% ============================================================================== % chapters/ch06_algorithmic_architecture/sec01_global_pipeline/content.tex % Section 6.1: Global Algorithmic Architecture and Six-Phase Execution Pipeline % ============================================================================== \subsection{Global Algorithmic Architecture and Six-Phase Execution Pipeline} \label{sec:global_pipeline} The mathematical formulation developed across Sections~\ref{ch:ch01_introduction} through~\ref{ch:ch05_higher_order_symplectic} establishes the exact geometric desingularization of the gravitational $N$-body problem on the compactified configuration manifold with corners $\mathbf{Q}[N]$ \cite{fulton1994compactification} and its shifted cotangent lift $T^*\mathbf{Q}[N]$ \cite{calaque2024deformation}. The objective of this section is to translate these differential geometric structures into an autonomous, certified algorithmic architecture with asynchronous shared-memory parallelism. In this introductory section, we formalize the \emph{Unified Symplectic Integration Architecture} --- here operating as the concrete algorithmic realization of the discrete symplectic splitting on the shifted cotangent lift of the normal cone deformation of collision diagonals ($T^*\mathbf{Q}[N]$) --- as a hierarchical six-phase execution pipeline (Phases~0 through~5). We establish the phase space representations, the immutable state structures, the asynchronous dependency graph governing parallel CPU execution without mutex contention, and the general operational complexity of the solver. % ------------------------------------------------------------------------------ \subsubsection{The Unified Symplectic Integration Architecture: A Six-Phase Stratification} \label{sub:five_tier_stratification} Rather than representing an eclectic assembly of preexisting numerical heuristics, the architecture formalized here is the rigorous, constructive operationalization of the analytical theorems established across Sections~\ref{ch:ch01_introduction} through~\ref{ch:ch05_higher_order_symplectic}. Every operational phase is strictly anchored to, and justified by, a proved mathematical necessity: desingularization on $\mathbf{Q}[N]$, smooth apex screening, spectral Hill confinement, homological corrector solvability, boundary synchronization, leaf/apex decoupling, and canonical gauge continuity. Our computational engine organizes these structures into six sequentially coupled operational phases (Phases~0 through~5, comprising an initial setup Phase~0 followed by five runtime integration phases), summarized in Table~\ref{tab:five_tier_pipeline_architecture}. The operational rationale of each phase is directly anchored to our theoretical theorems: \begin{enumerate} \item \textbf{Phase~0 (Topological and Geometric Initialization)}: At the onset of simulation ($t = 0$), ambient coordinates $(\mathbf{q}, \mathbf{p}) \in (\R^3)^N \times (\R^3)^N$ are projected onto the center-of-mass frame $M_{\mathrm{rel}} = M / \Delta_{\mathrm{trans}}$. An initial adapted binary tree $T_0$ is constructed by agglomerative hierarchical clustering on the Fulton--MacPherson configuration manifold with corners $\mathbf{Q}[N]$ (Theorem~\ref{thm:fulton_macpherson_snc}), exploiting the exact cotangent commutation of disjoint sub-clusters (Proposition~\ref{prop:disjoint_cotangent_commutation}). Strongly