BeeTheory · Foundations · Technical Note XIII

Input Data and the Three Test Corpora

The methodology of Note XII transforms five observational inputs into a complete set of geometric parameters per galaxy, ready for the wave-field convolution. This note presents those parameters explicitly for the three test corpora that will be used to evaluate the model: the Milky Way alone, the 22-galaxy calibration set, and the 94-galaxy blind sample. Each step extends the test by one order of magnitude in the number of galaxies.

1. The three-step protocol

Three corpora, three roles

Step 1 — Milky Way (1 galaxy). Reference point. Fixes the global wave-field coupling $\lambda$ from internal stellar surveys and 21-cm maps.

Step 2 — Calibration set (22 galaxies). The first twenty entries of the SPARC catalogue plus three extreme cases (dense, classical spiral, gas-rich). The model is applied with $\lambda$ frozen from Step 1, allowing one global re-calibration if needed.

Step 3 — Blind test (94 galaxies). All parameters are frozen from Step 2. No further adjustment. The rotation curves of the remaining SPARC galaxies are pure predictions.

2. Universal theory parameters (identical for all three corpora)

Five numbers, fixed once for all galaxies of all sizes and types. They define the wave kernel and the global coupling. They will not vary across the three steps.

Parameter	Symbol	Value	Role
Wave-mass amplitude	$K_0$	$0.3759$	Sets the dimensionless scale of the wave kernel
3D coherence ratio	$c_\text{sph}$	$0.41$	$\ell_b / r_b$ for the bulge
2D coherence ratio	$c_\text{disk}$	$3.17$	$\ell / R_\text{scale}$ for disks and gas ring
Spiral coherence ratio	$c_\text{arm}$	$2.0$	$\ell_\text{arm} / R_d$ for spiral arms
Stellar mass-to-light ratio	$\Upsilon_\star$	$0.5\,M_\odot/L_\odot$	Spitzer 3.6 µm (McGaugh 2014)

3. Step 1 — The Milky Way

3.1 Observational inputs

Quantity	Value	Source
Hubble type $T$	4 (Sbc)	de Vaucouleurs et al. 1991
Disk scale length $R_d$	$2.6$ kpc	Bovy & Rix 2013
Total stellar mass $M_\star$	$4.0 \times 10^{10}\,M_\odot$	Photometric surveys (Bland-Hawthorn & Gerhard 2016)
Total gas mass $M_\text{gas}$ (HI + He)	$1.06 \times 10^{10}\,M_\odot$	21-cm maps
Observed flat velocity $V_f$	$\approx 230$ km/s at $R_\odot$	Gaia DR3 (Ou et al. 2024)

3.2 Derived geometric parameters per component

Component	Mass ($10^{10}\,M_\odot$)	Spatial scale	Coherence length $\ell$	Profile
Bulge ($T \leq 4$ → activated)	1.240	$r_b$ = 0.61 kpc	$\ell_b$ = 0.25 kpc	3D Hernquist
Thin disk	2.070	$R_d$ = 2.60 kpc	$\ell_\text{thin}$ = 8.24 kpc	2D exponential
Thick disk	0.690	$1.5\,R_d$ = 3.90 kpc	$\ell_\text{thick}$ = 12.36 kpc	2D exponential
Gas ring	1.060	$R_g$ = 4.42 kpc	$\ell_\text{gas}$ = 14.01 kpc	2D exp. with hole
Spiral arms	0.2070	$R_d$ = 2.60 kpc	$\ell_\text{arm}$ = 5.20 kpc	2D azimuthal

Note on the Milky Way inputs: the Milky Way uses direct observational decompositions (Bland-Hawthorn & Gerhard 2016) rather than the photometric formula $M_star = 2pi R_d^2,Sigma_d,Upsilon_star$ used for SPARC galaxies. This is because the Milky Way is observed from within, and its mass components are measured by combining stellar surveys, microlensing, and dynamics rather than by a single integrated luminosity. The decomposition into components and the wave-field equations are otherwise identical.

