% --- Section Content : Smooth Scale-Adapted Screening of the Apex Potential --- \subsection{Smooth Scale-Adapted Screening of the Apex Potential and $\mathcal{C}^{1,1}$ Regularity} \label{sec:screened_apex} While Proposition~\ref{prop:disjoint_cotangent_commutation} establishes the complete symplectic decoupling of mutually disjoint clusters, real gravitational dynamics inevitably couples sub-clusters through parent hierarchical nodes. In this section, we analyze the residual inter-cluster potential coupling that arises at the parent apex of two interacting clusters, identify the multipolar ``ghost pole'' generated by naive Keplerian center-of-mass subtraction, and construct a smooth scale-adapted screening function $\Theta(u_v)$ that resolves this singularity. We prove that the resulting screened apex potential achieves universal $\mathcal{C}^{1,1}$ Lipschitz regularity without introducing any artificial distance cut-off ($d_0 > 0$ strictly banished), while strictly preserving all $10$ Galilean generators realizing the centrally extended Lie--Bargmann algebra and ensuring palindromic symplecticity. % ------------------------------------------------------------------------------ \subsubsection{Residual Inter-Cluster Potential Coupling and Multipolar Ghost Poles} \label{sub:apex_kinetic_coupling} Consider an internal node $v \in \mathcal{V}_{\mathrm{int}}(\mathcal{T})$ of a complete binary Jacobi tree $\mathcal{T}$, whose left and right children represent two disjoint clusters $L(v)$ and $R(v)$ with total masses $M_{L(v)} = \sum_{i \in L(v)} m_i$ and $M_{R(v)} = \sum_{j \in R(v)} m_j$. The relative position vector between the centers of mass of these two clusters is the Jacobi apex coordinate: \begin{equation} \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:apex_jacobi_vector} \end{equation} In Cartesian configuration space, the mutual distance between an arbitrary particle $i \in L(v)$ and an arbitrary particle $j \in R(v)$ decomposes linearly along the hierarchy: \begin{equation} \mathbf{q}_i - \mathbf{q}_j = \mathbf{r}_v + (\mathbf{q}_i - \mathbf{X}_{L(v)}) - (\mathbf{q}_j - \mathbf{X}_{R(v)}). \label{eq:interparticle_apex_decomposition} \end{equation} Although the kinetic energy is strictly diagonalized by the Jacobi momenta ($T_{\mathrm{kin}} = \frac{1}{2}\sum_u \|\mathbf{w}_u\|^2/\mu_u$ with $u \in \mathcal{V}_{\mathrm{int}}(\mathcal{T})$), the gravitational potential retains a non-decoupled inter-cluster interaction at the apex node. In Cartesian coordinates, the inverse mass metric $\mathbb{M}^{-1} = \operatorname{diag}(m_1^{-1} I_3, \dots, m_N^{-1} I_3)$ generates non-orthogonal cross-products between the gradients of mutual interparticle distances sharing a common mass point: \begin{equation} \langle \nabla_{\mathbf{q}} \|\mathbf{q}_i - \mathbf{q}_j\|, \nabla_{\mathbf{q}} \|\mathbf{q}_j - \mathbf{q}_k\| \rangle_{\mathbb{M}^{-1}} = - \frac{1}{m_j} \frac{\langle \mathbf{q}_i - \mathbf{q}_j, \mathbf{q}_j - \mathbf{q}_k \rangle}{\|\mathbf{q}_i - \mathbf{q}_j\| \, \|\mathbf{q}_j - \mathbf{q}_k\|} \ne 0 \quad (i \ne k). \label{eq:non_orthogonal_metric_crossproduct} \end{equation} As established in the classical analysis of multi-body collision manifolds and asymptotic configurations by Marchal and Saari~\cite{marchal1976general} and Waldvogel~\cite{waldvogel1972new}, the relative geometry of interacting clusters cannot be fully decoupled by linear coordinate changes alone. To separate the macroscopic Keplerian motion of the two clusters from their internal high-frequency dynamics, standard hierarchical approaches attempt to subtract the point-mass barycentric interaction $- G M_{L(v)} M_{R(v)} / \|\mathbf{r}_v\|$ from the cross-cluster Newtonian potential: \begin{equation} V_v(\mathbf{q}) = - \sum_{i \in L(v)} \sum_{j \in R(v)} \frac{G m_i m_j}{\|\mathbf{q}_i - \mathbf{q}_j\|}. \label{eq:cross_cluster_potential_def} \end{equation} However, the naive residual difference, \begin{equation} V_{\mathrm{res}, v}(\mathbf{q}) = V_v(\mathbf{q}) - \left( - \frac{G M_{L(v)} M_{R(v)}}{\|\mathbf{r}_v\|} \right) = - \sum_{i \in L(v)} \sum_{j \in R(v)} \frac{G m_i m_j}{\|\mathbf{q}_i - \mathbf{q}_j\|} + \frac{G M_{L(v)} M_{R(v)}}{\|\mathbf{r}_v\|}, \label{eq:naive_residual_potential} \end{equation} exhibits an unphysical, catastrophic singularity: when the centers of mass of the two clusters coincide ($\|\mathbf{r}_v\| \to 0$) while all internal interparticle separations remain strictly positive ($\rho_{L(v)} > 0$ and $\rho_{R(v)} > 0$), the added term $+ G M_{L(v)} M_{R(v)} / \|\mathbf{r}_v\|$ diverges as an artificial \emph{multipolar ghost pole} ($\sim \|\mathbf{r}_v\|^{-1}$). In conventional approaches, this divergence forces the introduction of an arbitrary distance threshold $d_0 > 0$, destroying scale invariance and introducing secular energy drifts. % ------------------------------------------------------------------------------ \subsubsection{The Smooth Scale-Adapted Screening Function $\Theta(u_v)$} \label{sub:screening_function_def} To neutralize the ghost pole without softening the gravitational law ($\varepsilon = 0$ strict), we construct a scale-invariant screening mechanism grounded in the geometry of the Fulton--MacPherson configuration manifold $\mathbf{Q}[N]$. From an architectural perspective, this screening function $\Theta(u_v)$ acts as an operational bridge connecting the continuous configuration geometry of Section~\ref{ch:ch04_fulton_macpherson_splitting} to the discrete symplectic numerical propagators developed in Section~\ref{ch:ch05_higher_order_symplectic}: it guarantees that the residual interaction remains uniformly Lipschitz continuous on cotangent fibers, providing the exact analytical regularity required by the Picard--Lindelöf theorem and formal backward error analysis to prevent algorithmic instability during discrete operator kicks. For an elementary binary leaf pair ($|L(v)| = |R(v)| = 1$), the cross-cluster potential $V_v(\mathbf{q}) = - G m_i m_j / \|\mathbf{q}_i - \mathbf{q}_j\|$ is already pure Keplerian with internal hyper-radii $\rho_{\{i\}} \equiv 0$ and $\rho_{\{j\}} \equiv 0$, so the residual potential vanishes identically ($V_{\mathrm{res}, v} \equiv 0$) and requires no screening. Accordingly, we restrict the apex screening to the set of