\newpage
\subsection{The Canonical 4-Step Normal Cone Splitting Recipe}
\label{sec:algorithmic_recipe}

Translating the derived geometry of Calaque and Safronov into an operational numerical algorithm requires a systematic four-step procedure. Rather than applying an off-the-shelf black-box integrator or heuristic regularizations, the normal cone methodology dictates where and how to partition the physical phase space into complementary sub-geometries: a stiff singular base manifold $Z$ and a flat, abelian transverse momentum fiber $N_{Z/M}^*$.

\subsubsection{The Four Invariant Steps of Normal Cone Splitting}
Given an autonomous Hamiltonian system $\dot{\mathbf{z}} = \mathbb{J} \nabla H(\mathbf{z})$ on a symplectic manifold $(T^*M, \omega_0)$, the normal cone geometric discretization proceeds systematically as follows:

\begingroup
\renewcommand{\descriptionlabel}[1]{\hspace\labelsep\upshape\bfseries\rlap{#1:}}%
\begin{description}[leftmargin=2em, itemsep=4pt, topsep=3pt]
    \item[\textbf{Step~1: Canonical Detection and Partition $(Z, N_{Z/M})$}] \leavevmode\par\nopagebreak
    Identify the geometric submanifold $Z \subset M$ governing the stiffness, constraints, or singular locus of the physical interaction (e.g., an algebraic collision divisor $\Delta_{ij}$, a rigid spherical constraint $\mathbb{S}^2$, or an attitude Lie group $\mathrm{SO}(3)$). On the full cotangent bundle $T^*M$, canonical Darboux coordinates partition the state into base variables $(\mathbf{x}_Z, \mathbf{p}_Z) \in T^*Z$ and transverse normal and cotangent variables $(\mathbf{n}, \mathbf{w}) \in N_{Z/M} \times_Z N_{Z/M}^*$. On reduced Lie--Poisson spaces $\mathfrak{g}^*$ (such as rigid body dynamics), this reduces to base orientation $\mathbf{x}_Z \in Z$ and conjugate transverse momentum $\mathbf{p}_N \in N_{Z/M}^*$:
    \begin{equation}
        \mathbf{z} = (\mathbf{x}_Z, \mathbf{p}_N) \in Z \times N_{Z/M}^*.
    \end{equation}

    \item[\textbf{Step~2: Exact Linear Integration in the Transverse Momentum Fiber ($t = 0$)}] \leavevmode\par\nopagebreak
    In the degenerated central fiber $t = 0$, the transverse dynamics is governed by an abelian vector field. The transverse normal kick $\mathbf{K}(\mathbf{x}_Z) = -\nabla_N V(\mathbf{x}_Z)$ induces an exact linear momentum translation along the flat transverse momentum fibers without matrix inversion or non-linear iteration:
    \begin{equation}
        \Phi_{h/2}^{NL} : \quad \mathbf{p}_N \longmapsto \mathbf{p}_N + \frac{h}{2} \mathbf{K}(\mathbf{x}_Z), \qquad \mathbf{x}_Z \longmapsto \mathbf{x}_Z.
    \end{equation}
    Because $\mathbf{K}(\mathbf{x}_Z)$ is strictly transverse and derives from the base geometry, it satisfies the orthogonality condition $\langle \nabla C, \mathbf{K} \rangle = 0$. When $C$ is affine with respect to fiber momenta, this guarantees unconditional preservation for all finite step sizes $h \in \mathbb{R}$ without numerical projection.

