\newpage
\subsection{Retrospective Case Studies: Unifying Known Splittings on \texorpdfstring{$\mathrm{SE}(3)$}{SE(3)} and \texorpdfstring{$\mathbb{S}^2$}{S2}}
\label{sec:retrospective_cases}

To establish that our geometric splitting methodology is not an ad-hoc heuristic but an overarching framework, we examine how our splitting recipe, grounded in the deformation to the normal cone of Fulton--Verdier and its cotangent lift due to Calaque--Safronov, naturally unifies and justifies classical symplectic algorithms developed for rigid bodies and fluid vortices. Originating in the Lie--Poisson geometric frameworks of Ge--Marsden~\cite{ge1988lie} and Channell--Scovel~\cite{channell1991integrators}, and the explicit splitting paradigm of McLachlan~\cite{mclachlan1993explicit}, Lie--Poisson splitting methods exploit subgroup geometries to conserve Casimirs.

\subsubsection{Case Study 1: The Heavy Top on the Lie--Poisson Space $\mathfrak{se}(3)^*$}
The dynamics of a heavy rigid top spinning about a fixed point is described on the dual Lie algebra $\mathfrak{se}(3)^* \cong \R^3 \times \R^3$ by the Hamiltonian:
\begin{equation}
    H(\mathbf{\Pi}, \mathbf{\Gamma}) = \frac{1}{2} \mathbf{\Pi} \cdot \mathbb{I}^{-1} \mathbf{\Pi} + m g l \, (\mathbf{\Gamma} \cdot \boldsymbol{\chi}),
\end{equation}
generating the classical Euler--Poisson equations of motion:
\begin{equation}
    \dot{\mathbf{\Pi}} = \mathbf{\Pi} \times \mathbf{\Omega} + m g l \, (\mathbf{\Gamma} \times \boldsymbol{\chi}), \qquad
    \dot{\mathbf{\Gamma}} = \mathbf{\Gamma} \times \mathbf{\Omega},
\end{equation}
where $\mathbf{\Pi} \in \R^3$ is the angular momentum in the body frame, $\mathbf{\Omega} = \mathbb{I}^{-1} \mathbf{\Pi}$ is the angular velocity, $\mathbf{\Gamma} \in \mathbb{S}^2$ is the unit vertical gravity vector, and $\boldsymbol{\chi}$ points toward the center of mass. The Lie--Poisson structure possesses two fundamental Casimir invariants:
\begin{equation}
    C_1(\mathbf{\Gamma}) = \|\mathbf{\Gamma}\|^2 = 1, \qquad C_2(\mathbf{\Pi}, \mathbf{\Gamma}) = \mathbf{\Pi} \cdot \mathbf{\Gamma} = \text{const}.
\end{equation}
\enlargethispage{1.5\baselineskip}
Identifying the zero-momentum sphere $Z \cong \mathbb{S}^2$ with the configuration base and $\mathfrak{so}(3)^*$ with the transverse fiber, our geometric base-fiber splitting recipe decomposes the dynamics into two explicit algebraic updates:
\begin{enumerate}[leftmargin=2em, itemsep=3pt, topsep=2pt]
    \item \textbf{Normal Fiber Step ($t = 0$):} Gravity exerts a torque $\mathbf{K}(\mathbf{\Gamma}) = m g l \, (\mathbf{\Gamma} \times \boldsymbol{\chi})$. The normal fiber update is an exact linear translation along the transverse momentum fiber $\mathfrak{so}(3)^* \cong N_Z \mathfrak{se}(3)^*$:
    \begin{equation}
        \mathbf{\Pi}^* = \mathbf{\Pi} + \frac{h}{2} m g l \, (\mathbf{\Gamma} \times \boldsymbol{\chi}), \qquad \mathbf{\Gamma}^* = \mathbf{\Gamma}.
    \end{equation}
    Crucially, because $\mathbf{K}(\mathbf{\Gamma}) \cdot \mathbf{\Gamma} \equiv 0$ by antisymmetry of the vector cross product, the transverse torque is strictly orthogonal to the vertical axis, guaranteeing that $C_2$ is unaffected by the kick step without any numerical projection.
    \item \textbf{Groupoid Exponential ($t = 1$):} The kinematic rotation on $Z = \mathbb{S}^2$ is integrated in $\mathrm{SO}(3)$ acting simultaneously on gravity and angular momentum via Rodrigues' exponential formula:
    \begin{equation}
        \mathbf{R}_h = \mathbf{I} + \frac{\sin(h\|\mathbf{\Omega}^*\|)}{\|\mathbf{\Omega}^*\|} \widehat{\mathbf{\Omega}}^* + \frac{1 - \cos(h\|\mathbf{\Omega}^*\|)}{\|\mathbf{\Omega}^*\|^2} (\widehat{\mathbf{\Omega}}^*)^2, \qquad \mathbf{\Gamma}^{**} = \mathbf{R}_h^T \mathbf{\Gamma}^*, \quad \mathbf{\Pi}^{**} = \mathbf{R}_h^T \mathbf{\Pi}^*.
    \end{equation}
    This single-exponential map integrates the free rigid-body sub-step either for an isotropic top ($\mathbb{I} = I_0\mathbf{I}_3$, where $\mathbf{\Omega}^*$ is constant) or along each individual principal inertia axis in McLachlan's explicit symmetric three-rotation splitting $\mathbf{R}_h = \mathbf{R}_{h/2}^{(1)}\mathbf{R}_{h/2}^{(2)}\mathbf{R}_h^{(3)}\mathbf{R}_{h/2}^{(2)}\mathbf{R}_{h/2}^{(1)}$ of $H_{\mathrm{base}} = \sum_{a=1}^3 \Pi_a^2/(2I_a)$~\cite{mclachlan1993explicit}. In either realization, $\mathbf{R}_h \in \mathrm{SO}(3)$ is an exact orthogonal isometry on $\mathbb{S}^2$ and $\R^3$, preserving $C_1 = \|\mathbf{\Gamma}\|^2 \equiv 1$ and $C_2 = \mathbf{\Pi} \cdot \mathbf{\Gamma}$ identically to floating-point precision ($10^{-16}$).
\end{enumerate}

