Quantum Localization Bounds from Schrödinger Wave-Packet Geometry: A Multiscale Kakeya-Inspired Framework

This preprint develops a mathematical framework connecting the space-time geometry of Schrödinger wave packets with time-averaged quantum localization probabilities. It introduces a dimensionless multiscale concentration coefficient designed to quantify tube clustering across scales and derives a geometric-to-quantum localization inequality separating geometric concentration, mean occupancy, incoherent packet energy, and controlled wave-packet tails. The framework is calibrated using a classical Córdoba-type estimate for a restricted direction-uniform tube family, providing a sharp benchmark within that model. The paper also discusses an application to freely propagating Gaussian matter-wave packets and finite-detector occupancy. This is a preprint version submitted for consideration for publication in Reports on Mathematical Physics.