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