\begin{propbox}[Exact Casimir Preservation of the Heavy Top Splitting]{prop:heavy_top_casimirs}
Following McLachlan~\cite{mclachlan1993explicit} and Hairer, Lubich, and Wanner~\cite[Sections~VII.5.3--VII.5.4]{hairer2006geometric}, let $\mathcal{S}_2(h) = \Phi_{h/2}^{\mathrm{normal}} \circ \Phi_h^{\mathrm{base}} \circ \Phi_{h/2}^{\mathrm{normal}}$ be the Strang splitting of the heavy rigid top on $\mathfrak{se}(3)^*$. Then $\mathcal{S}_2(h)$ unconditionally preserves both Casimirs:
\begin{equation}
    C_1\bigl(\mathcal{S}_2(h) (\mathbf{\Pi}, \mathbf{\Gamma})\bigr) = C_1(\mathbf{\Pi}, \mathbf{\Gamma}), \qquad
    C_2\bigl(\mathcal{S}_2(h) (\mathbf{\Pi}, \mathbf{\Gamma})\bigr) = C_2(\mathbf{\Pi}, \mathbf{\Gamma}),
\end{equation}
identically for all step sizes $h \in \mathbb{R}$.
\end{propbox}
\leanrefright{cas1_heavy_top}{HeavyTop.heavy_top_strang_casimirs_exact}{06_HeavyTop_Theorems.lean}

