^d;A)}
\ge c_\sigma e^{\gamma_\sigma t}\norm{Y}_{L^2(\R^d;A)}
\qquad
\text{for all } t\ge 0 \text{ and all } Y\in \mathfrak X_\sigma.
\]
Then every datum with $Y_{0,\sigma}\in \mathfrak X_\sigma$ satisfies
\[
\norm{Q_\sigma z(t)}_Z
=
\norm{z_\sigma(t)}_Z
\ge
m_\sigma c_\sigma e^{\gamma_\sigma t}\norm{Y_{0,\sigma}}_{L^2(\R^d;A)}
\qquad
\text{for all } t\ge 0.
\]
\end{corollary}

\begin{proof}
Combining the transported sensor-norm estimate in Lemma~\ref{thm:conditional-branch-norm-transfer} with the additional lower dynamical estimate on $\mathfrak X_\sigma$ gives
\[
\norm{Q_\sigma z(t)}_Z
\ge
m_\sigma\norm{\mathcal T(t)Y_{0,\sigma}}_{L^2(\R^d;A)}
\ge
m_\sigma c_\sigma e^{\gamma_\sigma t}\norm{Y_{0,\sigma}}_{L^2(\R^d;A)},
\]
which is the claimed lower bound.
\end{proof}


The preceding compactness obstruction used the closed-range consequence of a
bounded-below estimate in a special case; we record the same principle
separately for the branch-reconstruction statements below.

\begin{lemma}[Closed-range branch-sensor inverse]
\label{lem:closed-range-reconstruction-bounded-below}
Let $X$ and $Z$ be Banach spaces, and let
\[
O:X\to Z
\]
be bounded and linear. No Hilbert structure is used in this lemma; completeness
of the two normed spaces is the only ambient-space requirement. Suppose that there exists $m>0$ such that
\[
\norm{Ox}_Z\ge m\norm{x}_X
\qquad
\text{for all }x\in X.
\]
Then $O$ is injective, $\mathrm{Ran}(O)$ is closed in $Z$, and the inverse
\[
O^{-1}:\mathrm{Ran}(O)\to X
\]
is bounded with
\[
\norm{O^{-1}z}_X\le m^{-1}\norm{z}_Z
\qquad
\text{for all }z\in \mathrm{Ran}(O).
\]
No inverse on all of $Z$ is asserted unless surjectivity $\mathrm{Ran}(O)=Z$ is separately assumed.
\end{lemma}

\begin{proof}
This is the usual bounded-below-operator argument: the lower bound gives injectivity, makes preimages of Cauchy sequences Cauchy, closes the range, and gives the inverse norm bound \(\|O^{-1}\|\le m^{-1}\). The short verification is recorded in Appendix~\ref{app:routine-proof-ledger}.
\end{proof}

\begin{corollary}[Bookkeeping form of the conditional closed-range branch inverse]
\label{cor:branch-reconstruction-observation}
Assume the hypotheses of Proposition~\ref{prop:observation-space-branch-separation};
in particular, assume that \((a,b)\) satisfies the standing
multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}. The lower bound in this corollary is an input assumption on the separated observation operator, not a consequence of the dynamics generated by $\Acal$. No control-theoretic observability inequality is proved here; the statement records only the stable inverse-on-range consequence of an already assumed bounded-below branch sensor.

Fix a branch $\sigma\in\{+, -, c\}$ and suppose that the separated branch sensor
\[
\Ocal_\sigma:L^2_\sigma(\R^d;A)\to Q_\sigma Z
\]
satisfies the assumed bounded-below branch-sensor estimate on the full branch space $L^2_\sigma(\R^d;A)$ in the sense of Definition~\ref{def:boundedly-visible-branch-sensor}. In particular, there exists $m_\sigma>0$ such that
\[
\norm{\Ocal_\sigma Y}_{Q_\sigma Z}
\ge
m_\sigma\norm{Y}_{L^2(\R^d;A)}
\qquad
\text{for all }Y\in L^2_\sigma(\R^d;A).
\]
Then $\mathrm{Ran}(\Ocal_\sigma)$ is closed in $Q_\sigma Z$, and the inverse map
\[
\mathcal R_\sigma
:=
(\Ocal_\sigma)^{-1}
:
\mathrm{Ran}(\Ocal_\sigma)
\to
L^2_\sigma(\R^d;A)
\]
is bounded with the quantitative estimate
\[
\norm{\mathcal R_\sigma w}_{L^2(\R^d;A)}
\le
m_\sigma^{-1}\norm{w}_{Q_\sigma Z}
=
m_\sigma^{-1}\norm{w}_{Z}
\qquad
\text{for all }w\in\mathrm{Ran}(\Ocal_\sigma).
\]
Moreover, for every observed trajectory generated by the model and every $t\ge 0$,
\[
Q_\sigma z(t)\in \mathrm{Ran}(\Ocal_\sigma),
\]
and
\[
\mathcal T(t)Y_{0,\sigma}
=
\mathcal R_\sigma Q_\sigma z(t).
\]
In particular, the displayed formula is only an inverse-on-range statement: under the stated separated-sensor and bounded-below branch-sensor hypotheses, the branch state is obtained by applying the bounded inverse to data already known to lie in the closed observed range along model-generated trajectories, and only through the inverse on the actual closed range $\mathrm{Ran}(\Ocal_\sigma)$. No surjectivity of $\Ocal_\sigma$ onto the whole channel $Q_\sigma Z$ is asserted.
\end{corollary}

\begin{proof}
By Definition~\ref{def:separating-observation-projectors}, $Q_\sigma Z$ is a
closed Banach subspace of the real Banach space $Z$, equipped with the
inherited norm. Apply Lemma~\ref{lem:closed-range-reconstruction-bounded-below} with
\[
X=L^2_\sigma(\R^d;A),
\qquad
Z=Q_\sigma Z,
\qquad
O=\Ocal_\sigma,
\qquad
m=m_\sigma.
\]
This gives injectivity, closedness of $\mathrm{Ran}(\Ocal_\sigma)$ in $Q_\sigma Z$, boundedness of the inverse on this closed observed range, and the estimate
\[
\norm{\mathcal R_\sigma w}_{L^2(\R^d;A)}
\le
m_\sigma^{-1}\norm{w}_{Q_\sigma Z}
\qquad
(w\in\mathrm{Ran}(\Ocal_\sigma)).
\]
By Proposition~\ref{prop:observation-space-branch-separation}, one has
\[
Q_\sigma z(t)
=
\Ocal\mathcal T(t)Y_{0,\sigma}
=
\Ocal_\sigma\mathcal T(t)Y_{0,\sigma}.
\]
Since $\mathcal T(t)Y_{0,\sigma}\in L^2_\sigma(\R^d;A)$, the displayed identity shows that $Q_\sigma z(t)\in\mathrm{Ran}(\Ocal_\sigma)$. Applying the inverse $\mathcal R_\sigma$ to this identity yields
\[
\mathcal R_\sigma Q_\sigma z(t)=\mathcal T(t)Y_{0,\sigma},
\]
which proves the inverse-on-range formula and its stability estimate. The closed-range inverse and its norm bound come entirely from the assumed lower estimate for $\Ocal_\sigma$; the semigroup dynamics are used only to identify model-generated observed channels as elements of $\mathrm{Ran}(\Ocal_\sigma)$.
\end{proof}

\begin{remark}[Conditional closed-range sensor inversion versus observability and onto-channel inversion]
\label{rem:range-vs-onto-channel-reconstruction}
Corollary~\ref{cor:branch-reconstruction-observation} uses the bounded-below estimate as an input. Without it, the algebraic inverse on $\mathrm{Ran}(\Ocal_\sigma)$ may be unbounded, and small observation errors may correspond to large branch-state errors. Thus conditional exact branch recovery here means applying a stable inverse on the closed observed range of a separated branch sensor that has already been assumed bounded below.

If one strengthens the sensor hypothesis to
\[
\mathrm{Ran}(\Ocal_\sigma)=Q_\sigma Z,
\]
then the same inverse may be viewed as a bounded operator
\[
\mathcal R_\sigma:Q_\sigma Z\to L^2_\sigma(\R^d;A).
\]
Without this additional surjectivity assumption, the inverse is defined only on $\mathrm{Ran}(\Ocal_\sigma)$. This is nevertheless sufficient for observed trajectories generated by the present model, because $Q_\sigma z(t)\in\mathrm{Ran}(\Ocal_\sigma)$ for every such trajectory and every $t\ge0$.
\end{remark}

\begin{remark}[From state-induced splitting to observation-space splitting]\label{rem:state-vs-observation-splitting}
Theorem~\ref{thm:observation-pushforward} is a state-induced decomposition statement: it says that once the state has been split branchwise, the observation inherits an additive splitting. Proposition~\ref{prop:observation-space-branch-separation} and Lemma~\ref{thm:conditional-branch-norm-transfer} are stronger. They show that, under an additional intertwining structure on the sensor, the observation space itself carries enough explicitly assumed branch geometry to project the single measured signal back onto the three observed channels and to transport a pre-assumed two-sided sensor-side norm comparison to those separated channels on the chosen branch. Neither statement derives an intrinsic observability inequality from the branch dynamics alone.
\end{remark}