coupled subsystems are tagged as candidate synchronous supernodes via the spectral Hill Schmitt trigger (Corollary~\ref{cor:hill_hysteresis_anticipation}). \item \textbf{Phase~1 (Symplectic Pre-Processing and Regime Dichotomy)}: The initialization of the phase state branches according to the selected symplectic architecture: \begin{itemize} \item \emph{Processed Regime ($\mathrm{PBM}_4$)}: To unlock higher effective order in the hierarchical regime without intermediate force evaluations, the physical state $\mathbf{s}_{\mathrm{phys}}(0)$ is transferred into the symplectic shadow manifold via the negative time-step Lie corrector flow ($K_4 \in \mathcal{P}_1$): \[ \mathbf{s}_{\mathrm{shadow}}(0) = \mathcal{Z}_4^{\mathrm{BM4}}(-h_0)[\mathbf{s}_{\mathrm{phys}}(0)] = \bigl(\exp(-D_{K_{4}}) + \mathcal{O}(\varepsilon^2 h_0^4)\bigr)[\mathbf{s}_{\mathrm{phys}}(0)]. \] The homological cancellation of the linear tidal error and asymptotic effective order~6 on $\mathcal{D}_{\mathrm{reg}}$ (for $\varepsilon \ll h^2$) alongside the irreducible quadratic floor $\mathcal{O}(\varepsilon^2 h^4)$ are guaranteed by Theorems~\ref{thm:universal_order6_obstruction}, \ref{thm:c4_corrector_solvability}, \ref{thm:asymptotic_conormal_scaling}, and~\ref{thm:shadow_alignment}. \item \emph{Direct Composition Regime ($\mathrm{BM}_4^{(7)}$, $\mathrm{BM}_6^{(11)}$)}: Because direct Blanes--Moan compositions operate directly on the physical phase space without boundary conjugation, the boundary corrector is identically trivial ($\mathcal{C} \equiv \operatorname{Id}$). Consequently, physical phase space and shadow phase space coincide identically ($\mathbf{s}_{\mathrm{shadow}}(0) \equiv \mathbf{s}_{\mathrm{phys}}(0)$), requiring zero pre-processing overhead. \end{itemize} Exact Lie--Bargmann preservation across both regimes is certified by Theorem~\ref{thm:c4_corrector_solvability} and Proposition~\ref{prop:corrector_lie_bargmann}. \item \textbf{Phase~2 (Macro-Time Loop and Dyadic Scheduler)}: Time progresses through discrete macro-intervals $[t_n, t_n + \Delta t_{\mathrm{macro}}]$. In accordance with the Ge--Marsden theorem \cite{ge1988lie} on variable time-stepping obstructions, unconditional exact symplecticity is guaranteed by freezing the chart topology $\mathcal{T}_n$ and dyadic refinement level $k_n$ throughout each macroblock, as grounded in the Skeel--Gear boundary synchronization and dyadic time-reversibility proposition (Proposition~\ref{prop:skeel_gear_boundary_sync}). At each macro-step entry, the dyadic multirate scheduler scans pairwise dynamical timescales (Proposition~\ref{prop:dyadic_scale_invariance}) and applies our Leaf/Apex Decoupling Theorem (Theorem~\ref{thm:leaf_apex_decoupling}) to isolate regularized Keplerian pairs from singular pericenter anticipation, strictly preventing dyadic step-size saturation. \item \textbf{Phase~3 (High-Order Symplectic Sub-Cycle)}: Over each sub-step $h_k = \Delta t_{\mathrm{macro}} / 2^k$, the system executes a palindromic composition ($\mathrm{BM}_6^{(11)}$ or $\mathrm{BM}_4^{(7)}$) whose elementary stages interleave three geometric operators: \begin{itemize} \item \emph{Universal Kepler--Sundman Propagation}: analytic leaf-pair propagation (Proposition~\ref{prop:sundman_stumpff_diffeomorphism}) equipped with Battin symplectic projection and the unitary McGehee radial bounce $\mathcal{B}_C^2 = \mathrm{Id}$ (Lemma~\ref{thm:mcgehee_bounce_unitary}); \item \emph{Synchronous Supernode Evolution}: internal recursive sub-cycling of strongly coupled clusters with universal anti-Zeno regularization (Theorem~\ref{thm:anti_zeno_regularization}); \item \emph{Screened Apex Kick}: global multipolar/tidal momentum update whose non-explosion is ensured by the universal $\mathcal{C}^{1,1}$ Lipschitz regularity of $W_{\mathrm{apex}}^{\mathrm{screen}}$ on $\mathbf{Q}[N]$ (Theorem~\ref{thm:universal_c11_apex}). \end{itemize} \item \textbf{Phase~4 (Dynamical Atlas Transitions and Symplectic Gauge Connection)}: At integer macro-step boundaries, the active tree chart $T$ is tested against Delaunay--Jacobi geometric admissibility ($d_{\min} < \gamma r_{ij}$, with canonical Voronoi default $\gamma = 1.0$ or optional anti-chattering hysteresis $\gamma = 0.60$). Under the canonical Voronoi partition $\gamma = 1.0$, chart assignment is a memoryless single-valued function of configuration space that commutes natively with time reversal; when optional hysteresis $\gamma = 0.60$ is enabled, our anti-chattering stability theorem (Theorem~\ref{thm:atlas_transition_anti_chattering}) guarantees a spatial contraction factor $\gamma^2 = 0.36 < 1$. State transfer to the new chart $T'$ is governed by the symplectic gauge connection: under $\mathrm{PBM}_4$, the dedicated covariant gauge operator $T_{A \to B}^{(\mathrm{PBM4})}$ (Theorem~\ref{thm:pbm4_gauge_operator_exact}) eliminates shadow shock; under direct composition schemes ($\mathrm{BM}_4^{(7)}$, $\mathrm{BM}_6^{(11)}$), the transition collapses to the pure linear Darboux map $\mathbf{M}_{\mathrm{coord}} \in \mathrm{SL}(2, \mathbb{R}) \subset \operatorname{Sp}(6N-6, \mathbb{R})$ with strictly zero energy jump $\Delta H_{\mathrm{gauge}} \equiv 0$ (Proposition~\ref{prop:sl2_unimodularity}). \item \textbf{Phase~5 (Symplectic Post-Processing and Invariant Verification)}: Upon reaching the target integration epoch $t = t_{\mathrm{end}}$, the shadow state is mapped back to physical coordinates via the direct forward corrector $\mathbf{s}_{\mathrm{phys}}(t_{\mathrm{end}}) = \mathcal{Z}_4^{\mathrm{BM4}}(+h_{\mathrm{exit}})[\mathbf{s}_{\mathrm{shadow}}(t_{\mathrm{end}})]$ for $\mathrm{PBM}_4$, or identity mapping for direct composition schemes, delivering the final high-order state without secular drift (Proposition~\ref{prop:corrector_lie_bargmann}). This exit step is followed by an exhaustive audit of the 10 Lie--Bargmann conservation laws (Proposition~\ref{prop:lie_bargmann_closure}). \end{enumerate} \suppressfloats[t] \begin{table}[!htbp] \centering \small \caption{\textbf{The Six Operational Phases of the Unified Symplectic Integrator and their Monograph Foundations.}} \label{tab:five_tier_pipeline_architecture} \vspace{0.3em} \renewcommand{\arraystretch}{1.22} \resizebox{\textwidth}{!}{% \begin{tabular}{p{3.2cm}p{6.8cm}p{4.8cm}} \toprule \textbf{Operational Phase \& Cadence} & \textbf{Mathematical Objective \& Invariants} & \textbf{Monograph Foundations} \\ \midrule \textbf{Phase~0: Geometric Initialization} \newline \scriptsize Initial setup ($t = 0$) & Barycentric reduction ($\mathbf{P}_{\mathrm{tot}} = \mathbf{0}$, $\mathbf{R}_{\mathrm{cm}} = \mathbf{0}$), optimal Jacobi tree construction $T_0$, and spectral Hill identification of synchronous supernodes $\mathcal{S}_{\mathrm{sync}} = \{ S \mid \eta_S \le 