\begin{proofbox}[Proof of Proposition~\ref{prop:heavy_top_casimirs}]
Along the normal fiber kick $\Phi_\tau^{\mathrm{normal}}(\mathbf{\Pi}, \mathbf{\Gamma}) = (\mathbf{\Pi} + \tau mgl(\mathbf{\Gamma} \times \boldsymbol{\chi}), \mathbf{\Gamma})$, the gravity vector $\mathbf{\Gamma}$ is unchanged so $C_1 = \|\mathbf{\Gamma}\|^2$ is trivially preserved, while the scalar triple product identity $(\mathbf{\Gamma} \times \boldsymbol{\chi}) \cdot \mathbf{\Gamma} = 0$ yields $(\mathbf{\Pi} + \tau mgl(\mathbf{\Gamma} \times \boldsymbol{\chi})) \cdot \mathbf{\Gamma} = \mathbf{\Pi} \cdot \mathbf{\Gamma} = C_2$. Along the base groupoid update $\Phi_h^{\mathrm{base}}(\mathbf{\Pi}^*, \mathbf{\Gamma}^*) = (\mathbf{R}_h^T \mathbf{\Pi}^*, \mathbf{R}_h^T \mathbf{\Gamma}^*)$ with $\mathbf{R}_h \in \mathrm{SO}(3)$, orthogonality $\mathbf{R}_h \mathbf{R}_h^T = \mathbf{I}$ preserves the Euclidean inner product on $\mathbb{R}^3$, so $\|\mathbf{R}_h^T \mathbf{\Gamma}^*\|^2 = \|\mathbf{\Gamma}^*\|^2$ and $(\mathbf{R}_h^T \mathbf{\Pi}^*) \cdot (\mathbf{R}_h^T \mathbf{\Gamma}^*) = \mathbf{\Pi}^* \cdot \mathbf{\Gamma}^*$. Applying Theorem~\ref{thm:casimir_preservation} to $\mathcal{S}_2(h) = \Phi_{h/2}^{\mathrm{normal}} \circ \Phi_h^{\mathrm{base}} \circ \Phi_{h/2}^{\mathrm{normal}}$ concludes the proof.
\end{proofbox}

Every stage of the composition preserves both invariants identically, illustrating how the explicit Lie--Poisson splitting methods developed by Channell and Scovel~\cite{channell1991integrators} and McLachlan~\cite{mclachlan1993explicit} provide a Lie--Poisson counterpart to the normal cone base-fiber decoupling on $\mathfrak{se}(3)^*$.

\newpage
To synthesize this geometric architecture into an operational reference, we summarize the resulting structure in the dedicated specification below. This structural synopsis delineates the reduced phase-space geometry, the canonical Hamiltonian polarization into base and normal components, and the unconditionally preserved Lie--Poisson Casimirs, establishing the unified classification format adopted for constrained and singular systems throughout this treatise:

\begin{conormalbox}{Euler--Poisson Heavy Top on $\mathfrak{se}(3)^*$}
    \textbf{Geometric Base \& Normal Fiber:} The physical state describes a heavy rigid body spinning about a fixed pivot under uniform vertical gravity. The system is parameterized by the body angular momentum vector $\mathbf{\Pi} = (\Pi_1, \Pi_2, \Pi_3) \in \mathfrak{so}(3)^* \cong \R^3$ in the body frame, the unit Poisson vector $\mathbf{\Gamma} = (\Gamma_1, \Gamma_2, \Gamma_3) \in \mathbb{S}^2$ tracing the spatial gravity direction, the positive-definite inertia tensor $\mathbb{I} = \operatorname{diag}(I_1, I_2, I_3)$, and the constant center-of-mass displacement vector $\boldsymbol{\chi} \in \R^3$.
    
