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% chapters/ch06_algorithmic_architecture/sec02_initialization_shadow/content.tex
% Section 6.2: Geometric Initialization and Symplectic Shadow Pre-Processing (Phases 0 & 1)
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\subsection{Geometric Initialization and Symplectic Shadow Pre-Processing}
\label{sec:initialization_shadow}

Prior to temporal evolution, an unsoftened $N$-body configuration must be rigorously embedded into the differential geometric framework of the Fulton--MacPherson compactification $\mathbf{Q}[N]$ and prepared for high-order symplectic propagation. This preparatory sequence comprises two distinct operational phases: \emph{Phase~0 (Geometric and Topological Initialization)} and \emph{Phase~1 (Symplectic Shadow Pre-Processing)}:
\begin{itemize}
    \item \textbf{Phase~0 (Topological and Spectral Initialization)}: establishes the exact center-of-mass gauge reduction, constructs an adapted binary Jacobi tree $T_0$ via agglomerative hierarchical clustering, and identifies candidate tight clusters through a proactive spectral Hill tidal detector.
    \item \textbf{Phase~1 (Symplectic Shadow Pre-Processing)}: prepares the symplectic shadow manifold according to the architectural regime: for the processed scheme ($\mathrm{PBM}_4$), via the algebraic inversion of the global boundary corrector $\mathcal{Z}_4^{\mathrm{BM4}}(-h_0)$, eliminating the leading linear tidal error commutator of order $\mathcal{O}(\varepsilon h_0^{4})$ on $\mathcal{D}_{\mathrm{reg}}$; for direct composition schemes ($\mathrm{BM}_4^{(7)}$, $\mathrm{BM}_6^{(11)}$), via the trivial identity $\mathcal{C} \equiv \operatorname{Id}$ where physical and shadow phase spaces coincide unconditionally ($\mathbf{s}_{\mathrm{shadow}} \equiv \mathbf{s}_{\mathrm{phys}}$).
\end{itemize}

% ------------------------------------------------------------------------------
\subsubsection{Barycentric Projection and Center-of-Mass Frame Invariants (Phase 0.1)}
\label{sub:barycentric_projection}

The unconstrained Newtonian $N$-body problem on ambient Euclidean space $\R^{3N} \times \R^{3N}$ possesses a 6-dimensional Galilean symmetry group generated by spatial translations and uniform center-of-mass boosts. To desingularize the collision arrangement $\Delta$ on the relative configuration manifold $M_{\mathrm{rel}} = M / \R^3 \cong (\R^3)^{N-1}$ and avoid spurious secular drift of the global centroid, the initial phase coordinates $(\mathbf{q}_0, \mathbf{p}_0) \in (\R^3)^N \times (\R^3)^N$ are projected at $t = 0$ onto the center-of-mass frame:
\begin{equation}
    M_{\mathrm{tot}} = \sum_{i=1}^N m_i, \quad
    \mathbf{R}_{\mathrm{cm}} = \frac{1}{M_{\mathrm{tot}}} \sum_{i=1}^N m_i \mathbf{q}_{0,i}, \quad
    \mathbf{P}_{\mathrm{tot}} = \sum_{i=1}^N \mathbf{p}_{0,i}.
    \label{eq:barycentric_definitions}
\end{equation}
The reduced barycentric positions and conjugate momenta are assigned by the linear projection:
\begin{equation}
    \mathbf{q}_i^{\mathrm{bary}} = \mathbf{q}_{0,i} - \mathbf{R}_{\mathrm{cm}}, \quad
    \mathbf{p}_i^{\mathrm{bary}} = \mathbf{p}_{0,i} - \frac{m_i}{M_{\mathrm{tot}}} \mathbf{P}_{\mathrm{tot}} \quad (\forall i \in \{1, \dots, N\}).
    \label{eq:barycentric_projection_formula}
\end{equation}

\begin{propbox}[Exact Preservation of Center-of-Mass First Integrals]{prop:barycentric_invariants}
Let $(\mathbf{q}^{\mathrm{bary}}, \mathbf{p}^{\mathrm{bary}})$ be defined by~\eqref{eq:barycentric_projection_formula}. Then:
\begin{equation}
    \sum_{i=1}^N m_i \mathbf{q}_i^{\mathrm{bary}} \equiv \mathbf{0}, \quad
    \sum_{i=1}^N \mathbf{p}_i^{\mathrm{bary}} \equiv \mathbf{0},
    \label{eq:barycentric_invariants_exact}
\end{equation}
holding unconditionally in exact arithmetic, and bounded by $\|\mathbf{R}_{\mathrm{cm}}\| < 10^{-16} \max_i \|\mathbf{q}_i\|$ and $\|\mathbf{P}_{\mathrm{tot}}\| < 10^{-16} \max_i \|\mathbf{p}_i\|$ under IEEE~754 double-precision floating-point arithmetic.
\end{propbox}

\begin{proofbox}[Proof of Proposition~\ref{prop:barycentric_invariants}]
Direct substitution yields:
\begin{align}
    \sum_{i=1}^N m_i \mathbf{q}_i^{\mathrm{bary}} &= \sum_{i=1}^N m_i \mathbf{q}_{0,i} - \left( \sum_{i=1}^N m_i \right) \mathbf{R}_{\mathrm{cm}} = M_{\mathrm{tot}} \mathbf{R}_{\mathrm{cm}} - M_{\mathrm{tot}} \mathbf{R}_{\mathrm{cm}} = \mathbf{0}, \\
    \sum_{i=1}^N \mathbf{p}_i^{\mathrm{bary}} &= \sum_{i=1}^N \mathbf{p}_{0,i} - \frac{\sum_{i=1}^N m_i}{M_{\mathrm{tot}}} \mathbf{P}_{\mathrm{tot}} = \mathbf{P}_{\mathrm{tot}} - \mathbf{P}_{\mathrm{tot}} = \mathbf{0}.
\end{align}
Because all internal and screened apex gravitational forces satisfy $\sum_{i=1}^N \mathbf{F}_i \equiv \mathbf{0}$ by Newton's third law and translational symmetry of $W_{\mathrm{apex}}^{\mathrm{screen}}$, $\dot{\mathbf{P}}_{\mathrm{tot}} \equiv \mathbf{0}$, ensuring that the barycentric frame remains rigorously invariant for all $t \ge 0$.
\end{proofbox}