\begin{remark}[Three distinct levels of growth information]\label{rem:three-growth-levels}
The present framework requires a consistent separation between three kinds of statements:
\begin{enumerate}[label=\textup{(\roman*)}]
    \item the branchwise numbers $\omega_\sigma$, which are admissible operator-norm upper envelopes for the reduced semigroups, and are sharp in the fixed branch-adapted $L^2$ Hilbert norm only when chosen as $\operatorname*{ess\,sup}\Re\lambda_\sigma$;
    \item datum-specific branch behavior, which depends on the spectral support and amplitudes of the chosen initial branch component;
    \item observed branch behavior, which depends in addition on the bounded bulk observation map and may mask part or all of the branch content.
\end{enumerate}
Confusing these levels leads to overstatements. In particular, $\omega_\sigma>0$ does not by itself imply exponential growth for every datum, and even genuine branch amplification for a specific datum need not survive observation.
\end{remark}

\begin{corollary}[Auxiliary conditional lower-bound transfer]\label{cor:conditional-lower-bound}
Assume the hypotheses of Theorem~\ref{thm:observation-pushforward}; in particular,
assume that \((a,b)\) satisfies the standing multiplier-semigroup package of
Definition~\ref{def:standing-multiplier-semigroup-package}. Fix a branch $\sigma\in\{+, -, c\}$ and suppose there exists a constant $\kappa_\sigma>0$ such that
\[
\norm{\Ocal Y}_Z\ge \kappa_\sigma \norm{Y}_{L^2(\R^d;A)}
\qquad
\text{for all } Y\in L^2_\sigma(\R^d;A).
\]
Assume in addition that there exist a subset $\mathfrak X_\sigma\subseteq L^2_\sigma(\R^d;A)$ and constants $c_\sigma>0$ and $\gamma_\sigma\in\R$ such that
\[
\norm{\mathcal T(t)Y}_{L^2(\R^d;A)}
\ge c_\sigma e^{\gamma_\sigma t}\norm{Y}_{L^2(\R^d;A)}
\qquad
\text{for all } t\ge 0 \text{ and all } Y\in \mathfrak X_\sigma.
\]
Then every initial datum whose branch component satisfies $Y_{0,\sigma}\in \mathfrak X_\sigma$ obeys
\[
\norm{z_\sigma(t)}_Z
\ge \kappa_\sigma c_\sigma e^{\gamma_\sigma t}\norm{Y_{0,\sigma}}_{L^2(\R^d;A)}
\qquad
\text{for all } t\ge 0.
\]
In particular, this corollary does not manufacture a lower observation bound from the abstract framework itself: any genuine lower bound in the present paper arises only by combining an explicitly assumed observation-side nondegeneracy estimate with a lower dynamical estimate supplied on the same branch.
\end{corollary}

\begin{proof}
Since $Y_{0,\sigma}\in L^2_\sigma(\R^d;A)$ and the reconstructed flow preserves branch subspaces, one has $\mathcal T(t)Y_{0,\sigma}\in L^2_\sigma(\R^d;A)$ for all $t\ge 0$. Therefore
\[
\norm{z_\sigma(t)}_Z
= \norm{\Ocal \mathcal T(t)Y_{0,\sigma}}_Z
\ge \kappa_\sigma \norm{\mathcal T(t)Y_{0,\sigma}}_{L^2(\R^d;A)}
\ge \kappa_\sigma c_\sigma e^{\gamma_\sigma t}\norm{Y_{0,\sigma}}_{L^2(\R^d;A)},
\]
which is the claimed lower bound.
\end{proof}

\begin{remark}[Why Corollary~\ref{cor:conditional-lower-bound} is deliberately conditional]\label{rem:conditional-lower-bound-transfer}
Corollary~\ref{cor:conditional-lower-bound} remains useful even after Lemma~\ref{thm:conditional-branch-norm-transfer} and Corollary~\ref{cor:conditional-lower-bound-separated}. The latter requires the stronger existence of separating projectors on the observation space together with an assumed bounded-below branch-sensor estimate for the separated sensor on the chosen branch, whereas Corollary~\ref{cor:conditional-lower-bound} records only an auxiliary transfer wrapper that uses a branchwise lower bound for $\Ocal$ together with a model-specific dynamical lower bound supplied from outside the general framework on the same branch. Proposition~\ref{prop:frequency-window} in Section~7 supplies the clearest concrete source of such lower bounds in the present paper.
\end{remark}

\paragraph*{What is proved in Section~6.}
Theorem~\ref{thm:observation-pushforward} proves a branch-induced additive splitting of the observed signal, branchwise upper-envelope bounds obtained by pushing the semigroup estimates through a bounded bulk observation map, and an exact masking criterion when a branch is annihilated by the sensor. Proposition~\ref{prop:observation-space-branch-separation} then shows conditionally that, under an imposed sensor-projector intertwining structure on the sensor, the single full observed signal can be projected onto its plus, minus, and complex observed channels directly in the observation space. Definition~\ref{def:boundedly-visible-branch-sensor} isolates the extra bounded-below branch-sensor assumption, Lemma~\ref{thm:conditional-branch-norm-transfer} transfers that pre-assumed bounded-below branch-sensor estimate to the corresponding observed channel, and Proposition~\ref{prop:compact-sensors-not-globally-visible} records why compact sensors cannot satisfy such an assumed global lower bound on an infinite-dimensional branch space. Corollary~\ref{cor:conditional-lower-bound-separated} records the additional lower-bound consequence once an external lower dynamical estimate is supplied on the same branch, Corollary~\ref{cor:branch-reconstruction-observation} records conditional closed-range branch inverse on any branch where the separated sensor is bounded below, with an explicit inverse bound on its closed observed range; the inverse is applied only to projected observed trajectories, which automatically belong to that range. Corollary~\ref{cor:conditional-lower-bound} is kept only as an auxiliary wrapper for the weaker non-separated situation.

\paragraph*{Interpretation.}
The first theorem provides one-sided upper estimates together with an exact masking criterion. Lower information enters only through the additional hypotheses isolated later: an observation-space projector structure, a bounded-below separated branch sensor, and, when needed, an external lower dynamical estimate on the same branch. Proposition~\ref{prop:fourier-frame-separated-branch-sensors} gives a concrete multiplier-frame mechanism for verifying such a sensor lower bound in non-canonical separated branch channels, while Proposition~\ref{prop:compact-sensors-not-globally-visible} records the corresponding limitation for compact sensors.

Section~6 should therefore be read as a layered bounded-bulk observation bookkeeping formalism, not as a new observability theory. At the base level it gives transport and masking statements for arbitrary bounded bulk sensors. At the stronger level it proves conditional observation-space extraction of observed branch channels together with transported separated sensor norm comparisons for the subclass of sensors that are endowed a priori with projectors intertwining the branch decomposition, and it records only auxiliary conditional lower-bound wrappers once a lower dynamical estimate is supplied on the same branch. Observation lower bounds are never manufactured by the semigroup dynamics; they enter only through stated sensor hypotheses or through independently checkable conditions such as the Fourier-frame bound in Proposition~\ref{prop:fourier-frame-separated-branch-sensors}.


\subsection{Masking and observation dependence}

The masking criterion in Theorem~\ref{thm:observation-pushforward} shows that branch invisibility can be exact, not merely approximate. This is already enough to break any hope for a universal threshold law formulated purely at the level of the full generator. A branch may carry nontrivial growth while the observed signal remains identically blind to it, simply because the observation map annihilates that branch. Even when complete annihilation does not occur, the branch factors $\norm{\Ocal\Pplus}$, $\norm{\Ocal\Pminus}$, and $\norm{\Ocal\Pc}$ can differ greatly, so the same full distributed evolution may look stable, unstable, or nearly neutral depending on how it is observed.

It is useful to distinguish three levels of behavior.

\begin{enumerate}[label=\textup{(\alph*)}]
    \item \textbf{Intrinsic branch behavior.} This is determined by the reduced semigroups $T_+$, $T_-$, and $T_c$, and is quantified in the present paper by the growth exponents and constants from Proposition~\ref{prop:branchwise-growth}.

    \item \textbf{Observed branch behavior.} This is determined jointly by the intrinsic branch behavior and by the observation map $\Ocal$. The same branch semigroup can produce very different observed signals under different sensors.

    \item \textbf{Aggregated observed behavior.} The final observed signal is the sum of the three observed branch contributions. At the algebraic level, separate branchwise control alone does not preclude cancellation or domination among the summands; aggregate lower-bound statements therefore require additional observation-space separation or non-cancellation structure.
\end{enumerate}

This trichotomy is the proper distributed-setting version of the observation-dependent philosophy already present in the earlier papers. The novelty of the present multiplier setting is that masking can now arise not only from branch selectivity but also from the distributed geometry of the state: certain frequencies may be emphasized or suppressed by the observation device, and oscillatory components in the complex branch may be attenuated by averaging even when they are not identically annihilated.

For this reason Section~7 will treat explicit examples not merely to illustrate the formulas but to show how branch geometry and observation geometry interact. The point is no longer just whether a branch grows or decays. The point is whether that growth or decay survives the chosen observation protocol and, when relevant, whether any lower visibility can be justified under additional observation-side nondegeneracy assumptions.

\subsection{Bernoulli-readout compatibility}

The present paper remains consistent with the observation-layer philosophy of Paper~1. The distributed evolution itself is deterministic: the state $Y(t)=\mathcal T(t)Y_0$ is produced entirely by the reconstructed semigroup, with no stochastic forcing, no random switching, and no stochastic branch selection in the multiplier generator. Randomness, if introduced at all, should enter only after deterministic propagation and after the bounded observation map has been applied.

