Geometric Connection Between Quantum Bounce, QCD Mass Gap, and Planck Scale — A Trans-Scale Hypothesis in the Framework of Memory-Force Theory

Based on the framework of memory-force theory, this paper proposes a unified geometric hypothesis connecting the quantum bouncer, QCD mass gap, and Planck scale. A quantitatively testable trans-scale scaling relation is derived: > \left(\frac{\Lambda_{\text{QCD}}}{M_{\text{Pl}}}\right)^{\frac{3\pi}{8}} \approx 10^{-22.6} > where \Lambda_{\text{QCD}} \sim 200 MeV is the characteristic energy scale of the QCD mass gap, M_{\text{Pl}} \approx 1.22 \times 10^{19} GeV is the Planck mass, and the exponent \frac{3\pi}{8} originates from the vacuum D_8 geometric symmetry in memory-force theory, representing an intrinsic geometric constant of UV-IR duality. This paper demonstrates that gravitational strength, mass gap, and quantum bouncer energy levels are not independent phenomena, but unified responses of vacuum geometric stiffness at different energy scales. The hypothesis does not contradict any verified physical laws, is logically self-consistent, and yields falsifiable predictions. It provides a consistent unified explanation for why gravity is weak, how mass originates, and why spacetime curves.