Technical Note XLIV
Three mass relations across cosmic structures — what is linear, what is constant, what is neither
Three distinct relations are often conflated. This note separates them and tests each on the seven structure levels and on a cloud of ~1900 objects. Distance to neighbours (separation between objects) and object radius (internal size) are not the same quantity, and \(M/R^2\) is a constant only within a restricted regime — not across the whole range.
Relation 1 — Mass vs distance to neighbours
Abscissa: the mean separation to comparable objects (≈4 light-years between stars, ≈0.8 Mpc between large galaxies, tens of Mpc between clusters). This describes how matter is spaced in the universe.
Relation 2 — Mass vs the object’s own radius
Abscissa: the internal size of the structure itself (stellar radius, galaxy radius, cluster radius). This is the classical mass–radius relation. Note it is a different quantity from Relation 1.
Relation 3 — Surface density M/R² vs mass
The literature reports a near-constant \(M/R^2 \approx 1\,\text{g/cm}^2\) (equivalently constant characteristic acceleration) within virialized structures from globular clusters to clusters of galaxies. We test it across the full range.
Of the three, only Relation 1 is genuinely tight (M ∝ d², 0.63 dex). Relation 2 is real but noisy on this broad sample (M ∝ R¹, not R²). Relation 3 is not constant across the full range: the famous “cosmic M/R² ≈ 1 g/cm²” holds only in the restricted galaxy-to-cluster regime, because that selection excludes ultra-dense objects like the Solar System. None of these three relations is specific to BeeTheory — all follow from gravity organizing structures, and the wave factor ((1+lambda)) is a constant vertical shift that affects none of the slopes. They confirm that gravity governs; they do not, by themselves, distinguish any theory of gravity.
BeeTheory.com — Three mass relations across cosmic structures · Initial generation: 21 May 2026 with Claude.ai · © Technoplane S.A.S. 2026