ELECTRON SHELL FILLING IN THE TVM FRAMEWORK Geometric origin of the 2n^2 rule from the First Principles of the Theory of Time Modulation and the established initial st

Abstract

In standard quantum mechanics, the electron shell filling rule is explained using four quantum numbers (), the Pauli exclusion principle, and spin as a fundamental property of the electron. This is not an explanation – this is a description. The question remains: why exactly ? In the TVM framework, the rule ceases to be an empirical rule or a consequence of quantum postulates. It becomes a geometric necessity – a direct consequence of: The radius of the -th level: (derived in the main work) The wavelength at the fundamental level: (derived in the First supplementary work) The condition of wave stationarity – so that destructive interference does not occur Starting from these TVM results, the work derives the rule as follows: The circumference of the orbit at level is The minimum distance between two waves on the same orbit is The maximum number of waves that can fit on the circumference is The work also derives the geometric origin of the subshells, showing that they are not spatial orientations, but places where extrema (maxima and minima) of the stationary wave on a circular orbit are realized. The number of possible configurations (arrangements of extremum points on the circle, not spatial orientations) relative to zero points gives the capacities 2, 6, 10, 14..., and their sum yields . This demonstrates that: Spin does not exist as a phenomenon – the factor 2 originates from geometry, not from spin The Pauli principle is not a separate law – it is a description of a geometric constraint Orbital quantum numbers are not necessary – at the fundamental level, comes directly The periodic system is not empirical – it is geometrically necessary The rule is not a hypothesis – it is recognized in the TVM structure. It has always been there, in the geometry of the circle, the wave nature of the electron, and the condition of wave stationarity to avoid destructive interference. Keywords: TVM, shell filling, , geometry, wavelength, interference, Pauli principle, periodic system, s, p, d, f, stationary state, standing wave. NOTE: This is the second in a series of supplementary scientific papers that elaborate the application of the TVM formalism to specific physical phenomena. Starting from the results of the First supplementary work from the series (de Broglie relation), this work derives the electron shell filling rule from pure geometry, without introducing spin, the Pauli principle, or orbital quantum numbers. It is recommended that the reader first study the First supplementary work [2], as well as Chapters 3 and 4 of the main work [1].