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May 19, 20260 citationsOpen Access

From Quantum Potential to Atomic Structure: Seven Derived Results in D=3 Geometry

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MSMandeep Singh

Key Points

  • The aim is to derive fundamental results from the Clausius-Mossotti framework in three-dimensional geometry without free parameters.
  • Utilized a single complementarity principle of the Clausius-Mossotti framework.
  • Derivations included determining quantum potential and energy formulas across various hydrogen-like ions.
  • Analyze relationships through D=3 geometry and CM metric.
  • Quantum potential is shown to be zero for CM density.
  • An energy formula achieves 0.52% accuracy compared to NIST values across 34 hydrogen-like ions.
  • A testable prediction is established for wall-orbit crossing at Z=79 (Gold).

Abstract

The Clausius-Mossotti (CM) framework is built on a single complementarity principle: OE + W = 1. This paper derives seven results from D=3 geometry and the CM metric, using zero free parameters: (1) quantum potential Q=0 for CM density, (2) D=3 uniqueness via Qcoeff = - (D-1) (D-3) /4, (3) strip ratio t/w = 8π/9, (4) standing wave rule m×R = (4/3) πℏ/c giving proton radius 0. 881 fm (0. 7%), (5) particle=sphere confirmation, (6) CM metric produces atomic Klein-Gordon equation, (7) energy formula E = ½mec² (W⁻¹/³−1) achieves 0. 52% average across 34 hydrogen-like ions (Z=1-92) vs NIST measured values, beating Coulomb 30/34 and Dirac 9/34. Eighth result: wall-orbit crossing at Z=79=Gold, a testable CM prediction. Paper 2026z in the Speed Gap Framework series.

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Cite This Study

Mandeep Singh (2026) studied this question.

synapsesocial.com/papers/6a0bfdc7166b51b53d3790e3https://doi.org/10.5281/zenodo.20254897
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