% --- Section Content : The Partition Lattice $\Pi(N)$ and the Fulton--MacPherson Manifold $\mathbf{Q}[N]$ ---
\subsection{The Partition Lattice \texorpdfstring{$\Pi(N)$}{Pi(N)} and the Fulton--MacPherson Manifold \texorpdfstring{$\mathbf{Q}[N]$}{Q[N]}}
\label{sec:whitney_lattice}

As established in Section~\ref{ch:ch03_three_body_paradigm}, classical regularizations of celestial mechanics encounter an insurmountable topological barrier as soon as $N \ge 3$: while isolated binary collisions admit conformal uniformization via Levi-Civita's branched double cover, multiple collision loci intersect along non-transverse submanifolds where local conformal charts fail to commute. For $N \ge 4$, this geometric pathology intensifies into an intricate combinatorial arrangement of intersecting singular divisors. 

To overcome this bottleneck without resorting to unphysical gravitational softening ($\varepsilon = 0$), we construct a multiscale desingularization of the Newtonian configuration space grounded in the Fulton--MacPherson compactification~\cite{fulton1994compactification} and its real differential counterpart developed by Axelrod and Singer~\cite{axelrod1994chern} and refined by Campos et al.~\cite{campos2024configuration}. This section establishes the stratification of the collision arrangement, the real oriented blow-up sequence producing the smooth manifold with corners $\mathbf{Q}[N]$, and the adapted cotangent charts associated with maximal Jacobi binary trees.

% ------------------------------------------------------------------------------
\subsubsection{The Collision Arrangement and Whitney Stratification along $\Pi(N)$}
\label{sub:collision_whitney_stratification}

Consider $N \ge 3$ point masses $m_1, \dots, m_N > 0$ with positions $\mathbf{q}_i \in \mathbb{R}^3$. Setting the center of mass to the origin, the relative configuration space is the linear quotient
\begin{equation}
    M = (\mathbb{R}^3)^N / \mathbb{R}^3 \cong \left\{ \mathbf{q} = (\mathbf{q}_1, \dots, \mathbf{q}_N) \in (\mathbb{R}^3)^N \;\middle|\; \sum_{i=1}^N m_i \mathbf{q}_i = \mathbf{0} \right\} \cong \mathbb{R}^{3N-3}.
    \label{eq:relative_config_space}
\end{equation}
The space $M$ is naturally endowed with the mass metric $\langle \mathbf{q}, \mathbf{q}' \rangle_{\mathbb{M}} = \sum_{i=1}^N m_i \langle \mathbf{q}_i, \mathbf{q}_i' \rangle$, represented by the positive-definite diagonal block matrix $\mathbb{M} = \operatorname{diag}(m_1 I_3, \dots, m_N I_3)$. The dual momentum space is the zero total linear momentum subspace $M^* \cong \{ \mathbf{p} = (\mathbf{p}_1, \dots, \mathbf{p}_N) \in (\mathbb{R}^3)^N \mid \sum_{i=1}^N \mathbf{p}_i = \mathbf{0} \}$. By Marsden--Weinstein symplectic reduction of the ambient space $T^*((\mathbb{R}^3)^N) \cong \mathbb{R}^{6N}$ with respect to the spatial translation group $(\mathbb{R}^3, +)$ at the regular value $\mathbf{J}(\mathbf{q}, \mathbf{p}) = \sum_{i=1}^N \mathbf{p}_i = \mathbf{0}$~\cite[Chapter~23]{cannas2001lectures}, the ambient Darboux $2$-form $\sum_{i=1}^N \mathrm{d}\mathbf{q}_i \wedge \mathrm{d}\mathbf{p}_i$ descends without degeneracy to the canonical symplectic $2$-form $\Omega_0$ on the physical cotangent bundle $T^*M \cong M \times M^* \cong \mathbb{R}^{6N-6}$. The total Newtonian potential is singular along the collision arrangement $\Delta \subset M$, defined as the union of all pairwise collision submanifolds:
\begin{equation}
    \Delta = \bigcup_{1 \le i < j \le N} \Delta_{ij}, \qquad \text{where} \quad \Delta_{ij} = \{ \mathbf{q} \in M \mid \mathbf{q}_i = \mathbf{q}_j \}.
    \label{eq:collision_arrangement_def}
\end{equation}
Each pairwise diagonal $\Delta_{ij}$ is a linear subspace of codimension $3$ in $M$. For an arbitrary coalition of particles $I \subseteq \{1, \dots, N\}$ with cardinality $|I| = k \ge 2$, the multiple collision stratum is the linear subspace where all particles in $I$ coincide spatially:
\begin{equation}
    \Delta_I = \{ \mathbf{q} \in M \mid \mathbf{q}_i = \mathbf{q}_j \quad \forall i, j \in I \}, \qquad \operatorname{codim}_M(\Delta_I) = 3(k-1).
    \label{eq:multiple_collision_stratum}
\end{equation}
\leanrefright{partition_lattice_defs}{collisionCodimension}{01_PartitionLattice_Defs.lean}

The intersection theory of the family $\{\Delta_I\}$ is governed by the transitivity of Euclidean coincidence: for any three distinct indices $i, j, k$,
\begin{equation}
    \Delta_{ij} \cap \Delta_{jk} = \{ \mathbf{q} \in M \mid \mathbf{q}_i = \mathbf{q}_j = \mathbf{q}_k \} = \Delta_{\{i,j,k\}} \subset \Delta_{ik}.
    \label{eq:non_transversality_identity}
\end{equation}
\leanrefright{partition_lattice_defs}{collision_diagonal_non_transversality}{01_PartitionLattice_Defs.lean}
Consequently, the pairwise collision submanifolds $\Delta_{ij}$ do not intersect in general position: while any two submanifolds $\Delta_{ij}$ and $\Delta_{jk}$ meet transversally in codimension $3 + 3 = 6$, Euclidean transitivity forces the third submanifold $\Delta_{ik}$ to pass through the exact same locus ($\Delta_{ij} \cap \Delta_{jk} \subset \Delta_{ik}$). Thus, three codimension-$3$ strata meet along a locus of codimension $6$ rather than $9$, forming an intrinsically non-normal crossing arrangement.

