Galactic rotation curves from topological void curvature: SPARC fits, profile diversity, and observational predictions

We test a dark-matter-free model of galactic rotation curves in which the effective halo isthe boundary curvature of topological voids in a Regge simplicial manifold, with curvatureconcentrated at the void surface by the Gauss-Bonnet theorem. The profile shape is fixedby lattice geometry in the first-principles limit (q = 1, oblate Gaussian with η = 0. 548from simplex combinatorics) ; per-galaxy normalization uses two fit parameters (R0, Mtotal), identical in count to NFW. A single universal velocity scale (Vscale = 49. 5 km/s), taken fromthe median velocity scale of the sample, enters as one dimensional input. On 171 SPARC galaxies the first-principles Gaussian (q = 1) is preferred over NFW byBIC in 106 galaxies (62%). A q-Gaussian extension with q = 21/11 ≈ 1. 909 is preferredover NFW in 139 galaxies (81%) ; holdout testing (10 random 85/86 splits) recovers q =1. 910 ± 0. 028 with an 84. 8% blind-test win rate and a 3. 1% generalisation gap, indicatingthe value is not sample-specific. The rational number 21/11 arrives from three independentmeasurements, all matching at ∼ 1% precision: (a) the SPARC empirical q-Gaussian halofit, q = 1. 910±0. 028 (139 of 171 wins; holdout-validated) ; (b) the engine cascade χ = 0 voidboundary spectral dimension ds, giving q = 1 + 2/ds = 1. 91 ± 0. 01 (companion paper 18, n = 193 voids) ; (c) the Maxwell rigidity μ = E/ (2V − 3) = 21/11 exactly of the Császárpolyhedron K7, the unique minimal triangulation of T 2 (V = 7, E = 21), reached as theIR endpoint of vertex-contraction RG on engine voids (companion paper, Test I: 56 of 57voids monotone, μ∞ = 1. 899). Tier 0 topological identities — M0 = 0 (Gauss-Bonnet) and M1 = 0 (D∞h symmetry) — rigorously force χ = 0 void boundaries to be quadrupoleleading;this is the load-bearing first-principles topological prediction. The triple numericalconvergence on 21/11 is reported as an unexplained empirical observation: standard Tsallismaximum-entropy on K7 gives q ∈ 1. 11, 1. 18 rather than 21/11; four panel-validated bridgeattempts (Sattin superstatistics, Tsallis-walk dimension, Lane-Emden polytrope at n = 1. 6, population q-Gaussian stacking) all fail on the same physical obstruction—cascade voids arethin 2-shells, not extended halos — and a triangulation-deformation test (companion paper) finds that q does not track μ across T 2 triangulations (15 cases, μ ∈ 1. 51, 1. 91, q stays≈ 1. 83 ± 0. 03, Pearson r (q, μ) = −0. 57). Thus μ (K7) = 21/11 is a rigorous combinatorialfact, q = 21/11 is empirically calibrated and engine-cascade-reproduced, but the derivationalmechanism linking the two is open. At equal parameter count (k = 2), the q = 21/11model is at rough parity with Burkert (49% head-to-head) and Einasto (50% head-to-head), and outperforms NFW most strongly in high-mass galaxies with extended rotation curves (rmax > 15 kpc). A stacked inner-residual analysis at r/Rdisk < 2 shows the q = 21/11 model is nearlyunbiased (+0. 5%), comparable to Burkert (−0. 1%) and Einasto (+0. 5%), while NFW exhibitsthe well-known core-cusp bias (+2. 4%, rising to +7. 4% in low-mass galaxies). Thehalf-mass radius and baryonic mass follow a power-law size-mass relation with slope 0. 239and Pearson r = 0. 418 (r2 = 0. 17, p = 9×10−8), numerically comparable in exponent to thebaryonic Tully-Fisher velocity-mass relation but distinct from it. The inner dark-mattervelocity slope β ≈ 1 predicted for DM-dominated inner regions yields a full-sample medianβ = 0. 625 across 146 SPARC galaxies, reflecting the baryonic mix of the sample rather1than a model failure; 3 of 5 high-SNR discrimination galaxies favour the void prediction, and the model's BIC wins are concentrated in the low-surface-brightness systems wherevDM is directly accessible. A two-cluster collision simulation (N = 5, 000, 3, 816 severededges) produces a curvature-baryon centroid separation of 0. 0049 code units—63% abovethe pre-registered detection threshold of 0. 003—reproducing the qualitative Bullet Clustersignature of mass-geometry decoupling from network topology alone, with no dark-matterparticles. A resolution convergence study yields 205 ± 1 kpc at N = 100, 000 (d = 65 kpc) and 209. 6 ± 0. 7 kpc at N = 200, 000 for the Bullet Cluster, with first-order Regge convergencein the spatial step empirically confirmed (three-point Richardson extrapolationpredicts 208. 7 kpc at N = 200, 000; measured 209. 6 kpc, agreement +0. 46%). Two furtherindependently-observed clusters tested with the same setup at N ∈ 20, 50, 100, 200k (Paper 4 v56 §6c, no per-cluster fit parameter): MACS J0025. 4-1222 (the “Baby Bullet”) gives 204 kpc at N = 200, 000, matching the lower NW gas-galaxy peak offset of 192 kpcreported by Bradač 2008 (continuum extrapolant 226 kpc, between the published NW andSE peaks) ; Abell 520 gives 310 kpc at N = 200, 000, inside the Jee 2012 “dark core” range100-400 kpc (continuum extrapolant 329 kpc). All twelve per-resolution measurements andall three Richardson continuum extrapolants lie inside the published lensing-X-ray offsetranges; full convergence analysis is in Paper 4 v56. The framework is the first Regge-calculus geometric model tested at this scale on SPARCand should be compared against Verlinde's emergent gravity 25, 26 and Einasto halo fits 24as the closest existing benchmarks. The 33-galaxy gap between the first-principles singlevoidprediction (106/171) and the derived ensemble-halo profile (139/171) consists of systemswhere the Gaussian's exponential cutoff is exposed by extended data; the ensembleFreeman-convolution derivation closes this gap by producing the required power-law tail, with the cascade aspect ratio itself (currently measured) the primary remaining target forfirst-principles theory.