% ==============================================================================
% chapters/ch06_algorithmic_architecture/sec05_atlas_transitions_diagnostics/content.tex
% Section 6.5: Atlas Transitions, Gauge Connection, and Symplectic Post-Processing (Phases 4 & 5)
% ==============================================================================

\subsection{Atlas Transitions, Gauge Connection, and Symplectic Post-Processing}
\label{sec:atlas_transitions_diagnostics}

At macro-interval boundaries $t = n \Delta t_{\mathrm{macro}}$, the topological admissibility of the active Jacobi tree chart $T \in \mathcal{A}$ is audited against the evolving $N$-body spatial configuration. When an intruder particle penetrates a binary or hierarchical subsystem, the active chart ceases to isolate the dominant singular divisor on the Fulton--MacPherson configuration space $\mathbf{Q}[N]$~\cite{fulton1994compactification}, threatening the $\mathcal{C}^{1,1}$ Lipschitz regularity of the hierarchical screened apex potential $W_{\mathrm{apex}}^{\mathrm{screen}}$. 

This concluding section formalizes the final two operational phases of the symplectic integrator:
\begin{enumerate}
    \item \textbf{Phase~4 (Dynamic Atlas Transitions and Gauge Connection)}: Evaluating the three-regime Hill-weighted Delaunay--Jacobi admissibility criterion (operated by default at the canonical memoryless Voronoi threshold $\gamma = \gamma_{\mathrm{build}} = 1.0$, with optional spatial hysteresis $\gamma = 0.60, \gamma_{\mathrm{build}} = 0.70$), and executing the exact canonical symplectic gauge connection $\mathcal{J}_{A \to B} \in \mathrm{Sp}(6N-6, \mathbb{R})$ between adjacent charts.
    \item \textbf{Phase~5 (Symplectic Post-Processing and Metrological Diagnostics)}: Exiting the symplectic shadow manifold at $t = t_{\mathrm{end}}$ via the forward corrector $\mathcal{C}_{2p}(h)$ to reconstitute the physical phase state $\mathbf{s}_{\mathrm{phys}}$, and executing rigorous conservation diagnostics for the ten generators of the extended Galilean Lie--Bargmann group and the Keplerian Wronskian.
\end{enumerate}

% ------------------------------------------------------------------------------
\subsubsection{Delaunay--Jacobi Chart Admissibility and Boundary Detection (Phase 4.1)}
\label{sub:delaunay_jacobi_admissibility}

Let $\mathcal{A} \coloneqq \{ T_\alpha \}_{\alpha \in \mathcal{T}_N}$ denote the maximal atlas of rooted binary Jacobi trees on the compactified configuration manifold $\mathbf{Q}[N]$. Each chart $T \in \mathcal{A}$ diagonalizes the kinetic energy into $N-1$ uncoupled fictitious center-of-mass degrees of freedom. In particular, the subset of terminal leaf pairs $\operatorname{LeafPairs}(T) \subset \{ (i, j) \mid 1 \le i < j \le N \}$ corresponds to physically bound or strongly interacting binary subsystems whose relative motion is integrated analytically via the universal Kepler--Sundman flow (\S\ref{sec:symplectic_subcycle}).

For any leaf pair $(i, j) \in \operatorname{LeafPairs}(T)$ of combined mass $m_{ij} \coloneqq m_i + m_j$ with $m_{\min} \coloneqq \min(m_i, m_j)$ and $m_{\max} \coloneqq \max(m_i, m_j)$, let $r_{ij} \coloneqq \|\mathbf{q}_i - \mathbf{q}_j\|$ denote their mutual separation, and for each external third body $k \notin \{i, j\}$ of mass $m_k$, let $d_{ij, k} \coloneqq \min\bigl( \|\mathbf{q}_i - \mathbf{q}_k\|, \|\mathbf{q}_j - \mathbf{q}_k\| \bigr)$ and $\eta_{\mathrm{raw}}(i, j; k) \coloneqq \bigl((m_i + m_j)/(2\, m_k)\bigr)^{1/3}$. Depending on the leaf mass ratio and the threshold parameter $\gamma \in (0, 1]$, we define the three-regime Hill-weighted effective intruder distance:
\begin{equation}
\begin{aligned}
    d_{\mathrm{ext}}^{\mathrm{Hill}}(i, j; \gamma) &\coloneqq \min_{k \notin \{i, j\}} \Bigl( \eta_{\mathrm{Hill}}(i, j; k; \gamma) \, d_{ij, k} \Bigr), \\
    \eta_{\mathrm{Hill}}(i, j; k; \gamma) &\coloneqq \left\{\begin{aligned}
        &\eta_{\mathrm{raw}}(i, j; k), & &\text{if } m_{\min} < 0.01\, m_{\max}, \\
        &1, & &\text{if } m_{\min} \ge 0.01\, m_{\max} \text{ and } \gamma = 1, \\
        &\max\!\bigl(1, \, \eta_{\mathrm{raw}}(i, j; k)\bigr), & &\text{if } m_{\min} \ge 0.01\, m_{\max} \text{ and } \gamma \le 0.60,
    \end{aligned}\right.
\end{aligned}
\label{eq:external_intruder_distance}
\end{equation}
with linear interpolation $1 + \frac{1-\gamma}{0.40}\bigl(\max(1, \eta_{\mathrm{raw}}) - 1\bigr)$ for comparable-mass leaves when $\gamma \in (0.60, 1)$.

\begin{propbox}[Three-Regime Hill-Weighted Delaunay--Jacobi Admissibility]{prop:delaunay_jacobi_admissibility}
A Jacobi tree chart $T \in \mathcal{A}$ is topologically and dynamically admissible with respect to the phase configuration $\mathbf{q} \in \mathbf{Q}[N]$ if and only if:
\begin{enumerate}
    \item Every terminal leaf pair $(i, j) \in \operatorname{LeafPairs}(T)$ satisfies the Hill-weighted Delaunay isolation condition:
    \begin{equation}
    \begin{aligned}
        &d_{\mathrm{ext}}^{\mathrm{Hill}}(i, j; \gamma) \ge \gamma \, r_{ij} \\
        &\quad \iff \forall k \notin \{i, j\}, \quad
        \eta_{\mathrm{Hill}}(i, j; k; \gamma) \min\!\bigl( \|\mathbf{q}_i - \mathbf{q}_k\|, \|\mathbf{q}_j - \mathbf{q}_k\| \bigr) \ge \gamma \, r_{ij},
    \end{aligned}
    \label{eq:delaunay_jacobi_condition}
    \end{equation}
    where $\gamma = 1.0$ is the canonical parameter-free Voronoi default (\texttt{GAMMA\_HYSTERESIS = 1.0}, with $\gamma = \gamma_{\mathrm{monitor}} = 0.60$ and $\gamma_{\mathrm{build}} = 0.70$ available as an optional hysteresis deadband).
    \item No non-leaf pair of bodies $(p, b) \notin \operatorname{LeafPairs}(T)$ has entered a planetary Hill capture zone where $\min_{k \notin \{p, b\}} \eta_{\mathrm{raw}}(p, b; k) \, d_{pb, k} \ge \gamma_{\mathrm{build}} \, r_{pb}$ while violating the light body $b$'s current attachment to the dominant primary $D$: $r_{pb}\bigl((m_D + m_b)/(2 m_p)\bigr)^{1/3} < \gamma \, r_{Db}$.
\end{enumerate}
When inadmissibility is detected, agglomerative tree reconstruction (\texttt{build\_optimal\_tree}, Proposition~\ref{prop:tree_builder_algorithm}) pairs clusters by minimizing $\|\mathbf{R}_a - \mathbf{R}_b\|$ (with radial-velocity tie-breaking $\dot{r}_{a,b}$) under strict lexicographic priority to candidate leaf merges satisfying $\mathrm{Adm}_{\mathrm{leaf}}(b_i, b_j)$ (Eq.~\eqref{eq:build_optimal_tree_hill_guard}).
\end{propbox}

\begin{proofbox}[Proof of Proposition~\ref{prop:delaunay_jacobi_admissibility}]
On the Fulton--MacPherson compactification $\mathbf{Q}[N]$, the singular collision divisor $\Delta \coloneqq \bigcup_{i < j} \Delta_{ij}$ is resolved by iterated real blow-ups into a normal crossing divisor with smooth boundary faces (Theorem~\ref{thm:fulton_macpherson_snc}) \cite{fulton1994compactification}. Physically, a third body $k$ of mass $m_k$ disrupts the Keplerian dominance of a pair $(i, j)$ when the tidal acceleration $\sim G m_k r_{ij} / d_{ij, k}^3$ becomes comparable to the internal attraction $\sim G (m_i + m_j) / r_{ij}^2$, i.e., across the mutual Hill sphere $R_H = r_{ij} \bigl((m_i + m_j)/(2 m_k)\bigr)^{1/3}$.

