Technical Note XLIII
Total mass versus neighbour distance — a near-linear relation in log space
Plotting total mass \((1+\lambda)M_{vis}\) against the typical distance to neighbours of the same type, the seven levels of structure fall close to a single straight line in log–log: \( M \propto d^{2.05} \), i.e. \( d \propto M^{0.49} \approx \sqrt{M} \), with 0.63 dex scatter.
1. Total mass vs neighbour distance
For each level the abscissa is the mean separation to comparable objects (e.g. ~4 light-years between stars, ~0.78 Mpc between the Milky Way and Andromeda, tens of Mpc between clusters). The ordinate is the total mass with the central-source wave term, \((1+\lambda)M_{vis} = 13.7\,M_{vis}\).
2. Filling each mass decade
Around each power of ten in mass, from 10³⁰ to 10⁴⁸ kg, we place ~100 objects (50 just below, 50 just above 10ˣ). They cluster into vertical columns at the mass decades.
Two cautions. (a) In the second figure the neighbour distance of the ~1900 points is derived from the line of the first figure, so they necessarily fall on it — the figure illustrates the relation, it does not independently confirm it. The real information is the seven points of figure 1. (b) Mass and inter-object distance are correlated almost by construction in the universe: massive objects are rare and therefore widely spaced. The relation \(d \propto \sqrt{M}\) is primarily a statement about how matter occupies space, and the factor \((1+\lambda)\) only shifts the line vertically — so this plot does not by itself test the wave coupling \(\lambda\). Testing \(\lambda\) requires the rotation-curve and cluster calculations of the series.
BeeTheory.com — Total mass vs neighbour distance · Initial generation: 21 May 2026 with Claude.ai · © Technoplane S.A.S. 2026