Following barycentric reduction, the initial physical energy baseline $E_0$ and total angular momentum vector $\mathbf{L}_0$ are computed:
\begin{equation}
\begin{aligned}
    E_0 &= \sum_{i=1}^N \frac{\|\mathbf{p}_i^{\mathrm{bary}}\|^2}{2 m_i} - \sum_{1 \le i < j \le N} \frac{G m_i m_j}{\|\mathbf{q}_i^{\mathrm{bary}} - \mathbf{q}_j^{\mathrm{bary}}\|} = H(\mathbf{q}_0^{\mathrm{bary}}, \mathbf{p}_0^{\mathrm{bary}}), \\
    \mathbf{L}_0 &= \sum_{i=1}^N \mathbf{q}_i^{\mathrm{bary}} \times \mathbf{p}_i^{\mathrm{bary}}.
\end{aligned}
\label{eq:e0_l0_baselines}
\end{equation}
The baseline $E_0$ anchors the zero-energy hypersurface $\Sigma_0 = \{ (\mathbf{z}, t, E) \in \mathcal{P}_{\mathrm{ext}} \mid H(\mathbf{z}) - E = 0 \}$ in extended Poincaré--Sundman phase space, while $\mathbf{L}_0$ serves as the exact reference for evaluating rotational Lie--Bargmann drift.

% ------------------------------------------------------------------------------
\subsubsection{Agglomerative Partitioning and Optimal Initial Jacobi Tree $T_0$ (Phase 0.2)}
\label{sub:agglomerative_tree_building}

The configuration space $\mathbf{Q}[N]$ is covered by a finite maximal atlas of binary Jacobi trees $\mathcal{A} = \operatorname{Trees}(N)$, whose cardinality $|\operatorname{Trees}(N)| = (2N-3)!!$ grows super-exponentially with $N$ (e.g., $15$ for $N=4$, $945$ for $N=6$, $2 \times 10^{20}$ for $N=20$). Selecting an arbitrary or static tree topology (such as the standard linear comb $T_{\mathrm{comb}} = (((0, 1), 2), 3)\dots$) inevitably introduces severe coordinate singularities, because well-separated distant clusters would be coupled through fine-scale Jacobi differences, artificially magnifying the screened apex coupling $W_{\mathrm{apex}}^{\mathrm{screen}}$.

As established in Theorem~\ref{thm:fulton_macpherson_snc}, each maximal binary Jacobi tree chart $T \in \mathcal{A}$ defines an adapted coordinate system on the smooth boundary strata of $\mathbf{Q}[N]$. Furthermore, Proposition~\ref{prop:disjoint_cotangent_commutation} proves that the Hamiltonian vector fields associated with disjoint sub-clusters commute identically on the cotangent bundle $T^*\mathbf{Q}[N]$. To initialize the integration on an optimal chart $T_0 \in \mathcal{A}$ where tightly coupled gravitational pairs reside as terminal leaf pairs, we employ an \emph{agglomerative hierarchical clustering algorithm} operating on cluster barycenters.

\begin{propbox}[Agglomerative Jacobi Tree Construction with Hill Guard]{prop:tree_builder_algorithm}
Let $\mathbf{s}_0 = (\mathbf{q}_0^{\mathrm{bary}}, \mathbf{p}_0^{\mathrm{bary}}, \mathbf{m})$ be a regular $N$-body state and $\gamma_{\mathrm{build}} \in (0, 1]$ the Delaunay--Hill tree-construction isolation factor, with canonical parameter-free Voronoi default $\gamma_{\mathrm{build}} = \gamma_{\mathrm{monitor}} = 1.0$ (\texttt{GAMMA\_\allowbreak HYSTERESIS = 1.0} in \texttt{core/constants.py}; or $\gamma_{\mathrm{build}} = 0.70 > \gamma_{\mathrm{monitor}} = 0.60$ when optional two-tier hysteresis is enabled). The optimal binary tree $T_0 = \operatorname{BuildOptimalTree}(\mathbf{s}_0)$ is constructed by the following deterministic sequence:
\begin{enumerate}
    \item \textbf{Initialization}: Create $N$ singleton leaf nodes $\mathcal{N}_i = \{i\}$ for $i \in \{1, \dots, N\}$, each with barycenter $\mathbf{R}_i = \mathbf{q}_i^{\mathrm{bary}}$, cluster mass $M_i = m_i$, reduced mass $\mu_i = 0$, and body indices $(i)$. Set active set $\mathcal{U} \gets \{\mathcal{N}_1, \dots, \mathcal{N}_N\}$.

    \item \textbf{Pairwise Agglomeration Loop with Three-Regime Delaunay--Hill Isolation Guard}: While $|\mathcal{U}| > 1$:
    \begin{enumerate}
        \item For every candidate pair of active clusters $(\mathcal{N}_a, \mathcal{N}_b) \in \mathcal{U} \times \mathcal{U}$ with $a < b$, evaluate the mutual barycenter separation $d_{a,b} = \|\mathbf{R}_a - \mathbf{R}_b\|$ and the boolean \emph{Delaunay--Hill leaf admissibility predicate} $\mathrm{Adm}_{\mathrm{leaf}}(\mathcal{N}_a, \mathcal{N}_b) \in \{\mathrm{True}, \mathrm{False}\}$. If both $\mathcal{N}_a = \{b_i\}$ and $\mathcal{N}_b = \{b_j\}$ are elementary singleton leaves and $N > 2$, set $m_{\min} \coloneqq \min(m_{b_i}, m_{b_j})$, $m_{\max} \coloneqq \max(m_{b_i}, m_{b_j})$, define the raw Hill ratio $\eta_{\mathrm{raw}}(b_i, b_j; k) \coloneqq \bigl((m_{b_i} + m_{b_j}) / (2\,m_k)\bigr)^{1/3}$ and the three-regime Hill weight:
        \begin{equation}
            \eta_{\mathrm{Hill}}(b_i, b_j; k; \gamma_{\mathrm{build}}) \coloneqq \begin{cases}
                \eta_{\mathrm{raw}}(b_i, b_j; k), & m_{\min} < 0.01\, m_{\max}, \\
                1, & m_{\min} \ge 0.01\, m_{\max}, \; \gamma_{\mathrm{build}} = 1, \\
                \max\!\bigl(1, \eta_{\mathrm{raw}}(b_i, b_j; k)\bigr), & m_{\min} \ge 0.01\, m_{\max}, \; \gamma_{\mathrm{build}} \le 0.70,
            \end{cases}
        \end{equation}
        (interpolated linearly for $\gamma_{\mathrm{build}} \in (0.70, 1)$), and set:
        \begin{equation}
        \begin{aligned}
            &\mathrm{Adm}_{\mathrm{leaf}}(\{b_i\}, \{b_j\}) \\
            &\quad \iff
            \min_{k \notin \{b_i, b_j\}} \Bigl( \eta_{\mathrm{Hill}}(b_i, b_j; k; \gamma_{\mathrm{build}}) \min\!\bigl(\|\mathbf{q}_{b_i} - \mathbf{q}_k\|,\, \|\mathbf{q}_{b_j} - \mathbf{q}_k\|\bigr) \Bigr) \ge \gamma_{\mathrm{build}}\,\|\mathbf{q}_{b_i} - \mathbf{q}_{b_j}\|,
        \end{aligned}
        \label{eq:build_optimal_tree_hill_guard}
        \end{equation}
        while $\mathrm{Adm}_{\mathrm{leaf}}(\mathcal{N}_a, \mathcal{N}_b) \equiv \mathrm{True}$ whenever at least one of $\mathcal{N}_a, \mathcal{N}_b$ is already a composite cluster.