composite internal nodes: \begin{equation} \mathcal{V}_{\mathrm{comp}}(\mathcal{T}) = \{ v \in \mathcal{V}_{\mathrm{int}}(\mathcal{T}) \mid |L(v)| \ge 2 \text{ or } |R(v)| \ge 2 \}. \label{eq:composite_nodes_def} \end{equation} For each $v \in \mathcal{V}_{\mathrm{comp}}(\mathcal{T})$, let $\bar{\rho}_{\mathrm{int}}(v) \coloneqq \bar{\rho}_{L(v)} + \bar{\rho}_{R(v)} > 0$ denote the combined mass-normalized internal hyper-radius of the child clusters (where $\bar{\rho}_C \coloneqq \rho_C / \sqrt{M_C} \in [L]$). We define the dimensionless scale ratio coordinate: \begin{equation} u_v = \frac{\|\mathbf{r}_v\|}{\bar{\rho}_{\mathrm{int}}(v)} = \frac{\|\mathbf{r}_v\|}{\bar{\rho}_{L(v)} + \bar{\rho}_{R(v)}} \in [0, \infty). \label{eq:dimensionless_scale_ratio} \end{equation} The ratio $u_v$ monitors the relative hierarchy: when $u_v \to 0$, $\|\mathbf{r}_v\| \to 0$ corresponds to the internal transverse locus where cluster barycenters coincide in $\operatorname{Int}(\mathbf{Q}[N])$; conversely, when $u_v \to \infty$ (i.e., $u_v^{-1} \to 0$), the two clusters are asymptotically separated, reaching the boundary divisor $E_{L(v)} \cap E_{R(v)} \subset \partial \mathbf{Q}[N]$. In particular, in the point-particle limit where both child clusters shrink to point masses ($\bar{\rho}_{\mathrm{int}}(v) \to 0$ with $\|\mathbf{r}_v\| > 0$), we have $u_v \to \infty$, identically recovering the unscreened point-mass Keplerian attraction with $\Theta(u_v) = 1$. We introduce a smooth, monotonic scale-adapted screening function $\Theta : [0, \infty) \to [0, 1]$ satisfying the asymptotic partition: \begin{equation} \Theta(u) = \begin{cases} 0 & \text{for } u \le 1, \\ \sigma(u) & \text{for } 1 < u < 2, \\ 1 & \text{for } u \ge 2, \end{cases} \label{eq:apex_screening_function} \end{equation} where $\sigma : [1, 2] \to [0, 1]$ is a smooth, monotonic transition function with vanishing derivatives of all required orders at $u=1$ and $u=2$. The mathematical specification of $\sigma(u)$ reflects a deliberate architectural duality balancing theoretical rigor with practical computational efficiency: \begin{enumerate} \item \emph{Canonical Theoretical Standard ($\mathcal{C}^\infty$ Cauchy Mollifier)}: For high-order symplectic Baker--Campbell--Hausdorff expansions (Section~\ref{ch:ch05_higher_order_symplectic}), $\sigma(u)$ is defined as the normalized Cauchy mollifier. Letting $s = u - 1 \in [0, 1]$, it admits the exact closed-form evaluation: \begin{equation} \sigma_{\mathrm{Cauchy}}(s) = \frac{\exp\left(-\frac{1}{s}\right)}{\exp\left(-\frac{1}{s}\right) + \exp\left(-\frac{1}{1-s}\right)} \quad \text{for } s \in (0, 1), \label{eq:cauchy_mollifier_closed_form} \end{equation} with continuous boundary limits $\sigma_{\mathrm{Cauchy}}(0) = 0$ and $\sigma_{\mathrm{Cauchy}}(1) = 1$. This closed form is strictly $\mathcal{C}^\infty([0, 1])$ with vanishing derivatives of all orders at the boundaries ($\sigma^{(k)}(0) = \sigma^{(k)}(1) = 0$ for all $k \ge 1$), circumventing the computational burden of numerical quadratures. Its Gevrey-2 regularity guarantees that formal modified Hamiltonians $\widetilde{H}(h)$ exist asymptotically to all orders without truncation, enabling sub-exponential shadow energy conservation for high-order integrators. \item \emph{Order-Adapted Computational Proxies ($\mathcal{C}^{2k}$ Centered Hermite Splines)}: In high-performance scientific computing where transcendentals incur non-negligible cost, $\sigma(u)$ is algebraically realized via a family of odd-degree Hermite polynomial splines centered in $\xi = s - 0.5 \in [-0.5, 0.5]$ to prevent catastrophic floating-point subtractive cancellation: \begin{itemize}[noitemsep, topsep=2pt] \item $\mathcal{C}^2$ quintic spline (degree 5, $\kappa \sim 1$): for 2nd-order Strang splitting $\mathcal{S}_2$; \item $\mathcal{C}^4$ nonic spline (degree 9): $s^5 (126 - 420s + 540s^2 - 315s^3 + 70s^4)$, for 4th-order compositions ($\mathrm{BM}_4$, Yoshida $\mathcal{S}_4$); \item $\mathcal{C}^6$ tridecic spline (degree 13): with vanishing derivatives $1 \dots 6$ at boundaries, matched to 6th-order schemes ($\mathrm{BM}_6$, $\mathrm{PBM}_4$); \item $\mathcal{C}^8$ heptadecic spline (degree 17): derived from $\sigma_8'(s) = 218790 (s(1-s))^8$, matched to 8th-order effective schemes ($\mathrm{PBM}_6$). \end{itemize} Each spline degree is calibrated to match the order of the symplectic integrator, ensuring that the truncation error of the screening boundary does not degrade the asymptotic convergence slope. \end{enumerate} \begin{defbox}[Screened Hierarchical Apex Potential]{def:screened_apex_potential} For each internal composite node $v \in \mathcal{V}_{\mathrm{comp}}(\mathcal{T})$, the \emph{screened apex potential} is defined by \begin{equation} W_{\mathrm{apex}, v}^{\mathrm{screen}}(\mathbf{q}) = V_v(\mathbf{q}) + \Theta(u_v) \frac{G M_{L(v)} M_{R(v)}}{\|\mathbf{r}_v\|}. \label{eq:screened_apex_node_def} \end{equation} When the configuration contains compact sub-clusters promoted to synchronous supernodes (Section~\ref{sec:synchronous_supernodes}), let $\mathcal{S}_{\mathrm{act}}$ denote the collection of disjoint sub-clusters $S \subset \{1, \dots, N\}$ satisfying the compactness condition $\rho_S \le \delta_{\mathrm{cluster}}$ (whose intrinsic, scale-invariant dynamical determination $\delta_{\mathrm{cluster}}^{\mathrm{nat}}$ is established in Section~\ref{sub:intrinsic_compactness_threshold}). The set of free composite nodes is defined by \begin{equation} \mathcal{V}_{\mathrm{comp}}^{\mathrm{free}}(\mathcal{T}) = \mathcal{V}_{\mathrm{comp}}(\mathcal{T}) \setminus \bigcup_{S \in \mathcal{S}_{\mathrm{act}}} \mathcal{V}_{\mathrm{int}}(\mathcal{T}_S). \label{eq:free_composite_nodes_def} \end{equation} The total hierarchical screened apex potential across the tree is the sum over free composite nodes: \begin{equation} W_{\mathrm{apex}}^{\mathrm{screen}}(\mathbf{q}) = \sum_{v \in \mathcal{V}_{\mathrm{comp}}^{\mathrm{free}}(\mathcal{T})} W_{\mathrm{apex}, v}^{\mathrm{screen}}(\mathbf{q}). \label{eq:total_screened_apex_def} \end{equation} \end{defbox} The screened apex potential enables an exact five-term decomposition of the total physical Hamiltonian of the $N$-body system: \begin{equation} H = T_{\mathrm{kin}} + W_{\mathrm{apex}}^{\mathrm{screen}}(\mathbf{q}) + V_{\mathrm{kepler}}^{\mathrm{screened}}(\mathbf{q}) + V_{\mathrm{leaves}}^{\mathrm{free}}(\mathbf{q}) + \sum_{S \in \mathcal{S}_{\mathrm{act}}} H_S^{\mathrm{super}}, \label{eq:total_hamiltonian_exact_decomposition} \end{equation} where $V_{\mathrm{kepler}}^{\mathrm{screened}}(\mathbf{q}) = - \sum_{v \in \mathcal{V}_{\mathrm{comp}}^{\mathrm{free}}(\mathcal{T})} \Theta(u_v) \frac{G M_{L(v)} M_{R(v)}}{\|\mathbf{r}_v\|}$ is the screened macroscopic Keplerian interaction, $V_{\mathrm{leaves}}^{\mathrm{free}}(\mathbf{q})$ contains the uncoupled elementary leaf-pair potentials outside active supernodes, and $H_S^{\mathrm{super}}$ governs the internal multi-body dynamics of each synchronous supernode. When $u_v \le 1$, $\Theta(u_v) = 0$, completely extinguishing the Keplerian subtraction and leaving the exact, smooth physical potential $V_v(\mathbf{q})$. When $u_v \ge 2$, $\Theta(u_v) = 1$, restoring the full multipolar subtraction where the interaction decays rapidly. In the intermediate zone $1 < u_v < 2$, $\Theta'(u_v)$ is compactly supported, ensuring smooth, bounded transitions (see Figure~\ref{fig:ch04_screened_apex_regimes}). \begin{figure}[htbp] \centering \resizebox{\textwidth}{!}{% \begin{tikzpicture}[>=stealth, xscale=4.2, yscale=3.2, every node/.style={font=\small}] % Background shaded zones for the 3 regimes \fill[PythonGreen!7] (0, 0) rectangle (1.0, 1.58); \fill[NavyBlue!7] (1.0, 0) rectangle (2.0, 1.58); \fill[AcademicPurple!6] (2.0, 0) rectangle (3.5, 1.58); % Regime vertical dividers & y=1 asymptote \draw[dashed, thick, DarkSlate!70] (1.0, 0) -- (1.0, 1.58); \draw[dashed, thick, DarkSlate!70] (2.0, 0) -- (2.0, 1.58); \draw[dotted, thick, DarkSlate!60] (0, 1.0) -- (3.5, 1.0); % Axes \draw[->, thick, DarkSlate] (0, 0) -- (3.62, 0) node[right=2pt, font=\scriptsize\bfseries, text=DarkSlate] {$u_v = \dfrac{\|\mathbf{r}_v\|}{\bar{\rho}_{L(v)} + \bar{\rho}_{R(v)}}$}; \draw[->, thick, DarkSlate] (0, 0) -- (0, 1.68) node[above=2pt, font=\scriptsize\bfseries, text=DarkSlate] {Normalized Profile / Magnitude}; % Ticks on axes \draw[thick, DarkSlate] (0, 0.025) -- (0, -0.025) node[below=2pt, font=\scriptsize] {$0$}; \draw[thick, DarkSlate] (1.0, 0.025) -- (1.0, -0.025) node[below=2pt, font=\footnotesize\bfseries] {$1$}; \draw[thick, DarkSlate] (1.5, 0.025) -- (1.5, -0.025) node[below=2pt, font=\scriptsize] {$1.5$ ($\xi=0$)}; \draw[thick, DarkSlate] (2.0, 0.025) -- (2.0, -0.025) node[below=2pt, font=\footnotesize\bfseries] {$2$}; \draw[thick, DarkSlate] (3.0, 0.025) -- (3.0, -0.025) node[below=2pt, font=\scriptsize] {$3$}; \draw[thick, DarkSlate] (0.025, 1.0) -- (-0.025, 1.0) node[left=2pt, font=\footnotesize\bfseries] {$1$}; \draw[thick, DarkSlate] (0.025, 0.5) -- (-0.025, 0.5) node[left=2pt, font=\scriptsize] {$0.5$}; % Curve 1: Unscreened Ghost Pole +G M_L M_R / ||r_v|| (diverging as u_v -> 0) \draw[very thick, dashed, AcademicPurple, domain=0.29:1.65, samples=80, smooth] plot (\x, {0.35/\x}); \node[font=\scriptsize\bfseries, text=AcademicPurple, align=center] at (0.64, 1.02) {Ghost pole\\[1pt]$+G M_L M_R / \|\mathbf{r}_v\|$}; % Curve 2: Screening function \Theta(u_v) (0 on [0,1], C^\infty / C^{2k} S-curve on [1,2], 1 on [2, 3.5]) \draw[ultra thick, NavyBlue, line join=round] (0, 0) -- (1.0, 0) -- plot[domain=1.0:2.0, samples=100, smooth] (\x, {(\x - 1)^3 * (10 - 15*(\x - 1) + 6*(\x - 1)^2)}) -- (3.5, 1.0); \node[font=\scriptsize\bfseries, text=NavyBlue, fill=white, draw=BorderGray, inner sep=2pt, rounded corners=2pt] at (1.50, 0.86) {$\Theta(u_v) = \sigma(u_v)$}; % Curve 3: Screened Apex Potential magnitude |W_apex^screen| % Single unified C^\infty / C^{2k} analytical profile: (1 - \Theta(u_v)) V_mono(u_v) + V_quad(u_v) % Guaranteed identical value, slope, and curvature across u_v = 1 and u_v = 2 \draw[very thick, PythonGreen, line join=round, smooth] plot[domain=0.0:1.0, samples=50] (\x, {0.14 / (1.0 + 0.15*\x*\x) + 1.65 / ((\x*\x + 3.2)^1.5)}) -- plot[domain=1.0:2.0, samples=100] (\x, {(1.0 - ((\x - 1)^3 * (10 - 15*(\x - 1) + 6*(\x - 1)^2))) * (0.14 / (1.0 + 0.15*\x*\x)) + 1.65 / ((\x*\x + 3.2)^1.5)}) -- plot[domain=2.0:3.5, samples=60] (\x, {1.65 / ((\x*\x + 3.2)^1.5)}); \node[font=\scriptsize\bfseries, text=PythonGreen, align=left] at (2.78, 0.44) {Screened apex $|W_{\mathrm{apex},v}^{\mathrm{screen}}|$\\[1pt]$\sim \dfrac{G|\mathcal{Q}_v(\mathbf{s})|}{\|\mathbf{r}_v\|^3} = \mathcal{O}(u_v^{-3})$}; \node[font=\scriptsize\bfseries, text=PythonGreen, align=center] at (0.50, 0.16) {$\mathcal{C}^{1,1}$ core: $W_{\mathrm{apex},v}^{\mathrm{screen}} \equiv V_v$}; % Regime Headers at top \node[font=\scriptsize\bfseries, text=PythonGreen, align=center] at (0.50, 1.43) {Regime 1 ($u_v \le 1$)\\$\Theta(u_v) \equiv 0$ (No Ghost Pole)}; \node[font=\scriptsize\bfseries, text=NavyBlue, align=center] at (1.50, 1.43) {Regime 2 ($1 < u_v < 2$)\\Compact Annulus $\mathcal{A}_{[1,2]}$}; \node[font=\scriptsize\bfseries, text=AcademicPurple, align=center] at (2.75, 1.43) {Regime 3 ($u_v \ge 2$)\\$\Theta(u_v) \equiv 1$ (Quadrupolar Tail)}; \end{tikzpicture}% } \caption{Scale-adapted apex screening function $\Theta(u_v)$ and universal $\mathcal{C}^{1,1}$ regularity across the three scale regimes of Definition~\ref{def:screened_apex_potential} and Theorem~\ref{thm:universal_c11_apex}. \textbf{Regime~1 ($u_v \le 1$):} $\Theta(u_v) \equiv 0$ completely extinguishes the unphysical multipolar ghost pole $+G M_L M_R/\|\mathbf{r}_v\|$ (dashed purple curve) when cluster barycenters overlap ($\|\mathbf{r}_v\| \to 0$). \textbf{Regime~2 ($1 < u_v < 2$):} The transition function $\sigma(u_v)$ (solid blue curve, realized via the $\mathcal{C}^\infty$ Gevrey-2 Cauchy mollifier or centered $\mathcal{C}^{2k}$ Hermite splines around $\xi = u_v - 1.5 = 0$) interpolates smoothly across the compact projectivized annulus $\mathcal{A}_{[1,2]}$. \textbf{Regime~3 ($u_v \ge 2$):} $\Theta(u_v) \equiv 1$ restores the full Keplerian subtraction, where exact monopole and barycentric dipole cancellations leave a rapidly decaying trace-free quadrupolar interaction $W_{\mathrm{apex},v}^{\mathrm{screen}} = -G\mathcal{Q}_v(\mathbf{s})/\|\mathbf{r}_v\|^3 + \mathcal{O}(\|\mathbf{r}_v\|^{-4})$ (solid green curve).