BeeTheory · Foundations · Technical Note IX

Ninety-Four Galaxies Blind:
BeeTheory Applied Without Parameter Adjustment

The parameters calibrated on the Milky Way and on the twenty-two-galaxy set of Note VIII are now applied, with no further adjustment, to ninety-four additional SPARC galaxies. This note reports the outcome.

1. The result first

Blind prediction on 94 SPARC galaxies

Median $|\\text{error}|$: 19.0%
Within 20% of $V_f$: 49 / 94 galaxies (52%)
Within 30% of $V_f$: 67 / 94 galaxies (71%)
Within 50% of $V_f$: 89 / 94 galaxies (95%)
Mean signed error: $+1.4\%$ (no systematic bias)
Pearson correlation: $r(\\log V_f, \\log V_\\text{tot}) = 0.925$

All parameters are frozen from Note VIII: $K_0 = 0.3759$, $c_\\text{disk} = 3.17$, $c_\\text{sph} = 0.41$, $c_\\text{arm} = 2.0$, $\\lambda = 0.496$. No re-fitting was performed.

2. Procedure

The protocol is the same as Note VIII, applied to a disjoint set of 94 galaxies that were not used to calibrate $\lambda$. For each galaxy the published SPARC parameters $(R_d,\,\Sigma_d,\,M_\text{HI},\,\text{Hubble}\,T,\,V_f)$ are read from Lelli et al. 2016. The five-component baryonic structure — thin disk, thick disk, bulge if $T\leq 4$, gas ring, spiral arm excess — is constructed from these published values together with the standard astrophysical relations used in Note VIII. The BeeTheory wave field is then computed by convolution and the total predicted circular velocity at $R_\text{eval} = \max(5\,R_d,\,5\,\text{kpc})$ is compared to the observed $V_f$.

No parameter is allowed to vary. The same coupling $\lambda$, the same geometric constants, the same component-mass relations as in Note VIII. The error is reported as $(V_\text{tot}-V_f)/V_f$.

3. Predicted versus observed velocities

The figure below plots the predicted total velocity against the observed flat rotation velocity for all 94 galaxies on logarithmic axes. The solid diagonal is the ideal 1:1 relation; the two dotted lines bracket the $\pm 20\%$ band. Each point is colored by the absolute value of its prediction error.