The three-regime definition of $\eta_{\mathrm{Hill}}(i, j; k; \gamma)$ in~\eqref{eq:external_intruder_distance} and the free-body capture guard in Item~(2) simultaneously resolve all physical mass hierarchies without hidden hysteresis:
\begin{enumerate}
    \item \textbf{Hierarchical Leaf Pairs ($\min(m_i, m_j) < 0.01\max(m_i, m_j)$, e.g.\ Sun--comet or planet--satellite):} Leaving $\eta_{\mathrm{Hill}} = \eta_{\mathrm{raw}} = \bigl((m_i + m_j)/(2 m_k)\bigr)^{1/3}$ unclamped ensures both directions of planetary Hill capture: for a Sun--comet leaf $(0, c)$ approached by a giant planet $J$ ($m_0 + m_c \gg m_J$), $\eta_{\mathrm{raw}} \gg 1$ triggers the heliocentric $\to$ planetocentric flip precisely when $r_{Jc} < \gamma^{-1} r_{0c} \bigl(2 m_J / M_\odot\bigr)^{1/3} \sim R_{\mathrm{Hill},J}$; conversely, for a planet--comet leaf $(J, c)$ perturbed by the Sun ($m_J + m_c \ll M_\odot$), $\eta_{\mathrm{raw}} \ll 1$ triggers the planetocentric $\to$ heliocentric egress flip precisely when $r_{Jc}$ exits $R_{\mathrm{Hill},J}$.
    \item \textbf{Comparable-Mass Pairs at Canonical Voronoi Default ($\min(m_i, m_j) \ge 0.01\max(m_i, m_j)$ and $\gamma = 1.0$, e.g.\ equal-mass or $3\text{--}4\text{--}5$ democratic scattering):} Setting $\eta_{\mathrm{Hill}} \equiv 1$ at $\gamma = 1.0$ eliminates any mass-asymmetry overlap ($\eta_{\mathrm{raw}} \ne 1$) between adjacent charts, ensuring that chart selection defines an exact, memoryless Euclidean Voronoi partition on configuration space (while Item~(2) still prevents two co-orbital light fragments $(F_1, F_2)$ from remaining paired deep inside a massive primary's tidal zone).
    \item \textbf{Comparable-Mass Pairs under Optional Hysteresis ($\gamma \le 0.60$):} Setting $\eta_{\mathrm{Hill}} = \max(1, \eta_{\mathrm{raw}})$ enforces pure geometric Delaunay proximity ($d_{ij, k} \ge \gamma r_{ij}$) as a lower bound inside the hysteresis band.
\end{enumerate}
Condition~\eqref{eq:delaunay_jacobi_condition} defines the boundary of the adapted Jacobi chart domain $U_T \subset \mathbf{Q}[N]$ (see Figure~\ref{fig:ch06_voronoi_hill_chart_flip}).
\end{proofbox}

\begin{figure}[htbp]
\centering
\resizebox{0.98\textwidth}{!}{%
\begin{tikzpicture}[>=stealth, font=\small]

    % ==========================================================================
    % PANEL (a): Heliocentric Chart \mathcal{T}_{13} = ((1,3), 2) (Outside R_Hill)
    % ==========================================================================
    \begin{scope}[shift={(-5.35, 0)}]
        \node[font=\footnotesize\bfseries] at (0, 2.85) {(a) Heliocentric chart $\mathcal{T}_{13}$};

        % Mini binary tree \mathcal{T}_{13} at top
        \coordinate (Ra) at (0, 2.35);
        \coordinate (Ia) at (-0.42, 1.85);
        \coordinate (L1a) at (-0.78, 1.35);
        \coordinate (L3a) at (-0.06, 1.35);
        \coordinate (L2a) at (0.55, 1.35);
        \draw[line width=0.85pt, black] (Ra) -- (Ia);
        \draw[line width=0.85pt, black] (Ra) -- (L2a);
        \draw[line width=0.85pt, black] (Ia) -- (L1a);
        \draw[line width=0.85pt, black] (Ia) -- (L3a);
        \filldraw[fill=white, draw=black, line width=0.8pt] (Ra) circle (1.8pt);
        \filldraw[fill=white, draw=black, line width=0.8pt] (Ia) circle (1.8pt);
        \filldraw[black] (L1a) circle (1.8pt) node[below=1pt, font=\scriptsize] {$1$};
        \filldraw[black] (L3a) circle (1.8pt) node[below=1pt, font=\scriptsize] {$3$};
        \filldraw[black] (L2a) circle (1.8pt) node[below=1pt, font=\scriptsize] {$2$};

        % Physical configuration below: Primary 1 at (-1.55, -0.55), Planet 2 at (1.15, -0.55)
        \coordinate (P1a) at (-1.55, -0.55);
        \coordinate (P2a) at (1.15, -0.55);
        \coordinate (P3a) at (-0.35, 0.22);
        \coordinate (B13a) at (-1.35, -0.42);

        % Shaded Hill sphere around Planet 2 (radius 0.92)
        \fill[black!8] (P2a) circle (0.92);
        \draw[dashed, line width=0.7pt, black!70] (P2a) circle (0.92);
        \node[font=\scriptsize] at (1.15, -1.72) {$R_{\mathrm{Hill}}$};

        % Faint trajectory curve of body 3 approaching Planet 2's Hill sphere
        \draw[dotted, line width=0.7pt, black!55] (-0.95, 0.48) .. controls (-0.35, 0.22) and (0.15, -0.02) .. (0.85, -0.22);

        % Dashed cluster oval around active leaf pair {1, 3}
        \draw[densely dashed, line width=0.6pt, black!55, rotate around={33:(-0.95,-0.16)}] (-0.95, -0.16) ellipse (1.02 and 0.42);

        % Active leaf segment r_{13} (thick solid) and apex segment r_apex (thin solid)
        \draw[line width=1.2pt, black] (P1a) -- (P3a) node[midway, above left=0.5pt, font=\scriptsize] {$\mathbf{r}_{13}$};
        \draw[line width=0.7pt, black!75] (B13a) -- (P2a) node[pos=0.45, below=1pt, font=\scriptsize] {$\mathbf{r}_{\mathrm{apex}}$};

        % Bodies 1 (large), 2 (medium), 3 (small) and barycenter o_{13}
        \filldraw[black] (P1a) circle (3.4pt) node[below left=1pt, font=\scriptsize] {$m_1$};
        \filldraw[black] (P2a) circle (2.5pt) node[below right=1pt, font=\scriptsize] {$m_2$};
        \filldraw[black] (P3a) circle (1.9pt) node[above=2pt, font=\scriptsize] {$m_3$};
        \filldraw[fill=white, draw=black, line width=0.75pt] (B13a) circle (1.7pt);
    \end{scope}

    % Transition arrow (a) -> (b)
    \draw[->, line width=0.9pt, black] (-2.95, -0.45) -- (-2.15, -0.45);

    % ==========================================================================
    % PANEL (b): Hill Voronoi Boundary Crossing (r_{23} = R_Hill at t_n)
    % ==========================================================================
    \begin{scope}[shift={(0, 0)}]
        \node[font=\footnotesize\bfseries] at (0, 2.85) {(b) Hill boundary $\partial R_{\mathrm{Hill}}$ ($t_n$)};

        % Gauge transformation formula at top
        \node[font=\scriptsize] at (0, 1.95) {$\eta_{\mathrm{Hill}}\, r_{23} = r_{13}$};
        \node[font=\scriptsize] at (0, 1.48) {$\mathcal{J}_{13 \to 23} = T_{A \to B}^{(\mathrm{PBM4})}$};

        % Physical configuration at boundary crossing
        \coordinate (P1b) at (-1.55, -0.55);
        \coordinate (P2b) at (1.15, -0.55);
        % Point P3b lies exactly on the circle of radius 0.92 around (1.15, -0.55) at angle 142 deg:
        % (1.15 + 0.92*cos(142), -0.55 + 0.92*sin(142)) = (0.425, 0.016)
        \coordinate (P3b) at (0.425, 0.016);

        % Shaded Hill sphere around Planet 2
        \fill[black!8] (P2b) circle (0.92);
        \draw[dashed, line width=0.75pt, black] (P2b) circle (0.92);
        \node[font=\scriptsize] at (1.15, -1.72) {$r_{23} = R_{\mathrm{Hill}}$};

        % Faint trajectory curve of body 3 crossing the Hill circle at P3b
        \draw[dotted, line width=0.7pt, black!55] (-0.95, 0.48) .. controls (-0.15, 0.25) and (0.20, 0.12) .. (0.95, -0.25);

        % Both competing leaf segments r_{13} and r_{23} shown at the equidistant Hill threshold
        \draw[densely dashed, line width=0.85pt, black!75] (P1b) -- (P3b) node[midway, above left=0.5pt, font=\scriptsize] {$\mathbf{r}_{13}$};
        \draw[densely dashed, line width=0.85pt, black!75] (P2b) -- (P3b) node[midway, above right=0.5pt, font=\scriptsize] {$\mathbf{r}_{23}$};

        % Bodies 1, 2 and synchronization boundary circle at P3b
        \filldraw[black] (P1b) circle (3.4pt) node[below left=1pt, font=\scriptsize] {$m_1$};
        \filldraw[black] (P2b) circle (2.5pt) node[below right=1pt, font=\scriptsize] {$m_2$};
        \filldraw[fill=white, draw=black, line width=1.0pt] (P3b) circle (2.5pt) node[above=3pt, font=\scriptsize] {$m_3(t_n)$};
    \end{scope}

    % Transition arrow (b) -> (c)
    \draw[->, line width=0.9pt, black] (2.25, -0.45) -- (3.05, -0.45);

    % ==========================================================================
    % PANEL (c): Planetocentric Chart \mathcal{T}_{23} = ((2,3), 1) (Inside R_Hill)
    % ==========================================================================
    \begin{scope}[shift={(5.35, 0)}]
        \node[font=\footnotesize\bfseries] at (0, 2.85) {(c) Planetocentric chart $\mathcal{T}_{23}$};

        % Mini binary tree \mathcal{T}_{23} at top
        \coordinate (Rc) at (0, 2.35);
        \coordinate (Ic) at (0.42, 1.85);
        \coordinate (L1c) at (-0.55, 1.35);
        \coordinate (L2c) at (0.06, 1.35);
        \coordinate (L3c) at (0.78, 1.35);
        \draw[line width=0.85pt, black] (Rc) -- (L1c);
        \draw[line width=0.85pt, black] (Rc) -- (Ic);
        \draw[line width=0.85pt, black] (Ic) -- (L2c);
        \draw[line width=0.85pt, black] (Ic) -- (L3c);
        \filldraw[fill=white, draw=black, line width=0.8pt] (Rc) circle (1.8pt);
        \filldraw[fill=white, draw=black, line width=0.8pt] (Ic) circle (1.8pt);
        \filldraw[black] (L1c) circle (1.8pt) node[below=1pt, font=\scriptsize] {$1$};
        \filldraw[black] (L2c) circle (1.8pt) node[below=1pt, font=\scriptsize] {$2$};
        \filldraw[black] (L3c) circle (1.8pt) node[below=1pt, font=\scriptsize] {$3$};

        % Physical configuration below: body 3 inside Planet 2's Hill sphere
        \coordinate (P1c) at (-1.55, -0.55);
        \coordinate (P2c) at (1.15, -0.55);
        \coordinate (P3c) at (0.82, -0.15);
        \coordinate (B23c) at (1.08, -0.47);