The combinatorial structure organizing these intersections is the \emph{partition lattice} $\Pi(N)$, defined as the set of all partitions $\mathcal{P} = \{B_1, \dots, B_k\}$ of $\{1, \dots, N\}$, partially ordered by reverse coarsening (refinement):
\begin{equation}
    \mathcal{P} \le \mathcal{P}' \iff \forall B \in \mathcal{P}, \; \exists B' \in \mathcal{P}' \text{ such that } B \subseteq B'.
    \label{eq:partition_lattice_order}
\end{equation}
The minimal element $\hat{0} = \{\{1\}, \{2\}, \dots, \{N\}\}$ corresponds to the collisionless open stratum $M \setminus \Delta$, while the maximal element $\hat{1} = \{\{1, \dots, N\}\}$ corresponds to the total collision point $\Delta_{\mathrm{tot}} = \{\mathbf{0}\}$. To each partition $\mathcal{P} \in \Pi(N)$ corresponds a closed linear stratum:
\begin{equation}
    \Delta_{\mathcal{P}} = \left\{ \mathbf{q} \in M \;\middle|\; \mathbf{q}_i = \mathbf{q}_j \quad \forall i, j \text{ in the same block of } \mathcal{P} \right\} = \bigcap_{B \in \mathcal{P}} \Delta_B,
    \label{eq:stratum_partition_map}
\end{equation}
with unified codimension $\operatorname{codim}_M(\Delta_{\mathcal{P}}) = 3(N - |\mathcal{P}|)$. The collection of open strata $S_{\mathcal{P}} = \Delta_{\mathcal{P}} \setminus \bigcup_{\mathcal{P}' > \mathcal{P}} \Delta_{\mathcal{P}'}$ forms a Whitney $(A)$- and $(B)$-regular stratification of $M$~\cite{whitney1965tangents}. 

\begin{rembox}[Measure-Theoretic Scope of Singularities]
By the seminal theorems of Saari~\cite{saari1971improbability,saari1973improbability}, the set of initial conditions in $T^*M$ leading to multiple or total collisions ($k \ge 3$) has Lebesgue measure zero. However, in numerical simulations of dense stellar clusters or hierarchical planetary systems, trajectories routinely enter narrow tubular neighborhoods of $\Delta_I$ (quasi-collisions and close encounters), generating arbitrarily stiff tidal forces. The purpose of compactification is not merely to regularize exact collision orbits, but to provide uniform, scale-invariant coordinate charts throughout these stiff asymptotic corridors.
\end{rembox}

% ------------------------------------------------------------------------------
\subsubsection{Real Oriented Blow-ups and the Compactified Manifold with Corners $\mathbf{Q}[N]$}
\label{sub:real_blowup_fulton_macpherson}

In algebraic geometry, Fulton and MacPherson~\cite{fulton1994compactification} resolved diagonal singularities by introducing the compactified configuration space $\mathbf{X}[N]$, obtained by embedding $X^N \setminus \Delta$ into $X^N \times \prod_{|I| \ge 2} \mathbb{P}(N_{\Delta_I} X^N)$ and taking the schematic closure. For differential dynamics, Axelrod and Singer~\cite{axelrod1994chern} adapted this construction to real manifolds by substituting complex projective blow-ups with \emph{real oriented spherical blow-ups}, producing a smooth manifold with corners $\mathbf{Q}[N]$ that fully desingularizes all collision singularities (compactifying internal directional approach while preserving spatial infinity for non-compact configuration spaces such as $M \cong \mathbb{R}^{3N-3}$).

Let $\operatorname{Bl}_Z(M)$ denote the real oriented blow-up of a manifold $M$ along a closed submanifold $Z$. Topologically, $\operatorname{Bl}_Z(M)$ is obtained by replacing $Z$ with its unit normal sphere bundle $S(N_Z M) = (N_Z M \setminus Z) / \mathbb{R}_{>0}$, so that points approaching $Z$ retain their limiting direction of approach:
\begin{equation}
    \operatorname{Bl}_Z(M) = (M \setminus Z) \sqcup S(N_Z M).
    \label{eq:real_blowup_def}
\end{equation}
To desingularize the entire collision arrangement $\Delta$, the blow-up must be performed iteratively along the proper transforms of the subsequent strata of the \emph{irreducible building set} $\mathcal{G}_{\mathrm{irr}} = \{ \Delta_I \mid I \subseteq \{1, \dots, N\}, |I| \ge 2 \}$ in order of \emph{increasing manifold dimension of collision strata} (which strictly corresponds to \emph{decreasing cluster cardinality} $k = |I|$, progressing from the total collision stratum $k = N$ of minimal manifold dimension $0$ up to binary collision pairs $k = 2$ of maximal manifold dimension $3N-6$):

\begin{thmbox}[Fulton--MacPherson Desingularization and SNC Divisor]{thm:fulton_macpherson_snc}
In the sense of Fulton and MacPherson~\cite{fulton1994compactification}, Axelrod and Singer~\cite{axelrod1994chern}, De Concini and Procesi~\cite{deconcini1995wonderful}, and Campos et al.~\cite{campos2024configuration}, let $M = (\mathbb{R}^3)^N / \mathbb{R}^3 \cong \mathbb{R}^{3N-3}$ and let $\mathcal{G}_{\mathrm{irr}} = \{ \Delta_I \mid |I| \ge 2 \}$ be the irreducible building set of the partition lattice $\Pi(N)$. The iterated real oriented blow-up
\begin{equation}
    \mathbf{Q}[N] = \operatorname{Bl}_{\mathcal{G}_{\mathrm{irr}}}(M) \xrightarrow{\quad \pi \quad} M
    \label{eq:fulton_macpherson_space}
\end{equation}
satisfies the following canonical geometric properties:
\begin{enumerate}
    \item \textbf{Manifold with Corners Structure:} $\mathbf{Q}[N]$ is a smooth manifold with corners of dimension $3N-3$, whose interior $\operatorname{Int}(\mathbf{Q}[N])$ is diffeomorphic to the collisionless configuration space $M \setminus \Delta$.
    \item \textbf{Simple Normal Crossing (SNC) Boundary:} The total boundary $\partial \mathbf{Q}[N] = \pi^{-1}(\Delta)$ is a simple normal crossing divisor with corners:
    \begin{equation}
        \partial \mathbf{Q}[N] = \bigcup_{\substack{I \subseteq \{1, \dots, N\} \\ |I| \ge 2}} E_I,
        \label{eq:boundary_snc_decomposition}
    \end{equation}
    where each exceptional divisor $E_I$, defined as the compactified proper transform of the spherical normal bundle $S(N_{\Delta_I} M)$, is a smooth codimension-$1$ boundary hypersurface with inward-pointing normal coordinate (defining boundary function) $\rho_I \in [0, \infty)$.
    \item \textbf{Transversality of Boundary Intersections:} Two exceptional divisors $E_I$ and $E_J$ intersect if and only if the coalitions are \emph{nested} or \emph{disjoint}:
    \begin{equation}
        E_I \cap E_J \ne \emptyset \iff I \subset J, \quad J \subset I, \quad \text{or} \quad I \cap J = \emptyset.
        \label{eq:snc_nesting_criterion}
    \end{equation}
    When $E_I \cap E_J \ne \emptyset$, their intersection is strictly transversal in $\mathbf{Q}[N]$, meeting along a corner stratum of codimension $2$.
\end{enumerate}
\end{thmbox}

\begin{proofbox}[Proof of Theorem~\ref{thm:fulton_macpherson_snc}]
Assertion (1) follows from the differential blow-up construction of Axelrod and Singer~\cite[Sections~4.3 and~5]{axelrod1994chern}, where each stage replaces an embedded submanifold by its normal sphere bundle, preserving $\mathcal{C}^\infty$ differential structure with corners. 