One convenient way to express this compatibility is to insert a deterministic finite-dimensional post-processing stage after the distributed observation. Let $\Lambda:Z\to\R^m$ be a bounded linear map and define
\[
q(t):=\Lambda\Ocal Y(t)\in\R^m.
\]
Any Bernoulli or other finite-valued readout may then be applied to $q(t)$ at the same terminal observation layer used in the earlier papers, without changing the deterministic distributed multiplier architecture developed here. In other words, the present section provides the deterministic pre-readout signal, while the actual random readout---if one wishes to add it---still belongs strictly to the observation layer.

\begin{proposition}[Compatibility with post hoc finite-dimensional readout]
Let $\Ocal:L^2(\R^d;A)\to Z$ and $\Lambda:Z\to\R^m$ be bounded linear maps. Then the processed signal
\[
q(t):=\Lambda\Ocal\mathcal T(t)Y_0
\]
admits the same branch-induced additive splitting
\[
q(t)=q_+(t)+q_-(t)+q_c(t),
\qquad
q_\sigma(t):=\Lambda z_\sigma(t),
\]
and satisfies the corresponding one-sided bounds
\[
\norm{q_\sigma(t)}_{\R^m}
\le
\norm{\Lambda}\,\norm{z_\sigma(t)}_Z.
\]
Consequently any subsequent finite-valued readout applied to $q(t)$ leaves the distributed flow deterministic and uses randomness only at the terminal observation stage.
\end{proposition}

\begin{proof}
The decomposition follows immediately from linearity of $\Lambda$ together with Theorem~\ref{thm:observation-pushforward}. The norm estimate is simply the boundedness inequality for $\Lambda$.
\end{proof}

This proposition is structural: it confirms that the distributed multiplier realization remains faithful to the guiding principle of the series, namely that randomness should enter through observation after deterministic propagation rather than by being built directly into the evolution law.

\section{Three Generator Benchmarks, One Observation Benchmark, and a Domain-Gap Stress Test}

The preceding sections developed the branchwise multiplier architecture at a structural level. The purpose of the present section is twofold. First, it records three generator benchmarks and one observation benchmark in which the reduced symbols, branchwise semigroups, reconstruction formulas, and observation effects can all be written down explicitly. These examples are benchmark subclasses only because their symbols satisfy the standing multiplier hypotheses; no statement in this section should be read as saying that every polynomial constant-coefficient symbol is admissible. Second, it ends with a diagnostic stress test that probes the strongest raw-domain distinction in the paper: the componentwise strong prototype $\Gcal$ may have a strictly smaller domain than the branchwise reconstructed generator $\Acal$, even though
\[
\overline{\Gcal}^{\,\mathrm{graph}}=\Acal
\]
in the graph sense. The first two benchmark computations are intentionally elementary; the third generator benchmark is more diagnostic and displays transport, parabolic damping, and complex oscillatory-diffusive behavior in a single admissible pair. Together they make the branch geometry and the observation-dependent masking mechanism concrete before the final stress test isolates the pre-closure domain issue that is invisible in the gentler examples.

\subsection{Shifted heat amplification benchmark}

We begin with a benchmark constant-coefficient model
\begin{equation}\label{eq:heat-amp-symbols}
 a(\xi)=-\abs{\xi}^2+\alpha,
 \qquad
 b(\xi)=\beta,
 \qquad \alpha,\beta\in\R.
\end{equation}
The prototype operator therefore takes the form
\[
\Gcal
=
I_A\otimes (\alpha+\Delta)+\Lu\otimes \beta,
\]
where $\Delta$ is the Laplacian on $\R^d$. This is a shifted heat benchmark rather than a claim that the unshifted heat equation amplifies. The possible growth below is caused by the zero-order shifts $\alpha\pm\beta$ on top of the dissipative Laplacian multiplier $-\abs{\xi}^2$; diffusion still dominates at high frequency. This is the cleanest benchmark because diffusion enters through the common term $-\abs{\xi}^2$, while branch selection enters through the constant shift $\beta$.

The reduced branch symbols are immediately
\begin{equation}\label{eq:heat-amp-reduced-symbols}
\lambda_+(\xi)=-\abs{\xi}^2+\alpha+\beta,
\qquad
\lambda_-(\xi)=-\abs{\xi}^2+\alpha-\beta,
\qquad
\lambda_c(\xi)=-\abs{\xi}^2+\alpha+i\beta.
\end{equation}
Hence the corresponding reduced semigroups are the Fourier multipliers
\begin{align*}
\widehat{T_+(t)f}(\xi)&=e^{t(-\abs{\xi}^2+\alpha+\beta)}\widehat f(\xi),\\
\widehat{T_-(t)f}(\xi)&=e^{t(-\abs{\xi}^2+\alpha-\beta)}\widehat f(\xi),\\
\widehat{T_c(t)g}(\xi)&=e^{t(-\abs{\xi}^2+\alpha+i\beta)}\widehat g(\xi).
\end{align*}
Thus the branchwise growth exponents can be chosen as
\[
\omega_+=\alpha+\beta,
\qquad
\omega_-=\alpha-\beta,
\qquad
\omega_c=\alpha,
\]
so, in this benchmark, these choices equal the sharp values $\operatorname*{ess\,sup}\Re\lambda_\bullet$ in the fixed branch-adapted $L^2$ Hilbert norm, and the corresponding $L^2$ multiplier envelopes are $e^{\omega_+ t}$, $e^{\omega_- t}$, and $e^{\omega_c t}$. Even here, the actual behavior of a specific datum depends on where its Fourier mass is located inside each branch; sharp semigroup envelopes are not datum-specific growth laws.

The model separates the low-frequency and high-frequency mechanisms very cleanly. On the plus branch, frequencies satisfying
\[
\abs{\xi}^2<\alpha+\beta
\]
are amplified whenever $\alpha+\beta>0$, whereas high frequencies still decay because diffusion dominates there. On the minus branch the same discussion applies with $\alpha-\beta$ in place of $\alpha+\beta$. On the complex branch the parameter $\beta$ does not change the real growth envelope at all; it produces only a uniform oscillatory phase factor $e^{i\beta t}$ on top of the diffusive envelope $e^{t(-\abs{\xi}^2+\alpha)}$.

This is already enough to illustrate a basic visibility lesson. Suppose that $\Ocal:L^2(\R^d;A)\to Z$ is a bounded observation map whose restriction to the plus branch is nonzero, and let the initial datum be purely plus-branch, say $Y_0=f_0\Pplus$. Then
\[
Y(t)=\bigl(T_+(t)f_0\bigr)\Pplus,
\qquad
z(t)=\Ocal\bigl((T_+(t)f_0)\Pplus\bigr).
\]
If $\widehat f_0$ is concentrated near $\xi=0$ and $\alpha+\beta>0$, the specific plus-branch datum exhibits low-frequency amplification. Whether that amplification is actually visible in $z(t)$ depends entirely on whether the observation map sees those frequencies through its action on the plus branch. The model therefore makes the logic of Section~6 completely explicit: instability is branchwise and frequency-local, while visibility is observation-dependent.


\begin{proposition}[Low-frequency plus-branch growth caused by zero-order shift]
\label{prop:heat-plus-low-frequency-amplification}
Assume the shifted heat benchmark \eqref{eq:heat-amp-symbols} and suppose
\(\alpha+\beta>0\). Let
\[
r_+(\xi):=-|\xi|^2+\alpha+\beta.
\]
If \(E\subset\R^d\) is measurable and there exists \(\delta>0\) such that
\[
0<\int_E |\widehat f_0(\xi)|^2\,d\xi<\infty,
\qquad
r_+(\xi)\ge\delta
\quad\text{for a.e. }\xi\in E,
\]
then the plus-branch solution satisfies
\[
\int_E |\widehat{T_+(t)f_0}(\xi)|^2\,d\xi
\ge
 e^{2\delta t}\int_E |\widehat f_0(\xi)|^2\,d\xi
\qquad\text{for every }t\ge0.
\]
In particular, any nonzero portion of the datum supported in a positive-growth
frequency window grows at least exponentially at rate \(\delta\) on that window.
\end{proposition}

\begin{proof}
For the shifted heat benchmark,
\[
\widehat{T_+(t)f_0}(\xi)=e^{t r_+(\xi)}\widehat f_0(\xi).
\]
Therefore
\[
\int_E |\widehat{T_+(t)f_0}(\xi)|^2\,d\xi
=
\int_E e^{2t r_+(\xi)}|\widehat f_0(\xi)|^2\,d\xi
\ge
 e^{2\delta t}\int_E |\widehat f_0(\xi)|^2\,d\xi.
\]
\end{proof}

This lower bound is frequency-local and datum-dependent. It is not a uniform
lower bound on the whole plus branch, and it is not a property of the unshifted
heat equation. It only records the growth forced by Fourier mass lying inside a
region where the shifted reduced exponent is positive.

\subsection{Mixed decay-oscillation model}

A second benchmark is obtained by choosing
\begin{equation}\label{eq:mixed-decay-osc-symbols}
 a(\xi)=-\abs{\xi}^2,
 \qquad
 b(\xi)=\gamma\abs{\xi},
 \qquad \gamma>0.
\end{equation}
The symbol $b(\xi)=\gamma\abs{\xi}$ corresponds to the nonlocal multiplier $\gamma\abs{D}$. This choice is intentional: it shows that the present theorem package is slightly broader than the admissible local constant-coefficient differential-operator benchmarks and is more accurately described as a translation-invariant Fourier-multiplier realization theorem. Locality of $b(D)$ is not used in the branch theorem: the framework requires only measurable finite-a.e. Fourier symbols satisfying the Hermitian reality condition and the reduced real-part upper bounds from the standing multiplier-semigroup package. The local shifted heat benchmark from Section~7.1 remains the main constant-coefficient differential-operator benchmark, while the present example serves to test the branch architecture against a simple nonlocal oscillatory branch.

