Threshold-Crossing Collapse: A Quantitative Formulation and Experimental Confrontation

This paper formulates the threshold-crossing collapse mechanism of the canvas programme as a phenomenological two-parameter objective-collapse model of the Ghirardi-Rimini-Weber (GRW) type, with parameters (p, r_C): the threshold-exceedance probability per carrier cycle and the localization length. It computes the spontaneous electromagnetic radiation spectrum produced by discrete Poissonian collapse events, and confronts the model with underground \gamma-spectroscopy, anomalous-heating limits, and a stated macroscopic-definiteness criterion. What was found: (i) The sudden-hit radiation spectrum coincides analytically with the known CSL 1/E spectrum at low photon energies, so existing germanium bounds import rigorously; they supersede the heating bound by eleven orders of magnitude and confine the model to a narrow island r_C \gtrsim 10^{-8} m with, at r_C = 10^{-7} m, only about two decades of allowed collapse rate, \lambda_n \in [10^{-16}, 10^{-14}] s^{-1}. (ii) Under the width reading of the localization postulate, voxel-scale localization conflicts with these bounds by \sim27 orders of magnitude in r_C; the postulate requires reinterpretation at the envelope scale. (iii) Formulating the trigger via Rice's level-crossing statistics for a narrowband Gaussian process fixes the rate formula to \lambda = \nu_c e^{-\Theta^2/2\sigma_I^2}; if the sub-voxel fluctuations are Gaussian, the threshold is pinned to \Theta = (13.1\text{--}13.5)\sigma_I, i.e. \Theta/\sigma_I = 13.3 \pm 0.2. What it means: The model carries two structural signatures that are independent of its two fitted parameters and distinguish it from both GRW and CSL: · Linear mass scaling of the collapse rate, \lambda \propto m (versus GRW's mass-independent rate and mass-proportional CSL's \lambda \propto m^2)· A spectral rolloff of the spontaneous radiation within or adjacent to the germanium analysis band (position 40–160 keV for O(1) hit duration, versus CSL's 1/E shape throughout) The top of the surviving parameter window sits at present germanium sensitivity, so next-generation experiments (LEGEND-scale) will either detect the predicted excess or, under the stated definiteness criterion, close most of the remaining window at small r_C. Keywords: objective collapse, threshold crossing, GRW, CSL, spontaneous radiation, germanium spectroscopy, collapse rate, localization length, Rice's formula, mass scaling, quantum foundations