The Gravitational Constant from Coherence Geometry: A Swarm Theory Derivation

New Version 10FEB2026

This paper presents a first-principles derivation of Newton’s gravitational constant from coherence geometry using three microscopic scales: a coherence aperture, a coherence cycle time, and the quantum of action.

Starting from the weak-field limit of a coherence-based Unified Field Equation (UFE), the paper shows that the weak-field coupling must have units of metres per kilogram and therefore admits a unique dimensional decomposition in terms of the underlying lattice parameters. Matching the linearised UFE to the Newton–Poisson equation identifies a dimensionless geometry factor that depends only on the long-wavelength normalisation of the UFE Green function and on lattice projection geometry.

The resulting expression shows that the gravitational constant is structurally determined by the geometry and update dynamics of the coherence lattice rather than being a fundamental parameter. Using the standard coherence scales adopted in the accompanying Swarm-theory framework, the predicted value of the gravitational constant agrees with the CODATA value to within approximately 0.3 percent without the use of fitted physical parameters.

The derivation is fully reproducible and falsifiable. Python scripts are provided to verify the algebraic relations and to demonstrate how the geometry factor can, in principle, be obtained from a discrete lattice Green-function calculation.

⸻

Version 2 – what changed

This version corrects and clarifies the weak-field matching and Green-function normalisation used to define the gravitational coupling.

Specifically:

* all weak-field matching relations to the Newton–Poisson equation are written explicitly with division by the square of the speed of light, removing earlier sign and dimensional ambiguities;

* the definition of the UFE weak-field coupling and of the dimensionless geometry factor is made explicit in terms of the long-wavelength normalisation of the linearised Green function;

* the rescaled geometry factor used in the final expression for the gravitational constant is defined consistently throughout the paper;

* Appendix B is rewritten to show explicitly how the one-over-k-squared pole of the linearised UFE operator produces a one-over-r potential and how the geometry factor is extracted from the far-field Green-function behaviour;

* the Python lattice scaffold is aligned with the corrected Green-function normalisation and coupling definitions.

No new physical assumptions are introduced in this revision.
The changes are algebraic, notational and definitional, and remove inconsistencies in the previous treatment of the weak-field coupling and geometry factor.

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