4. Step 2 — Twenty-two calibration galaxies

The first twenty entries of the SPARC catalogue (Lelli et al. 2016), augmented by three extreme cases that test the limits of the model: NGC 2841 (massive dense early-type), NGC 3198 (classical grand-design spiral), DDO 154 (gas-dominated dwarf).

For each galaxy, the five observational inputs $(T, R_d, \Sigma_d, M_\text{HI}, V_f)$ are taken from SPARC. From these, the masses and coherence lengths of the five components are computed using the formulas of Note XII. The table below lists all derived quantities.

Galaxy	Type	$R_d$ (kpc)	$\Sigma_d$ ($L_\odot/$pc$^2$)	$M_\text{gas}$ $(10^{10})$	$M_\star$ $(10^{10})$	$f_\text{gas}$	$M_b$ $(10^{10})$	$r_b$ (kpc)	$M_\text{thin}$ $(10^{10})$	$M_\text{thick}$ $(10^{10})$	$\ell_\text{thin}$ (kpc)	$\ell_\text{gas}$ (kpc)
CamB	Im	0.47	66	0.002	0.005	0.32	—	—	0.003	0.001	1.49	2.53
D631-7	Im	0.70	115	0.051	0.018	0.74	—	—	0.013	0.004	2.22	3.77
DDO064	Im	0.33	120	0.023	0.004	0.85	—	—	0.003	0.001	1.05	1.78
DDO154	Im (gas)	0.60	45	0.063	0.005	0.92	—	—	0.004	0.001	1.90	3.23
DDO161	Im	1.10	35	0.109	0.013	0.89	—	—	0.010	0.003	3.49	5.93
DDO168	Im	0.69	100	0.028	0.015	0.65	—	—	0.011	0.004	2.19	3.72
DDO170	Im	1.10	25	0.051	0.010	0.84	—	—	0.007	0.002	3.49	5.93
ESO116-G012	Sd	2.10	115	0.160	0.159	0.50	—	—	0.119	0.040	6.66	11.32
ESO444-G084	Im	0.55	60	0.016	0.006	0.74	—	—	0.004	0.001	1.74	2.96
F561-1	Im	2.50	30	0.120	0.059	0.67	—	—	0.044	0.015	7.92	13.47
F563-1	Im	2.70	20	0.160	0.046	0.78	—	—	0.034	0.011	8.56	14.55
F563-V1	Im	1.20	25	0.040	0.011	0.78	—	—	0.008	0.003	3.80	6.47
F563-V2	Im	1.10	30	0.047	0.011	0.80	—	—	0.009	0.003	3.49	5.93
F565-V2	Im	1.00	18	0.027	0.006	0.82	—	—	0.004	0.001	3.17	5.39
F567-2	Im	1.80	15	0.080	0.015	0.84	—	—	0.011	0.004	5.71	9.70
F568-1	Sd	3.20	40	0.239	0.129	0.65	—	—	0.097	0.032	10.14	17.24
F568-3	Sd	3.00	35	0.200	0.099	0.67	—	—	0.074	0.025	9.51	16.17
F568-V1	Im	2.10	20	0.106	0.028	0.79	—	—	0.021	0.007	6.66	11.32
F571-8	Sd	4.50	50	0.293	0.318	0.48	—	—	0.239	0.080	14.27	24.25
F574-1	Sd	3.60	30	0.253	0.122	0.67	—	—	0.092	0.031	11.41	19.40
NGC2841	Sb	3.50	605	1.104	2.328	0.32	0.466	1.75	1.397	0.466	11.09	18.86
NGC3198	Sc	3.14	153	1.144	0.474	0.71	—	—	0.355	0.118	9.95	16.92

All 22 galaxies of the calibration set. Bulge masses and radii appear only for $T \leq 4$ (the last two rows, NGC 2841 and NGC 3198). The coherence lengths $\ell_\text{thick} = 1.5\,\ell_\text{thin}$ and $\ell_\text{arm} = (2.0/3.17)\,\ell_\text{thin}$ are omitted from the table to keep it compact.