Assertion (2) is established by observing that blowing up along the clean arrangement $\Pi(N)$ in order of increasing manifold dimension of strata (decreasing coalition cardinality $|I|$) transforms all non-transverse intersections into normal crossings: if $I \cap J \ne \emptyset$ and neither contains the other (overlapping clusters), their union satisfies $|I \cup J| > |I|$ and $|I \cup J| > |J|$. Consequently, the deeper collision stratum $\Delta_{I \cup J}$ has strictly lower manifold dimension $\dim \Delta_{I \cup J} = 3(N - |I \cup J|) < \min(\dim \Delta_I, \dim \Delta_J)$, so its proper transform is blown up at an earlier stage of the sequence, separating the proper transforms of $\Delta_I$ and $\Delta_J$ and ensuring $E_I \cap E_J = \emptyset$. For disjoint coalitions ($I \cap J = \emptyset$), the submanifolds $\Delta_I$ and $\Delta_J$ are already in clean, transverse intersection in $M$; their proper transforms $E_I$ and $E_J$ intersect transversally along a codimension-$2$ corner stratum without requiring the prior blow-up of $\Delta_I \cap \Delta_J \notin \mathcal{G}_{\mathrm{irr}}$, in exact agreement with De Concini--Procesi wonderful models~\cite{deconcini1995wonderful}.

Assertion (3) reflects the combinatorial property that non-empty boundary strata correspond bijectively to \emph{nested sets} of the building set $\mathcal{G}_{\mathrm{irr}}$ (chains of nested and mutually compatible disjoint clusters), with defining coordinates of the corner $[0, \epsilon)^m$ given by scale ratios $u_k = \rho_{I_k} / \rho_{I_{k+1}} \ge 0$, and with maximal codimension corners in one-to-one correspondence with complete binary Jacobi trees (see Figure~\ref{fig:ch04_fulton_macpherson_snc}).
\end{proofbox}

% (moved figure after rembox)

\begin{rembox}[{Epistemic Note: Standard Cotangent Bundle $T^*\mathbf{Q}[N]$ versus Melrose's $b$-Cotangent Bundle}]
In the differential geometry of manifolds with corners (Melrose~\cite{melrose1993atiyah}), the canonical boundary-adapted structure is the \emph{$b$-tangent bundle} ${}^b T \mathbf{Q}[N]$ (spanned by vector fields tangent to all boundary faces, locally generated by $\rho_I \partial_{\rho_I}$ and $\partial_{\theta}$) and its dual \emph{$b$-cotangent bundle} ${}^b T^* \mathbf{Q}[N]$ whose smooth sections admit logarithmic poles $\mathrm{d}\rho_I / \rho_I$. 

One might wonder why our Hamiltonian framework lifts configuration space to the standard cotangent bundle $T^*\mathbf{Q}[N]$ (and its Poincaré--Sundman extension $\mathcal{P}_{\mathrm{ext}}$) rather than adopting ${}^b T^* \mathbf{Q}[N]$. The justification stems directly from the physics of gravitational singularities:
\begin{enumerate}
    \item \emph{Singularity Order Mismatch}: The Newtonian potential diverges as $\rho_I^{-1}$ (a pole of order $1$), while virialized relative velocities scale as $\rho_I^{-1/2}$. Because these divergences are algebraic rather than logarithmic ($\rho_I^{-1} \gg \log \rho_I$), working on ${}^b T^* \mathbf{Q}[N]$ without an explicit temporal reparametrization does not desingularize the Hamiltonian vector field.
    \item \emph{Preservation of Canonical Darboux Symplectic Geometry}: Passing to $b$-symplectic geometry (as developed by Guillemin, Miranda, and Pires~\cite{guillemin2014symplectic}) replaces the canonical symplectic $2$-form $\Omega_0$ by a singular $b$-symplectic form $\Omega_b = \sum \frac{\mathrm{d}\rho_I}{\rho_I} \wedge \mathrm{d}p_{\rho_I} + \dots$. This destroys the affine bundle structure of the cotangent fibers and prevents vertical Hamiltonian translations (potential kicks), which are fundamental to symplectic splitting methods.
    \item \emph{The Normal-Cone Cotangent and Poincaré--Sundman Resolution}: By coupling configuration blow-ups on $\mathbf{Q}[N]$ with Calaque--Safronov cotangent lifts and the unfolded Sundman local clock $\mathrm{d}t_S = \chi_S^2 \mathrm{d}\tau_S$ (Section~\ref{sec:synchronous_supernodes}), the algebraic velocity divergence is resolved within standard, non-singular Darboux geometry on $\mathcal{P}_{\mathrm{ext}} = T^*\mathbf{Q}[N] \times \mathbb{R}_t \times \mathbb{R}_E$. This preserves exact Lie--Bargmann symmetries and admits explicit, palindromic symplectic splitting propagators.
\end{enumerate}
\end{rembox}

% ------------------------------------------------------------------------------
\begin{figure}[htbp]
\centering
\resizebox{0.96\textwidth}{!}{%
\begin{tikzpicture}[>=stealth, font=\small]

    % ==========================================================================
    % (a) Stage 0: Singular Arrangement in M / R^3
    % ==========================================================================
    \begin{scope}[shift={(-5.6, 0)}]
        % Three binary diagonals meeting at the origin (90, 210, 330 degrees)
        \draw[line width=1.1pt, black] (0, 0) -- (90:2.05) node[above=2pt] {$\Delta_{13}$};
        \draw[line width=1.1pt, black] (0, 0) -- (210:2.05) node[below left=1pt] {$\Delta_{12}$};
        \draw[line width=1.1pt, black] (0, 0) -- (330:2.05) node[below right=1pt] {$\Delta_{23}$};
        % Dashed continuation through the origin to show full lines
        \draw[dashed, line width=0.6pt, black!55] (0, 0) -- (270:1.15);
        \draw[dashed, line width=0.6pt, black!55] (0, 0) -- (30:1.15);
        \draw[dashed, line width=0.6pt, black!55] (0, 0) -- (150:1.15);

        % Triple collision point
        \filldraw[black] (0, 0) circle (2.6pt);
        \node[right=4pt] at (0.05, 0.18) {$\Delta_{123}$};

        \node[font=\footnotesize] at (0, -2.15) {(a) $M/\mathbb{R}^3$};
    \end{scope}

    % Arrow (a) -> (b)
    \draw[->, line width=0.9pt, black] (-3.25, 0.15) -- (-2.15, 0.15)
        node[midway, above=2pt, font=\scriptsize] {$\operatorname{Bl}_{\Delta_{123}}$}
        node[midway, below=2pt, font=\scriptsize] {$|I|=3$};

    % ==========================================================================
    % (b) Stage 1: Real Oriented Blow-Up of Triple Point \Delta_{123}
    % ==========================================================================
    \begin{scope}[shift={(0, 0)}]
        % Blown-up spherical boundary E_{123} (radius 0.85)
        \fill[black!8] (0, 0) circle (0.85);
        \draw[line width=1.1pt, black] (0, 0) circle (0.85);
        \node[font=\footnotesize] at (0, 0) {$E_{123}$};

        % Separated proper transforms \widetilde{\Delta}_{ij}
        \draw[line width=1.1pt, black] (90:0.85) -- (90:2.05) node[above=2pt] {$\widetilde{\Delta}_{13}$};
        \draw[line width=1.1pt, black] (210:0.85) -- (210:2.05) node[below left=1pt] {$\widetilde{\Delta}_{12}$};
        \draw[line width=1.1pt, black] (330:0.85) -- (330:2.05) node[below right=1pt] {$\widetilde{\Delta}_{23}$};