        % Shaded Hill sphere around Planet 2
        \fill[black!8] (P2c) circle (0.92);
        \draw[dashed, line width=0.7pt, black!70] (P2c) circle (0.92);
        \node[font=\scriptsize] at (1.15, -1.72) {$R_{\mathrm{Hill}}$};

        % Faint trajectory curve of body 3 deflecting around Planet 2
        \draw[dotted, line width=0.7pt, black!55] (-0.95, 0.48) .. controls (0.15, 0.15) and (0.65, -0.02) .. (1.35, -0.12);

        % Active leaf segment r_{23} (thick solid) inside R_Hill and new apex segment r'_apex (thin solid)
        \draw[line width=1.2pt, black] (P2c) -- (P3c) node[midway, right=2pt, font=\scriptsize] {$\mathbf{r}_{23}$};
        \draw[line width=0.7pt, black!75] (P1c) -- (B23c) node[midway, below=1pt, font=\scriptsize] {$\mathbf{r}_{\mathrm{apex}}'$};

        % Bodies 1, 2, 3 and new barycenter o_{23}
        \filldraw[black] (P1c) circle (3.4pt) node[below left=1pt, font=\scriptsize] {$m_1$};
        \filldraw[black] (P2c) circle (2.5pt) node[below right=1pt, font=\scriptsize] {$m_2$};
        \filldraw[black] (P3c) circle (1.9pt) node[above left=1pt, font=\scriptsize] {$m_3$};
        \filldraw[fill=white, draw=black, line width=0.75pt] (B23c) circle (1.7pt);
    \end{scope}

\end{tikzpicture}%
}
\caption{\textbf{Sequential geometry of a Hill-weighted Voronoi--Delaunay chart transition (Proposition~\ref{prop:delaunay_jacobi_admissibility}).} \textbf{(a)} Heliocentric chart $\mathcal{T}_{13} = ((1,3), 2)$ outside $R_{\mathrm{Hill}}$ with leaf $\mathbf{r}_{13}$ and apex $\mathbf{r}_{\mathrm{apex}}$; \textbf{(b)} macro-step boundary $t_n$ on the Hill Voronoi bisector $\eta_{\mathrm{Hill}} r_{23} = r_{13}$ triggering $\mathcal{J}_{13 \to 23} = T_{A \to B}^{(\mathrm{PBM4})}$; \textbf{(c)} planetocentric chart $\mathcal{T}_{23} = ((2,3), 1)$ inside $R_{\mathrm{Hill}}$ with regularized leaf $\mathbf{r}_{23}$ and apex $\mathbf{r}_{\mathrm{apex}}'$.}
\label{fig:ch06_voronoi_hill_chart_flip}
\end{figure}

Whenever condition~\eqref{eq:delaunay_jacobi_condition} is violated at a macro-step boundary $t_n$, the active chart $T$ is deemed inadmissible, triggering an agglomerative tree restructuring (Proposition~\ref{prop:tree_builder_algorithm}) to construct an adapted tree $T_{\mathrm{new}} \in \mathcal{A}$ satisfying~\eqref{eq:delaunay_jacobi_condition}. Crucially, because both the synchronous supernode detector $\mathcal{S}_{\mathrm{sync}}(T)$ (Definition~\ref{def:synchronous_supernode}) and the dyadic step-size scheduler $h_k(T)$ (Theorem~\ref{thm:leaf_apex_decoupling}) inspect the internal composite nodes and leaf pairs of the active tree, the order of operations at each macro-step boundary must obey a strict causal hierarchy:

\begin{propbox}[Causal Macro-Boundary Ordering and Supernode Refresh]{prop:causal_macro_boundary_ordering}
At the entrance of each macro-step $[t_n, t_n + \Delta t_{\mathrm{macro}}]$ from prior chart $T_{\mathrm{old}}$, supernode partition $\mathcal{S}_{\mathrm{old}}$, and sub-step scale $h_{\mathrm{old}}$, the engine (\texttt{NBodySystem.step}) executes the four boundary stages in strict causal sequence:
\begin{equation}
\begin{aligned}
    &\text{(1) Atlas Adaptation } T_{\mathrm{act}} \longrightarrow \text{(2) Supernode Refresh } \mathcal{S}_{\mathrm{act}}(T_{\mathrm{act}}) \\
    &\qquad \longrightarrow \text{(3) Dyadic Scale } h_{\mathrm{new}}(T_{\mathrm{act}}) \longrightarrow \text{(4) Gauge Connection},
\end{aligned}
\label{eq:causal_macro_boundary_sequence}
\end{equation}
where $T_{\mathrm{act}} \coloneqq T_{\mathrm{new}}$ if a chart transition occurs and $T_{\mathrm{act}} \coloneqq T_{\mathrm{old}}$ otherwise. This causal ordering guarantees that the active leaf-pair set $\operatorname{LeafPairs}(T_{\mathrm{act}})$ and the synchronous supernode family $\mathcal{S}_{\mathrm{act}} = \mathcal{S}_{\mathrm{sync}}(T_{\mathrm{act}})$ define a mutually disjoint partition of all two-body Keplerian channels at every sub-step, eliminating the one-step stale-tree double-drift anomaly.
\end{propbox}

\begin{proofbox}[Proof of Proposition~\ref{prop:causal_macro_boundary_ordering}]
Suppose instead that supernode detection were evaluated on the prior chart $T_{\mathrm{old}}$ \emph{before} executing the atlas transition $T_{\mathrm{old}} \to T_{\mathrm{new}}$ (as occurs under a naive ordering $\mathcal{S}_{\mathrm{old}}(T_{\mathrm{old}}) \to T_{\mathrm{new}}$). Consider a multi-fragment cometary cluster $S = \{F_1, F_2, F_3\}$ approaching a massive primary $J$ across the Roche tidal limit (such as Comet Shoemaker--Levy~9 in Section~\ref{sub:historical_encounters_validation}):
\begin{enumerate}
    \item In chart $T_{\mathrm{old}}$, the three fragments form a composite internal sub-tree $v_S = ((F_1, F_2), F_3)$, which is registered as an active synchronous supernode $S \in \mathcal{S}_{\mathrm{old}}(T_{\mathrm{old}})$.
    \item Upon crossing the tidal disruption boundary at $t_n$, the atlas transition replaces $T_{\mathrm{old}}$ with $T_{\mathrm{new}} = (((J, F_1), F_2), F_3)$, dissolving the composite node $v_S$ and promoting $(J, F_1)$ to an external terminal leaf pair of $T_{\mathrm{new}}$.
    \item If $\mathcal{S}_{\mathrm{old}}(T_{\mathrm{old}})$ is retained during step $n$ alongside $T_{\mathrm{new}}$, the sub-cycle propagator $\Phi_h$ integrates the fragment $F_1$ \emph{twice} within the same step: first inside the stale supernode solver $\Phi_h^{\mathrm{SN}}(S)$ and simultaneously along the newly formed Jovicentric Keplerian leaf channel $(J, F_1) \in \operatorname{LeafPairs}(T_{\mathrm{new}})$, injecting an unphysical $\mathcal{O}(h)$ spatial elongation jump ($\Delta S \approx 5.5 \times 10^{-7}$) and a discrete energy discontinuity ($\Delta E/E_0 \approx 7.8 \times 10^{-15}$).
\end{enumerate}
By enforcing the causal sequence~\eqref{eq:causal_macro_boundary_sequence}, $\mathcal{S}_{\mathrm{act}} = \mathcal{S}_{\mathrm{sync}}(T_{\mathrm{act}})$ is evaluated directly on the post-adaptation tree $T_{\mathrm{act}}$ prior to both dyadic timescale selection and shadow gauge conjugation, ensuring exact partition consistency across all steps.
\end{proofbox}

% ------------------------------------------------------------------------------
\subsubsection{Canonical Voronoi Bijectivity ($\gamma = 1.0$) vs.\ Anti-Chattering Hysteresis ($\gamma = 0.60$) and Time-Reversal Non-Injectivity (Phase 4.2)}
\label{sub:anti_chattering_safeguard}

In the vicinity of near-triple close encounters or Lagrange equilateral configurations where the mutual distances between three bodies $i, j, k$ become comparable ($r_{ij} \approx r_{jk} \approx r_{ik}$), one may theoretically introduce an asymmetric hysteresis threshold $\gamma < 1$ (such as $\gamma = 0.60, \gamma_{\mathrm{build}} = 0.70$) to prevent hypothetical back-and-forth chart chattering across the Voronoi bisector $r_{ij} = r_{jk}$. As proven below, $\gamma = 0.60$ enforces a strict spatial contraction factor $\gamma^2 = 0.36 < 1$.

In practical $N$-body integrations, however, sustained boundary chattering is exceedingly rare because generic trajectories intersect codimension-1 Voronoi bisectors transversely with non-zero relative radial velocity ($\dot{r}_{ij} \ne \dot{r}_{jk}$) at discrete macro-step intervals $\Delta t_{\mathrm{macro}} > 0$. More fundamentally, our retrospective analysis of time-reversal symmetry revealed a deep structural drawback of any strict hysteresis deadband $\gamma < 1$: inside the overlapping co-admissible deadband
\[
    \mathrm{Adm}_\gamma(T_A) \cap \mathrm{Adm}_\gamma(T_B) = \bigl\{\gamma < d_{\mathrm{ext}}^{\mathrm{Hill}}(i, j)/r_{ij} < \gamma^{-1}\bigr\} \ne \emptyset,
\]
the chart selector $\mathbf{q} \mapsto T(\mathbf{q})$ is \emph{non-injective} (multi-valued), since the active chart is no longer a pure function of the instantaneous configuration $\mathbf{q}$ but depends on the trajectory's prior history.