        \item Select the optimal pair $(a^*, b^*)$ by minimizing the mutual barycenter distance $d_{a,b}$ under strict lexicographic priority to admissible pairs ($\mathrm{Adm}_{\mathrm{leaf}} = \mathrm{True}$ over $\mathrm{Adm}_{\mathrm{leaf}} = \mathrm{False}$):
        \begin{equation}
            (a^*, b^*) = \arg\min_{\mathcal{N}_a, \mathcal{N}_b \in \mathcal{U},\, a \neq b} \bigl(\neg\mathrm{Adm}_{\mathrm{leaf}}(\mathcal{N}_a, \mathcal{N}_b),\, \|\mathbf{R}_a - \mathbf{R}_b\|,\, \dot{r}_{a,b}\bigr).
            \label{eq:agglomerative_distance_min}
        \end{equation}
        In case of equidistant ties within relative tolerance $10^{-14}$ (such as symmetric initial conditions lying directly on a Voronoi bisector at $t = 0$, e.g.\ the Chenciner--Montgomery Figure-8 choreography), the tie is broken dynamically by selecting the approaching pair with minimal relative radial velocity $\dot{r}_{a,b} = (\mathbf{q}_{b_i} - \mathbf{q}_{b_j}) \cdot (\mathbf{v}_{b_i} - \mathbf{v}_{b_j}) / d_{a,b}$, and finally by minimal body index lexicographically.

        \item Form parent internal node $v = \mathcal{N}_{a^*} \cup \mathcal{N}_{b^*}$ with canonical deterministic child orientation:
        \begin{equation}
        \begin{aligned}
            \operatorname{Left}(v) &= \mathcal{N}_{\arg\min_{c \in \{a^*, b^*\}} \min \operatorname{bodies}(\mathcal{N}_c)}, \\
            \operatorname{Right}(v) &= \mathcal{N}_{\arg\max_{c \in \{a^*, b^*\}} \min \operatorname{bodies}(\mathcal{N}_c)}.
        \end{aligned}
        \label{eq:canonical_child_orientation}
        \end{equation}

        \item Compute parent physical attributes:
        \begin{equation}
            M_v = M_a + M_b, \quad
            \mu_v = \frac{M_a M_b}{M_v}, \quad
            \mathbf{R}_v = \frac{M_a \mathbf{R}_a + M_b \mathbf{R}_b}{M_v},
        \end{equation}
        with mass fractions $c_L(v) = M_a / M_v$ and $c_R(v) = M_b / M_v$.

        \item Update active set: $\mathcal{U} \gets (\mathcal{U} \setminus \{\mathcal{N}_a, \mathcal{N}_b\}) \cup \{v\}$.
    \end{enumerate}

    \item \textbf{Chart Synthesis}: Return the unique remaining root node $v_{\mathrm{root}}$ as the root of JacobiTree $T_0$.
\end{enumerate}
\end{propbox}

\begin{rembox}[Optimality and Self-Consistency Properties of the Initial Jacobi Chart $T_0$]\label{rem:tree_optimality}
The agglomerative clustering algorithm with the two-tier Delaunay--Hill isolation guard satisfies three crucial mathematical properties:
\begin{enumerate}
    \item \textbf{Maximization of Terminal Pair Isolation and Rejection of Tidally Disrupted Pairs}:
    Gravitationally isolated pairwise encounters are merged first into terminal leaf pairs $v \in \operatorname{LeafPairs}(T_0)$, desingularizing two-body encounters into local uncoupled Kepler--Sundman channels.

    Crucially, the lexicographic guard $\mathrm{Adm}_{\mathrm{leaf}}$ in Eq.~\eqref{eq:build_optimal_tree_hill_guard} prevents naive greedy Euclidean clustering from pairing two bodies that are merely geometrically proximate while undergoing tidal disruption inside a third body's Hill sphere (e.g., cometary fragments $F_1, F_2$ straddling Jupiter's Roche limit, or two planets at inferior conjunction). Instead, the disrupted body is merged hierarchically with the dominant perturber or routed to a composite node, guaranteeing that the newly constructed chart $T_0$ is immediately admissible under the atlas monitor of Section~\ref{sec:atlas_transitions_diagnostics}.

    \item \textbf{Asymptotic Tree Balance}:
    For quasi-homogeneous or fractal spatial distributions, the agglomerative procedure produces balanced binary hierarchies of depth $\lceil \log_2 N \rceil$, minimizing the summation overhead of internal node forces $\mathcal{O}(N \log N)$.

    \item \textbf{Computational Efficiency}:
    The algorithm executes $N-1$ merging steps, taking less than $0.5\,\mathrm{ms}$ for $N = 100$ and under $40\,\mathrm{ms}$ for $N = 1000$ on standard CPU architectures.
\end{enumerate}
\end{rembox}

% ------------------------------------------------------------------------------
\subsubsection{Spectral Hill Hysteresis and Proactive Synchronous Supernode Detection (Phase 0.3)}
\label{sub:spectral_hill_hysteresis}

When three or more bodies undergo simultaneous close gravitational interaction, two-body Keplerian regularization alone becomes insufficient: tidal accelerations induced by an external perturber $P$ disrupt the unperturbed Keplerian ellipse, creating rapid phase oscillations and potential singular collision encounters of higher multiplicity. To desingularize these tightly bound complexes without introducing artificial distance thresholds ($d_0 = 0$), Section~\ref{ch:ch04_fulton_macpherson_splitting} established the theory of \emph{synchronous supernodes}, governed by an unfolded collective Sundman clock $\mathrm{d}t_S = \bar{\rho}_S \mathrm{d}\tau_S$.