2^{-1/3} \}$. & Theorem~\ref{thm:fulton_macpherson_snc}, Proposition~\ref{prop:disjoint_cotangent_commutation}, Corollary~\ref{cor:hill_hysteresis_anticipation} \\ \textbf{Phase~1: Symplectic Pre-Processing} \newline \scriptsize Initial setup ($t = 0$) & \emph{Processed ($\mathrm{PBM}_4$)}: canonical shadow entry $\mathbf{s}_{\mathrm{shadow}} = \mathcal{Z}_4^{\mathrm{BM4}}(-h_0)[\mathbf{s}_{\mathrm{phys}}]$, aligning the trajectory with $\widetilde{H}_{\mathrm{eff}} = H + \mathcal{O}(\varepsilon^2 h^4 + \varepsilon h^6)$ on $\mathcal{D}_{\mathrm{reg}}$. \newline \emph{Direct ($\mathrm{BM}_4^{(7)}, \mathrm{BM}_6^{(11)}$)}: $\mathcal{C} \equiv \operatorname{Id}$, so $\mathbf{s}_{\mathrm{shadow}} \equiv \mathbf{s}_{\mathrm{phys}}$ unconditionally with zero overhead. & Theorems~\ref{thm:universal_order6_obstruction}, \ref{thm:c4_corrector_solvability}, \ref{thm:asymptotic_conormal_scaling}, \ref{thm:shadow_alignment} \\ \textbf{Phase~2: Macro-Time Scheduling} \newline \scriptsize Each macro-step ($t_n$) & Dynamical interaction timescale evaluation, dyadic subdivision $h_k = \Delta t_{\mathrm{macro}} / 2^k$ ($k \in [0, k_{\max}]$), bit-exact timestamping, and leaf/apex decoupling. & Proposition~\ref{prop:skeel_gear_boundary_sync}, Theorem~\ref{thm:leaf_apex_decoupling}, Proposition~\ref{prop:dyadic_scale_invariance} \\ \textbf{Phase~3: Symplectic Sub-Cycle} \newline \scriptsize $2^k$ micro-steps ($h_k$) & Palindromic composition ($\mathrm{BM}_6^{(11)}$ or $\mathrm{BM}_4^{(7)}$) interleaving analytic Kepler--Sundman flow, McGehee bounce ($\mathcal{B}_C^2 = \mathrm{Id}$), Battin projection ($f\dot{g} - \dot{f}g \equiv 1$), and screened apex kick. & Lemma~\ref{thm:mcgehee_bounce_unitary}, Theorems~\ref{thm:universal_c11_apex}, \ref{thm:anti_zeno_regularization}, Proposition~\ref{prop:sundman_stumpff_diffeomorphism} \\ \textbf{Phase~4: Dynamical Atlas Transition} \newline \scriptsize Macro-step boundary (conditional) & Delaunay--Jacobi geometric admissibility ($d_{\mathrm{ext}}^{\mathrm{Hill}} < \gamma r_{ij}$, guaranteeing \emph{a posteriori} anti-chattering dwell time $\Delta t_{\mathrm{dwell}} \ge \tau_{\min} > 0$ via spatial hysteresis $\gamma^2 = 0.36 < 1$) and canonical gauge connection $\mathcal{J}_{T \to T'}$ ($\det \mathbf{M} = 1$). \newline \emph{Processed ($\mathrm{PBM}_4$)}: covariant gauge operator $T_{A \to B}^{(\mathrm{PBM4})}$. \newline \emph{Direct ($\mathrm{BM}_4^{(7)}, \mathrm{BM}_6^{(11)}$)}: $\mathcal{J}_{T \to T'} \equiv \mathbf{M}_{\mathrm{coord}} \in \mathrm{SL}(2, \mathbb{R})$ with $\Delta H_{\mathrm{gauge}} \equiv 0$. & Theorems~\ref{thm:atlas_transition_anti_chattering}, \ref{thm:pbm4_gauge_operator_exact}, Proposition~\ref{prop:sl2_unimodularity} \\ \textbf{Phase~5: Symplectic Post-Processing} \newline \scriptsize Terminal epoch ($t = t_{\mathrm{end}}$) & \emph{Processed ($\mathrm{PBM}_4$)}: shadow exit $\mathbf{s}_{\mathrm{phys}} = \mathcal{Z}_4^{\mathrm{BM4}}(+h_{\mathrm{exit}})[\mathbf{s}_{\mathrm{shadow}}]$ (asymptotic effective order $6$ on $\mathcal{D}_{\mathrm{reg}}$ when $\varepsilon \ll h^2$). \newline \emph{Direct ($\mathrm{BM}_4^{(7)}, \mathrm{BM}_6^{(11)}$)}: identity mapping $\mathbf{s}_{\mathrm{phys}} \equiv \mathbf{s}_{\mathrm{shadow}}$. \newline Followed by Lie--Bargmann audit ($\Delta \mathbf{P}, \Delta \mathbf{L}, \Delta \mathbf{G} < 10^{-14}$; $\Delta H = \mathcal{O}(h_{\mathrm{macro}}^{p_{\mathrm{eff}}})$). & Proposition~\ref{prop:lie_bargmann_closure}, Theorem~\ref{thm:c4_corrector_solvability} \\ \bottomrule \end{tabular}% } \end{table} % ------------------------------------------------------------------------------ \subsubsection{Phase Space Representations and Fundamental Data Structures} \label{sub:data_structures} To maintain strict isomorphism between the mathematical monograph and software realization, we identify three coordinate representations of the instantaneous $N$-body state, mapped directly to immutable Python data structures: \begin{enumerate} \item \textbf{Cartesian Ambient State and Barycentric Projection}: The physical configuration of the $N$ bodies is stored in Cartesian arrays $\mathbf{q} \in \R^{N \times 3}$ and $\mathbf{p} \in \R^{N \times 3}$, with strictly positive diagonal masses $\mathbf{m} \in \R_{>0}^N$. The barycentric reduction onto $M_{\mathrm{rel}} \cong (\R^3)^{N-1}$ is performed by: \begin{equation} \begin{aligned} \mathbf{R}_{\mathrm{cm}} &= \frac{1}{\Mtot} \sum_{i=1}^N m_i \mathbf{q}_i, &\qquad \mathbf{P}_{\mathrm{tot}} &= \sum_{i=1}^N \mathbf{p}_i, \\ \mathbf{q}_i^{\mathrm{bary}} &= \mathbf{q}_i - \mathbf{R}_{\mathrm{cm}}, &\qquad \mathbf{p}_i^{\mathrm{bary}} &= \mathbf{p}_i - \frac{m_i}{\Mtot} \mathbf{P}_{\mathrm{tot}}. \end{aligned} \label{eq:barycentric_reduction_def} \end{equation} In the software engine, this corresponds to the immutable frozen dataclass \texttt{Extended\allowbreak Phase\allowbreak State} (defined in \texttt{core/state.py}), which enforces read-only array flags (\texttt{flags.writeable = False}) to ensure mathematical immutability across parallel threads. \item \textbf{Hierarchical Jacobi Coordinates on $\mathbf{Q}[N]$}: Let $T$ be an adapted binary tree in the dynamical atlas $\mathcal{A}$. Each internal node $v \in \mathrm{Internal}(T)$ defines a clustering partition of particle indices into left and right sub-clusters $S_L(v)$ and $S_R(v)$, with cumulative masses $M_L(v) = \sum_{i \in S_L} m_i$ and $M_R(v) = \sum_{j \in S_R} m_j$. The relative Jacobi separation, reduced mass, and conjugate cotangent momentum are given by: \begin{equation} \begin{aligned} \mathbf{r}_v &= \mathbf{X}_L(v) - \mathbf{X}_R(v) = \frac{1}{M_L(v)} \sum_{i \in S_L} m_i \mathbf{q}_i - \frac{1}{M_R(v)} \sum_{j \in S_R} m_j \mathbf{q}_j, \\ \mu_v &= \frac{M_L(v) M_R(v)}{M_L(v) + M_R(v)}, \qquad \mathbf{w}_v = \mu_v \dot{\mathbf{r}}_v \in \R^3. \end{aligned} \label{eq:jacobi_coordinates_def} \end{equation} Terminal leaf pairs $v \in \operatorname{LeafPairs}(T)$ correspond to isolated binary pairs $S(v) = \{i, j\}$ with $\mathbf{r}_v = \mathbf{q}_i - \mathbf{q}_j$ and $\mu_v = m_i m_j / (m_i + m_j)$. In the code, this topology is modeled by the \texttt{JacobiTree} class (in \texttt{core/jacobi.py}). \item \textbf{Extended Poincaré--Sundman State Space}: To govern multirate time re-scaling and guarantee exact conservation of the energy hypersurface $\Sigma_0$, the physical state is embedded into the extended phase space: \begin{equation} \mathcal{P}_{\mathrm{ext}} = T^*\mathbf{Q}[N] \times \R_t \times \R_E, \quad \Omega_{\mathrm{ext}} = \sum_{i=1}^N \mathrm{d}\mathbf{q}_i \wedge \mathrm{d}\mathbf{p}_i + \mathrm{d}E \wedge \mathrm{d}t, \label{eq:extended_poincare_sundman_phase_space} \end{equation} where $E$ represents the extended conjugate energy variable satisfying the canonical Poisson bracket $\{t, E\} = -1$. The conserved baseline $E_0 = H(\mathbf{q}_0, \mathbf{p}_0)$ anchors the zero-energy