% --- Section Content : Synchronous Supernodes and the $\mathcal{O}(\delta^{-5/2})$ Commutator Resolution --- \subsection{Synchronous Supernodes and the $\mathcal{O}(\delta^{-5/2})$ Commutator Resolution} \label{sec:synchronous_supernodes} While Section~\ref{sec:whitney_lattice} establishes the exact cotangent commutation of disjoint clusters and Section~\ref{sec:screened_apex} constructs a smooth scale-adapted apex potential ensuring universal $\mathcal{C}^{1,1}$ regularity, gravitational dynamics inevitably concentrates matter into tightly bound, dense sub-clusters. In this section, we analyze the analytical catastrophe that arises when multi-rate asynchronous time-stepping is naively applied inside a compact sub-system of $k \ge 3$ bodies, demonstrate the resulting $\mathcal{O}(\delta^{-5/2})$ commutator divergence, and resolve this obstruction by introducing the \emph{synchronous supernode} governed by Sundman's unfolded local clock $\mathrm{d}t_S = \chi_S^2 \mathrm{d}\tau_S$. We prove the universal Anti-Zeno property precluding algorithmic deadlock and establish the symplectic continuity of Jacobi atlas transitions in $\operatorname{Sp}(6N-6, \mathbb{R})$ with guaranteed elimination of topological chattering under hysteresis $\gamma = 0.6$. % ------------------------------------------------------------------------------ \subsubsection{The Excluded Middle Dilemma and the $\mathcal{O}(\delta^{-5/2})$ Commutator Barrier} \label{sub:excluded_middle_dilemma} In classical celestial mechanics and astrophysical stellar dynamics, gravitational contraction and resonant multi-body scattering systematically harden close binaries (Heggie's law~\cite{heggie1975binary}) and generate dense multi-body sub-clusters ($k \ge 3$) undergoing chaotic close encounters~\cite{aarseth2003gravitational}. Consider a compact sub-cluster $S \subset \{1, \dots, N\}$ with $|S| = k \ge 3$, characterized by its internal McGehee hyper-radius $\rho_S = \sqrt{\sum_{i < j \in S} m_i m_j \|\mathbf{q}_i - \mathbf{q}_j\|^2 / M_S} \le \delta_{\text{cluster}}$, where $\delta_{\text{cluster}} > 0$ denotes a cluster compactness threshold (whose intrinsic, scale-invariant determination $\delta_{\mathrm{cluster}}^{\mathrm{nat}}$ is established in Section~\ref{sub:intrinsic_compactness_threshold}). In an asynchronous multi-rate scheme, one would ideally wish to assign independent, local time steps $h_u \ne h_w$ to distinct internal Jacobi sub-nodes $u, w \in \mathcal{V}_{\mathrm{int}}(\mathcal{T}_S)$ within the cluster. However, unlike disjoint clusters where the Poisson bracket vanishes identically by Proposition~\ref{prop:disjoint_cotangent_commutation}, internal Jacobi nodes within the same compact cluster interact through non-orthogonal kinetic cross-terms and strongly coupled Newtonian potentials. Let $\mathcal{H}_u = \frac{\|\boldsymbol{\pi}_u\|^2}{2 \mu_u} + V_u$ and $\mathcal{H}_w = \frac{\|\boldsymbol{\pi}_w\|^2}{2 \mu_w} + V_w$ (where $\boldsymbol{\pi}_u \equiv \mathbf{w}_u$ denotes the canonical momentum conjugate to the Jacobi vector $\mathbf{r}_u$) be the local Hamiltonians associated with two overlapping internal nodes of $\mathcal{T}_S$. Differentiating their Newtonian potentials $V_u \sim -G m^2 / \rho_S$ with respect to relative positions and momenta yields the Poisson bracket scaling: \begin{equation} \{\mathcal{H}_u, \mathcal{H}_w\} = \sum_{v \in \mathcal{V}_{\mathrm{int}}(\mathcal{T}_S)} \left( \frac{\partial \mathcal{H}_u}{\partial \mathbf{r}_v} \cdot \frac{\partial \mathcal{H}_w}{\partial \boldsymbol{\pi}_v} - \frac{\partial \mathcal{H}_u}{\partial \boldsymbol{\pi}_v} \cdot \frac{\partial \mathcal{H}_w}{\partial \mathbf{r}_v} \right) \sim \frac{G m^2}{\rho_S^2} \cdot \frac{\|\boldsymbol{\pi}\|}{\mu} \sim \mathcal{O}\left( \rho_S^{-5/2} \right), \label{eq:commutator_divergence_scaling} \end{equation} since virialized relative velocities scale as $\|\boldsymbol{\pi}\|/\mu \sim \sqrt{G m / \rho_S} \sim \mathcal{O}(\rho_S^{-1/2})$. In the Baker--Campbell--Hausdorff expansion of any split asynchronous operator, the local truncation error per step contains the commutator contribution: \begin{equation} \Delta \mathcal{H}_{\mathrm{async}} = \frac{h_u h_w}{2} \{\mathcal{H}_u, \mathcal{H}_w\} + \mathcal{O}(h^3) \sim \mathcal{O}\left( h^2 \, \delta_{\text{cluster}}^{-5/2} \right). \label{eq:bch_commutator_catastrophe} \end{equation} As the cluster contracts ($\rho_S \to 0$), the commutator error diverges as $\mathcal{O}(\rho_S^{-5/2})$. Any attempt to absorb this divergence by shrinking the asynchronous step size $h \to 0$ leads directly to algorithmic Zeno paralysis. This analytical obstruction constitutes the \emph{asymptotic commutator barrier} for split gravitational operators (commonly referred to as the \emph{Excluded Middle Dilemma}): \begin{propbox}[Asymptotic Commutator Barrier for Split Operators]{prop:excluded_middle_dilemma} Let $S \subset \{1, \dots, N\}$ be an interacting cluster in a hierarchical $N$-body splitting scheme. Then: \begin{enumerate} \item \emph{Macroscopic Regime ($u_v > 1$)}: If the cluster is separated from external bodies such that $u_v = \|\mathbf{r}_v\| / \bar{\rho}_{\mathrm{int}}(v) > 1$, the screened apex interaction $W_{\mathrm{apex}}^{\mathrm{screen}}$ is uniformly of class $\mathcal{C}^{1,1}$ (Theorem~\ref{thm:universal_c11_apex}), permitting exact macroscopic asynchrony between clusters without secular drift. \item \emph{Microscopic Compact Regime ($\rho_S \le \delta_{\text{cluster}}$, $|S| \ge 3$)}: Any internal desynchronization of sub-steps within the cluster triggers an unremovable Lie commutator divergence of order $\mathcal{O}(\delta_{\text{cluster}}^{-5/2})$. \end{enumerate} Consequently, there exists no consistent intermediate splitting regime: dense sub-clusters of cardinality $k \ge 3$ must be dynamically unified and integrated strictly synchronously. \end{propbox} \begin{proofbox}[Proof of Proposition~\ref{prop:excluded_middle_dilemma}] Item~(1) follows directly from Theorem~\ref{thm:universal_c11_apex}: for $u_v > 1$, the scale-adapted screening function $\Theta(u_v)$ guarantees that the screened apex remainder $W_{\mathrm{apex}}^{\mathrm{screen}}$ belongs to $\mathcal{C}^{1,1}(U_{\mathcal{T}})$ with uniformly bounded force and Hessian norms $M_{\mathrm{apex}}, L_{\mathrm{apex}} < \infty$, while disjoint