    The reduced phase space is the dual Lie algebra $\mathfrak{se}(3)^* \cong \mathfrak{so}(3)^* \times \R^3$, whose symplectic leaves are coadjoint orbits equipped with the Kirillov--Kostant--Souriau 2-form:
    \begin{align}
        \mathcal{O}_{c_1, c_2} &= \left\{(\mathbf{\Pi}, \mathbf{\Gamma}) \in \mathfrak{se}(3)^* \;\middle|\; \|\mathbf{\Gamma}\|^2 = c_1 > 0, \; \mathbf{\Pi} \cdot \mathbf{\Gamma} = c_2\right\}, \\
        \omega_{\mathrm{KKS}}(\delta u, \delta v) &= -\langle (\mathbf{\Pi}, \mathbf{\Gamma}), [u, v]\rangle.
    \end{align}
    The kinematic configuration base corresponding to zero rotational momentum is the unit sphere:
    \begin{equation}
        Z = \left\{(\mathbf{\Pi}, \mathbf{\Gamma}) \in \mathfrak{se}(3)^* \;\middle|\; \mathbf{\Pi} = \mathbf{0}, \; \|\mathbf{\Gamma}\|^2 = 1\right\} \cong \mathbb{S}^2, \qquad \mathcal{I}_Z = \langle \Pi_1, \Pi_2, \Pi_3, \|\mathbf{\Gamma}\|^2 - 1 \rangle \subset \mathcal{O}_{\mathfrak{se}(3)^*}.
    \end{equation}

    \textbf{Hamiltonian Polarization:} Polarizing the physical energy into base kinematic rotation and transverse gravitational torque:
    \begin{equation}
        H(\mathbf{\Pi}, \mathbf{\Gamma}) = \underbrace{\frac{1}{2}\mathbf{\Pi} \cdot \mathbb{I}^{-1}\mathbf{\Pi}}_{H_{\mathrm{base}}} + \underbrace{mgl\,(\mathbf{\Gamma} \cdot \boldsymbol{\chi})}_{H_{\mathrm{normal}}}.
    \end{equation}
    \begin{itemize}[leftmargin=1.5em, itemsep=1pt, topsep=1pt]
        \item \emph{Base Kinematic Hamiltonian $H_{\mathrm{base}}$}: Generates free Euler rigid body rotation along the symmetry base $Z \cong \mathbb{S}^2$, integrated via Rodrigues' formula in $\mathrm{SO}(3)$ acting simultaneously on $\mathbf{\Gamma}$ and $\mathbf{\Pi}$ (or via McLachlan's 3-rotation splitting~\cite{mclachlan1993explicit}), preserving both Casimirs $C_1$ and $C_2$ strictly without truncation error.
        \item \emph{Normal Potential Hamiltonian $H_{\mathrm{normal}}$}: Generates transverse gravitational torque $\mathbf{K}(\mathbf{\Gamma}) = mgl(\mathbf{\Gamma} \times \boldsymbol{\chi})$, integrated as an exact abelian translation $\mathbf{\Pi} \mapsto \mathbf{\Pi} + h \mathbf{K}(\mathbf{\Gamma})$ along the transverse momentum fiber $\mathfrak{so}(3)^* \cong N_Z \mathfrak{se}(3)^*$.
    \end{itemize}

    \textbf{Preserved Invariants:} Under the symmetric Strang splitting $\mathcal{S}_2(h) = \Phi_{h/2}^{\mathrm{normal}} \circ \Phi_h^{\mathrm{base}} \circ \Phi_{h/2}^{\mathrm{normal}}$, both quadratic Lie--Poisson Casimirs are conserved identically along trajectories:
    \begin{align}
        \{C_1, H_{\mathrm{base}}\} = \{C_1, H_{\mathrm{normal}}\} \equiv 0 &\implies \frac{\mathrm{d}C_1}{\mathrm{d}t} \equiv 0, \\
        \{C_2, H_{\mathrm{base}}\} = \{C_2, H_{\mathrm{normal}}\} \equiv 0 &\implies \frac{\mathrm{d}C_2}{\mathrm{d}t} \equiv 0,
    \end{align}
    guaranteeing unconditional confinement to the coadjoint orbit $\mathcal{O}_{1, c_2}$.\par\vspace{3pt}
    \leanrefright{cas1_heavy_top}{HeavyTop.heavy_top_strang_casimirs_exact}{06_HeavyTop_Theorems.lean}
\end{conormalbox}