Under phase-space time reversal $\mathcal{R}(\mathbf{q}, \mathbf{p}) = (\mathbf{q}, -\mathbf{p})$, a state-only forward rule switches $T_A \to T_B$ upon exiting $\mathrm{Adm}_\gamma(T_A)$ at ratio $\gamma$, whereas the reversed trajectory starting in $T_B$ remains admissible in $\mathrm{Adm}_\gamma(T_B)$ across the entire deadband and only switches back to $T_A$ at ratio $\gamma^{-1} \ne \gamma$. This non-injectivity shifts the forward and backward chart transition timestamps, breaking pure state-based time-reversibility unless a stateful history stack is maintained. Setting $\gamma = \gamma_{\mathrm{build}} = 1.0$ as the canonical default collapses the multi-valued deadband to the codimension-1 Voronoi boundary, making chart selection a memoryless single-valued bijection on generic configurations and ensuring intrinsic time-reversal symmetry across all regimes.

\begin{thmbox}[Anti-Chattering Stability via Geometric Contraction under Optional $\gamma = 0.60$]{thm:anti_chattering_stability}
As established in Theorem~\ref{thm:atlas_transition_anti_chattering}, when the optional hysteresis mode $\gamma = 0.60$ is enabled, an instantaneous cyclic bifurcation $T_A \to T_B \to T_A$ between adjacent Jacobi trees is geometrically impossible. Any chart transition incurs a strict spatial contraction factor bounded by:
\begin{equation}
    \gamma^2 = 0.36 < 1,
    \label{eq:geometric_contraction_gamma2}
\end{equation}
rendering infinite Zeno chattering topologically impossible.
\end{thmbox}

\begin{proofbox}[Proof of Theorem~\ref{thm:anti_chattering_stability}]
Consider three bodies $i, j, k$. In chart $T_A$, the pair $(i, j)$ is grouped as a leaf, with separation $r_{ij}$. A transition to chart $T_B$, where $(j, k)$ is grouped as the leaf with separation $r_{jk}$, is triggered only if body $k$ penetrates the isolation zone of $(i, j)$:
\begin{equation}
    r_{jk} < \gamma \, r_{ij}.
    \label{eq:chattering_step1}
\end{equation}
Once chart $T_B$ is established, the active leaf pair is $(j, k)$. For an immediate reverse transition back to $T_A$ to occur, body $i$ must in turn penetrate the isolation zone of the new leaf $(j, k)$, which requires:
\begin{equation}
    r_{ij} < \gamma \, r_{jk}.
    \label{eq:chattering_step2}
\end{equation}
Multiplying inequalities~\eqref{eq:chattering_step1} and~\eqref{eq:chattering_step2} yields:
\begin{equation}
    r_{ij} r_{jk} < \gamma^2 \, r_{ij} r_{jk} \implies 1 < \gamma^2 = 0.36,
\end{equation}
which is an absolute arithmetic contradiction. Thus, a minimum finite physical displacement is strictly required before the reverse transition can become admissible, eliminating instantaneous high-frequency switching.
\end{proofbox}

\begin{rembox}[Analytical Dwell-Time Diagnostic $\tau_{\min}$ from Spatial Hysteresis]\label{rem:dynamical_dwell_time}
As proven in Theorem~\ref{thm:atlas_transition_anti_chattering} (Eqs.~\eqref{eq:hysteresis_gap_width}--\eqref{eq:minimal_dwell_time}), any two-chart cyclic transition $T_A \to T_B \to T_A$ must traverse the dimensionless spatial hysteresis gap $\Delta \lambda = \gamma^{-1} - \gamma = \frac{5}{3} - \frac{3}{5} = \frac{16}{15}$ and any three-chart permutation $T_A \to T_B \to T_C$ must traverse at least $\Delta \mu = 1 - \gamma = \frac{2}{5}$ at quotient-rule rates bounded by:
\[
    |\dot{\lambda}(t)| \le \frac{2(1+\lambda) v_{\max}}{r_{\min}} \le \frac{16}{3} \frac{v_{\max}}{r_{\min}}, \qquad
    |\dot{\mu}(t)| \le 4 \frac{v_{\max}}{r_{\min}},
\]
yielding the exact \emph{a posteriori} physical dwell-time bounds:
\begin{equation}
    \Delta t_{\mathrm{dwell}}^{(2\text{-chart})} \ge \frac{1}{5} \, \frac{r_{\min}}{v_{\max}}, \qquad \Delta t_{\mathrm{dwell}}^{(3\text{-chart})} \ge \frac{1}{10} \, \frac{r_{\min}}{v_{\max}}, \qquad \tau_{\min}^{\mathrm{diag}}(\mathbf{s}) \coloneqq \frac{16}{15} \, \frac{r_{\min}}{v_{\max}},
    \label{eq:tau_min_formula}
\end{equation}
where $r_{\min} \coloneqq \min_{i < j} \|\mathbf{q}_i - \mathbf{q}_j\|$, $v_{\max} \coloneqq \max_{1 \le i \le N} \|\mathbf{v}_i\|$, and $\tau_{\min}^{\mathrm{diag}}(\mathbf{s})$ denotes the nominal hysteresis-gap diagnostic exposed by \texttt{compute\_dynamic\_dwell\_time} in \texttt{system/atlas\_manager.py}.

Because the spatial hysteresis band $(\gamma, \gamma_{\mathrm{build}}) = (0.60, 0.70)$ already enforces the strict geometric contraction $\gamma^2 = 0.36 < 1$, anti-chattering stability ($\Delta t_{\mathrm{dwell}} > 0$) holds automatically as an \emph{a posteriori} property of the trajectory without imposing a redundant \emph{a priori} temporal lock (\texttt{dynamic\_tau\_min = False} by default), thereby preserving exact geometric locality and time-reversal symmetry across all mass regimes.
\end{rembox}

\begin{rembox}[Time-Reversal-Covariant Adjoint Hysteresis Law under $\Delta t \mapsto -\Delta t$]\label{rem:adjoint_time_reversal_hysteresis}
Crucially, the spatial hysteresis band $[r_{\mathrm{in}}, r_{\mathrm{out}}]$ ($r_{\mathrm{in}} < r_{\mathrm{out}}$, $\gamma^2 = 0.36 < 1$) is a directional topological memory along the trajectory rather than an irreversible thermodynamic arrow of time. In forward time ($\Delta t > 0$), a macro-step $\Psi_{+\Delta t} = \mathcal{B}_{+\Delta t} \circ \mathcal{A}_{+}$ evaluates the forward chart adaptation $\mathcal{A}_{+}$ at the step entrance $t_n$---switching $T_{n-1} \to T_n = \operatorname{Opt}_+(\mathbf{z}_n)$ whenever $\mathbf{z}_{n-1} \in \mathrm{Adm}(T_{n-1})$ and $\mathbf{z}_n \notin \mathrm{Adm}(T_{n-1})$ (entering a planetary leaf at the inner capture threshold $r_{\mathrm{in}}$ and exiting at the outer breakup threshold $r_{\mathrm{out}}$)---followed by the palindromic symplectic sub-step block $\mathcal{B}_{+\Delta t} = \Phi_{+h_n, T_n}^{2^{k_{\mathrm{sym}}}}$ on $[t_n, t_{n+1}]$.

Under time reversal ($\Delta t \mapsto -\Delta t$), exact group inversion requires:
\begin{equation}
    \bigl(\Psi_{+\Delta t}\bigr)^{-1} = \bigl(\mathcal{B}_{+\Delta t} \circ \mathcal{A}_{+}\bigr)^{-1} = \mathcal{A}_{-} \circ \mathcal{B}_{-\Delta t},
    \label{eq:adjoint_time_reversal_hysteresis}
\end{equation}
where $\mathcal{B}_{-\Delta t} = \Phi_{-h_n, T_n}^{2^{k_{\mathrm{sym}}}}$ integrates backward from $\mathbf{z}_{n+1} = \mathbf{z}(t_{n+1})$ to $\mathbf{z}_n = \mathbf{z}(t_n)$ on chart $T_n$, and on the subsequent backward step $[t_n \to t_{n-1}]$ the \emph{adjoint hysteresis operator} $\mathcal{A}_{-}$ (\texttt{evaluate\_reverse\_tree\_adaptation} in \texttt{system/atlas\_manager.py}) inspects the trial backward endpoint $\mathbf{z}_{n-1}$ with inverted entry and exit thresholds ($r_{\mathrm{in}} \leftrightarrow r_{\mathrm{out}}$).

Specifically, whenever the backward segment $[\mathbf{z}_n, \mathbf{z}_{n-1}]$ crosses $\partial\mathrm{Adm}(T_{n-1})$ from $\mathbf{z}_n \notin \mathrm{Adm}(T_{n-1})$ into $\mathbf{z}_{n-1} \in \mathrm{Adm}(T_{n-1})$ with $\operatorname{Opt}_+(\mathbf{z}_n) = T_n$, \texttt{NBodySystem.step} switches $T_n \to T_{n-1}$ at $t_n$, applies the inverse shadow gauge connection $(T_{T_{n-1}\to T_n}^{(\mathrm{PBM4})})^{-1} = T_{T_n\to T_{n-1}}^{(\mathrm{PBM4})}$, and re-integrates $[t_n \to t_{n-1}]$ on $T_{n-1}$ with the two-endpoint symmetric level $k_{\mathrm{sym},n-1} = \max(k(\mathbf{z}_n), k(\mathbf{z}_{n-1}))$. Consequently, any forward--backward integration across multiple Hill-sphere entries and exits traverses the exact reversed chart sequence on every interval $[t_m, t_{m+1}]$ at the identical transition timestamps while preserving the full anti-chattering band $r_{\mathrm{in}} < r_{\mathrm{out}}$ (see Figure~\ref{fig:ch06_causal_boundary_and_adjoint_hysteresis}).
\end{rembox}

\begin{figure}[htbp]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
    >=stealth,
    font=\small,
    stage_blue/.style={draw=NavyBlue, line width=1.05pt, fill=ThmBg, text=DarkSlate, rounded corners=3pt, minimum width=3.55cm, minimum height=1.35cm, align=center},
    stage_purple/.style={draw=AcademicPurple, line width=1.05pt, fill=PropBg, text=DarkSlate, rounded corners=3pt, minimum width=3.75cm, minimum height=1.35cm, align=center},
    stage_green/.style={draw=PythonGreen, line width=1.05pt, fill=DefBg, text=DarkSlate, rounded corners=3pt, minimum width=3.75cm, minimum height=1.35cm, align=center},
    warn_box/.style={draw=AcademicPurple, line width=0.9pt, dashed, fill=PropBg, text=DarkSlate, rounded corners=3pt, inner sep=4pt, align=center},
    arrow/.style={->, line width=1.05pt, color=DarkSlate}
]
    % === Top Panel (a): 4-Stage Causal Macro-Boundary Pipeline ===
    \node[font=\bfseries\footnotesize, text=NavyBlue] at (0.0, 4.05) {(a) Four-Stage Causal Macro-Boundary Ordering at Entrance of $I_n = [t_n, t_n + \Delta t_{\mathrm{macro}}]$ (Proposition~\ref{prop:causal_macro_boundary_ordering})};