sub-clusters commute identically on phase space by Proposition~\ref{prop:disjoint_cotangent_commutation}. For Item~(2), within a single compact cluster $S$ with $|S| \ge 3$ and $\rho_S \le \delta_{\mathrm{cluster}}$, overlapping internal Jacobi nodes $u, w \in \mathcal{V}_{\mathrm{int}}(\mathcal{T}_S)$ share common particle positions, so their Newtonian gradients scale as $\|\partial_{\mathbf{r}} \mathcal{H}_u\| \sim G m^2 / \rho_S^2$ while virialized relative velocities scale as $\|\partial_{\boldsymbol{\pi}} \mathcal{H}_w\| \sim \sqrt{G m / \rho_S}$. Substituting these scalings into the canonical Poisson bracket~\eqref{eq:commutator_divergence_scaling} yields $\{\mathcal{H}_u, \mathcal{H}_w\} \sim \mathcal{O}(\rho_S^{-5/2})$, which enters the leading Baker--Campbell--Hausdorff error~\eqref{eq:bch_commutator_catastrophe} of any asynchronous multi-rate splitting as $\mathcal{O}(h^2 \delta_{\mathrm{cluster}}^{-5/2})$. \end{proofbox} % ------------------------------------------------------------------------------ \subsubsection{Unified Synchronous Supernodes and the Universal Anti-Zeno Property} \label{sub:synchronous_supernode_anti_zeno} To resolve the $\mathcal{O}(\delta^{-5/2})$ commutator catastrophe without compromising symplectic conservation, we introduce the concept of a \emph{synchronous supernode}: \begin{defbox}[Synchronous Supernode]{def:synchronous_supernode} Let $\mathcal{T} \in \mathrm{Trees}(N)$ be a binary Jacobi tree. A composite node $S \in \mathcal{V}_{\mathrm{comp}}(\mathcal{T})$ with $|S| \ge 3$ is designated a \emph{synchronous supernode} whenever its internal hyper-radius satisfies the compactness condition: \begin{equation} \operatorname{isCompactCluster}(\rho_S, \delta_{\text{cluster}}) \iff \rho_S \le \delta_{\text{cluster}}, \label{eq:compact_cluster_predicate} \end{equation} where $\delta_{\text{cluster}} > 0$ is a scale threshold, whose canonical, scale-covariant value $\delta_{\mathrm{cluster}}^{\mathrm{nat}}(S)$ is dynamically determined by the gravitational tidal tensor and Marchal--Sundman energy scaling in Section~\ref{sub:intrinsic_compactness_threshold}. The full $N$-body Hamiltonian then decomposes into supernode blocks and a regularized inter-cluster remainder: \begin{equation} H = \sum_{S \in \mathcal{S}_{\mathrm{act}}} H_S^{\mathrm{super}} + H_{\mathrm{rem}}, \label{eq:global_supernode_decomposition} \end{equation} where the internal sub-tree $\mathcal{T}_S$ is dynamically collapsed into a single entity governed by the full, unsoftened internal Hamiltonian: \begin{equation} H_S^{\mathrm{super}}(\mathbf{q}_S, \mathbf{p}_S) = \sum_{u \in \mathcal{V}_{\mathrm{int}}(\mathcal{T}_S)} \frac{\|\boldsymbol{\pi}_u\|^2}{2 \mu_u} - \sum_{\substack{i < j \\ i, j \in S}} \frac{G m_i m_j}{\|\mathbf{q}_i - \mathbf{q}_j\|}, \label{eq:supernode_hamiltonian_def} \end{equation} while $H_{\mathrm{rem}}$ contains the spectator kinetic terms and the screened inter-cluster apex potentials governed by Theorem~\ref{thm:universal_c11_apex}. All internal degrees of freedom of $S$ advance synchronously under the single, unified unfolded Sundman clock~\cite{sundman1912memoire}: \begin{equation} \mathrm{d}t_S = \chi_S^2 \, \mathrm{d}\tau_S = \rho_S \, \mathrm{d}\tau_S \quad (\text{with } \chi_S = \sqrt{\rho_S} > 0). \label{eq:unfolded_sundman_clock} \end{equation} \end{defbox} \leanrefright{unified_super_node_theorems}{super_node_time_step_sq_pos}{01_UnifiedSuperNodeDef_Theorems.lean} This formulation articulates the essential duality between continuous vector field desingularization and discrete numerical time-stepping: McGehee's continuous homothety scaling $\mathrm{d}t_S = \rho_S^{3/2} \mathrm{d}\tau_S$~\cite{mcgehee1974triple} is indispensable for eliminating the algebraic velocity divergence $\rho_S^{-1/2}$ and extending the Hamiltonian vector field smoothly to the collision manifold $\{\rho_S = 0\}$ in phase space; conversely, for discrete multi-step sub-stepping, Sundman's unfolded local clock $\Delta t_S = \chi_S^2 \Delta \tau_S = \rho_S \Delta \tau_S$ with non-singular state $\chi_S > 0$ guarantees strict algebraic positivity $(\Delta t_S)^2 = \chi_S^4 (\Delta \tau_S)^2 > 0$ and establishes the discrete Anti-Zeno finite sub-stepping property without artificial time freezing. \noindent The accompanying formal certification badge verifies the strict algebraic positivity $(\Delta t_S)^2 > 0$ and the cumulative forward progression of time in the Constructive 1-Jet Kernel. The central analytical property of the synchronous supernode is that the unfolded Sundman transformation completely eliminates the algorithmic Zeno effect across multiple close encounters: \begin{thmbox}[Universal Anti-Zeno Regularization of Synchronous Supernodes]{thm:anti_zeno_regularization} Let $S$ be a synchronous supernode governed by the unfolded Sundman clock $\mathrm{d}t_S = \chi_S^2 \mathrm{d}\tau_S$ on the extended phase space $\mathcal{P}_{\mathrm{ext}}$. Then: \begin{enumerate} \item \emph{Strict Temporal Progression}: For any regularized time step $\Delta \tau_S > 0$ and any non-singular state $\chi_S > 0$, the induced physical time increment is strictly positive: \begin{equation} \Delta t_S = \operatorname{physicalTimeIncrement}(\chi_S, \Delta \tau_S) = \chi_S^2 \, \Delta \tau_S > 0. \label{eq:strict_time_increment_pos} \end{equation} \item \emph{Orbit-Wise Anti-Zeno Finiteness for Almost All Initial Conditions}: On any finite physical macro-step $\Delta t_{\mathrm{phys}} > 0$, for all non-collisional initial states $\mathbf{z}_0 \in \mathcal{P}_{\mathrm{ext}} \setminus \mathcal{N}_{\mathrm{coll}}$ belonging to the full Lebesgue-measure complement of Saari's collision set ($\mu_{\mathrm{Leb}}(\mathcal{N}_{\mathrm{coll}}) = 0$~\cite{saari1971improbability}), the total number of regularized sub-steps $N_{\mathrm{sub}}(\mathbf{z}_0)$ required to advance through the encounter is strictly finite along the orbit: \begin{equation} N_{\mathrm{sub}}(\mathbf{z}_0) \le \left\lceil \frac{\Delta t_{\mathrm{phys}}}{C_{\mathrm{anti\text{-}zeno}}(\mathbf{z}_0)} \right\rceil < \infty, \label{eq:anti_zeno_finite_bound} \end{equation} where $C_{\mathrm{anti\text{-}zeno}}(\mathbf{z}_0) = \rho_{\min}(\mathbf{z}_0) \Delta \tau_S > 0$ with $\rho_{\min}(\mathbf{z}_0) = \min_{t \in [0, \Delta t_{\mathrm{phys}}]} \rho_S(\mathbf{z}(t)) > 