The reduced symbols are then
\begin{equation}\label{eq:mixed-decay-osc-reduced}
\lambda_+(\xi)=-\abs{\xi}^2+\gamma\abs{\xi},
\qquad
\lambda_-(\xi)=-\abs{\xi}^2-\gamma\abs{\xi},
\qquad
\lambda_c(\xi)=-\abs{\xi}^2+i\gamma\abs{\xi}.
\end{equation}
Several distinct branch behaviors now coexist in the same distributed evolution model. The sharp real-part envelopes in this example are
\[
\omega_+^{\mathrm{sh}}=\frac{\gamma^2}{4},
\qquad
\omega_-^{\mathrm{sh}}=0,
\qquad
\omega_c^{\mathrm{sh}}=0.
\]
The zero values for the minus and complex branches are operator-norm envelopes, not claims of uniform exponential decay.

First, the minus branch is non-expansive because
\[
\lambda_-(\xi)\le 0
\qquad\text{for all }\xi\in\R^d,
\]
with strict spectral damping away from $\xi=0$. On the full real scalar space $\LtwoR$ this is not a uniform exponential decay statement: the essential supremum of $\lambda_-$ is $0$, because arbitrarily low frequencies approach the origin. Second, the complex branch has purely diffusive real part and a nontrivial frequency-dependent oscillatory phase:
\[
 e^{t\lambda_c(\xi)}=e^{-t\abs{\xi}^2}e^{i\gamma t\abs{\xi}}.
\]
Thus the complex branch carries oscillation under a non-expansive heat envelope, with datum-dependent decay for data whose Fourier mass is separated from zero frequency. Third, the plus branch can exhibit band-limited exponential amplification on the annulus
\[
0<\abs{\xi}<\gamma,
\]
because there
\[
-\abs{\xi}^2+\gamma\abs{\xi}>0.
\]
The branch geometry is therefore mixed in a very concrete sense: one branch is non-expansive with damping away from zero frequency, one branch oscillates under the same non-expansive diffusive envelope, and one branch has positive exponential growth rate on a frequency band.

This model is useful because it shows why observed damping does not imply full-state damping. Consider a bounded observation map $\Ocal$ such that
\[
\Ocal\vert_{L^2_+(\R^d;A)}=0.
\]
Then by Theorem~\ref{thm:observation-pushforward} the entire plus branch is exactly masked. If the initial datum contains a plus component supported in the amplifying band $0<\abs{\xi}<\gamma$, that specific component may amplify, yet it contributes nothing to the observed signal. The observed dynamics are then built only from the minus and complex branches, both of which are non-expansive and are damped away from zero frequency by the diffusive factor $e^{-t\abs{\xi}^2}$. In particular, for data whose observed Fourier mass avoids a neighborhood of the origin, an observer may record genuinely decaying or oscillatory-decaying behavior even though the full state contains an unstable hidden channel. Without such a spectral-gap restriction, the correct full-branch statement is non-expansiveness and datum-dependent decay, not uniform exponential decay on all of $\LtwoR$.

The mixed decay-oscillation model therefore isolates a second conceptual point. Observation dependence is not just a nuisance added after the forward theory is finished. It changes what one is even allowed to infer about the full distributed evolution from measured data. The hidden branch can be dynamically significant while remaining observationally absent.


\subsection{Three-regime cancellation-coupled generator benchmark}
\label{subsec:three-regime-generator-benchmark}

The preceding two generator examples are deliberately simple. We now record a
more diagnostic admissible pair in which the three reduced branches have visibly
different operator characters. Fix a vector $c\in\R^d$ and set
\begin{equation}\label{eq:three-regime-symbols}
 a(\xi)=-|\xi|^2+i\,c\cdot\xi,
 \qquad
 b(\xi)=|\xi|^2+i\,c\cdot\xi.
\end{equation}
Both symbols are finite a.e. and satisfy the Hermitian symmetry condition. The
reduced symbols are
\begin{equation}\label{eq:three-regime-reduced-symbols}
\lambda_+(\xi)=2i\,c\cdot\xi,
\qquad
\lambda_-(\xi)=-2|\xi|^2,
\qquad
\lambda_c(\xi)=-|\xi|^2-c\cdot\xi+i\bigl(|\xi|^2+c\cdot\xi\bigr).
\end{equation}
Thus the plus branch is a skew-adjoint transport-type multiplier, the minus
branch is genuinely parabolic, and the complex branch is an oscillatory
multiplier under a shifted heat envelope. The real-part envelopes are
\[
\operatorname*{ess\,sup}\Re\lambda_+=0,
\qquad
\operatorname*{ess\,sup}\Re\lambda_-=0,
\qquad
\operatorname*{ess\,sup}\Re\lambda_c=\frac{|c|^2}{4},
\]
because
\[
\Re\lambda_c(\xi)=-|\xi|^2-c\cdot\xi
=\frac{|c|^2}{4}-\left|\xi+\frac{c}{2}\right|^2.
\]
This single pair therefore places three different behaviors inside one
branch-algebraic flow: transport on $+$, heat damping on $-$, and oscillatory
shifted heat behavior on $c$.

\begin{proposition}[Three-regime admissible multiplier pair]
\label{prop:three-regime-admissible-pair}
For the symbols \eqref{eq:three-regime-symbols}, the standing
multiplier-semigroup package holds. The reduced semigroups are given by
\[
\widehat{T_+(t)f}(\xi)=e^{2itc\cdot\xi}\widehat f(\xi),
\qquad
\widehat{T_-(t)g}(\xi)=e^{-2t|\xi|^2}\widehat g(\xi),
\]
and
\[
\widehat{T_c(t)h}(\xi)
=
\exp\{t[-|\xi|^2-c\cdot\xi+i(|\xi|^2+c\cdot\xi)]\}\widehat h(\xi).
\]
In the fixed branch-adapted Hilbert norm their sharp operator-norm envelopes are
\[
\|T_+(t)\|=1,
\qquad
\|T_-(t)\|=1,
\qquad
\|T_c(t)\|=e^{|c|^2t/4}.
\]
\end{proposition}

\begin{proof}
The Hermitian identities follow from
\[
a(-\xi)=-|\xi|^2-i\,c\cdot\xi=\overline{a(\xi)},
\qquad
b(-\xi)=|\xi|^2-i\,c\cdot\xi=\overline{b(\xi)}.
\]
The reduced-symbol formulas in \eqref{eq:three-regime-reduced-symbols} are
immediate. The displayed real-part bounds verify \textup{(A2)}. The semigroup
formulas then follow from the branchwise Fourier-multiplier construction of
Section~4, and the coordinate-exact operator-norm envelopes follow from the multiplier
norm identities for the reduced branches in the fixed branch-adapted Hilbert norm.
\end{proof}

\begin{remark}[Common-symbol origin of the three-regime benchmark]
\label{rem:three-regime-not-toy}
This benchmark should not be read as escaping the direct-sum nature of the
branch reduction. After branch reduction the model is still a direct sum of
scalar multiplier evolutions. The narrower point is that it is not a randomly
prescribed direct sum: the three reduced scalar regimes arise from one common
pair \((a,b)\) before branch reduction. The branch maps cancel and recombine
that common pair differently: the plus branch loses the parabolic part and
retains only transport, the minus branch loses the transport part and retains a
parabolic heat multiplier, while the complex branch keeps a shifted heat envelope
with an oscillatory phase. The final domain-gap stress test below is the
one-dimensional raw-domain version of the same cancellation mechanism.
\end{remark}

\subsection{Frequency-window observation example}

The third benchmark fixes not a new generator, but a new observation geometry. We return to the shifted heat benchmark \eqref{eq:heat-amp-symbols} and assume for definiteness that
\[
\alpha+\beta>0.
\]
Then the plus branch possesses an unstable low-frequency region
\[
\mathcal U_+:=\{\xi\in\R^d:\abs{\xi}^2<\alpha+\beta\}.
\]
We now define a bounded frequency-window observation that sees only a prescribed measurable set $B\subset\R^d$ on the plus branch. All lower-bound or reconstruction language in this subsection is therefore restricted to the $B$-windowed branch class or to the observed $B$-windowed subcomponent. No statement below gives a lower bound, inverse map, or recovery result for the full plus branch unless the datum is restricted to the windowed range of $\Pi_B$ or unless $B$ is the whole frequency space up to null sets. Initially we assume only symmetry up to null sets, namely
\[
\mathbf 1_B(-\xi)=\mathbf 1_B(\xi)
\quad\text{for a.e. }\xi\in\R^d.
\]
By Lemma~\ref{lem:hermitian-representatives-symmetric-cutoffs}, we replace $B$ by an exactly symmetric representative, still denoted by $B$, satisfying
\[
B=-B.
\]
This replacement does not change the Fourier multiplier projection on any $L^2$ space, because the characteristic functions agree almost everywhere. Throughout the rest of this subsection, $B$ denotes this fixed exactly symmetric representative; remaining measure-zero qualifications concern only the usual equivalence of $L^2$ functions.
For $f\in \LtwoR$, let
\[
\Pi_B f:=\F^{-1}(\mathbf 1_B\widehat f).
\]
Because $B=-B$ in this fixed representative, the multiplier $\mathbf 1_B$ preserves the Fourier reality symmetry of real-valued signals, so $\Pi_B$ maps $\LtwoR$ into itself and is the orthogonal Fourier multiplier projection onto the band $B$. We then set
\begin{equation}\label{eq:freq-window-observation}
\Ocal_B(Y):=\Pi_B\mathcal I_+(\Pplus Y),
\qquad
Y\in L^2(\R^d;A).
\end{equation}
Then $\Ocal_B:L^2(\R^d;A)\to \LtwoR$ is bounded, with
\[
\norm{\Ocal_B}
\le
\norm{\Pi_B}\,\norm{\mathcal I_+\Pplus}
=\norm{\mathcal I_+\Pplus}.
\]
Thus the bounded-bulk observation and masking parts of Section~6 apply directly to symmetric frequency windows. However, this example should not be read as an instance of the bounded-below branch-sensor hypothesis from Definition~\ref{def:boundedly-visible-branch-sensor}. The map \(\Pi_B\) is an orthogonal projection. Unless the admissible plus-branch data are restricted to \(\operatorname{Ran}\Pi_B\), or unless \(B\) covers the full frequency space up to null sets, \(\Pi_B\) has a nontrivial kernel and hence cannot be bounded below on the full plus branch.