\newpage
\subsubsection{Case Study 2: The Bogomolov--Kirchhoff Point Vortex System on $\mathbb{S}^2$}
A parallel realization occurs for the spherical point vortex system formulated by Bogomolov~\cite{bogomolov1977dynamics} (extending Helmholtz's vortex principles and Kirchhoff's planar Hamiltonian dynamics~\cite{kirchhoff1876}), with Hamiltonian:
\begin{equation}
    H(\mathbf{x}_1, \dots, \mathbf{x}_K) = -\frac{1}{4\pi} \sum_{1 \le i < j \le K} \Gamma_i \Gamma_j \ln \|\mathbf{x}_i - \mathbf{x}_j\|^2 = \sum_{1 \le i < j \le K} H_{ij}, \qquad \mathbf{x}_i \in \mathbb{S}^2,
\end{equation}
where $\Gamma_i \in \R \setminus \{0\}$ denotes the vortex circulation and $\|\mathbf{x}_i - \mathbf{x}_j\| = 2\sin(\theta_{ij}/2)$ is the Euclidean chord distance in $\R^3$, which yields the exact Green's function on $\mathbb{S}^2$ up to an additive constant. Here, the phase space is the compact manifold $(\mathbb{S}^2)^K$ equipped with the symplectic form $\omega = \sum_{i=1}^K \Gamma_i \, \omega_{\mathbb{S}^2}^{(i)}$ (Kähler when all $\Gamma_i > 0$). The singular loci are pairwise collision strata $\Delta_{ij} = \{\mathbf{x} \in (\mathbb{S}^2)^K \mid \mathbf{x}_i = \mathbf{x}_j\}$.

Rather than separating a kinetic cotangent drift and a potential kick, the normal cone philosophy of isolating singular divisors adapts to first-order vortex dynamics via the Lie--Poisson splitting framework of McLachlan~\cite{mclachlan1993explicit}, instantiated for spherical point vortex pairs by Myerscough and Frank~\cite{myerscough2016explicit}:
\begin{enumerate}[leftmargin=2em, itemsep=2pt, topsep=2pt]
    \item \textbf{Pairwise Lie--Poisson Foliation:} The Hamiltonian is decomposed into $K(K-1)/2$ autonomous pairwise stages $H = \sum_{i<j} H_{ij}$. Under each isolated Hamiltonian $H_{ij}$, the mutual chord distance $\|\mathbf{x}_i - \mathbf{x}_j\|$ is an exact invariant, and the vortex pair $(i, j)$ undergoes an exact, non-singular rigid rotation around their pairwise momentum vector $\mathbf{J}_{ij} = \Gamma_i \mathbf{x}_i + \Gamma_j \mathbf{x}_j \in \mathfrak{so}(3)^*$ with constant angular velocity $\boldsymbol{\omega}_{ij} = \frac{\mathbf{J}_{ij}}{2\pi \|\mathbf{x}_i - \mathbf{x}_j\|^2}$, computed analytically via Rodrigues' formula, while spectator vortices $k \notin \{i, j\}$ remain stationary.
    \item \textbf{Symmetric Strang Composition:} The complete flow is integrated via the symmetric composition $\mathcal{S}_2(h) = \prod_{i<j} \Phi_{h/2}^{H_{ij}} \circ \prod_{i<j}^{\mathrm{rev}} \Phi_{h/2}^{H_{ij}}$.
\end{enumerate}
Because each pairwise flow $\Phi_h^{H_{ij}}$ is an exact $\mathrm{SO}(3)$ isometry preserving sphere radii and commuting with the diagonal $\mathrm{SO}(3)$ action, this integrator unconditionally preserves the individual Casimirs $\|\mathbf{x}_i\|^2 = 1$ on $(\mathfrak{so}(3)^*)^{\oplus K}$ and the global $\mathrm{SO}(3)$ momentum map $\mathbf{J} = \sum_{i=1}^K \Gamma_i \mathbf{x}_i$, hence its quadratic Casimir $\|\mathbf{J}\|^2$, analytically without projection.