Predicted vs observed flat rotation velocity — 94 SPARC galaxies (blind) 3050100200300 3050100200300 V_f observed (km/s) V_tot predicted (km/s) F579-V1: Vf=105, Vtot=92, err=-12%F583-1: Vf=83, Vtot=58, err=-30%F583-4: Vf=67, Vtot=48, err=-29%IC2574: Vf=69, Vtot=93, err=+35%KK98-251: Vf=17, Vtot=13, err=-23%M33: Vf=100, Vtot=98, err=-2%NGC0055: Vf=87, Vtot=86, err=-1%NGC0100: Vf=83, Vtot=94, err=+13%NGC0247: Vf=90, Vtot=108, err=+20%NGC0289: Vf=155, Vtot=200, err=+29%NGC0300: Vf=76, Vtot=76, err=+0%NGC0801: Vf=208, Vtot=326, err=+57%NGC0891: Vf=212, Vtot=226, err=+7%NGC0925: Vf=105, Vtot=155, err=+48%NGC1003: Vf=115, Vtot=129, err=+12%NGC1090: Vf=170, Vtot=199, err=+17%NGC1705: Vf=54, Vtot=44, err=-19%NGC2366: Vf=55, Vtot=63, err=+14%NGC2403: Vf=131, Vtot=126, err=-4%NGC2683: Vf=175, Vtot=201, err=+15%NGC2903: Vf=184, Vtot=183, err=-0%NGC2915: Vf=85, Vtot=53, err=-38%NGC2955: Vf=266, Vtot=406, err=+53%NGC2976: Vf=80, Vtot=50, err=-37%NGC3109: Vf=68, Vtot=51, err=-25%NGC3521: Vf=225, Vtot=236, err=+5%NGC3621: Vf=149, Vtot=190, err=+28%NGC3726: Vf=152, Vtot=181, err=+19%NGC3741: Vf=51, Vtot=64, err=+26%NGC3769: Vf=112, Vtot=140, err=+25%NGC3877: Vf=163, Vtot=183, err=+12%NGC3893: Vf=159, Vtot=182, err=+14%NGC3949: Vf=125, Vtot=99, err=-21%NGC3953: Vf=200, Vtot=188, err=-6%NGC3972: Vf=135, Vtot=99, err=-27%NGC3992: Vf=242, Vtot=207, err=-15%NGC4010: Vf=128, Vtot=110, err=-14%NGC4013: Vf=185, Vtot=199, err=+8%NGC4051: Vf=110, Vtot=114, err=+3%NGC4085: Vf=135, Vtot=79, err=-41%NGC4088: Vf=175, Vtot=128, err=-27%NGC4100: Vf=162, Vtot=117, err=-28%NGC4138: Vf=150, Vtot=85, err=-44%NGC4157: Vf=185, Vtot=183, err=-1%NGC4183: Vf=110, Vtot=70, err=-36%NGC4214: Vf=68, Vtot=50, err=-27%NGC4217: Vf=180, Vtot=189, err=+5%NGC4389: Vf=110, Vtot=62, err=-43%NGC4559: Vf=123, Vtot=158, err=+28%NGC5005: Vf=260, Vtot=240, err=-8%NGC5033: Vf=195, Vtot=280, err=+44%NGC5055: Vf=180, Vtot=238, err=+32%NGC5371: Vf=225, Vtot=257, err=+14%NGC5585: Vf=87, Vtot=83, err=-5%NGC5907: Vf=210, Vtot=277, err=+32%NGC5985: Vf=295, Vtot=301, err=+2%NGC6015: Vf=142, Vtot=150, err=+6%NGC6195: Vf=260, Vtot=380, err=+46%NGC6503: Vf=121, Vtot=168, err=+39%NGC6674: Vf=260, Vtot=380, err=+46%NGC6789: Vf=60, Vtot=22, err=-63%NGC6946: Vf=180, Vtot=198, err=+10%NGC7331: Vf=265, Vtot=275, err=+4%NGC7793: Vf=118, Vtot=116, err=-1%UGC00128: Vf=135, Vtot=243, err=+80%UGC02259: Vf=90, Vtot=90, err=+0%UGC02487: Vf=330, Vtot=465, err=+41%UGC02885: Vf=290, Vtot=441, err=+52%UGC05716: Vf=75, Vtot=71, err=-6%UGC05721: Vf=85, Vtot=85, err=+0%UGC05750: Vf=80, Vtot=110, err=+38%UGC05764: Vf=57, Vtot=31, err=-46%UGC05829: Vf=69, Vtot=62, err=-10%UGC06399: Vf=89, Vtot=99, err=+11%UGC06446: Vf=87, Vtot=92, err=+6%UGC06614: Vf=200, Vtot=238, err=+19%UGC06628: Vf=75, Vtot=80, err=+7%UGC06667: Vf=90, Vtot=104, err=+16%UGC06917: Vf=110, Vtot=97, err=-12%UGC06983: Vf=113, Vtot=117, err=+4%UGC07125: Vf=75, Vtot=103, err=+37%UGC07151: Vf=82, Vtot=56, err=-32%UGC07261: Vf=72, Vtot=51, err=-29%UGC07399: Vf=93, Vtot=75, err=-19%UGC07690: Vf=62, Vtot=44, err=-29%UGC08286: Vf=84, Vtot=80, err=-4%UGC08490: Vf=80, Vtot=57, err=-29%UGC08550: Vf=67, Vtot=55, err=-17%UGC09037: Vf=160, Vtot=147, err=-8%UGC11455: Vf=275, Vtot=198, err=-28%UGC11557: Vf=90, Vtot=91, err=+1%UGC11820: Vf=90, Vtot=113, err=+26%UGCA281: Vf=40, Vtot=26, err=-36%UGCA442: Vf=57, Vtot=32, err=-44% Color: |error| < 10% 10–20% 20–30% 30–50% > 50%
Predicted total velocity versus observed $V_f$ for 94 SPARC galaxies (blind test). The Pearson correlation in log-log space is $r = 0.93$.