The one-channel sensor \(\Ocal_B\) should not, by itself, be read as an instance
of the full separated observation-space projector package from Section~6: it
observes only the plus branch. The following enlarged version records the exact
way to embed the same frequency-window idea into the separated projector
architecture.

\begin{proposition}[Separated frequency-window output on windowed branch classes]
\label{prop:frequency-window-separated-output}
Let \(B\subset\R^d\) be the fixed exactly symmetric measurable representative from the preceding convention. Let the same
symbol \(\Pi_B\) denote the Fourier projection with multiplier \(\mathbf 1_B\) on
both real and complex \(L^2\) spaces, and set
\[
L^2_{\R,B}:=\{f\in L^2_{\R}(\R^d):\Pi_B f=f\},
\qquad
L^2_{\C,B}:=\{h\in L^2_{\C}(\R^d):\Pi_B h=h\}.
\]
Define the separated windowed observation space
\[
Z_B:=L^2_{\R,B}\oplus L^2_{\R,B}\oplus L^2_{\C,B}.
\]
We regard \(Z_B\) as a real Hilbert direct sum, where the third summand is the underlying real
Hilbert space of \(L^2_{\C,B}\). All coordinate projections on \(Z_B\), including
\(Q_c\), are real-linear projections; no complex-linear structure is imposed on
the full observation space. Define
\[
\widetilde{\Ocal}_B Y
:=
\bigl(
\Pi_B\mathcal I_+(\Pplus Y),\,
\Pi_B\mathcal I_-(\Pminus Y),\,
\Pi_B\mathcal I_c(\Pc Y)
\bigr).
\]
Then
\[
\widetilde{\Ocal}_B:L^2(\R^d;A)\to Z_B
\]
is a bounded bulk observation map. If \(Q_+\), \(Q_-\), and \(Q_c\) denote the
coordinate projections on \(Z_B\), then
\[
Q_\sigma\widetilde{\Ocal}_B
=
\widetilde{\Ocal}_B P_{\sigma},
\qquad
\sigma\in\{+,-,c\}.
\]
Thus the enlarged windowed sensor satisfies the sensor-projector intertwining
architecture of Section~6.

Moreover, the separated sensor on a branch is bounded below only on the
corresponding windowed branch class
\[
\begin{aligned}
L^2_{+,B}
&:=\{f\Pplus:\Pi_B f=f\},\\
L^2_{-,B}
&:=\{g\Pminus:\Pi_B g=g\},\\
L^2_{c,B}
&:=\{Y_c\in L^2_c(\R^d;A):
      \Pi_B\mathcal I_cY_c=\mathcal I_cY_c\}.
\end{aligned}
\]
On these restricted classes the lower bound is isometric. On the full branch
spaces it fails whenever \(\R^d\setminus B\) has positive measure.
\end{proposition}

\begin{proof}
Boundedness follows from the boundedness of the branch projections, the branch
coordinate isometries, and the orthogonal projection property of \(\Pi_B\). The
identity
\[
Q_+\widetilde{\Ocal}_B Y
=(\Pi_B\mathcal I_+(\Pplus Y),0,0)
=
\widetilde{\Ocal}_B(\Pplus Y)
\]
is immediate, and the minus and complex branches are identical. Hence the
coordinate projections \(Q_+\), \(Q_-\), and \(Q_c\) supply the separated
observation-space projectors required in Section~6.

If \(Y=f\Pplus\in L^2_{+,B}\), then
\(Q_+\widetilde{\Ocal}_B Y=(f,0,0)\), so the branch sensor is isometric on the
restricted plus class. The same argument applies to the minus and complex
windowed classes. Conversely, if \(\R^d\setminus B\) has positive measure, one
can choose nonzero branch data with Fourier support in \(B^c\); the corresponding
windowed observation is zero, so no positive lower bound can hold on the full
branch space.
\end{proof}

For a purely plus-branch initial datum $Y_0=f_0\Pplus$, the observed signal is
\[
z_B(t)=\Ocal_B\mathcal T(t)Y_0
=\Pi_B T_+(t)f_0,
\]
and therefore
\[
\widehat{z_B(t)}(\xi)=\mathbf 1_B(\xi)e^{t(-\abs{\xi}^2+\alpha+\beta)}\widehat f_0(\xi).
\]
This formula displays how frequency-selective observation dependence works in this model.

\begin{remark}[Frequency windows versus globally visible separated sensors]
\label{rem:frequency-window-not-global-visibility}
The map \(\Ocal_B\) is intentionally a one-channel sensor: it observes only the plus branch and only the frequencies selected by \(B\). It therefore illustrates branch masking and frequency-window detection under the bounded observation framework, but it does not instantiate the full sensor-projector architecture of Definition~\ref{def:separating-observation-projectors} and it does not satisfy the bounded-below hypothesis of Definition~\ref{def:boundedly-visible-branch-sensor} on the full plus branch unless the omitted frequency set has measure zero.

The correct lower-bound statement for this example is restricted. After the admissible plus-branch class is reduced to the band-limited class
\[
L^2_{+,B}(\R^d;A)
:=
\{\,f\Pplus\in L^2_+(\R^d;A):\Pi_B f=f\,\},
\]
the window becomes bounded below on that restricted class. Without such a restriction, any nonzero plus-branch state with Fourier support in \(B^c\) is completely invisible to \(\Ocal_B\).
\end{remark}

\begin{proposition}[Frequency windows are not globally visible]
\label{prop:frequency-window-not-globally-visible}
Let \(B\subset\R^d\) be the fixed exactly symmetric measurable representative from the preceding convention, and let \(\Ocal_B\) be defined by \eqref{eq:freq-window-observation}. If
\[
|\R^d\setminus B|>0,
\]
then the restricted plus-branch sensor
\[
\Ocal_B\vert_{L^2_+(\R^d;A)}:
L^2_+(\R^d;A)\to \LtwoR
\]
is not bounded below on \(L^2_+(\R^d;A)\). In particular, there is no constant \(m>0\) such that
\[
m\norm{Y}_{L^2(\R^d;A)}
\le
\norm{\Ocal_B Y}_{\LtwoR}
\qquad
\text{for all }Y\in L^2_+(\R^d;A).
\]
\end{proposition}

\begin{proof}
Set \(C:=\R^d\setminus B\). Since the fixed representative satisfies \(B=-B\), the complement satisfies \(C=-C\) exactly. Choose a nonzero \(\widehat f\in \LtwoC\) supported in \(C\) and satisfying the Hermitian symmetry
\[
\widehat f(-\xi)=\overline{\widehat f(\xi)}.
\]
Then \(f=\F^{-1}\widehat f\) is real-valued and nonzero. Define
\[
Y:=f\Pplus\in L^2_+(\R^d;A).
\]
By construction,
\[
\Ocal_B Y=\Pi_B f=0,
\]
whereas \(\norm{Y}_{L^2(\R^d;A)}>0\). Hence no positive lower bound can hold on the full plus branch.
\end{proof}

\begin{proposition}[Visibility only on the $B$-windowed plus-branch class]
\label{prop:frequency-window-restricted-visibility}
Let \(B\subset\R^d\) be the fixed exactly symmetric measurable representative from the preceding convention. Define
\[
L^2_{+,B}(\R^d;A)
:=
\{\,f\Pplus\in L^2_+(\R^d;A):\Pi_B f=f\,\}.
\]
Then \(\Ocal_B\) is bounded below on \(L^2_{+,B}(\R^d;A)\). More precisely, for every \(Y=f\Pplus\in L^2_{+,B}(\R^d;A)\),
\[
\norm{\Ocal_B Y}_{\LtwoR}
=
\norm{f}_{\LtwoR}
=
\norm{Y}_{L^2(\R^d;A)}.
\]
Thus the frequency-window sensor is bounded below only after the admissible plus-branch state class is restricted to the windowed range \(\operatorname{Ran}\Pi_B\), or in the degenerate case where \(B\) is the full frequency space up to null sets.
\end{proposition}

\begin{proof}
If \(Y=f\Pplus\in L^2_{+,B}(\R^d;A)\), then \(\Pi_B f=f\). Therefore
\[
\Ocal_B Y
=
\Pi_B\mathcal I_+(\Pplus Y)
=
\Pi_B f
=
f.
\]
Taking norms gives the displayed identity, using the branch-coordinate isometry of \(\mathcal I_+\).
\end{proof}

\begin{proposition}[Frequency-window subcomponent growth on detected unstable support]\label{prop:frequency-window}
Assume the shifted heat benchmark \eqref{eq:heat-amp-symbols} with $\alpha+\beta>0$, let $B\subset\R^d$ be the fixed exactly symmetric measurable representative from the preceding convention, and let $\Ocal_B$ be the bounded observation \eqref{eq:freq-window-observation}. Write
\[
r(\xi):=-\abs{\xi}^2+\alpha+\beta.
\]
This proposition concerns only the observed $B$-windowed component
\(z_B(t)=\Pi_B T_+(t)f_0\). It does not give a lower bound for the full plus-branch state, does not recover frequency mass outside \(B\), and does not imply a global bound of the form \(\|z_B(t)\|\gtrsim e^{\delta t}\|f_0\|\) for arbitrary data. Any lower estimate below is only a subcomponent estimate on an explicitly displayed positive-measure subwindow inside \(B\) on which the initial datum has nonzero Fourier mass.
\begin{enumerate}[label=\textup{(\roman*)}]
    \item If $|B\cap \mathcal U_+|=0$, then every observed frequency satisfies
    \[
    r(\xi)\le 0
    \qquad\text{for a.e. }\xi\in B,
    \]
    and therefore
    \[
    \norm{z_B(t)}_{\LtwoR}
    \le
    \norm{\Pi_B f_0}_{\LtwoR}
    \qquad \text{for all } t\ge 0.
    \]
    In particular, the observation window detects no exponential growth even if $f_0$ has unstable plus-branch mass outside $B$.

