Evidence Gap CurvatureProjective Geometry and the Atomic Structure Path DependenceThe Hexagonal Atom скачать в хорошем качестве

Evidence Gap CurvatureProjective Geometry and the Atomic Structure Path DependenceThe Hexagonal Atom

This research explores how path dependence and order sensitivity arise in systems where only the relative scale of a state matters, such as in quantum mechanics or statistical inference. By applying projective geometry, the authors demonstrate that discrepancies in these systems are essentially geometric phases or "holonomies" created by moving through a state space. A key finding is that these discrepancies cannot exist in simple, commuting square structures, as those remain flat and consistent regardless of the path taken. Instead, the paper proves that non-commuting braid relations serve as the fundamental source of curvature, making the hexagon the smallest possible unit where path dependence can occur. Ultimately, the study establishes a discrete Stokes-type decomposition that identifies these hexagonal cells as the universal "atomic" carriers of curvature across various scientific disciplines.