Spacetime Phase Transition and Sawtooth Collapse Dynamics: Resolving the Radiation Paradox via Spin-Network Cut-offs in Matter-Wave Interferometry

The Diósi-Penrose (DP) objective collapse model faces a severe empirical crisis, as recent underground experiments have definitively ruled out spontaneous bremsstrahlung radiation predicted by its stochastic noise mechanism. In this paper, we resolve this radiation paradox by proposing "Geometric Yielding"—a deterministic spacetime phase transition. Operating within the weak-field limit, we posit that spatial superposition collapses instantaneously when its quantum expansion pressure exceeds the structural threshold of the local linearized metric (Lc), dissipating energy into spacetime curvature without electromagnetic coupling. Crucially, to resolve the singularity of point-like collapse, we anchor the ultimate lower bound of this localization (L₀) in the discrete geometry of Loop Quantum Gravity (LQG). We propose that the collapse is stopped at the minimum volume eigenvalue of a spin network node, scaling with the Barbero-Immirzi parameter () and the Planck length (P). This fundamental reduction in spectral dimension establishes a deterministic sawtooth-like dynamical cycle (). For macroscopic objects, this cycle frequency is infinitely rapid, trapping them in a classical state via a self-induced Quantum Zeno effect. As a definitive empirical signature, we predict that for a 10⁶ AMU silicon nanosphere, this Geometric Yielding will manifest itself as a Heaviside step-like suppression of interference visibility at a critical superposition distance (9. 5), fundamentally distinguishing our model from the exponential decay predicted by continuous spontaneous localization (CSL) models.