    \item If there exists a measurable subset
    $E\subset B\cap \mathcal U_+$ of positive measure, after discarding a null subset if necessary, such that
    \[
    \int_E \abs{\widehat f_0(\xi)}^2\,d\xi>0,
    \]
    then there exist a measurable subset $E'\subset E\subset B\cap \mathcal U_+$ of positive measure and a constant $\delta>0$ such that
    \[
    r(\xi)\ge \delta
    \qquad\text{for a.e. }\xi\in E',
    \]
    and hence
    \[
    \int_{E'} \abs{\widehat{z_B(t)}(\xi)}^2\,d\xi
    =
    \int_{E'} e^{2t r(\xi)}\abs{\widehat f_0(\xi)}^2\,d\xi
    \ge
    e^{2\delta t}\int_{E'} \abs{\widehat f_0(\xi)}^2\,d\xi.
    \]
    In particular, the observation window captures a uniformly exponentially growing observed $B$-windowed subcomponent, not a globally reconstructed plus-branch state.
\end{enumerate}
\end{proposition}

\begin{proof}
Part \textup{(i)} follows immediately from the Fourier formula for $z_B(t)$. If $|B\cap \mathcal U_+|=0$, then the exponent $r(\xi)$ is nonpositive for a.e. $\xi\in B$, so
\[
\abs{\widehat{z_B(t)}(\xi)}
\le
\mathbf 1_B(\xi)\abs{\widehat f_0(\xi)}
\qquad\text{for a.e. }\xi,
\]
and Plancherel yields the displayed $L^2$ bound.

For part \textup{(ii)}, the fixed representative convention lets us discard a
Lebesgue null subset of the witnessing set and then work with
\[
E\subset B\cap\mathcal U_+
\]
without changing any $L^2$ quantity. On this set the factor \(\mathbf 1_B\) in
\(\widehat{z_B(t)}\) equals one a.e., and the explicit Fourier formula gives
\[
\int_E \abs{\widehat{z_B(t)}(\xi)}^2\,d\xi
=
\int_E e^{2t r(\xi)}\abs{\widehat f_0(\xi)}^2\,d\xi.
\]
Set
\[
F:=\{\xi\in E:\ \abs{\widehat f_0(\xi)}>0\}.
\]
Because
\[
\int_E \abs{\widehat f_0(\xi)}^2\,d\xi>0,
\]
the set $F$ has positive measure. Since $E\subset \mathcal U_+$ after the harmless null-set adjustment, one has $r(\xi)>0$ for a.e. $\xi\in E$. For each $n\in\mathbb N$, define
\[
E_n:=\{\xi\in E:\ r(\xi)\ge 1/n\}.
\]
Because $r(\xi)>0$ on $E$, we have
\[
E=\bigcup_{n=1}^\infty E_n,
\qquad\text{hence}\qquad
F=\bigcup_{n=1}^\infty (F\cap E_n).
\]
Since $F$ has positive measure, there exists $n_0\in\mathbb N$ such that $F\cap E_{n_0}$ has positive measure. Set
\[
E':=F\cap E_{n_0},
\qquad
\delta:=1/n_0.
\]
Then $E'\subset E$ has positive measure, $r(\xi)\ge \delta$ for a.e. $\xi\in E'$, and $\abs{\widehat f_0(\xi)}>0$ for a.e. $\xi\in E'$. Therefore
\[
\int_{E'} \abs{\widehat f_0(\xi)}^2\,d\xi>0.
\]
Restricting the preceding identity to $E'$ yields
\[
\int_{E'} \abs{\widehat{z_B(t)}(\xi)}^2\,d\xi
=
\int_{E'} e^{2t r(\xi)}\abs{\widehat f_0(\xi)}^2\,d\xi
\ge
 e^{2\delta t}\int_{E'} \abs{\widehat f_0(\xi)}^2\,d\xi,
\]
which is the claimed lower exponential bound.
\end{proof}

\begin{remark}[Why the symmetry condition is needed]
If a raw representative of $B$ is not symmetric, then $\Pi_B f$ need not be real-valued even when $f$ is real-valued. Thus the present example first imposes symmetry up to null sets and then fixes an exactly symmetric representative, without changing the associated $L^2$ multiplier. This keeps the observation inside the real framework of Section~6 while avoiding repeated modulo-null-set qualifications.
\end{remark}

This proposition expresses the sharpest distributed-model lesson of the section, but only at the observed-subcomponent level. Visibility is determined not only by the branch seen by the observation, but also by the frequencies seen within that branch. A bounded sensor may therefore miss an instability by projecting onto the wrong spectral window. Conversely, when the observation window captures unstable frequencies carrying nontrivial data, the result gives a lower exponential bound on a positive-measure detected subwindow. It is not a full observed-norm growth theorem for arbitrary data and not a reconstruction statement for the whole plus branch. The global observation exclusions are those recorded in Subsection~\ref{subsec:scope-and-nonclaims}.


\begin{proposition}[Raw-domain gap stress test]\label{prop:stress-test-cancellation}
In the one-dimensional specialization \(d=1\), consider the multiplier pair
\[
 a(\xi)=-\xi^2+i\xi,
 \qquad
 b(\xi)=\xi^2+i\xi.
\]
This pair satisfies \textup{(A1)}--\textup{(A2)}. Indeed,
\[
a(-\xi)=\overline{a(\xi)},
\qquad
b(-\xi)=\overline{b(\xi)},
\]
and the reduced symbols computed below have sharp real-part upper envelopes
\[
\operatorname*{ess\,sup}_{\xi\in\R}\Re\lambda_+(\xi)=0,
\qquad
\operatorname*{ess\,sup}_{\xi\in\R}\Re\lambda_-(\xi)=0,
\qquad
\operatorname*{ess\,sup}_{\xi\in\R}\Re\lambda_c(\xi)=\frac14.
\]
Then the reduced symbols are
\[
 \lambda_+(\xi)=2i\xi,
 \qquad
 \lambda_-(\xi)=-2\xi^2,
 \qquad
 \lambda_c(\xi)=(-\xi^2-\xi)+i(\xi^2+\xi).
\]
By the Sobolev convention fixed in Remark~\ref{rem:sobolev-convention}, on the plus branch one has
\[
\Dom(G_+) = H^1(\R),
\]
whereas the componentwise strong prototype imposes second-order regularity there. Such real-valued examples are concrete: one may prescribe the Hermitian Fourier transform
\[
\widehat f(\xi)=(1+\xi^2)^{-1},
\]
which gives a nonzero real-valued function with $f\in H^1(\R)\setminus H^2(\R)$. More precisely, for any real-valued $f\in H^1(\R)\setminus H^2(\R)$ and $Y=f\Pplus$, one has
\[
Y\in \Dom(\Acal)\setminus \Dom(\Gcal).
\]
In particular, the raw-domain inclusion $\Gcal\subseteq \Acal$ is strict for this benchmark pair, even though Proposition~\ref{prop:strong-prototype-core} shows that $\Gcal$ is graph-dense in $\Acal$. Thus the example detects a strict pre-closure domain mismatch, not a failure of graph-density.
\end{proposition}

\begin{proof}
The Hermitian identities for \(a\) and \(b\) are immediate from the displayed formulas, so \textup{(A1)} holds. The formulas for $\lambda_+$, $\lambda_-$, and $\lambda_c$ are immediate. Their real parts are
\[
\Re\lambda_+(\xi)=0,
\qquad
\Re\lambda_-(\xi)=-2\xi^2,
\qquad
\Re\lambda_c(\xi)=-\xi^2-\xi=\frac14-\Bigl(\xi+\frac12\Bigr)^2.
\]
Hence \textup{(A2)} holds with the sharp fixed-Hilbert-norm envelopes stated above. Since $\lambda_+(\xi)=2i\xi$, the plus-branch reduced generator is the Fourier multiplier with symbol $2i\xi$, so by Remark~\ref{rem:sobolev-convention} its domain is exactly $H^1(\R)$. The displayed example $\widehat f(\xi)=(1+\xi^2)^{-1}$ is even and real-valued, hence Hermitian; Plancherel gives
\[
\int_\R (1+\xi^2)\abs{\widehat f(\xi)}^2\,d\xi<\infty,
\qquad
\int_\R (1+\xi^2)^2\abs{\widehat f(\xi)}^2\,d\xi=\infty,
\]
so its inverse Fourier transform is real-valued and lies in $H^1(\R)\setminus H^2(\R)$. Now let $f\in H^1(\R)\setminus H^2(\R)$ be real-valued and set $Y=f\Pplus$. Since $f\in H^1(\R)$, the plus-branch reduced multiplier $\lambda_+(D)=2\partial_x$ is defined on $f$; the minus and complex components vanish. Hence $Y\in\Dom(\Acal)$ by Theorem~\ref{thm:prescribed-multiplier-flow-reconstruction}.

