A free inner mass removes 10 of 16 slope-1 failures in 141 SPARC curves with fixed geometry, bounded masses, and speed boxes of three times the listed errors

We test how an unmeasured inner density condition changes spherical halo limits in 141 SPARC rotation curves. We require nonnegative density rho and nonincreasing r^gamma rho. In u=r^(3-gamma), this gives a concave enclosed mass. We compare a condition from the origin with a condition that starts at the first measured radius and leaves the inner mass free. Both use every retained speed point. The primary case fixes distance and inclination and bounds gas and stellar masses. Speed boxes use three times each listed error. The gamma=1 failure count falls from 16 to six. Thus ten failures depend on the inner condition. The count falls from four to two in the 72-galaxy quality-1, inclination-below-80 group. We publish paired slope limits for all curves under four sets of assumptions. We search gamma from 0 to 2.875. Finite brackets have width 1/128. A passing cap gives only a lower bound. Exact rational witnesses and integer bounds on fractional powers permit verification without a numerical optimizer. These deterministic shape tests do not measure an asymptotic cusp slope or give a model-failure probability. A gamma=1 pass does not establish an NFW fit. The earlier NFW exclusions remain valid because NFW also requires the inner shape condition. The package contains source data, provenance, code, tables, figures, and reproduction instructions.

SPARC · rotation curves · dark matter · spherical halos · density slope · inner mass · shape constraints · exact certificates · reproducible astronomy