Registration Balance, Radial Propagation, and the Emergence of the 1/r Field: A Black-Hole–Motivated Derivation in Registration Theory

This research note records and timestamps a derivational sequence developed within Registration Theory (Physical Foundations) from a black-hole-motivated space-time registration thought experiment to a conditional derivation of a 1/r registration field. The analysis begins from the Registration Theory balance variable χ = log(S/cT), where S and T denote the space-facing and time-facing components of a local registration. Common scaling of S and T leaves χ invariant, while local unequal scaling produces a measurable registration imbalance. A reciprocal space-time bridge is then introduced to preserve the locally measured propagation speed c while allowing different registration regimes to exhibit different effective temporal and spatial scalings. The note next examines whether a localized time-dominant registration concentration can generate the radial behavior required for gravity without inserting the Newtonian potential by assumption. Under the conditions that the exterior balance mode is effectively massless, that the relevant registration flux is conserved, and that the emergent propagation volume is three-dimensional, spherical boundary growth gives Q_reg = 4πR² dχ/dR, which integrates to χ(R) = χ∞ + C/R. The corresponding registration gradient therefore falls as 1/R². When combined with the previously derived weak-field relation between acceleration and the registration gradient, this produces an inverse-square acceleration law. The note also tests the result on a globally expanding manifold with R = a(t)r and shows that common expansion is absorbed into the physical radius, preserving the 1/R dependence. A closed three-sphere geometry is examined separately, yielding a cotangent solution whose local limit reproduces 1/R. By contrast, a literal two-dimensional Riemann-sphere propagation geometry does not produce the required falloff, providing a useful dimensional constraint. The result is explicitly conditional and does not constitute a completed derivation of gravity from first principles. In particular, the source normalization relating the conserved registration charge Q_reg to physical mass, the derivation of Newton's constant G, the emergence of three spatial dimensions, and the microscopic conservation law for radial registration flux remain open problems. This note is intended as a timestamped derivation record preserving the mathematical path, assumptions, intermediate failures, and resulting theorem targets before proceeding to source normalization.