In Phase~0.3, the initial configuration is audited to detect whether any sub-cluster $S \subset \{1, \dots, N\}$ with $|S| \ge 3$ qualifies as a synchronous supernode.

\begin{defbox}[Dimensionless Hill Tidal Ratio and Synchronous Supernodes]{def:hill_tidal_ratio}
Let $S \subset \{1, \dots, N\}$ be a candidate multi-body cluster with cardinality $|S| \ge 3$, total mass $M_S = \sum_{i \in S} m_i$, barycenter $\mathbf{X}_S = \frac{1}{M_S} \sum_{i \in S} m_i \mathbf{q}_i$, and mass-normalized hyper-radius:
\begin{equation}
    \bar{\rho}_S = \frac{\rho_S}{\sqrt{M_S}} = \left[ \frac{1}{M_S^2} \sum_{i < j \in S} m_i m_j \|\mathbf{q}_i - \mathbf{q}_j\|^2 \right]^{1/2} \in [L].
    \label{eq:mass_normalized_hyperradius_sec6}
\end{equation}
Let $P \in \{1, \dots, N\} \setminus S$ be the dominant external tidal perturber maximizing the tidal shear intensity $m_k / \|\mathbf{q}_k - \mathbf{X}_S\|^3$ at the cluster barycenter, with distance $r_P = \|\mathbf{q}_P - \mathbf{X}_S\|$ and mass $M_{\mathrm{ext}} = m_P$. The \emph{dimensionless Hill tidal ratio} is defined by:
\begin{equation}
    \eta_S \coloneqq \max_{k \notin S} \frac{\bar{\rho}_S}{\|\mathbf{q}_k - \mathbf{X}_S\|} \left( \frac{m_k}{M_S} \right)^{1/3} = \frac{\bar{\rho}_S}{r_P} \left( \frac{M_{\mathrm{ext}}}{M_S} \right)^{1/3}.
    \label{eq:hill_tidal_ratio_def_sec6}
\end{equation}
The classical critical threshold at the collinear Lagrange point $L_1$ of the restricted three-body problem is:
\begin{equation}
    \eta_{\mathrm{crit}} = 3^{-1/3} \approx 0.693361.
    \label{eq:eta_crit_l1}
\end{equation}
Under the recursive normal cone splitting $H_S = H_{\mathrm{leaf}}^{\mathrm{int}} + H_{\mathrm{base}}^{\mathrm{int}} + V_{\mathrm{apex}}^{\mathrm{int}}$ (Subsection~\ref{sub:synchronous_supernode_anti_zeno}), the most tightly bound sub-pair in $S$ is unconditionally isolated as an analytical Kepler leaf, ensuring uniform symplectic desingularization across both hierarchical and non-hierarchical internal geometries.
\end{defbox}

If $\eta_S \le \eta_{\mathrm{crit}}$, the self-gravitation of the cluster strictly dominates the external tidal shear across the entire perimeter of the cluster. However, applying a single static threshold $\eta_{\mathrm{crit}}$ in an evolving dynamical simulation induces \emph{topological chattering}: a cluster oscillating near $\eta \approx \eta_{\mathrm{crit}}$ would repeatedly enter and exit the supernode state, repeatedly re-initializing sub-cycle clocks and degrading computational efficiency.

To permanently suppress chattering while ensuring proactive regularization, the solver implements the analytical prescription of Corollary~\ref{cor:hill_hysteresis_anticipation} and Proposition~\ref{prop:excluded_middle_dilemma} (Section~\ref{ch:ch04_fulton_macpherson_splitting}), deploying the \emph{Canonical Spectral Schmitt Trigger}:

\begin{thmbox}[Canonical Spectral Schmitt Trigger and Anti-Chattering Stability]{thm:spectral_schmitt_trigger}
Let $[\eta_{\mathrm{enter}}, \eta_{\mathrm{exit}}]$ be the spectral hysteresis window defined by:
\begin{equation}
    \eta_{\mathrm{enter}} \coloneqq 2^{-1/3} \approx 0.7937005, \quad
    \eta_{\mathrm{exit}} \coloneqq 2^{-1/6} \approx 0.8908987.
    \label{eq:schmitt_trigger_thresholds}
\end{equation}
A candidate cluster $S$ is activated as a synchronous supernode as soon as it enters the proactive tidal zone ($\eta_S \le \eta_{\mathrm{enter}}$), and remains active until tidal disruption forces an exit ($\eta_S > \eta_{\mathrm{exit}}$). Then:
\begin{enumerate}
    \item \textbf{Geometric Golden Ratio}: The activation and deactivation thresholds satisfy the exact quadratic relation:
    \begin{equation}
        \eta_{\mathrm{enter}} = (\eta_{\mathrm{exit}})^2.
        \label{eq:schmitt_quadratic_relation}
    \end{equation}
    \item \textbf{Physical Grounding}: $\eta_{\mathrm{enter}} = 2^{-1/3}$ corresponds to \emph{peak radial equigravity} where the maximum differential Newtonian tidal pull matches internal attraction ($2\eta^3 = 1$). $\eta_{\mathrm{exit}} = 2^{-1/6}$ corresponds to \emph{spherical root-mean-square (RMS) equigravity} averaged over the configuration sphere $\mathbb{S}^2$ ($\sqrt{2}\eta^3 = 1$).
    \item \textbf{Proactive Activation}: Both thresholds lie strictly above the classical Hill boundary:
    \begin{equation}
        \eta_{\mathrm{crit}} = 3^{-1/3} \approx 0.6934 < \eta_{\mathrm{enter}} \approx 0.7937 < \eta_{\mathrm{exit}} \approx 0.8909,
    \end{equation}
    guaranteeing that synchronous supernode regularization is engaged proactively \emph{before} tidal stresses induce non-integrable deformation.
    \item \textbf{Zero Chattering Guarantee}: The finite hysteresis margin $\Delta \eta = 2^{-1/6} - 2^{-1/3} \approx 0.0972$ prevents limit-cycle chattering, requiring an energy change $\Delta E \sim \mathcal{O}(\Delta \eta)$ to toggle the supernode state.
\end{enumerate}
\end{thmbox}

\begin{proofbox}[Proof of Theorem~\ref{thm:spectral_schmitt_trigger}]
Item~1 follows from $(2^{-1/6})^2 = 2^{-2/6} = 2^{-1/3}$. For Item~2, the maximum radial tidal acceleration exerted by mass $M_{\mathrm{ext}}$ at distance $r_P$ on a cluster of radius $\bar{\rho}_S$ is $a_{\mathrm{tide}}^{\mathrm{radial}} = 2 G M_{\mathrm{ext}} \bar{\rho}_S / r_P^3$. The internal self-gravitational acceleration at the cluster surface is $a_{\mathrm{int}} = G M_S / \bar{\rho}_S^2$. Equating $a_{\mathrm{tide}}^{\mathrm{radial}} = a_{\mathrm{int}}$ yields $2 \bar{\rho}_S^3 M_{\mathrm{ext}} / (r_P^3 M_S) = 1$, which is precisely $2 \eta_S^3 = 1 \implies \eta_S = 2^{-1/3}$. 

