The Three-Body Problem in Canvas Temporal Mathematics: A Complete Spectral Framework

This volume presents a complete reformulation of the three-body problem within Canvas Temporal Mathematics (CTM). Across five papers, we develop the mathematical foundations, derive observable predictions for astronomical systems, reformulate the problem in terms of spacetime voxel deformation, establish universal critical scaling near the escape threshold, and provide a detailed numerical methodology for testing the framework. Paper I: A Spectral Framework with a Rigorous Discrete Cheeger–Plank Theorem The gravitational three-body problem is a paradigm of deterministic chaos. Long-term statistical properties are encoded not in individual trajectories (which are unpredictable) but in the spectrum of the Koopman generator—a linear operator acting on phase-space functions. We prove the Discrete Cheeger–Plank Theorem for the voxel-lattice-regularized Sitnikov problem, establishing the rigorous inequality \gamma \geq h^2/2 for the spectral gap. The four-tether decomposition classifies the constraints on three-body dynamics into spatial tethers (bounded phase space), parameter tethers (masses, coupling constants), symmetry tethers (conservation laws), and intersection tethers (close encounters as threshold-crossing events). This decomposition follows from the Canvas Periodic Table of Tethers and provides a systematic classification of the system's discrete structure. Paper II: Escape Time Distributions, Hierarchical Stability, and Observable Predictions for Astronomical Systems We derive observable predictions from the spectral framework. For unbounded systems (E > 0), the escape time distribution is exponential: P(T > t) = e^{-\gamma_{\text{esc}} t}, providing a first-principles derivation of the previously empirical exponential law. For hierarchical triples with separation ratio \epsilon = a_{\text{outer}}/a_{\text{inner}} \gg 1, the stability probability decays as P_{\text{stable}}(t) \propto e^{-\gamma_{\text{stab}} t} with \gamma_{\text{stab}} \propto \epsilon^{-3/2}. Near the escape threshold E = 0^+, the spectral gap, Lyapunov exponent, and Cheeger constant all scale as \sqrt{E}, giving universal critical exponent \nu = 1/2. The Cheeger–Plank inequality \gamma \geq h^2/2 provides a geometric lower bound; the observed scaling satisfies this bound but is not tight. Paper III: A Spectral Formulation via Spacetime Voxel Deformation We propose a paradigm shift grounded in the Canvas Model. Instead of tracking point masses, we track the deformation of the spacetime voxel lattice—the discrete structure of spacetime itself. The three bodies are localized concentrations of lattice compression. Their motion is a propagating wave of deformation. The lattice deformation field \Phi(\mathbf{x}, t) satisfies a wave equation with nonlinear source terms from the mass concentrations. This formulation reveals that the three-body problem is structurally identical to other spectral problems in CTM: the Riemann zeta function (TAC operator on the prime lattice), the Yang–Mills mass gap (Cheeger constant of the gauge lattice), and the Sitnikov problem (Koopman generator on phase space). All are instances of a single principle: spectral analysis on a lattice with a threshold condition. Paper IV: Evidence for \nu = 1/2 and a Conjectured Renormalization Group Fixed Point The three-body escape problem exhibits a critical transition at E = 0. Near this threshold, the Lyapunov exponent and escape rate are expected to scale as power laws: \lambda(E) \sim A (E - E_c)^{\nu} and \gamma_{\text{esc}}(E) \sim B (E - E_c)^{\mu}. We present four independent lines of evidence that \nu = 1/2 and \mu = 1/2: (1) geometric bottleneck scaling (quadratic tangency in the Poincaré map), (2) renormalization group analysis of the saddle-node bifurcation (relevant eigenvalue \delta_{\text{SN}} = 2), (3) the Cheeger–Plank inequality (a lower bound satisfied by the observed scaling), and (4) numerical evidence from the Sitnikov problem, the standard map at KAM breakup, and existing three-body scattering experiments. Paper V: A Methodology for Testing the Cheeger–Plank Mechanism and Critical Exponents We present a detailed numerical methodology to compute \gamma_{\text{esc}}(E) for the equal-mass, zero-angular-momentum three-body problem and determine the scaling exponent \mu. We provide: a complete algorithm for microcanonical sampling of initial conditions on the constant-energy manifold; a high-precision integration scheme (IAS15) for the three-body equations of motion; an escape criterion based on pairwise distance and energy; a maximum likelihood method for fitting the escape time distribution to an exponential, including treatment of censored data; a protocol for extracting \gamma_{\text{esc}}(E) and its scaling exponent \mu; predicted scaling relations and expected numerical values; and a simplified test case—the Sitnikov problem—as a cheaper preliminary validation. The methodology is designed to be implemented in existing N-body codes (REBOUND, HNBody, or a custom Python/C integrator). Key results across all five papers: · The spectral gap \gamma of the Koopman generator controls the decay of correlations, escape times, and hierarchical stability. The discrete Cheeger–Plank theorem proves \gamma \geq h^2/2 for the voxel-lattice Sitnikov problem.· The escape time distribution is exponential: P(T > t) \propto e^{-\gamma_{\text{esc}} t}. This matches numerical experiments (Heggie 1975, Hut 1983) and is now derived from first principles.· Hierarchical stability decays exponentially with rate \gamma_{\text{stab}} \propto \epsilon^{-3/2}, matching perturbation theory. The stability can be understood through isochronous potentials: the inner binary remains stable as long as it is approximately isochronous, and the stability boundary is where this approximation fails.· Near the escape threshold, critical exponents are \nu = 1/2 and \mu = 1/2, placing the three-body escape problem in the saddle-node universality class.· The Cheeger–Plank inequality provides a lower bound, not a scaling relation. The observed scaling satisfies the bound but is not tight; the actual escape rate is larger than the geometric lower bound due to chaotic mixing.