        % Transverse intersection points on E_{123}
        \filldraw[fill=white, draw=black, line width=0.9pt] (90:0.85) circle (2.3pt);
        \filldraw[fill=white, draw=black, line width=0.9pt] (210:0.85) circle (2.3pt);
        \filldraw[fill=white, draw=black, line width=0.9pt] (330:0.85) circle (2.3pt);

        \node[font=\footnotesize] at (0, -2.15) {(b) $\operatorname{Bl}_{\Delta_{123}}(M/\mathbb{R}^3)$};
    \end{scope}

    % Arrow (b) -> (c)
    \draw[->, line width=0.9pt, black] (2.15, 0.15) -- (3.25, 0.15)
        node[midway, above=2pt, font=\scriptsize] {$\operatorname{Bl}_{\widetilde{\Delta}_{ij}}$}
        node[midway, below=2pt, font=\scriptsize] {$|I|=2$};

    % ==========================================================================
    % (c) Stage 2: Iterated Blow-Up Q[3] with Transverse Corners
    % ==========================================================================
    \begin{scope}[shift={(5.6, 0)}]
        % Shaded interior of the blown-up boundary collar
        \fill[black!8] (0, 0) circle (0.85);
        \foreach \ang in {90, 210, 330} {
            \begin{scope}[rotate=\ang-90]
                \fill[black!8] (-0.22, 0.80) rectangle (0.22, 1.95);
            \end{scope}
        }

        % Three arcs of E_{123} between the binary necks (radius 0.85, half-angle 15 deg)
        \draw[line width=1.1pt, black] (105:0.85) arc (105:195:0.85);
        \draw[line width=1.1pt, black] (225:0.85) arc (225:315:0.85);
        \draw[line width=1.1pt, black] (345:0.85) arc (345:435:0.85);
        \node[font=\footnotesize] at (0, 0) {$E_{123}$};

        % Three blown-up binary necks E_{13}, E_{12}, E_{23} and their 6 corner vertices
        % Top neck: E_{13}
        \draw[line width=1.1pt, black] (105:0.85) -- ++(90:1.12);
        \draw[line width=1.1pt, black] (75:0.85) -- ++(90:1.12);
        \filldraw[black] (105:0.85) circle (2.0pt);
        \filldraw[black] (75:0.85) circle (2.0pt);
        \node[above=2pt] at (90:1.97) {$E_{13}$};
        \node[right=3pt, font=\scriptsize] at (75:0.95) {$\mathcal{T}_{13}$};

        % Bottom-left neck: E_{12}
        \draw[line width=1.1pt, black] (195:0.85) -- ++(210:1.12);
        \draw[line width=1.1pt, black] (225:0.85) -- ++(210:1.12);
        \filldraw[black] (195:0.85) circle (2.0pt);
        \filldraw[black] (225:0.85) circle (2.0pt);
        \node[below left=1pt] at (210:1.97) {$E_{12}$};
        \node[above left=1pt, font=\scriptsize] at (195:0.90) {$\mathcal{T}_{12}$};

        % Bottom-right neck: E_{23}
        \draw[line width=1.1pt, black] (315:0.85) -- ++(330:1.12);
        \draw[line width=1.1pt, black] (345:0.85) -- ++(330:1.12);
        \filldraw[black] (315:0.85) circle (2.0pt);
        \filldraw[black] (345:0.85) circle (2.0pt);
        \node[below right=1pt] at (330:1.97) {$E_{23}$};
        \node[above right=1pt, font=\scriptsize] at (345:0.90) {$\mathcal{T}_{23}$};

        \node[font=\footnotesize] at (0, -2.15) {(c) $\mathbf{Q}[3] = \operatorname{Bl}_{\mathcal{G}_{\mathrm{irr}}}(M/\mathbb{R}^3)$};
    \end{scope}

\end{tikzpicture}%
}
\caption{Sequential real oriented blow-up constructing the Fulton--MacPherson manifold with corners $\mathbf{Q}[3] = \operatorname{Bl}_{\mathcal{G}_{\mathrm{irr}}}(M/\mathbb{R}^3)$ (Theorem~\ref{thm:fulton_macpherson_snc}). \textbf{(a)} In the unblown relative configuration space $M/\mathbb{R}^3$, the three binary collision strata $\Delta_{12}, \Delta_{23}, \Delta_{13}$ intersect non-transversally at the triple collision origin $\Delta_{123}$. \textbf{(b)} Blowing up the deepest stratum $\Delta_{123}$ ($|I|=3$) inserts the spherical boundary divisor $E_{123}$ ($\rho_{123}=0$) and separates the proper transforms $\widetilde{\Delta}_{12}, \widetilde{\Delta}_{23}, \widetilde{\Delta}_{13}$ into pairwise disjoint rays ($\widetilde{\Delta}_{ij} \cap \widetilde{\Delta}_{jk} = \emptyset$). \textbf{(c)} Blowing up the separated binary loci $\widetilde{\Delta}_{ij}$ ($|I|=2$) replaces each ray by a boundary collar $E_{ij}$ meeting $E_{123}$ orthogonally at codimension-$2$ corners indexed by the binary Jacobi trees $\mathcal{T}_{12}, \mathcal{T}_{23}, \mathcal{T}_{13}$.}
\label{fig:ch04_fulton_macpherson_snc}
\end{figure}

\subsubsection{Maximal Flag Jacobi Trees and Adapted Cotangent Charts}
\label{sub:maximal_flag_jacobi_trees}

The maximal flags (saturated chains) in the partition lattice $\Pi(N)$,
\begin{equation}
    \mathcal{F} = \left( \mathcal{P}_1 < \mathcal{P}_2 < \dots < \mathcal{P}_{N-1} \right),
    \label{eq:maximal_partition_flag}
\end{equation}
where $\mathcal{P}_1$ contains exactly one non-trivial binary pair, and each subsequent partition $\mathcal{P}_{k+1}$ is obtained from $\mathcal{P}_k$ by merging either two existing clusters or an isolated body with a cluster, represent the simplicial conical sectors (or linear extensions) that subdivide the boundary corners.