Coverage of parameter space

The 22 calibration galaxies span $R_d$ from $0.33$ to $4.5$ kpc (factor 14), $Sigma_d$ from 15 to 605 $L_odot/text{pc}^2$ (factor 40), and stellar mass from $4 times 10^7$ to $2.3 times 10^{10},M_odot$ (factor 500). The Milky Way ($R_d = 2.6$ kpc, $M_star = 4 times 10^{10}$) sits at the upper-massive end of the range, making it a stringent calibration anchor for the dwarfs that dominate the sample.

5. Step 3 — Blind test on 94 SPARC galaxies

The blind test set consists of 94 galaxies drawn from the SPARC catalogue, distinct from the 22 calibration galaxies. They span the entire range of disk galaxies — from compact dwarfs to giant spirals — and were never used in the calibration of any parameter.

For brevity, only twelve representative galaxies are shown in the table below. The full list of 94 is given in Appendix A.

Galaxy	Type	$R_d$ (kpc)	$\Sigma_d$	$M_\text{gas}$ $(10^{10})$	$M_\star$ $(10^{10})$	$f_\text{gas}$	$M_b$ $(10^{10})$	$r_b$ (kpc)	$M_\text{thin}$ $(10^{10})$	$M_\text{thick}$ $(10^{10})$	$\ell_\text{thin}$ (kpc)	$\ell_\text{gas}$ (kpc)
F583-1	Im	1.80	22	0.093	0.022	0.81	—	—	0.017	0.006	5.71	9.70
IC2574	Sm	2.80	18	0.293	0.044	0.87	—	—	0.033	0.011	8.88	15.09
M33	Sc	1.40	190	0.146	0.117	0.56	—	—	0.088	0.029	4.44	7.54
NGC0801	Sc	5.80	190	0.931	2.008	0.32	—	—	1.506	0.502	18.39	31.26
NGC2403	Sc	1.80	186	0.279	0.189	0.60	—	—	0.142	0.047	5.71	9.70
NGC3521	Sbc	2.80	327	1.144	0.805	0.59	0.161	1.40	0.483	0.161	8.88	15.09
NGC5055	Sbc	3.50	250	0.998	0.962	0.51	0.192	1.75	0.577	0.192	11.09	18.86
UGC02885	Sc	8.50	150	2.394	3.405	0.41	—	—	2.554	0.851	26.95	45.81
UGC11455	Sc	5.50	40	1.064	0.380	0.74	—	—	0.285	0.095	17.43	29.64
NGC6503	Sc	2.40	210	0.466	0.380	0.55	—	—	0.285	0.095	7.61	12.93
NGC2915	Im	0.50	160	0.064	0.013	0.84	—	—	0.009	0.003	1.58	2.69
UGC02487	S0	7.50	300	1.596	5.301	0.23	1.060	3.75	3.181	1.060	23.77	40.42

Representative subset (12 of 94 blind galaxies). The complete list spans Hubble types S0–Im and includes the most massive observed disk galaxies (UGC 02885, UGC 11455) as well as ultra-compact dwarfs (NGC 6789, UGC 05764).

Test scope

The 94 blind galaxies extend the parameter space well beyond the calibration set. $R_d$ ranges from $0.30$ to $8.50$ kpc, surface density from $12$ to $605$ $L_\odot/\text{pc}^2$, and observed flat velocity from $17$ to $330$ km/s. The Milky Way calibration anchor at $R_d = 2.6$ kpc sits roughly at the geometric median of this distribution.

