Vacuum Permittivity from Coherence Geometry: A Swarm Theory Derivation V2

Description

This version (V2.0, 12 March 2026) updates the paper “Vacuum Permittivity from Coherence Geometry: A Swarm Theory Derivation.” The manuscript presents a geometric derivation of vacuum permittivity within the Swarm Field Theory framework, in which spacetime is modeled as a lattice of tension-bearing coherence membranes enclosing zero nodes. Electromagnetic fields arise as small oscillations of a coherence displacement field on this lattice, and vacuum permittivity is interpreted as the linear polarization susceptibility of the membrane network.

The derivation expresses vacuum permittivity as a function of the geometric lattice parameters: coherence aperture, update interval, membrane tension, and projected propagation speed. Using these parameters derived in the Swarm Theory geometric appendices, the resulting value reproduces the observed SI value of vacuum permittivity within the precision of the lattice parameters.

Changes in Version 2.0 include:

• Reorganization of the derivation into a clearer step-by-step structure describing lattice assumptions, membrane susceptibility, characteristic frequency, and the resulting permittivity relation.

• Clarified physical interpretation of vacuum permittivity as the mechanical polarization response of the coherence lattice.

• Added constant consistency table showing agreement between predicted electromagnetic constants and CODATA values.

• Added sensitivity analysis figures showing how the predicted permittivity depends on lattice parameters such as coherence aperture and membrane impedance.

• Added Appendix A showing the scaling structure of the permittivity relation when combined with the projected wave speed relation of the lattice.

• Minor corrections to wording, formatting, and internal references.

The central derivation and predicted numerical value of vacuum permittivity remain unchanged. This revision focuses on improving clarity, documentation, and internal consistency of the geometric framework.`

• Reorganization of the derivation into a clearer step-by-step structure describing lattice assumptions, membrane susceptibility, characteristic frequency, and the resulting permittivity relation.

• Clarified physical interpretation of vacuum permittivity as the mechanical polarization response of the coherence lattice.

• Added constant consistency table showing agreement between predicted electromagnetic constants and CODATA values.

• Added sensitivity analysis figures showing how the predicted permittivity depends on lattice parameters such as coherence aperture and membrane impedance.

• Added Appendix A showing the scaling structure of the permittivity relation when combined with the projected wave speed relation of the lattice.

• Minor corrections to wording, formatting, and internal references.

The central derivation and predicted numerical value of vacuum permittivity remain unchanged. This revision focuses on improving clarity, documentation, and internal consistency of the geometric framework.

Additional materials and related work are available on GitHub: https://github.com/crumpler4/Swarm-Theory-Unified-Field-Equation-Series