0$. Consequently, algorithmic Zeno stagnation is strictly impossible along any physical trajectory. \end{enumerate} \end{thmbox} \leanrefright{super_node_anti_zeno_theorems}{super_node_anti_zeno_regularization}{02_SuperNodeAntiZenoRegularization_Theorems.lean} \pythonrefright{py_hierarchy}{potential/supernode_detector.py, solvers/{supernode, jit_supernode}.py}{Float64 / Float128} \noindent The formal certification badge denotes the verification in Lean~4 of the Archimedean property ensuring the existence of the finite uniform bound $N_{\mathrm{max}} \in \mathbb{N}$ on the required sub-steps. \begin{proofbox}[Proof of Theorem~\ref{thm:anti_zeno_regularization}] Item~(1) follows immediately from the product of strictly positive real numbers: since $\chi_S > 0$ and $\Delta \tau_S > 0$, we have $\chi_S^2 > 0$ and $\Delta t_S = \chi_S^2 \Delta \tau_S > 0$. For Item~(2), consider a physical macro-step of duration $\Delta t_{\mathrm{phys}} > 0$. In McGehee coordinates $(\chi_S, \mathbf{s}_S, \mathbf{w}_S)$ on the blown-up boundary manifold with corners $\mathbf{M}_S = \{\chi_S = 0\} \times \mathbb{S}^{3|S|-4} \times \mathbb{R}^{3|S|-3}$, the equations of motion are analytic in regularized time $\tau_S$~\cite{mcgehee1974triple}. In the asymptotic approach to a multiple quasi-collision, the hyper-radius scales according to Kepler's generalized law $\rho_S(t) \sim c \, t^{2/3}$ ($c > 0$), which implies by integration of Sundman's relation $\mathrm{d}t_S = \rho_S \mathrm{d}\tau_S$ ($t^{-2/3} \mathrm{d}t = c \, \mathrm{d}\tau_S \implies 3 t^{1/3} = c \tau_S$) that $t(\tau_S) \sim \frac{1}{27} c^3 \tau_S^3$ and $\mathrm{d}t/\mathrm{d}\tau_S \sim \frac{1}{9} c^3 \tau_S^2$. By the classical theorems of Saari~\cite{saari1971improbability,saari1973improbability}, exact multiple collisions ($\rho_S = 0$) form a subset of Lebesgue measure zero in phase space. For every individual non-collisional trajectory $\mathbf{z}(t)$ belonging to the full-measure complement ($\mu_{\mathrm{Leb}} > 0$), the hyper-radius $t \mapsto \rho_S(\mathbf{z}(t))$ is continuous over the compact physical interval $[0, \Delta t_{\mathrm{phys}}]$ and strictly positive. By the Weierstrass extreme value theorem, it attains a strictly positive minimum: \begin{equation} \rho_{\min}(\mathbf{z}_0) = \min_{t \in [0, \Delta t_{\mathrm{phys}}]} \rho_S(\mathbf{z}(t)) > 0. \label{eq:saari_orbit_minimum} \end{equation} Setting $C_{\mathrm{anti\text{-}zeno}}(\mathbf{z}_0) = \rho_{\min}(\mathbf{z}_0) \Delta \tau_S > 0$, each sub-step advances physical time by at least $\Delta t_S \ge C_{\mathrm{anti\text{-}zeno}}(\mathbf{z}_0) > 0$. By the Archimedean property of the real field $\mathbb{R}$, there exists an integer $N \in \mathbb{N}$ such that $N \cdot C_{\mathrm{anti\text{-}zeno}}(\mathbf{z}_0) \ge \Delta t_{\mathrm{phys}}$. Taking $N_{\mathrm{sub}}(\mathbf{z}_0) = \lceil \Delta t_{\mathrm{phys}} / C_{\mathrm{anti\text{-}zeno}}(\mathbf{z}_0) \rceil < \infty$ guarantees algorithmic completion. We emphasize that this bound is orbit-wise: while $\rho_{\min}(\mathbf{z}_0) > 0$ holds strictly along any individual non-collisional path, no strictly positive infimum $\inf_{\mathbf{z}_0} \rho_{\min}(\mathbf{z}_0) > 0$ exists across the full phase space since initial conditions may approach the measure-zero collision set $\mathcal{N}_{\mathrm{coll}}$ arbitrarily closely. Furthermore, even in the idealized limit of an exact collision, the Sundman regularized time integral $\Delta \tau_S = \int_0^{\Delta t} \rho_S(t)^{-1} \mathrm{d}t \sim \int_0^{\Delta t} c^{-1} t^{-2/3} \mathrm{d}t = 3 c^{-1} (\Delta t)^{1/3} < \infty$ remains strictly finite, precluding any infinite accumulation of regularized steps. \end{proofbox} % ------------------------------------------------------------------------------ \subsubsection{Symplectic Atlas Transitions in $\operatorname{Sp}(6N-6, \mathbb{R})$ and Hysteresis Anti-Chattering} \label{sub:symplectic_atlas_anti_chattering} As a dense multi-body system evolves, close encounters alter the mutual hierarchy of particle distances, necessitating a transition between distinct adapted Jacobi binary trees in the atlas $\{(\mathcal{T}, U_{\mathcal{T}})\}_{\mathcal{T} \in \mathrm{Trees}(N)}$. We now demonstrate that these coordinate transitions are exact symplectomorphisms and that topological chattering (infinite alternating flips across transition boundaries) is strictly eliminated: \begin{thmbox}[Symplectic Atlas Transitions and Unconditional Anti-Chattering]{thm:atlas_transition_anti_chattering} Let $\mathbf{Q}[N] = \bigcup_{\mathcal{T}} U_{\mathcal{T}}$ be the Fulton--MacPherson configuration manifold endowed with the atlas of adapted binary Jacobi trees. \begin{enumerate} \item \emph{Exact Symplectomorphism in $\operatorname{Sp}(6N-6, \mathbb{R})$}: For any two adjacent Jacobi trees $\mathcal{T}_A$ and $\mathcal{T}_B$ differing by an elementary tree rotation $((S_1, S_2), S_3) \longleftrightarrow (S_1, (S_2, S_3))$ on overlapping charts $U_{\mathcal{T}_A} \cap U_{\mathcal{T}_B} \ne \emptyset$, the relative coordinate transformation is governed by the $2 \times 2$ block matrix: \begin{equation} \mathbf{M}_{\mathrm{coord}} = \begin{pmatrix} -\frac{m_1}{m_1 + m_2} & 1 \\ -\frac{m_2 M_{\mathrm{tot}}}{(m_1 + m_2)(m_2 + m_3)} & -\frac{m_3}{m_2 + m_3} \end{pmatrix} \in \mathrm{SL}(2, \mathbb{R}), \quad \det \mathbf{M}_{\mathrm{coord}} = +1, \label{eq:jacobi_transition_matrix_sl2} \end{equation} where $M_{\mathrm{tot}} = m_1 + m_2 + m_3$. Its cotangent lift induces a global linear symplectomorphism on phase space: \begin{align} \Psi_{A \to B} &= \Psi_{\mathrm{rot}} \oplus \mathbf{I}_{6N-18} \in \operatorname{Sp}(6N-6, \mathbb{R}), \nonumber \\ \text{with} \quad \Psi_{\mathrm{rot}} &= \begin{pmatrix} \mathbf{M}_{\mathrm{coord}} \otimes \mathbf{I}_3 & \mathbf{0} \\ \mathbf{0} & (\mathbf{M}_{\mathrm{coord}}^{-1})^T \otimes \mathbf{I}_3 \end{pmatrix} \in \operatorname{Sp}(12, \mathbb{R}), \label{eq:cotangent_lift_sp} \end{align} acting as the identity on all spectator Jacobi coordinates and preserving the canonical extended Darboux form $\Psi_{A \to B}^* \Omega_{\mathrm{ext}} = \Omega_{\mathrm{ext}}$. Furthermore, because $\Psi_{\mathrm{rot}}$ acts as a scalar $2 \times 2$ block tensor product with $\mathbf{I}_3$, it commutes with any global 3D spatial rotation: $\Psi_{\mathrm{rot}} \circ \mathcal{R}_{\mathrm{spatial}} = \mathcal{R}_{\mathrm{spatial}} \circ \Psi_{\mathrm{rot}}$. To preserve numerical time-reversibility and prevent shadow Hamiltonian discontinuities (Skeel \& Gear~\cite{skeel1992does}), chart transitions $\Psi_{A \to B}$ are evaluated strictly at discrete global synchronization boundaries $t = n \Delta t_{\mathrm{phys}}$. The physical Cartesian state $(\mathbf{q}, \mathbf{p})$ and extended energy $E$ are strictly $\mathcal{C}^0$-continuous across chart boundaries. \item \emph{Unconditional Anti-Chattering under Optional Hysteresis $\gamma = 0.6$ vs.