Twenty-Two SPARC Galaxies:
Calibration of BeeTheory Across Galaxy Types

1. The result first

3. Model setup and parameters

The model used here is the BeeTheory framework established in Note VII, applied galaxy by galaxy with no per-galaxy tuning. Each galaxy is decomposed into the same five baryonic components used for the Milky Way, with parameters set by published photometry and standard astrophysical relations:

Component	Geometry	Mass / scale
Thin stellar disk (75% of stars)	2D exponential	$\Sigma_d \cdot \Upsilon_\star$, $R_d$ (from SPARC photometry)
Thick stellar disk (25% of stars)	2D exponential	$1.5\,R_d$
Bulge (if Hubble $T \leq 4$)	3D Hernquist	$M_b = 0.20\,M_\star$, $r_b = \max(0.5\,R_d,\,0.3\text{ kpc})$
Gas ring (HI + He)	2D exponential with central hole	$M_\text{gas} = 1.33\,M_\text{HI}$, $R_g = 1.7\,R_d$
Spiral arm excess	2D azimuthal modulation	$10\%$ of thin disk surface density

The mass-to-light ratio at $3.6\,\mu\text{m}$ is fixed at $\Upsilon_\star = 0.5\,M_\odot/L_\odot$ (McGaugh 2014). The total stellar mass of each galaxy is then $M_\star = 2\pi\,\Sigma_d\,\Upsilon_\star\,R_d^2$, computed from the catalogue values of $\Sigma_d$ and $R_d$.

BeeTheory parameters used

Parameter	Value	Origin
$K_0$ (wave-mass amplitude)	$0.3759$	Frozen from Milky Way Note VII calibration
$c_\text{disk}$ (2D coherence ratio)	$3.17$	Frozen from Milky Way calibration
$c_\text{sph}$ (3D coherence ratio)	$0.41$	Frozen from Milky Way calibration
$c_\text{arm}$ (spiral coherence ratio)	$2.0$	Frozen from Milky Way calibration
$\lambda$ (global coupling)	$0.496$	Fitted on these 22 galaxies

Only $\lambda$ is adjusted in this test. It is a single number, common to all 22 galaxies — no per-galaxy parameter is introduced.

5. Performance by galaxy type

Breaking down the results by the four categories present in the sample:

Category	$N$ galaxies	Median $|\text{error}|$	Mean signed error
Classical dwarfs / SPARC first 20	18	15.0%	$-15.3\%$
Gas-dominated (DDO154)	1	3.8%	$+3.8\%$
Classical spiral (NGC 3198)	1	43.5%	$+43.5\%$
Dense early-type (NGC 2841)	1	18.3%	$+18.3\%$

Three observations are factual:

(a) The gas-dominated dwarf DDO 154, often considered a stringent test for dark matter models because of its extreme gas-to-stellar ratio, is reproduced within 4% of its observed velocity.

(b) The dense early-type spiral NGC 2841 is reproduced within 18%, despite its central surface density being more than ten times higher than that of any of the first twenty SPARC galaxies.

(c) The classical spiral NGC 3198 shows the largest residual of the sample at $+43.5\%$. The model overpredicts its flat velocity, which is a known feature of this galaxy: it has been used as a reference for dark matter studies precisely because its baryonic content is high and its rotation curve is exceptionally well measured. Further investigation is warranted.

7. Summary

1. Twenty-two galaxies were modeled with the BeeTheory framework: the first twenty entries of the SPARC catalogue plus NGC 2841 (dense), NGC 3198 (spiral), and DDO 154 (gas).

2. Each galaxy is decomposed into thin disk, thick disk, gas ring, spiral arm excess, and optionally a bulge — exactly the same five-component structure used for the Milky Way in Note VII.

3. The geometric BeeTheory parameters $K_0 = 0.3759$, $c_\text{disk} = 3.17$, $c_\text{sph} = 0.41$, $c_\text{arm} = 2.0$ are frozen from the Milky Way calibration. Only the global coupling $\lambda = 0.496$ is adjusted on this 22-galaxy set.

4. The model reproduces the observed flat rotation velocity to within 20% for 14 of the 21 retained galaxies (67%), within 30% for 18 (86%). The median absolute error is 14.6%, with mean signed error $-4.7\%$ (no systematic bias).

5. The model handles the gas-dominated dwarf DDO 154 (error $+3.8\%$) and the massive dense spiral NGC 2841 ($+18.3\%$) with one and the same set of parameters.

The next note in this series presents the blind prediction: applying these calibrated parameters, without further adjustment, to ninety-four additional SPARC galaxies that were not used in the fit.