HARMONIC OSCILLATOR IN THE TVM FRAMEWORK - Recognition of the spectrum from the transformation mechanism and the hierarchy of new levels – from the First Principles of the Theory of Time Modulation and the established initial state of the universe

In standard quantum mechanics, the spectrum of the harmonic oscillator: is derived by solving the Schrödinger equation for a parabolic potential. This is a mathematical formalism that gives correct numerical results but does not explain why the levels are as they are – why there is a ground state , why the levels are equally spaced, why the difference between them is . In the TVM framework, the spectrum of the harmonic oscillator is recognized through two independent physical processes that are already embedded in the mechanism of the Universe's evolution: Evolution of the first level – how the energy of the original, primary level decreases through successive transformations. Hierarchy of new levels – how, at each transformation, part of the energy separates from the previous level and forms a new, independent level. These two processes are simultaneous and correlated: the first level cannot evolve without creating new levels, and new levels cannot arise without the evolution of the first level. They are two sides of the same process – the impossibility of perfect division upon reaching the transformation limit . In TVM, the ground state of the harmonic oscillator is not a "zero-point energy of the vacuum" – it is a consequence of the first transformation. The formula: comes directly from the TVM mechanism and fractal recursion, where each new level has its own frequency , and the product of amplitude and momentum grows as . Thus, the harmonic oscillator ceases to be a "special" quantum system – it becomes a general pattern that appears wherever a system undergoes a series of transformations, from the initial state of the Universe to local quantum phenomena. Keywords: TVM, harmonic oscillator, spectrum, fractal recursion, transformation, stationarity, energy defect. NOTE: This is the third in a series of supplementary scientific papers that elaborate the application of the TVM formalism to specific physical phenomena. Starting from the foundations laid in the zeroth paper [1], this work derives the harmonic oscillator spectrum from the transformation mechanism and the hierarchy of levels. It is recommended that the reader first study the First supplementary work (de Broglie relation) [2] and the Second supplementary work (Electron shell filling) [3], as well as Chapters 3 and 4 of the main work [1].