· The three-body problem is not separate from the Canvas Model. It is another configuration of the same eight primitives—occupying a specific row in the periodic table of tethers, alongside the Riemann zeros, the Yang–Mills mass gap, the Sitnikov problem, and the tautochrone. Keywords: three-body problem, Koopman operator, spectral gap, Cheeger constant, Cheeger–Plank mechanism, escape time distribution, hierarchical stability, critical scaling, saddle-node bifurcation, renormalization group, spacetime voxel lattice, Canvas Temporal Mathematics, periodic table of tethers本卷完整重构了画布时序数学（Canvas Temporal Mathematics，CTM）框架下的三体问题。本研究通过五篇论文，依次构建其数学基础、推导天文系统的可观测预言、以时空体素变形为基础重构该问题、确立逃逸阈值附近的普适临界标度律，并提供用于验证该框架的精细化数值方法。 ## 论文一：带严格离散Cheeger–Plank定理的谱框架 引力三体问题是确定性混沌的典型范式。其长期统计特性并非编码于不可预测的单条轨迹中，而是编码于库普曼生成器（Koopman generator）的谱中——该线性算子作用于相空间函数。我们针对体素晶格正则化的西特尼科夫问题（Sitnikov problem）证明了离散Cheeger–Plank定理，确立了谱隙满足$gamma geq h^2/2$的严格不等式。四Tether分解将三体动力学的约束划分为空间Tether（有界相空间）、参数Tether（质量、耦合常数）、对称性Tether（守恒律）与交叠Tether（作为阈值穿越事件的近距交会）。该分解源自画布Tether周期表（Canvas Periodic Table of Tethers），为系统的离散结构提供了系统性分类。 ## 论文二：逃逸时间分布、层级稳定性与天文系统可观测预言 我们从该谱框架中推导出可观测预言。对于无界系统（总能量$E>0$），逃逸时间分布服从指数律：$P(T > t) = e^{-gamma_{ ext{esc}} t}$，首次从第一性原理推导出此前仅为经验性的指数分布规律。对于分离比$epsilon = a_{ ext{外}}/a_{ ext{内}} gg 1$的层级三星系统，其稳定概率随时间衰减满足$P_{ ext{稳定}}(t) propto e^{-gamma_{ ext{稳}} t}$，其中$gamma_{ ext{稳}} propto epsilon^{-3/2}$。在逃逸阈值$E = 0^+$附近，谱隙、李雅普诺夫指数（Lyapunov exponent）与Cheeger常数（Cheeger constant）均按$sqrt{E}$标度，由此得到普适临界指数$ u = 1/2$。Cheeger–Plank不等式$gamma geq h^2/2$给出了几何下界；观测到的标度律满足该下界但并不紧致。 ## 论文三：基于时空体素变形的谱表述 我们提出了基于画布模型（Canvas Model）的范式革新。相较于追踪质点运动，我们转而追踪时空体素晶格——即时空本身的离散结构——的变形。三体为晶格压缩的局域化聚集，其运动即为变形的传播波。晶格变形场$Phi(mathbf{x}, t)$满足带非线性源项的波动方程，其中源项源自质量聚集效应。该表述表明，三体问题在结构上与CTM框架下的其他谱问题完全一致：黎曼ζ函数（素数晶格上的TAC算子）、杨-米尔斯质量隙（规范晶格的Cheeger常数）以及西特尼科夫问题（相空间上的库普曼生成器）。上述所有问题均遵循同一原理：带阈值条件的晶格谱分析。 ## 论文四：$ u=1/2$的实验证据与猜想的重整化群不动点 三体逃逸问题在总能量$E=0$处存在临界相变。在该阈值附近，李雅普诺夫指数与逃逸速率可按幂律标度：$lambda(E) sim A (E - E_c)^{ u}$与$gamma_{ ext{esc}}(E) sim B (E - E_c)^{mu}$。我们提供了四条独立证据以证明$ u=1/2$且$mu=1/2$：(1) 几何瓶颈标度（庞加莱映射中的二次相切）；(2) 鞍结分岔的重整化群分析（相关本征值$delta_{ ext{SN}}=2$）；(3) Cheeger–Plank不等式（观测标度满足该下界）；(4) 来自西特尼科夫问题、KAM破裂时的标准映射以及现有三体散射实验的数值证据。 ## 论文五：验证Cheeger–Plank机制与临界指数的数值方法 我们提出了一套精细化数值方法，用于计算等质量、零角动量三体问题的$gamma_{ ext{esc}}(E)$并确定标度指数$mu$。该方法包含：常数能量流形上初始条件的微正则采样完整算法；三体运动方程的高精度积分格式（IAS15）；基于粒子间距与能量的逃逸判定准则；将逃逸时间分布拟合为指数分布的极大似然方法（含删失数据处理）；提取$gamma_{ ext{esc}}(E)$及其标度指数$mu$的流程；预测的标度关系与预期数值结果；以及简化的测试案例——西特尼科夫问题——用于低成本的初步验证。本方法可兼容现有N体模拟代码（如REBOUND、HNBody或自研Python/C积分器）。 ## 五篇论文核心结论 · 库普曼生成器的谱隙$gamma$控制着关联衰减、逃逸时间与层级稳定性。针对体素晶格西特尼科夫问题，离散Cheeger–Plank定理证明$gamma geq h^2/2$。 · 逃逸时间分布服从指数律：$P(T > t) propto e^{-gamma_{ ext{esc}} t}$。该结果与数值实验（Heggie 1975, Hut 1983）相符，且首次从第一性原理推导得出。 · 层级稳定性以速率$gamma_{ ext{stab}} propto epsilon^{-3/2}$指数衰减，与微扰理论结果一致。该稳定性可通过等时势理解：内双星系统在近似等时的条件下保持稳定，稳定性边界对应该近似失效之处。 · 在逃逸阈值附近，临界指数为$ u=1/2$与$mu=1/2$，将三体逃逸问题归入鞍结分岔普适类。 · Cheeger–Plank不等式仅提供下界而非标度关系。观测标度满足该下界但并不紧致；由于混沌混合效应，实际逃逸速率大于几何下界。 · 三体问题并非独立于画布模型，而是同一套八种基元的另一种组态——占据画布Tether周期表的特定一行，与黎曼零点、杨-米尔斯质量隙、西特尼科夫问题以及等时摆问题同属一类。 ## 关键词 三体问题、库普曼算子、谱隙、Cheeger常数（Cheeger constant）、Cheeger–Plank机制、逃逸时间分布、层级稳定性、临界标度、鞍结分岔、重整化群、时空体素晶格、画布时序数学（Canvas Temporal Mathematics，CTM）、画布Tether周期表（Canvas Periodic Table of Tethers）