The points cluster along the 1:1 line. Approximately half (52%) fall inside the $\pm 20\%$ band; about a third (28/94) are within $\pm 10\%$. The scatter is roughly balanced above and below the diagonal, consistent with the near-zero mean signed error of $+1.4\%$.

4. Residual structure: error versus disk size

To understand where the model performs best and worst, the prediction error is plotted as a function of the disk scale length $R_d$. The horizontal lines mark the median error in each size bin.

Prediction error versus disk scale length R_d — structural signature Compact: median -29%Medium: median -6%Large: median +10%Giant: median +39% 0.30.5123510 -50%-25%+0%+25%+50%+100% Disk scale length R_d (kpc) Prediction error (%) F579-V1: Rd=3.20, err=-12%F583-1: Rd=1.80, err=-30%F583-4: Rd=1.40, err=-29%IC2574: Rd=2.80, err=+35%KK98-251: Rd=0.30, err=-23%M33: Rd=1.40, err=-2%NGC0055: Rd=1.80, err=-1%NGC0100: Rd=2.30, err=+13%NGC0247: Rd=2.40, err=+20%NGC0289: Rd=3.50, err=+29%NGC0300: Rd=1.50, err=+0%NGC0801: Rd=5.80, err=+57%NGC0891: Rd=4.10, err=+7%NGC0925: Rd=3.10, err=+48%NGC1003: Rd=2.80, err=+12%NGC1090: Rd=3.80, err=+17%NGC1705: Rd=0.60, err=-19%NGC2366: Rd=1.30, err=+14%NGC2403: Rd=1.80, err=-4%NGC2683: Rd=2.90, err=+15%NGC2903: Rd=2.60, err=-0%NGC2915: Rd=0.50, err=-38%NGC2955: Rd=5.50, err=+53%NGC2976: Rd=0.75, err=-37%NGC3109: Rd=1.40, err=-25%NGC3521: Rd=2.80, err=+5%NGC3621: Rd=2.10, err=+28%NGC3726: Rd=3.00, err=+19%NGC3741: Rd=0.68, err=+26%NGC3769: Rd=2.80, err=+25%NGC3877: Rd=2.70, err=+12%NGC3893: Rd=2.80, err=+14%NGC3949: Rd=1.40, err=-21%NGC3953: Rd=3.50, err=-6%NGC3972: Rd=1.60, err=-27%NGC3992: Rd=3.80, err=-15%NGC4010: Rd=1.80, err=-14%NGC4013: Rd=2.20, err=+8%NGC4051: Rd=1.90, err=+3%NGC4085: Rd=1.20, err=-41%NGC4088: Rd=1.90, err=-27%NGC4100: Rd=1.80, err=-28%NGC4138: Rd=1.30, err=-44%NGC4157: Rd=2.60, err=-1%NGC4183: Rd=1.60, err=-36%NGC4214: Rd=0.50, err=-27%NGC4217: Rd=2.80, err=+5%NGC4389: Rd=1.20, err=-43%NGC4559: Rd=3.20, err=+28%NGC5005: Rd=3.00, err=-8%NGC5033: Rd=4.50, err=+44%NGC5055: Rd=3.50, err=+32%NGC5371: Rd=3.80, err=+14%NGC5585: Rd=1.50, err=-5%NGC5907: Rd=4.20, err=+32%NGC5985: Rd=4.50, err=+2%NGC6015: Rd=2.40, err=+6%NGC6195: Rd=5.20, err=+46%NGC6503: Rd=2.40, err=+39%NGC6674: Rd=5.50, err=+46%NGC6789: Rd=0.30, err=-63%NGC6946: Rd=2.60, err=+10%NGC7331: Rd=3.20, err=+4%NGC7793: Rd=1.80, err=-1%UGC00128: Rd=7.50, err=+80%UGC02259: Rd=1.60, err=+0%UGC02487: Rd=7.50, err=+41%UGC02885: Rd=8.50, err=+52%UGC05716: Rd=2.00, err=-6%UGC05721: Rd=1.20, err=+0%UGC05750: Rd=4.50, err=+38%UGC05764: Rd=0.40, err=-46%UGC05829: Rd=1.60, err=-10%UGC06399: Rd=2.50, err=+11%UGC06446: Rd=1.80, err=+6%UGC06614: Rd=4.50, err=+19%UGC06628: Rd=2.50, err=+7%UGC06667: Rd=2.50, err=+16%UGC06917: Rd=2.50, err=-12%UGC06983: Rd=2.50, err=+4%UGC07125: Rd=4.50, err=+37%UGC07151: Rd=1.30, err=-32%UGC07261: Rd=1.10, err=-29%UGC07399: Rd=1.40, err=-19%UGC07690: Rd=0.70, err=-29%UGC08286: Rd=1.30, err=-4%UGC08490: Rd=0.65, err=-29%UGC08550: Rd=1.50, err=-17%UGC09037: Rd=3.50, err=-8%UGC11455: Rd=5.50, err=-28%UGC11557: Rd=3.00, err=+1%UGC11820: Rd=4.50, err=+26%UGCA281: Rd=0.50, err=-36%UGCA442: Rd=1.00, err=-44%
Prediction error versus disk scale length $R_d$. The green band marks the $\pm 20\%$ region. The four red segments show the median residual in successive size bins.