\ Canonical Voronoi Bijectivity $\gamma = 1.0$}: Let $D_{12} = \|\mathbf{X}_{S_1} - \mathbf{X}_{S_2}\|$ and $D_{23} = \|\mathbf{X}_{S_2} - \mathbf{X}_{S_3}\|$ denote the mutual Euclidean distances between the barycenters of the interacting clusters, and let $\gamma \in (0, 1]$ denote the admissibility threshold ratio. In chart $\mathcal{T}_A$ where $(S_1, S_2)$ is the internal close pair, a transition to $\mathcal{T}_B$ (where $(S_2, S_3)$ becomes the internal pair) is triggered when $D_{23} < \gamma D_{12}$; conversely, the return transition from $\mathcal{T}_B$ back to $\mathcal{T}_A$ requires $D_{12} < \gamma D_{23}$. When operated with the strict asymmetric hysteresis ratio $\gamma = 3/5 = 0.6$, closed back-and-forth chattering cycles across the transition boundary are strictly forbidden, and on every compact physical time interval $[0, T]$, the total number of atlas transitions is uniformly finite: $N_{\mathrm{trans}}(T) \le \lceil T / \tau_{\min} \rceil < \infty$, with strictly positive \emph{a posteriori} minimal dwell time $\Delta t_{\mathrm{dwell}}^{(2\text{-chart})} \ge \frac{16}{15 \, \Omega_{\max}} \ge \frac{1}{5}\frac{r_{\min}}{v_{\max}} > 0$ for two-chart cycles and $\Delta t_{\mathrm{dwell}}^{(3\text{-chart})} \ge \frac{1}{10}\frac{r_{\min}}{v_{\max}} > 0$ for three-chart permutations. However, while spatial hysteresis ($\gamma < 1$) was theoretically introduced as a worst-case safeguard against pathological boundary chattering, genuine chattering along physical $N$-body trajectories is exceedingly rare in practice (since generic non-collinear trajectories cross codimension-1 Voronoi bisectors $\Sigma_{AB} = \{D_{12} = D_{23}\}$ transversely at non-zero radial velocity). Furthermore, any strict hysteresis deadband $\gamma < 1$ creates an overlapping co-admissible region $\mathrm{Adm}_\gamma(\mathcal{T}_A) \cap \mathrm{Adm}_\gamma(\mathcal{T}_B) = \{\gamma < D_{23}/D_{12} < \gamma^{-1}\} \ne \emptyset$ inside which the chart assignment map $\mathbf{q} \mapsto \mathcal{T}(\mathbf{q})$ is \emph{non-injective} (multi-valued, depending on past trajectory history). Under pure phase-space time reversal $\mathcal{R}(\mathbf{q}, \mathbf{p}) = (\mathbf{q}, -\mathbf{p})$, this history-dependent non-injectivity breaks state-only time-reversal symmetry (as forward entry occurs at $\lambda = \gamma^{-1}$ whereas backward return switches late at $\lambda = \gamma$ unless an explicit history stack is maintained). Consequently, the canonical operational default sets $\gamma = 1.0$, restoring a memoryless, single-valued Voronoi partition where chart selection is a pure bijection on generic configurations and time-reversal symmetry holds intrinsically without history dependence. \end{enumerate} \end{thmbox} \leanrefright{atlas_transition_symplectic_theorems}{atlas_transition_symplectic_anti_chattering}{03_AtlasTransitionSymplecticAntiChattering_Theorems.lean} \pythonrefright{py_system}{core/transitions.py, system/atlas_manager.py}{Float64 / Float128} \noindent The formal certification badge verifies in Lean~4 that any $\mathbf{M} \in \mathrm{SL}(2, \mathbb{R})$ preserves the canonical symplectic area $x_1 y_2 - x_2 y_1$ and proves by contradiction that the hysteresis conditions cannot be simultaneously satisfied. \begin{proofbox}[Proof of Theorem~\ref{thm:atlas_transition_anti_chattering}] For Item~(1), let $\mathbf{q}_1, \mathbf{q}_2, \mathbf{q}_3$ be the barycentric positions of three interacting sub-clusters of masses $m_1, m_2, m_3$. In chart $\mathcal{T}_A$, the relative Jacobi coordinates are $\mathbf{r}_{A, 1} = \mathbf{q}_2 - \mathbf{q}_1$ and $\mathbf{r}_{A, 2} = \mathbf{q}_3 - \frac{m_1 \mathbf{q}_1 + m_2 \mathbf{q}_2}{m_1 + m_2}$. In chart $\mathcal{T}_B$, the new coordinates are $\mathbf{r}_{B, 1} = \mathbf{q}_3 - \mathbf{q}_2$ and $\mathbf{r}_{B, 2} = \mathbf{q}_1 - \frac{m_2 \mathbf{q}_2 + m_3 \mathbf{q}_3}{m_2 + m_3}$. Expressing $(\mathbf{r}_{B, 1}, \mathbf{r}_{B, 2})$ in terms of $(\mathbf{r}_{A, 1}, \mathbf{r}_{A, 2})$ yields: \begin{align} \mathbf{r}_{B, 1} &= \mathbf{r}_{A, 2} - \frac{m_1}{m_1 + m_2} \mathbf{r}_{A, 1}, \label{eq:coord_trans_1} \\ \mathbf{r}_{B, 2} &= - \frac{m_2 M_{\mathrm{tot}}}{(m_1 + m_2)(m_2 + m_3)} \mathbf{r}_{A, 1} - \frac{m_3}{m_2 + m_3} \mathbf{r}_{A, 2}. \label{eq:coord_trans_2} \end{align} The determinant of the $2 \times 2$ matrix $\mathbf{M}_{\mathrm{coord}}$ evaluates identically to: \begin{align} \det \mathbf{M}_{\mathrm{coord}} &= \left( -\frac{m_1}{m_1 + m_2} \right) \left( -\frac{m_3}{m_2 + m_3} \right) - (1) \left( -\frac{m_2 M_{\mathrm{tot}}}{(m_1 + m_2)(m_2 + m_3)} \right) \nonumber \\ &= \frac{m_1 m_3 + m_2(m_1 + m_2 + m_3)}{(m_1 + m_2)(m_2 + m_3)} \nonumber \\ &= +1. \label{eq:det_m_coord_proof} \end{align} Thus $\mathbf{M}_{\mathrm{coord}} \in \mathrm{SL}(2, \mathbb{R})$. In cotangent coordinates, the canonical momenta transform via the inverse transpose $\boldsymbol{\pi}_B = (\mathbf{M}_{\mathrm{coord}}^{-1})^T \boldsymbol{\pi}_A$. By the canonical theorem of cotangent lifts (Marsden \& West~\cite{marsden2001discrete}, Cannas da Silva~\cite{cannas2001lectures}), $\Psi_{\mathrm{rot}}^* (\mathrm{d}\boldsymbol{\pi} \wedge \mathrm{d}\mathbf{r}) = \mathrm{d}\boldsymbol{\pi} \wedge \mathrm{d}\mathbf{r}$. Since the remaining $N-3$ Jacobi vectors and momenta are spectators, embedding $\Psi_{A \to B} = \Psi_{\mathrm{rot}} \oplus \mathbf{I}_{6N-18}$ yields a linear symplectomorphism in $\operatorname{Sp}(6N-6, \mathbb{R})$. Since the transformation operates purely on relative spatial degrees of freedom, time $t$ and extended energy $E$ remain strictly invariant, ensuring $\Psi_{A \to B}^* \Omega_{\mathrm{ext}} = \Omega_{\mathrm{ext}}$. For Item~(2), suppose for contradiction that an instantaneous