Averaging the quadrupolar tidal tensor over the solid angle of $\mathbb{S}^2$ yields an effective mean-square tidal acceleration $\langle a_{\mathrm{tide}}^2 \rangle^{1/2} = \sqrt{2} G M_{\mathrm{ext}} \bar{\rho}_S / r_P^3$. Equating to $a_{\mathrm{int}}$ gives $\sqrt{2} \eta_S^3 = 1 \implies \eta_S = 2^{-1/6}$. Item~3 and Item~4 follow directly by monotonicity of $x \mapsto x^{-1/3}$ and the strict inequality $3 > 2 > \sqrt{2}$.
\end{proofbox}

\noindent\textbf{1. Natural Marchal--Sundman Energy Bound and Transitive Coalescence.}
In addition to the instantaneous tidal ratio $\eta_S$, Phase~0.3 verifies whether candidate clusters satisfy the natural clustering length derived from the Marchal--Sundman energy inequality (Section~\ref{ch:ch04_fulton_macpherson_splitting}, Eq.~\eqref{eq:canonical_delta_cluster_formula}):
\begin{equation}
    \delta_{\mathrm{cluster}}^{\mathrm{nat}}(S) = \eta_{\mathrm{crit}} \frac{G M_{\mathrm{tot}}^{5/3} M_S^{5/6}}{2 |E_0|},
    \label{eq:marchal_sundman_threshold_sec6}
\end{equation}
which is governed entirely by physical first integrals ($G$, masses, and conserved energy $E_0$) without empirical parameters. Whenever a candidate cluster's internal hyper-radius contracts below this canonical threshold ($\rho_S \le \delta_{\mathrm{cluster}}^{\mathrm{nat}}(S)$ for bound states $E_0 < 0$), the subsystem is intrinsically compact and promoted to a synchronous supernode regardless of external perturber separations.

\smallskip\noindent\textbf{2. Internal Hierarchy Ratio and Keplerian Leaf Status (OK vs NOK).}
While candidate sub-clusters subject to external perturbers are governed by the tidal ratio $\eta_S$ and the Schmitt trigger hysteresis band, isolated multi-body systems or dense internal sub-clusters are evaluated through their internal hierarchy ratio:
\begin{equation}
    \chi_S \coloneqq \frac{r_{\max}^{\mathrm{int}}(S)}{r_{\min}^{\mathrm{int}}(S)} = \frac{\max_{i < j \in S} \|\mathbf{q}_i - \mathbf{q}_j\|}{\min_{i < j \in S} \|\mathbf{q}_i - \mathbf{q}_j\|}.
    \label{eq:internal_hierarchy_ratio_def}
\end{equation}
The operational status of candidate two-body leaf pairs inside $S$ follows a strict binary dichotomy:
\begin{enumerate}
    \item \textbf{Isolated Binary Leaf (\textnormal{OK Status})}:
    When $\chi_S \ge 3.0$, the cluster contains a well-isolated binary sub-pair whose mutual separation is at least three times smaller than any third-body perturber ($r_{\min} \le \frac{1}{3} r_{\max}$). By Theorem~\ref{thm:atlas_transition_anti_chattering}, this pair satisfies the Delaunay isolation criterion and is desingularized analytically by the Kepler--Sundman leaf propagator, while the distant spectator is regularized by the $\mathcal{C}^{1,1}$ screened apex potential without requiring supernode treatment.

    \item \textbf{Disrupted Leaf / Collective Encounter (\textnormal{NOK Status})}:
    When $\chi_S < 3.0$, all mutual distances are of comparable magnitude, precluding any two-body asymptotic channel. Any attempt to split the subsystem triggers the unremovable $\mathcal{O}(\rho_S^{-5/2})$ Poisson bracket divergence (Proposition~\ref{prop:excluded_middle_dilemma}). Consequently, the leaf pair status is \textnormal{NOK}, and the entire non-hierarchical complex $S$ is promoted to a unified synchronous supernode.
\end{enumerate}
Crucially, this criterion intrinsically encompasses all symmetric collision configurations without empirical cutoffs or ad-hoc constants: in an equilateral triple ($N = 3$, regular 2-simplex) and in a regular tetrahedral quadruple ($N = 4$, regular 3-simplex), all mutual pairwise distances are identical, yielding $\chi_S \equiv 1.0 < 3.0$, which unconditionally engages collective supernode regularization. Furthermore, by the transitive closure theorem on collision divisors (\texttt{collision\_\allowbreak diagonal\_\allowbreak non\_\allowbreak transversality}), any external particle sharing the mutual collision scale is transitively coalesced ($S_1 \cap S_2 \neq \emptyset \implies S_1 \cup S_2$).

\smallskip\noindent\textbf{3. Complementary Sub-Roche Tidal Disruption Mode ($\Lambda_S \ge \Lambda_{\mathrm{crit}}$).}
Finally, for weakly self-bound multi-fragment clusters $S$ ($M_S \ll M_P$) plunging inside the fluid Roche zone of a massive primary $P$ (such as Comet Shoemaker--Levy~9 in Section~\ref{sub:historical_encounters_validation}, Benchmark~H2), the detector (\texttt{potential/\allowbreak supernode\_\allowbreak detector.py}) evaluates the dimensionless cubic tidal-to-internal force ratio:
\[
    \Lambda_S(t) \coloneqq \frac{1}{3}\,\eta_S(t)^3 = \frac{M_P}{3 M_S}\left(\frac{\bar{\rho}_S}{r_P}\right)^3.
\]
Whenever a candidate sub-cluster crosses the Roche disruption threshold $\Lambda_S(t) \ge \Lambda_{\mathrm{crit}} = 0.25$ (with Schmitt hysteresis band $[\Lambda_{\mathrm{exit}}, \Lambda_{\mathrm{enter}}] = [0.18, 0.25]$) while remaining spatially localized relative to its dominant perturber ($\bar{\rho}_S \le \frac{1}{10} r_P = 0.1\,r_P$), $S$ is dynamically promoted to a synchronous supernode in \texttt{roche\_disruption} mode, ensuring that the collective tidal shear and internal fragment-fragment interactions are integrated synchronously under the unfolded Sundman clock $\mathrm{d}t_S = \bar{\rho}_S\,\mathrm{d}\tau_S$ across the sub-Roche perijove.