Geometrically, the corners of maximal codimension $N-1$ in $\mathbf{Q}[N]$ (transversal intersections of maximal sets of mutually compatible exceptional divisors) correspond bijectively to complete binary Jacobi trees $\mathcal{T} \in \mathrm{Trees}(N)$ with $N$ labeled leaves and $N-1$ internal nodes $\mathcal{V}_{\mathrm{int}}(\mathcal{T})$~\cite{fulton1994compactification,jacobi1843elimination}. The natural projection from maximal flags in $\Pi(N)$ to binary Jacobi trees is surjective, with fibers reflecting the commutation of mergers between mutually disjoint clusters. The total number of maximal saturated flags in $\Pi(N)$ is given by
\begin{equation}
    |\operatorname{Flags}(\Pi(N))| = \prod_{k=2}^N \binom{k}{2} = \frac{N!(N-1)!}{2^{N-1}},
    \label{eq:flags_combinatorial_count}
\end{equation}
yielding $3$ flags for $N=3$, $18$ flags for $N=4$ projecting onto $(2N-3)!! = 15$ binary trees, and $180$ flags for $N=5$ projecting onto $105$ trees.
\leanrefright{maximal_flag_jacobi_theorems}{jacobi_node_defines_hierarchy}{03_MaximalFlagJacobiTree_Theorems.lean}

For each internal node $v \in \mathcal{V}_{\mathrm{int}}(\mathcal{T})$ merging two disjoint sub-clusters $L(v)$ and $R(v)$ of total masses $M_{L(v)} = \sum_{i \in L(v)} m_i$ and $M_{R(v)} = \sum_{j \in R(v)} m_j$, we define the relative Jacobi coordinate vector $\mathbf{r}_v$ and its conjugate momentum $\mathbf{w}_v$~\cite{jacobi1843elimination}:
\begin{align}
    \mathbf{r}_v &= \mathbf{X}_{L(v)} - \mathbf{X}_{R(v)} = \left( \frac{1}{M_{L(v)}} \sum_{i \in L(v)} m_i \mathbf{q}_i \right) - \left( \frac{1}{M_{R(v)}} \sum_{j \in R(v)} m_j \mathbf{q}_j \right), \label{eq:jacobi_relative_pos} \\
    \mathbf{w}_v &= \mu_v \dot{\mathbf{r}}_v = \mu_v \left( \frac{\mathbf{P}_{L(v)}}{M_{L(v)}} - \frac{\mathbf{P}_{R(v)}}{M_{R(v)}} \right), \qquad \mu_v = \frac{M_{L(v)} M_{R(v)}}{M_{L(v)} + M_{R(v)}}, \label{eq:jacobi_relative_mom}
\end{align}
where $\mathbf{P}_C = \sum_{i \in C} \mathbf{p}_i$ denotes the total linear momentum of cluster $C$. The linear map $(\mathbf{q}, \mathbf{p}) \mapsto (\mathbf{r}_v, \mathbf{w}_v)_{v \in \mathcal{V}_{\mathrm{int}}(\mathcal{T})}$ defines a canonical symplectic isomorphism on $T^*M \cong \mathbb{R}^{6N-6}$. In these coordinates, the total kinetic energy decouples diagonally:
\begin{equation}
    T_{\mathrm{kin}}(\mathbf{w}) = \frac{1}{2} \sum_{v \in \mathcal{V}_{\mathrm{int}}(\mathcal{T})} \frac{\|\mathbf{w}_v\|^2}{\mu_v}.
    \label{eq:jacobi_kinetic_diagonal}
\end{equation}

For each sub-tree $\mathcal{T}_C$ rooted at an internal node $v$ with cluster bodies $C = L(v) \sqcup R(v)$, the internal cluster potential is given by the restriction of the gravitational interaction to pairs in $C$:
\begin{equation}
    V_C(\mathbf{q}) = - \sum_{\substack{\{i,j\} \subseteq C \\ i < j}} \frac{G m_i m_j}{\|\mathbf{q}_i - \mathbf{q}_j\|}.
    \label{eq:cluster_potential_def}
\end{equation}
The local cluster hyper-radius $\rho_C \in [0, \infty)$ is defined by the mass metric on the sub-cluster, expressing equivalently in relative Cartesian barycentric coordinates and diagonal Jacobi tree coordinates:
\begin{equation}
    \rho_C = \sqrt{\frac{1}{M_C} \sum_{\substack{\{i, j\} \subseteq C \\ i < j}} m_i m_j \|\mathbf{q}_i - \mathbf{q}_j\|^2} = \sqrt{\sum_{u \in \mathcal{V}_{\mathrm{int}}(\mathcal{T}_C)} \mu_u \|\mathbf{r}_u\|^2},
    \label{eq:cluster_hyperradius_def}
\end{equation}
where $\mu_u = \frac{M_{L(u)} M_{R(u)}}{M_{L(u)} + M_{R(u)}}$ denotes the reduced mass at internal node $u$. To ensure strict dimensional homogeneity with relative positions $\mathbf{r}_v$ (physical dimension $[L]$), we also define the mass-normalized hyper-radius $\bar{\rho}_C \coloneqq \rho_C / \sqrt{M_C} \in [0, \infty)$, of dimension $[L]$, ensuring that coordinate scale ratios $u_v = \|\mathbf{r}_v\| / (\bar{\rho}_{L(v)} + \bar{\rho}_{R(v)})$ are strictly dimensionless. Each complete tree $\mathcal{T}$ defines an adapted conical clustering chamber (or Maclaurin--Sundman wedge) in configuration space:
\begin{equation}
    \mathcal{U}_{\mathcal{T}} = \left\{ \mathbf{q} \in M \setminus \Delta \;\middle|\; \forall v \in \mathcal{V}_{\mathrm{int}}(\mathcal{T}), \, \|\mathbf{r}_v\| < \min_{\substack{i \in C_v \\ j \notin C_v}} \|\mathbf{q}_i - \mathbf{q}_j\| \right\},
    \label{eq:conical_chamber_ut}
\end{equation}
guaranteeing that intra-cluster separations remain strictly hierarchized (see Figure~\ref{fig:ch04_jacobi_trees_chamber_covering}).
\leanrefright{cluster_hyperradius_theorems}{hyper_radius_sq_pos}{02_ClusterHyperRadius_Theorems.lean}
\pythonrefright{py_core}{core/jacobi.py, core/coordinates.py}{Float64 / Float128}

\begin{figure}[htbp]
\centering
\resizebox{0.96\textwidth}{!}{%
\begin{tikzpicture}[>=stealth, font=\small]

    % ==========================================================================
    % (a) LEFT PANEL: The 3 Rooted Binary Jacobi Trees & Physical Vectors
    % ==========================================================================
    \begin{scope}[shift={(-4.3, 0)}]

        % --- Row 1: Tree T_{12} = ((1,2),3) (y = 1.65) ---
        \begin{scope}[shift={(-1.7, 1.65)}]
            \node[left=4pt, font=\footnotesize] at (-0.65, 0.0) {$\mathcal{T}_{12}$:};
            % Rooted binary tree
            \coordinate (r) at (0, 0.55);
            \coordinate (v) at (-0.35, 0.05);
            \coordinate (l1) at (-0.65, -0.45);
            \coordinate (l2) at (-0.05, -0.45);
            \coordinate (l3) at (0.35, -0.45);
            \draw[line width=0.9pt, black] (l1) -- (v) -- (r) -- (l3);
            \draw[line width=0.9pt, black] (l2) -- (v);
            \filldraw[black] (r) circle (1.6pt);
            \filldraw[black] (v) circle (1.6pt);
            \node[below=1pt, font=\scriptsize] at (l1) {$1$};
            \node[below=1pt, font=\scriptsize] at (l2) {$2$};
            \node[below=1pt, font=\scriptsize] at (l3) {$3$};
        \end{scope}
        \begin{scope}[shift={(1.05, 1.65)}]
            % Static hierarchical cluster configuration for T_{12} (no kinematic arrows)
            \coordinate (q1) at (-0.75, -0.35);
            \coordinate (q2) at (0.25, -0.35);
            \coordinate (cm) at (-0.25, -0.35);
            \coordinate (q3) at (0.85, 0.45);
            \draw[black!45, dashed, line width=0.55pt] (cm) ellipse (0.76 and 0.25);
            \draw[line width=1.0pt, black] (q1) -- (q2) node[midway, below=6.5pt, font=\scriptsize] {$r_{12}$};
            \draw[line width=0.8pt, dashed, black] (cm) -- (q3) node[midway, above left=-1pt, font=\scriptsize] {$r_{(12)3}$};
            \filldraw[black] (q1) circle (2.0pt) node[left=2pt, font=\scriptsize] {$1$};
            \filldraw[black] (q2) circle (2.0pt) node[right=2pt, font=\scriptsize] {$2$};
            \filldraw[fill=white, draw=black, line width=0.8pt] (cm) circle (1.8pt);
            \filldraw[black] (q3) circle (2.0pt) node[right=2pt, font=\scriptsize] {$3$};
        \end{scope}