\newpage
\enlargethispage{2.5\baselineskip}
To synthesize this pairwise geometric architecture into an operational reference, we summarize the resulting structure in the dedicated specification below:

\begin{conormalbox}{Bogomolov--Kirchhoff Point Vortex System on $\mathbb{S}^2$}
    \textbf{Compact Kähler Phase Space \& Symplectic Collision Diagonals:} The physical system describes $K$ interacting ideal point vortices on the unit two-sphere $\mathbb{S}^2 \subset \R^3$. The state is specified by the vortex positions $\mathbf{x}_i = (x_{i,1}, x_{i,2}, x_{i,3}) \in \mathbb{S}^2$ and their non-vanishing vortex circulations $\Gamma_i \in \R \setminus \{0\}$ for $i \in \{1, \dots, K\}$.
    
    Unlike mechanical cotangent bundles $T^*Q$, the geometric phase space is a compact boundaryless Kähler product manifold equipped with the Kirillov--Kostant--Souriau symplectic form:
    \begin{equation}
        \mathcal{M} = (\mathbb{S}^2)^K, \qquad \omega = \sum_{i=1}^K \Gamma_i \, \omega_{\mathbb{S}^2}^{(i)}, \qquad \omega_{\mathbb{S}^2}^{(i)}(\delta \mathbf{u}, \delta \mathbf{v}) = \mathbf{x}_i \cdot (\delta \mathbf{u} \times \delta \mathbf{v}).
    \end{equation}
    The singular locus is the union of pairwise collision diagonals (which are codimension-2 symplectic submanifolds whenever $\Gamma_i + \Gamma_j \neq 0$, rather than Lagrangian conormal submanifolds):
    \begin{equation}
        \Delta = \bigcup_{1 \le i < j \le K} \Delta_{ij}, \qquad \Delta_{ij} = \left\{(\mathbf{x}_1, \dots, \mathbf{x}_K) \in (\mathbb{S}^2)^K \;\middle|\; \mathbf{x}_i = \mathbf{x}_j\right\}, \qquad \mathcal{I}_{\Delta_{ij}} = \langle \mathbf{x}_i - \mathbf{x}_j \rangle.
    \end{equation}
    \vspace{1pt}

    \textbf{Pairwise Lie--Poisson Chordwise Splitting:} Because $(\mathbb{S}^2)^K$ carries no cotangent base-fiber fibration ($H_{\mathrm{kin}} \equiv 0$), the singular divisor isolation principle transposes analogically into a pairwise Lie--Poisson splitting across the $K(K-1)/2$ chordal logarithmic interactions:
    \begin{equation}
        H(\mathbf{x}_1, \dots, \mathbf{x}_K) = \sum_{1 \le i < j \le K} H_{ij}(\mathbf{x}_i, \mathbf{x}_j), \qquad H_{ij}(\mathbf{x}_i, \mathbf{x}_j) = -\frac{\Gamma_i \Gamma_j}{4\pi} \ln \|\mathbf{x}_i - \mathbf{x}_j\|^2.
    \end{equation}
    \begin{itemize}[leftmargin=1.5em, itemsep=1.5pt, topsep=1.5pt]
        \item \emph{Absence of Inertial Base Drift}: Reflecting the first-order, non-inertial nature of point-vortex dynamics on compact Kähler manifolds, coordinates and conjugate momenta are intrinsically coupled on each $\mathbb{S}^2$ factor without free kinetic translation.
        \item \emph{Exact Pairwise $\mathrm{SO}(3)$ Isometry Stages $H_{ij}$}: Each isolated pair Hamiltonian $H_{ij}$ generates an exact rigid $\mathrm{SO}(3)$ rotation around the pair angular momentum vector $\mathbf{J}_{ij} = \Gamma_i \mathbf{x}_i + \Gamma_j \mathbf{x}_j \in \mathfrak{so}(3)^*$ with constant angular velocity $\boldsymbol{\omega}_{ij} = \frac{\mathbf{J}_{ij}}{2\pi \|\mathbf{x}_i - \mathbf{x}_j\|^2}$ via Rodrigues' formula, leaving the singular chordal distance $\|\mathbf{x}_i - \mathbf{x}_j\|$ strictly invariant along each stage.
    \end{itemize}
    \vspace{1pt}

