Geometry–Flow Gravity: From Slow Geometry to Poisson, PPN, and Solar–System Bounds

Geometry–Flow (GF) is a modified–gravity framework in which the helicoid–catenoid associate family encodes the fast, unitary geometric sector, while matter is interpreted as the slow envelope of the same geometric flow. Critiques of the original formulation have correctly pointed out the absence of (i) an explicit matter–coupled field equation, (ii) a clear Newtonian limit and Poisson equation, and (iii) a Parameterized Post–Newtonian (PPN) analysis with quantitative constraints. In this paper I address these points directly. First, I introduce a four–field system (ρ, S1, S2, S3) describing the two–timescale geometry fluid and write down an effective action for the slow sector. Varying this action yields a continuity equation, an Euler/Hamilton–Jacobi equation, and—crucially—the Poisson equation ∇^2 Φ = 4πG_eff ρ in the weak–field, slow–geometry regime. Second, I construct the asymptotic GF metric, write it in isotropic form, and derive general formulas for the PPN parameters γ and β in terms of the asymptotic coefficients. I then compare these with Solar–System bounds from Cassini tracking and Lunar Laser Ranging. The result is that GF has a standard Newtonian limit provided the slow–geometry action is adopted and Geff is identified with the measured Newton constant. Its PPN structure can be tuned to match current constraints. The strong–field throat and helicoid–catenoid structure then sit on top of a phenomenologically acceptable weak–field sector. I close by outlining the remaining stability and causality questions that GF must answer.