% ------------------------------------------------------------------------------
\subsubsection{Symplectic Shadow Pre-Processing and Boundary Corrector Inversion (Phase 1)}
\label{sub:shadow_preprocessing}

High-order geometric composition schemes (such as $\mathrm{BM}_4^{(7)}$ and $\mathrm{BM}_6^{(11)}$) achieve their formal order by eliminating low-order commutators through symmetric palindromic factorizations. However, advancing from order 4 to effective order 6 by direct composition increases the number of force evaluations from 7 to 11 stages per macro-step.

In the asymptotic hierarchical regime $\varepsilon \sim (r/R)^3 \ll h^2$ on $\mathcal{D}_{\mathrm{reg}}$, our solver can elevate the effective convergence order of $\mathrm{BM}_4^{(7)}$ at zero marginal interior force evaluation cost through \emph{processed symplectic integration} (boundary correctors) \cite{wisdom1996correctors, blanes2000processing, laskar2001large, hairer2006geometric}. A processed fourth-order symplectic integrator ($\mathrm{PBM}_4$) conjugates the base composition $\mathcal{S}_{4}^{\mathrm{BM}}(h)$ with the global boundary symplectomorphism $\mathcal{Z}_4^{\mathrm{BM4}}(h)$:
\begin{equation}
    \Psi_{6}^{\mathrm{PBM4}}(h) = \mathcal{Z}_4^{\mathrm{BM4}}(+h) \circ \mathcal{S}_{4}^{\mathrm{BM}}(h) \circ \mathcal{Z}_4^{\mathrm{BM4}}(-h).
    \label{eq:processed_conjugation_principle}
\end{equation}
Composing $M$ consecutive steps results in complete interior telescoping:
\begin{equation}
    \bigl(\Psi_{6}^{\mathrm{PBM4}}(h)\bigr)^M = \mathcal{Z}_4^{\mathrm{BM4}}(+h) \circ \bigl(\mathcal{S}_{4}^{\mathrm{BM}}(h)\bigr)^M \circ \mathcal{Z}_4^{\mathrm{BM4}}(-h),
    \label{eq:processed_telescoping_chain}
\end{equation}
requiring the corrector operator to be evaluated \emph{strictly once at the beginning ($t = 0$) and strictly once at the end ($t = t_{\mathrm{end}}$)} of any constant-chart, constant-step macro-trajectory.

In Phase~1, the initial phase state is prepared for propagation according to the architectural regime:
\begin{enumerate}
    \item \textbf{Processed Symplectic Regime ($\mathrm{PBM}_4$)}: The physical state $\mathbf{s}_{\mathrm{phys}}(0) \in T^* M_{\mathrm{rel}}$ is transferred into the \emph{symplectic shadow manifold} via the exact algebraic inverse corrector:
    \begin{equation}
        \mathbf{s}_{\mathrm{shadow}}(0) = \mathcal{Z}_4^{\mathrm{BM4}}(-h_0)[\mathbf{s}_{\mathrm{phys}}(0)] = \exp(-D_{K_{4}})[\mathbf{s}_{\mathrm{phys}}(0)] + \mathcal{O}(\varepsilon^2 h_0^4),
        \label{eq:shadow_entry_operator}
    \end{equation}
    where $h_0 = \Delta t_{\mathrm{sub}}(0) = \Delta t_{\mathrm{macro}} / 2^{k_0}$ is the initial dyadic sub-step size determined by the local timescale scan.
    \item \textbf{Direct Symplectic Composition Regime ($\mathrm{BM}_4^{(7)}$, $\mathrm{BM}_6^{(11)}$)}: By the Universal Obstruction Theorem (Theorem~\ref{thm:universal_order6_obstruction}), no single-commutator boundary corrector can eliminate the quadratic apex self-interaction floor globally on $T^*\mathbf{Q}[N]$. Direct compositions therefore operate with $\mathcal{C} \equiv \operatorname{Id}$, so that the physical state coincides identically with the shadow manifold:
    \begin{equation}
        \mathbf{s}_{\mathrm{shadow}}(0) \equiv \mathbf{s}_{\mathrm{phys}}(0),
        \label{eq:aba_shadow_identity}
    \end{equation}
    providing uniform nominal order $2p \in \{4, 6\}$ across all five strata of $T^*\mathbf{Q}[N]$ without pre-processing overhead.
\end{enumerate}

\begin{thmbox}[Exact Shadow Hamiltonian Alignment across Symplectic Regimes]{thm:shadow_alignment}
Let $\mathcal{S}(h)$ be a symplectic integrator operating on the Fulton--MacPherson hierarchical splitting $H = A_{\mathrm{JK}} + W_{\mathrm{apex}}^{\mathrm{screen}}$ on $T^*\mathbf{Q}[N]$ with tidal perturbation parameter $\varepsilon \sim (r/R)^3 \ll 1$ (Proposition~\ref{prop:bigraded_bch_conormal}). The numerical trajectory aligns with an effective shadow Hamiltonian $\widetilde{H}_{\mathrm{eff}}$ according to the architectural regime:
\begin{enumerate}
    \item \textbf{Processed Fourth-Order Regime ($\mathrm{PBM}_4$):}
    By Theorems~\ref{thm:c4_corrector_solvability} and~\ref{thm:asymptotic_conormal_scaling}, the cotangent-lifted homological tidal generator ($c_4^{\mathrm{BM4}} = +1.7600050100410129 \times 10^{-4}$):
    \[
        K_4 = c_4^{\mathrm{BM4}} h^4 D_A^3 B = \sum_{\lambda \in \operatorname{LeafPairs}(T)} \operatorname{Tr}\bigl(\mathbf{T}_{\mathrm{apex},\lambda}(\mathbf{R}) \, \mathbf{M}_{\mathrm{sym}}(\mathbf{r}_\lambda, \mathbf{w}_\lambda)\bigr) \in \mathcal{P}_1,
    \]
    realized non-perturbatively by the 2-stage symmetric Lie conjugator $\mathcal{Z}_4^{\mathrm{BM4}}(h)$, cancels the leading linear tidal error commutator $H_5^{\mathrm{linear}} = c_4^{\mathrm{BM4}} D_A^4 B \in \mathcal{L}_{5, 1}$ identically on $\mathcal{D}_{\mathrm{reg}}$. Consequently, the shadow state $\mathbf{s}_{\mathrm{shadow}} = \mathcal{Z}_4^{\mathrm{BM4}}(-h)[\mathbf{s}_{\mathrm{phys}}]$ evolves under:
    \begin{equation}
        \widetilde{H}_{\mathrm{eff}} = H + \mathcal{O}(\varepsilon^2 h^4) + \mathcal{O}(\varepsilon h^6),
        \label{eq:pbm4_shadow_eff}
    \end{equation}
    which confers authentic pre-asymptotic effective order~6 ($\widetilde{H}_{\mathrm{eff}} = H + \mathcal{O}(\varepsilon h^6)$, octave ratio $64$) on the regular hierarchical domain $\mathcal{D}_{\mathrm{reg}}$ where $\varepsilon \ll h^2$, with an irreducible quadratic apex error floor $\mathcal{O}(\varepsilon^2 h^4)$ when $h \lesssim \sqrt{\varepsilon}$ (suppressing the fourth-order error amplitude by $\mathcal{O}(\varepsilon)$ relative to $\mathrm{BM}_4$) in complete agreement with the Universal Obstruction Theorem (Theorem~\ref{thm:universal_order6_obstruction}).

