Step 2 — The Twenty-Three Galaxies:
Yukawa Kernel Applied, Three Normalisations

Twenty-three galaxies, three normalisations

22 SPARC galaxies (λ = 0.496): Median |err| = 14.6%, mean signed err = −4.7%, 18/21 within 30%, 14/21 within 20%.

Milky Way (λ = 0.189): err = +14.9% at $R = 5R_d$, consistent with the same structural over-prediction pattern documented in Note XIV.

Normalised rotation curves: when scaled by $R/R_d$, the predicted curves of all 23 galaxies overlap into a single band, with the dispersion driven primarily by surface density (consistent with Note XI).

$$x \;=\; R/R_d, \qquad y \;=\; V_\text{tot}(R)\,/\,V_\text{dyn} \quad\text{with}\quad V_\text{dyn} = \sqrt{G\,M_\text{bar}/R_d}$$

Under this normalisation, low-mass dwarfs (blue, Sd–Im) lie at high $y$ — their observed rotation greatly exceeds the dynamical velocity their visible mass would produce, by factors of 2 to 4. Massive spirals (red, Sb–Sbc) sit closer to $y \sim 1$. The Milky Way (thick red line) sits in the lower half, consistent with its high baryonic mass. The vertical spread at fixed $R/R_d$ reflects the well-known fact that low-mass galaxies need proportionally more dark matter relative to their baryons.

$$x \;=\; R/R_d, \qquad y \;=\; V_\text{tot}(R) \;\text{ in km/s}$$

This view separates the galaxies vertically by their absolute rotation. The massive spirals (NGC 2841 at the top, then the Milky Way and NGC 3198) occupy the upper band. The Sd–Im dwarfs cluster in the lower third. All curves rise from low $R/R_d$ to their flat regime around $R/R_d \approx 3$–$5$, and the BeeTheory prediction follows the same morphology across all galaxies — the curves do not cross, indicating that no class of galaxy is qualitatively mishandled by the model.

$$x \;=\; R/R_d, \qquad y \;=\; V_\text{tot}(R)\,/\,V_f^\text{obs}$$

The strongest test of the model

At $R/R_d = 5$, where the rotation curve has reached its flat regime, the predicted-over-observed ratio is the prediction error itself. The bulk of galaxies cluster between $y = 0.7$ and $y = 1.2$ — confirming the 14.6% median error. The few outliers stretching to $y \sim 1.4$ are the over-predicted massive spirals (NGC 3198 at $y = 1.43$); those near $y \sim 0.6$ are the under-predicted low-density disks. This view confirms that the residual structure is not a random scatter but a systematic envelope, identifiable across morphological types.

A unified prediction across six decades in mass

The Milky Way ($M_\text{bar} \sim 5 \times 10^{10}\,M_\odot$, $V_f \sim 230$ km/s) and the smallest dwarf in the calibration set, DDO 064 ($M_\text{bar} \sim 4 \times 10^{8}\,M_\odot$, $V_f = 26$ km/s) are separated by more than five orders of magnitude in baryonic mass and an order of magnitude in rotation amplitude. The same Yukawa kernel with the same geometric constants $(c_\text{sph}, c_\text{disk}, c_\text{arm})$ describes both, with a median deviation of 14.6%.

Residual structure remains

As shown in view 3, the residuals are not random: they form a systematic envelope between $0.6$ and $1.4$ around $y = 1$. The signature is identical to the one identified in Note XI — disks with high $\Sigma_d$ are over-predicted, disks with low $\Sigma_d$ are under-predicted. The Milky Way ($\Sigma_d^\text{eff} \sim 800\,L_\odot/\text{pc}^2$, much denser than the SPARC dwarfs) is among the over-predicted galaxies. This consistency between the MW behaviour and the SPARC sample reinforces the conclusion that surface density is the missing variable.

Ready for the blind step

With the formalism explicit, the geometric integration verified, and the residual signature characterised, the next step is to apply the same machinery — same kernel, same parameters, same procedure — to the 94 SPARC galaxies that were never used in calibration. This is the subject of Step 3.