The 1/3 Geometric Constant: Scale Invariance and the Origin of 'Missing Energy' in 3D Quantum Fragmentation

Abstract

We report the discovery of a universal geometric constraint on the detection of kinetic energy release (KER) in three-dimensional quantum fragmentation. By analyzing the dissociation of localized wavepackets, we demonstrate that the $4πr^2$ radial volume element acts as a topological filter that inherently masks a significant portion of a system's energy budget, imposing a fundamental peak-to-mean bound of $R_E < 0.5$. We introduce an invariant scaling law, $α= MQ/ζ$, and prove that the resulting energy detection ratio is scale-invariant across twelve orders of magnitude, bridging attosecond molecular science and nuclear physics. We identify a universal \textbf{geometric landmark} at $R_E \approx 0.33$, which precisely replicates the 7~eV discrepancy in $H_2^+$ fragmentation. Furthermore, we show that the population of excited-state manifolds and the increase in nuclear localization ($ζ$) provide a definitive geometric mechanism for the \textbf{spectral broadening} observed across atomic and subatomic scales. Remarkably, the spectral morphology derived from our scaling law aligns with the universal 1/3 energy landmark of historical beta decay, while the high-mass limit naturally accounts for the sharpening of alpha spectra. Our results suggest that ``missing energy'' is often a topological artifact of 3D geometry rather than an exclusive signature of undetected particles. This work establishes a universal master curve for energy reconstruction and identifies a \textbf{``detection crisis''} in highly localized systems, where the true interaction energy becomes effectively invisible to peak-centric calorimetry.