    \item \textbf{Direct Sixth-Order Composition Regime ($\mathrm{BM}_6^{(11)}$):}
    By Theorem~\ref{thm:bm6_shadow_confinement}, the 11-stage palindromic composition $\mathcal{S}_6^{\mathrm{BM}}(h)$ satisfies all 8 RKN order conditions up to degree~6 directly via its stage weights without boundary correctors ($\mathcal{C} \equiv \operatorname{Id}$). The shadow manifold coincides identically with physical phase space:
    \begin{equation}
        \mathbf{s}_{\mathrm{shadow}} \equiv \mathbf{s}_{\mathrm{phys}}, \quad \widetilde{H}_{\mathrm{eff}} = H + \mathcal{O}(\varepsilon h^6 + \varepsilon^2 h^6),
        \label{eq:aba84_shadow_eff}
    \end{equation}
    eliminating the $\mathcal{O}(\varepsilon^2 h^4)$ quadratic floor globally while requiring zero Phase~1 pre-processing or Phase~5 post-processing overhead and zero gauge transition jump ($\Delta H_{\mathrm{gauge}} \equiv 0$).
\end{enumerate}
Furthermore, by Theorem~\ref{thm:c4_corrector_solvability} and Proposition~\ref{prop:corrector_lie_bargmann}, all 10 Galilean Lie--Bargmann conservation laws are preserved to machine precision across both regimes.
\end{thmbox}

\begin{rembox}[Engineering Realization IEEE 754 \& Float128: Symmetric Lie Conjugator $\mathcal{Z}_4^{\mathrm{BM4}}(h)$ and Torque-Free Cotangent Lift]\label{rem:ieee754_leibniz_constants}
The numerical realization of the global fourth-order symplectic boundary corrector $\mathcal{Z}_4^{\mathrm{BM4}}(h)$ (\texttt{integrators/correctors.py}, class \texttt{Palindromic\allowbreak CorrectorZ4}) satisfies three fundamental mathematical and numerical properties:
\begin{enumerate}
    \item \textbf{Exact Duhamel Calibration ($c_4^{\mathrm{BM4}}$), Antipodal Half-Shell Symmetrization, and Algebraic Reversibility of $\mathcal{Z}_4^{\mathrm{BM4}}(h)$}:
    Using the exact centered/uncentered Duhamel quartic moment identity of $\mathrm{BM}_4$:
    \[
        c_4^{\mathrm{BM4}} = \frac{1}{24}\!\left(\sum_{k=1}^8 b_k\Bigl(\tau_k - \frac{1}{2}\Bigr)^{\!4} - \frac{1}{80}\right) = \frac{1}{24}\!\left(\sum_{k=1}^8 b_k \tau_k^4 - \frac{1}{5}\right) = +1.7600050100410129 \times 10^{-4},
    \]
    the 2-stage symmetric palindromic Lie conjugation operator $\mathcal{Z}_4^{\mathrm{BM4}}(h) = S_{\mathrm{sym}}(\alpha_1, \beta_1; |h|) \circ S_{\mathrm{sym}}(\alpha_2, \beta_2; |h|)$ with $\alpha_1 = 1/4$, $\alpha_2 = 1/8$, $\beta_1 = 256\,c_4^{\mathrm{BM4}}$, $\beta_2 = -512\,c_4^{\mathrm{BM4}}$ pairs each stage into two antipodal half-shells $S_{\mathrm{sym}}(\alpha_i, \beta_i; |h|) = S(+\alpha_i, +\tfrac{1}{2}\beta_i; |h|) \circ S(-\alpha_i, -\tfrac{1}{2}\beta_i; |h|)$ (where $S(\alpha, \beta; \tau) = \Phi_A(-\alpha\tau)\circ\Phi_B(-\beta\tau)\circ\Phi_A(2\alpha\tau)\circ\Phi_B(+\beta\tau)\circ\Phi_A(-\alpha\tau)$).

    This antipodal pairing cancels the $\mathcal{O}(\alpha_i\beta_i^2 |h|^3 [D_B, [D_A, D_B]])$ quadratic self-commutator identically, realizing $\exp(\pm D_{K_4})$ non-perturbatively across arbitrary $N$-body hierarchies (including composite barycentric drift $\dot{\mathbf{R}} \ne \mathbf{0}$ and all multipoles $l \ge 2$ of $W_{\mathrm{apex}}^{\mathrm{screen}}$):
    \begin{equation}
    \begin{aligned}
        \mathcal{Z}_4^{\mathrm{BM4}}(-h) &= \bigl(\mathcal{Z}_4^{\mathrm{BM4}}(+h)\bigr)^{-1} = \exp(-D_{K_4}) + \mathcal{O}(\varepsilon^2 h^4), \\
        \mathcal{Z}_4^{\mathrm{BM4}}(+h) &= \exp(+D_{K_4}) + \mathcal{O}(\varepsilon^2 h^4),
    \end{aligned}
    \label{eq:exact_kappa_constants}
    \end{equation}
    guaranteeing exact algebraic involution $\mathcal{Z}_4^{\mathrm{BM4}}(-h) \circ \mathcal{Z}_4^{\mathrm{BM4}}(+h) = \operatorname{Id}$ down to machine precision ($< 10^{-16}$ in \texttt{float64} and $< 10^{-35}$ in 35-digit \texttt{float128}).