} \label{fig:ch04_screened_apex_regimes} \end{figure} % ------------------------------------------------------------------------------ \subsubsection{Universal $\mathcal{C}^{1,1}$ Lipschitz Regularity without Distance Thresholds} \label{sub:universal_c11_regularity} The foundational analytical property of the screened apex potential is its uniform $\mathcal{C}^{1,1}$ regularity (i.e., belonging to the Sobolev--Lipschitz space $\mathcal{W}^{2,\infty}$) on each adapted chart of the configuration manifold: \begin{thmbox}[Universal $\mathcal{C}^{1,1}$ Lipschitz Regularity of the Screened Apex Potential]{thm:universal_c11_apex} Let $\mathcal{T} \in \mathrm{Trees}(N)$ be a complete binary Jacobi tree, and let $U_{\mathcal{T}} \subset \mathbf{Q}[N]$ be its adapted atlas chart. Endow the relative configuration space $M$ with the Riemannian mass metric induced by $\mathbb{M}$ (and its cotangent bundle $T^*M$ with the dual metric $\mathbb{M}^{-1}$). Then, on the domain of validity of the adapted chart $U_{\mathcal{T}}$ and without requiring any artificial distance threshold ($d_0 = 0$ strictly), there exist finite strictly positive constants $M_{\mathrm{apex}} > 0$ and $L_{\mathrm{apex}} > 0$ such that for all configurations $\mathbf{q} \in U_{\mathcal{T}}$ and all relative Jacobi coordinates $\mathbf{r} = (\mathbf{r}_u)_{u \in \mathcal{V}_{\mathrm{int}}(\mathcal{T})}$: \begin{equation} \|\nabla_{\mathbf{r}} W_{\mathrm{apex}}^{\mathrm{screen}}(\mathbf{q})\|_{\mathbb{M}^{-1}} \le M_{\mathrm{apex}} < \infty, \qquad \|\operatorname{Hess}_{\mathbf{r}} W_{\mathrm{apex}}^{\mathrm{screen}}(\mathbf{q})\|_{\mathbb{M}^{-1}} \le L_{\mathrm{apex}} < \infty. \label{eq:universal_lipschitz_bounds} \end{equation} Consequently, $W_{\mathrm{apex}}^{\mathrm{screen}}$ is uniformly of class $\mathcal{C}^{1,1}(U_{\mathcal{T}}) = \mathcal{W}^{2,\infty}(U_{\mathcal{T}})$, so its gradient force field $\nabla_{\mathbf{r}} W_{\mathrm{apex}}^{\mathrm{screen}}$ is globally Lipschitz continuous ($\mathcal{C}^{0,1}$) on $U_{\mathcal{T}}$. \end{thmbox} \leanrefright{apex_universal_lipschitz_theorems}{apex_universal_lipschitz_bound}{01_ApexUniversalLipschitzBound_Theorems.lean} \pythonrefright{py_kernels}{potential/{screened_apex, optimized_screened_apex, screening}.py}{Float64 / Float128} \noindent The accompanying formal certification badge denotes the verification of the \emph{Constructive 1-Jet Algebraic Kernel} (formalized in Lean~4 over an arbitrary commutative ring), certifying the constructive algebra of the Lipschitz bounds, with the global topological compactness and the smooth transitions established by the differential proof below. \begin{rembox}[Epistemic Scope: $\mathcal{C}^{1,1}$ Regularity and Atlas Chart Domain $U_{\mathcal{T}}$] \label{rem:c11_regularity_and_chart_domain} Two essential structural remarks must be highlighted: \begin{enumerate} \item \emph{Functional Class $\mathcal{C}^{1,1}$ vs $\mathcal{C}^\infty$ (Algorithmic Interface)}: Qualifying $W_{\mathrm{apex}}^{\mathrm{screen}}$ as $\mathcal{C}^{1,1}$ designates the minimal regularity guaranteed by the algebraic $\mathcal{C}^2$ quintic Hermite spline proxy. Under this minimal Sobolev--Lipschitz regularity ($\mathcal{W}^{2,\infty}$), the Hamiltonian vector field $X_{W_{\mathrm{apex}}^{\mathrm{screen}}}$ is Lipschitz continuous, satisfying the classical Picard--Lindelöf theorem and ensuring strict convergence of the second-order Strang splitting $\mathcal{S}_2(h)$ (Section~\ref{sec:strang_propagator}). When our canonical theoretical baseline (the $\mathcal{C}^\infty$ Cauchy mollifier) is activated, $W_{\mathrm{apex}}^{\mathrm{screen}}$ is locally $\mathcal{C}^\infty$ with uniformly bounded derivatives of all orders $\le 2$ and Gevrey-2 sub-exponential stability, fulfilling the analytical prerequisites for all-order formal backward error analysis in high-order composition schemes. \item \emph{Domain of Validity $U_{\mathcal{T}}$ and Synchronous Supernodes}: The Lipschitz bounds $M_{\mathrm{apex}}$ and $L_{\mathrm{apex}}$ are established on the adapted chart $U_{\mathcal{T}} \subset \mathbf{Q}[N]$. While $W_{\mathrm{apex}, v}^{\mathrm{screen}}(\mathbf{q})$ eliminates the multipolar ghost pole upon barycentric coincidence ($\mathbf{r}_v \to \mathbf{0}$), it does not protect against an internal collision between a particle $i \in L(v)$ and a particle $j \in R(v)$ if the hierarchy of $\mathcal{T}$ ceases to be adapted. Under such dynamics, either the distance hierarchy reverses (triggering an $\mathrm{SL}(2, \mathbb{R})$ Jacobi tree transition via the $\gamma = 0.6$ hysteresis band of Theorem~\ref{thm:atlas_transition_anti_chattering}) or the entire sub-cluster collapses into a compact $k$-body quasi-collision ($k \ge 3$), in which case the sub-tree is subsumed into a synchronous supernode governed by Sundman's unfolded local clock $\mathrm{d}t_S = \chi_S^2 \mathrm{d}\tau_S$ (Section~\ref{sec:synchronous_supernodes}). \end{enumerate} \end{rembox} \begin{proofbox}[Proof of Theorem~\ref{thm:universal_c11_apex}] It suffices to establish the bounds for each node $v \in \mathcal{V}_{\mathrm{comp}}(\mathcal{T})$ across three complementary scale regimes covering $[0, \infty)$: \textbf{Regime 1: Compact Sub-Apex Regime ($u_v \le 1$).} In this regime, $\Theta(u_v) = 0$, and Definition~\ref{def:screened_apex_potential} reduces identically to the restricted interaction $W_{\mathrm{apex}, v}^{\mathrm{screen}}(\mathbf{q}) = V_v(\mathbf{q}) = - \sum_{i \in L(v), j \in R(v)} \frac{G m_i m_j}{\|\mathbf{q}_i - \mathbf{q}_j\|}$. By definition of an adapted Jacobi chart $U_{\mathcal{T}}$, when $u_v \le 1$, interparticle collisions between $L(v)$ and $R(v)$ are either bounded away from zero by pairwise separation within the tree hierarchy or absorbed into a synchronous supernode (analyzed in Section~\ref{sec:synchronous_supernodes}). Hence, on the regular locus of $U_{\mathcal{T}}$, the pairwise distances $\|\mathbf{q}_i - \mathbf{q}_j\|$ remain uniformly bounded below away from zero, ensuring $\|\nabla V_v\| \le B_1$ and $\|\operatorname{Hess} V_v\| \le B_1'$ for finite constants $B_1, B_1' > 0$. The artificial ghost pole is completely eliminated since its coefficient $\Theta(u_v)$ is identically zero. \textbf{Regime 2: Compact Transition Annulus ($1 < u_v < 2$).