BeeTheory · Foundations · Technical Note XXIV

The Milky Way with the Corrected Kernel:
Dimensionally Clean, Physically Coherent

The Milky Way rotation curve is recomputed with the normalised kernel of Note XXII, where $\lambda$ is now the dimensionless wave-mass fraction and $\ell_0$ is the coherence length. The result is the cleanest fit yet — $\chi^2/\text{dof} = 0.89$ — with $\lambda$ now of order unity, consistent with the magnitude of the “missing mass” in galactic dynamics. The improved framework also exposes a previously-hidden factor in the geometric projection that must be calibrated.

2. The corrected formulation, recalled

From Note XXII, the BeeTheory wave kernel is normalised so that a point mass $m$ generates an asymptotic wave mass $lambda m$:

For a galaxy treated as an axisymmetric distribution in the plane, the total baryonic surface density is summed over the four components, and the wave-field surface density is obtained by an azimuthally-averaged convolution:

with the azimuthally averaged kernel $\langle\mathcal{K}\rangle(R,R’) = \frac{1}{4\pi^2 \ell_0^2}\int_0^\pi \frac{e^{-D(\phi)/\ell_0}}{D(\phi)}\,d\phi$, where $D(\phi) = \sqrt{R^2 + R’^2 – 2RR’\cos\phi}$.

5. The geometric factor: why $M_\text{wave} \neq \lambda M_\text{bar}$ exactly

From the calculation, the total wave mass integrated out to $R = 40$ kpc is $M_\text{wave}(<40) = 10.5 \times 10^{10}\,M_\odot$, while $\lambda M_\text{bar} = 1.02 \times 5.27 \times 10^{10} = 5.37 \times 10^{10}\,M_\odot$. The ratio is $\sim 2$, not $1$.

This means the fit value $\lambda_\text{plane} = 1.02$ corresponds to a 3D physical coupling of approximately $\lambda_\text{3D} \approx 0.5$. The exact ratio could be derived analytically by carrying the disk thickness explicitly. For now, we keep $\lambda$ as a phenomenological 2D-projected parameter, noting that its physical interpretation is “wave fraction in the plane”.

7. Implications

7.1 Coherence length is sub-kpc

$ell_0 approx 500$ pc is approximately the thickness of the Milky Way disk. The wave field of a star deploys over the disk thickness, not over the entire galaxy. This means the wave mass of a star is essentially “above and below” its position — confined to a column $\sim 1$ kpc tall, $\sim 1$ kpc wide.

7.2 Wave mass is comparable to visible mass

$\lambda \approx 1$ means: as much wave mass as visible mass, locally. For the Earth, the same coupling implies that out of the total $5.97 \times 10^{24}$ kg measured locally, only $\approx 50\%$ is “atomic mass” in the BeeTheory interpretation, the rest being delocalised wave mass over $\sim 500$ pc. This is a dramatic reinterpretation — but it is invisible to all local experiments (Note XXIII).

7.3 The remaining factor 5–10 in galactic dynamics

The standard model requires roughly $5$–$10$ times the visible mass to explain galactic rotation curves. Here, BeeTheory with $\lambda = 1.02$ contributes a factor of $\sim 2$. The remaining factor $3$–$5$ would need to come from a more sophisticated mechanism — possibly a non-linear amplification of the wave field in regions of high baryonic concentration, or a longer coherence length component contributing diffuse background. These directions are open for further investigation.