surface $\Sigma_0 = \{ (\mathbf{z}, t, E) \mid H(\mathbf{z}) - E = 0 \}$. \item \textbf{Unified Simulation Orchestrator (\texttt{NBodySystem})}: The high-level orchestrator \texttt{NBodySystem} (in \texttt{system/nbody\_system.py}) encapsulates the coupled interaction between the instantaneous state, the active tree topology, the symplectic integrator engine, the dyadic multirate scheduler, and the diagnostic history tracker. \end{enumerate} \pythonrefright{py_system}{core/state.py, system/nbody_system.py}{Float64 / Float128} % ------------------------------------------------------------------------------ \subsubsection{Asynchronous Shared-Memory Task Scheduling} \label{sub:asynchronous_task_scheduling} A central architectural decision of the present solver is its strict specialization for \emph{multithreaded CPU architectures}, explicitly excluding GPU / CUDA hardware accelerators. This choice is grounded on fundamental dynamical and computational considerations: \begin{enumerate} \item \textbf{High Algorithmic Branching Factor and Warp Divergence}: In unsoftened gravitational integration ($\varepsilon = 0$), particles occupy widely separated dynamical regimes. A tight binary pair near pericenter requires high dyadic refinement ($k \ge 5$) and multiple cubic Halley iterations for Stumpff series inversion, whereas wide outer clusters evolve under coarse macro-steps ($k = 0$). In a SIMD/SIMT GPU execution environment, threads within the same warp executing divergent branches tend to be serialized, which can substantially reduce effective GPU arithmetic throughput. \item \textbf{Irregular and Dynamic Tree Topologies}: Evaluating the screened apex potential $W_{\mathrm{apex}}^{\mathrm{screen}}$ involves traversal of hierarchical Jacobi trees with non-uniform clustering depths. Tree transformations ($\mathcal{J}_{A \to B}$) alter memory locality at run-time, causing irregular indirect memory accesses that defeat GPU cache hierarchies. \item \textbf{Microscopic Symplectic Reversibility and Float64 Guardrails}: Exact preservation of the time-reversibility involution ($\Phi_{-h} \circ \Phi_h = \mathrm{Id}$ to $10^{-16}$) requires bit-exact scalar IEEE~754 float64 determinism, avoiding non-associative parallel floating-point reductions that corrupt formal invariants. \end{enumerate} To achieve high deterministic throughput on modern CPU processors, the computational workload within each symplectic sub-step (Phase~3) is organized into decoupled, zero-allocation kernels exploiting spatial independence: \begin{itemize} \item \textbf{Operator Independence of Leaf Pairs}: Each terminal leaf pair $v \in \operatorname{LeafPairs}(T)$ corresponds to a disjoint particle index partition $S(v) = \{i, j\}$. The associated Kepler--Sundman flows commute exactly in Poisson bracket and operate on mutually disjoint slices of the configuration and momentum arrays. \item \textbf{Vectorized Spatial Reductions and \texttt{@njit} Compilation}: The global screened apex force field $\mathbf{F}_i = -\nabla_{\mathbf{q}_i} W_{\mathrm{apex}}^{\mathrm{screen}}$ is evaluated via cache-aligned NumPy broadcasted reductions with pre-allocated thread-local scratchpads (\texttt{Optimized\allowbreak Screened\allowbreak Apex\allowbreak Potential}). Concurrently, synchronous supernode sub-cycles are compiled to native machine code via Numba \texttt{@njit(fastmath=False)} (\texttt{jit\_supernode.py}), eliminating