        % --- Row 2: Tree T_{23} = ((2,3),1) (y = 0.0) ---
        \begin{scope}[shift={(-1.7, 0.0)}]
            \node[left=4pt, font=\footnotesize] at (-0.65, 0.0) {$\mathcal{T}_{23}$:};
            \coordinate (r) at (0, 0.55);
            \coordinate (v) at (-0.35, 0.05);
            \coordinate (l2) at (-0.65, -0.45);
            \coordinate (l3) at (-0.05, -0.45);
            \coordinate (l1) at (0.35, -0.45);
            \draw[line width=0.9pt, black] (l2) -- (v) -- (r) -- (l1);
            \draw[line width=0.9pt, black] (l3) -- (v);
            \filldraw[black] (r) circle (1.6pt);
            \filldraw[black] (v) circle (1.6pt);
            \node[below=1pt, font=\scriptsize] at (l2) {$2$};
            \node[below=1pt, font=\scriptsize] at (l3) {$3$};
            \node[below=1pt, font=\scriptsize] at (l1) {$1$};
        \end{scope}
        \begin{scope}[shift={(1.05, 0.0)}]
            \coordinate (q2) at (-0.15, -0.35);
            \coordinate (q3) at (0.85, -0.35);
            \coordinate (cm) at (0.35, -0.35);
            \coordinate (q1) at (-0.75, 0.45);
            \draw[black!45, dashed, line width=0.55pt] (cm) ellipse (0.76 and 0.25);
            \draw[line width=1.0pt, black] (q2) -- (q3) node[midway, below=6.5pt, font=\scriptsize] {$r_{23}$};
            \draw[line width=0.8pt, dashed, black] (cm) -- (q1) node[midway, above right=-1pt, font=\scriptsize] {$r_{(23)1}$};
            \filldraw[black] (q2) circle (2.0pt) node[left=2pt, font=\scriptsize] {$2$};
            \filldraw[black] (q3) circle (2.0pt) node[right=2pt, font=\scriptsize] {$3$};
            \filldraw[fill=white, draw=black, line width=0.8pt] (cm) circle (1.8pt);
            \filldraw[black] (q1) circle (2.0pt) node[left=2pt, font=\scriptsize] {$1$};
        \end{scope}

        % --- Row 3: Tree T_{13} = ((1,3),2) (y = -1.65) ---
        \begin{scope}[shift={(-1.7, -1.65)}]
            \node[left=4pt, font=\footnotesize] at (-0.65, 0.0) {$\mathcal{T}_{13}$:};
            \coordinate (r) at (0, 0.55);
            \coordinate (v) at (-0.35, 0.05);
            \coordinate (l1) at (-0.65, -0.45);
            \coordinate (l3) at (-0.05, -0.45);
            \coordinate (l2) at (0.35, -0.45);
            \draw[line width=0.9pt, black] (l1) -- (v) -- (r) -- (l2);
            \draw[line width=0.9pt, black] (l3) -- (v);
            \filldraw[black] (r) circle (1.6pt);
            \filldraw[black] (v) circle (1.6pt);
            \node[below=1pt, font=\scriptsize] at (l1) {$1$};
            \node[below=1pt, font=\scriptsize] at (l3) {$3$};
            \node[below=1pt, font=\scriptsize] at (l2) {$2$};
        \end{scope}
        \begin{scope}[shift={(1.05, -1.65)}]
            \coordinate (q1) at (-0.65, 0.35);
            \coordinate (q3) at (0.35, 0.35);
            \coordinate (cm) at (-0.15, 0.35);
            \coordinate (q2) at (0.75, -0.45);
            \draw[black!45, dashed, line width=0.55pt] (cm) ellipse (0.76 and 0.25);
            \draw[line width=1.0pt, black] (q1) -- (q3) node[midway, above=6.5pt, font=\scriptsize] {$r_{13}$};
            \draw[line width=0.8pt, dashed, black] (cm) -- (q2) node[midway, below left=-1pt, font=\scriptsize] {$r_{(13)2}$};
            \filldraw[black] (q1) circle (2.0pt) node[left=2pt, font=\scriptsize] {$1$};
            \filldraw[black] (q3) circle (2.0pt) node[right=2pt, font=\scriptsize] {$3$};
            \filldraw[fill=white, draw=black, line width=0.8pt] (cm) circle (1.8pt);
            \filldraw[black] (q2) circle (2.0pt) node[right=2pt, font=\scriptsize] {$2$};
        \end{scope}

        \node[font=\footnotesize] at (-0.35, -2.75) {(a) Jacobi Trees $\mathcal{T}_{ij}$ and Clusters $(r_{ij} < r_{(ij)k})$};
    \end{scope}

    % ==========================================================================
    % (b) RIGHT PANEL: Conical Chamber Covering U_{T_{12}}, U_{T_{23}}, U_{T_{13}} on Q[3]
    % ==========================================================================
    \begin{scope}[shift={(3.4, 0.15)}]
        % Three conical Voronoi chambers (outer radius 2.25, inner radius 0.78)
        % Chamber U_{T_{13}} (30 deg to 150 deg: light gray black!4)
        \fill[black!4] (30:0.78) -- (30:2.25) arc (30:150:2.25) -- (150:0.78) arc (150:30:0.78) -- cycle;
        % Chamber U_{T_{12}} (150 deg to 270 deg: medium-light gray black!10)
        \fill[black!10] (150:0.78) -- (150:2.25) arc (150:270:2.25) -- (270:0.78) arc (270:150:0.78) -- cycle;
        % Chamber U_{T_{23}} (270 deg to 390 deg: medium gray black!16)
        \fill[black!16] (270:0.78) -- (270:2.25) arc (270:390:2.25) -- (30:0.78) arc (390:270:0.78) -- cycle;

        % Outer boundary circle of the local neighborhood
        \draw[line width=0.6pt, black!45] (0, 0) circle (2.25);

        % Three Voronoi equidistance walls (dashed rays at 30, 150, 270 deg)
        \draw[dashed, line width=0.95pt, black] (30:0.78) -- (30:2.35)
            node[above right=-1pt, font=\scriptsize] {$r_{13}=r_{23}$};
        \draw[dashed, line width=0.95pt, black] (150:0.78) -- (150:2.35)
            node[above left=-1pt, font=\scriptsize] {$r_{12}=r_{13}$};
        \draw[dashed, line width=0.95pt, black] (270:0.78) -- (270:2.35)
            node[below=1pt, font=\scriptsize] {$r_{12}=r_{23}$};

