Inward Fixed-Point Gravity: A Natural Solution to the Cosmological Coincidence Problem

Inward Fixed-Point Gravity (IFG) is a new scalar–tensor extension of general relativity in which the scalar field is volume-normalized: \(μ̃ ≡ μ / √-g\). This geometric construction automatically produces an exact tracker solution in cosmology: the scalar tracks the dominant background (radiation → matter) and naturally transitions to dark-energy domination today, solving the long-standing cosmological coincidence problem without fine-tuning of potential parameters or initial conditions. Key breakthroughs:• Automatic tracker mechanism induced purely by volume normalization (no ad-hoc potential tuning required)• Smooth combined experimental constraint envelope using soft-min p-norm (Cassini PPN + MICROSCOPE EP + Eöt-Wash torsion balance)• Realistic numerical FLRW background evolution with wide initial-condition scans yielding w₀ ≈ −0.85 and wₐ ≈ −0.21• Benchmark point at m ∼ 10⁻³ eV, ℓ ∼ 0.2 mm, β² ∼ 10⁻⁶ lying inside the sub-millimeter sensitivity window The theory recovers Einstein gravity with an emergent cosmological constant in the static limit, predicts finite-range Yukawa corrections, and satisfies all current local gravity constraints while naturally producing dynamical dark energy. Figures:• Figure 1: Smooth combined exclusion envelope (black curve β²_max(m))• Figure 2: Numerical cosmological evolution (Ω_μ(a), w(a) with CPL fit, \(μ̃\)(a)) IFG is fully falsifiable with current sub-millimeter gravity experiments (Eöt-Wash, MICROSCOPE) and upcoming cosmological surveys (DESI, Euclid, CMB-S4). The volume-normalization mechanism offers a novel geometric route to dynamical dark energy that simultaneously satisfies all local tests. Keywords: Inward Fixed-Point Gravity, IFG, volume-normalized scalar, cosmological coincidence problem, tracker quintessence, dynamical dark energy, scalar-tensor gravity, modified gravity, cosmological constant problem, Yukawa corrections, post-Newtonian constraints, equivalence principle, torsion balance, Cassini, MICROSCOPE, Eöt-Wash