} In this domain, the ratio $u_v = \|\mathbf{r}_v\| / \bar{\rho}_{\mathrm{int}}(v)$ lies in the compact interval $[1, 2]$. Consequently, $\|\mathbf{r}_v\| \ge \bar{\rho}_{\mathrm{int}}(v) > 0$, guaranteeing that cluster centers cannot coincide. On the set of free composite nodes $\mathcal{V}_{\mathrm{comp}}^{\mathrm{free}}(\mathcal{T})$, any sub-cluster satisfying $\bar{\rho}_{\mathrm{int}}(v) \le \delta_{\mathrm{cluster}}$ is subsumed into an active synchronous supernode $S \in \mathcal{S}_{\mathrm{act}}$ (Section~\ref{sec:synchronous_supernodes}); therefore, $\bar{\rho}_{\mathrm{int}}(v) \ge c_0 \delta_{\mathrm{cluster}} > 0$ holds uniformly on $\mathcal{V}_{\mathrm{comp}}^{\mathrm{free}}$, ensuring that $\bar{\rho}_{\mathrm{int}}^{-1} \le (c_0 \delta_{\mathrm{cluster}})^{-1} < \infty$ is strictly bounded away from infinity (with $\delta_{\mathrm{cluster}} = \delta_{\mathrm{cluster}}^{\mathrm{nat}}(E_0)$ determined dynamically by the total energy $E_0$ and masses via Proposition~\ref{prop:intrinsic_compactness_threshold}, eliminating any arbitrary metric regulator). Furthermore, since $\Theta(u_v) = \sigma(u_v)$ is a smooth function with compact support of its derivatives in $[1, 2]$, both $\Theta(u_v)$ and all its derivatives are uniformly bounded. We emphasize that while the Euclidean hyper-radius $\rho_C$ is non-differentiable at the origin in naive Cartesian coordinates (due to the singularity of $r \mapsto \|r\|$ at $r=0$), on the Fulton--MacPherson compactified manifold with corners $\mathbf{Q}[N]$ (or in McGehee polar blow-up coordinates $\rho_C \in [0, \infty[$ and $\mathbf{s}_C \in \mathbb{S}^{3k-4}$), the total collision locus $\Delta_C$ is replaced by the smooth boundary divisor $\{\rho_C = 0\}$. The radial coordinate $\rho_C$ is a defining boundary coordinate of the manifold with corners, and the radial vector field $\partial_{\rho_C}$ is smooth up to the boundary stratum $\rho_C = 0$ within each adapted chart $U_{\mathcal{T}}$. By the chain rule, $\nabla_{\mathbf{q}_i} u_v = \bar{\rho}_{\mathrm{int}}^{-1} \nabla_{\mathbf{q}_i} \|\mathbf{r}_v\| - \|\mathbf{r}_v\| \bar{\rho}_{\mathrm{int}}^{-2} \nabla_{\mathbf{q}_i} \bar{\rho}_{\mathrm{int}}$; since $\|\nabla_{\mathbf{q}_i} \bar{\rho}_{\mathrm{int}}\| \le \sqrt{m_i / M_C}$ uniformly by Cauchy--Schwarz on cluster hyper-radii and $\bar{\rho}_{\mathrm{int}} \ge c_0 \delta_{\mathrm{cluster}} > 0$, the spatial gradients and second derivatives of $u_v$ remain uniformly bounded on $[1, 2]$. By Weierstrass's extreme value theorem on the compact projectivized annulus $\mathcal{A}_{[1,2]} \subset S(N_{\Delta_v} M)$ (or on the compact slice in the coordinates with corners of $\mathbf{Q}[N]$), the gradient and Hessian of $W_{\mathrm{apex}, v}^{\mathrm{screen}}$ are continuous functions on a compact domain, hence bounded by finite constants $B_2, B_2' > 0$. \textbf{Regime 3: Asymptotic Multipolar Regime ($u_v \ge 2$).} In this regime, $\Theta(u_v) = 1$, and $W_{\mathrm{apex}, v}^{\mathrm{screen}}(\mathbf{q}) = V_v(\mathbf{q}) + \frac{G M_{L(v)} M_{R(v)}}{\|\mathbf{r}_v\|}$. Let $\mathbf{d}_{ij} = (\mathbf{q}_i - \mathbf{X}_{L(v)}) - (\mathbf{q}_j - \mathbf{X}_{R(v)})$ be the physical internal displacement vector, such that $\mathbf{q}_i - \mathbf{q}_j = \mathbf{r}_v + \mathbf{d}_{ij}$. For any particle $i \in C$, the distance to the cluster barycenter is bounded by $\|\mathbf{q}_i - \mathbf{X}_C\| \le \sqrt{M_C / m_i} \, \bar{\rho}_C \le \kappa(C) \bar{\rho}_C$, where $\kappa(C) = \max_{k \in C} \sqrt{M_C / m_k} \ge 1$ is the finite mass disparity factor of the cluster, and setting $\kappa_v = \max(\kappa(L(v)), \kappa(R(v)))$ yields $\|\mathbf{d}_{ij}\| \le \kappa_v (\bar{\rho}_L + \bar{\rho}_R) = \kappa_v \bar{\rho}_{\mathrm{int}}(v)$. On the compact intermediate annulus $2 \le u_v \le 2\kappa_v$, the adapted chart condition on $U_{\mathcal{T}}$ maintains pairwise separations uniformly bounded below away from zero, so the gradient and Hessian are bounded by projective compactness on $\mathcal{A}_{[2, 2\kappa_v]}$ exactly as in Regime~2. In the far-field multipolar domain $u_v \ge 2\kappa_v$, we have strictly $\|\mathbf{d}_{ij}\| \le \kappa_v \bar{\rho}_{\mathrm{int}}(v) \le \frac{1}{2} \|\mathbf{r}_v\|$. Since $\kappa_v < \infty$ is a fixed constant determined exclusively by the mass spectrum $\{m_i\}$, expanding the pairwise distance $\|\mathbf{q}_i - \mathbf{q}_j\|^{-1}$ in multipole Legendre--Taylor series on $u_v \ge 2\kappa_v$ yields: \begin{equation} \frac{1}{\|\mathbf{q}_i - \mathbf{q}_j\|} = \frac{1}{\|\mathbf{r}_v + \mathbf{d}_{ij}\|} = \frac{1}{\|\mathbf{r}_v\|} - \frac{\langle \mathbf{r}_v, \mathbf{d}_{ij} \rangle}{\|\mathbf{r}_v\|^3} + \frac{3 \langle \mathbf{r}_v, \mathbf{d}_{ij} \rangle^2 - \|\mathbf{r}_v\|^2 \|\mathbf{d}_{ij}\|^2}{2 \|\mathbf{r}_v\|^5} + \mathcal{O}\left( \frac{\kappa_v^3 \bar{\rho}_{\mathrm{int}}^3}{\|\mathbf{r}_v\|^4} \right). \label{eq:multipole_taylor_expansion} \end{equation} Summing over all $i \in L(v)$ and $j \in R(v)$ weighted by $G m_i m_j$: \begin{enumerate} \item \emph{Monopole Cancellation}: $\sum_{i \in L(v), j \in R(v)} \frac{G m_i m_j}{\|\mathbf{r}_v\|} = \frac{G M_{L(v)} M_{R(v)}}{\|\mathbf{r}_v\|}$, exactly cancelling the Keplerian subtraction $+ \frac{G M_{L(v)} M_{R(v)}}{\|\mathbf{r}_v\|}$. \item \emph{Dipole Cancellation}: By definition of the barycenters $\mathbf{X}_L$ and $\mathbf{X}_R$, $\sum_{i \in L(v)} m_i (\mathbf{q}_i - \mathbf{X}_{L(v)}) = \mathbf{0}$ and $\sum_{j \in R(v)} m_j (\mathbf{q}_j - \mathbf{X}_{R(v)}) = \mathbf{0}$. Therefore, the linear dipole term $\sum_{i,j} m_i m_j \mathbf{d}_{ij} = M_R \sum_i m_i (\mathbf{q}_i - \mathbf{X}_L) - M_L \sum_j m_j (\mathbf{q}_j - \mathbf{X}_R) = \mathbf{0}$ vanishes identically! \item \emph{Quadrupole Leading Order}: The leading non-vanishing term is purely quadrupolar: \begin{equation} W_{\mathrm{apex}, v}^{\mathrm{screen}}(\mathbf{q}) = - \frac{G}{2 \|\mathbf{r}_v\|^5} \sum_{i \in L(v), j \in R(v)} m_i m_j \left( 3 \langle \mathbf{r}_v, \mathbf{d}_{ij} \rangle^2 - \|\mathbf{r}_v\|^2 \|\mathbf{d}_{ij}\|^2 \right) + \mathcal{O}\left( \frac{G M_{L(v)} M_{R(v)} \bar{\rho}_{\mathrm{int}}^3}{\|\mathbf{r}_v\|^4} \right), \label{eq:quadrupole_leading_order} \end{equation} which contracts to: \begin{equation} W_{\mathrm{apex}, v}^{\mathrm{screen}}(\mathbf{q}) = - \frac{G}{\|\mathbf{r}_v\|^3} \mathcal{Q}_v(\mathbf{s}) + \mathcal{O}\left( \frac{G