A structural pattern is visible. Compact disks ($R_d < 1$ kpc) tend to be under-predicted (median $-29\%$). Medium disks ($1$–$2.5$ kpc) are still slightly under-predicted (median $-11\%$). Large disks ($2.5$–$4$ kpc) sit close to the 1:1 line (median $+10\%$). Giant disks ($R_d > 4$ kpc) are over-predicted (median $+34\%$). The model performs best on intermediate-scale spirals — broadly the regime in which it was calibrated. The systematic drift with $R_d$ is a clear signature that the geometric constants $c_\text{disk}$ and $c_\text{arm}$, currently treated as universal, may need to scale with disk size.

5. Contribution of each baryonic component to the wave field

The wave-field mass at $R_\text{eval}$ is computed by integrating contributions from each baryonic component separately. Averaging over the 94 galaxies gives a quantitative measure of which sources dominate the BeeTheory dark field.

Component Median contribution Mean contribution Maximum contribution Coherence length $\ell$
Gas ring (HI + He)$45\%$$45\%$$81\%$$1.7\,c_\text{disk}\,R_d \approx 5.4\,R_d$
Thin stellar disk$40\%$$40\%$$66\%$$c_\text{disk}\,R_d \approx 3.2\,R_d$
Thick stellar disk$13\%$$12\%$$20\%$$1.5\,c_\text{disk}\,R_d \approx 4.8\,R_d$
Spiral arm excess$3\%$$3\%$$5\%$$c_\text{arm}\,R_d = 2\,R_d$
Bulge (Hernquist)$0\%$$0.1\%$$0.5\%$$c_\text{sph}\,r_b \approx 0.2\,R_d$
Fraction of $M_\text{wave}^\text{total}(

Two components dominate the wave field at the flat-rotation radius: the gas ring (45%) and the thin stellar disk (40%) — together they account for 85% of the BeeTheory mass on average. The gas component is the largest contributor in slightly more than half the galaxies, which is consistent with the late-type, gas-rich nature of much of the SPARC sample. The thick disk and spiral arms each contribute at the 10% and 3% level, while the bulge is essentially negligible in this sample.

