# Bulk properties and non-unique interiors

Run `python reproduce.py` with numpy and matplotlib. Standalone synthetic mathematical study; no observations or planet identity are asserted. There are603 CSVrows:441 mass/radius combinations,101 radius/uncertainty points and61 interior alternatives. Runtime versions and generator SHA256 are recorded.

Bulk sphere: G=6.67430e-11m³kg⁻¹s⁻², g=GM/R², mean density=3M/(4piR³). Mass grid0.1–10 in10²⁴kg (21 values); radius2000–12000km (21 values). Radius sweep has101 values at fixed6e24kg. Neutral default is6e24kg and6000km. Inputs are assumed spherical, nonrotating; no atmospheric, altitude or flattening correction. G is held exact for this exercise.

Assigned independent standard uncertainties are5% in mass and2% in radius. First-order relative standard uncertainty is sqrt(.05²+4*.02²) for g and sqrt(.05²+9*.02²) for density. Analytic partial derivatives are checked with independent central differences at the default. These are local propagated standard uncertainties, not exact confidence intervals.

Separately, interpreting ±5% mass and ±2% radius as hard input bounds gives extrema: minimum uses0.95M and1.02R, maximum1.05M and0.98R. These bounds and the standard-uncertainty assumptions are distinct hypothetical scenarios. The figure shades only bounded-input extrema, not statistical uncertainty intervals. At fixed mass, radius affects gravity quadratically and density cubically.

Interior model: two uniform-density concentric layers with outer radius normalized to1, mean density5500kg/m³, core radius fraction x from.2 to.8 by.01 and default mantle3500kg/m³. Core density=[5500-mantle*(1-x³)]/x³. Normalized inertia I/(MR²)=(2/5)[core*x⁵+mantle*(1-x⁵)]/5500. The UI-domain validation additionally checks all46 mantle densities1000..5500 by100 at all61 x values. A uniform density5500 gives2/5 exactly within floating-point tolerance.

Independent checks compute mass from core and shell volumes and inertia from radial shell integrals (8pi/3 integral rho*r⁴dr). All2,806 interior combinations close the mass balance and agree with the normalized formula. Very large inferred core densities at small x are mathematical consequences, not plausible materials or identified planets. No equation of state, pressure, temperature, compressibility or stability criterion is supplied. This demonstrates non-uniqueness from mean density alone, not a reconstruction.

Validation includes mass-volume closure, uniform-sphere inertia, finite-difference derivatives (maxrelative error3.34e-10) and six rejected invalid inputs. Two original figures are provided as SVG/PDF/PNG. No legacy import, network access or private data is needed to reproduce them.