chattering cycle exists between trees $\mathcal{T}_A$ and $\mathcal{T}_B$. In chart $\mathcal{T}_A$, the pair $(S_1, S_2)$ is internal, so a forward transition $\mathcal{T}_A \to \mathcal{T}_B$ requires $D_{23} < \gamma \, D_{12}$. In chart $\mathcal{T}_B$, $(S_2, S_3)$ is internal, so an immediate return transition $\mathcal{T}_B \to \mathcal{T}_A$ requires $D_{12} < \gamma \, D_{23}$. Combining these inequalities with $\gamma = 3/5 > 0$: \begin{equation} D_{12} < \gamma \, D_{23} < \gamma (\gamma \, D_{12}) = \gamma^2 D_{12} = \frac{9}{25} D_{12} = 0.36 \, D_{12}. \label{eq:chattering_absurd_derivation} \end{equation} Since physical distances are strictly positive ($D_{12} > 0$), dividing by $D_{12}$ yields $1 < 0.36$, a blatant contradiction. To bridge this static geometric exclusion to dynamic finiteness $N_{\mathrm{trans}}(T) < \infty$, consider the dimensionless distance ratio $\lambda(t) = D_{12}(t) / D_{23}(t)$. In tree $\mathcal{T}_A$, a transition to $\mathcal{T}_B$ occurs when $D_{23} \le \gamma \, D_{12}$, which corresponds to $\lambda \ge \gamma^{-1} = 5/3$. To trigger a subsequent return transition from $\mathcal{T}_B$ back to $\mathcal{T}_A$, the system must satisfy $D_{12} \le \gamma \, D_{23}$, which is equivalent to $\lambda \le \gamma = 3/5$. Thus, between any two consecutive transitions in a two-chart cycle $\mathcal{T}_A \to \mathcal{T}_B \to \mathcal{T}_A$, the continuous trajectory must traverse the finite hysteresis gap: \begin{equation} \Delta \lambda = \gamma^{-1} - \gamma = \frac{5}{3} - \frac{3}{5} = \frac{16}{15} \approx 1.067. \label{eq:hysteresis_gap_width} \end{equation} On any compact energy hypersurface $\Sigma_{E_0} = \{ (\mathbf{q}, \mathbf{p}) \in \mathcal{P}_{\mathrm{ext}} \mid H(\mathbf{q}, \mathbf{p}) = E_0 \}$ with negative energy $E_0 < 0$ within the compactified manifold with corners $\mathbf{Q}[N]$, the time derivative of the dimensionless ratio $\lambda(t)$ is a transition rate bounded by a finite frequency: $\sup_{t \in [0, T]} |\dot{\lambda}(t)| \le \Omega_{\max} < \infty$ with $[\Omega_{\max}] = [T]^{-1}$. Explicitly, differentiating $\lambda(t) = D_{12}(t) / D_{23}(t)$ by the quotient rule and bounding $|\dot{D}_{ij}| \le 2 v_{\max}$ and $D_{23} \ge r_{\min}$ yields $|\dot{\lambda}(t)| \le \frac{2(1 + \lambda(t)) v_{\max}}{r_{\min}} \le \frac{16}{3} \frac{v_{\max}}{r_{\min}}$ across the hysteresis interval $\lambda \in [3/5, 5/3]$. Consequently, the minimal physical dwell time for any two-chart cycle $\mathcal{T}_A \to \mathcal{T}_B \to \mathcal{T}_A$ is bounded below \emph{a posteriori}: \begin{equation} \Delta t_{\mathrm{dwell}}^{(2\text{-chart})} \ge \frac{\gamma^{-1} - \gamma}{\Omega_{\max}} = \frac{16}{15 \, \Omega_{\max}} \ge \frac{16/15}{(16/3)(v_{\max}/r_{\min})} = \frac{1}{5} \frac{r_{\min}}{v_{\max}} > 0. \label{eq:minimal_dwell_time} \end{equation} Similarly, if the trajectory transitions from $\mathcal{T}_B$ (where $(S_2, S_3)$ was selected at $t_1$ as the shortest pair, so $\mu(t_1) \coloneqq D_{13}(t_1) / D_{23}(t_1) \ge 1$) to a third distinct chart $\mathcal{T}_C$ at $t_2$ (requiring $\mu(t_2) \le \gamma = 3/5$), the ratio $\mu(t)$ must traverse at least $\Delta \mu_{3\text{-chart}} = 1 - \gamma = 2/5$ at rate $|\dot{\mu}(t)| \le \frac{2(1+\mu) v_{\max}}{r_{\min}} \le 4 \frac{v_{\max}}{r_{\min}}$, enforcing $\Delta t_{\mathrm{dwell}}^{(3\text{-chart})} \ge \frac{2/5}{4(v_{\max}/r_{\min})} = \frac{1}{10} \frac{r_{\min}}{v_{\max}} > 0$. Therefore, on any compact physical time interval $[0, T]$, spatial hysteresis alone guarantees $\Delta t_{\mathrm{dwell}} \ge \tau_{\min} > 0$ \emph{a posteriori} and bounds the total number of atlas transitions by $N_{\mathrm{trans}}(T) \le \lceil T / \tau_{\min} \rceil < \infty$, eliminating all Zeno-type topological chattering without any \emph{a priori} temporal lock. \end{proofbox} % ------------------------------------------------------------------------------ \subsubsection{Intrinsic Scale Invariance and Dynamical Determination of the Compactness Threshold} \label{sub:intrinsic_compactness_threshold} A critical foundational inquiry naturally arises regarding the status of the cluster compactness threshold $\delta_{\mathrm{cluster}} > 0$ introduced in Definition~\ref{def:synchronous_supernode}: if $\delta_{\mathrm{cluster}}$ were treated merely as an arbitrary, user-specified dimensioned parameter ($[\delta_{\mathrm{cluster}}] = [L] \cdot [M]^{1/2}$), it would appear to reintroduce through the back door an artificial metric cut-off akin to the banished softening length $d_0 > 0$. Furthermore, unsoftened Newtonian gravity ($\varepsilon = 0$) possesses an exact scale symmetry under the homothety group: \begin{equation} \mathbf{q} \longmapsto \lambda \mathbf{q}, \qquad t \longmapsto \lambda^{3/2} t, \qquad H \longmapsto \lambda^{-1} H \quad (\lambda > 0). \label{eq:gravitational_homothety_scaling} \end{equation} While the apex screening variable $u_v = \|\mathbf{r}_v\| / (\bar{\rho}_{L(v)} + \bar{\rho}_{R(v)})$ (Section~\ref{sub:screening_function_def}) is strictly dimensionless and invariant under homotheties ($\mathbf{q} \mapsto \lambda \mathbf{q} \implies u_v \mapsto u_v$), a rigid dimensioned constant $\delta_{\mathrm{cluster}}$ would break this scale covariance across disparate astronomical systems (from compact stellar triples to hierarchical planetary systems or galactic cores). We now prove that $\delta_{\mathrm{cluster}}$ is not an independent or arbitrary numerical regulator, but an intrinsic, dynamically determined observable dictated by the gravitational tidal tensor on $\mathbf{Q}[N]$ and the fundamental first integrals of motion: \begin{propbox}[Dynamical Determination of the Compactness Threshold]{prop:intrinsic_compactness_threshold} Let $\mathcal{T} \in \mathrm{Trees}(N)$ be an adapted binary Jacobi tree, and let $S \in \mathcal{V}_{\mathrm{comp}}(\mathcal{T})$ with $|S| \ge 3$ be an interacting sub-cluster whose parent node is $P = \operatorname{parent}(S) \in \mathcal{V}_{\mathrm{int}}(\mathcal{T})$ with relative Jacobi vector $\mathbf{r}_P = \mathbf{X}_S - \mathbf{X}_{S'}$ separating $S$ from its sibling cluster $S'$. \begin{enumerate} \item \emph{Dimensionless Gravitational Tidal Ratio}: The dynamical susceptibility of $S$ to tidal disruption by its external environment is governed by the dimensionless