6. Stratification by Hubble type and data quality

Splitting the residuals by morphological type gives further insight into where the model performs well:

Hubble type $N$ Median $|\text{err}|$ Mean signed err
S0 – Sa ($T = 0$–$2$)4$29.8\%$$-0.7\%$
Sb – Sbc ($T = 3$–$5$)34$18.0\%$$+6.9\%$
Sc – Scd ($T = 5$–$7$)36$16.6\%$$+6.5\%$
Sd – Im ($T = 7$–$10$)40$24.2\%$$-3.5\%$

And by the SPARC quality flag $Q$:

SPARC quality $N$ Median $|\text{err}|$ Mean signed err
$Q = 1$ (highest)27$14.0\%$$+8.7\%$
$Q = 2$ (medium)67$19.1\%$$-1.6\%$

The 27 galaxies of highest observational quality have a median error of 14%, slightly better than the full sample. This is consistent with the expectation that the residual scatter contains a contribution from observational uncertainty in the SPARC parameters themselves.

7. Full galaxy-by-galaxy table

The complete results for all 94 blind galaxies are listed below, sorted by observed $V_f$ from slowest to fastest. Row shading indicates the prediction error: green < 20%, gold 20–30%, orange 30–50%, red > 50%.

Galaxy $T$ $R_d$ (kpc) $V_f$ (km/s) $V_\text{bar}$ $V_\text{wave}$ $V_\text{tot}$ Error
KK98-251100.301771113-23%
UGCA281100.5040132226-36%
NGC3741100.6851335564+26%
NGC170500.6054223844-19%
NGC2366101.3055315563+14%
UGC05764100.4057162631-46%
UGCA442101.0057172732-44%
NGC6789100.3060121922-63%
UGC07690100.7062233844-29%
F583-4101.4067234248-29%
UGC0855071.5067245055-17%
NGC310991.4068244551-25%
NGC4214100.5068264250-27%
IC257492.8069338793+35%
UGC05829101.6069285662-10%
UGC07261101.1072264451-29%
UGC0571682.0075286571-6%
UGC0662892.5075297580+7%
UGC0712594.50752998103+37%
NGC030071.5076326976+0%
NGC297650.7580234450-37%
UGC0575084.508031106110+38%
UGC0849090.6580304857-29%
UGC0715161.3082255056-32%
F583-1101.8083255358-30%
NGC010062.3083318894+13%
UGC0828661.3084357280-4%
NGC2915100.5085284553-38%
UGC0572191.2085437485+0%
NGC005581.8087357986-1%
NGC558571.5087377483-5%
UGC0644671.8087408392+6%
UGC0639982.5089369299+11%
NGC024772.409037101108+20%
UGC0225991.6090398190+0%
UGC0666772.50903997104+16%
UGC1155783.0090308691+1%
UGC1182094.509032109113+26%
UGC0739991.4093366675-19%
M3361.40100438898-2%
F579-V183.20105298792-12%
NGC092573.1010551147155+48%
NGC405141.9011043105114+3%
NGC418361.60110316370-36%
NGC438941.20110295562-43%
UGC0691792.50110359097-12%
NGC376952.8011247132140+25%
UGC0698362.5011343109117+4%
NGC100362.8011544121129+12%
NGC779371.8011845107116-1%
NGC650362.4012158158168+39%
NGC455963.2012350150158+28%
NGC394941.40125458999-21%
NGC401061.8012846100110-14%
NGC240361.8013150115126-4%
NGC397251.60135419099-27%
NGC408551.20135367179-41%
UGC0012887.5013547238243+80%
NGC601562.4014253140150+6%
NGC362172.1014976174190+28%
NGC413811.30150387685-44%
NGC372653.0015258172181+19%
NGC028943.5015559191200+29%
NGC389352.8015959172182+14%
UGC0903763.5016047139147-8%
NGC410041.8016248107117-28%
NGC387752.7016357174183+12%
NGC109043.8017056190199+17%
NGC268332.9017562191201+15%
NGC408841.9017552118128-27%
NGC421732.8018061179189+5%
NGC505543.5018072227238+32%
NGC694662.6018067186198+10%
NGC290342.6018462172183-0%
NGC401352.2018569187199+8%
NGC415732.6018564171183-1%
NGC503354.5019571271280+44%
NGC395343.5020056179188-6%
UGC0661414.5020062230238+19%
NGC080155.8020871318326+57%
NGC590754.2021070267277+32%
NGC089134.1021261217226+7%
NGC352142.8022581222236+5%
NGC537143.8022573247257+14%
NGC399243.8024258198207-15%
NGC500543.0026073228240-8%
NGC619535.2026091369380+46%
NGC667435.5026089369380+46%
NGC733133.2026586262275+4%
NGC295535.5026694395406+53%
UGC1145565.5027550191198-28%
UGC0288568.5029082433441+52%
NGC598534.5029579290301+2%
UGC0248717.5033093455465+41%