On the other hand,
\[
a(\xi)\widehat Y(\xi)=(-\xi^2+i\xi)\widehat f(\xi)\Pplus,
\qquad
b(\xi)\widehat Y(\xi)=(\xi^2+i\xi)\widehat f(\xi)\Pplus.
\]
If both quantities belonged to $L^2(\R;\AC)$, then both $\xi^2\widehat f$ and $\xi\widehat f$ would lie in $L^2(\R)$, forcing $f\in H^2(\R)$, contrary to the choice of $f$. Hence $Y\notin\Dom(\Gcal)$.
\end{proof}

\begin{remark}[Raw-domain gap versus graph-closure equality]
Proposition~\ref{prop:stress-test-cancellation} should be read only as a raw-domain diagnostic. It shows that the componentwise strong prototype can impose more separate regularity on $a(D)Y$ and $b(D)Y$ than the reconstructed branchwise generator requires after cancellation in the reduced branch symbol. Thus
\[
\Dom(\Gcal)\subsetneq\Dom(\Acal)
\]
may occur. This does not contradict, and should not be confused with, the graph-core identity
\[
\overline{\Gcal}^{\,\mathrm{graph}}=\Acal=\Mss
\]
from Proposition~\ref{prop:strong-prototype-core}. The stress test detects a strict pre-closure domain mismatch, not a failure of graph-density.
\end{remark}

\begin{remark}[Exact role of the raw-domain gap stress test]
\label{rem:exact-role-raw-domain-gap-stress-test}
For the benchmark symbols in Proposition~\ref{prop:stress-test-cancellation},
the plus-branch symbol-sum cancellation gives
\[
(a+b)(\xi)=2i\xi,
\]
so the maximal plus-branch symbol-sum domain is
\[
\Dom((a+b)(D))=H^1(\R).
\]
The raw componentwise prototype, however, requires the two separate conditions
\[
a(D)f\in L^2(\R),\qquad b(D)f\in L^2(\R),
\]
which force quadratic Fourier control and hence
\[
\Dom(a(D))\cap\Dom(b(D))=H^2(\R)
\]
for the plus-branch polynomial pair used in the stress test. Therefore
\[
\Dom(a(D))\cap\Dom(b(D))
\subsetneq
\Dom((a+b)(D))
\]
on the plus branch. This is the precise diagnostic content of the example. It
is not a counterexample to the graph-core theorem; it explains why the raw
componentwise strong prototype should not be identified with the maximal
branchwise generator domain before taking graph closure.
\end{remark}

Taken together, the three generator benchmarks, the observation benchmark, and the final stress test show that the theorem package from Sections~3--6 is not merely formal. The reduced branch symbols can be computed explicitly, the reconstructed flow can be written down concretely, one admissible pair can carry transport, parabolic, and oscillatory-diffusive regimes at once, observation geometry can hide or reveal instability in a controlled way, and the cancellation example detects a genuine pre-closure raw-domain gap between $\Gcal$ and $\Acal$, not a graph-closure gap. Section~8 uses this combined picture as the background against which the limitations of the present multiplier realization become easiest to read: once variable coefficients, nonlinearities, or boundary phenomena are introduced, the exact intrinsic branch decomposition can no longer be expected to admit the same global Fourier-side representation unchanged.




\section{Limitations and Outlook}

The preceding sections identify a concrete translation-invariant Fourier-multiplier realization of the branchwise semigroup program, including both the prescribed branch-coordinate reconstruction and the equivalent full matrix-symbol multiplier formula. The result should be read as an exact branch-algebraic assembly theorem inside a restricted translation-invariant multiplier class, not as a new scalar Fourier-multiplier or $C_0$-semigroup theory. Its novelty is program-relative: it supplies one concrete $L^2(\R^d;A)$ realization in this series where branch reduction, full reconstruction, graph-core identification, coordinate-exact branch identities in the fixed branch-adapted Hilbert norm, and bounded-bulk observation pushforward are present in one controlled theorem package. The point of this final section is twofold. First, it records exactly what the present paper has established and therefore what may now be used as firm infrastructure in subsequent work. Second, it makes equally clear where the present framework stops. The restricted Fourier-multiplier setting was chosen precisely because it allows one theorem-bearing multiplier-realization bridge to be completed without conflating it with several harder neighboring problems. The value of the paper therefore depends as much on the sharpness of its limitations as on the strength of its positive results.

\subsection{What Paper~5 has established}

At the most basic level, Paper~5 shows that the abstract branchwise semigroup architecture of Paper~4 is not merely formal. There exists a nontrivial and mathematically natural sufficient-hypothesis class of spatially distributed linear evolutions on which that architecture has a concrete multiplier realization. This is an assembly and realization statement for the present branch-algebraic program, not a priority claim over the surrounding multiplier or semigroup literature. This is one basic point that should be retained from the paper as a whole.

More precisely, the paper has established five structural facts.

\begin{enumerate}[label=\textup{(\roman*)}]
    \item \textbf{Intrinsic branch decomposition on $L^2(\R^d;A)$.} Section~3 proved that the state space $L^2(\R^d;A)$ decomposes into invariant branch subspaces determined by the constant algebraic projectors. Fourier transform then represents that intrinsic decomposition through reduced branch-coordinate symbols, with the complex branch interpreted through the real-linear coordinate map $\mathcal I_c$ induced by $J^2=-P_c$. Thus the scalar notation $\lambda_c(D)$ is a transported global real-coordinate complex multiplier, not an independent complex-linear fibre reduction of the physical state space.

    \item \textbf{Reduced multiplier semigroup package.} Section~4 showed that, under explicit sufficient reduced-symbol real-part bounds, the plus, minus, and complex branches generate $C_0$-semigroups on their natural reduced model spaces. These bounds are the admissibility conditions for the branch-adapted $L^2$ theorem proved here, not a minimality theorem for all multiplier realizations. This gives a concrete reduced semigroup package at the multiplier level, rather than a merely symbolic reduction.

    \item \textbf{Relative symbol-sum realization and graph-core domain theorem.} Section~5 first isolated the closed relative symbol-sum multiplier \(\Mss\) and proved the main operator-domain identity
    \[
    \overline{\Gcal}^{\,\mathrm{graph}}=\Mss.
    \]
    This is the non-tautological domain content of the paper: the raw componentwise domain may be strictly smaller because it asks for \(a(D)\) and \(b(D)\) separately, but the finite-symbol cutoff sequence shows that the raw prototype still determines the closed symbol-sum realization by graph closure.

    \item \textbf{Prescribed branch-coordinate semigroup reconstruction and separate internal uniqueness.} Section~5 then proved that the full $A$-valued distributed flow is exactly the direct sum of the prescribed reduced branch semigroups in branch coordinates. The reconstruction theorem is stated without a global uniqueness claim; the separate internal-uniqueness corollary forces a competing $C_0$-semigroup to coincide with $\mathcal T(t)$ only after one has already imposed preservation of the same branch subspaces and the same prescribed reduced branch actions. In the multiplier class treated in the paper, this yields the canonical reconstructed semigroup $\mathcal T(t)$, identifies its generator as the branchwise realization $\Acal$, and identifies that generator with the same relative symbol-sum realization \(\Mss\). For arbitrary equivalent coefficient norms, the corresponding norm statements are equivalent estimates with constants rather than literal isometric identities; the exactness is a feature of the chosen branch-adapted Hilbert coordinate norm. The non-tautological realization content is the symbol-sum and graph-core identification, not a global classification of all possible full semigroups.

    \item \textbf{Distributed observation pushforward, masking, and conditional range bookkeeping for bounded bulk sensors.} Section~6 proved that bounded bulk observation maps yield a branch-induced additive splitting at the observed-signal level and produce one-sided observation bounds and masking statements. For the subclass of sensors whose observation spaces are endowed a priori with projectors intertwining the branch decomposition, it further showed that the full observed signal can be projected directly onto its plus, minus, and complex observed channels and that those observed channels satisfy transported separated sensor norm comparisons only after the corresponding lower sensor estimate has been assumed or independently verified. A Fourier-frame branch-sensor example shows one concrete way to verify the bounded-below hypothesis without using only the canonical branch projector. The section then separately obtained a conditional closed-range sensor inverse only on branches where the restricted sensor is bounded below, with an explicit inverse bound on the actual closed range unless surjectivity onto the whole observation channel is separately assumed, and a further auxiliary conditional lower-bound transfer once an external lower dynamical estimate is supplied on the same branch. Section~7 then made those statements explicit in three generator benchmarks, an observation benchmark, and a cancellation stress test, showing how branch geometry, observation geometry, multiple branch regimes, and strict raw-domain mismatch interact in practice while preserving the graph-core identity $\overline{\Gcal}^{\,\mathrm{graph}}=\Acal$.
\end{enumerate}

Taken together, these results justify describing Paper~5 as a function-space realization within the present program in a translation-invariant Fourier-multiplier class. More concretely, the paper identifies one class of generators for which branch splitting, reduced semigroup generation, exact prescribed branch-coordinate reconstruction, relative symbol-sum multiplier realization, graph-core identification of the componentwise strong prototype, separate internal direct-sum uniqueness, distributed multiplier-flow reconstruction, and a layered bounded-bulk observation bookkeeping formalism coexist in a single framework. The component multiplier, cutoff, multiplier-frame, and closed-range arguments used to establish this package are standard; their role is to make the branch-adapted domains, graph-core closure, and observation hypotheses explicit inside the branch-coordinate architecture. The non-claims governing generation, classification, and observability have been centralized in Subsection~\ref{subsec:scope-and-nonclaims}.