    \item[\textbf{Step~3: Symplectic Groupoid Exponential on the Base ($t = 1$)}] \leavevmode\par\nopagebreak
    On the base manifold $Z$, the motion is non-linear and governed by the Lie algebroid $\operatorname{Lie}(\mathcal{G})$ of configuration symmetries (or the Lie algebra $\mathfrak{g}$ of a symmetry group acting transitively on $Z$). Rather than applying an unconstrained additive Euler-type update that would depart from the manifold, the flow is integrated via the exact exponential map of the symplectic groupoid $\exp_{\mathcal{G}} : \mathfrak{g} \to \mathcal{G}$, accompanied by the conjugate coadjoint action on momenta:
    \begin{equation}
        \Phi_h^{\mathcal{G}} : \quad \mathbf{x}_Z \longmapsto \exp_{\mathcal{G}}\bigl(h \mathbf{v}(\mathbf{p}_N)\bigr) \cdot \mathbf{x}_Z, \qquad \mathbf{p}_N \longmapsto \mathrm{Ad}^*_{\exp_{\mathcal{G}}(h \mathbf{v})}(\mathbf{p}_N).
    \end{equation}
    For Lie group actions, this is computed via closed-form analytical expressions (e.g., Rodrigues' rotation formula on $\mathrm{SO}(3)$ or great-circle geodesics on $\mathbb{S}^2$), preserving algebraic manifold invariants with exact geometric closure (bounded in floating-point realization solely by IEEE~754 roundoff). Crucially, by operating globally on the Lie group action rather than through local chart parameterizations, this groupoid update is intrinsically immune to coordinate singularities, entirely circumventing artificial coordinate breakdowns such as gimbal lock in Euler angle formulations or polar coordinate singularities at coordinate axes.

    \item[\textbf{Step~4: Symmetric Canonical Splitting and High-Order Compositions}] \leavevmode\par\nopagebreak
    Compose the transverse normal kick and the groupoid exponential into a time-reversible, second-order Strang splitting:
    \begin{equation}
        \mathcal{S}_{2}(h) = \Phi_{h/2}^{NL} \circ \Phi_h^{\mathcal{G}} \circ \Phi_{h/2}^{NL}.
    \end{equation}
    Higher-order symmetric compositions (such as the Yoshida 4th-order scheme~\cite{yoshida1990construction}, Suzuki's fractal composition~\cite{suzuki1990fractal}, or the optimized partitioned Runge--Kutta schemes of Blanes and Moan~\cite{blanes2002practical}) are constructed by recursive substeps $\prod_{j} \mathcal{S}_2(w_j h)$, preserving exact symplecticity and time reversibility (see Figure~\ref{fig:ch02_canonical_4step_strang_recipe}).
\end{description}
\endgroup

\begin{figure}[htbp]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[scale=1.0, every node/.style={font=\small}]
    % --- Background Casimir Level Manifold C(z) = c_0 ---
    \draw[fill=ThmBg, draw=NavyBlue!65, thick, rounded corners=5pt] (-1.2, 0.30) rectangle (17.4, 6.45);
    \node[NavyBlue, font=\footnotesize\bfseries] at (8.1, 6.08) {Invariant Casimir Foliation Leaf $\mathcal{O}_{c_0} = \{\mathbf{z} = (\mathbf{x}_Z, \mathbf{p}_N) \in Z \times N_{Z/M}^* \mid C(\mathbf{z}) = c_0\}$};
    \node[DarkSlate, font=\scriptsize\bfseries] at (8.1, 5.68) {\textbf{Step 1:} Canonical Phase-Space Partition $\mathbf{z} = (\mathbf{x}_Z, \mathbf{p}_N) \in Z \times N_{Z/M}^*$};

    % --- Base Manifold Z at Bottom ---
    \draw[very thick, DarkSlate] (1.6, 1.25) .. controls (5.8, 1.50) and (10.4, 1.50) .. (14.6, 1.25);
    \node[DarkSlate, font=\footnotesize\bfseries] at (8.1, 0.72) {Base Manifold $Z$ (Symmetry Groupoid $\mathcal{G} \rightrightarrows Z$, $t = 1$)};

    % --- Vertical Transverse Fibers N_{Z/M}^* at x_Z^{(n)} and x_Z^{(n+1)} ---
    \draw[dashed, NavyBlue!70, thick] (3.4, 1.32) -- (3.4, 5.25);
    \fill[DarkSlate] (3.4, 1.32) circle (2.0pt);
    \node[NavyBlue, font=\footnotesize\bfseries, anchor=east] at (3.22, 1.18) {$\mathbf{x}_Z^{(n)}$};
    \node[NavyBlue, font=\scriptsize, align=right, anchor=east] at (3.20, 4.98) {Abelian Fiber\\[-1pt]$N_{Z/M}^*|_{\mathbf{x}_Z^{(n)}}$};

