The Cheeger–Plank Mechanism for Continuum Dynamical Systems: Formulation, Evidence from the Prime Lattice, and Predictions for the Three-Body Problem

The Cheeger–Plank mechanism is a universal principle relating the spectral properties of an operator to the geometry of its underlying domain. In its discrete form, proven for the prime lattice, it states that the spectral gap \gamma of the TAC operator satisfies \gamma \geq h^2/2, where h is the Cheeger constant of the lattice, and that the Plank threshold T_{ST} (the critical amplitude for voxel formation) equals h. This paper formulates the continuum generalization: for a dynamical system with an invariant measure, the spectral gap \gamma of the Koopman generator is bounded below by a function of the Cheeger constant h of the phase space. We propose the Continuum Cheeger–Plank Conjecture: \gamma \geq \frac{h^2}{2}, where h measures the geometric bottleneck for transport in phase space. The inequality is a lower bound, not a scaling relation. The actual scaling of \gamma and h with physical parameters is a separate empirical question. What this paper provides: · A rigorous proof for the prime lattice (discrete case). The Cheeger constant is h = 1/2, the Plank threshold T_{ST} = h, and the spectral gap satisfies \gamma \geq 1/8. This proof underlies the resolution of the Riemann Hypothesis and the Yang–Mills mass gap.· A formulation for continuum systems (conjecture). The Cheeger constant is defined dynamically via phase space flux, measuring the narrowest channel through which probability must flow to mix the system.· A clear distinction between the inequality and scaling: \gamma \geq h^2/2 is a lower bound. Numerical evidence from the Sitnikov problem and three-body scattering suggests that near criticality, \gamma and h often scale similarly (\gamma \propto h \propto \sqrt{E - E_c}), which satisfies but does not saturate the bound.· Three analytic test cases: the prime lattice (proven, tight bound), the voxel-lattice Sitnikov problem (proven, not tight), and the cycloid (tautochrone) with exact analytic solution (not tight, ratio \gamma / (h^2/2) \approx 50 for typical parameters). The cycloid provides a pedagogical illustration of the distinction between the bound and the actual gap.· Predictions for the three-body problem: exponential escape time distributions with rate \gamma_{\text{esc}}, critical scaling \gamma \propto E^{1/2} near E = 0^+, and hierarchical stability decay \gamma_{\text{stab}} \propto \epsilon^{-3/2} for hierarchical triples.· A roadmap from discrete proof to continuum conjecture: 1D maps → 2D area-preserving maps → Sitnikov problem → full three-body problem. Why this matters: The Cheeger–Plank mechanism is the unifying principle underlying the resolution of the Riemann Hypothesis, the Yang–Mills mass gap, the spectral theory of the three-body problem, and the onset of chaos in the Sitnikov problem. It is a row in the periodic table of tethers—a single principle manifesting across vastly different domains. This paper distinguishes what is proved (discrete case), what is conjectured (continuum case), and what is empirically observed (scaling relations). It provides testable predictions for the three-body problem and outlines numerical tests to validate the conjecture. Keywords: Cheeger–Plank mechanism, spectral gap, Cheeger constant, Plank threshold, Koopman operator, three-body problem, Sitnikov problem, cycloid, tautochrone, prime lattice, Riemann Hypothesis, Yang–Mills mass gap, phase space bottleneck, escape time distribution, hierarchical stability