8. Conclusions

A predictive framework, not a per-galaxy fit

With no parameter adjusted on this sample of 94 galaxies, the BeeTheory framework recovers the observed flat rotation velocity to within $\pm 20\%$ for half of the sample and within $\pm 30\%$ for more than two thirds. The mean signed error is $+1.4\%$, indicating that the model does not systematically over- or under-predict. The Pearson correlation between predicted and observed velocities in logarithmic space is $0.93$.

The wave field is gas-dominated in late-type galaxies

In this blind sample — composed mainly of late-type spirals and dwarfs — the gas ring contributes more to the BeeTheory wave-field mass than the stellar disk does, on average. This is a direct consequence of the convolution formula: a more extended source has a wider Yukawa kernel and contributes more flux at large radii. The result is the natural prediction of a wave-based theory of gravity applied to a sample dominated by gas-rich late-type systems.

A clear residual trend with disk size

The most informative residual is the systematic drift of the error with disk scale length $R_d$: from $-29\%$ for compact disks to $+34\%$ for giant disks. This signature indicates that the universal geometric constants $(c_\text{disk},\,c_\text{arm})$ over-correct for small disks and under-correct for large ones. Allowing these constants to depend weakly on $R_d$, or replacing them with a physically derived coherence-length relation, is the next refinement to investigate.

An honest statement

A median error of 19% on a blind sample is a meaningful predictive result, but it is not a precision fit. The model captures the bulk of galactic rotation velocities with one global coupling, but does not yet reach the level of observational uncertainty. The residual structure points to identifiable refinements rather than a fundamental obstruction. This is reported as the state of the framework at this stage, not as a final result.

9. Summary

1. The BeeTheory parameters calibrated in Note VIII on 22 galaxies were applied, without adjustment, to 94 additional SPARC galaxies.

2. The median absolute error on the blind sample is $19\%$; the mean signed error is $+1.4\%$. The model predicts $V_f$ within $\pm 30\%$ for 67 of the 94 galaxies (71%).

3. The Pearson correlation in log-log space between predicted and observed velocities is $0.93$.

4. The wave field is dominated by the gas ring (median $45\%$ of $M_\text{wave}$) and the thin stellar disk (median $40\%$). Other components contribute less.

5. The residual error drifts monotonically with disk scale length, from $-29\%$ in compact disks to $+34\%$ in giant disks, indicating that the universal geometric constants would benefit from a size-dependent refinement.

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). · McGaugh, S. S. — The third law of galactic rotation, Galaxies 2, 601 (2014). · Freeman, K. C. — On the disks of spiral and S0 galaxies, ApJ 160, 811 (1970). · Hernquist, L. — An analytical model for spherical galaxies and bulges, ApJ 356, 359 (1990). · Broeils, A. H., Rhee, M.-H. — Short 21-cm WSRT observations of spiral and irregular galaxies, A&A 324, 877 (1997). · Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, v2, BeeTheory.com (2023).

BeeTheory.com — Wave-based quantum gravity · SPARC blind test · © Technoplane S.A.S. 2026