    \draw[dashed, NavyBlue!70, thick] (12.8, 1.32) -- (12.8, 5.25);
    \fill[DarkSlate] (12.8, 1.32) circle (2.0pt);
    \node[NavyBlue, font=\footnotesize\bfseries, anchor=west] at (12.98, 1.18) {$\mathbf{x}_Z^{(n+1)}$};
    \node[NavyBlue, font=\scriptsize, align=left, anchor=west] at (13.00, 4.98) {Abelian Fiber\\[-1pt]$N_{Z/M}^*|_{\mathbf{x}_Z^{(n+1)}}$};

    % --- 4 States of the Symmetric Strang Bridge ---
    \node[circle, fill=DarkSlate, inner sep=2.2pt, label={[left=4pt, font=\footnotesize]$\mathbf{z}_n = (\mathbf{x}_Z^{(n)}, \mathbf{p}_N^{(n)})$}] (z0) at (3.4, 2.30) {};
    \node[circle, fill=PythonGreen, inner sep=2.2pt, label={[left=4pt, font=\footnotesize]$\mathbf{z}_1 = (\mathbf{x}_Z^{(n)}, \mathbf{p}_N^*)$}] (z1) at (3.4, 4.25) {};
    \node[circle, fill=AcademicPurple, inner sep=2.2pt, label={[right=4pt, font=\footnotesize]$\mathbf{z}_2 = (\mathbf{x}_Z^{(n+1)}, \mathbf{p}_N^{**})$}] (z2) at (12.8, 4.25) {};
    \node[circle, fill=NavyBlue, inner sep=2.2pt, label={[right=4pt, font=\footnotesize]$\mathbf{z}_{n+1} = \mathcal{S}_2(h)\mathbf{z}_n$}] (z3) at (12.8, 2.30) {};

    % --- (I) First Half-Kick \Phi_{h/2}^{NL} (Step 2, t = 0) ---
    \draw[->, >=latex, very thick, PythonGreen] (z0) -- (z1);
    \node[PythonGreen, font=\scriptsize, align=right, anchor=east] at (3.20, 3.28) {\textbf{(I) First Half-Kick}\\[-1pt][\textbf{Step 2}, $t=0$]: $\Phi_{h/2}^{NL}$\\[-1pt]$\mathbf{p}_N^* = \mathbf{p}_N^{(n)} + \frac{h}{2}\mathbf{K}(\mathbf{x}_Z^{(n)})$\\[-1pt]$\langle \nabla C, \mathbf{K}\rangle = 0$};

    % --- (II) Base Groupoid Exponential \Phi_h^{\mathcal{G}} (Step 3, t = 1) ---
    \draw[->, >=latex, very thick, AcademicPurple] (z1) .. controls (6.4, 4.65) and (9.8, 4.65) .. (z2);
    \node[AcademicPurple, font=\scriptsize, align=center] at (8.1, 5.05) {\textbf{(II) Base Groupoid Exponential $\Phi_h^{\mathcal{G}}$} [\textbf{Step 3}, $t=1$]\\[-1pt]$\mathbf{x}_Z^{(n+1)} = \exp_{\mathcal{G}}\bigl(h\mathbf{v}(\mathbf{p}_N^*)\bigr)\cdot\mathbf{x}_Z^{(n)}, \quad \mathbf{p}_N^{**} = \mathrm{Ad}^*_{\exp_{\mathcal{G}}(h\mathbf{v})}(\mathbf{p}_N^*)$};