    \textbf{Preserved Invariants:} Under the symmetric Strang pairwise composition $\mathcal{S}_2(h) = \prod_{i<j} \Phi_{h/2}^{H_{ij}} \circ \prod_{i<j}^{\mathrm{rev}} \Phi_{h/2}^{H_{ij}}$, all sphere Casimirs and the total Lie--Poisson angular momentum are preserved identically:
    \begin{equation}
        \frac{\mathrm{d}}{\mathrm{d}t}\|\mathbf{x}_i\|^2 = 2 \mathbf{x}_i \cdot \dot{\mathbf{x}}_i \equiv 0 \quad (\forall i \in \{1, \dots, K\}), \qquad \{\mathbf{J}, H\} \equiv \mathbf{0}, \qquad \{\|\mathbf{J}\|^2, H\} \equiv 0,
    \end{equation}
    where $\mathbf{J} = \sum_{i=1}^K \Gamma_i \mathbf{x}_i \in \mathfrak{so}(3)^*$ is the total vorticity momentum map, ensuring unconditional confinement to the coadjoint $\mathrm{SO}(3)$ orbit on $(\mathfrak{so}(3)^*)^{\oplus K}$.
\end{conormalbox}

\subsubsection{Conceptual Bridge to the Gravitational $N$-Body Problem}
These retrospective case studies illustrate the core thesis: whenever classical mechanics has successfully designed stable, long-term geometric integrators for constrained or singular systems, those integrators implicitly leveraged the abelian structure of the normal cone. 

In the gravitational $N$-body problem, however, singular configurations do not form isolated smooth submanifolds. Instead, pairwise collision diagonals $\Delta_{ij}$ and multiple collision strata $\Delta_S$ intersect along a non-transverse conical stratification. While an isolated groupoid/normal-cone splitting suffices when singular loci are disjoint (as in $\mathrm{SE}(3)$ or $\mathbb{S}^2$), handling intersecting collision cones requires a systematic desingularization of the arrangement. To extend this methodology systematically to gravitation, we proceed in two stages:
\begin{itemize}[leftmargin=1.5em, itemsep=2pt]
    \item \textbf{Low-dimensional prototype (Section~\ref{ch:ch03_three_body_paradigm}):} We first examine the planar circular restricted three-body problem (PCR3BP), where the singular locus $Z = \{\mathbf{q}_{(1)}, \mathbf{q}_{(2)}\}$ consists of two isolated primary points, allowing an unsoftened base-fiber treatment of binary close encounters.
    \item \textbf{Multiscale hierarchical desingularization (Section~\ref{ch:ch04_fulton_macpherson_splitting}):} We then lift the full $N$-body collision arrangement to the Fulton--MacPherson configuration space $\mathbf{Q}[N]$. By iteratively blowing up the collision diagonals along the Whitney lattice of collision partitions, intersecting collision cones are desingularized into a simple normal crossing (SNC) divisor with corners, enabling synchronous, multiscale normal cone desingularization across all cluster hierarchies.
\end{itemize}