Horizon Resonance vs SPARC – full test

Prediction tested: V_obs^2 = V_b^2 + sqrt(V_b^2 * a_kappa * r), equivalently delta = 1 + x with delta = V_obs^2/V_b^2, x = sqrt(a_kappa*r)/V_b. Zero free parameters.

quantity	value
sample	2681 points / 144 galaxies (of 3391 / 175) after Qual<3, i>30deg, e_Vobs/Vobs<0.10
Upsilon_*	disk 0.5, bulge 0.7 (Lelli+2016), fixed globally, no per-galaxy tuning
a_kappa	7.02530e-11 m/s^2 = c*H0/(3pi), H0=68.15 km/s/Mpc derived from T_CMB and alpha
median delta_obs/(1+x)	0.9924 IQR [0.824, 1.193]
velocity MedAE / bias	12.62 / -0.43 km/s F10 41.7% F20 68.2%
chi2/N (random errors only)	70.5 – excludes Upsilon*, inclination, distance systematics
free weighted fit	slope 1.1517 +/- 0.0016, intercept 0.7034 +/- 0.0018 (stat. only)
through-origin slope	0.948 weighted / 1.049 unweighted [V_obs^2-V_b^2 vs V_b*sqrt(a_k r)]
Upsilon* degeneracy broken	Ups=0.35 flattens trend (rho=+0.057) but median -> 1.216; Ups=0.5 centres median but rho=+0.298
trend is within-galaxy	median per-galaxy spearman(x,ratio) = +0.327 (76/130 positive); vs V_flat: -0.049
exponent p, all radii	0.5863 (prediction 0.5)
exponent p, drop 2 inner	0.5227
exponent p, Qual=1 only	0.5248
exponent p, r>R_disk	0.5245
exponent p, r>2R_disk	0.5065 <- geometric mean recovered
prefactor (all subsamples)	0.9988 - 1.0001 – a_kappa normalisation confirmed to <0.2%
outer sample r>2R_disk	n=1639 / 137 gal: median ratio 1.0339, MedAE 10.75 km/s, bias +1.85, F10 47.5%, F20 74.2%
UNRESOLVED	highest-x bin (x>5) sits at ratio ~1.26 in BOTH samples: prediction runs low in the deepest-deficit regime

Data: Lelli, McGaugh & Schombert 2016, AJ 152, 157 (SPARC), tables 1 and 2. V_b^2 = |V_gas|V_gas + 0.5V_disk^2 + 0.7*V_bulge^2 (sign of V_gas preserved).