# Inverted Buoyancy Origin: Numerical Verification
# Motion Postulate: a = c^2 d(dtau/dt)/dr, at the body location
# Test body: M = 5.9720e+24 kg   evaluated at r = 6.3710e+06 m
#
# CLAIM 1  GRADIENT OF THE MEDIUM GIVES a = GM/r^2
#   dtau/dt at r          = 9.99999999303958442e-01
#   weak-field GM/(rc^2)  = 6.960415e-10
#   numeric gradient      = 1.092515317111e-16 1/s
#   analytic gradient     = 1.092515316283e-16 1/s
#   a from gradient       = 9.8195320403 m/s^2
#   analytic GM/r^2       = 9.8195320328 m/s^2
#   relative error        = 7.580e-10
#   non-linearization     = 6.960e-10 relative
#   RESULT: PASS
#
# CLAIM 2  COUPLING INDEPENDENT OF COMPOSITION
#   gradient at r          = 2.731288e-17 1/s (r = 1.2742e+07 m)
#   hydrogen (H2)        rho=8.9900e-08  a=2.7312882907e-17 1/s
#   water (H2O)          rho=1.0000e+03  a=2.7312882907e-17 1/s
#   silicate rock        rho=2.5000e+03  a=2.7312882907e-17 1/s
#   iron core            rho=7.8700e+03  a=2.7312882907e-17 1/s
#   lead                 rho=1.3400e+04  a=2.7312882907e-17 1/s
#   degenerate matter    rho=1.0000e+12  a=2.7312882907e-17 1/s
#   spread across bodies  = 0.000e+00
#   RESULT: PASS
#
# SCOPE
#   Geometry and algebra of the gradient-following origin only. Does
#   not address the five-sector extension, the derivation of the
#   entropy arrow, any anomaly dataset, or coupling magnitudes.
#
# CHECKS: 2/2 PASS
#
# FORMULATION NOTE
#   The Motion Postulate is a gradient-following rule evaluated at the
#   body's location. No volume integral over the body is posited, so no
#   extended-body force law or shell theorem is claimed or tested.
#   Inverted buoyancy is the intuition for the direction and sign of the
#   gradient, not the mechanism.