Python interpreter overhead and heap allocation during sub-cycles. \item \textbf{Immutable State Flow}: Because \texttt{ExtendedPhaseState} instances enforce read-only memory views (\texttt{flags.writeable = False}), diagnostic and orchestration layers access consistent phase-space snapshots without side effects. \end{itemize} \begin{rembox}[Architectural Paradigm: Deterministic CPU Execution vs GPU SIMD Bottlenecks]\label{rem:cpu_lock_free_paradigm} Unsoftened geometric symplectic integration constitutes an \emph{irregular, multirate, branch-heavy dynamical system}. Unlike smoothed particle hydrodynamics (SPH) or softened cosmological particle-mesh solvers where uniform kernel loops dominate, the exact geometric solver concentrates its numerical effort on localized close encounters requiring deep dyadic refinement and variable Halley/Sundman sub-stepping. Modern CPUs with out-of-order execution, wide L2/L3 caches, and strict IEEE~754 ordering provide the optimal hardware substrate for executing the hierarchical Jacobi splitting with bit-exact symplecticity. \end{rembox} % ------------------------------------------------------------------------------ \subsubsection{Operational Flowchart and Computational Complexity} \label{sub:flowchart_and_complexity} The end-to-end execution flow of the symplectic integrator is governed by the state transition machine illustrated in Figure~\ref{fig:conormal_execution_flowchart}. The theoretical complexity and memory footprint of the five execution phases are characterized as follows: \begin{itemize} \item \textbf{Kinematic and Kepler--Sundman Drift (Phase 3.1)}: For an adapted tree $T$ with $N_{\mathrm{leaves}} \le \lfloor N/2 \rfloor$ terminal leaf pairs, solving the Kepler--Sundman equations requires $N_{\mathrm{leaves}}$ independent invocations of the Halley cubic iterator. Because Stumpff series evaluation converges within $3$ to $5$ iterations for all eccentrically bounded orbits, the computational cost is strictly $\mathcal{O}(N_{\mathrm{leaves}}) = \mathcal{O}(N)$. \item \textbf{Screened Apex Force Field (Phase 3.3)}: Evaluating $W_{\mathrm{apex}}^{\mathrm{screen}}$ requires summing non-leaf pairwise and composite node interactions across $\operatorname{Internal}(T)$, with exact $\mathcal{O}(N^2)$ direct evaluation in the reference and vectorized engines (extendable to $\mathcal{O}(N \log N)$ via multipolar tree-code truncation for large-$N$ clustered configurations). \item \textbf{Dyadic Scheduling and Timescale Scanning (Phase 2)}: Scanning pairwise kinematic, dynamical, and pericenter timescales $\tau_{ij}$ entails an exact $\mathcal{O}(N^2)$ scan over all $N(N-1)/2$ pairs once per macro-step boundary, maintaining a negligible scheduling overhead relative to the $2^k \times s$ sub-cycle force evaluations ($< 2\%$ of macro-step runtime). \item \textbf{Memory Footprint and Cache Locality}: The spatial state data structures ($\mathbf{q}, \mathbf{p}, \mathbf{m}, \mathbf{r}_v, \mathbf{w}_v$) require strictly $\mathcal{O}(N)$ memory storage: exactly $6N$ floating-point scalars for Cartesian coordinates, plus $(2N-2) \times 3$ scalars for Jacobi coordinates. For $N = 10^3$ bodies, the entire working dataset occupies less than $250\,\mathrm{kB}$, fitting comfortably within the fast L2/L3 cache of modern CPU architectures and eliminating main-memory latency bottlenecks. \end{itemize} \begin{figure}[!htbp] \centering \begin{tikzpicture}[ font=\small\rmfamily, box/.style={draw=black, line width=0.8pt, fill=blue!5, rounded corners=2pt, minimum height=0.58cm, inner sep=3.5pt, text centered}, decision/.style={draw=black, line width=0.8pt, fill=yellow!15, rounded corners=3pt, minimum height=0.58cm, inner sep=3.5pt, text centered}, arrow/.style={->, >=stealth, line width=0.8pt} ] % Nodes \node[box, fill=green!10] (init) {\textbf{Phase 0}: Geometric Initialization $(T_0, \mathbf{P}_{\mathrm{cm}}=\mathbf{0})$}; \node[box, below=0.45cm of init] (prep) {\textbf{Phase 1}: Shadow Entry $\mathbf{s}_{\mathrm{shadow}} = \mathcal{Z}_4^{\mathrm{BM4}}(-h_0)[\mathbf{s}_{\mathrm{phys}}]$}; \node[decision, below=0.45cm of prep] (admissible) {\begin{tabular}{c}\textbf{Stage (1): Topological Adaptation?}\\ \scriptsize ($d_{\mathrm{ext}}^{\mathrm{Hill}} < \gamma_{\mathrm{monitor}} r_{ij} \implies T_{\mathrm{act}}$, $\Delta t_{\mathrm{dwell}} \ge \tau_{\min}$)\end{tabular}}; \node[box, fill=orange!10, right=0.45cm of admissible] (transition) {\begin{tabular}{c}\textbf{Phase 4}: Reconstruct $T_{\mathrm{act}}$\\ \scriptsize \& Gauge $\mathcal{J}_{T \to T_{\mathrm{act}}}$\end{tabular}}; \node[box, below=0.45cm of admissible] (sched) {\begin{tabular}{c}\textbf{Phase 2}: Stage (2) Supernodes $\mathcal{S}_{\mathrm{sync}}(T_{\mathrm{act}})$\\ \scriptsize \& Stage (3) Dyadic Step $h_k(T_{\mathrm{act}})$\end{tabular}}; \node[box, below=0.45cm of sched] (subcycle) {\begin{tabular}{c}\textbf{Phase 3}: Symplectic Sub-Cycle\\ \scriptsize ($2^k$ micro-steps of $h_k$ on $T_{\mathrm{act}}$)\end{tabular}}; \node[decision, below=0.45cm of subcycle] (sim_end) {\begin{tabular}{c}\textbf{Simulation Complete?}\\ \scriptsize ($t \ge t_{\mathrm{end}}$)\end{tabular}}; \node[box, fill=purple!10, below=0.45cm of sim_end] (postp) {\textbf{Phase 5}: Shadow Exit $\mathbf{s}_{\mathrm{phys}} = \mathcal{Z}_4^{\mathrm{BM4}}(+h_{\mathrm{exit}})[\mathbf{s}_{\mathrm{shadow}}]$}; \node[box, fill=green!10, below=0.42cm of postp] (done) {\begin{tabular}{c}\textbf{Metrological Diagnostics}\\ \scriptsize $\Delta\mathbf{P},\mathbf{L},\mathbf{G} < 10^{-14}, \; \Delta H = \mathcal{O}(h_{\mathrm{macro}}^{p_{\mathrm{eff}}})$\end{tabular}}; % Connections \draw[arrow] (init) -- (prep); \draw[arrow] (prep) -- (admissible); \draw[arrow] (admissible) -- node[above, font=\scriptsize] {Yes} (transition); \draw[arrow] (admissible) -- node[right, font=\scriptsize] {No ($T_{\mathrm{act}} = T$)} (sched); \draw[arrow] (transition) |- (sched.east); \draw[arrow] (sched) -- (subcycle); \draw[arrow] (subcycle) -- (sim_end); % Return loop for next macro-step: routes cleanly around the outside of all boxes \coordinate (looptrack) at ([xshift=-1.0cm]sched.west); \draw[arrow] (sim_end.west) -- node[pos=0.45, above, font=\scriptsize] {No} (sim_end.west -| looptrack) -- node[pos=0.5, above, sloped, font=\scriptsize] {$t \gets t + \Delta t_{\mathrm{macro}}$} (admissible.west -| looptrack) -- (admissible.west); \draw[arrow] (sim_end) -- node[right, font=\scriptsize] {Yes} (postp); \draw[arrow] (postp) -- (done); \end{tikzpicture} \caption{\textbf{Operational flowchart of the six-phase Unified Symplectic Integrator pipeline.} Causal macro-boundary execution sequence linking topological adaptation (Phase~4), dyadic multirate scheduling (Phase~2), and the palindromic symplectic sub-cycle (Phase~3).} \label{fig:conormal_execution_flowchart} \end{figure}