    % --- (III) Second Half-Kick \Phi_{h/2}^{NL} (Step 2, t = 0) ---
    \draw[->, >=latex, very thick, PythonGreen] (z2) -- (z3);
    \node[PythonGreen, font=\scriptsize, align=left, anchor=west] at (13.00, 3.28) {\textbf{(III) Second Half-Kick}\\[-1pt][\textbf{Step 2}, $t=0$]: $\Phi_{h/2}^{NL}$\\[-1pt]$\mathbf{p}_N^{(n+1)} = \mathbf{p}_N^{**} + \frac{h}{2}\mathbf{K}(\mathbf{x}_Z^{(n+1)})$\\[-1pt]$\langle \nabla C, \mathbf{K}\rangle = 0$};

    % --- Step 4: Direct Palindromic Strang Map S_2(h) ---
    \draw[->, >=latex, thick, dashed, NavyBlue] (z0) -- (z3);
    \node[NavyBlue, font=\scriptsize, align=center, above=2pt] at (8.1, 2.30) {\textbf{Step 4:} Palindromic Composition $\mathcal{S}_2(h) = \Phi_{h/2}^{NL} \circ \Phi_h^{\mathcal{G}} \circ \Phi_{h/2}^{NL}$};
    \node[NavyBlue, font=\scriptsize, align=center, below=2pt] at (8.1, 2.30) {$C(\mathbf{z}_{n+1}) = C(\mathbf{z}_n) = c_0$ \quad \& \quad $\mathcal{S}_2(-h)\circ\mathcal{S}_2(h) = \mathrm{id}$};
\end{tikzpicture}%
}
\caption{Geometric architecture of the canonical 4-step normal cone Strang splitting $\mathcal{S}_2(h)$ on $Z \times N_{Z/M}^*$. Following the canonical partition (\textbf{Step 1}), the first and second transverse half-kicks (\textbf{I} and \textbf{III}, \textbf{Step 2} at $t=0$) act along the vertical fibers $N_{Z/M}^*$ orthogonally to $\nabla C$, while the central base groupoid exponential (\textbf{II}, \textbf{Step 3} at $t=1$) transports the state along the symmetry orbit without local chart singularities. Their palindromic composition (\textbf{Step 4}) preserves all Casimirs $C(\mathbf{z}) = c_0$ unconditionally (Theorem~\ref{thm:casimir_preservation}) and ensures exact time-reversibility (Proposition~\ref{prop:strang_time_reversibility}).}
\label{fig:ch02_canonical_4step_strang_recipe}
\end{figure}

\begin{thmbox}[Exact Invariant and Casimir Preservation]{thm:casimir_preservation}
Following McLachlan~\cite[Section~2]{mclachlan1993explicit} and Hairer, Lubich, and Wanner~\cite[Section~VII.4.2]{hairer2006geometric}, let $\mathcal{S}_2(h)$ be constructed according to Steps~1--4. If both the transverse kick $\Phi^{NL}$ and the groupoid flow $\Phi^{\mathcal{G}}$ preserve a scalar functional $C : Z \times N_{Z/M}^* \to \mathbb{R}$, then the symmetric Strang composition unconditionally preserves $C$ for all step sizes $h \in \mathbb{R}$:
\begin{equation}
    C\bigl(\mathcal{S}_2(h) \mathbf{z}\bigr) = C(\mathbf{z}).
\end{equation}
In particular, all Casimirs of the Poisson bracket belonging to the kernel of the groupoid Lie algebroid are preserved to floating-point machine precision without numerical drift.
\end{thmbox}
\leanrefright{strang_casimir_preservation}{ConormalSplitting.strang_exact_casimir_preservation}{04_CanonicalSplitting_Theorems.lean}

