The Structural Law of Dark Matter Cores: A Pressure‐Like Scaling and a Constraint on an Effective Equation of State

ABSTRACT Dark‐sector models that fit the same rotation curves can be forced into mutually exclusive coefficients for how halo core size scales with velocity and central density. An isothermal pressure‐supported core predicts one specific pair through its Jeans length, constant surface density predicts another, and disk‐feedback cores a third, near zero. We measure this law from Burkert fits to the SPARC sample, using outer‐curve velocities with the baryons subtracted, so no model's own fitter enters the regression. The measured coefficients, a = 0.86 and b = −0.67 from 131 galaxies, exclude all three predictions, but unequally: surface‐density and feedback are strongly excluded, the isothermal point only mildly disfavoured. A closure test on synthetic isothermal populations, passed through the identical pipeline, recovers the injected coefficients unbiased: the velocity exponent matches +1, and the deviation is physical, residing in the density exponent. Calibrating the same pipeline on polytropic halos gives an effective equation‐of‐state index of 1.39, excluding the isothermal value. The pressure model's invariant combination scatters at 0.19 dex against 0.35 dex for the surface‐density invariant, and out‐predicts both NFW and Burkert on held‐out data, with cosmologically compliant NFW concentrations. These are structural‐law coefficients, a first instrument‐calibrated equation‐of‐state constraint, not a detection of pressure.