6. The structure of the three corpora — comparative summary

Property	Step 1 — Milky Way	Step 2 — 22 calibration galaxies	Step 3 — 94 blind galaxies
Number of galaxies	1	22	94
Role	Anchor	Calibration / global fit of $\lambda$	Prediction
$R_d$ range	2.6 kpc (fixed)	$0.33$ – $4.5$ kpc	$0.30$ – $8.5$ kpc
$\Sigma_d$ range	(direct masses)	15 – 605 $L_\odot/\text{pc}^2$	12 – 605 $L_\odot/\text{pc}^2$
$M_\star$ range	$4 \times 10^{10}\,M_\odot$	$4 \times 10^7$ – $2.3 \times 10^{10}$	$3 \times 10^7$ – $5.3 \times 10^{10}$
$V_f$ range	230 km/s	2 – 278 km/s	17 – 330 km/s
Hubble types covered	Sbc	S0a, Sb, Sc, Sd, Im	S0, Sa, Sb, Sbc, Sc, Sd, Im, Sm
Bulges activated ($T \leq 4$)	Yes	2 of 22	$\sim$30 of 94
What is fitted	$\lambda$ (global coupling)	$\lambda$ may be re-fitted globally	Nothing — fully blind

7. What this note establishes

Inputs fully specified before any computation

For each of the 117 galaxies (1 + 22 + 94), the five observational inputs $(T, R_d, \Sigma_d, M_\text{HI}, \Upsilon_\star)$ and the resulting geometric decomposition are fixed before the wave-field computation begins. The wave-field equations of Note XII operate on these inputs without any galaxy-specific tuning beyond the universal parameters $(K_0, c_\text{sph}, c_\text{disk}, c_\text{arm}, \lambda)$.

A graduated test of generalisation

The three steps form a natural cascade of increasing test severity. Step 1 establishes that the framework can describe the Milky Way using its observed baryonic content. Step 2 verifies that the calibration generalises to a small heterogeneous sample including extreme cases. Step 3 places the framework in a true predictive mode, with no further parameter adjustment, on a sample large enough for the residual statistics to be meaningful.

Unidirectional throughout

At every step, the rotation curve is computed from the baryonic inputs, never the reverse. The comparison with observation is a test, not a calibration loop. The single number $\lambda$ is fixed once on the Milky Way (Step 1), possibly refined globally on the 22 calibration galaxies (Step 2), and then frozen for the blind prediction on the 94 remaining galaxies (Step 3).

8. Summary

1. The BeeTheory framework will be applied in three successive steps: 1 galaxy (Milky Way), then 22 (calibration), then 94 (blind).

2. For each galaxy, the five observational inputs $(T, R_d, \Sigma_d, M_\text{HI}, \Upsilon_\star)$ produce a five-component decomposition with explicit masses, scales, and coherence lengths, computed once via the formulas of Note XII.

3. The five universal theory parameters $(K_0, c_\text{sph}, c_\text{disk}, c_\text{arm}, \Upsilon_\star)$ apply identically to all 117 galaxies. The global coupling $\lambda$ is fitted in Step 2 at the latest and frozen for Step 3.

4. The calibration set covers a factor of 14 in $R_d$, 40 in $\Sigma_d$, and 500 in $M_\star$. The blind set extends these ranges further. The Milky Way anchor sits within both.

5. Each step is a test of generalisation of the model. The blind step is purely predictive: no rotation-curve information from the 94 galaxies enters the computation at any stage.

References. Lelli, F., McGaugh, S. S., Schombert, J. M. — SPARC: Mass Models for 175 Disk Galaxies with Spitzer Photometry and Accurate Rotation Curves, AJ 152, 157 (2016). Catalogue source. · Bland-Hawthorn, J., Gerhard, O. — The Galaxy in Context, ARA&A 54, 529 (2016). Milky Way structural parameters. · Bovy, J., Rix, H.-W. — A direct dynamical measurement of the Milky Way’s disk surface density profile, ApJ 779, 115 (2013). · McGaugh, S. S. — The third law of galactic rotation, Galaxies 2, 601 (2014). $\Upsilon_\star$ at 3.6 µm. · Ou, X. et al. — The dark matter profile of the Milky Way, MNRAS 528, 693 (2024). Gaia 2024 rotation curve. · Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, v2, BeeTheory.com (2023).