    \item \textbf{5-Strata Partition Compliance ($\mathcal{D}_{\mathrm{trans}}$ and $\mathcal{D}_{\mathrm{super}}$)}:
    In accordance with Theorem~\ref{thm:asymptotic_conormal_scaling}, whenever a terminal leaf pair $\lambda = (a, b)$ belongs to an active synchronous supernode ($\{a, b\} \subset S \in \mathcal{S}_{\mathrm{sync}}$ on $\mathcal{D}_{\mathrm{super}}$), its corrector contribution is bypassed. For active leaf pairs, each external spectator contribution $k \notin \{a, b\}$ is smoothly attenuated by the order-matched Hermite screening factor $\Theta(u_k)$ on $\mathcal{D}_{\mathrm{trans}}$.

    \item \textbf{Torque-Free Dynamic Quadrupole Tensor $\mathbf{M}_{\mathrm{sym}}$ and $\mathrm{SO}(3)$ Angular Momentum Conservation}:
    For each active leaf pair $\lambda = (a, b)$ with relative vector $\mathbf{r} = \mathbf{q}_b - \mathbf{q}_a$, relative momentum $\mathbf{w}$, barycenter $\mathbf{R}_{\mathrm{cm}} = (m_a \mathbf{q}_a + m_b \mathbf{q}_b)/M_{ab}$, and reduced mass $\mu = m_a m_b / M_{ab}$, the analytical cotangent lift $K_4 = \sum_\lambda \operatorname{Tr}(\mathbf{T}_{\mathrm{apex},\lambda}\,\mathbf{M}_{\mathrm{sym}}(\mathbf{r}_\lambda, \mathbf{w}_\lambda)) \in \mathcal{P}_1$ uses the symmetric dynamic quadrupole tensor:
    \begin{equation}
        \mathbf{M}_{\mathrm{sym}}(\mathbf{r}, \mathbf{w}) = c_4^{\mathrm{BM4}} h^4 \, \frac{G M_{ab}}{r^3} \left[ 2\left(\mathbf{r} \otimes \mathbf{w} + \mathbf{w} \otimes \mathbf{r}\right) - \frac{3(\mathbf{r}\cdot\mathbf{w})}{r^2}\,\mathbf{r} \otimes \mathbf{r} \right] \in \operatorname{Sym}(3),
        \label{eq:corrector_verlet_step}
    \end{equation}
    whose symmetry $\mathbf{M}_{\mathrm{sym}}^T = \mathbf{M}_{\mathrm{sym}}$ guarantees strictly zero internal binary torque ($\boldsymbol{\tau}_{\mathrm{int}} = \mathbf{r} \times \Delta\mathbf{w} + \Delta\mathbf{r} \times \mathbf{w} \equiv \mathbf{0}$), while the octupolar gradient $-\nabla_{\mathbf{R}_k} K_4$ on spectator bodies $k \notin \{a, b\}$ and its equal and opposite barycentric reaction on $\{a, b\}$ preserve total linear momentum $\mathbf{P}_{\mathrm{tot}}$, center of mass $\mathbf{R}_{\mathrm{cm}}$, and total angular momentum $\mathbf{L}_{\mathrm{tot}}$ to machine precision.
\end{enumerate}
\end{rembox}

\begin{propbox}[Exact Lie--Bargmann Invariance and Shadow Shift under $\mathcal{Z}_4^{\mathrm{BM4}}(\pm h)$]{prop:corrector_lie_bargmann}
The boundary symplectic corrector $\mathcal{Z}_4^{\mathrm{BM4}}(-h)$ preserves all $9$ kinematic first integrals of the Galilean group identically:
\begin{equation}
    \Delta \mathbf{P} \equiv \mathbf{0}, \quad
    \Delta \mathbf{L} \equiv \mathbf{0}, \quad
    \Delta \mathbf{G} \equiv \mathbf{0},
    \label{eq:corrector_kinematic_invariants}
\end{equation}
holding unconditionally in exact arithmetic, and bounded by floating-point roundoff ($< 10^{-15}$ in \texttt{float64} and $< 10^{-34}$ in \texttt{float128}) in numerical realizations. Entering the shadow manifold imparts a controlled Hamiltonian energy shift $\Delta E_{\mathrm{shadow}} = H(\mathbf{s}_{\mathrm{shadow}}) - H(\mathbf{s}_{\mathrm{phys}}) = -D_{K_{4}} H + \mathcal{O}(\varepsilon^2 h_0^{4})$ aligning the state with $\widetilde{H}_{\mathrm{eff}}$, while physical energy is conserved over the complete closed cycle $\mathcal{Z}_4^{\mathrm{BM4}}(+h) \circ (\mathcal{S}_{4}^{\mathrm{BM}})^M \circ \mathcal{Z}_4^{\mathrm{BM4}}(-h)$.
\end{propbox}

\begin{proofbox}[Proof of Proposition~\ref{prop:corrector_lie_bargmann}]
Because every elementary flow $\Phi_A$ (Kepler--Sundman leaf flow and barycentric drift) and $\Phi_B$ (screened apex interaction) in $\mathcal{Z}_4^{\mathrm{BM4}}(\pm h)$ --- as well as the analytical generator $K_4 = \sum_\lambda \operatorname{Tr}(\mathbf{T}_{\mathrm{apex},\lambda}\,\mathbf{M}_{\mathrm{sym}})$ --- is invariant under the full Galilean group, Noether's theorem on $T^*\mathbf{Q}[N]$ guarantees that $\mathcal{Z}_4^{\mathrm{BM4}}(-h)$ preserves total linear momentum $\mathbf{P}_{\mathrm{tot}}$, total angular momentum $\mathbf{L}_{\mathrm{tot}}$, and the Galilean boost $\mathbf{G}_{\mathrm{tot}}$ identically, proving~\eqref{eq:corrector_kinematic_invariants}. For the energy integral, the pullback by $\mathcal{Z}_4^{\mathrm{BM4}}(-h_0)$ shifts the raw Hamiltonian value by $-D_{K_4} A_{\mathrm{JK}} = -c_4^{\mathrm{BM4}} h_0^4 D_A^4 B$, precisely compensating for the linear tidal commutator $H_{5}^{\mathrm{linear}}$ according to Theorem~\ref{thm:c4_corrector_solvability}. Upon trajectory exit, the forward corrector $\mathcal{Z}_4^{\mathrm{BM4}}(+h_{\mathrm{exit}})$ inverts this shift identically.
\end{proofbox}


\pythonrefright{py_core}{core/jacobi.py, potential/supernode_detector.py, integrators/correctors.py}{F64 / F128}