gravitational tidal ratio: \begin{equation} \eta_S(\mathbf{q}) \coloneqq \frac{\bar{\rho}_S}{\|\mathbf{r}_P\|} \left( \frac{M_{S'}}{M_S} \right)^{1/3} = \frac{\rho_S}{\sqrt{M_S} \|\mathbf{r}_P\|} \left( \frac{M_{S'}}{M_S} \right)^{1/3}, \label{eq:gravitational_tidal_ratio_def} \end{equation} which is strictly scale-invariant ($\eta_S(\lambda \mathbf{q}) = \eta_S(\mathbf{q})$ for all $\lambda > 0$). An interacting cluster is dynamically uncoupled from external tidal shear, forming a gravitationally self-bound entity whose internal evolution triggers the $\mathcal{O}(\rho_S^{-5/2})$ commutator catastrophe if integrated asynchronously, if and only if: \begin{equation} \eta_S(\mathbf{q}) \le \eta_{\mathrm{crit}} \coloneqq 3^{-1/3} \approx 0.69336, \label{eq:intrinsic_hill_criterion} \end{equation} where $\eta_{\mathrm{crit}}$ is the universal Hill stability constant corresponding to the collinear $L_1$ Lagrange point. \item \emph{Canonical Energy-Scale Threshold (Marchal--Sundman)}: For any isolated, gravitationally bound $N$-body system with negative total energy $E_0 < 0$ and center of mass at rest, the configuration is constrained by the macroscopic Lagrange radius $R_{\mathrm{sys}}(E_0, M_{\mathrm{tot}}) \coloneqq \frac{G M_{\mathrm{tot}}^2}{2 |E_0|}$. The canonical, scale-covariant compactness threshold for cluster $S$ is uniquely given by: \begin{equation} \delta_{\mathrm{cluster}}^{\mathrm{nat}}(S) \coloneqq \eta_{\mathrm{crit}} \left( \frac{M_S}{M_{\mathrm{tot}}} \right)^{1/3} R_{\mathrm{sys}}(E_0, M_{\mathrm{tot}}) \sqrt{M_S} = \eta_{\mathrm{crit}} \frac{G M_{\mathrm{tot}}^{5/3} M_S^{5/6}}{2 |E_0|}. \label{eq:canonical_delta_cluster_formula} \end{equation} Under homothety $\mathbf{q} \mapsto \lambda \mathbf{q}$, $\delta_{\mathrm{cluster}}^{\mathrm{nat}}$ scales covariantly ($\delta_{\mathrm{cluster}}^{\mathrm{nat}} \mapsto \lambda \delta_{\mathrm{cluster}}^{\mathrm{nat}}$), ensuring that the compactness ratio $\rho_S / \delta_{\mathrm{cluster}}^{\mathrm{nat}}$ is an absolute conformal invariant. \item \emph{Universal Closure of the Lipschitz Bounds}: On every energy hypersurface $\Sigma_{E_0} = \{ (\mathbf{q}, \mathbf{p}) \in \mathcal{P}_{\mathrm{ext}} \mid H(\mathbf{q}, \mathbf{p}) = E_0 \}$, the predicate $\operatorname{isCompactCluster}(S)$ is entirely determined by the first integrals $(G, \{m_i\}_{i=1}^N, E_0)$, and the uniform Lipschitz constants $M_{\mathrm{apex}}$ and $L_{\mathrm{apex}}$ of Theorem~\ref{thm:universal_c11_apex} depend exclusively on $(G, \{m_i\}_{i=1}^N, E_0)$, completing the rigorous elimination of all free parameters. \end{enumerate} \end{propbox} \begin{proofbox}[Proof of Proposition~\ref{prop:intrinsic_compactness_threshold}] For Item~(1), let $\mathbf{q}_i = \mathbf{X}_S + \boldsymbol{\delta}_i$ be the position of an arbitrary particle $i \in S$, with $\sum_{i \in S} m_i \boldsymbol{\delta}_i = \mathbf{0}$. The gravitational force exerted by the sibling cluster $S'$ on particle $i$ expands in multipole Taylor series about the barycenter $\mathbf{X}_S$: \begin{equation} \mathbf{F}_{i \leftarrow S'} = - \sum_{j \in S'} \frac{G m_i m_j (\mathbf{q}_i - \mathbf{q}_j)}{\|\mathbf{q}_i - \mathbf{q}_j\|^3} = - \frac{G m_i M_{S'} \mathbf{r}_P}{\|\mathbf{r}_P\|^3} - G m_i M_{S'} \mathbb{T}(\mathbf{r}_P) \boldsymbol{\delta}_i + \mathcal{O}\left( \frac{G m_i M_{S'} \bar{\rho}_S^2}{\|\mathbf{r}_P\|^4} \right), \label{eq:tidal_force_expansion} \end{equation} where $\mathbb{T}(\mathbf{r}_P) = \|\mathbf{r}_P\|^{-3} (\mathbf{I}_3 - 3 \hat{\mathbf{r}}_P \hat{\mathbf{r}}_P^T)$ is the classical symmetric trace-free tidal tensor. The leading term accelerates the center of mass $\mathbf{X}_S$ uniformly, while the differential tidal acceleration disrupting the internal geometry of $S$ is $\mathbf{a}_{\mathrm{tide}, i} = - G M_{S'} \mathbb{T}(\mathbf{r}_P) \boldsymbol{\delta}_i$. Under the mass-weighted Riemannian metric $\mathbb{M}_S = \operatorname{diag}(m_i \mathbf{I}_3)_{i \in S}$, the root-mean-square norm of the internal tidal acceleration satisfies: \begin{align} \|\mathbf{a}_{\mathrm{tide}}\|_S^2 &\coloneqq \frac{1}{M_S} \sum_{i \in S} m_i \|\mathbf{a}_{\mathrm{tide}, i}\|^2 \le \frac{G^2 M_{S'}^2}{\|\mathbf{r}_P\|^6} \|\mathbb{T}(\hat{\mathbf{r}}_P)\|^2 \left( \frac{1}{M_S} \sum_{i \in S} m_i \|\boldsymbol{\delta}_i\|^2 \right) \nonumber \\ &\le \frac{4 G^2 M_{S'}^2}{\|\mathbf{r}_P\|^6} \bar{\rho}_S^2 \implies \|\mathbf{a}_{\mathrm{tide}}\|_S \le \frac{2 G M_{S'}}{\|\mathbf{r}_P\|^3} \bar{\rho}_S. \label{eq:rms_tidal_acceleration_bound} \end{align} Concurrently, the characteristic internal gravitational binding acceleration holding cluster $S$ together scales as $\|\mathbf{a}_{\mathrm{int}}\|_S = G M_S / \bar{\rho}_S^2$. The ratio of disruptive tidal acceleration to cohesive self-gravitational acceleration is therefore: \begin{equation} \frac{\|\mathbf{a}_{\mathrm{tide}}\|_S}{\|\mathbf{a}_{\mathrm{int}}\|_S} \le 2 \frac{M_{S'}}{M_S} \left( \frac{\bar{\rho}_S}{\|\mathbf{r}_P\|} \right)^3 = 2 \, \eta_S(\mathbf{q})^3. \label{eq:tidal_to_internal_ratio} \end{equation} By classical Roche--Hill stability theory (Hill~\cite{hill1878researches}, Marchal~\cite{marchal1990threebody}), an $N$-body sub-system retains dynamical autonomy without tidal stripping if its mean density exceeds three times the Roche density of the perturber, which translates directly to $2 \eta_S^3 \le 2/3 \iff \eta_S \le 3^{-1/3} = \eta_{\mathrm{crit}}$. Whenever $\eta_S \le \eta_{\mathrm{crit}}$, the external tidal field is a minor perturbation ($< 1/3$) of internal gravity, and the internal degrees of freedom are strongly coupled through non-orthogonal kinetic cross-terms, inducing the $\mathcal{O}(\rho_S^{-5/2})$ Poisson bracket divergence (Proposition~\ref{prop:excluded_middle_dilemma}) and necessitating synchronous Sundman integration. Because both $\bar{\rho}_S$ and $\|\mathbf{r}_P\|$ carry the dimension of length $[L]$ while $M_{S'} / M_S$ is dimensionless, $\eta_S(\mathbf{q})$ is strictly scale-invariant under $\mathbf{q} \mapsto \lambda \mathbf{q}$. For Item~(2), consider a bound system with total energy $E_0 = T + V < 0$. By the Sundman--Smale inequality~\cite{sundman1912memoire,smale1970topology} and the Lagrange identity $\ddot{I} = 4 E_0 - 2 V = 4 E_0 + 2 U > 0$, the moment of inertia $I = \sum_{i=1}^N m_i \|\mathbf{q}_i - \mathbf{X}_{\mathrm{cdm}}\|^2$ is bounded in mean by the virial equilibrium $2 \langle T \rangle = -\langle V \rangle = |E_0|$. The total gravitational potential satisfies $U(\mathbf{q}) \ge G M_{\mathrm{tot}}^2 / (2 R_{\mathrm{sys}})$, which establishes $R_{\mathrm{sys}}(E_0, M_{\mathrm{tot}}) = G M_{\mathrm{tot}}^2 / (2 |E_0|)$ as the maximal macroscopic bounding scale of the system (Marchal \& Saari~\cite{marchal1976general}). For any internal sub-cluster $S$, the maximum separation from the remaining mass $M_{\mathrm{tot}} - M_S$ is bounded by $R_{\mathrm{sys}}$, so that the tidal condition $\bar{\rho}_S \le \eta_{\mathrm{crit}} (M_S / M_{\mathrm{tot}})^{1/3} \|\mathbf{r}_P\|$ yields the canonical scale threshold $\delta_{\mathrm{cluster}}^{\mathrm{nat}}(S) = \eta_{\mathrm{crit}} (M_S / M_{\mathrm{tot}})^{1/3} R_{\mathrm{sys}} \sqrt{M_S}$. Under homothety $\mathbf{q} \mapsto \lambda \mathbf{q}$, the potential scales as $V \mapsto \lambda^{-1} V$ and the energy as $E_0 \mapsto \lambda^{-1} E_0$; hence $R_{\mathrm{sys}} \mapsto \lambda R_{\mathrm{sys}}$ and $\delta_{\mathrm{cluster}}^{\mathrm{nat}}(S) \mapsto \lambda \delta_{\mathrm{cluster}}^{\mathrm{nat}}(S)$, demonstrating strict covariance. For Item~(3), substituting $\delta_{\mathrm{cluster}}^{\mathrm{nat}}(S)$ into the lower bound $\bar{\rho}_{\mathrm{int}}(v) \ge c_0 \delta_{\mathrm{cluster}}$ of Theorem~\ref{thm:universal_c11_apex} (Regime 2) expresses the infimum $\bar{\rho}_{\mathrm{int}}^{-1} \le (c_0 \delta_{\mathrm{cluster}}^{\mathrm{nat}})^{-1}$ as a function of $E_0$, $G$, and $\{m_i\}$. Consequently, the Lipschitz bounds $M_{\mathrm{apex}}$ and $L_{\mathrm{apex}}$ are closed, finite functions of the physical first integrals alone, without any external parameter choice. \end{proofbox} \begin{corbox}[Predictive Dyadic Anticipation and Spectral Schmitt Trigger]{cor:hill_hysteresis_anticipation} In practical numerical integration, the boundary crossing of the Hill tidal domain $\{\eta_S \le \eta_{\mathrm{crit}}\}$ is governed by a two-tier coupled temporal-topological protocol: \begin{enumerate} \item \emph{Upstream Predictive Dyadic Anticipation}: For any approaching external perturber ($v_{\mathrm{rad}} = \langle \mathbf{v}_P, \mathbf{r}_P \rangle / \|\mathbf{r}_P\| < 0$), the dyadic multirate scheduler (formally defined in Section~\ref{sec:dyadic_multirate_scheduling} and algorithmically implemented in Sections~\ref{sec:initialization_shadow}--\ref{sec:dyadic_scheduler} and Appendix~\ref{app:dyadic_scheduler}) continuously estimates the ballistic encounter timescale $t_{\mathrm{approach}} = - \|\mathbf{r}_P\| / v_{\mathrm{rad}}$. Whenever $t_{\mathrm{approach}} \le 2 \Delta t_{\mathrm{macro}}$, the dyadic exponent $k$ is refined proactively ($h_k = \Delta t_{\mathrm{macro}} / 2^k$) \emph{prior} to crossing the tidal boundary. This guarantees that the system enters the stiff Hill interaction zone with an already refined temporal grid, preventing sudden truncation shock and energy error spikes. \item \emph{Canonical Spectral Schmitt Trigger Hysteresis ($[\eta_{\mathrm{enter}}, \eta_{\mathrm{exit}}] = [2^{-1/3}, 2^{-1/6}] \approx [0.7937, 0.8909]$)}: To eliminate high-frequency numerical chattering without delaying activation, both thresholds of the Schmitt trigger band are derived analytically from the spectral norms of the trace-free tidal tensor $\mathbb{T}(\hat{\mathbf{r}}_P) = \mathbf{I}_3 - 3 \hat{\mathbf{r}}_P \hat{\mathbf{r}}_P^T$ and positioned strictly \emph{above} the Hill stability limit $\eta_{\mathrm{crit}} = 3^{-1/3} \approx 0.6934$: \begin{enumerate} \item \emph{Proactive Activation at Radial Equigravity ($\eta_{\mathrm{enter}} \coloneqq 2^{-1/3} \approx 0.7937$)}: Along the line of centers $\hat{\mathbf{r}}_P$, the maximum stretching tidal eigenvalue is $|\lambda_{\mathrm{rad}}| = 2$. The exact balance where peak external tidal acceleration equals internal self-gravitational binding acceleration ($a_{\mathrm{tide}, \max} = a_{\mathrm{int}} \iff 2 \eta_S^3 = 1$) uniquely defines: \begin{equation} \eta_{\mathrm{enter}} = \left(\frac{1}{2}\right)^{1/3} = 2^{-1/3} \approx 0.7937005. \label{eq:eta_enter_spectral_def} \end{equation} Cluster $S$ is promoted to an active synchronous supernode as soon as $\eta_S \le \eta_{\mathrm{enter}}$, anticipating entry into the physical Hill sphere by a scale factor of $(3/2)^{1/3} \approx 1.145$ ($14.5\%$ geometric safety margin). \item \emph{Guaranteed Deactivation at Spherical RMS Equigravity ($\eta_{\mathrm{exit}} \coloneqq 2^{-1/6} \approx 0.8909$)}: Averaged uniformly over the unit sphere $\mathbb{S}^2$, the root-mean-square norm of the tidal tensor is $\|\mathbb{T}\|_{\mathrm{rms}} = \sqrt{\frac{1}{3} \operatorname{Tr}(\mathbb{T}^2)} = \sqrt{2}$. The threshold where the mean spherical tidal pull equals internal binding ($\sqrt{2} \eta_S^3 = 1$) uniquely yields: \begin{equation} \eta_{\mathrm{exit}} = \left(\frac{1}{\sqrt{2}}\right)^{1/3} = 2^{-1/6} \approx 0.8908987. \label{eq:eta_exit_spectral_def} \end{equation} $S$ remains an active supernode until $\eta_S > \eta_{\mathrm{exit}}$. \end{enumerate} The canonical spectral ordering: \begin{equation} \eta_{\mathrm{crit}} = 3^{-1/3} \approx 0.6934 < \eta_{\mathrm{enter}} = 2^{-1/3} \approx 0.7937 < \eta_{\mathrm{exit}} = 2^{-1/6} \approx 0.8909, \label{eq:spectral_hysteresis_hierarchy} \end{equation} satisfies the exact dyadic squaring identity $\eta_{\mathrm{enter}} = (\eta_{\mathrm{exit}})^2$. Because both thresholds reside strictly above the physical Hill limit, orbital breathing and roundoff fluctuations around $\eta_{\mathrm{crit}}$ occur deep inside the locked active supernode regime, completely precluding topological chattering while ensuring zero activation latency. \end{enumerate} \end{corbox} \begin{proofbox}[Proof of Corollary~\ref{cor:hill_hysteresis_anticipation}] By Proposition~\ref{prop:intrinsic_compactness_threshold}, the dimensionless tidal tensor $\mathbb{T}(\hat{\mathbf{r}}_P) = \mathbf{I}_3 - 3\hat{\mathbf{r}}_P \hat{\mathbf{r}}_P^T$ has eigenvalues $(-2, 1, 1)$. Its operator norm (realized along $\pm \hat{\mathbf{r}}_P$) is $\|\mathbb{T}\|_{\mathrm{op}} = 2$, while its spherical root-mean-square norm over $\mathbb{S}^2$ is $\|\mathbb{T}\|_{\mathrm{rms}} = \sqrt{\frac{1}{3}((-2)^2 + 1^2 + 1^2)} = \sqrt{2}$. Equating the corresponding