Technical Note XLV

Equations and coefficients — reference sheet

BeeTheory.com · Wave-based quantum gravity · 21 May 2026

All quantities in SI units: mass \(M\) in kg, distances \(d, R, r\) in metres. Universal parameters: \(\lambda = 12.696\), \(c = 0.163\), \(\ell_{floor} = 3.00\) kpc \(= 9.26\times10^{19}\) m.

1. Model definitions (exact, not fitted)

Wave mass and total mass

\[ M_{wave} = \lambda\,M_{vis} = 12.696\,M_{vis} \] \[ M_{total} = (1+\lambda)\,M_{vis} = 13.696\,M_{vis} \]

Enclosed wave-mass profile

\[ M_{wave}(

Wave density (exponential kernel)

\[ \rho_{wave}(s) = \frac{\lambda\,M_{vis}}{8\pi\,\ell^3}\,e^{-s/\ell} \]

Rotation velocity

\[ V^2(R) = V^2_{baryon}(R) + V^2_{wave}(R), \qquad V^2_{wave}(R) = G\,\lambda\,\frac{\Sigma M_{wave}(

2. Three linear relations (log–log, 7 cosmic levels)

Each relation is a straight line in log–log coordinates — i.e. a power law of the form \(\log_{10} y = a\,\log_{10} x + b\). Least-squares fits on the seven representative levels; M is the total mass \((1+\lambda)M_{vis}\). The scatter \(\sigma\) measures how tightly the points lie on each line; it does not affect the linearity.

Relation 1 — Mass vs distance to neighbours \(d\)

\[ \log_{10} M = 2.049\,\log_{10} d – 3.117 \] \[ M = 7.65\times10^{-4}\;d^{\,2.05} \quad (\sigma = 0.63\ \text{dex}) \]

Equivalently \(d \propto M^{0.49} \approx \sqrt{M}\). Tightest of the three.

Relation 2 — Mass vs object radius \(R\)

\[ \log_{10} M = 1.052\,\log_{10} R + 20.780 \] \[ M = 6.03\times10^{20}\;R^{\,1.05} \quad (\sigma = 1.49\ \text{dex}) \]

Close to \(M \propto R\) on this broad sample (not \(R^2\)).

Relation 3 — Surface density \(M/R^2\) vs mass \(M\)

\[ \log_{10}\!\left(\frac{M}{R^2}\right) = -0.743\,\log_{10} M + 32.995 \] \[ \frac{M}{R^2} = 9.89\times10^{32}\;M^{\,-0.74} \quad (\sigma = 2.71\ \text{dex}) \]

A straight line of slope \(-0.74\) — linear in log–log, but with a non-zero slope, so \(M/R^2\) is not constant across the full range (a true cosmic constant would mean slope \(0\)). This relation is also algebraically implied by Relations 1–2, since \(M/R^2\) is built from \(M\) and \(R\). Median over the sample: \(M/R^2 \approx 8.4\ \text{kg/m}^2 \approx 0.84\ \text{g/cm}^2\) (meaningful only within the restricted galaxy-to-cluster regime).

3. Coefficient summary

All three are linear in log–log (\(\log_{10} y = a\,\log_{10} x + b\)); the columns give the prefactor \(10^{b}\) and the slope \(a\).

Equation	Prefactor	Exponent	Scatter
M vs d	7.65×10⁻⁴	+2.05	0.63 dex
M vs R	6.03×10²⁰	+1.05	1.49 dex
M/R² vs M	9.89×10³²	−0.74	2.71 dex
M_wave / M_vis	λ = 12.696	—	exact
M_total / M_vis	1+λ = 13.696	—	exact

Note: the factor \((1+\lambda)\) enters only the prefactors (a constant +1.14 dex vertical shift), never the exponents. The fitted slopes are unchanged whether one plots visible or total mass.

BeeTheory.com — Equations & coefficients reference sheet · Initial generation: 21 May 2026 with Claude.ai · © Technoplane S.A.S. 2026