One Geometry, Three Frameworks

Three independently developed frameworks — the Quarter Circle (QC), the Zeta-Filter (ZF), and the Harmonic Framework (HF) — are unified here. All three arrive at the same boundary σ = ½, the same constant L(1,χ₄) = π/4, and the same structural role for the number 2. The central result is the Master Equation: ζ(r*) / ζ(1−r*) = r* / (2π(1−π/4)) which determines K = Ω_visible/Ω_dark from the Riemann zeta function and π alone, with no free parameters and no cosmological measurement as input. The equation arises from the Timeless-Temporal Crossing: the −0 side of the Zeta Line is atemporal (outside time entirely), while the +1 side acquires time at the moment of crossing. At t = 0 — the Big Bang — the timeless −0 transmission amplitude χ(r*) = ζ(r*)/ζ(1−r*) is converted to the temporal +1 crossing fraction F_GUE = r*/(1−π/4) by the factor 2π, the circumference of the full circle. The equation predicts K = 0.18800, Ω_vis = 0.05038, consistent with Planck 2018 at 1.4σ. The gap δp = 0.060 between the two frameworks decomposes exactly into a pre-existing quantum asymmetry (2−π)/4 and the post-Big-Bang order parameter δ = ½−r*. Eight precision predictions follow, all within 1σ of observation. Three mathematical results — the existence and uniqueness proof for r*, the 80-digit numerical verification, and the analytic proof that 2π arises exactly from the structure of the functional equation — are established in Addenda A and B.