The paper has also deliberately replaced the search for a universal scalar threshold by a regime language based on operator-norm envelopes, datum-specific branch amplification, masking, and observation-dependent visibility. Once the state is spatially distributed and the observation map may see only selected branches or selected frequency windows, the mathematically meaningful question is no longer whether a single scalar threshold controls everything, but which growth envelopes belong to the semigroup itself, which amplification mechanisms are activated by the chosen datum, which survive the chosen observation protocol, and which remain hidden. Frequency-window lower bounds and inverse-on-window statements therefore remain windowed statements, as specified locally where they are used.

\subsection{What remains open}

The restricted success of Paper~5 leaves several major directions open. Each one is natural, but each introduces a new layer of difficulty that was intentionally excluded from the theorem-bearing core.

\paragraph{Variable coefficients.}
The present paper relies crucially on translation invariance. The Fourier-side representation of the intrinsic branch decomposition works because the prototype generator is a constant-coefficient Fourier multiplier. For that reason the paper should be read, strictly speaking, as a Fourier-multiplier realization theorem and only then as a source of benchmark consequences for admissible constant-coefficient differential-operator subclasses where the polynomial multipliers also satisfy the standing multiplier-semigroup package of Definition~\ref{def:standing-multiplier-semigroup-package}. Once coefficients depend on space, one can no longer expect an exact global Fourier-side decomposition. Any extension in that direction would likely require a microlocal or symbolic branch analysis rather than the exact spectral splitting used here.

\paragraph{Semilinear perturbations.}
The full reconstruction theorem is linear. If a nonlinear or semilinear term is added, then the branch subspaces need no longer remain invariant and the exact recombination formula can fail. The main question is no longer merely whether solutions exist, but whether the branchwise observation-bound and masking structures developed here persist, deform, or collapse under controlled nonlinear coupling. This is the most natural immediate continuation of the present paper.

\paragraph{True inverse multiplier-identification questions.}
Section~6 developed only bounded-operator observation pushforward, masking, transported separated sensor norm comparisons, conditional range-inverse statements, and conditional lower-bound bookkeeping statements. It explains how branchwise information propagates through a bounded observation map, but it does not ask whether one can reconstruct the underlying multiplier symbols or branch parameters from observed data. An inverse multiplier-identification paper would need to combine the forward branch architecture developed here with an identification strategy analogous in spirit to the inverse program of Paper~2, but in a genuinely distributed setting.

\paragraph{Boundary observations and trace geometry.}
The present observation class consists of bounded bulk linear maps on $L^2(\R^d;A)$. This excludes boundary traces, point evaluations, local fluxes, time-dependent sensors, and the regularity theory needed to justify them. Yet many physically natural measurements for distributed evolution problems are boundary-based rather than bulk-based. Extending the present observation-pushforward and masking framework to such sensors would require a different functional-analytic setting and likely a more delicate compatibility theory between branch splitting and trace operators.

\paragraph{Stronger spectral or microlocal theories.}
Even within linear function-space evolution, the present paper remains deliberately coarse. It tracks branch structure and observation dependence at the level of semigroup operator-norm envelopes, explicit branch symbols, and benchmark examples. It does not yet attempt refined spectral decompositions, resonance analysis, or microlocal transport of branch content. Such theories are potentially important, especially once variable coefficients or boundary effects are introduced, but they belong to a later stage.

What unifies these open directions is that they all threaten the exactness that made Paper~5 possible. Variable coefficients threaten the global Fourier-side splitting, nonlinearities threaten invariant branches, inverse problems threaten the purely forward architecture, and boundary sensors threaten the simplicity of bounded observation maps. Each extension is therefore a real research problem, not a routine elaboration.

\subsection{Toward Paper~6}

Among the open directions just listed, the most natural next paper is the semilinear one. The reason is structural. Paper~5 has now established a complete forward linear multiplier branch architecture on a function space: branch splitting, reduced semigroup generation, prescribed branch-coordinate reconstruction, separate internal uniqueness inside the branch-compatible class, and observation-dependent pushforward/masking statements. The immediate next question is therefore whether this architecture is stable under controlled nonlinear perturbation.

A plausible form of Paper~6 would be to study semilinear equations of the type
\[
\partial_t Y = \Acal Y + \mathcal{N}(Y),
\]
where $\Acal$ is the reconstructed Fourier-multiplier generator treated here and $\mathcal{N}$ is a nonlinear term small enough, regular enough, or structured enough to permit a perturbative theory. The central issue would not simply be existence of mild solutions. Rather, the issue would be whether one can retain any meaningful analogue of the branchwise picture from the linear theory.

Several concrete questions arise immediately.
\begin{enumerate}[label=\textup{(\arabic*)}]
    \item Under what assumptions on $\mathcal{N}$ do branch-supported initial data remain approximately branch-supported for short or intermediate times?
    \item Can one derive perturbative formulas in which the linear branch semigroups still dominate the nonlinear dynamics, at least up to controlled error terms?
    \item Do the observation pushforward, masking, observation-space branch-separation, transported separated sensor norm comparisons, and conditional lower-bound statements from Section~6 persist in a stable form, or can nonlinear interaction create visible growth from an intrinsically masked branch or destroy an otherwise separating sensor geometry?
    \item Can one isolate structural classes of nonlinearities that respect the observation-layer philosophy of the series, so that randomness still enters only after deterministic propagation?
\end{enumerate}

These questions explain why the semilinear problem is the right successor to Paper~5. It does not discard the multiplier landing achieved here; it stress-tests it. In that sense Paper~6 would play for the multiplier-realization framework developed here the same role that later refinement papers played for the earlier scalar and operator-valued stages of the program.

We therefore end with the following perspective. Paper~5 is a controlled multiplier-realization theorem. It proves that the branchwise semigroup architecture survives one nontrivial passage into a concrete translation-invariant Fourier-multiplier setting; admissible local constant-coefficient differential-operator models are benchmark subclasses of that setting, and the bounded-bulk observation consequences remain explicitly conditional at the separated-branch level. Precisely because that landing has now been made exact, the next stage can ask the more difficult question of persistence: once semilinear coupling, variable geometry, inverse aims, or boundary sensing are introduced, which parts of the branchwise picture endure and which parts break.



\appendix
\section{Routine proof ledger}\label{app:routine-proof-ledger}
This appendix records the standard arguments that are used in compressed form in the main text. It is included to keep the theorem-bearing part of the paper focused on branch normal form, reduced-generator packaging, prescribed reconstruction, graph-core identity, and bounded-observation limitations.

\subsection*{Reduced multiplier domains}
For Lemma~\ref{lem:reduced-multipliers-closed-dense}, density is obtained by cutting off the Fourier transform on finite-symbol sets
\[
E_n^\bullet=\{\xi:\ |\lambda_\bullet(\xi)|\le n\}.
\]
On the real branches one replaces \(E_n^\pm\) by the exactly symmetric representative \(E_n^\pm\cap(-E_n^\pm)\), which preserves Hermitian Fourier symmetry. Dominated convergence gives \(f_n\to f\), and boundedness of \(\lambda_\bullet\) on the cutoff gives \(f_n\in\Dom(G_\bullet)\). Closedness is the ordinary closedness of a maximal multiplication operator: if \(f_k\to f\) and \(\lambda_\bullet\widehat f_k\to \widehat g\) in \(L^2\), a subsequence converges a.e., hence \(\lambda_\bullet\widehat f=\widehat g\) a.e.

\subsection*{Bounded observation pushforward and masking}
For Theorem~\ref{thm:observation-pushforward}, the decomposition follows from linearity and branch invariance:
\[
\mathcal T(t)Y_0=\sum_\sigma \mathcal T(t)Y_{0,\sigma},
\qquad
z(t)=\sum_\sigma \Ocal\mathcal T(t)Y_{0,\sigma}.
\]
The estimates follow from
\[
\|\Ocal P_\sigma\mathcal T(t)Y_{0,\sigma}\|_Z
\le
\|\Ocal P_\sigma\|\,\|\mathcal T(t)Y_{0,\sigma}\|_{L^2}
\le
\|\Ocal P_\sigma\|e^{\omega_\sigma t}\|Y_{0,\sigma}\|,
\]
and masking is immediate if \(\Ocal\) vanishes on the corresponding branch.

\subsection*{Compact sensors and closed-range inverse}
If a compact operator \(\mathcal B:X\to Z\) on an infinite-dimensional Banach space were bounded below, then it would be injective, its range would be closed, and \(\mathcal B^{-1}:\mathrm{Ran}(\mathcal B)\to X\) would be bounded. The identity would factor as \(I_X=\mathcal B^{-1}\mathcal B\), hence would be compact, which is impossible on an infinite-dimensional Banach space. This proves Proposition~\ref{prop:compact-sensors-not-globally-visible}. The same bounded-below estimate gives Lemma~\ref{lem:closed-range-reconstruction-bounded-below}: if \(z_n=Ox_n\) is Cauchy in \(\mathrm{Ran}(O)\), then \((x_n)\) is Cauchy by \(\|x_n-x_k\|\le m^{-1}\|z_n-z_k\|\), so the range is closed and \(\|O^{-1}\|\le m^{-1}\).

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