    \node[stage_blue] (S1) at (-6.85, 2.75) {
        \textbf{\color{NavyBlue}(1) Atlas Adaptation}\\[-1pt]
        \scriptsize $T_{\mathrm{act}} \in \mathcal{A}$ ($\gamma = 1.0$)\\[-2pt]
        \scriptsize Delaunay--Jacobi Hill Check
    };

    \node[stage_purple] (S2) at (-2.30, 2.75) {
        \textbf{\color{AcademicPurple}(2) Supernode Refresh}\\[-1pt]
        \scriptsize $\mathcal{S}_{\mathrm{act}} = \mathcal{S}_{\mathrm{sync}}(T_{\mathrm{act}})$\\[-2pt]
        \scriptsize Disjoint from $\operatorname{LeafPairs}(T_{\mathrm{act}})$
    };

    \node[stage_green] (S3) at (2.30, 2.75) {
        \textbf{\color{PythonGreen}(3) Dyadic Scale}\\[-1pt]
        \scriptsize $h_{\mathrm{new}}(T_{\mathrm{act}}) = \Delta t_n / 2^{k_n}$\\[-2pt]
        \scriptsize Leaf/Apex Decoupled $k_n$
    };

    \node[stage_green, minimum width=3.95cm] (S4) at (6.95, 2.75) {
        \textbf{\color{PythonGreen}(4) Gauge Connection}\\[-1pt]
        \scriptsize $T_{(A,\mathcal{S}_A,h_A)\to(B,\mathcal{S}_B,h_B)}^{(\mathrm{PBM4})}$\\[-2pt]
        \scriptsize Shadow Re-Alignment
    };

    \draw[arrow, NavyBlue] (S1.east) -- (S2.west);
    \draw[arrow, AcademicPurple] (S2.east) -- (S3.west);
    \draw[arrow, PythonGreen] (S3.east) -- (S4.west);

    % Stale-tree warning box below S1-S2
    \node[warn_box, font=\scriptsize] (stale) at (-4.55, 1.25) {
        \textbf{\color{AcademicPurple}Eliminates Stale-Tree Double-Drift Anomaly:} Evaluating $\mathcal{S}_{\mathrm{old}}(T_{\mathrm{old}})$ \emph{before} $T_{\mathrm{old}}\to T_{\mathrm{new}}$ would integrate\\[-1pt]
        disrupted Roche fragments twice (both in $\Phi_h^{\mathrm{SN}}(S)$ and in $\operatorname{LeafPairs}(T_{\mathrm{new}})$)
    };
    \draw[->, dashed, line width=0.85pt, AcademicPurple] (S1.south) -- (-6.85, 1.68);
    \draw[->, dashed, line width=0.85pt, AcademicPurple] (S2.south) -- (-2.30, 1.68);

    % === Bottom Panel (b): Voronoi Bijectivity vs Adjoint Hysteresis ===
    \node[font=\bfseries\footnotesize, text=NavyBlue] at (0.0, 0.25) {(b) Canonical Voronoi Bijection ($\gamma = 1.0$) vs.\ Time-Reversal-Covariant Adjoint Hysteresis $\mathcal{A}_\pm$ (Theorem~\ref{thm:anti_chattering_stability} \& Remark~\ref{rem:adjoint_time_reversal_hysteresis})};

    % Deadband shaded region
    \fill[PropBg] (-3.6, -3.25) rectangle (3.6, -0.55);
    \draw[dashed, line width=0.85pt, AcademicPurple] (-3.6, -3.40) -- (-3.6, -0.45);
    \draw[line width=1.1pt, NavyBlue] (0.55, -3.40) -- (0.55, -0.45);
    \draw[dashed, line width=0.85pt, AcademicPurple] (3.6, -3.40) -- (3.6, -0.45);

    \node[above=2pt, font=\scriptsize\bfseries, text=AcademicPurple] at (0.0, -0.45) {Co-Admissible Hysteresis Deadband $\mathrm{Adm}_\gamma(T_A)\cap\mathrm{Adm}_\gamma(T_B)$ ($\gamma^2 = 0.36 < 1$)};

    % Horizontal Axis
    \draw[->, line width=0.95pt, DarkSlate] (-7.6, -3.25) -- (7.8, -3.25) node[right=2pt, font=\scriptsize\bfseries, text=DarkSlate] {$d_{\mathrm{ext}}^{\mathrm{Hill}}(i,j) / r_{ij}$};

    \node[below=2pt, font=\scriptsize\bfseries, text=AcademicPurple, align=center] at (-3.6, -3.38) {$r_{\mathrm{in}} = \gamma$\\[-2pt]\scriptsize($= 0.60$)};
    \node[below=2pt, font=\scriptsize\bfseries, text=NavyBlue, align=center] at (0.55, -3.38) {$\gamma = 1.0$ (Voronoi)\\[-2pt]\scriptsize Memoryless Bijection};
    \node[below=2pt, font=\scriptsize\bfseries, text=AcademicPurple, align=center] at (3.6, -3.38) {$r_{\mathrm{out}} = \gamma^{-1}$\\[-2pt]\scriptsize($= 1.67$)};

    % Upper Branch (Chart T_A) and Lower Branch (Chart T_B)
    \draw[line width=1.5pt, NavyBlue] (-3.6, -1.05) -- (7.2, -1.05);
    \node[above=2pt, font=\scriptsize\bfseries, text=NavyBlue] at (5.6, -1.05) {Chart $T_A$ ($\mathrm{Adm}_\gamma(T_A)$)};

    \draw[line width=1.5pt, PythonGreen] (-7.2, -2.65) -- (3.6, -2.65);
    \node[below=2pt, font=\scriptsize\bfseries, text=PythonGreen] at (-5.6, -2.65) {Chart $T_B$ ($\mathrm{Adm}_\gamma(T_B)$)};

    % Forward Adaptation A_+ at r_in
    \draw[->, line width=1.15pt, NavyBlue] (-3.6, -1.05) -- node[left=3pt, font=\scriptsize\bfseries, text=NavyBlue, align=right] {Forward $\mathcal{A}_+$\\[-2pt]\scriptsize($\Delta t > 0$, switches at $r_{\mathrm{in}}$)} (-3.6, -2.65);

    % Adjoint Backward Adaptation A_- at r_in
    \draw[->, line width=1.15pt, AcademicPurple] (-3.15, -2.65) -- node[right=2pt, font=\scriptsize\bfseries, text=AcademicPurple, align=left] {Adjoint $\mathcal{A}_-$ ($\Delta t < 0$)\\[-2pt]\scriptsize Inverts $r_{\mathrm{in}}\leftrightarrow r_{\mathrm{out}}$\\[-2pt]\scriptsize$(\Psi_{+\Delta t})^{-1} = \mathcal{A}_- \circ \mathcal{B}_{-\Delta t}$} (-3.15, -1.05);

    % Naive state-only backward switch at r_out
    \draw[->, line width=1.0pt, dashed, AcademicPurple] (3.6, -2.65) -- node[right=3pt, font=\scriptsize, text=AcademicPurple, align=left] {Naive state-only ($\Delta t < 0$)\\[-2pt]\scriptsize Switches late at $\gamma^{-1} \neq \gamma$\\[-2pt]\scriptsize\textbf{Breaks timestamp symmetry}} (3.6, -1.05);
    \node[font=\normalsize\bfseries, text=AcademicPurple] at (3.6, -1.85) {$\times$};

\end{tikzpicture}%
}
\caption{\textbf{Macro-boundary causal pipeline and time-reversal hysteresis geometry.} \textbf{(a)} Four-stage causal ordering $T_{\mathrm{act}} \to \mathcal{S}_{\mathrm{act}}(T_{\mathrm{act}}) \to h_{\mathrm{new}}(T_{\mathrm{act}}) \to T_{(A,\mathcal{S}_A,h_A)\to(B,\mathcal{S}_B,h_B)}^{(\mathrm{PBM4})}$ (Proposition~\ref{prop:causal_macro_boundary_ordering}); \textbf{(b)} canonical Voronoi bijection ($\gamma = 1.0$) versus optional anti-chattering hysteresis ($\gamma = 0.60$, $\gamma^2 = 0.36 < 1$) with adjoint reversal $(\Psi_{+\Delta t})^{-1} = \mathcal{A}_-\circ\mathcal{B}_{-\Delta t}$ (Theorem~\ref{thm:anti_chattering_stability} and Remark~\ref{rem:adjoint_time_reversal_hysteresis}).}
\label{fig:ch06_causal_boundary_and_adjoint_hysteresis}
\end{figure}

% ------------------------------------------------------------------------------
\subsubsection{Canonical Symplectic Gauge Connection (Phase 4.3)}
\label{sub:canonical_gauge_connection}

When a chart transition from $T_A$ to $T_B$ is validated at macro-boundary $t_n$, the numerical state resides in the shadow phase space $\mathcal{P}_{\mathrm{shadow}}^{(A)}$ associated with chart $T_A$. Directly rotating Jacobi coordinates in shadow space without gauge conjugation would induce an $\mathcal{O}(\varepsilon h_0^{4})$ step-size shock, corrupting the chart-dependent shadow Hamiltonian invariant $\widetilde{H}_A$.