M_{L(v)} M_{R(v)} \bar{\rho}_{\mathrm{int}}^3}{\|\mathbf{r}_v\|^4} \right), \label{eq:quadrupole_contracted_potential} \end{equation} with unit direction vector $\mathbf{s} = \mathbf{r}_v / \|\mathbf{r}_v\|$ and trace-free quadrupole interaction tensor contraction: \begin{equation} \mathcal{Q}_v(\mathbf{s}) = \frac{1}{2} \left[ M_{R(v)} (\mathbf{s}^T \mathbf{Q}_{L(v)} \mathbf{s}) + M_{L(v)} (\mathbf{s}^T \mathbf{Q}_{R(v)} \mathbf{s}) \right], \label{eq:quadrupole_tensor_contraction} \end{equation} where $\mathbf{Q}_{L(v)} = \sum_{i \in L(v)} m_i \left( 3 (\mathbf{q}_i - \mathbf{X}_{L(v)})(\mathbf{q}_i - \mathbf{X}_{L(v)})^T - \|\mathbf{q}_i - \mathbf{X}_{L(v)}\|^2 I_3 \right)$ and $\mathbf{Q}_{R(v)} = \sum_{j \in R(v)} m_j \left( 3 (\mathbf{q}_j - \mathbf{X}_{R(v)})(\mathbf{q}_j - \mathbf{X}_{R(v)})^T - \|\mathbf{q}_j - \mathbf{X}_{R(v)}\|^2 I_3 \right)$ are the trace-free quadrupole inertia tensors of the child clusters. \end{enumerate} Differentiating with respect to $\mathbf{r}_v$, the gradient decays asymptotically as $\|\nabla_{\mathbf{r}_v} W_{\mathrm{apex}, v}^{\mathrm{screen}}\| \sim \mathcal{O}(\|\mathbf{r}_v\|^{-4})$, and the Hessian decays as $\mathcal{O}(\|\mathbf{r}_v\|^{-5})$. Although the spatial domain $\{ u_v \ge 2 \}$ is closed but unbounded (extending to spatial infinity $\|\mathbf{r}_v\| \to \infty$), the gradient $\nabla_{\mathbf{r}_v} W_{\mathrm{apex}, v}^{\mathrm{screen}}$ and Hessian $\operatorname{Hess}_{\mathbf{r}_v} W_{\mathrm{apex}, v}^{\mathrm{screen}}$ are continuous functions on this closed domain that vanish asymptotically at infinity ($\mathcal{C}_0$ regularity) due to their fast multipolar decay. Consequently, both fields attain their finite supremum on $\{ u_v \ge 2 \}$, bounding them by finite positive constants $B_3, B_3' < \infty$ without requiring spatial compactness. Summing across the three regimes and all composite nodes, we obtain $M_{\mathrm{apex}} = \sum_v (B_{1,v} + B_{2,v} + B_{3,v}) < \infty$ and $L_{\mathrm{apex}} = \sum_v (B_{1,v}' + B_{2,v}' + B_{3,v}') < \infty$, completing the proof (see Figure~\ref{fig:ch04_quadrupolar_screening_annulus}). \end{proofbox} \begin{figure}[htbp] \centering \resizebox{0.78\textwidth}{!}{% \begin{tikzpicture}[>=stealth, font=\small] % ========================================================================== % Concentric Screening Zones around Binary Barycenter X_C = (0, 0) % Exact 2:1 scale ratio: R_C = 1.35 cm, 2 R_C = 2.70 cm % ========================================================================== % Shaded transition annulus [R_C, 2 R_C] (1 < u_v < 2, \mathcal{A}_{[1,2]}) \fill[black!9, even odd rule] (0, 0) circle (2.70) (0, 0) circle (1.35); % Boundary circles u_v = 1 (r = R_C) and u_v = 2 (r = 2 R_C) \draw[dashed, line width=0.75pt, black!75] (0, 0) circle (1.35); \draw[dashed, line width=0.75pt, black!75] (0, 0) circle (2.70); % Radial scale ticks and dimension arrows (R_C and 2 R_C) along negative/vertical directions \draw[<->, line width=0.55pt, black!65] (0, 0) -- (-1.35, 0) node[midway, above=1pt, font=\scriptsize] {$R_C$}; \draw[<->, line width=0.55pt, black!65] (0, 0) -- (0, -2.70) node[pos=0.76, right=2pt, font=\scriptsize] {$2 R_C$}; % ========================================================================== % Internal Binary Cluster C = {i, j} inside the core disk u_v <= 1 % ========================================================================== % Faint internal Keplerian orbit of the binary around X_C \draw[dotted, line width=0.6pt, black!55, rotate=58] (0, 0) ellipse (0.88 and 0.48); % Binary particles q_i and q_j (asymmetric masses around barycenter X_C = (0,0)) \coordinate (XC) at (0, 0); \coordinate (qi) at (0.42, 0.67); \coordinate (qj) at (-0.42, -0.67); % Internal leaf segment r_C = q_i - q_j \draw[line width=1.1pt, black] (qj) -- (qi) node[pos=0.28, left=2pt, font=\scriptsize] {$\mathbf{r}_C$}; % Barycenter X_C (hollow circle) and particles q_i, q_j (solid disks) \filldraw[fill=white, draw=black, line width=0.85pt] (XC) circle (2.1pt) node[below right=1pt, font=\scriptsize] {$\mathbf{X}_C$}; \filldraw[black] (qi) circle (2.4pt) node[above left=1pt, font=\scriptsize] {$\mathbf{q}_i$}; \filldraw[black] (qj) circle (2.4pt) node[below left=1pt, font=\scriptsize] {$\mathbf{q}_j$}; % ========================================================================== % External Perturber q_k in the Quadrupolar Zone u_v >= 2 % ========================================================================== \coordinate (qk) at (4.45, 0.85); % Exact pairwise interparticle lines q_k - q_i and q_k - q_j (multipolar rays) \draw[densely dotted, line width=0.75pt, black!70] (qi) -- (qk) node[pos=0.56, above=2pt, sloped, font=\scriptsize, fill=white, inner sep=1pt] {$\mathbf{q}_k - \mathbf{q}_i$}; \draw[densely dotted, line width=0.75pt, black!70] (qj) -- (qk) node[pos=0.58, below=2pt, sloped, font=\scriptsize, fill=white, inner sep=1pt] {$\mathbf{q}_k - \mathbf{q}_j$}; % Jacobi apex vector r_v = q_k - X_C \draw[line width=1.15pt, black] (XC) -- (qk) node[pos=0.75, above=1.5pt, sloped, font=\scriptsize, fill=white, inner sep=1pt] {$\mathbf{r}_v$}; % External body q_k \filldraw[black] (qk) circle (2.6pt) node[right=3pt, font=\scriptsize] {$\mathbf{q}_k$}; % ========================================================================== % Minimalist Mathematical Regime Symbols in the 3 Zones % ========================================================================== \node[font=\scriptsize] at (-0.58, 0.56) {$\Theta = 0$}; \node[font=\scriptsize] at (-1.35, 1.48) {$\mathcal{A}_{[1,2]}$}; \node[font=\scriptsize] at (0.0, 2.02) {$0 < \Theta(u_v) < 1$}; \node[font=\scriptsize] at (3.65, -1.55) {$\Theta = 1$}; \node[font=\scriptsize] at (3.65, -2.05) {$W_{\mathrm{apex},v}^{\mathrm{screen}} \sim -\dfrac{G\mathcal{Q}_v(\mathbf{s})}{\|\mathbf{r}_v\|^3}$}; \end{tikzpicture}% } \caption{Spatial geometry of the scale-adapted apex screening around a binary cluster $C = \{i, j\}$ with barycenter $\mathbf{X}_C$, internal leaf separation $\mathbf{r}_C = \mathbf{q}_i - \mathbf{q}_j$, and characteristic scale $R_C = \bar{\rho}_{\mathrm{int}}(v)$. Inside the core disk $\|\mathbf{r}_v\| \le R_C$ ($u_v \le 1$), $\Theta(u_v) \equiv 0$ extinguishes the barycentric ghost pole $+G M_C m_k / \|\mathbf{r}_v\|$. Across the shaded projectivized annulus $\mathcal{A}_{[1,2]} = \{R_C < \|\mathbf{r}_v\| < 2R_C\}$ ($1 < u_v < 2$), $\Theta(u_v) = \sigma(u_v)$ interpolates smoothly. In the exterior domain $\|\mathbf{r}_v\| \ge 2R_C$ ($u_v \ge 2$, $\Theta \equiv 1$), subtracting the barycentric monopole along $\mathbf{r}_v = \mathbf{q}_k - \mathbf{X}_C$ from the exact pairwise interactions along $\mathbf{q}_k - \mathbf{q}_i$ and $\mathbf{q}_k - \mathbf{q}_j$ cancels both monopole and dipole terms, leaving the trace-free quadrupolar tide $-G\mathcal{Q}_v(\mathbf{s})/\|\mathbf{r}_v\|^3$.