tidal-to-self-gravity acceleration ratios $2\eta_S^3$ and $\sqrt{2}\eta_S^3$ to unity yields $\eta_{\mathrm{enter}} = 2^{-1/3}$ and $\eta_{\mathrm{exit}} = 2^{-1/6} = \sqrt{\eta_{\mathrm{enter}}}$. Since $3 > 2 > \sqrt{2} > 1$, taking the inverse cube root reverses the strict inequalities to yield $\eta_{\mathrm{crit}} = 3^{-1/3} < \eta_{\mathrm{enter}} = 2^{-1/3} < \eta_{\mathrm{exit}} = 2^{-1/6}$, establishing a non-empty hysteresis deadband of width $\eta_{\mathrm{exit}} - \eta_{\mathrm{enter}} \approx 0.0972$ strictly above the physical Roche--Hill stability limit. \end{proofbox} % ------------------------------------------------------------------------------ \subsubsection{Synthesis of Section 4: The Complete Hierarchical Symplectic Architecture} \label{sub:ch04_synthesis} With the establishment of Theorems~\ref{thm:fulton_macpherson_snc}, \ref{thm:universal_c11_apex}, \ref{thm:anti_zeno_regularization}, and~\ref{thm:atlas_transition_anti_chattering}, the geometric and analytical construction of the Fulton--MacPherson hierarchical splitting on $\mathbf{Q}[N]$ is complete: \begin{enumerate} \item \emph{Spatial Regularization without Distance Softening}: The collision arrangement $\Delta$ is blown up into the compact manifold with corners $\mathbf{Q}[N]$, transforming physical collision singularities into regular boundary divisors with simple normal crossings, without modifying Newton's gravitational law ($\varepsilon = 0$ strictly preserved). \item \emph{Disjoint Symplectic Decoupling}: On maximal Jacobi trees, disjoint sub-clusters commute exactly on phase space ($[\mathcal{X}_A, \mathcal{X}_B] = 0$), allowing fully independent local integration. \item \emph{Universal $\mathcal{C}^{1,1}$ Apex Screening}: The scale-adapted screening function $\Theta(u_v)$ eliminates the multipolar ghost pole at cluster barycenter coincidence, providing uniform Lipschitz bounds ($M_{\mathrm{apex}}, L_{\mathrm{apex}} < \infty$) without introducing any artificial distance cut-off ($d_0 = 0$). \item \emph{Exact Symmetry and Palindromic Invariance}: The screened apex flow strictly conserves all 10 generators of the centrally extended Lie--Bargmann algebra, while symmetric Strang splitting eliminates all odd-order dissipative errors. \item \emph{Anti-Zeno Supernode Synchrony}: Dense sub-clusters ($k \ge 3$) are unified into synchronous supernodes under Sundman's unfolded local clock $\mathrm{d}t_S = \chi_S^2 \mathrm{d}\tau_S$, eliminating the $\mathcal{O}(\delta^{-5/2})$ commutator catastrophe and guaranteeing finite crossing time across multiple quasi-collisions. \item \emph{Symplectic Atlas Continuity}: Transitions between adapted Jacobi charts belong to $\operatorname{Sp}(6N-6, \mathbb{R})$ under canonical parameter-free Voronoi partitioning ($\gamma = 1.0$, or optional asymmetric hysteresis $\gamma = 0.6$ guaranteeing positive dwell time without topological chattering). \end{enumerate} \begin{conormalbox}{Unsoftened 3D Newtonian $N$-Body Problem on $\mathbf{Q}[N]$} \textbf{Geometric Base \& Normal Fiber:} The physical system describes $N \ge 3$ point masses $m_1, \dots, m_N > 0$ in $\R^3$ interacting through unsoftened Newtonian gravity ($\varepsilon = 0$). In the center-of-mass frame, the relative configuration space is $M = (\R^3)^N / \R^3 \cong \R^{3N-3}$, equipped with the mass Riemannian metric $\mathbb{M} = \operatorname{diag}(m_1 I_3, \dots, m_N I_3)$ and physical momentum space $M^* = \{\mathbf{p} \in (\R^3)^N \mid \sum \mathbf{p}_i = \mathbf{0}\}$. The geometric phase space is the extended Poincaré--Sundman manifold equipped with the extended Darboux 2-form: \begin{equation} \mathcal{P}_{\mathrm{ext}} = T^*\mathbf{Q}[N] \times \R_t \times \R_E, \qquad \Omega_{\mathrm{ext}} = \Omega_0 + \mathrm{d}E \wedge \mathrm{d}t = \mathrm{d}\mathbf{q} \wedge \mathrm{d}\mathbf{p} + \mathrm{d}E \wedge \mathrm{d}t, \end{equation} where $(t, -E) = (q_t, p_t)$ form a canonical pair with $\{t, -E\} = 1$. The collision locus is the Whitney-stratified arrangement organized along the partition lattice $\Pi(N)$: \begin{equation} \Delta = \bigcup_{1 \le i < j \le N} \Delta_{ij}, \qquad \Delta_I = \{\mathbf{q} \in M \mid \mathbf{q}_i = \mathbf{q}_j \; \forall i, j \in I\}, \qquad \operatorname{codim}_M(\Delta_I) = 3(|I| - 1). \end{equation} The configuration base is the smooth manifold with corners $\mathbf{Q}[N] = \operatorname{Bl}_{\mathcal{G}_{\mathrm{irr}}}(M)$ obtained by iterative real oriented spherical blow-ups in order of increasing stratum dimension (decreasing cluster cardinality), whose boundary $\partial \mathbf{Q}[N] = \bigcup_{|I| \ge 2} E_I$ is a simple normal crossing (SNC) divisor with corners with defining normal coordinates $\rho_I = \sqrt{\frac{1}{M_I}\sum_{i 0$, guaranteeing finite sub-steps $N_{\mathrm{sub}} \le \lceil \Delta t_{\mathrm{phys}} / (\rho_{\min} \Delta \tau_S) \rceil < \infty$ across all multiple close encounters. \item \emph{Symplectic Atlas Transitions \& Anti-Chattering}: Inter-chart Jacobi transformations $\Psi_{A \to B} = \Psi_{\mathrm{rot}} \oplus \mathbf{I}_{6N-18} \in \operatorname{Sp}(6N-6, \R)$ with $\mathbf{M}_{\mathrm{coord}} \in \mathrm{SL}(2, \R)$ preserve $\Omega_{\mathrm{ext}}$ under canonical parameter-free Voronoi partitioning ($\gamma = 1.0$), while optional asymmetric hysteresis ($\gamma = 0.6$) strictly forbids topological chattering ($1 < 0.36$ impossibility), ensuring a finite dwell time $\tau_{\min} = \frac{16}{15 \Omega_{\max}} > 0$ and bounded transitions $N_{\mathrm{trans}}(T) < \infty$. \end{itemize} \end{conormalbox} \paragraph{Transition to High-Order Numerical Integrators.} The continuous Hamiltonian decomposition established in Section~\ref{ch:ch04_fulton_macpherson_splitting} provides the exact mathematical substrate required for discrete algorithmics. In Section~\ref{ch:ch05_higher_order_symplectic}, we operationalize this structure into computable high-order numerical propagators: we construct the 7-stage palindromic Strang scheme $\mathcal{S}_2(h)$, analyze the Suzuki barrier forbidding positive-coefficient integrators beyond order 2, prove that McGehee bounce involution guarantees unconditional stability under negative sub-steps ($w_i < 0$), and formulate the optimized 4th-order Blanes--Moan propagator $\mathrm{BM}_4$ minimizing the shadow Hamiltonian error on $\mathbf{Q}[N]$.