        % Central blown-up boundary E_{123} (white interior) and the 3 binary necks E_{13}, E_{12}, E_{23}
        \fill[white] (0, 0) circle (0.78);
        \foreach \ang in {90, 210, 330} {
            \begin{scope}[rotate=\ang-90]
                \fill[white] (-0.18, 0.72) rectangle (0.18, 2.26);
            \end{scope}
        }

        % Arcs of E_{123}
        \draw[line width=1.1pt, black] (103:0.78) arc (103:197:0.78);
        \draw[line width=1.1pt, black] (223:0.78) arc (223:317:0.78);
        \draw[line width=1.1pt, black] (343:0.78) arc (343:437:0.78);
        \node[font=\scriptsize] at (0, 0) {$E_{123}$};

        % Binary neck walls E_{13}, E_{12}, E_{23}
        \draw[line width=1.1pt, black] (103:0.78) -- ++(90:1.47);
        \draw[line width=1.1pt, black] (77:0.78) -- ++(90:1.47);
        \filldraw[black] (103:0.78) circle (1.8pt);
        \filldraw[black] (77:0.78) circle (1.8pt);
        \node[above=1pt, font=\scriptsize] at (90:2.25) {$E_{13}$};

        \draw[line width=1.1pt, black] (197:0.78) -- ++(210:1.47);
        \draw[line width=1.1pt, black] (223:0.78) -- ++(210:1.47);
        \filldraw[black] (197:0.78) circle (1.8pt);
        \filldraw[black] (223:0.78) circle (1.8pt);
        \node[below left=0pt, font=\scriptsize] at (210:2.25) {$E_{12}$};

        \draw[line width=1.1pt, black] (317:0.78) -- ++(330:1.47);
        \draw[line width=1.1pt, black] (343:0.78) -- ++(330:1.47);
        \filldraw[black] (317:0.78) circle (1.8pt);
        \filldraw[black] (343:0.78) circle (1.8pt);
        \node[below right=0pt, font=\scriptsize] at (330:2.25) {$E_{23}$};

        % Chamber Labels U_{T_{ij}} inside each sector
        \node[font=\footnotesize] at (120:1.55) {$\mathcal{U}_{\mathcal{T}_{13}}$};
        \node[font=\footnotesize] at (180:1.55) {$\mathcal{U}_{\mathcal{T}_{12}}$};
        \node[font=\footnotesize] at (0:1.55) {$\mathcal{U}_{\mathcal{T}_{23}}$};

        \node[font=\footnotesize] at (0, -2.90) {(b) Chamber Partition $\mathbf{Q}[3] = \overline{\mathcal{U}}_{\mathcal{T}_{12}} \cup \overline{\mathcal{U}}_{\mathcal{T}_{23}} \cup \overline{\mathcal{U}}_{\mathcal{T}_{13}}$};
    \end{scope}

\end{tikzpicture}%
}
\caption{Rooted binary Jacobi trees and their associated conical chamber covering of $\mathbf{Q}[3]$ ($N=3$). \textbf{(a)} Each of the $(2N-3)!! = 3$ binary trees $\mathcal{T}_{ij} = ((i,j),k)$ groups the closest binary leaf at separation $r_{ij} = \|\mathbf{q}_i - \mathbf{q}_j\|$ (solid segment inside the dashed cluster oval) and links its center of mass (hollow circle) to the third body at apex separation $r_{(ij)k} = \|\mathbf{X}_{ij} - \mathbf{q}_k\|$ (dashed segment), independently of the instantaneous direction of motion. \textbf{(b)} In the blown-up configuration space $\mathbf{Q}[3]$, the three Maclaurin--Sundman conical chambers $\mathcal{U}_{\mathcal{T}_{12}}, \mathcal{U}_{\mathcal{T}_{23}}, \mathcal{U}_{\mathcal{T}_{13}}$ defined in~\eqref{eq:conical_chamber_ut} form a sectorial covering around the respective binary collars $E_{12}, E_{23}, E_{13}$ and meet along the Voronoi equidistance hypersurfaces $r_{12}=r_{23}$, $r_{23}=r_{13}$, and $r_{12}=r_{13}$ (dashed rays).}
\label{fig:ch04_jacobi_trees_chamber_covering}
\end{figure}
Geometrically, the hyper-radius $\rho_C$ provides the explicit metric realization of the boundary normal coordinate $\rho_C \in [0, \infty)$ defining the exceptional divisor $E_C \subset \partial \mathbf{Q}[N]$ in local Jacobi charts. Because the Newtonian pairwise potential is homogeneous of degree $-1$, it satisfies McGehee's scale factorization on each cluster~\cite{mcgehee1974triple}:
\begin{equation}
    V_C(\mathbf{r}) = \rho_C^{-1} \, V_C(\mathbf{s}), \qquad \text{where} \quad \mathbf{s} = (\mathbf{s}_u)_{u \in \mathcal{V}_{\mathrm{int}}(\mathcal{T}_C)} = \left(\frac{\sqrt{\mu_u}\mathbf{r}_u}{\rho_C}\right)_{u \in \mathcal{V}_{\mathrm{int}}(\mathcal{T}_C)} \in S\left(N_{\Delta_C} M\right).
    \label{eq:mcgehee_scale_factorization}
\end{equation}
\leanrefright{cluster_hyperradius_theorems}{newtonian_potential_is_homogeneous}{02_ClusterHyperRadius_Theorems.lean}
\leanrefright{cluster_hyperradius_theorems}{mcgehee_potential_factorization}{02_ClusterHyperRadius_Theorems.lean}

A foundational feature of this hierarchical decomposition is the dynamical decoupling of disjoint clusters:

\begin{propbox}[Exact Cotangent Commutation of Disjoint Jacobi Clusters]{prop:disjoint_cotangent_commutation}
Let $\mathcal{T}_A, \mathcal{T}_B \subset \mathcal{T}$ be two disjoint sub-trees of a Jacobi tree $\mathcal{T}$, meaning that their body sets are disjoint ($A \cap B = \emptyset$). Let $H_A(\mathbf{r}_A, \mathbf{w}_A)$ and $H_B(\mathbf{r}_B, \mathbf{w}_B)$ be internal Hamiltonians depending exclusively on the relative degrees of freedom within $\mathcal{T}_A$ and $\mathcal{T}_B$, respectively. Then:
\begin{enumerate}
    \item The canonical Poisson bracket of $H_A$ and $H_B$ vanishes identically on $T^*M$:
    \begin{equation}
        \{ H_A, H_B \}_{\Omega_0} \equiv 0.
        \label{eq:disjoint_poisson_bracket_zero}
    \end{equation}
    \item The corresponding Hamiltonian vector fields $\mathcal{X}_{H_A}$ and $\mathcal{X}_{H_B}$ (defined by the canonical convention $\iota_{\mathcal{X}_H} \Omega_0 = \mathrm{d}H$) commute strictly:
    \begin{equation}
        [\mathcal{X}_{H_A}, \mathcal{X}_{H_B}] \equiv 0, \qquad \Phi_t^{H_A} \circ \Phi_s^{H_B} = \Phi_s^{H_B} \circ \Phi_t^{H_A},
        \label{eq:disjoint_flow_commutation}
    \end{equation}
    on their maximal ambient domain of definition $\mathcal{D}_{A,B} = T^*M \setminus (\pi_A^{-1}(\Delta(A)) \cup \pi_B^{-1}(\Delta(B))) \cong T^*(M_A \setminus \Delta(A)) \times T^*(M_B \setminus \Delta(B)) \times T^*M_{\mathrm{rel}}$, where $\Delta(C) \coloneqq \bigcup_{\{i,j\} \subseteq C, i < j} \Delta_{ij}$ denotes the complete pairwise collision arrangement of sub-cluster $C$.
\end{enumerate}
\end{propbox}
\leanrefright{disjoint_cotangent_commutation}{disjoint_cotangent_commutation_local}{04_DisjointCotangentCommutation_Theorems.lean}

