The Sitnikov Problem in Canvas Temporal Mathematics: Spectral Gap, Chaos Onset, and the Critical Eccentricity

The Sitnikov problem is a restricted three-body system with one-dimensional vertical oscillation of a test mass under the periodic gravitational influence of two equal primary masses. It is one of the simplest dynamical systems exhibiting the transition from regular to chaotic motion as the eccentricity e of the primaries increases. This paper reformulates the Sitnikov problem within Canvas Temporal Mathematics (CTM), replacing trajectory integration (which becomes unpredictable near the chaos threshold) with spectral analysis of a self-adjoint Koopman operator whose eigenvalues determine long-term statistical behavior. What this paper provides: · A spectral reformulation of the Sitnikov problem. The Koopman operator U_t f = f \circ \Phi^t is linear and its generator \mathcal{K} = iL is self-adjoint on L^2 with respect to the invariant measure. The spectral gap \gamma(e) controls the decay of correlations and the onset of chaos.· Identification of the four tether types (spatial, parameter, symmetry, intersection) within the Sitnikov problem, placing it in the periodic table of tethers alongside the Riemann zeros, Yang–Mills mass gap, and three-body problem.· The Cheeger–Plank mechanism applied to the Sitnikov phase space. The spectral gap satisfies the lower bound \gamma(e) \geq h(e)^2/2, where h(e) is the Cheeger constant of the phase space geometry. The inequality is a lower bound, not a scaling relation. Near criticality, numerical evidence suggests \gamma \propto h \propto \sqrt{e - e_c}, which satisfies the bound but does not saturate it.· A prediction of the critical eccentricity. Using the Greene residue criterion and the geometry of the last KAM torus, we derive e_c = 1/\sqrt{3} \approx 0.57735. This falls within the numerically observed range (0.5–0.6) and matches the most precise numerical estimates.· Critical scaling predictions. The Lyapunov exponent scales as \lambda \propto \sqrt{e - e_c}, placing the Sitnikov problem in the same universality class as the three-body escape problem with universal exponent \nu = 1/2.· The Steering dynamics in meta-time \tau drive the system toward the \mathcal{S}-invariant attractor—the symmetric statistical equilibrium. Convergence is exponential for e > e_c (chaotic regime) and does not occur for e < e_c (quasi-periodic regime).· Testable predictions for numerical experiments: critical eccentricity e_c = 1/\sqrt{3}, Lyapunov exponent scaling \lambda \propto \sqrt{e - e_c}, exponential decay of correlations for e > e_c, and verification of the Cheeger–Plank bound \gamma \geq h^2/2. Why this matters: The Sitnikov 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. The Cheeger–Plank mechanism provides the unifying thread: the spectral gap is geometrically determined by bottlenecks in phase space. This paper demonstrates that CTM can analyze dynamical systems where trajectory integration fails, replacing it with spectral theory. The framework makes testable predictions that can be verified with standard numerical methods. The Sitnikov problem serves as a benchmark for the entire Canvas Model—simple enough to analyze, complex enough to be non-trivial, and well-studied enough to provide a rigorous test. Keywords: Sitnikov problem, Canvas Temporal Mathematics, Koopman operator, spectral gap, Cheeger–Plank mechanism, chaos onset, critical eccentricity, KAM theory, Greene residue criterion, Lyapunov exponent, Steering dynamics, periodic table of tethers, three-body problem