\medskip
\noindent\textbf{Covariant Shadow-to-Shadow Conjugation vs.\ Uncorrected Rotation.}
Because the hierarchical splitting $H = A_{\mathrm{JK}}^{(T)} + W_{\mathrm{apex}}^{\mathrm{screen},(T)}$ depends on the active Jacobi tree $T$, the boundary symplectomorphisms $\mathcal{Z}_4^{\mathrm{BM4},(A)}(h_0)$ and $\mathcal{Z}_4^{\mathrm{BM4},(B)}(h_0)$ map onto distinct shadow manifolds ($\mathbf{M}_{\mathrm{coord}}^* K_4^{(B)} \ne K_4^{(A)}$):
\begin{enumerate}
    \item \textbf{Uncorrected Rotation Shock}: Applying $\mathbf{M}_{\mathrm{coord}}$ directly to the shadow state $\widetilde{\mathbf{s}}_A$ without exiting and re-entering the shadow manifold injects an uncompensated discrete energy jump $\Delta H_{\mathrm{gauge}} \sim \mathcal{O}(\varepsilon h_0^{4})$ at every chart flip, degrading secular energy conservation.
    \item \textbf{Exact Covariant Symplectic Gauge Connection}: Conjugating the linear chart transition $\mathbf{M}_{\mathrm{coord}}$ by the exact forward exit $\mathcal{Z}_4^{\mathrm{BM4},(A)}(+h_0)$ and backward re-entry $\mathcal{Z}_4^{\mathrm{BM4},(B)}(-h_0)$ transfers the shadow state seamlessly from $\mathcal{P}_{\mathrm{shadow}}^{(A)}$ to $\mathcal{P}_{\mathrm{shadow}}^{(B)}$ with machine-zero physical discontinuity.
\end{enumerate}

\begin{thmbox}[Covariant Symplectic Gauge Operator $T_{(A, h_A) \to (B, h_B)}^{(\mathrm{PBM4})}$]{thm:pbm4_gauge_operator_sec6}
For the processed fourth-order symplectic scheme $\mathrm{PBM}_4$ (Theorem~\ref{thm:pbm4_gauge_operator_exact}), let $\mathcal{Z}_4^{\mathrm{BM4},(A, \mathcal{S}_A)}(h_A)$ and $\mathcal{Z}_4^{\mathrm{BM4},(B, \mathcal{S}_B)}(h_B)$ denote the boundary symplectomorphisms generated by $K_4^{(A)}, K_4^{(B)} \in \mathcal{P}_1$ for chart-supernode-scale states $(T_A, \mathcal{S}_A, h_A)$ and $(T_B, \mathcal{S}_B, h_B)$:

\begin{enumerate}
    \item \textbf{Exact Coupled Spatio-Temporal Gauge Conjugation}: Whenever a macro-step boundary triggers either a Jacobi chart transition ($T_B \neq T_A$, altering $\mathcal{S}_B = \mathcal{S}_{\mathrm{sync}}(T_B)$) \emph{or} a dyadic sub-step scale transition ($h_B \neq h_A$ with $h_A > 0$), the shadow state is transported by the coupled covariant gauge connection:
    \begin{equation}
        T_{(A, \mathcal{S}_A, h_A) \to (B, \mathcal{S}_B, h_B)}^{(\mathrm{PBM4})} \coloneqq \mathcal{Z}_4^{\mathrm{BM4},(B, \mathcal{S}_B)}(-h_B) \circ \mathbf{M}_{\mathrm{coord}} \circ \mathcal{Z}_4^{\mathrm{BM4},(A, \mathcal{S}_A)}(+h_A),
    \label{eq:symplectic_gauge_connection_naive}
    \end{equation}
    with $\mathbf{M}_{\mathrm{coord}} = \mathbf{I}$ when $T_B = T_A$ and $h_B \neq h_A$. At simulation startup ($t = t_0$), the prior sub-step register is initialized to the neutral sentinel $h_{\mathrm{prev}}(t_0) = 0$ (\texttt{\_last\_dt\_substep = 0.0}), ensuring that the initial shadow entry $\mathcal{Z}_4^{\mathrm{BM4},(T_0, \mathcal{S}_0)}(-h_0)$ is executed exactly once without triggering a spurious scale-transition kick on step~$0$.
    \item \textbf{Abelian Poisson Collapse on $\mathcal{P}_0$}: In the unipotent specialization to configuration-only potentials ($K^{(A)}, K^{(B)} \in \mathcal{P}_0$, $\nabla_{\mathbf{w}} K \equiv \mathbf{0}$), the cross-Poisson bracket vanishes identically on $T^*\mathbf{Q}[N]$:
    \begin{equation}
        \{K^{(B)}, \mathbf{M}_{\mathrm{coord}}^* K^{(A)}\} \equiv 0,
        \label{eq:pbm4_abelian_cross_bracket}
    \end{equation}
    collapsing the Baker--Campbell--Hausdorff series to the closed-form unipotent momentum kick:
    \begin{equation}
        \mathbf{q}_{\mathrm{new}} = \mathbf{q}_{\mathrm{old}}, \quad \mathbf{p}_{\mathrm{new}} = \mathbf{p}_{\mathrm{old}} + \Delta \mathbf{p}_{\mathrm{kick}}^{(A)}(\mathbf{q}, h_A) - \Delta \mathbf{p}_{\mathrm{kick}}^{(B)}(\mathbf{q}, h_B).
        \label{eq:pbm4_exact_unipotent_kick}
    \end{equation}
\end{enumerate}
The operator $T_{(A, \mathcal{S}_A, h_A) \to (B, \mathcal{S}_B, h_B)}^{(\mathrm{PBM4})}$ satisfies the following six canonical axioms:
\begin{itemize}
    \item \textbf{Axiom 1 (Exact Symplecticity)}: $T_{(A, \mathcal{S}_A, h_A) \to (B, \mathcal{S}_B, h_B)}^{(\mathrm{PBM4})} \in \operatorname{Symp}(T^*\mathbf{Q}[N])$ as a composition of exact symplectomorphisms.
    \item \textbf{Axiom 2 (Physical State Continuity)}: $\mathcal{Z}_4^{\mathrm{BM4},(B, \mathcal{S}_B)}(+h_B)\bigl(T_{(A, \mathcal{S}_A, h_A) \to (B, \mathcal{S}_B, h_B)}^{(\mathrm{PBM4})}[\widetilde{\mathbf{s}}_A]\bigr) = \mathbf{M}_{\mathrm{coord}}\,\mathcal{Z}_4^{\mathrm{BM4},(A, \mathcal{S}_A)}(+h_A)[\widetilde{\mathbf{s}}_A]$ identically down to machine precision.
    \item \textbf{Axiom 3 (Lie--Bargmann Invariance)}: Conserves all ten extended Galilean generators: total linear momentum ($\Delta \mathbf{P}_{\mathrm{tot}} = \mathbf{0}$), center-of-mass boost ($\Delta \mathbf{G}_{\mathrm{tot}} = \mathbf{0}$), and total $\mathrm{SO}(3)$ angular momentum ($\Delta \mathbf{L}_{\mathrm{tot}} = \mathbf{0}$).
    \item \textbf{Axiom 4 (Zero Physical Gauge Jump)}: The discrete physical energy jump across any combined chart and scale transition vanishes to machine roundoff:
    \begin{equation}
    \begin{aligned}
        \Delta H_{\mathrm{gauge}} &\coloneqq |H(\mathbf{s}_{B,\mathrm{phys}}) - H(\mathbf{s}_{A,\mathrm{phys}})| \le \mathcal{O}(\epsilon_{\mathrm{mach}}) \\
        &\qquad (\sim 2.07 \times 10^{-35}\text{ in Float128, } 2.06 \times 10^{-16}\text{ in Float64}).
    \end{aligned}
    \end{equation}
    \item \textbf{Axiom 5 (\v{C}ech 1-Cocycle Groupoid Transitivity)}: On any chart-scale triple $((A, h_A), (B, h_B), (C, h_C))$,
    \[
        T_{(B, h_B) \to (C, h_C)}^{(\mathrm{PBM4})} \circ T_{(A, h_A) \to (B, h_B)}^{(\mathrm{PBM4})} = T_{(A, h_A) \to (C, h_C)}^{(\mathrm{PBM4})},
    \]
    and on any closed cycle $(A, h_A) \to (B, h_B) \to (C, h_C) \to (A, h_A)$, the holonomy is the identity ($\sim 5.30 \times 10^{-36}$ algebraically and $4.08 \times 10^{-35}$ dynamically in Float128), completely suppressing atlas and multirate hysteresis drift.
    \item \textbf{Axiom 6 (Time-Reversibility)}: Inversion satisfies $(T_{(A, h_A) \to (B, h_B)}^{(\mathrm{PBM4})})^{-1} = T_{(B, h_B) \to (A, h_A)}^{(\mathrm{PBM4})}$ via $\mathcal{Z}_4^{\mathrm{BM4}}(-h) = (\mathcal{Z}_4^{\mathrm{BM4}}(+h))^{-1}$.
\end{itemize}
\end{thmbox}
\leanrefright{pbm4_gauge_transition_order6}{pbm4_gauge_transition_exact}{06_ConormalGaugeTransitionPBM4_Order6.lean}

\begin{proofbox}[Proof of Theorem~\ref{thm:pbm4_gauge_operator_sec6}]
By definition~\eqref{eq:symplectic_gauge_connection_naive}, $T_{(A, \mathcal{S}_A, h_A) \to (B, \mathcal{S}_B, h_B)}^{(\mathrm{PBM4})} = \mathcal{Z}_4^{\mathrm{BM4},(B, \mathcal{S}_B)}(-h_B) \circ \mathbf{M}_{\mathrm{coord}} \circ \mathcal{Z}_4^{\mathrm{BM4},(A, \mathcal{S}_A)}(+h_A)$ is a composition of exact Galilean-invariant symplectomorphisms, establishing Axioms~1--4. On any chart-scale triple, the intermediate boundary correctors telescope identically via $\mathcal{Z}_4^{\mathrm{BM4},(B, \mathcal{S}_B)}(+h_B) \circ \mathcal{Z}_4^{\mathrm{BM4},(B, \mathcal{S}_B)}(-h_B) = \operatorname{Id}$ (Axioms~5--6). In the unipotent specialization to $\mathcal{P}_0$, $\{F, G\} = 0$ for all $F, G \in \mathcal{P}_0$ collapses the BCH series to~\eqref{eq:pbm4_exact_unipotent_kick} with symmetric Hessian Jacobian $\mathbf{D} = \begin{psmallmatrix} \mathbf{I} & \mathbf{0} \\ \mathbf{H} & \mathbf{I} \end{psmallmatrix} \in \mathrm{Sp}(6N-6, \mathbb{R})$.
\end{proofbox}


\begin{propbox}[Exact Unimodularity and Area Preservation of $\mathbf{M}_{\mathrm{coord}}$]{prop:sl2_unimodularity}
Let $A, B, C$ denote three contiguous clusters undergoing the elementary tree rotation $((A, B), C) \mapsto (A, (B, C))$, with respective cluster masses $m_A, m_B, m_C > 0$. Let $m_{AB} \coloneqq m_A + m_B$, $m_{BC} \coloneqq m_B + m_C$, and $m_{ABC} \coloneqq m_A + m_B + m_C$.

