Paper 2: Candidate Gravitational Signatures of the Four-Dimensional Scale Space Framework: Asymmetric Force, Wave Polarisations, and Scale Dragging

We explore phenomenological consequences of the four-dimensional scale-space framework introduced in a companion paper 1, in which physical scale is treated as a genuine geometric coordinate s with metric dσ² = e^ (2s/L) (dx² + dy² + dz²) + α²ds² (anti-de Sitter space AdS4). The framework is kinematic: it treats (x, y, z, s) as a fixed background and does not yet constitute a complete dynamical theory of gravity. Under minimal assumptions about a possible dynamical extension, we examine how motion in the scale dimension may modify effective gravitational interactions and give rise to additional degrees of freedom. Three classes of potential consequence are considered. First, a heuristic two-body construction shows that bodies of differing compactness at different scale positions may experience an asymmetric effective force in scale-stationary projection; scale-covariant four-momentum is rigorously conserved (Proposition 1, the paper’s one fully derived result), establishing this asymmetry as a projection effect rather than a violation of Newton’s third law. Order-of-magnitude considerations suggest the effect may be non-negligible for neutron star systems, but quantitative assessment requires a full dynamical model. Second, mode-counting and gauge-fixing of the (x, y, z, s) perturbation tensor indicate five physical polarisation classes — two tensor modes identical to GR, two vector modes coupling spatial and scale perturbations, and one pure scale mode hss — as potential signatures of a dynamical completion. Third, the Christoffel structure of the (x, y, z, s) manifold implies scale dragging analogous to frame dragging in GR; the exact rotating metric is identified as a priority open problem. All considerations are consistent with current LIGO/Virgo/KAGRA observations and are offered as guidance for future theoretical and observational development.