} \label{fig:ch04_quadrupolar_screening_annulus} \end{figure} % ------------------------------------------------------------------------------ \subsubsection{Invariance of the Lie--Bargmann Algebra and Palindromic Symplecticity} \label{sub:lie_bargmann_and_palindromy} The introduction of the scale-adapted screening factor $\Theta(u_v)$ must not compromise the fundamental conservation laws of Newtonian mechanics. We now demonstrate that the screened apex flow rigorously preserves the $10$ classical generators of the Galilean group and preserves extended symplecticity. \begin{propbox}[Exact Invariance of the Extended Lie--Bargmann Algebra]{prop:lie_bargmann_preservation} Let $H = T_{\mathrm{kin}} + W_{\mathrm{apex}}^{\mathrm{screen}} + V_{\mathrm{kepler}}^{\mathrm{screened}} + V_{\mathrm{leaves}}^{\mathrm{free}} + \sum_{S \in \mathcal{S}_{\mathrm{act}}} H_S^{\mathrm{super}}$ be the Hamiltonian on the extended phase space $\mathcal{P}_{\mathrm{ext}} = T^*M \times \mathbb{R}_t \times \mathbb{R}_E$ equipped with the canonical configuration-momentum conjugate pair $(q_t, p_t) = (t, -E)$, extended Liouville $1$-form $\lambda_{\mathrm{ext}} = \mathbf{p} \cdot \mathrm{d}\mathbf{q} + p_t\,\mathrm{d}t = \mathbf{p} \cdot \mathrm{d}\mathbf{q} - E\,\mathrm{d}t$, and canonical extended $2$-form $\Omega_{\mathrm{ext}} \coloneqq -\mathrm{d}\lambda_{\mathrm{ext}} = \Omega_0 - \mathrm{d}p_t \wedge \mathrm{d}t = \Omega_0 + \mathrm{d}E \wedge \mathrm{d}t = \Omega_0 - \mathrm{d}t \wedge \mathrm{d}E$ (with $\Omega_0 = -\mathrm{d}(\mathbf{p}\cdot\mathrm{d}\mathbf{q}) = \mathrm{d}\mathbf{q} \wedge \mathrm{d}\mathbf{p}$). Then the $10$ classical generators of the Galilean group, realizing the $11$-dimensional centrally extended Lie--Bargmann algebra~\cite{bargmann1954unitary} with total mass $M_{\mathrm{tot}}$ as central charge ($\{G_\alpha, P_\beta\} = M_{\mathrm{tot}} \delta_{\alpha\beta}$ for spatial indices $\alpha, \beta \in \{1, 2, 3\}$): \begin{equation} \mathbf{P} = \sum_{i=1}^N \mathbf{p}_i, \quad \mathbf{L} = \sum_{i=1}^N \mathbf{q}_i \times \mathbf{p}_i, \quad \mathbf{G} = M_{\mathrm{tot}} \mathbf{R}_{\mathrm{cdm}} - t \mathbf{P}, \quad \mathcal{H}_{\mathrm{ext}} = H - E, \label{eq:bargmann_generators_def} \end{equation} where $\mathbf{R}_{\mathrm{cdm}} := \frac{1}{M_{\mathrm{tot}}} \sum_{i=1}^N m_i \mathbf{q}_i$ is the center of mass with total mass $M_{\mathrm{tot}} = \sum_{i=1}^N m_i$, satisfy exact discrete conservation under the screened apex time step for all spatial invariants: \begin{equation} \Delta \mathbf{P} = \mathbf{0}, \qquad \Delta \mathbf{L} = \mathbf{0}, \qquad \Delta \mathbf{G} = \mathbf{0}, \label{eq:bargmann_discrete_invariance} \end{equation} while the energy defect $\Delta \mathcal{H}_{\mathrm{ext}}$ undergoes zero secular drift, oscillating within an $\mathcal{O}(h^2)$ energy band of the shadow invariant energy surface $\widetilde{\mathcal{H}}_{\mathrm{ext}} = \mathcal{H}_{\mathrm{ext}} + \mathcal{O}(h^2)$ (and on shadow invariant tori in quasi-periodic hierarchical regimes). \end{propbox} \leanrefright{lie_bargmann_invariance_theorems}{lie_bargmann_exact_invariance}{02_LieBargmannExactInvariance_Theorems.lean} \noindent The formal certification badge establishes the algebraic closure of the Lie--Bargmann invariants and the exact Poisson bracket compensation of the Galilean boost in the Constructive 1-Jet Kernel. \begin{proofbox}[Proof of Proposition~\ref{prop:lie_bargmann_preservation}] \textbf{Linear Momentum ($\mathbf{P}$):} The screened apex potential $W_{\mathrm{apex}}^{\mathrm{screen}}$ depends exclusively on interparticle vectors $\mathbf{q}_i - \mathbf{q}_j$ and Jacobi relative vectors $\mathbf{r}_v$. Both are invariant under uniform spatial translations $\mathbf{q}_i \mapsto \mathbf{q}_i + \mathbf{a}$. By Noether's theorem, $\sum_{i=1}^N \nabla_{\mathbf{q}_i} W_{\mathrm{apex}}^{\mathrm{screen}} = \mathbf{0}$. The momentum kick $\Delta \mathbf{p}_i = - \Delta t \nabla_{\mathbf{q}_i} W_{\mathrm{apex}}^{\mathrm{screen}}$ therefore sums to zero: $\Delta \mathbf{P} = \sum \Delta \mathbf{p}_i = \mathbf{0}$. \textbf{Angular Momentum ($\mathbf{L}$):} The scale ratio $u_v = \|\mathbf{r}_v\| / (\bar{\rho}_{L(v)} + \bar{\rho}_{R(v)})$ is constructed entirely from Euclidean norms invariant under spatial rotations $\mathrm{SO}(3)$. Since each constituent term in $W_{\mathrm{apex}, v}^{\mathrm{screen}}$ is a scalar function of rotationally invariant norms and scalar products, $W_{\mathrm{apex}}^{\mathrm{screen}}(\mathcal{R} \mathbf{q}) = W_{\mathrm{apex}}^{\mathrm{screen}}(\mathbf{q})$ for all $\mathcal{R} \in \mathrm{SO}(3)$. The internal torques vanish identically: $\sum_{i=1}^N \mathbf{q}_i \times \nabla_{\mathbf{q}_i} W_{\mathrm{apex}}^{\mathrm{screen}} = \mathbf{0}$, ensuring $\Delta \mathbf{L} = \mathbf{0}$. \textbf{Galilean Boost ($\mathbf{G}$):} Under a translation drift sub-step $\Delta t$, the center of mass advances by $\mathbf{R}_{\mathrm{cdm}}(t + \Delta t) = \mathbf{R}_{\mathrm{cdm}}(t) + \frac{\Delta t}{M_{\mathrm{tot}}} \mathbf{P}$. The updated boost generator is: \begin{align} \mathbf{G}(t + \Delta t) &= M_{\mathrm{tot}} \mathbf{R}_{\mathrm{cdm}}(t + \Delta t) - (t + \Delta t) \mathbf{P} \nonumber \\ &= M_{\mathrm{tot}} \left( \mathbf{R}_{\mathrm{cdm}}(t) + \frac{\Delta t}{M_{\mathrm{tot}}} \mathbf{P} \right) - t \mathbf{P} - \Delta t \mathbf{P} \nonumber \\ &= M_{\mathrm{tot}} \mathbf{R}_{\mathrm{cdm}}(t) - t \mathbf{P} = \mathbf{G}(t). \label{eq:boost_algebraic_cancellation} \end{align} During the apex kick sub-step, positions $\mathbf{q}_i$ (hence the barycenter $\mathbf{R}_{\mathrm{cdm}}$) and time $t$ remain frozen, while total linear momentum $\mathbf{P}$ is conserved ($\Delta_{\mathrm{kick}} \mathbf{P} = \mathbf{0}$), ensuring $\Delta_{\mathrm{kick}} \mathbf{G} = \mathbf{0}$ identically. In the Poisson bracket formulation on $\mathcal{P}_{\mathrm{ext}}$, with temporal momentum $p_t = -E$ conjugate to $t$ inducing $\{t, p_t\} = \{t, -E\} = 1 \iff \{t, E\} = -1$, the boost generator satisfies: \begin{align} \{\mathbf{G}, \mathcal{H}_{\mathrm{ext}}\} &= \{M_{\mathrm{tot}} \mathbf{R}_{\mathrm{cdm}} - t \mathbf{P}, H - E\} \nonumber \\ &= \{M_{\mathrm{tot}} \mathbf{R}_{\mathrm{cdm}}, H\} - \{t \mathbf{P}, -E\} - \{M_{\mathrm{tot}} \mathbf{R}_{\mathrm{cdm}}, E\} - \{t \mathbf{P}, H\} \nonumber \\ &= \mathbf{P} - \mathbf{P} - \mathbf{0} - \mathbf{0} = \mathbf{0}, \label{eq:boost_poisson_full_cancellation} \end{align} where $\{M_{\mathrm{tot}} \mathbf{R}_{\mathrm{cdm}}, E\} = \mathbf{0}$ because the barycentric position is independent of the conjugate energy $E$, and $\{t \mathbf{P}, H\} = t \{\mathbf{P}, H\} = \mathbf{0}$ because the total physical potential depends exclusively on translation-invariant pairwise vectors, ensuring exact Poisson compensation without secular drift. \end{proofbox} \begin{propbox}[Extended Symplecticity and Palindromic Parity]{prop:extended_palindromic_symplecticity} Let $\mathcal{R}_{\mathrm{ext}} : (\mathbf{q}, \mathbf{p}, t, E) \mapsto (\mathbf{q}, -\mathbf{p}, -t, E)$ be the extended time-reversal involution on $\mathcal{P}_{\mathrm{ext}}$, satisfying $\mathcal{R}_{\mathrm{ext}}^2 = \mathrm{id}$ and anti-symplectic reflection $\mathcal{R}_{\mathrm{ext}}^* \Omega_{\mathrm{ext}} = -\Omega_{\mathrm{ext}}$. Then: \begin{enumerate}[noitemsep, topsep=2pt] \item The elementary apex kick $\Phi_h^{\mathrm{apex}}$ and kinetic drift $\Phi_h^{\mathrm{drift}}$ are exact symplectomorphisms on $(\mathcal{P}_{\mathrm{ext}}, \Omega_{\mathrm{ext}})$. \item The symmetric Strang splitting~\cite{strang1968construction} $\mathcal{S}_2(h) = \Phi_{h/2}^{\mathrm{drift}} \circ \Phi_h^{\mathrm{apex}} \circ \Phi_{h/2}^{\mathrm{drift}}$ is strictly palindromic: \begin{equation} \mathcal{R}_{\mathrm{ext}} \circ \mathcal{S}_2(h) \circ \mathcal{R}_{\mathrm{ext}} = \mathcal{S}_2(-h) = \mathcal{S}_2(h)^{-1}. \label{eq:palindromic_strang_property} \end{equation} Consequently, its formal Lie generator $Z(h) = \log \mathcal{S}_2(h)$ is strictly odd in $h$ ($Z(-h) = -Z(h)$), and the shadow Hamiltonian $\widetilde{H}(h)$ contains exclusively even powers of $h$, eliminating all odd-order energy errors. \end{enumerate} \end{propbox} \leanrefright{extended_palindromic_theorems}{extended_palindromic_symplecticity}{03_ExtendedPalindromicSymplecticity_Theorems.lean} \noindent The Lean~4 certification badge verifies the palindromic algebraic involution identity $\mathcal{R} \circ \mathcal{S}_2(h) \circ \mathcal{R} = \mathcal{S}_2(-h)$ and the symplecticity of the elementary 1-jet updates. \begin{proofbox}[Proof of Proposition~\ref{prop:extended_palindromic_symplecticity}] Item (1) follows from the fact that the kick $\mathbf{p} \mapsto \mathbf{p} - h \nabla W_{\mathrm{apex}}^{\mathrm{screen}}$ has symmetric Jacobian $\operatorname{Hess} W_{\mathrm{apex}}^{\mathrm{screen}} = (\operatorname{Hess} W_{\mathrm{apex}}^{\mathrm{screen}})^T$ (Schwarz symmetry), preserving the Darboux symplectic $2$-form $\Omega_0 = \mathrm{d}\mathbf{q} \wedge \mathrm{d}\mathbf{p}$, while the time advancement $t \mapsto t + h/2$ under the kinetic drift is a pure translation that trivially preserves $\mathrm{d}E \wedge \mathrm{d}t$ (with frozen $(t, E)$ under the apex kick). Furthermore, because all potential operators depend exclusively on base configuration positions $\mathbf{q} \in \mathbf{Q}[N]$, their corresponding Hamiltonian vector fields commute identically in the vertical cotangent fibers ($[X_{V_a}, X_{V_b}] = 0$), allowing them to be evaluated simultaneously or nested into the 7-stage palindromic Strang scheme formalized in Section~\ref{ch:ch05_higher_order_symplectic}. Item (2) is verified by direct composition: under momentum and time negation, $\Phi_{h/2}^{\mathrm{drift}}$ maps to $\Phi_{-h/2}^{\mathrm{drift}}$ and $\Phi_h^{\mathrm{apex}}$ maps to $\Phi_{-h}^{\mathrm{apex}}$, reversing the operator sequence and inverting the composite map: $\mathcal{R}_{\mathrm{ext}} \circ \mathcal{S}_2(h) \circ \mathcal{R}_{\mathrm{ext}} = \mathcal{S}_2(-h) = \mathcal{S}_2(h)^{-1}$. In the Baker--Campbell--Hausdorff expansion for $Z(h) = \log(\exp(\frac{h}{2} A) \exp(h B) \exp(\frac{h}{2} A))$, this time-reversal symmetry enforces $Z(-h) = -Z(h)$. Hence the shadow Hamiltonian $\widetilde{H}(h) = h^{-1} Z(h) = H + h^2 H_3 + h^4 H_5 + \dots$ satisfies $\widetilde{H}(-h) = \widetilde{H}(h)$, which forbids all odd error powers $h, h^3, h^5, \dots$ and confines the energy error to $\mathcal{O}(h^2)$. Under the canonical theoretical $\mathcal{C}^\infty$ Cauchy mollifier $\Theta(u_v)$, backward error analysis~\cite[Theorem~IX.8.1]{hairer2006geometric} ensures the asymptotic existence of $\widetilde{H}$ to arbitrary order, endowing the transition with Gevrey-2 regularity ($G^2$) and guaranteeing sub-exponential shadow Hamiltonian confinement in $\exp(-c / h^{1/2})$ following Marco \& Sauzin~\cite{marco2003stability}, or arbitrary polynomial order $\mathcal{O}(h^M)$ for any $M \in \mathbb{N}$, completely bypassing the $\mathcal{C}^2$ jump discontinuity of polynomial splines. \end{proofbox} \paragraph{Transition to Synchronous Supernodes.} Theorem~\ref{thm:universal_c11_apex} ensures that whenever clusters are separated by $u_v > 1$, the residual apex interaction is bounded and smooth, enabling efficient asynchronous multi-rate integration where the apex potential acts as a slow, smooth tidal perturbation across synchronous supernodes. However, while the apex interaction remains globally bounded across $\mathbf{Q}[N]$, the internal dynamical frequencies $\omega_C \sim \sqrt{G M_C / \rho_C^3}$ and tidal forces inside compact sub-clusters ($\rho_C \le \delta_{\text{cluster}}$) diverge as $\mathcal{O}(\rho_C^{-3/2})$ and $\mathcal{O}(\rho_C^{-2})$ respectively. In Section~\ref{sec:synchronous_supernodes}, we resolve this high-frequency singularity by transitioning compact sub-clusters into \emph{synchronous supernodes} governed by Sundman's unfolded local clock $\mathrm{d}t_S = \chi_S^2 \mathrm{d}\tau_S$, establishing the finite Anti-Zeno crossing property across multiple quasi-collisions.