The linear coordinate transformation on relative spatial Jacobi vectors $(\mathbf{r}_1, \mathbf{r}_2) \mapsto (\mathbf{r}'_1, \mathbf{r}'_2)$ is given by:
\begin{equation}
    \begin{pmatrix} \mathbf{r}'_1 \\ \mathbf{r}'_2 \end{pmatrix} = \mathbf{M}_{\mathrm{coord}} \begin{pmatrix} \mathbf{r}_1 \\ \mathbf{r}_2 \end{pmatrix}, \quad \text{with} \quad \mathbf{M}_{\mathrm{coord}} \coloneqq \begin{pmatrix} -\frac{m_A}{m_{AB}} & 1 \\[6pt] -\frac{m_B m_{ABC}}{m_{AB} m_{BC}} & -\frac{m_C}{m_{BC}} \end{pmatrix} \in \mathrm{SL}(2, \mathbb{R}).
    \label{eq:sl2_matrix_exact}
\end{equation}
The conjugate canonical momenta $(\mathbf{w}_1, \mathbf{w}_2) \mapsto (\mathbf{w}'_1, \mathbf{w}'_2)$ transform contragrediently via $(\mathbf{M}_{\mathrm{coord}}^{-1})^T$:
\begin{equation}
    \begin{pmatrix} \mathbf{w}'_1 \\ \mathbf{w}'_2 \end{pmatrix} = \begin{pmatrix} d & -c \\ -b & a \end{pmatrix} \begin{pmatrix} \mathbf{w}_1 \\ \mathbf{w}_2 \end{pmatrix} = \begin{pmatrix} -\frac{m_C}{m_{BC}} & \frac{m_B m_{ABC}}{m_{AB} m_{BC}} \\[6pt] -1 & -\frac{m_A}{m_{AB}} \end{pmatrix} \begin{pmatrix} \mathbf{w}_1 \\ \mathbf{w}_2 \end{pmatrix}.
    \label{eq:sl2_momenta_exact}
\end{equation}
Both transformations strictly preserve the canonical Darboux 2-form:
\begin{equation}
    \mathrm{d}\mathbf{r}'_1 \wedge \mathrm{d}\mathbf{w}'_1 + \mathrm{d}\mathbf{r}'_2 \wedge \mathrm{d}\mathbf{w}'_2 = \mathrm{d}\mathbf{r}_1 \wedge \mathrm{d}\mathbf{w}_1 + \mathrm{d}\mathbf{r}_2 \wedge \mathrm{d}\mathbf{w}_2.
    \label{eq:darboux_preservation_sl2}
\end{equation}
\end{propbox}

\begin{proofbox}[Proof of Proposition~\ref{prop:sl2_unimodularity}]
The determinant of $\mathbf{M}_{\mathrm{coord}} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ evaluates algebraically to:
\begin{align}
    \det(\mathbf{M}_{\mathrm{coord}}) &= ad - bc = \left( -\frac{m_A}{m_{AB}} \right) \left( -\frac{m_C}{m_{BC}} \right) - (1) \left( -\frac{m_B m_{ABC}}{m_{AB} m_{BC}} \right) \nonumber \\
    &= \frac{m_A m_C + m_B(m_A + m_B + m_C)}{m_{AB} m_{BC}} \nonumber \\
    &= \frac{m_A m_C + m_A m_B + m_B^2 + m_B m_C}{(m_A + m_B)(m_B + m_C)} = \frac{(m_A + m_B)(m_B + m_C)}{(m_A + m_B)(m_B + m_C)} = 1.
\end{align}
Because $\det(\mathbf{M}_{\mathrm{coord}}) = 1$, the matrix belongs to $\mathrm{SL}(2, \mathbb{R})$. In the 12-dimensional phase subspace spanned by the two active 3D Jacobi vectors $(\mathbf{r}_1, \mathbf{r}_2, \mathbf{w}_1, \mathbf{w}_2)$, the full transformation matrix $\mathbf{\Psi} \coloneqq \mathbf{M}_{\mathrm{coord}} \otimes \mathbf{I}_3 \oplus (\mathbf{M}_{\mathrm{coord}}^{-1})^T \otimes \mathbf{I}_3$ belongs to $\mathrm{Sp}(12, \mathbb{R})$ and satisfies $\mathbf{\Psi}^T \mathbb{J}_{12} \mathbf{\Psi} = \mathbb{J}_{12}$, where $\mathbb{J}_{12} \coloneqq \begin{pmatrix} \mathbf{0} & \mathbf{I}_6 \\ -\mathbf{I}_6 & \mathbf{0} \end{pmatrix}$, proving exact symplecticity.
\end{proofbox}

\begin{rembox}[IEEE~754 Orientation Convention and Sign Invariance]\label{rem:ieee754_orientation_convention}
Certain historical celestial mechanics conventions~\cite{marchal1990threebody} configure the Jacobi rotation matrix with opposite signs $\mathbf{M}_{\mathrm{Marchal}} = -\mathbf{M}_{\mathrm{coord}}$:
\begin{equation}
    a_{\mathrm{Marchal}} = +\frac{m_A}{m_{AB}}, \quad b_{\mathrm{Marchal}} = -1.0, \quad c_{\mathrm{Marchal}} = +\frac{m_B m_{ABC}}{m_{AB} m_{BC}}, \quad d_{\mathrm{Marchal}} = +\frac{m_C}{m_{BC}}.
\end{equation}
In the reference implementation (\texttt{core/transitions.py}, lines 104--225), the matrix $\mathbf{M}_{\mathrm{coord}} = -\mathbf{M}_{\mathrm{Marchal}}$ defined in~\eqref{eq:sl2_matrix_exact} is adopted, while \texttt{core/coordinates.py} defines relative Jacobi separation vectors systematically as left child minus right child ($\mathbf{r}_v \coloneqq \mathbf{X}_L(v) - \mathbf{X}_R(v)$), accompanied by the explicit orientation parity signs $(\sigma_1, \sigma_2) = (+1, -1)$ in \texttt{rotate\_jacobi\_tree\_coordinates} so that $\mathbf{r}_{B,1} = \mathbf{X}_B - \mathbf{X}_C$ and $\mathbf{r}_{B,2} = \mathbf{X}_A - \mathbf{X}_{BC}$ coincide bit-for-bit with the direct bottom-up recursion on chart $T_B = (A, (B, C))$.

Crucially, because the matrix dimension is $2 \times 2$, we have:
\begin{equation}
    \det(-\mathbf{M}_{\mathrm{Marchal}}) = (-1)^2 \det(\mathbf{M}_{\mathrm{Marchal}}) = +1.0,
\end{equation}
so that both conventions belong strictly to $\mathrm{SL}(2, \mathbb{R})$ and preserve the canonical Darboux 2-form. Under IEEE~754 double-precision arithmetic (and 35-digit quadruple precision), the determinant evaluates to $1.0$ with zero floating-point error ($|\det - 1.0| < 10^{-16}$ in \texttt{float64} and $< 10^{-34}$ in \texttt{float128}), certifying exact gauge invariance. The executable $\mathrm{SL}(2, \mathbb{R})$ transition and invariant audit pipeline is colocalized in Technical Appendix~\ref{app:atlas_transitions_diagnostics}.
\end{rembox}

% ------------------------------------------------------------------------------
\subsubsection{Final Shadow Exit and Metrological Lie--Bargmann Diagnostics (Phase 5)}
\label{sub:final_shadow_exit_diagnostics}

Upon reaching the final simulation time $t = t_{\mathrm{end}}$ (or at any intermediate visualization checkpoint where physical observables are requested), the integrator executes \textbf{Phase~5 : Symplectic Post-Processing}.

\medskip
\noindent\textbf{1. Definitive Physical Trajectory Reconstruction.}
Throughout the time evolution under $\mathrm{PBM}_4$, the phase coordinates $(\mathbf{q}, \mathbf{p})$ advance along the shadow Hamiltonian $\widetilde{H}$. To eliminate the leading linear tidal truncation error of order $4$ on $\mathcal{D}_{\mathrm{reg}}$ and recover the physical trajectory on $\mathbf{Q}[N]$, the final state is mapped through the forward corrector $\mathcal{Z}_4^{\mathrm{BM4}}(+h_{\mathrm{exit}})$:
\begin{equation}
    \mathbf{s}_{\mathrm{phys}}(t_{\mathrm{end}}) = \mathcal{Z}_4^{\mathrm{BM4}}(+h_{\mathrm{exit}}) \left[ \mathbf{s}_{\mathrm{shadow}}(t_{\mathrm{end}}) \right],
    \label{eq:final_shadow_exit_formula}
\end{equation}
where $h_{\mathrm{exit}}$ is the elementary sub-step size of the final sub-cycle.

By virtue of the homological equation $D_A K_{4} = h_{\mathrm{exit}}^4 H_{5}^{\mathrm{linear}}$ established in Theorem~\ref{thm:c4_corrector_solvability}, mapping through $\mathcal{Z}_4^{\mathrm{BM4}}(+h_{\mathrm{exit}})$ removes the leading linear tidal perturbation on $\mathcal{D}_{\mathrm{reg}}$:
\begin{equation}
    H(\mathbf{s}_{\mathrm{phys}}(t_{\mathrm{end}})) - H(\mathbf{s}_0) = \mathcal{O}(\varepsilon^2 h_{\mathrm{exit}}^{4}) + \mathcal{O}(\varepsilon h_{\mathrm{exit}}^{6}),
\end{equation}
producing authentic pre-asymptotic effective order $p_{\mathrm{eff}} = 6$ on $\mathcal{D}_{\mathrm{reg}}$ when $\varepsilon \ll h_{\mathrm{exit}}^2$ (for $\mathrm{BM}_4^{(7)} + \mathcal{Z}_4^{\mathrm{BM4}}$, as established in Theorems~\ref{thm:c4_corrector_solvability} and~\ref{thm:asymptotic_conormal_scaling}), while direct sixth-order composition $\mathrm{BM}_6^{(11)}$ achieves uniform order $6$ globally across all strata of $T^*\mathbf{Q}[N]$.

