A Topological Geometrodynamics of Wave Functions: On the Stationary State Wave Functions, Their Partial Time Derivative and Resultant Theoretical Implications

Update (v205): Expanded Edition. Includes geometric proofs for the stability of composite nuclear structures (Ch. 98-99) and why the topological knots hold in $3+\epsilon$ dimension.

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This paper proposes an extension of General Relativity by replacing the standard isotropic Riemannian manifold with a Generalized Finsler Geometry of fractional dimension $D = 3 + |\epsilon|$. We argue that the isotropic nature of standard General Relativity is not complete when describing the stability of localized energy densities. By introducing an anisotropic Finsler metric, we demonstrate that Intrinsic Spin and Rest Mass arise naturally as geometric necessities of the spacetime manifold itself, rather than external quantum parameters. Founded on the Principle of the Holistic Quantum State, this framework extends the domain of quantum coherence to macroscopic and cosmological scales, suggesting that the universe operates as a single, self-contained quantum object. Within this geometric framework, we show derivation of the Dirac Equation as the boundary limit of a null-geometry wave propagating through a Finslerian vacuum, effectively unifying the descriptions of fermions and spacetime curvature. We define Mass topologically as the Winding Number ($k_{\epsilon}$) of a wave function knotted within the fractional $\epsilon$-dimension. We derive a universal scaling law, $\epsilon \propto M^{0.38}$, which links the geometric thickness of the vacuum to the mass of the topological defect. When applied to the Standard Model, this law reveals that fundamental particles correspond to quantized geometric harmonics: Leptons and Hadrons map to discrete integer or half-integer winding numbers (e.g., Electron $k=1$, Muon $k \approx 1564.5$, Proton $k \approx 32,483$).Confinement is explained as a topological constraint where half-integer "open strings" (quarks) must combine to form integer "closed loops" (baryons) to maintain geometric stability. This work offers a consistent Topological Geometrodynamics, resolving the Wave-Particle Duality paradox by identifying "Particles" as closed topological knots and "Waves" as open geometric twists within a dynamic, anisotropic vacuum.

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Invitation to Collaboration

I invite anyone who finds the concepts in this framework plausible to dedicate time to developing them further. By doing so, we may reach a deeper understanding of the cosmos.

It is time for physics to move beyond proprietary constraints and embrace an Open Source ethos. Much like in software development, when a contributor identifies a glitch in the code, the goal should not be to dismiss the program, but to write a patch that fixes the error.

We should begin working on our "Physical Linux"—a shared, evolving understanding of the cosmos for the benefit of all.