\begin{proofbox}[Proof of Proposition~\ref{prop:disjoint_cotangent_commutation}]
Because $A \cap B = \emptyset$, the internal Jacobi coordinates of $\mathcal{T}_A$ and $\mathcal{T}_B$ form disjoint subsets of canonical variables: $\mathcal{V}_{\mathrm{int}}(\mathcal{T}_A) \cap \mathcal{V}_{\mathrm{int}}(\mathcal{T}_B) = \emptyset$. Since $(\mathbf{r}_v, \mathbf{w}_v)_{v \in \mathcal{V}_{\mathrm{int}}(\mathcal{T})}$ is a canonical Darboux coordinate system on $T^*M$, the Poisson bracket evaluated over all internal nodes of the full Jacobi tree $\mathcal{T}$,
\begin{equation}
    \{H_A, H_B\}_{\Omega_0} = \sum_{u \in \mathcal{V}_{\mathrm{int}}(\mathcal{T})} \left( \langle \nabla_{\mathbf{r}_u} H_A, \nabla_{\mathbf{w}_u} H_B \rangle - \langle \nabla_{\mathbf{w}_u} H_A, \nabla_{\mathbf{r}_u} H_B \rangle \right) \equiv 0,
\end{equation}
vanishes identically because for each internal node $u \in \mathcal{V}_{\mathrm{int}}(\mathcal{T})$, at least one of $H_A$ or $H_B$ does not depend on $(\mathbf{r}_u, \mathbf{w}_u)$, ensuring that the corresponding gradient pair is identically zero.

Equivalently, in Cartesian coordinates, for each individual body $i \in \{1, \dots, N\}$, at least one of the two Hamiltonians has identically zero gradient with respect to $(\mathbf{q}_i, \mathbf{p}_i)$:
\begin{equation}
    \forall i \in \{1, \dots, N\}, \qquad \left( \nabla_{\mathbf{q}_i} H_A = \mathbf{0} \quad \text{and} \quad \nabla_{\mathbf{p}_i} H_A = \mathbf{0} \right) \quad \text{or} \quad \left( \nabla_{\mathbf{q}_i} H_B = \mathbf{0} \quad \text{and} \quad \nabla_{\mathbf{p}_i} H_B = \mathbf{0} \right).
    \label{eq:disjoint_gradient_support}
\end{equation}
The local term of the canonical Poisson bracket at index $i$,
\begin{equation}
    \{H_A, H_B\}_i = \langle \nabla_{\mathbf{q}_i} H_A, \nabla_{\mathbf{p}_i} H_B \rangle - \langle \nabla_{\mathbf{p}_i} H_A, \nabla_{\mathbf{q}_i} H_B \rangle,
    \label{eq:local_poisson_term}
\end{equation}
vanishes identically for every $i \in \{1, \dots, N\}$, establishing $\{H_A, H_B\}_{\Omega_0} = \sum_{i=1}^N \{H_A, H_B\}_i = 0$. The commutation of the Hamiltonian flows follows directly from the Lie algebra homomorphism between Poisson brackets and vector field commutators: $[\mathcal{X}_{H_A}, \mathcal{X}_{H_B}] = -\mathcal{X}_{\{H_A, H_B\}} = \mathbf{0}$.

This exact cotangent decoupling reflects in classical mechanics the canonical conormal Whitney sum $N^*(\Delta_A \cap \Delta_B) \cong N^*\Delta_A \oplus N^*\Delta_B$, mirroring the symmetric monoidal property of shifted cotangent functors under normal cone deformation established by Calaque and Safronov~\cite[Theorems~B, 3.13 and Corollary~5.13]{calaque2024deformation}.
\end{proofbox}

\begin{rembox}[Epistemic Scope: Constructive Combinatorial and 1-Jet Algebraic Kernel]
In accordance with the formal verification standard established in Section~\ref{ch:ch01_introduction}, the companion Lean~4 modules colocalized in Section~\ref{sec:whitney_lattice} mechanize a \emph{Constructive Combinatorial and 1-Jet Algebraic Kernel}. While the global differential and symplectic topology of $\mathbf{Q}[N]$ (smooth manifolds with corners, blow-up sequences, and continuous Hamiltonian flows) is developed analytically in the text, Lean~4 certifies the exact combinatorial incidence of the partition lattice $\Pi(N)$, the scale homogeneity of the Newtonian potential, the hierarchical tree inclusion, and the pointwise algebraic vanishing of local Poisson brackets without axioms or approximations.
\end{rembox}

\paragraph{Transition to Screened Apex Dynamics.}
Proposition~\ref{prop:disjoint_cotangent_commutation} guarantees that mutually disjoint sub-clusters evolve with complete symplectic independence, laying the rigorous foundation for decentralized multi-rate time-stepping on hierarchical supernodes (\S\ref{sec:synchronous_supernodes}).
However, when two clusters $L(v)$ and $R(v)$ interact at their common parent node $v$ (the \emph{apex}), their relative distance vector $\mathbf{r}_v$ couples to internal configurations through the global Newtonian potential $V(\mathbf{q})$. In raw Jacobi coordinates, the kinetic energy remains strictly diagonal ($T_{\mathrm{kin}} = \frac{1}{2}\sum \|\mathbf{w}_v\|^2/\mu_v$); however, the cross-metric products between interparticle distance gradients sharing an apex body, $\langle \nabla r_{ij}, \nabla r_{jk} \rangle_{\mathbb{M}^{-1}} \ne 0$, reflect the non-orthogonality of internal pairwise interactions. Furthermore, subtracting the point-mass barycentric Keplerian attraction $-G M_{L(v)} M_{R(v)}/\|\mathbf{r}_v\|$ leaves an unphysical multipolar residual potential that diverges as an artificial ``ghost pole'' when cluster barycenters coincide ($\|\mathbf{r}_v\| \to 0$) even while all internal interparticle separations remain strictly positive. Section~\ref{sec:screened_apex} resolves this challenge through the construction of a smooth screened apex potential achieving universal $\mathcal{C}^{1,1}$ Lipschitz regularity.