\medskip
\noindent\textbf{2. Metrological Audit of the Ten Lie--Bargmann Invariants.}
In the extended phase space $\mathcal{P}_{\mathrm{ext}}$, the fundamental symmetries of Newtonian gravitation correspond to the ten-dimensional Galilean Lie--Bargmann group (Propositions~\ref{prop:lie_bargmann_closure} and~\ref{prop:lie_bargmann_preservation}). The integrity of the numerical simulation is continuously monitored by auditing the exact conservation of these generators:

\begin{propbox}[Conservation Spectrum of the Ten Lie--Bargmann Generators]{prop:lie_bargmann_conservation_spectrum}
Under the complete symplectic integrator pipeline (Phases~0 through~5), the ten generators of the Lie--Bargmann algebra satisfy the following rigorous conservation guarantees:
\begin{enumerate}
    \item \textbf{Total Linear Momentum} ($\mathbf{P}_{\mathrm{tot}} \in \mathbb{R}^3$, 3 invariants):
    \begin{equation}
        \mathbf{P}_{\mathrm{tot}}(\mathbf{s}) \coloneqq \sum_{i=1}^N \mathbf{p}_i, \quad \|\mathbf{P}_{\mathrm{tot}}(t) - \mathbf{P}_{\mathrm{tot}}(0)\| < 10^{-15},
    \end{equation}
    preserved exactly due to spatial translation invariance and Newton's third law in both the Kepler--Sundman flow and the screened apex kick.
    \item \textbf{Total Angular Momentum} ($\mathbf{L}_{\mathrm{tot}} \in \mathbb{R}^3$, 3 invariants):
    \begin{equation}
        \mathbf{L}_{\mathrm{tot}}(\mathbf{s}) \coloneqq \sum_{i=1}^N \mathbf{q}_i \times \mathbf{p}_i, \quad \|\mathbf{L}_{\mathrm{tot}}(t) - \mathbf{L}_{\mathrm{tot}}(0)\| < 10^{-14},
    \end{equation}
    preserved exactly due to $\mathrm{SO}(3)$ rotational invariance and the radial symmetry of screening mollifiers.
    \item \textbf{Galilean Center-of-Mass Boost} ($\mathbf{G}_{\mathrm{tot}} \in \mathbb{R}^3$, 3 invariants):
    \begin{equation}
        \mathbf{G}_{\mathrm{tot}}(\mathbf{s}, t) \coloneqq M_{\mathrm{tot}} \mathbf{R}_{\mathrm{cm}}(t) - t \, \mathbf{P}_{\mathrm{tot}}(t), \quad \|\mathbf{G}_{\mathrm{tot}}(t) - \mathbf{G}_{\mathrm{tot}}(0)\| < 10^{-14},
    \end{equation}
    preserved exactly because the center of mass advances linearly ($\dot{\mathbf{R}}_{\mathrm{cm}} = \mathbf{P}_{\mathrm{tot}} / M_{\mathrm{tot}}$) under the analytic drift operator $\Phi_h^{\mathrm{drift}}$.
    \item \textbf{Classical Hamiltonian Energy} ($H \in \mathbb{R}$, 1 invariant):
    \begin{equation}
        H(\mathbf{q}, \mathbf{p}) \coloneqq \frac{1}{2} \sum_{i=1}^N \frac{\|\mathbf{p}_i\|^2}{m_i} - \sum_{1 \le i < j \le N} \frac{G m_i m_j}{\|\mathbf{q}_i - \mathbf{q}_j\|}, \quad |\Delta H(t)| \le C \, h_{\mathrm{macro}}^{p_{\mathrm{eff}}},
    \end{equation}
    governed by backward error analysis and Nekhoroshev--Gevrey exponential stability \cite{benettin1994hamiltonian, hairer2006geometric}, which guarantees that the shadow energy remains confined within an $\mathcal{O}(h_{\mathrm{macro}}^{p_{\mathrm{eff}}})$ band over exponentially long timescales $T \sim \exp((h_*/h_{\mathrm{macro}})^{1/\sigma})$ where $\sigma$ is the Gevrey regularity index of the desingularized vector field on $\mathbf{Q}[N]$, preventing secular energy drift even in non-integrable chaotic regimes.
\end{enumerate}
\end{propbox}

\medskip
\noindent\textbf{3. Keplerian Wronskian Unimodularity Diagnostic.}
In addition to the global Lie--Bargmann invariants, every active terminal leaf pair $(i, j) \in \operatorname{LeafPairs}(T)$ is monitored at each substep by auditing the unimodularity of its universal Lagrange multipliers:
\begin{equation}
    \mathcal{W}_{ij}(\tau) \coloneqq \left| f(\tau) \dot{g}(\tau) - \dot{f}(\tau) g(\tau) - 1.0 \right|.
    \label{eq:wronskian_unimodularity_metric}
\end{equation}
Under the Battin unimodular projection~\eqref{eq:battin_projection}~\cite{battin1999introduction}, the metric satisfies $\mathcal{W}_{ij} \le 10^{-15}$ unconditionally, certifying that the two-body leaf flow remains strictly symplectic and area-preserving down to the IEEE~754 floating-point noise floor.

% ------------------------------------------------------------------------------
\subsubsection{Synthesis of Section~\ref{ch:ch06_algorithmic_architecture} and Algorithmic Closure}
\label{sub:chapter6_synthesis_closure}

With the completion of Section~\ref{sec:atlas_transitions_diagnostics}, the complete geometric symplectic integration pipeline is fully realized:
\begin{itemize}
    \item \textbf{Phase~0 (\S\ref{sec:initialization_shadow})}: Barycentric reduction, agglomerative tree clustering $T_0$ with the two-tier unilateral Delaunay--Hill leaf isolation guard $\mathrm{Adm}_{\mathrm{leaf}}$, and Hill tidal Schmitt trigger $[\eta_{\mathrm{enter}}, \eta_{\mathrm{exit}}] = [2^{-1/3}, 2^{-1/6}]$.
    \item \textbf{Phase~1 (\S\ref{sec:initialization_shadow})}: Pre-processing into shadow space $\mathbf{s}_{\mathrm{shadow}} = \mathcal{Z}_{4}^{\mathrm{BM4}}(-h_0)[\mathbf{s}_{\mathrm{phys}}]$ via the cotangent-lifted Lie generator:
    \[
        K_4 = c_4^{\mathrm{BM4}} h_0^4 D_A^3 B \in \mathcal{P}_1 \qquad \bigl(c_4^{\mathrm{BM4}} = +1.7600050100410129 \times 10^{-4}\bigr),
    \]
    and neutral scale initialization $h_{\mathrm{prev}}(t_0) = 0$.
    \item \textbf{Phase~2 (\S\ref{sec:dyadic_scheduler})}: Dynamical timescale triad $(\tau_{\mathrm{kin}}, \tau_{\mathrm{dyn}}, \tau_{\mathrm{peri}})$, leaf/apex decoupling, and bit-exact timestamp partitioning under Skeel--Gear topological freezing.
    \item \textbf{Phase~3 (\S\ref{sec:symplectic_subcycle})}: Universal Kepler--Sundman flow with Halley root-finding and 60-step bisection fallback, McGehee bounce $\mathcal{B}_C$, synchronous supernode unfolding $\mathrm{d}t_S = \bar{\rho}_S \mathrm{d}\tau_S$, and screened apex kick with order-matched polynomial Hermite mollifier $\Theta(u)$ and exact monopole cancellation.
    \item \textbf{Phase~4 (\S\ref{sec:atlas_transitions_diagnostics})}: Three-regime Hill-weighted Delaunay--Jacobi admissibility testing ($d_{\mathrm{ext}}^{\mathrm{Hill}}(i, j; \gamma) \ge \gamma r_{ij}$, operated by default at the canonical memoryless Voronoi threshold $\gamma = \gamma_{\mathrm{build}} = 1.0$, with optional hysteresis $\gamma = 0.60, \gamma_{\mathrm{build}} = 0.70$) and anti-chattering dwell time $\tau_{\min}$; causal macro-boundary ordering
    \[
        T_{\mathrm{act}} \longrightarrow \mathcal{S}_{\mathrm{sync}}(T_{\mathrm{act}}) \longrightarrow h_k(T_{\mathrm{act}}) \longrightarrow T_{(A, \mathcal{S}_A, h_A) \to (B, \mathcal{S}_B, h_B)}^{(\mathrm{PBM4})}
    \]
    (or exact linear coordinate rotation $\mathbf{M}_{\mathrm{coord}} \in \mathrm{SL}(2, \mathbb{R})$ under direct compositions).
    \item \textbf{Phase~5 (\S\ref{sec:atlas_transitions_diagnostics})}: Definitive shadow exit $\mathbf{s}_{\mathrm{phys}} = \mathcal{Z}_{4}^{\mathrm{BM4}}(+h_{\mathrm{exit}})[\mathbf{s}_{\mathrm{shadow}}]$ (under $\mathrm{PBM}_4$, or transparent identity $\mathbf{s}_{\mathrm{phys}} \equiv \mathbf{s}_{\mathrm{shadow}}$ under direct compositions) and metrological verification of the ten Lie--Bargmann invariants.
\end{itemize}

This asynchronous, cache-aligned, 100\% CPU task-graph architecture bridges the theoretical foundations of the deformation to the normal cone (and its shifted cotangent lift) to a production-grade scientific engine, ready for the astronomical validation benchmarks of Section~\ref{ch:ch07_benchmarks_and_lean4}.

\pythonrefright{py_system}{core/{transitions,invariants}.py, system/atlas_manager.py, diagnostics/metrics.py}{F64/F128}