What question did this study set out to answer?

The aim is to investigate the Klein-Gordon wave equation in the Clausius-Mossotti metric for quark position analysis.

April 17, 2026Open Access

Self-Consistent Quantization, the Three-Dimensional Kepler Law, and Standing-Wave Quark Positions from the Clausius-Mossotti Metric

Key Points

The aim is to investigate the Klein-Gordon wave equation in the Clausius-Mossotti metric for quark position analysis.
Analyzed the KG equation coupled with the CM field equation in both exterior and interior settings.
Tested seven coupling strengths to assess self-consistency and back-reaction effects on binding energies.
Solved the KG equation within a finite sphere to identify standing-wave node positions.
Binding energies modified by 7–20% due to back-reaction effects.
Found that proper volume produces 2.2–2.8× more probability shells than coordinate volume, aligning with a 3D Kepler law.
Identified standing-wave nodes at volume fractions f = 1/3 and f = 2/3, corresponding to quark positions, with minimal deviation.

Abstract

We study the Klein-Gordon wave equation in the Clausius-Mossotti (CM) metric in two settings. Part I (Exterior): The KG equation coupled to the CM field equation converges self-consistently at all seven coupling strengths tested, with back-reaction modifying binding energies by 7–20%. The proper volume (W^-1/3) produces 2. 2–2. 8× more equal probability shells than coordinate volume — a 3D Kepler law. Part II (Interior): Solving KG inside a finite sphere with hard wall at the proton surface, standing-wave nodes appear at volume fractions f = 1/3 (0. 2% deviation) and f = 2/3 (0. 4% deviation), matching d-quark and u-quark positions from the Damru geometry (Singh 2026d), outperforming JLAB pressure measurement (3. 9%). Zero free parameters throughout. Paper 2026s in the Speed Gap Framework series (papers 2026a–r published).

KI fragen

Bookmark

View Full Paper