\begin{proofbox}[Proof of Theorem~\ref{thm:casimir_preservation}]
By hypothesis, $C(\Phi_\tau^{NL}(\mathbf{z})) = C(\mathbf{z})$ and $C(\Phi_\tau^{\mathcal{G}}(\mathbf{z})) = C(\mathbf{z})$ hold for all $\tau \in \mathbb{R}$ and all $\mathbf{z} \in Z \times N_{Z/M}^*$. Evaluating $C$ along the three sequential sub-steps $\mathbf{z}_1 = \Phi_{h/2}^{NL}(\mathbf{z})$, $\mathbf{z}_2 = \Phi_h^{\mathcal{G}}(\mathbf{z}_1)$, and $\mathcal{S}_2(h)\mathbf{z} = \Phi_{h/2}^{NL}(\mathbf{z}_2)$ yields by composition
\begin{equation}
    C\bigl(\mathcal{S}_2(h)\mathbf{z}\bigr) = C\bigl(\Phi_{h/2}^{NL}(\mathbf{z}_2)\bigr) = C(\mathbf{z}_2) = C\bigl(\Phi_h^{\mathcal{G}}(\mathbf{z}_1)\bigr) = C(\mathbf{z}_1) = C\bigl(\Phi_{h/2}^{NL}(\mathbf{z})\bigr) = C(\mathbf{z}),
\end{equation}
and similarly by induction for any multi-stage symmetric composition $\prod_j \mathcal{S}_2(w_j h)$.
\end{proofbox}

\begin{propbox}[Time-Symmetry of the Normal Cone Splitting]{prop:strang_time_reversibility}
In accordance with Strang~\cite{strang1968construction} and Hairer et al.~\cite[Section~V.1]{hairer2006geometric}, let $\Phi^{NL}$ and $\Phi^{\mathcal{G}}$ satisfy the one-parameter flow group property $\Phi(h_1 + h_2) = \Phi(h_1) \circ \Phi(h_2)$ and $\Phi(0) = \mathrm{id}$. Then the symmetric Strang composition $\mathcal{S}_2(h)$ is an exact time-symmetric integrator:
\begin{equation}
    \mathcal{S}_2(-h) \circ \mathcal{S}_2(h) = \mathrm{id}.
\end{equation}
In particular, its formal Lie generator $Z(h) = \ln \mathcal{S}_2(h)$ is strictly odd in $h$ ($Z(-h) = -Z(h)$), eliminating all even powers in its Baker--Campbell--Hausdorff expansion and all odd dissipative error orders in the shadow Hamiltonian $\widetilde{H}(h)$.
\end{propbox}
\leanrefright{strang_time_reversibility}{ConormalSplitting.strang_time_reversibility}{04_CanonicalSplitting_Theorems.lean}

\begin{proofbox}[Proof of Proposition~\ref{prop:strang_time_reversibility}]
By direct evaluation under inversion of the numerical step size $h \mapsto -h$:
\begin{equation}
    \mathcal{S}_2(-h) = \Phi_{-h/2}^{NL} \circ \Phi_{-h}^{\mathcal{G}} \circ \Phi_{-h/2}^{NL} = \left(\Phi_{h/2}^{NL}\right)^{-1} \circ \left(\Phi_h^{\mathcal{G}}\right)^{-1} \circ \left(\Phi_{h/2}^{NL}\right)^{-1} = \mathcal{S}_2(h)^{-1}.
\end{equation}
Consequently, the formal Lie generator $Z(h) = \ln \mathcal{S}_2(h)$ satisfies $Z(-h) = \ln \mathcal{S}_2(-h) = \ln(\mathcal{S}_2(h)^{-1}) = -Z(h)$. In the Baker--Campbell--Hausdorff expansion:
\begin{equation}
    Z(h) = h X_H + h^3 Z_3 + h^5 Z_5 + \mathcal{O}(h^7),
\end{equation}
all even powers of $h$ vanish identically. Equivalently, the formal shadow Hamiltonian $\widetilde{H}(h) = Z(h)/h$ satisfies time reversibility $\widetilde{H}(-h) = \widetilde{H}(h)$, expanding exclusively in even powers:
\begin{equation}
    \widetilde{H}(h) = H + h^2 H_2 + h^4 H_4 + \mathcal{O}(h^6).
\end{equation}
This parity cancellation ensures that the local truncation error is $\mathcal{O}(h^3)$, yielding an exact second-order geometric integrator with global error $\mathcal{O}(h^2)$ free from odd-order secular drift.
\end{proofbox}